Handbook of Computational Chemistry 9789400761698

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Computational Chemistry: From the Hydrogen Molecule to Nanostructures Lucjan Piela

Contents Exceptional Status of Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Hypothetical Perfect Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Does Predicting Mean Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orbital Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rules of the Nanoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power of Computer Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Quantum chemical calculations rely on a few fortunate circumstances like usually small relativistic and negligible electrodynamic (QED) corrections and large nuclei-to-electron mass ratio. The fast progress in computer technology revolutionized theoretical chemistry and gave birth to computational chemistry. The computational quantum chemistry provides for experimentalists the readyto-use tools of new kind offering powerful insight into molecular internal structure and dynamics. It is important for the computational chemistry to elaborate methods, which look at molecule in a multiscale way, which provide first of all its global and synthetic description such as shape and charge distribution, and compare this description with those for other molecules. Only such a picture can free researchers from seeing molecules as a series of case-bycase studies. Chemistry represents a science of analogies and similarities, and

L. Piela () Department of Chemistry, University of Warsaw, Warsaw, Poland e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_1-2

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computational chemistry should provide tools for seeing this. This is especially useful for the supramolecular chemistry, which allow chemists to planify and study intermolecular interactions. Some of them, involving concave and convex moieties, represent molecular recognition, which assures a perfect fitting of the two molecular shapes. A sequence of molecular recognitions often leads to selfassembling and self-organization, typical to nanostructures.

Exceptional Status of Chemistry Contemporary science fails to explain the largest-scale phenomena taking place in the Universe, such as speeding up the galaxies (supposedly due to the undefined “black energy”) and the nature of the lion share of Universe’s matter (also unknown “dark matter,” estimated roughly as about 90 % of the existing matter). Quantum chemistry is in by far better position, which may be regarded even as exceptional in the whole domain of sciences. The chemical phenomena are explainable by contemporary theories down to individual molecules (which represent the subject of quantum chemistry). It turned out, by comparing theory and experiment, that the solution to the Schrödinger equation (Schrödinger 1926) offers in most cases a quantitatively correct picture. Only molecules with very heavy atoms, due to the relativistic effects becoming important, need to be treated in a special way based on the Dirac theory (Dirac 1928). This involves an approximate Hamiltonian in the form of sum of Dirac Hamiltonians for individual electrons and the electronelectron interactions in the form of the (nonrelativistic) Coulomb terms, a common and computational successful practice ignoring, however, the resulting resonance character of all the eigenvalues (Brown and Ravenhall 1951; Pestka et al. 2008). When, very rarely, a higher accuracy is needed, one may eventually include the quantum electrodynamics (QED) corrections, a procedure currently far for routine application, but still feasible for very small systems (Łach et al. 2004). This success of computational quantum chemistry is based on a few quite fortunate circumstances (for references, see, e.g., Piela 2014): • Atoms and molecules are built of only two kinds of particles: nuclei and electrons. • The constituents of atoms and molecules are treated routinely as point charges. Although nuclei have nonzero size (electrons are regarded as point-like particles), the size is so small that its influence is much below chemical accuracy (Łach et al. 2004). • The QED corrections are much smaller than energy changes in chemical phenomena (like 1:10,000,000) and may be safely neglected in most applications (Łach et al. 2004). • The nuclei are thousand times heavier than electrons, and therefore, except some special situations, they move thousand times slower than electrons. This makes possible to solve the Schrödinger equation for electrons, assuming that the nuclei do not move, i.e., their positions are fixed in space (“clamped nuclei”). This

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concept is usually presented within the so-called adiabatic approximation. In this approximation, the motion of the nuclei is considered in the next step, in which the electronic energy (precalculated for any position of the nuclei), together with a usually small diagonal correction for coupling the nuclei-electron motion, plays the role of the potential energy surface (PES). The total wave function is assumed to be a product of the electronic wave function and of the function describing the motion of the nuclei. The commonly used Born-Oppenheimer (BO) (1927) approximation is less accurate than the adiabatic one, because it neglects the abovementioned diagonal correction, making the PES independent of nuclear masses. Using the PES concept, one may introduce the crucial idea of the spatial structure of molecule, defined as those positions of the nuclei that assure a minimum of the PES. This concept may be traced back to Hund (1927). Moreover, this structure corresponds to a certain ground-state electron density distribution that exhibits atomic cores, atom-atom bonds, and atomic lone pairs. It is generally believed that the exact analytical solution to the Schrödinger equation for any atom (except the hydrogen-like atom) or molecule is not possible. Instead, some reasonable approximate solutions can be obtained, practically always involving calculation of a large number of molecular integrals, and some algebraic manipulations on matrices built of these integrals. The reason for this is efficiency of what is known as the algebraic approximation (“algebraization”) of the Schrödinger equation. The algebraization is achieved by postulating a certain finite basis set of functions fˆi giDM iD1 and expanding the unknown wave function as a linear combination of the ˆi ’s with unknown expansion coefficients. Such an expansion can be encountered in the one-electron case (e.g., linear combination of atomic orbitals, LCAO, introduced by Bloch (1928)) and/or in the many-electron case, e.g., the total wave function expansion in Slater determinants, related to configurations (Slater 1930), or in the explicitly correlated many-electron functions (Hylleraas 1929). It is assumed for good-quality calculations (arguments are as a rule of numerical character only) that a finite M chosen is large enough to produce sufficient accuracy, with respect to what would be with M D 1 (exact solution). The abovementioned integrals appear, because, after the expansion is inserted into the Schrödinger equation, one makes the scalar products (they represent the integrals, which should be easy to calculate) of the expansion with ˆ1 ; ˆ2 : : : ˆM ; consecutively: In this way the task of finding the wave function by solving the Schrödinger equation is converted into an algebraic problem of finding the expansion coefficients, usually by solving some matrix equation. It remains to take care of the choice of the basis set fˆi giDM iD1 . The choice represents a technical problem, but unfortunately it contains a lot of arbitrariness and, at the same time, is one of the most important factors influencing cost and quality of the computed approximate solution. Application of functions ˆi based on the Gaussian-type one-electron orbitals (GTO) (Boys et al. 1956) provides a low cost/quality ratio, and this fact is considered as one of the most important factors, which made the contemporary computational chemistry efficient.

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Algebraization involves as a rule a large M and therefore the whole procedure requires fast computing facilities.1 These facilities changed over time, from very modest manual mechanical calculators at the beginning of the twentieth century to what we consider now as powerful supercomputers. Almost immediately after formulation of quantum mechanics in 1926, Douglas Hartree published several papers (Hartree 1927) presenting his manual calculator-based solutions for atoms of rubidium and chlorine. However amazing it looks now, these were self-consistent ab initio2 computations. In 1927 Walter Heitler and Fritz Wolfgang London clarified the origin of the covalent chemical bond (Heitler and London 1927), the concept crucial for chemistry. The authors demonstrated, in numerical calculations, that the nature of the covalent chemical bond in H2 is of quantum character, because the (semiquantitatively) correct description of H2 emerged only after inclusion the exchange of electrons 1 and 2 between the nuclei in the formula a.1/b.2/ (a; b are the 1s atomic orbitals centered on nucleus a and nucleus b, respectively) resulting in the wave function a.1/b.2/ C a.2/b.1/. Thus, taking into account also the contribution of Hund, 1927 is therefore the year of birth of computational chemistry. Perhaps the most outstanding manual calculator computations were performed in 1933 by Hubert James and Albert Coolidge for the hydrogen molecule (James and Coolidge 1933). This variational result has been the best one in the literature over the period of 27 years. The 1950s mark the beginning of a new era – the time of programmable computers. Apparently, just another tool for number crunching became available. In fact, however, the idea of programming made a revolution because: • It liberated humans from tedious manual calculations. • It offered large speed of computation, incomparable to that of any manual calculator. • The new data storage tools soon became of massive character. • It resulted in more and more efficient programs, based on earlier versions (like staying “on the shoulders of the giants”), offering possibilities to calculate dozens of molecular properties. • These programs and data soon became transportable, and also they could be copied and diffused worldwide. • It allowed the dispersed, parallel, and remote calculations. • It resulted in the new branch of chemistry: computational chemistry.3

1 Computational chemistry contributed significantly to applied mathematics, because new methods had to be invented in order to treat the algebraic problems of the previously unknown scale (like for M of the order of billions); see, e.g., reference Roos (1972). 2 That is, derived from the first principles of the (nonrelativistic) quantum mechanics 3 It is difficult to define what computational chemistry is. Obviously, whatever involves calculations in chemistry might be treated as part of it. This, however, sounds like a pure banality. The same is with the idea that computational chemistry means chemistry, which uses computers.

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• It allowed performing calculations by anyone, even not trained in chemistry, quantum chemistry, mathematics, etc. These achievements appeared gradually in the chemical practice. The first ab initio Hartree-Fock calculations (based on the ideas of Douglas Hartree (1927) and Vladimir Fock (1930)) on programmable computer for diatomic molecules have been performed at the Massachusetts Institute of Technology in 1956, using basis set of Slater-type orbitals. The first calculations with Gaussian-type orbitals have been carried out by Boys and coworkers also in 1956. An unprecedented spectroscopic accuracy has been obtained for the hydrogen molecule in 1960 by Kołos and Roothaan. In the early 1970s began the era of gigantic programs offering computation of many physical quantities at various levels of approximation. We currently live in this era with computational possibilities growing exponentially (the notorious “Moore law” of doubling the computer power every 2 years).4 This enormous progress revolutionized our civilization in a very short time. The revolution in computational quantum chemistry changed chemistry in general, because computations became feasible for molecules of interest for experimental chemists. The progress has been accompanied by achievements in theory, however mainly of the character related to computational needs. Nowadays fair accuracy computations are possible for molecules composed of several hundreds of atoms, spectroscopic accuracy is achievable for molecules with a dozen of atoms, while the QED calculations can be performed for the smallest molecules only (few atoms). In many instances, we now reach the stage, when computing a molecular property is faster, less expensive, and more informative than to measure it.

A Hypothetical Perfect Computer Suppose we have at our disposal a computer, which is able to solve the Schrödinger equation exactly for any system and in negligible time.5 Thus, we have free access to the absolute detailed picture of any molecule. This means we may predict with high accuracy and confidence the value of any property of any molecule. We might be tempted to say that being able to give such predictions is the ultimate goal of science: We know everything about our system. If you want to know more about the world, take other molecules and just compute, you will know.

It is questionable whether this problem needs any solution at all. If yes, the author sticks to the opinion that computational chemistry means quantitative description of chemical phenomena at the molecular/atomic level. 4 The speed as well as the capacity of computer’s memory increased about 100 billion times over the period of 40 years. This means that what now takes an hour of computations would require in 1960 about 10;000 years of computing. 5 In addition, we assume the computer is so clever that it automatically rejects those solutions, which are not square integrable or do not satisfy the requirements of symmetry for fermions and bosons. Thus, all nonphysical solutions are rejected.

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Let us consider a system composed of 6 carbon nuclei, 6 protons, and 42 electrons. Suppose we want to know the geometry of the system for the ground state. Our computer answers that it does not know what we mean by the term “geometry.” We are forced to be more precise now and say that we are interested in the carbon-carbon (CC) and carbon-hydrogen (CH) distances. The computer answers that it is possible to compute only the mean distances and provides them together with the proton-proton, carbon-electron, proton-electron, and electronelectron distances, because it treats all the particles on equal footing basis. We look at the CC and CH distances and see that they are much larger than we expected for the CC and CH bonds in benzene. The reason is, that in our perfect wave function, the permutational symmetry is correctly included. This means that the average carbon-proton distance takes into account all carbons and all protons, the same with other distances. To deduce more, we may ask for computing other quantities like angles, involving three nuclei. Here too we will be confronted with numbers including averaging over identical particles. These difficulties do not necessarily mean that the molecule has no spatial structure at all, although this can also happen. The numbers produced would be extremely difficult to translate into a useful 3D picture even for quite small molecules, not speaking about such a floppy molecule as a protein. In many cases we would obtain a 3D picture we did not expect. This is because many molecular structures, we are familiar with, represent higher-energy metastable electronic states (isomers). This is the case in our example. When solving the time-dependent Schrödinger equation, we are confronted with this problem. Let us use as a starting wave function the one corresponding to the benzene molecule. In time evolution, we will stay probably with a similar geometry for a long time. However, there is a chance that after a long period, the wave function changes to that corresponding to three interacting acetylene molecules (3 times HCCH). The Born-Oppenheimer optimized ground electronic state corresponds to the benzene [230:703 a.u. in the Hartree-Fock approximation for the 6-311-G(d) basis set]. The three isolated acetylene molecules (in the same approximation) have the energy 230:453 a.u. and the molecule (also with the same formula C6 H6 / H3 CC  C  C  C  CH3  230:592 a.u. Thus, the benzene molecule seems to be a stable ground state, while the three acetylenes and the diacetylene are metastable states within the same ground electronic state of the system. All the three physically observed realizations of the system 6C + 6H are separated by barriers, and this is the reason why they are observable. What is, therefore, the most stable electronic ground state corresponding to the flask of benzene? This is a quite different question, which pertains to systems larger than a single molecule. If we multiply the number of atoms in a single molecule of benzene by a natural number N , we are confronted with new possibilities of combining atoms into molecules, not necessary of the same kind and possibly larger than C6 H6 . For a large N , we are practically unable to find all the possibilities. In some cases, when based on chemical intuition and limiting to simple molecules, we may guess particular solutions. For example, to lower the energy for the flask of benzene, we may allow formation of the methane molecules and the graphite (the

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most stable form of carbon). Therefore, the flask of benzene represents a metastable state. Suppose we wish to know the dipole moment of, say, the HCl molecule, the quantity that tells us important information about the charge distribution. We look up the computer output and we do not find anything about dipole moment. The reason is that all molecules have the same dipole moment in any of their stationary state ‰, and this dipole moment equals to zero; see, e.g., reference Piela (2014), p. 735. Indeed, the dipole moment is calculated ˝ Pas the mean ˛ value of the dipole O moment operator, i.e.,  D h‰j‰i D ‰j ‰ ; index i runs over all q r i i i electrons and nuclei. This integral can be calculated very easily: the integrand is antisymmetric with respect to inversion and therefore  D 0: Let us stress that our conclusion pertains to the total wave function, which has to reflect the space invariance with respect to inversion, leading to the zero dipole moment. If one applied the transformation r ! r only to some particles in the molecule (e.g., electrons), and not to the other ones (e.g., the nuclei), then the wave function will show no parity (it would be neither symmetric nor antisymmetric). We do this in the adiabatic or Born-Oppenheimer approximation, where the electronic wave function depends on the electronic coordinates only. This explains why the integral O e (the integration is over electronic coordinates only) does not equal  D h‰j‰i to zero for some molecules (which we call polar ones). Thus, to calculate the dipole moment, we have to use the adiabatic or the Born-Oppenheimer approximation. Now we decide to quit the perfect computer and the perfect picture of the molecule it produces, because the computer output looks like almost useless, paradoxically because it is “too exact,” similarly as a map of a terrain with the scale 1:1. We desperately need an approximation (a model), which will tell us the most important features of the molecule under consideration. This is why we introduce the Born-Oppenheimer approximation. Its first consequence is the molecular geometry, this one which leads to a minimum of the electronic energy. There is a problem though that usually we have many such minima of different energy, each minimum corresponding to its own electronic density distribution. Any such distribution corresponds to some particular chemical bond pattern.6 In most cases the user of computers does not even think of these minima, because he performs the calculations for a predefined configuration of the nuclei and forces the system (usually not being aware of it) to stay in its vicinity. This is especially severe for large molecules, such as proteins. They have an astronomic number of stable conformations,7 but often we take one of them and perform the calculations for this one. It is even difficult to say that we select this one, because we rarely even consider the other conformations. In this situation, we usually take as the starting point a crystal structure conformation (we believe in its relevance). 6

Bond patterns are the same for different conformers. For a dipeptide, one has something like ten energy minima, counting only the backbone conformations (and not counting the side-chain conformations for simplicity). For very small protein of, say, a hundred amino acids, the number of conformations is therefore of the order of 10100 , a very large number exceeding the estimated number of atoms in the Universe. 7

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Moreover, usually one starts calculations by setting a starting electronic density distribution. The choice of this density distribution may influence the final electronic density and the final geometry of the molecule. In routine computations, one guesses the starting density according to the starting nuclear configuration chosen. This may look as a reasonable choice, except when small deformation of the nuclear framework leads to large changes in the electronic density. In conclusion, in practice computer gives the solution, which is close to what the computing person considers as “reasonable” and sets as the starting point.

Does Predicting Mean Understanding The existing commercial programs allow us to make calculations for molecules, treating each molecule as a new task, as if every molecule represented a new world, which has nothing to do with other molecules. We might not be satisfied with such a picture. We might be curious: • Whether, living in the 3D space, the system has or has not a certain shape (“geometry”) • If yes, why the shape is of this particular kind • If the shape is fairly rigid or rather flexible • If it is reasonable to assign the system some external surface • If there are some characteristic substructures in the system • How these substructures interact one another interact • How they influence the calculated global properties, etc. • If the same substructures are present in other molecules • If the presence of the same substructures does determine similar properties It is of fundamental importance for chemistry that we do not study particular cases, case by case, but derive some general rules. Strictly speaking these rules are false for, because of approximations made, they are valid to some extent only. However, despite this, they enable chemists to operate, to understand, and to be efficient. If we relied uniquely on exact solutions of the Schrödinger equation, there would be no chemistry at all, and people would lose the power of rationalizing this branch of science, in particular to design syntheses of new molecules. Chemists rely on molecular spatial structure (nuclear framework), on the concepts of valence electrons, chemical bonds, electronic lone pairs, etc. All these notions have no rigorous definition, but they still are of great importance describing a model of molecule. A chemist predicts that two OH bonds have similar properties, wherever they are. Moreover, chemists are able to predict differences in the OH bonds by considering what the neighboring atoms are in each case. It is of fundamental importance in chemistry that a group of atoms with a certain bond pattern (functional group) represents an entity that behaves similarly, when present in different molecules. We have at our disposal various scales at which we can look at details of the molecule under study. In the crudest approach, we may treat the molecule as a point

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mass, which contributes to the gas pressure. Next we might become interested in the shape of the molecule, and we may approximate it first as a rigid rotator and get an estimation of rotational levels we can expect. Then we may leave the rigid body model and allow the atoms of the molecule to vibrate about their equilibrium positions. In such a case, we need to know the corresponding force constants. This requires either to choose the structural formula (chemical bond pattern) of the molecule together with taking the corresponding empirical force constants or to apply the normal mode analysis, first solving the Schrödinger equation in the BornOppenheimer approximation (we have a wide choice of the methods of solution). In the first case, we obtain an estimation of the vibrational levels; in the second, we get more reliable vibrational analysis, especially for larger atomic orbital expansions. If we wish we may consider anharmonicity of vibrations.8 At the same time, we obtain the electronic density distribution from the wave function ‰ for N electrons9

.r/ D N

1 Z 2 X

d2 d3 : : : dN j‰.r; 1 ; r2 ; 2 ; : : : ; rN ; N /j2 :

1 D 12

According to the Hellmann-Feynman theorem (Hellmann 1937; Feynman 1939)  is sufficient to compute the forces acting on the nuclei. We may compare the resulting  calculated at different levels of approximation. The density distribution  can be analyzed in the way known as Bader analysis (Bader 1994). First, we find all the critical points, in which r D 0. Then, one analyzes the nature of each critical point by diagonalizing the Hessian matrix calculated at the point10 : • If the three eigenvalues are negative, the critical point corresponds to a maximum of .

8 The low-frequency vibrations may be used as indicators to look at possible instabilities of the molecule, such as dissociation channels, formation of new bonds, etc. Moving all atoms, first according to a low-frequency normal mode vibration and continuing the atomic displacements according to the maximum gradient decrease, we may find the saddle point and then, sliding down, detect the products of a reaction channel. 9 The integration of j‰j2 is over the coordinates (space and spin ones) of all the electrons except one (in our case the electron 1 with the coordinates r; 1 ) and in addition the summation over its spin coordinate .1 /. As a result one obtains a function of the position of the electron 1 in space: .r/. The wave function ‰ is antisymmetric with respect to exchange of the coordinates of any two electrons, and, therefore, j‰j2 is symmetric with respect to such an exchange. Hence, the definition of  is independent of the label of the electron we do not integrate over. According to this definition,  represents nothing else but the density of the electron cloud carrying N electrons and is proportional to the probability density of finding an electron at position r. 10 Strictly speaking the nuclear attractors do not represent critical points, because of the cusp condition (Kato 1957). This worry represents rather a pure theoretical problem, when, what is a common practice, one uses Gaussian atomic orbitals as the basis set.

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• If two are negative and one positive, the critical point corresponds to a covalent bond. • If one is negative and two positive, the critical point corresponds to a center of an atomic ring. • If all three are positive, the critical point corresponds to an atomic cavity. The chemical bond critical points correspond to some pairs of atoms, and there are other pairs of atoms, which do not form bonds.11 The Bader analysis enables chemists to see molecules in a synthetic way, nearly independent of the level of theory that has been used to describe it, focusing on the ensemble of critical points. We may compare this density with the density of other molecules, similar to our one, to look whether one can see some local similarities. We may continue this, getting a more and more detailed picture down to the almost exact solution of the Schrödinger equation. It is important in chemistry to follow such way, because at its beginning as well as at its end, we know very little about chemistry. We learn chemistry traveling along the way.

Orbital Model The wave function for identical fermions has to be antisymmetric with respect to exchange of coordinates (space and spin ones) of any two of them. This means that two electrons having the same spin coordinate cannot occupy the same position in space. Since wave functions are continuous, this means that electrons of the same spin coordinate avoid each other (“Fermi hole” or “exchange hole”). This Pauli exclusion principle does not pertain to two electrons of opposite spin. However, electrons repel one another (Coulombic force) at any finite distance, i.e., they have to avoid one another because of their charge (“Coulomb hole” or “correlation hole”). It turned out (references in Piela (2014), p. 710) that the Fermi hole is by far more important than the Coulomb hole. A high-quality wave function has to reflect the Fermi and the Coulomb holes. The corresponding mathematical expression should have the antisymmetrization operator in front, and this will take care of the Pauli principle (and introduce a Fermi hole). Besides this, it should have some parameters or mathematical structure controlling somehow the distance between any pair of electrons (this will introduce the Coulomb repulsion). Since the Fermi hole is much more important, it is reasonable to consider first a wave function that takes care of the Fermi hole only. The simplest way to take the Fermi hole into account is the orbital model (approximation). Within the orbital model, the most advanced is the Hartree-Fock method. In this method, the Fermi hole is taken into account

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Although this is only after assuming a reasonable threshold as a criterion. This is a common situation in chemistry – strictly speaking there is no such thing as chemical bond in a polyatomic molecule.

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by construction (antisymmetrizer). The Coulomb hole is not present, because the Coulomb interaction is calculated through averaging the positions of the electrons. The orbital model is incorrect (i.e., represents an approximation), because it neglects the Coulomb hole. Due to its simplicity, it gains, however, enormous scientific power, because: • It allows to see the electronic structure as contributions of individual electrons, with their own “wave functions,” i.e., orbitals, mutually orthogonal, with a definite mathematical form, symmetry, energy (“orbital energy levels”), etc. • We take the Pauli exclusion principle into account by not allowing occupations of an orbital by more than two electrons (if two, then of the opposite spin coordinates). The occupation of all orbital levels is known as the all-important orbital diagram. • The orbital energy may be interpreted as the energy needed to remove an electron from the orbital (assuming that all the orbitals do not change during the removing, Koopmans theorem (Koopmaans 1933/1934)). • Molecular electron excitations may be often identified with changing the electron occupancy in the orbital diagram. • We may even consider electronic correlation (still incorporating a concept of the Coulomb hole) either by allowing different orbitals for electrons of different spin or considering a wave function expansion composed of electron diagrams with various occupations. • One may trace the molecular perturbations to changes in the orbital diagram. • One may describe chemical reactions as a continuous change from a starting to a final molecular diagram. Theory and computational experience bring some rules, like that only those orbitals of the molecular constituents mix, which have similar orbital energies and have the same symmetry. This leads to important symmetry selection rules for chemical reactions (Woodward and Hoffmann 1965; Fukui and Fujimoto 1968) and for optical excitations (Cotton 1990). • The orbital model provides a set of fundamental keywords, which create a chemical vocabulary – a tool to communicate among chemists (including quantum chemists). This language and the numerical experience supported by theory create a kind of quantum mechanical intuition coupled to experiment, which allows to compare molecules, to classify them, and to predict their properties in a descriptive way. A majority of theoretical terms in general chemistry stem from the orbital theory.

Molecular Surface Many ideas of chemistry, although extremely helpful and sometimes indispensable, are more or less arbitrary. The all-important concept of molecular surface belongs to this class. The molecular surface, also representing a kind of synthetic view of molecule, can be defined in several distinct ways, like:

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• The isosurface corresponding to a given electron density contour. • The isosurface corresponding to a given interaction energy with an atom like helium atom. • The surface (van der Waals surface) of the sum of the van der Waals spheres of all the atoms of the molecule, with the van der Waals radii assigned (by an empirical consensus). • The accessible surface area (Lee and Richards 1971) is the area of the solvent accessible surface (SAS) that corresponds to rolling a ball representing a water molecule (its radius usually taken as 1:4 Å) over the van der Waals surface, i.e., extending the latter by the radius of the water molecule. At any of these definitions, some molecules may possess multiple surfaces (external and internal), due to possible cavities. It is true that each of the definitions leads to a different surface. However, the role of such a molecular surface is in fact to define the molecular shape, and the resulting shapes do not differ much. In the all-important intermolecular interactions, the overall molecular shape belongs to primary and synthetic characteristics of the molecules involved and often decides about the strength of the interaction. The molecular shape is a function of the quantum mechanical state of the molecule. The most important in chemistry and physics are the low-energy lying states, in most cases the conformational states separated by some rather low energy barriers and all corresponding to the ground electronic state. How to find these lowest-energy states among myriads of others (their number sometimes of the order of 10100 ) represents a very serious problem of chemistry (Piela 2002). The most promising way is to use the concept of pseudoatoms (like united atoms), the appropriate force field, and the Monte Carlo approach to overcome the energy barriers (Koli´nski and Skolnick 2004).

Molecular Recognition The interaction energy of two molecules in the vacuum is dominated by several physically distinct contributions (Jeziorski and Kołos 1977). The first two result from the first-order perturbation theory: the electrostatic interaction energy and the valence repulsion energy. The electrostatic interaction represents a Coulomb interaction of the two molecular charge distributions, which are identical (“frozen”) to those corresponding to the noninteracting molecules. The valence repulsion, a consequence of the Pauli exclusion principle, increases very strongly (exponentially) with shortening of the intermolecular distance, when the two molecular charge distributions begin to overlap. In the second order, one has the induction interaction (an electrostatic effect of relaxing the charge distribution) and the dispersion interaction (the intermolecular electron-electron Coulomb correlation effect) with some small electron-exchange corrections of these quantities. Among these effects the most important are the electrostatic interaction and the valence repulsion. The first one is of the most long-range character and

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represents the only interaction that may change the sign (attraction or repulsion) upon rotation of one of the molecules by 180ı (large anisotropy). This means the electrostatic interaction usually predefines to a large extent the mutual orientation of the approaching molecules. The two molecules when approaching just try to fit electrostatically one another. This electrostatic fitting may be lost by an ill fitting of the inappropriate shapes of the molecules (which results in an excessive valence repulsion energy; this is why this effect may be critically important) or may be enhanced by a perfect fitting of the shapes over a large contact area of the interacting molecules (molecular recognition). The induction and dispersion interactions always represent attraction and have much smaller anisotropy. This is why in most cases they do not decide about the resulting structure, while always enhancing its stability. The molecular surface plays a role of determining a no-access zone for other molecules or atoms. A violation of the zone’s border is possible, but at a large energy price (valence repulsion). In addition to this repulsive effect, there is an energy gain or expense for approaching the molecular surface at a given surface point. This is related to the (long-range) electrostatic potential created by the charge distribution of the molecule. The electrostatic potential represents the interaction energy (attraction or repulsion) of the unit charge C1 a.u. with the electrostatic field created by the molecule. The sign and the absolute value of the electrostatic potential calculated at each point of the molecular surface represent an important information about accessibility of this part of the molecular surface by other molecules (with their own surface and their own electrostatic potential calculated on it). A large area of the molecule-molecule contact (fitting) together with the opposite signs of the corresponding electrostatic potentials (over the contact area) means a perfect fitting: of the shapes and of the electrostatic characteristics. Thus, the molecular surface “colored” by the electrostatic potential provides an essential (and synthetic) information about the ability of the molecule to interact with the other ones. The above described contributions are responsible for some specific interactions like: the ubiquitous hydrogen bonds, the coordination interaction (interaction of the lone electron pair of the electron donor with an empty orbital of the electron acceptor), and, in aqueous solutions, the hydrophobic effect (hydrocarbon, or “fatty,” residues having tendency to keep together in water). The hydrogen bond is dominated by the electrostatic energy, the coordination interaction (e.g., the coordination of a positive ion by a crown ether of suitable size) by the electrostatic attraction and the valence repulsion, and the hydrophobic effect by the hydrogen bond network of the liquid water. These three types of intermolecular interaction determine the majority of the appearing supramolecular structures, typically of the size of nanometers. In the case of two interacting molecules, the resulting molecular recognition is characterized by one of the following scenarios differing by the role played by molecular flexibility of the interacting moieties: • Keylock (both molecules are rigid and fit together just because the complementary molecular shapes in their ground conformational states)

14

L. Piela

• Template (a higher-energy conformation of a flexible molecule makes a perfect fit with the ground conformational state of a rigid molecule, the latter serving as a template) • Hand glove (two flexible molecules do not fit one another unless they acquire some particular higher-energy conformational states) In the two last cases, the expense of the necessary conformational excitation is more than compensated by the resulting interaction energy of the two moieties.

Rules of the Nanoscale Note that all three types of the molecular recognition stem from the same general principle that a molecule with a convex surface (guest) may find a perfect fitting with a certain molecule with a concave surface (host). This phenomenon, which may now look even trivial, has been discovered as late as in the 1960s of the twentieth century. The very reason of why it was so late is that chemists had to learn how to synthesize large molecules and only such molecules can have a concave surface. Another reason is that the progress in organic chemistry has been limited in the past by the possibilities of computational chemistry, which finally enabled chemists to plan the complex molecular geometries and check whether they fit as expected. This convex-concave paradigm represents a basis of the molecular selfassembling and self-organization. This type of interaction has some unique features, which make possible precise human operation in the nanoscale by planning the molecules to be synthesized using the tools of computational chemistry: • A convex molecule of a particular shape may fit a specific concave one, much more precisely than two convex molecules are able to do (molecular recognition). This may be used to get a sure and direct contact of some particular functional groups to make a particular chemical reaction to proceed with a high reaction yield. • Many interatomic contacts distributed over a large convex-concave contact area may assure a sufficiently strong intermolecular binding. • A concave reactant (“host”) means also preventing some potential and unwanted reactants (“guest molecules”) to enter the cavity, just because these molecules do not recognize each other (a large energy penalty by the valence repulsion, a result of ill fitting). • A sufficiently weak intermolecular binding in order to allow for an easy reorganization of the complex, if needed. • Leaving the reaction cavity by the products, because they may have not enough space there.

Computational Chemistry: From the Hydrogen Molecule to Nanostructures

15

Power of Computer Experiments In experiments we always see quantities averaged over all molecules in the specimen, all orientations allowed, all states available for a given temperature, etc. In some experiments, we are able to specify the external conditions in such a way as to receive the signal from molecules in a given state. Even in such a case, the results are averaged over molecular vibrations, which introduce (usually quite small) uncertainty for the positions of the nuclei, close to the minimum of the electronic energy (in the adiabatic or Born-Oppenheimer approximations). This means that in almost all cases, the experimenters investigate molecules close to the minimum of the electronic energy (minimum of PES). What happens to the electronic structure for other configurations of the nuclei is a natural question, sometimes of great importance (e.g., for chemical reactions). Only computational chemistry opens the way to see what would happen to the energy and to the electronic density distribution if: • Some internuclear distances increased, even to infinity (dissociation). • Some internuclear distances shortened, and the shortening may correspond even to collapsing the nuclei into a united nucleus, or approaching two atoms which in the minimum of PES form or do not form a chemical bond. This allows us to investigate what happens to the molecule under a gigantic pressure. • Some nuclei changed their mass or charge (beyond what one knows from experiment). • We apply to the system an electric field, whose character is whatever we imagine as appropriate. Sometimes such a field may mimic the influence of charge distributions in neighboring molecules. This makes from computational chemistry a quite unique tool allowing to give the answer about the energy and electronic density distribution (bond pattern) for any system and for any deformation of the system we imagine. This powerful feature can be used not only to see, what happens for a particular experimental situation, but also what would happen if we were able to set the conditions much beyond any imaginable experiment. This uniqueness is probably the most exciting feature of computational chemistry for it carries us to discoveries everyday.

Concluding Remarks What counts in computational chemistry is looking at molecules at various scales (using various models) and comparing the results for different molecules. If one could only obtain an exact picture of the molecule without comparing the results for other molecules, we would be left with no chemistry. The power of chemistry comes from analogies and similarities, as well as from trends rather than from the ability of predicting properties. Such ability is certainly important for being efficient

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L. Piela

in any particular case, but predicting by computation does not mean understanding. We need computers with their impressive speed, capacity, and possibility to give us precise predictions, but also we need a language to speak about the computations, a model which simplifies the reality, but allows us to understand what we are playing with in chemistry. Acknowledgements The author is very indebted to Professor Leszek Z. Stolarczyk, for joy to be with him, discussing all exciting aspects of chemistry, science, and beyond, a part of them included in the present paper.

Bibliography Bader, R. F. W. (1994). Atoms in molecules. A quantum theory. Oxford: Clarendon Press. Bloch, F. (1928). Uber die Quantenmechanik der Electronen in Kristallgittern. PhD Thesis. University of Leipzig. Born, M., & Oppenheimer, J. R. (1927). Zur Quantentheorie der Molekeln, Annalen der Physik, 84, 457. Boys, S. F., Cook, G. B., Reeves, C. M., & Shavitt, I. (1207). Automatic Fundamental Calculations of Molecular Structure. Nature, 178, 1956. Brown, G. E., & Ravenhall, D. G. (1951). On the interaction of two electrons. Proceedings of the Royal Society A, 208, 552. Cotton, F. A. (1990). Chemical applications of group theory (3rd ed.). New York: Wiley. Dirac, P. A. M. (1928). The quantum theory of the electron. Proceedings of the Royal Society of London, A117, 610; (1928) ibid., A118, 351. Feynman, R. P. (1939). Forces in Molecules. Physical Review, 56, 340. Fock, V. (1930). Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Zeitschrift für Physik, 61, 126; (1930) „Selfconsistent field“ mit Austausch für Natrium. ibid., 62, 795. Fukui, K., & Fujimoto, H. (1968). An MO-theoretical interpretation of nature of chemical reactions. I. Partitioning Analysis of Interaction Energy. Bulletin of the Chemical Society of Japan, 41, 1989. Hartree, D. R. (1927). The wave mechanics of an atom with a non-coulomb central field. Part I. Theory and methods. Proceedings of the Cambridge Philosophical Society, 24, 89. Heitler, W., & London, F. W. (1927). Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Zeitschrift für Physik, 44, 455. Hellmann, H. (1937). Einführung in die Quantenchemie. Leipzig: Deuticke. Hund, F. (1927). Zur Deutung der Molekelspektren. I- III. Zeitschrift für Physik, 40, 742; (1927) ibid., 42, 93; (1927) ibid., 43, 805. Hylleraas, E. A. (1929). Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium. Zeitschrift für Physik, 54, 347. James, H. M., & Coolidge, A. S. (1933). The ground state of the hydrogen molecule. Journal of Chemical Physics, 1 825. Jeziorski, B., & Kołos, W. (1977). On Symmetry Forcing in the Perturbation Theory of Weak Intermolecular Forces. International Journal of Quantum Chemistry, 12, 91. Kato, T. (1957). On the eigenfunctions of many-particle systems in quantum mechanics. Communications on Pure and Applied Mathematics, 10, 151. Kołos, W., & Roothaan, C. C. J. (1960). Accurate electronic wave functions for the H2 molecule. Reviews of Modern Physics, 32, 219. Koli´nski, A., & Skolnick, J. (2004). Reduced models of proteins and their applications. Polymer, 45, 511.

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Koopmaans, T. C. (1933/1934). Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms. Physica, 1, 104. Łach, G., Jeziorski, B., & Szalewicz, K. (2004). Radiative corrections to the polarizability of helium. Physical Review Letters, 92, 233001. Lee, B., & Richards, F. M. (1971). The interpretation of protein structures. Estimation of static accessibility. Journal of Molecular Biology, 55, 379. Pestka, G., Bylicki, M., & Karwowski, J. (2008). Dirac- Coulomb equation: Playing with artifacts In S. Wilson, P. J. Grout, J. Maruani, G. Delgado-Berrio, & P. Piecuch (Eds.), Frontiers in quantum systems in chemistry and physics (pp. 215–238). Heidelberg: Springer. Piela, L. (2002). Deformation methods of global optimization in chemistry and physics. In P. M. Pardalos & H. E. Romeijn (Eds.), Handbook of global optimization (Vol. 2, pp. 461–488). Dordrecht/Boston: Kluwer Academic Publishers. Piela, L. (2014). Ideas of quantum chemistry (2nd ed.). Amsterdam: Elsevier. Roos, B. O. (1972). A new method for large-scale CI calculations. Chemical Physics Letters, 15, 153. Schrödinger, E. (1926). Quantisierung als Eigenwertproblem.I-IV. Annalen der Physik, 79, 361; (1926) ibid., 79, 489; (1926) ibid., 80, 437; (1926) ibid., 81, 109. Slater, J. (1930). Cohesion in monovalent metals. Physical Review, 35, 509. Woodward, R. B., & Hoffmann, R. (1965). Selection rules for sigmatropic reactions. Journal of the American Chemical Society, 87, 395, 2511.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics Brian Sutcliffe and R. Guy Woolley

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Clamped Nuclei Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Separation of Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choosing Electronic and Nuclear Variables in the Translationally Invariant Hamiltonian. . . Which Is the “Correct” Clamped Nuclei Hamiltonian? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Symmetries of the Clamped Nuclei Electronic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Point Groups and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin and Point Group Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Construction of Approximate Eigenfunctions of the Clamped NucleiHamiltonian . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 6 8 18 22 24 29 43 45 51 52

Abstract

Arguments are advanced to support the view that at present it is not possible to derive molecular structure from the full quantum mechanical Coulomb Hamiltonian associated with a given molecular formula that is customarily regarded as representing the molecule in terms of its constituent electrons and nuclei. However molecular structure may be identified provided that some additional chemically motivated assumptions that lead to the clamped nuclei Hamiltonian are added to the quantum mechanical account.

B. Sutcliffe () • R.G. Woolley Service de Chimie Quantique et Photophysique, Université Libre de Bruxelles, Bruxelles, Belgium School of Science and Technology, Nottingham Trent University, Nottingham, UK e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_2-2

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B. Sutcliffe and R.G. Woolley

Introduction The traditional specification of a molecule in classical chemistry is in terms of atoms joined by bonds, and this accounts for the central fact of chemistry that the generic molecular formula is associated with the occurrence of isomers. Such an approach does not provide a useful basis for a physical theory since we do not know the general laws of interaction between atoms. Instead a more abstract description in terms of the particle constituents of a molecule, electrons and nuclei, is employed. We shall confine the discussion to the nonrelativistic level of theory; with this proviso the interactions between electrons and nuclei are assumed to be fully specified by Coulomb’s law, and this makes possible the explicit formulation of a molecular Hamiltonian. This so-called Coulomb Hamiltonian will be given explicitly (Eq. 1) in the next section; it forms the starting point of the chapter. We concentrate on two broad themes. It is obvious that the whole collection of isomers supported by a given molecular formula share the same Coulomb Hamiltonian. The first part of the chapter is concerned with how this fundamental fact has been treated in quantum chemistry through the introduction of the clamped nuclei Hamiltonian. This involves two crucial assumptions: (1) the nuclei can be treated as fixed (“clamped”) classical particles that merely provide a classical external potential for the electrons and (2) formally identical nuclei can be treated as distinguishable. The second part of the chapter discusses in a general way the basic quantum mechanical theory of the clamped nuclei Hamiltonian, concentrating particularly on its symmetry properties.

The Clamped Nuclei Approximation The conventional nonrelativistic Hamiltonian for a system of N electrons with position variables, xei , and a set of A nuclei with position variables xni may be written as 2

¯ H .xn ; xe / D  2m



¯2 2

N X   r 2 xei C iD1 A X kD1

r 2 .xnk / mk

C

0

e2 8 –0

e2 8 –0

N X i;j D1 A0 X

i;j D1

ˇ 1 ˇ ˇ e eˇ ˇxi xj ˇ



e2 4 –0

A X N X iD1 j D1

ˇ Zi ˇ ˇ e ˇ ˇxj xni ˇ

(1)

ZZ ˇ i j ˇ: ˇ n nˇ ˇxi xj ˇ

This is the Coulomb Hamiltonian for the electrons and nuclei specified by a given molecular formula. We use a Schrödinger representation in which the operators are simple time-independent multiplicative operators acting on functions of the coordinate variables (“wavefunctions”). Kato (1951) established that the Coulomb Hamiltonian, H, is essentially self-adjoint.1 This property, which is stronger than Hermiticity, guarantees that the time evolution 1

The work was completed in 1944 and was actually received by the journal in October 1948.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics



i Ht ‰.t / D exp ¯

3

 ‰.0/

of a Schrödinger wavefunction is unitary, and so conserves probability. Furthermore the eigenvalues of H are associated with a complete set of eigenfunctions. This is not necessarily true for operators that are Hermitian but not self-adjoint.2 It was pretty obvious to applied mathematicians that the kinetic energy operator alone is indeed self-adjoint because of their classical mechanical experience. It was shown by Stone in the 1930s that multiplicative operators of the kind specified above are also self-adjoint but it was entirely unobvious that the sum of the operators would be self-adjoint because the sum of the operators is defined only on the intersection of their domains. What Kato showed was that for a range of potentials including Coulomb ones, and for any function f in the domain D0 of the full kinetic energy operator T0 , the domain of the full problem DV contains D0 and there are two constants a and b such that jj Vf jj  a jj T0 f jj C b jj f jj ; where a can be taken as small as is liked. This result is often summarized by saying that the Coulomb potential is small compared to the kinetic energy. Given this result he then proved that the usual operator is indeed, for all practical purposes, selfadjoint and is bounded from below. Why worry about this? Well if the operator is not self-adjoint it could support solutions interpretable as a particle falling into a singularity or getting to infinity in a finite time and these are unacceptable as physical solutions. Such pathologies occur in, for example, the classical mechanics of three bodies in a Coulomb field. The practical significance of Kato’s proof is the guarantee that such unphysical solutions will not arise from solving the quantum mechanical eigenvalue problem for the Coulomb Hamiltonian. It is easily established that the Coulomb Hamiltonian is invariant under the coordinate transformations that correspond to uniform translations, rotation-reflections, and permutations of particles with identical masses and charges. Because of the symmetry of the Coulomb Hamiltonian its eigenfunctions will be basis functions for irreducible representations (irreps) of the translation group in three dimensions, the orthogonal group in three dimensions, and for the various symmetric groups corresponding to the sets of identical particles. Quantum mechanical molecular structure calculations are most commonly attempted by first clamping the nuclei at fixed positions and then performing electronic structure calculations treating the clamped nuclei as providing a potential

An elementary example is afforded by the momentum operator b p D i ¯d =dq, which is Hermitian on an appropriately defined class of L2 functions (q); for these functions it is selfadjoint on 1  q  C1 but this property is lost if either of the 1 limits is replaced by any finite value a – see, for example, Thirring (1981).

2

4

B. Sutcliffe and R.G. Woolley

field for the electronic motion. With the nuclei clamped at a particular fixed geometry specified by the constant vectors xni D ai ; i D 1; 2; : : : ; A, this modified Hamiltonian takes the form cn

e

H .a; x / D

¯2  2m

N X   r 2 xei  iD1 A0 X

e2 8 –0

C

i;j D1

e2 4 –0

A X N X iD1 j D1

0

ˇ Zi ˇ ˇ e ˇ ˇxj ai ˇ

C

N X

e2 8 –0

i;j D1

ˇ 1 ˇ ˇ e eˇ ˇxi xj ˇ

(2)

Zi Zj : jai aj j

It is customary to incorporate the nuclear repulsion energy into Eq. 2; the nuclear repulsion term is merely an additive constant and so does not affect the form of the electronic wavefunctions. Its inclusion modifies the spectrum of the clamped nuclei Hamiltonian only trivially by changing the origin of the energy. The eigenvalue equation for the clamped nuclei Hamiltonian is then Hcn .a; xe / §pcn .a; xe / D Epcn .a/ §pcn .a; xe / ;

(3)

in which the eigenvalues (“electronic energies”) have a parametric dependence on the constant nuclear position vectors fai g. It is sometimes asserted that the clamped nuclei Hamiltonian can be obtained from the Coulomb Hamiltonian by letting the nuclear masses increase without limit. The Hamiltonian that would result if this were done would be nn

n

e

H .x ; x / D

¯2  2m

C

N X   r 2 xei 

e2

iD1 A0 X

8 –0 i;j D1

e2 4 –0

A X N X iD1 j D1

0

ˇ Zi ˇ ˇ e ˇ ˇxj xni ˇ

C

e2 8 –0

N X i;j D1

ˇ 1 ˇ ˇ e eˇ ˇxi xj ˇ

(4)

Z i Zj

ˇ ˇ ˇ n nˇ; ˇxi xj ˇ

with the formal Schrödinger equation, by analogy with Eq. 3, Hnn .xn ; xe / §pnn .xn ; xe / D Epnn .xn / §pnn .xn ; xe / :

(5)

Given that the Coulomb Hamiltonian has eigenstates such that H .xn ; xe / § .xn ; xe / D E§ .xn ; xe / ;

(6)

if the solutions of Eq. 5 were well defined, it would seem that the eigenstates in Eq. 6 could be expanded as a sum of products of the form § .xn ; xe / D

X p

ˆp .xn / §pnn .xn ; xe / ;

(7)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

5

where the fˆg play the role of “nuclear wavefunctions.” In the Hamiltonian (Eq. 4), the nuclear variables are free and not constant and there are no nuclear kinetic energy operators to dominate the potential operators involving these free nuclear variables. The Hamiltonian thus specified cannot be self-adjoint in the Kato sense. The Hamiltonian can be made self-adjoint by clamping the nuclei because the electronic kinetic energy operators can dominate the potential operators which involve only electronic variables. The Hamiltonian (Eq. 2) is thus a proper one and the solutions Eq. 3 are a complete set. But since the Hamiltonian (Eq. 4) is not self-adjoint it is not at all clear that the hoped for eigensolutions of Eq. 5 form a complete set suitable for the expansion (Eq. 7). However that may be, it was observed more than 30 years ago (Woolley and Sutcliffe 1977) that the arguments for an expansion (Eq. 7) are quite formal because the Coulomb Hamiltonian has a completely continuous spectrum arising from the possibility of uniform translational motion and so its solutions cannot be properly approximated by a sum of this kind. This means too that the arguments of Born and Oppenheimer (1927), and of Born and Huang (1955) for their later approach to representations of this kind, are also quite formal. As a basis for the Born–Oppenheimer and the Born approach, it is commonly assumed that it is possible to construct an analytic potential function V .xn / such that Epcn .a/ D V .a/ ; for some p and for all a

(8)

and that this potential forms an adequate starting point for a discussion of the nuclear motion part of the full problem. Examination of the form of Eq. 2 makes it clear, however, that Ecn p (a) takes the same value for all choices of a that differs from a given choice merely by a uniform translation. Similarly it remains unchanged if the a differ only by a constant orthogonal transformation. Thus any potential formed according to Eq. 8 will have some variables under any change of which no change in the potential will be described. In the context of calculations of molecular spectra, these variables are often referred to as redundant ones. It is also the case that Ecn p (a) is invariant under the permutation of any nuclei with the same charge (nuclear mass does not enter into Eq. 3). This means that the potential in Eq. 8 will have the same value for all geometries that can be obtained from a given geometry by means of a permutation of nuclei with the same charge. Should the potential have any minima at all, it always has as many as there are permutations of the nuclei with the same charge. This would seem to make the assumption of a single isolated minimum in the potential, which is essential to the usual account of the Born–Oppenheimer approximation, a rather too restrictive one for comfort, except perhaps in the case of the diatomic system. It is thus not at all clear to precisely which question the clamped nuclei Hamiltonian provides the answer and a further discussion of the properties of the Coulomb Hamiltonian is required before the clamped nuclei problem can be put into an appropriate form for yielding a potential. There are two main ways in which such

6

B. Sutcliffe and R.G. Woolley

a discussion can be attempted. If it is desired to stay with the Coulomb Hamiltonian in its laboratory-fixed form then the solutions must be expressed in coherent state (wave-packet) form to allow for their continuum nature. If the solutions are required as L2 -normalizable wavefunctions, then the translational motion must be separated from the Coulomb Hamiltonian and the solutions of the remaining translationally invariant part must be sought. It is in this second approach that it is easiest to make contact with the standard arguments and this will be considered in the following section.

The Separation of Translational Motion All that is needed to remove the center-of-mass motion from the molecular Coulomb Hamiltonian is a linear point transformation symbolized by .t/ D xV:

(9)

In Eq. 9, t is a 3  NT  1 matrix .NT D N C A/ and  is a 3  1 matrix, so that the combined (bracketed) matrix on the left of Eq. 9 is 3  NT . V is an NT  NT matrix which, from the structure of the left side of Eq. 9, has a special last column whose elements are ViNT D MT 1 mi ;

MT D

NT X mi :

(10)

iD1

Hence  is the standard center-of-mass coordinate  D MT 1

NT X mi xi :

(11)

iD1

As the coordinates tj ; j D 1; 2; : : : NT  1 are to be translationally invariant we require the condition, NT X Vij D 0;

j D 1; 2; ::::NT  1

(12)

iD1

on each remaining column of V and it is easy to see that Eq. 12 forces tj ! tj as xi ! xi C a, all i. The ti are independent if the inverse transformation x D .t/ V1

(13)

exists. The structure of the right side of Eq. 13 shows that the bottom row of V1 is special and, without loss of generality, its elements may be required to be

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

 1  V NT i D 1;

7

i D 1; 2 : : : ::NT :

(14)

The inverse requirement on the remainder of V1 implies that NT X  1  V j i mi D 0;

j D 1; 2; ::::NT  1:

(15)

iD1

The Coulomb Hamiltonian (Eq. 1) in the new coordinates becomes

H .t; / D

2  ¯2

NX T 1

1 i i

2

r .ti / 

¯2 2

0 NX T 1

!   1 ! r .ti /  r ij

  tj C

0

e2 8–o

i;j D1

iD1

NT X ZZ i

j

rij .t/ i;j D1

2

¯  2M r 2 ./ T ¯2 r 2 ./ : D H0 .t/  2M T

(16) Here the positive constants 1/ij are given by: N

T X 1 D m1 k Vki Vkj ; ij

i; j D 1; 2; : : : NT  1:

(17)

kD1

The operator rij is the interparticle distance operator expressed as a function of the ti . Thus 0 rij .t / D @

X

NX T 1

˛

kD1

     V1 kj  V1 ki t˛k

!2 11=2 A

:

(18)

!  In Eq. 16, the r .ti / are gradient operators expressed in the Cartesian components of the ti and the last term represents the center-of-mass kinetic energy operator. Since the center-of-mass coordinate does not enter the potential energy term, the center-of-mass motion may be separated off completely so that the eigenfunctions of H are of the form T ./ ‰ .t/ ;

(19)

where ‰(t) is a wavefunction for the Hamiltonian H0 .t/ ; Eq. 16, which will be referred to as the translationally invariant Hamiltonian. The eigenfunctions of this Hamiltonian will be basis functions for irreps of the orthogonal group in three dimensions and for the various symmetric groups of the sets of identical particles. It should be emphasized that different choices of V are unitarily equivalent so that the spectrum of the translationally invariant Hamiltonian is independent of the

8

B. Sutcliffe and R.G. Woolley

particular form chosen for V, provided that it is consistent with Eqs. 10 and 12. In particular it is perfectly possible to put the kinetic energy operator into diagonal form by choosing an orthogonal matrix U that diagonalizes the positive definite symmetric matrix of dimension NT  1 formed from the 1/ij and then replacing elements of the originally chosen V according to Vij !

NX T 1

Vik Ukj ;

j D 1; 2 : : : NT 1 :

kD1

As can be seen from Eq. 18, the practical problem with any choice of V is the complicated form given to the potential operator.

Choosing Electronic and Nuclear Variables in the Translationally Invariant Hamiltonian In order to identify the electrons, let the translationally invariant electronic coordinates be chosen with respect to the center-of-nuclear mass tei D xei  X; X D M 1

X

mi xni ; M D

iD1

A X

mi :

iD1

In the case of the atom A D 1 and the origin is the nucleus. Other coordinate choices are possible, but this is the only choice that avoids a term in the kinetic energy operator coupling the electronic and nuclear variables and which allows the electronic part of the potential to be written in terms of the electronic variables and the clamped nuclei positions (see Mohallem and Tostes 2002; Sutcliffe 2000). There is no need to specify the proposed A  1 translationally invariant nuclear variables tn other than to say that they are expressed entirely in terms of the laboratory nuclear coordinates by means of a matrix Vn exactly like V in Eq. 9, but with side A and with M in place of M T and X in place of . It is also sometimes useful to define a set of redundant Cartesian coordinates xni D xni  X;

i D 1; 2; : : : A;

so that

A X mi xni D 0:

(20)

iD1

Of course the laboratory nuclear variable xni cannot be completely written in terms of the A  1 translationally invariant coordinates arising from the nuclei, but in the electron-nucleus attraction and in the nuclear repulsion terms the center-ofnuclear mass X cancels out. For ease of writing xni will continue to be used in those terms but it should be remembered that the nuclear potentials are functions of the translationally invariant coordinates defined by the nuclear coordinates. On making this choice of electronic coordinates the electronic part of Eq. 16 is

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

2

¯ H0 .xn ; te / D  2m

N X   r 2 tei  iD1

C

e2 8–0

¯2 2M

i;j D1 0

0

N X i;j D1

N X    !   e ! r ti  r tej 

ˇ 1 ˇ ˇ e eˇ ˇti tj ˇ

C

A X

i;j D1

e2 4–0

A X N X iD1 j D1

9

ˇ Zi ˇ ˇe ˇ ˇtj xni ˇ

ZZ ˇ i j ˇ: ˇ n nˇ ˇxi xj ˇ

(21) This electronic Hamiltonian is translationally invariant and would yield the usual form were the nuclear masses to increase without limit. It has been noted (Kutzelnigg 2007) that to take (Eq. 21) as the electronic Hamiltonian is inconsistent with a consideration of the solution to the full problem being expressed in a series in terms of powers of the inverse total nuclear mass, since this Hamiltonian already contains a term involving the inverse of this mass to the first power. There is, however, no need to consider this term at the first stage of development of a solution to the full problem and it can be included at the point where terms of similar magnitude are considered. The remaining part of Eq. 21 is then exactly the same as the clamped nuclei form. The clamped nuclei form can be deployed consistently in an account of solutions to the full problem only if a uniform translational factor is included in the full solution. In the work of Nakai et al. (2005) (see also Sutcliffe 2005) the translational motion of the center-of-mass is subtracted to yield a Schrödinger eigenvalue problem from which the translational part of any continuous spectrum has been removed. Of course the spectrum of the resulting operator can have a continuous spectral range, as can the translationally invariant form itself, for reasons quite other than translational motion. The nuclear part of Eq. 16 involves only kinetic energy operators and has the form: Kn .tn / D 

A1 ¯2 X 1 !   n   n ! n r ti : r tj ; 2 i;j D1 ij

(22)

with the inverse mass matrix defined as a special case of Eq. 17 involving only the original nuclear variables. Both (Eqs. 21 and 22) are invariant under any orthogonal transformation of both the electronic and nuclear variables. If the nuclei are clamped in Eq. 21 then invariance remains only under those orthogonal transformations of the electronic variables that can be reexpressed as changes in the positions of nuclei with identical charges while maintaining the same nuclear geometry. The form (Eq. 21) remains invariant under all permutations of the electronic variables and is invariant under permutations of the variables of those nuclei with the same charge. Thus if an electronic energy minimum is found at some clamped nuclei geometry there will be as many minima as there are permutations of identically charged nuclei. The kinetic energy operator (Eq. 22) is invariant under all orthogonal transformations of the nuclear variables and under all permutations of the variables of nuclei with the same mass.

10

B. Sutcliffe and R.G. Woolley

The splitting of the translationally invariant Hamiltonian H0 .t/ into two parts breaks its symmetry, since each part exhibits only a sub-symmetry of the full problem. If wavefunctions derived from approximate solutions to Eq. 21 are to be used to construct solutions to the full problem (Eq. 6) utilizing (Eq. 22) care will be needed to couple the sub-symmetries to yield solutions with full symmetry.

Atoms For the atom there is no nuclear kinetic energy part and, denoting the nuclear mass by mn , the full Hamiltonian is simply the electronic Hamiltonian 2

¯ H0 e .te / D  2m

C

N X

  r 2 tei 

iD1 N0 X

e2 8–0

i;j D1

¯2 2mn

N X    !   e ! r ti  r tej  i;j D1

ˇ 1 ˇ ˇ e eˇ ˇti tj ˇ

e2 4–0

N X j D1

ˇZiˇ ˇ eˇ ˇtj ˇ

(23)

:

The electronic problem for the atom (Eq. 23) has exactly the same form and symmetry as the full problem and meets the requirements for Kato self-adjointness, for there is a kinetic energy operator in all of the variables that are used to specify the potential terms. This would continue to be the case were the nuclear mass to increase without limit. The atom is sometimes used as an illustration when considering the original form of the Born–Oppenheimer approximation (as in Deshpande and Mahanty 1969), but the only aspect of the approximation that can be thus illustrated is the translational motion part and that is easily considered in first order by treating the second term in Eq. 23 as a perturbation to the solution obtained using an infinite nuclear mass. The inclusion of this term in this way is analogous to making the usual diagonal Born– Oppenheimer correction and it can be made exactly in the case of any one-electron atom (see Handy and Lee 1996). As noted in Hinze et al. (1998) it is usually made approximately simply by including the diagonal part of the mass polarization term (the second term in Eq. 23 above) to produce an electronic reduced mass 1 1 1 D C e mn m in place of 1/ m. The Hamiltonian (Eq. 23) maintains full symmetry and is invariant under electronic permutations and under rotation-reflections of the electronic coordinates. Trial functions are usually constructed from atomic orbitals and from their spinorbitals. Permutational antisymmetry is achieved by forming Slater determinants from the spin-orbitals. Rotational symmetry is usually realized by vector coupling

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

11

of orbitals that form bases for representations of the rotation group SO(3). Spineigenfunctions too are achieved by vector coupling.3

Molecules Even after separating the translational motion, for a molecule there is always at least one nuclear variable in the kinetic energy part of the operator, and self-adjointness cannot be achieved if such terms are neglected while the potential terms involving the nuclear variables are included except by clamping the nuclei. The treatment of molecules is, thus, technically much more difficult than is the treatment of atoms. Although the discussion that follows is, for the most part, quite general, explicit consideration is confined to the diatomic case in order to avoid overburdening the exposition with details. However here too, there are certain technical features which simplify the diatomic case and which cannot be transferred to the polyatomic case so care will be taken in the following discussion not to make the diatomic the general case. For a system with two nuclei the natural nuclear coordinate is the internuclear vector which will be denoted here simply as t. When needed to express the electronnuclei attraction terms, xni is simply of the form ˛ i t where ˛ i is a signed ratio of the nuclear mass to the total nuclear mass. In the case of a homonuclear system ˛i D ˙ 12 . The di-nuclear electronic Hamiltonian is 0e

e

H .t / D

¯2  2m



N X

  r tei  2

iD1 N X

e2

4–0

C

e2 8–0

j D1 N0 X

¯2 2.m1 Cm2 /

N X    !   e ! r ti  r tej i;j D1

!

ˇ Z1 ˇ ˇ ˇe ˇtj C˛1 tˇ

ˇ 1 ˇ ˇ e eˇ ˇt t ˇ i;j D1 i j

C

C

ˇ Z2 ˇ ˇ ˇe ˇtj C˛2 tˇ

Z1 Z2 ; R

(24)

R D jtj ;

while the nuclear kinetic energy part is: 

¯2 2



1 1 C m1 m2



r 2 .t/  

¯2 2 r .t/ : 2

(25)

The electronic part is not self-adjoint in the manner prescribed by Kato because it contains no kinetic energy terms involving the nuclear variable which would dominate the potential energy terms. The full Hamiltonian would not be Kato selfadjoint if both nuclear masses were to increase without limit either. It is seen from Eq. 25, however, that if only one nuclear mass increases without limit then the

3 Some specifics of the implementation of permutational and rotational symmetry in quantum mechanics are discussed in section “The Symmetries of the Clamped Nuclei Electronic Hamiltonian.”

12

B. Sutcliffe and R.G. Woolley

kinetic energy term in the nuclear variable remains in the full problem and so the Hamiltonian remains self-adjoint in the Kato sense. The di-nuclear case has been considered numerically by Frolov (1999) in a study of the hydrogen molecular ion. In extremely accurate calculations on the discrete states of this system, he investigated what happened when first one and then two nuclear masses are increased without limit. He showed that when one mass increased without limit, any discrete spectrum persisted but when two masses were allowed to increase without limit, the Hamiltonian ceased to be well-defined and this failure led to what he called adiabatic divergence in attempts to compute discrete eigenstates. This sort of behavior would certainly be anticipated from the present discussion. Irrespective of any choices made for the nuclear masses, the electronic Hamiltonian (Eq. 24) becomes self-adjoint in the Kato sense if the nuclei are clamped for then the nuclear variables in the potential terms become constants and the only variables are the electronic ones. So the clamped nuclei potential is dominated by the electronic kinetic energy. Thus the usual practice of clamped nuclei electronic structure calculations is a consistent one. Writing the variable t in spherical polar coordinates, R, ˇ, and ˛ where tz D R cos ˇ, were the clamped nuclei Hamiltonian to be used to define a potential it is easily seen that for t D a, R D a then E cn .a/ D V.a/;

(26)

so that the potential would have the form V.R/. But the potential is not just a curve, it is a series of spherical shells of rotation swept out by the curve by all choices of ˇ and ˛. It is thus a genuine central-field potential. If the internuclear distance is fixed but a allowed to rotate or invert, then Ecn (a) is a sphere of constant energy as swept out by the variables ˇ and ˛ at radius a. If a is oriented so as to define a z-axis then Ecn (a) will take the same value at Caz and az so that if there is a minimum at Caz there will be another at az . The electronic Hamiltonian is not invariant under inversion of the nuclear variables alone unless the two nuclei have identical charges in which case inversion and permutation will have identical effects. In differential geometry terms, the potential is homeomorphic to S 2 . The Hamiltonian (Eq. 24) is invariant under all rotations of the electronic coordinates about the internuclear axis and all reflections in a plane containing the internuclear axis. The electronic states can be labeled by a quantum number m which takes the values 0; ˙1; ˙2 : : : corresponding to the eigenvalues of the z- component of the electronic angular momentum about the internuclear axis. It is easily seen that the potential will tend to increase without limit as R ! 0 but the behavior as R ! 1 presents a problem. To see this consider the asymptotic behavior of the electron-nucleus potential terms in the case of the one-electron homonuclear di-hydrogen molecule ion. The electronic coordinate is te D x 

 1 n x1 C xn2 ; 2

(27)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

13

where x is the laboratory coordinate of the electron. As the internuclear distance becomes very large, the nuclear repulsion term becomes very small and one would expect the trial wavefunction to approach the wavefunction for a one-electron ion corresponding to one of the atoms. Thus one might expect the lowest energy wavefunction to be of the form Ne cr ;

r D x  xn1 ;

r D jrj ;

for instance. However working in the chosen coordinate set 1 r D te  t; 2 so that the expected asymptotic electronic solution could be expressed only in terms of both the electronic and nuclear variables. This does not, of course, mean that the potential cannot approach the required value. It simply means that it cannot do so in any calculation in which the trial functions are confined to electronic functions whose variable origin is at the center-of-nuclear mass. This sort of difficulty is a general one and obviously not confined simply to oneelectron diatomic molecules. It would clearly be unwise to attempt to approximate solutions for molecules at energies close to their dissociation limits in terms of electronic coordinates with the origin at the center-of-nuclear mass. A trial function for the general case of the Born–Huang form .tn ; te / D

X

ˆp .tn /

nn p

.tn ; te /;

(28)

p

where the te have an origin at the center-of-nuclear mass, could, therefore, approximate only a limited region of the spectrum of the full problem. This difficulty cannot be got round by working in the laboratory frame. The solution to the full problem would be defined in terms of a three-dimensional subspace expressed in terms of a translation variable and a 3(NT 1)-dimensional subspace expressible in terms of translationally invariant variables. Translationally invariant variables must involve at least a pair of variables and so there must be at least one such variable which involves a laboratory frame electron and a laboratory frame nuclear variable. All this can be easily illustrated by considering the exact ground-state wavefunction of the hydrogen atom, as is seen in Kutzelnigg (2007). This point is developed in more detail by Hunter (1981) in a paper considering to what extent a separation of electronic and nuclear motion would be possible if the exact solution to the Schrödinger equation for the Coulomb Hamiltonian was actually known. Were the exact solution known Hunter argues (1975) that it could be written in the form .tn ; te / D  .tn /  .tn ; te / ;

(29)

14

B. Sutcliffe and R.G. Woolley

with a nuclear wavefunction defined by means of j .tn /j 2 D

Z

.tn ; te /

.tn ; te / d te :

Then, providing this function has no nodes,4 an “exact” electronic wavefunction could be constructed as  .tn ; te / D if the normalization choice Z

.tn ; te / ; ¦ .tn /

(30)

 .tn ; te /  .tn ; te / d te D 1

is made. In fact it is possible (Hunter 1981) to show that ¦ must be nodeless even though the usual approximate nuclear wavefunctions for vibrationally excited states do have nodes. The electronic wavefunction (Eq. 30) is, therefore, properly defined and a potential energy surface could be defined in terms of it as U .tn / D

Z

.tn ; te / H0 .tn ; te /  .tn ; te / d te :

(31)

Although no exact solutions to the full problem are known for a molecule, some extremely good approximate solutions for excited vibrational states of H2 have been computed and Czub and Wolniewicz (1978) took such an accurate approximation for an excited vibrational state in the J D 0 rotational state of H2 and computed U.R/. They found strong spikes in the potential at close to two positions at which the usual wavefunction would have nodes. To quote Czub and Wolniewicz (1978): This destroys completely the concept of a single internuclear potential in diatomic molecules because it is not possible to introduce on the basis of nonadiabatic potentials a single, approximate, mean potential that would describe well more than one vibrational level. It is obvious that in the case of rotations the situation is even more complex. Bright Wilson suggested (1979) that using the clamped nuclei Hamiltonian instead of the full one in Eq. 31 to define the potential might avoid the spikes but Hunter (1981) showed why this was unlikely to be the case and Cassam-Chenai (2006) repeated the work of Czub and Wolniewicz using an electronic Hamiltonian and showed that exactly the same spiky behavior occurred. However CassamChenai showed, as Hunter had anticipated, that if one simply ignored the spikes,

4

A similar requirement must be placed on the denominator in Eq. 12 of Kutzelnigg uc2007) for the equation to provide a secure definition.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

15

the potential was almost exactly the same as would be obtained by deploying the electronic Hamiltonian in the usual way. Although the spiky nature of an “exact” potential has been demonstrated explicitly only for J D 0 states of a small diatomic molecule, there is no reason to suppose that its occurrence is not general. Matters would be further complicated by rotational motion. Thus the demonstrably smooth potentials generated by solving an electronic problem cannot be approximations to any exact form but are simply computationally useful intermediates in a solution to the full problem. It would therefore seem unwise to assign too much weight to them in explaining chemical structure. In the standard approach to solving the nuclear motion part of the diatomic problem, the potential V.R/ is specified and the nuclear motion Hamiltonian becomes 

¯2 2 r .t/ C V.R/: 2

(32)

Expressing this Hamiltonian in spherical polar coordinates one obtains the usual form   1 @ 2 @ 1 ¯2 C  L2 C V.R/; (33) R 2 R2 @R @R 2R2 where L is the operator for the angular momentum of the nuclear motion. The angular part of the solution is known analytically and the solution of the nuclear motion problem can be reduced to one in the single variable R. The eigensolutions to this problem are quite naturally eigenfunctions of the nuclear angular momentum and can easily be chosen with the required permutational symmetry. But things are not quite so clear for the electronic part of the problem because one does not in practice have a form which is explicit in the nuclear variables as it is computed only at fixed nuclear geometries. It is easy to achieve the correct permutational symmetry for the electronic part of the function at each and every nuclear geometry, but it would be not at all easy to make each function an eigenfunction of the electronic angular momentum. To try to deal with the rotational motion it is possible to reformulate the diatomic problem to exhibit explicitly the angular symmetry of the Hamiltonian. As shown in Kołos and Wolniewicz (1963) and, in a somewhat more general way in Sutcliffe (2007), it is possible to define an internal coordinate system by a transformation that makes the internuclear vector t the z-axis in a right-handed coordinate system and in this system the electronic Hamiltonian (Eq. 24) becomes 

with

N N X ¯2 ¯2 X 2 !    !  r .ri /  r .ri /  r rj C V .R; r/ ; 2m iD1 2 .m1 C m2 / i;j D1

(34)

16

B. Sutcliffe and R.G. Woolley

 N N  Z1 e2 X 1 e 2 Z1 Z2 e2 X Z2 ; V .R; r/ D C  C 8 –0 i;j D1 rij 4 –0 R 4 –0 j D1 ri1 .R/ ri2 .R/ 0

(35) where the ri are the electronic variables expressed in the transformed system. In this formulation the vector t orients freely and “clamping the nuclei” comes down to simply choosing R D a. A clamped nuclei solution of Eq. 34 would lead to a clamped nuclei energy E cn .a/ D V.a/;

(36)

rather than the form given by Eq. 26. Thus any minima in V.a/ would not be duplicated by the requirement of rotational invariance. Inversion is achieved by means of ˇ !    ˇ;

˛ ! C˛

and thus involves just the angular part of the formulation. An identical operation achieves the nuclear permutation. The electronic Hamiltonian is invariant under neither operation unless the nuclei have identical charge. The nuclear kinetic energy operator (Eq. 25) becomes 

1 ¯2 @ 2 @ R C D1 .˛; ˇ; r/ ; 2R2 @R @R 2R2

(37)

with   2 ¯   D1 .˛; ˇ; r/ D .Jx  lx /2 C Jy  ly C cot ˇ Jy  ly ; i where the electronic angular momentum is lD

N X iD1

N

l.i / D

¯X @ ri  ; i iD1 @ri

(38)

and the total angular momentum operator is denoted J and involves both the electronic and nuclear variables in such a way that Jz D lz . The Jacobian for this transformation is R2 sin ˇ: It is seen that in this formulation any solution of the clamped nuclei form of the electronic Hamiltonian (Eq. 34) will give rise to a potential which is simply a curve

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

17

and not a surface of rotation. However the angular part of the nuclear kinetic energy operator now involves the electronic angular momentum so that the electronic motion and the overall rotational motion are coupled. The electronic wavefunction in this case has the same axial symmetry as in the previous case and is characterized by the quantum number m. However the states can no longer be regarded as occurring in degenerate pairs for m > 0 since lz is the z-component of the total angular momentum so m can take integer values lying between J and  J, where J is the total angular momentum. The coupling of the electronic and nuclear angular momenta lifts the m degeneracy as the Hamiltonian becomes a system of 2 J C 1 coupled partial differential equations. In the first of the two possible ways of looking at the diatomic, one remains in the Cartesian product space R3  R3N and it is thus necessary to give some explicit consideration to the angular properties of solutions to the electronic part of the problem. If the usual approach were taken to approximating solutions to the nuclear motion Hamiltonian using sums of products of electronic and nuclear parts, a typical term in the sum used as trial function for the form (Eq. 33) would be pm .te ; R/L ˆpm .R/‚Lm .ˇ; ˛/ ;

(39)

where p denotes an electronic state and L the nuclear angular momentum quantum number. There are no operators in the nuclear motion part of the problem which explicitly couple the electronic and nuclear motions. It is thus possible to represent for any electronic state, any number of rotational states specified by values of L, without considering any coupling. L in Eq. 24 is not the total angular momentum operator and so a description of rotational motion given in these terms yields only an approximate quantum number. If one transforms to the manifold RC  S 2  R3N then one can consider explicitly the rotational coupling of electronic and angular motions. The fact that the transformation is to a manifold rather than a vector space means that any operator built using coordinates defined on the manifold will be well defined only where the Jacobian for the transformation does not vanish. This does not cause great problems here because the only places where the Jacobian vanishes are when R D 0 and where ˇ D 0 and ˇ D  . The region around R D 0 is inaccessible because of the nuclear repulsion term and the exact angular wavefunctions take care of the problem with ˇ. It would not be consistent to use only a single term in a product approximation for a trial for the form (Eq. 37) with a potential V.R/ except in considering a J D 0 state. Here the minimum consistent product approximation is J X

J

 pm .r; R/J ˆpm .R/‚J m .ˇ; ˛/ :

(40)

mDJ

It is only for J D 0 states that the forms (Eqs. 39 and 40) are the same. Of course the angular momentum coupling in Eq. 40 implies a coupling of different electronic states, between † and … states or † and  states, for example.

18

B. Sutcliffe and R.G. Woolley

To allow explicitly for that possibility (Eq. 40) should really be extended to J X XJ

pm .r; R/J ˆpm .R/‚J m .ˇ; ˛/;

(41)

mDJ pm

where the electronic state is denoted pm to indicate that the state must have quantum number m. Thus for J D 2 one would need at least five electronic states.

Which Is the “Correct” Clamped Nuclei Hamiltonian? There is clearly a choice between the form (Eqs. 24 and 34) and although in the clamped nuclei approximation both would yield the same energies for any chosen internuclear separation a, the resulting energy would be a potential for two quite different situations. To generalize from the diatomic case, if the usual approach were taken to approximating solutions to the nuclear motion Hamiltonian using sums of products of electronic and nuclear parts a typical term in the sum used as trial function for the form (Eq. 33) would be p .te ; tn / ˆp .tn / ;

(42)

where p denotes an electronic state. The solutions are on the Cartesian product space R3A3  R3N . There is again no explicit coupling of the nuclear motion and electronic motions and it is thus possible to represent for any electronic state, any number of rotational states. It is only in the diatomic case that the nuclear angular momentum can be realized explicitly as part of the nuclear kinetic energy so it is not generally possible to choose ˆ directly as an eigenfunction of the nuclear angular momentum, neither is it possible to choose  directly as an eigenfunction of the electronic angular momentum.  as usually computed belongs to the totally symmetric representation of the symmetric group of each set of nuclei with identical charges. ˆ could then be a basis function for an irreducible representation of the symmetric group for each set of particles with identical masses if the permutational symmetry were properly considered in solving the nuclear motion problem. Clamped nuclei calculations are usually undertaken so as to yield a potential that involves no redundant coordinates. Thus a translationally invariant electronic Hamiltonian like (Eq. 24) would actually generate a more general potential than this. A clamped nuclei potential is, therefore, more properly associated with the electronic Hamiltonian after the separation of rotational motion like (Eq. 34) than with the merely translationally invariant one. With this choice again, the minimum consistent product approximation is

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics J X

J

pm .r; R/J ˆpm .R/ jJM mi ;

19

(43)

mDJ

where R represents the 3 A  6 internal coordinates invariant under all orthogonal transformations of the tn and jJM mi is an angular momentum eigenfunction. The general solutions are on the manifold R3A6  S 3  R3N though for triatomic molecules the internal coordinate part of the manifold is confined to RC  R2 because the three nuclear positions define a plane. It is only in the diatomic case that the electronic variables play a direct part in the specification of the angular momentum eigenfunctions and where there is therefore only one internal coordinate. However the Coriolis coupling terms in the angular part of the Hamiltonian contain terms in the electronic angular momentum so coupling of the electronic motion to the angular motion would still be anticipated, see sections V and VI of Sutcliffe (2000). Although the angular momentum coupling could imply a coupling of different electronic states, as it certainly does in a diatomic, it is not obviously implied in the general case. To achieve permutational symmetry in the nuclear motion part of the wavefunction would in the general case be very tricky. The nuclei are identified in the process of defining a body-fixed frame to describe the rotational motion, even if they are identical. If only a subset of a set of identical nuclei were used in such a definition, some permutation of the nuclear variables would induce a change in the definition of the body-fixed frame and thus spoil the rotational separation. Thus permutations of identical nuclei are considered usually only if such permutations correspond to point group operations which leave the body-fixing choices invariant. If one considers the clamped nuclei Hamiltonian as providing input for the full Hamiltonian in which the rotational motion is made explicit, the basic nuclear motion problem should be treated as a 2 J C 1 dimensional problem. If this is done then the translational and rotational symmetries of the full problem are properly dealt with. However the solutions are not generally basis functions for irreps of the symmetric groups of sets of identical nuclei except for such subgroups as constitute the point groups used in frame fixing. This restriction of the permutations is usually assumed justified by appealing to the properties of the potential surface. The idea here was introduced by Longuet-Higgins (1963) and is widely used in interpreting molecular spectra. As noted earlier, the original attempts to justify the Born–Oppenheimer and the Born approaches from the full Coulomb Hamiltonian lack rigorous mathematical foundations. So far there have been no attempts to make the foundations of the Born approach mathematically secure. However the coherent states approach has been used to give mathematically rigorous accounts of surface crossings and a review of this work can be found in Hagedorn and Joye (2007). It seems very unlikely that it would be possible to provide a secure foundation for the Born approach in anything like the manner in which it is usually presented. The Born–Oppenheimer approximation, whose validity depends on there being a deep enough localized potential well in the electronic energy, has, however, been

20

B. Sutcliffe and R.G. Woolley

extensively treated. The mathematical approaches depend upon the theory of fiber bundles and the electronic Hamiltonian in these approaches is defined in terms of a fiber bundle. It is central to these approaches, however, that the fiber bundle should be trivial, that is that the base manifold and the basis for the fibers be describable as a direct product of Cartesian spaces. This is obviously possible with the decomposition choice made for Eq. 42 but not obviously so in the choice made for Eq. 43. The Born–Oppenheimer approach has been put on a rigorous foundation for diatomics with solutions of the form (Eq. 39) in work which is described in a helpful context in Combes and Seiler (1980). For solutions like (Eq. 40), it is possible that more than one vector (coordinate) space can be constructed on it because the transformation is to a manifold. In fact two coordinate spaces are possible on S 2 a trivial one and a twisted one, the latter associated with the possibility of an electronic wavefunction with a Berry phase and the “twisted” solutions are accounted for in Herrin and Howland (1997). A mathematically satisfactory account of the Born–Oppenheimer approximation for polyatomics in an approach based on Eq. 43 has not yet been provided but it has proved possible to provide one based on Eq. 42 (see Klein et al. 1992). Because the nuclear kinetic energy operator in the space R3A3 cannot be expressed in terms of the nuclear angular momentum, it is not possible in this formulation to separate the rotational motion from the other internal motions. This work also considers the possibility that there are two minima in the potential as indeed there would be because of inversion symmetry if the potential minimum were at other than a planar geometry. It does not, however, consider the possibility of such multiple minima as might be induced by permutational symmetry. It might be possible to extend the two minima arguments to the multiple minima case and perhaps provide a mathematically secure account of the Longuet-Higgins approach to ignoring some of the inconvenient permutations. This has not so far been attempted. For a secure account to be given in terms of the separation (Eq. 43), which is what is really required if one is to use the clamped nuclei electronic Hamiltonian, it would be necessary to consider more than one coordinate space. On the manifold S 3 at least two coordinate spaces are required to span the whole manifold. The internal coordinates within any coordinate space are such that it is possible to construct two distinct molecular geometries at the same internal coordinate specification, so that a potential expressed in the internal coordinates cannot be analytic everywhere (Collins and Parsons 1993). It would therefore seem to be a very tricky job. But even if it were to be accomplished it seems very unlikely that a multiple minima argument could be constructed to account for point group symmetry in this context. It is possible to show (see section IV of Sutcliffe (2000)) that in the usual Eckart form of the Hamiltonian for nuclear motion, permutations can be such as to cause the body-fixed frame definition to fail completely. If it is wished to perform a clamped nuclei calculation on a molecule containing three or more nuclei, avoiding translations and rigid rotations, it is necessary to fix the values of six of the 3 A nuclear variables. In practice this is usually done

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

21

by choosing one nucleus, xn1 , at the origin, one nucleus, xn2 , defining an axis and a third, xn3 , defining a plane. Every possible geometry of the molecule, except where the three particles become collinear, can be specified with this choice but not every component of the 3 A variables will appear in the clamped nuclei electronic energy as six of them have been chosen to be zero. This means that even though the clamped nuclei electronic energy can be specified in terms of a molecular geometry in which the positions of A points can be given, the energy itself is a function of only the relative positions of a subset of the nuclei. Thus performing clamped nuclei calculations will not make possible the expression of the electronic energy in anything other than internal coordinates and the electronic energy when expressed in any set of internal coordinates cannot be analytic everywhere. Thus there cannot really be a “global” potential energy surface. In any case, as has been seen, the potential, even locally, cannot be regarded as an approximation to anything in particular and thus should be treated simply as a convenient peg on which to hang further calculations. From this perspective, the further calculations should properly be ones in which the electronic Hamiltonian results from the full Hamiltonian in which the rotational motion has been made explicit. Such Hamiltonians have only a local validity and can be defined only where the Jacobian for the transformation to the rotational variables does not vanish. However at present there is no satisfactory account of how nuclear permutational symmetry should be treated from this perspective, neither is there any secure mathematical justification of the Born–Oppenheimer approximation or of the Born approach. Naturally any extension of the trial wavefunction for the full Coulomb Hamiltonian problem from a single term to a many term form must be welcomed as an advance; it is, however, simply a technical advance and it might prove premature to load that technical advance with too much physical import. At present it is not possible to place properly the clamped nuclei electronic Hamiltonian in the context of the full problem, including nuclear motion. However if the nuclei were treated as distinguishable particles, even when formally identical, then some of difficulties that arise from the consideration of nuclear permutations would not occur. But it would still be necessary to be able to justify the choice of subsets of permutations among identical particles when such seem to be required to explain experimental results. A particular difficulty arises here for it is not possible to distinguish between isomers nor is it possible to specify a molecular geometry, unless it is possible to distinguish between formally identical particles. But regarding the nuclei as distinguishable would not avoid the difficulty of constructing total angular momentum eigenfunctions from the nuclear and electronic parts. Such treatment of the nuclei would not make the traditional demonstrations of the Born–Oppenheimer or the Born approximations mathematically sound either. However it would ensure that the mathematically sound presentations of the Born– Oppenheimer approximation mentioned earlier need no further extension to include permutations of identical nuclei. There is, unfortunately, little good to be said, from a mathematical point of view, of the traditional Born argument. This is troubling because the Born approach is assumed to provide the basis for the consideration of chemical reactions on and between potential energy surfaces. However it is clear

22

B. Sutcliffe and R.G. Woolley

that the clamped nuclei electronic Hamiltonian can be usefully deployed in nuclear motion calculations if the nuclei are considered identifiable and in the next sections this Hamiltonian will be considered in detail.

The Symmetries of the Clamped Nuclei Electronic Hamiltonian Before considering the symmetry under permutations of identical particles it is necessary first to say a little about the spin of particles. Each particle is specified not only by space variables but also by spin variables. These have not been considered so far because there are no spin operators in the Hamiltonians discussed in the previous sections. Nevertheless spin is, indirectly, very important in the construction of approximate wavefunctions. Spin ideas are usually developed in terms of a vector operator S with three components S˛ ; ˛ D x; y; z and functions ‚N S;Ms ;k such that 2 N S2 ‚N S;Ms ;k D ¯ S .S C 1/ ‚S;Ms ;k ;

S2 D S2x C S2y C S2z ;

and 2 N Sz ‚N S;Ms ;k D ¯ M ‚S;Ms ;k ;

with, S D 0; 1; 2; : : : ;

S  Ms  S:

(44)

The ‚N S;Ms ;k are called spin eigenfunctions and the k index denotes a particular member of a possible set. In the simple case of a single electron, S D 12 ; Ms D ˙ 12 , and k D 1 so usually ‚11 ; 1 ;1 is written as ˛ and ‚11 ; 1 ;1 as ˇ. 2 2 2 2 It can be shown that under the operations of the full rotation-reflection group in three dimensions, O(3):     Sx Sy Sz ! Sx Sy Sz jRj R;

(45)

where R is the matrix representation of the rotation inversion and is thus an orthogonal matrix with determinant jRj which is C1 for a rotation and 1 for a reflection. Clearly S2 is an invariant operator. Thus, its eigenfunctions provide a basis for irreducible representations of O(3) and in general we label these representations by the S value to which they correspond, using k to distinguish between different basis functions for the same representation. Obviously the dimension of these is 2S C 1 since the 2S C 1 functions with MS D S; S  1; : : : ; S are degenerate in the sense of having the same S eigenvalue. It is useful to treat the spin functions as having variables si though these can take only discrete values and from now on to label the Cartesian space variables as ri so

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

23

that xi can be used to denote the totality of variables. To construct a many-particle function to describe space and spin for a collection of N identical particles it is apparently simply necessary to form the products: ‰n .x1 ; x2 : : : ; xN / D ‰n .r1 ; r2 ; : : : rN / ‚N S;Ms ;k .sl ; s2 ; : : : ; sN /

(46)

at will, and the generalization to groups of sets of identical particles is obvious. However there is a snag. It turns out that there is an extra rule that must be obeyed. The Hamiltonian is invariant under the operators that permute the variable designations (space and spin) of sets of identical particles. For any one set, these operators form a group and the full invariance group of H is the direct product of the groups for all the sets of identical particles. This means that the eigenfunctions of H are a basis for the irreps of this group. The extra rule is related to what irreps can actually occur and it is usually called the Pauli principle. If the set of identical particles individually have spin S D 0; 1; 2; 3, etc., then the only irreps that can arise are the totally symmetric ones. Thus every wavefunction must be invariant under permutations that interchange the variable designations of particles with integer spins (bosons). If the particles individually have spin S D 12 ; 32 ; 53 etc., then the only irreps are the antisymmetric ones. Thus every wavefunction must change sign if the permutation is odd or be invariant if the permutation is even for permutations that interchange the variable designations of particles with odd half-integer spin (fermions). What this means is that not every function of the form (Eq. 46) actually corresponds to a physical state, but it can be shown that in any given problem there are enough eigenfunctions of S2 with any given Ms value (i.e., k is sufficiently large) to form a basis for irreps of the permutation group of N particles. Similarly it can be shown that there are enough space functions to provide irreps of the same group. It is then possible to derive rules for combinations of the space and spin functions such that the resulting function obeys the Pauli principle. Since in this and the following sections only the clamped nuclei Hamiltonian will be considered, it will be convenient to simplify the notation somewhat and to drop the superscripts on the electronic variables because, from now on, these will be the only variables to be considered. Thus Eq. 2 is rewritten as 2

¯ H .a; r/ D  2m

N X r 2 .ri /  iD1 A0 X

e2 C 8 – 0

i;j D1

e2 4 –0

A X N X j D1 iD1

0

Zj jri aj j

C

N X

e2 8 –0

i;j D1

1

jri rj j (47)

Zi Zj ; jai aj j

while Eq. 3 is rewritten as H .a; r/

p

.a; r/ D Ep .a/

p

.a; r/ :

(48)

24

B. Sutcliffe and R.G. Woolley

For later purposes it is useful to group the first two terms by writing h.i / D 

A e 2 X Zj ¯2 2 ˇ ˇ  k.i / C V.i /; r .ri /  2m 4 –0 j D1 ˇri  aj ˇ

(49)

and to express the third term with g .i; j / D

1 e2 ˇ ˇ: 4 –0 ˇri  rj ˇ

(50)

The electronic part of the Hamiltonian for any given set of nuclear positions may then be written as 0

N N X 1X HD h.i / C g .i; j / : 2 i;j D1 iD1

(51)

The operator h is self-adjoint and a mathematically simple extension of the hydrogen atom operator. It therefore has some discrete one-particle eigenfunctions h.i / .ri / D – .ri / :

(52)

Such one-particle functions are called orbitals and the associated energies – are orbital energies. If there is only one nucleus, they are called atomic orbitals (AOs) and in the many nuclei case, molecular orbitals (MOs). The orbitals may be extended to become spin-orbitals by multiplying them with an ˛ or ˇ spin function and the resulting spin-orbitals are therefore denoted as (xi ) to indicate the inclusion of spin. Clearly every orbital can generate two spin-orbitals. It is usual in the case of electrons to work with functions that obey the Pauli principle from the start. This is possible when using an orbital basis through the use of Slater determinants of spin-orbitals which, by definition, change sign under odd permutations but are invariant under even permutations as is required for fermions. These will be explicitly considered later; first the permutational symmetry will be considered more generally.

Permutational Symmetry The clamped nuclei Hamiltonian is invariant under the permutation of electrons so that its eigenfunctions must be basis functions for irreducible representations (irreps) of the symmetric group of degree N. However, because of the Pauli principle, not all irreducible representations of this group are allowed. As noted above for electrons there are only two kinds of spin functions so that in any product of spin functions for N spin variables, only a limited set of permutations

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

25

of the spin variables will lead to distinct product functions. Since there are no spin terms in the electronic Hamiltonian, the operators for total spin commute with the Hamiltonian and so its eigenfunctions must be spin eigenfunctions as well as basis functions for an irrep of the symmetric group. To see how these two requirements can be combined consider permutations rather generally. The abstract permutation operator on N objects 1, 2, 3..... N is always denoted by  P D

1 2 3 ::: N i1 i2 i3 : : : iN

 ;

(53)

where the ir are the set of objects 1, 2, 3..... N at most in some new order. The effect of the operator on the original set of objects is P .1; 2; 3::::N / D .i1 i2 :::::iN / :

(54)

Clearly the product of any two permutations is a permutation, the products are associative, and there is an identity permutation, and every permutation possesses a unique inverse so the permutations form a group as required. The order of the group is obviously N!, because there are N!distinct permutations of N objects. This is the symmetric group of degree N, usually denoted as SN . A permutation like 

123 231

 is often

written as

.1 2 3/ ;

(55)

on the interpretation that 2 replaces 1, 3 replaces 2, and 1 replaces 3. The right-hand expression in Eq. 55 is said to be in a cycle form and is a cycle of length 3. Any permutation can be decomposed uniquely into cycles of disjoint elements, thus for example: 

123456 245136



   D 1 2 4 3 5 .6/;

(56)

where the order in which the individual cycles are written down does not matter and where, of course, each cycle is invariant under cyclic permutations of its elements. A cycle of length two is called a transposition and is an involutary operation, that is: .i1 i2 / D .i1 i2 /1 :

(57)

A cycle can always be written as a product of transpositions, thus 

    123 D 12 23 ;

(58)

26

B. Sutcliffe and R.G. Woolley

where the transposition on the right operates first, but the decomposition is obviously not unique nor are the transpositions always of disjoint elements. However the number of transpositions into which a cycle can be decomposed is obviously unique and this number is said to be the parity of the permutation, even if the number is even and odd if the number is odd. It follows, therefore, that we can classify the permutation operators themselves as either even or odd. Parity is usually indicated as .1/p where p is the number of transpositions. The product of two even permutations is an even permutation so that the even permutations constitute a subgroup (counting the identity permutation as even) of the symmetric group of order N!. This subgroup is called the alternating group. It is obvious then that the set of all distinct transpositions are the generators of the symmetric group. It is reasonably easy to show that all permutations in the same class have the same cycle structure.5 Thus if the possible cycle structuresare known, then the number of classes and hence the number of irreducible representations is known. But the number of cycle structures is obviously the number of possible ways that there is of dividing N objects in integer partitions. Thus for four objects the possible cycle structures are .i1 i2 i3 i4 / ; .i1 i2 i3 / .i4 / ; .i1 i2 / .i3 i4 / ; .i1 i2 / .i3 / .i4 / ; .i1 / .i2 / .i3 / .i4 / :

(59)

If there are m cycles in any partition, the cycle lengths are usually written symbolically as i , where 1 C 2 C 3 C ::::: C m D N;

(60)

1  2  3  :::::  m  0:

(61)

and by convention

Thus in Eq. 59 above in the first decomposition there is only one cycle and 1 D 4, in the second there are only two cycles 1 D 3 and 2 D 1. It is a custom to denote the totality of for any partition by the vector Π 4 Π1 ; 2 : : : : m and to indicate r repeats of a cycle of given length by r . Thus in Eq. 59 above the partitions should be denoted as [4], [3 1], [22 ], [2 12 ], and [14 ]. The notation (Eq. 59) is not especially well adapted for present purposes and instead a notation invented by Young, the so-called Young diagram, is used. The Young diagrams equivalent to Eq. 59 are

5 This means that the permutation and its inverse are always in the same class. A group with this property is said to be an ambivalent group.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

[4]

2

[ 3 1]

[2 ]

2

[2 1 ]

4

[1 ]

27

(62)

where the boxes in each row stand for a cycle and different cycles form different rows in the diagram. Since there are as many irreducible representations of a group as it has classes and there is one Young diagram for each class, Young diagrams can obviously be used to label each of the different irreducible representations. The dimensionality of the irreducible representation must pretty clearly be associated with the number of ways it is possible to put the objects 1, 2.... N into the boxes in the Young diagram, in a unique way. In fact it turns out that if the numbers 1, 2.... N are put into the boxes in a diagram so that when they are read across any row they are increasing and when they are read down any column they are also increasing, the totality of all such assignments is the dimension of the irreducible representation. Thus the first diagram in Eq. 62 can only yield

1

2

3

4

(63)

A Young diagram with numbers in it like (Eq. 63) is called a Young tableau and it is seen at once that there is only one tableau for the last Young diagram in Eq. 62, namely:

1 2 3 4

(64)

so that both [4] and [14 ] correspond to one-dimensional irreducible representations. The representation [3 1] yields

1 2

3

4

1 3

2

4

1 4

2

3 (65)

and in these circumstances it is clearly nice to have some sort of standard order; the one often adopted is to read the tableau across each row as if it were an ordinary number and then to arrange the tableaux in ascending order of these numbers, but other orders are used. Thus the numbers in Eq. 65 are

28

B. Sutcliffe and R.G. Woolley

1342

1243

1234

(66)

so that the standard order for the tableaux would be

1 4

2

3

1 3

2

4

1 2

3

4 (67)

This representation is therefore three-dimensional. One can obviously get the tableaux for [2 12 ] simply by transposing the tableaux for [3 1] and a little reflection will show that this is generally the case, simply because of the structure of the Young diagrams. The representations [2 12 ] and [3 1] are said to be conjugate representations (or sometimes adjoint, or dual or associated). In the present case the representation [22 ] is self-conjugate. Generally if the representation is denoted by [ ] then the conjugate representation is denoted Q and of course both representations are of the same dimension. by [ ], Since there is obviously freedom in which is associated with which, the choice is made in the two one-dimensional cases that [N] is associated with the symmetric representation and [1N ] with the antisymmetric representation. Thus if one has a Q is just Œ  1N . representation [ ], then the conjugate representation [ ] The [ ] symbols are also used to denote classes from time to time and in that context the correspondences established between Eqs. 62 and 59 are appropriate. Thus [14 ] represents the last partition in Eq. 59 which is clearly the class that just contains the unit operator, [2 12 ] stands for the class containing all operators involving only one transposition and so on. Now it can be shown that if one can find a basis for a representation [ ] for the h i Q symmetric group and another basis which is a representation for then the direct product of these two bases is a basis for the antisymmetric representation [1N ]. So to construct antisymmetric wavefunctions from space-spin products, as in constructing electronic wavefunctions, one can do so as N ‰ Œ1 D

X ΠQ

Œ

ˆi .r/ ‚S;MS ;i .s/;

(68)

i

where the sum goes over all functions in the basis for the irrep and N identical particles are assumed involved. They can have any spins at all and although in the expression above they are assumed to be spin eigenfunctions this is obviously not necessary just as long as a suitable set of spin products is used. However if they are spin- 12 particles then the only possible representations that can be constructed in the spin space are of the form [ 1 , 2 ]. No other irreps are possible. Furthermore in the case of N electrons properly coupled to a spin state S then 1 C 2 D N;

1  2 D 2S;

(69)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

29

so that 1 D

N C S; 2

2 D

N  S; 2

(70)

where

λ1 ~ ~ ~ ~ ~ ~ λ2 Thus any allowed space functions, such as those formed from orbital products, h wave i Q must form a basis for . This result arises basically because there are only two kinds of spin functions for spin- 12 particles. Thus lots of permutations that could arise do not arise in practice because they are the same if they involve only, say, ˛ spin particles. However it can also be shown that if the spin basis consists of spin eigenfunctions then the reps provided of SN are actually irreps and that they are the same irreps for any value of MS . That is, there are exactly as many spin eigenfunctions with the same S value as the dimension of the irrep [ 1 , 2 ] of SN . This is an enormous blessing and makes life much easier in electronic structure calculations than it might otherwise be. For fermions of spin greater than 12 it is not the case and it makes, for example, the problem of calculating statistical weights in molecular spectroscopy, or nuclear spin states in nuclear physics, much more complicated.

Point Groups and Transformations In the Cartesian coordinate system, the Laplacian operator for the ith electron is written as r 2 .i / D

@2 @2 @2 C 2 C 2; 2 @rxi @ryi @rzi

(71)

and the inter-electron distance as X 2 r˛j  r˛i rij D

!1=2 ;

(72)

˛

where here and hereafter sums over ˛ run over x, y, and z. The electron-nucleus distance will be written as

30

B. Sutcliffe and R.G. Woolley

rai D

X

!1=2 .rai  a˛ /

2

;

(73)

˛

where the ˛th coordinate of the nucleus a is written as a˛ to emphasize that it is a pure number and not a variable, since the nuclear positions are assumed to be fixed. It is easy to show that T    rj  ri ; rij2 D rj  ri

(74)

2 rai D .ri  a/T .ri  a/ ;

(75)

and

where the superscript T denotes the matrix transpose. Now let it be supposed that r is subject to positive rotation about the z-axis through an angle  to yield the vector r0 . Positive rotations are in the direction of a right-handed screw-twist when the point of the screw is moving in the axis direction. It is then easy to show that the new and old vectors are related by an orthogonal matrix r0 D R ./ r;

(76)

with 0

cos   sin  R ./ D @ sin  cos  0 0

1 0 0A: 1

(77)

In this section we shall be concerned only with transformations like (Eq. 77), that is, with point transformations and, in the case of the clamped nuclei Hamiltonian, consideration may be confined to finite point groups, specifically the crystallographic point groups. However from time to time it will be convenient to think a bit more generally and to recognize, for example, that the full rotation-reflection group in three dimensions is a point group, though an infinite one, and that some Hamiltonians are invariant under the operators of this group. Now it should be understood quite clearly as to what happens in a transformation of variables in a fixed basis. It is supposed that every point in space goes over into its image according to the appropriate transformation rule r0 D Rr;

(78)

where R is a constant orthogonal matrix.6 6 By “constant” here is meant simply that the elements of the matrix are not themselves dependent on the variables.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

31

Thus imagining the situation appropriate to the Schrödinger equation with Hamiltonian (Eq. 47), the assertion of a transformation of variables realized by R would imply r0i D Rri ;

i D 1; 2; : : : ; N:

(79)

However, in the fixed nuclei approximation, it would not imply that a0 D Ra;

a D any nucleus:

(80)

This is not for any sinister reason but simply because of the definition chosen of the clamped nuclei approximation. In this approximation the nuclear positions are triplets of numbers which take particular but fixed values once a particular basis is chosen.7 It is now very easy to show that the Laplacian and the inter-electronic distance are invariant under any constant orthogonal transformation of variables but that in general the electron-nucleus distances are not invariant. It is perhaps sensible to show how this comes about for it illustrates one or two points and makes the idea of invariance a bit clearer. First the Laplacian. Using the chain rule for differentiation 0

X @rˇi @ @ D 0 ; @r˛i @r˛i @rˇi

(81)

ˇ

since the transformation does not “mix” ri and rj so that X @ @ D Rˇ˛ 0 ; @r˛i @rˇi

(82)

X @ @2 D Rˇ Rˇ˛ 0 0 ; 2 @ri @rˇi @r˛i ˇ

(83)

ˇ

and hence

but r 2 .i / D

X @2 ; 2 @r˛i ˛

and

7 It is sometimes convenient to think of the nuclear positions as defining a particular embedding for the basis vectors or coordinate frame.

32

B. Sutcliffe and R.G. Woolley

X

  R˛ Rˇ˛ D ıˇ RT R D E3 ;

(84)

˛

therefore r 2 .i / D

X ˇ

ıˇ

X @2   @2 D D r2 i 0 : 0 0 2 0 @ri @rˇi @rˇi

(85)

ˇ

But clearly if r 2 .i /f .ri / D g .ri / ;

(86)

      r 2 i 0 f r0i D g r0i ;

(87)

then

so that r 2 .i / is an invariant operator under the transformation since its effect on an arbitrary function is unchanged (except for the variable names which, in this context, are irrelevant). Notice that if r 2 had been set up in the primed system as r 2 .i 0 / the same argument would have shown that r 2 .i 0 / D r 2 .i /, as required. Next rij . From Eq. 78 it follows that r0i D Rri

ri D RT r0i :

and

(88)

Combining the form (Eqs. 74 with 88) we get T      RT r0j  r0i rij2 D RT r0j  r0i  T   D r0j  r0i RRT r0j  r0i T    r0j  r0i ; D r0j  r0i so that 2

rij2 D rij0 ; showing that r2ij and hence rij is invariant in the same sense as above. For the electron-nucleus distance one gets  T  T 0  2 R r a rai D RT r0i  a  T 0 T  T  0i  R ri  Ra ; D R ri  Ra so that

(89)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

 T  0  2 ri  Ra ; rai D r0i  Ra

33

(90)

0

and this in general is obviously not rai 2 . However, if the transformation R is such that b D Ra

and

a D RT b;

(91)

where b is the position of another nucleus in the problem with the same charge as the nucleus whose position is a, then it is clear that even though individual terms in the electron-nucleus attraction operator are not invariant, nevertheless the operator as a whole is invariant. The generalization of this result is obvious. If the nuclear positions are divided into sets corresponding to nuclei of the same charge, the clamped nuclei Hamiltonian is invariant under all operators represented by orthogonal matrices R such that if b D Ra;

(92)

b is to be found in the same set as a was found.8 To look at it another way, were the coordinates of a set of equivalent nuclei to be written as a row matrix of the coordinates a1 a2 : : : ap (so that the “row” is actually a 3  p matrix), the effect of an invariance preserving operator (a symmetry operator) could be symbolized in the partitioned matrix form     b1 b2 : : : bp D a1 a2 : : : ap P;

(93)

where P is a p  p permutation matrix, that is, one of the form 0

1 0100:::0 B1 0 0 0 : : : 0C B C B C PDB: : : : C; B C @: : : : A 0010:::0

(94)

where the permutation corresponds to the effective rearrangement of the nuclei in the electron-nucleus attraction operator. Permutation matrices have just one nonzero entry in any row or column and that entry is always 1 so that such matrices are always orthogonal matrices. Their determinant can, however, be ˙1 depending on the parity of the permutation.

8 Notice that in this approximation the mass of the nucleus is of no consequence, only the charge matters.

34

B. Sutcliffe and R.G. Woolley

Thus, to all intents and purposes, when considering the symmetry of a molecule to be represented by a clamped nuclei Hamiltonian, all we need to look at is those orthogonal operators that let equivalent nuclear positions carry a permutation representation of the operator in question. Any such operator is a symmetry operator of the problem. We thus never need to think about the electrons and it is usual in elementary books to start from the point reached here, as if the operations were carried out on the nuclear positions. From here one can go on to show, in the usual way, that the collection of symmetry operators form a group and that the group is a point group essentially corresponding to the geometrical figure formed by the equivalent nuclei in the nuclear framework. Functions can always be visualized in terms of a fixed Cartesian basis and under the change (assumed orthogonal) r0 D Rr or r D RT r0 the change induced in a function f (r) is   OR f .r/ D f .r/  f RT r ;

(95)

meaning that f .r/ is constructed from f (r) by substituting (RT r)˛ wherever it occurs in the prescription for f (r). OR symbolizes the operation in function space. Considering the transformation of operators again, assuming that one specifies the operator form in a fixed basis, consider an operator A (assumed linear and Hermitian) such that Af .r/ D g .r/ ;

(96)

and by formal operator algebra we can rewrite this to exhibit the effect of the operator OR as OR Af .r/ D OR g .r/ ; OR Af .r/ D OR AO1 R OR f .r/ D OR g .r/ ; OR Af .r/ D OR AO1 R f .r/ D g .r/ ;

(97)

so that we can say that if R induces OR then it induces a change in the operators acting on the original function space such that A D OR AO1 R ;

(98)

where the functions on which the operators act are implicit in the formalism. Furthermore if the induced change is such that A D A, that is, A is an invariant operator, then A D OR AO1 R ;

(99)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

35

or OR A D AOR ; or OR A  AOR  ŒOR A D 0; and we say that the OR commute with A. We can thus characterize the symmetry operators in the function space that constitutes the domain of A as those operators which commute with A. In particular we are interested in the case when A is the Hamiltonian for the problem. Now, in practice, (Eq. 98) is not of much use for actually checking on the transformation properties of operators since it does not actually provide a specific rule for the construction of A. However as seen before, the actual variable names in an operator relation do not matter, so that if we regard RT r as just a new variable name, say y, then it follows immediately from Eqs. 96 and 97 that A is just the operator A made up in the same basis as A but with (RT r)˛ for every occurrence of r˛ in the operator. This prescription is usable in an obvious way for multiplicative terms in the operator and, for derivative terms, a little thought shows that the replacement X  @ X @ @ RT ˛ ! D R˛ @r˛ @r @r  

(100)

is the appropriate one for the new operator set up on the original basis. (Compare with Eqs. 81 and 82.) Thus we see that the momentum operator, for example, is not generally an invariant operator since from (Eq. 100) p˛ D

¯ i

X @ D R˛ p ; @r     p˛ D RT p ˛;

X

R˛

(101)

and (Eq. 101) is the equation that relates the elements of the transformed operator to those of the untransformed operator on a fixed basis. In fact it is easy to see from (Eq. 101) that in this case the operators actually provide a basis for the representation of R by R as p D pR;

(102)

where the operators are written as a row matrix. The transformation of operators is a matter to which we shall return later, but for the moment let it simply be noted that the arguments presented earlier in this section show that the clamped nuclei

36

B. Sutcliffe and R.G. Woolley

Hamiltonian is an invariant operator if R is a point group operation and we shall now look at the behavior of its eigenfunctions under OR . Any set of degenerate eigenfunctions of H carry a matrix representation of OR . Thus if we denote the set of degenerate eigenfunctions as f1 , f2 , : : : fn and treat these as a row matrix f, then we can write OR f D fOR ;

(103)

and we say that OR provides a matrix representation of OR in the basis fi so that a set of such functions in a function space play exactly the same role as do the unit vectors in a coordinate space and are, therefore, sometimes called basis functions of the representation. The representations provided in the basis of degenerate eigenfunctions are usually irreducible and can be chosen to be unitary matrices (in fact usually orthogonal matrices if the functions are real functions). In practice, what one is usually faced with is a collection of functions which have arbitrary (but known) transformation properties and what one actually wants to do is to adapt these functions so that they actually transform like the true eigenfunctions of the problem. This can be done by means of the group theoretical projection operator. If the unitary matrix irreducible representation of the point group is known and denoted by D() (R), where  labels the irreducible representation, then the projection operator is ./

Pij D

 n X  ./ D .R/ OR ; ij g R

(104)

where g is the order of the group and n the dimension of D() (R).9 This projection operator is such that ./

Pij f D fi

./

or a null result; ()

where f is an arbitrary function and f i such that

(105)

is the ith function in the row matrix f()

OR f./ D f./ D./ .R/:

(106)

The null result occurs in Eq. 105 if the arbitrary function has no component that lies in the subspace defined by the f() . In practice one often makes do with the character projection operator based on the characters () of the irreducible representation,

9 The operator in this form is clearly only possible for finite groups but similar operators are constructible for most infinite groups of interest.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

./ .R/ D

 X D./ .R/ ; i

ii

37

(107)

so that the character projection operator P./ is P./ D

 ./ X  ./  .R/ OR ; g R

(108)

and produces from an arbitrary function (if not a null result) then a function which is said to belong to the th irreducible representation but which in general transforms like a linear combination of the basis functions for D() . In practice, therefore, everything depends on knowing the representations of the group in question (and the irreducible representations are usually tabulated) and on knowing how the chosen functions actually transform under the operations of the group. It is helpful at this stage to think of constructing MOs as a linear combination of atomic orbitals (LCAOs) and as a preliminary to this the idea of an MO and of an AO need to be developed a bit further. It might be thought that orbitals could be best determined by solving the relevant one-electron problem specified by h, (Eq. 49), for each molecule. This is not so easily done and if a wavefunction is constructed as a determinant of spin-orbitals so constructed it yields a very poor approximation to the electronic energy. This is because the electron interaction term is large and has not been considered in constructing the orbital. Thus to get useful orbitals the fact that the electrons interact strongly must somehow be incorporated into the single particle potential through some sort of effective field process. This was first attempted for atoms in the 1920s by W. Hartree and D. R. Hartree (1927; 1936) but without considering antisymmetry. Their method was developed by Fock (1930) and by Slater (1930) in a properly antisymmetric form. Molecules resisted this approach because it was extremely difficult to solve the one-electron problem with a potential that was not centrosymmetric. However since it was possible to get atomic orbitals, two approaches using them developed, one at the hands of Pauling (1939) using atomic orbitals directly and the other at the hands of Mulliken (1931) by writing molecular orbitals as a linear combination of atomic orbitals. The Pauling approach was called the Valence Bond (VB) method and the Mulliken one the Linear Combination of Atomic Orbitals Molecular Orbital (LCAO MO) method. Although it was the VB method that was used in the first successful molecular electronic structure calculation, on H2 by Heitler and London (1927), it was the LCAO MO method that became standard in quantum chemical calculations following the work of Roothaan (1951) in the 1950s. Although these methods will be considered in great detail elsewhere in this volume, their basis can best be considered here. The most familiar atomic orbitals are those for the hydrogen atom which are usually designated according to their angular parts as s for l D 0, p for l D 1, d for l D 2, and so on where there are 2 l C 1 orbitals of each type. For calculational

38

B. Sutcliffe and R.G. Woolley

purposes Slater proposed that the radial part of the orbital, while still centered upon the nucleus, should be of the form N r n1 exp r;

n D 1; 2; : : : ;

(109)

where N is a normalizing factor, and  may be chosen at will or used as a variational parameter. It is customary to use a real form for the angular parts of the orbital so that, for example, the three p orbitals are specified as x exp r;

y exp r;

z exp r:

(110)

The most widespread atomic orbital form in use at present is the Gaussian which has the radial part N r n1 exp r 2 ;

n D 1; 2; : : : :

(111)

When it is not necessary to specify atomic orbitals in detail they will be denoted i (r). In elementary books it is often assumed that you can “eye-ball” the transformation properties from the “pictures” of the orbitals stuck on the nuclei. (Such pictures usually look a bit like twisted sausages and doughnuts.) Such an approach is a bit risky and knowing how to do this sort of thing algebraically avoids getting into the nasty mess that often happens if one relies on the pictures. Indeed, it is clear that it is quite impossible to draw a picture of an orbital. The sausages and doughnuts, therefore, must be at most projection representations in three (or two) dimensions and it is not clear that the transformation of the picture in three (or two) dimensions really corresponds correctly to the induced transformation of the orbital in the function space. If the atomic orbital (r) is situated at the origin of coordinates and the origin is the invariant point of the point group operators, then any orbital constructed about this origin simply transforms under OR to a new functional form in the space   OR .r/ D .r/ D RT r :

(112)

If, however, the orbital is developed about another origin, say a, so that the functional form is ..r  a// ;

(113)

then the situation is a little more involved. The coordinate point a is not a variable of the problem, but a parameter in the function construction. It does not change under the transformation so that

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

  OR ..r  a// D  RT r  a  D RT .r  Ra/ ;

39

(114)

but clearly Ra is some other fixed parameter point in the problem b, say, so that    RT .r  Ra/ D RT .r  b/   D RT rb :

(115)

Effectively the origin of the function is changed, but about that new origin, the transformed function is constructed by the usual rules in terms of the variable rb D .r  b/. Clearly if (RT r) is .r/ then (RT rb ) will be .rb / so that if one has worked out the transformation properties of an orbital at the origin .a D 0/ then the functional form of the transformed orbital carries over immediately to an orbital at arbitrary origin a given that b D Ra is known. Thus if one chooses a set of AOs which are identical but on different centers then they carry a permutation representation of the operation, much like (Eq. 94) but with appropriate transformation matrices in place of the 0 s and 1 s there. So if, for example, one had a set of identical s-orbitals with one on each center, the transformation matrices under any point group operation would just be 1, so that the matrix would remain just as in Eq. 94. But if one had a set of three equivalent p-functions on each center the 0 s would become 3  3 null matrices and the 1 s would become those three-dimensional transformation matrices which represent the operation on a set of three equivalent p-functions at the origin and so on. Once these transformation properties have been determined it is an easy matter to apply the projection operator either to construct functions that transform like irreducible representations or like a “special” representation. Thus in practice we can always get one-particle symmetry functions relatively easily, but the symmetry problem is clearly more involved when it comes to considering many-particle functions. The paradigm for an approximate many-particle function is a product of oneparticle functions (orbitals) ˆ .r1 ; r2 ; r3 ::::rN / D 1 .r1 / 2 .r2 / :::: N .rN / :

(116)

In thinking of approximate solutions to the clamped nuclei problem it is natural to think of the  r as molecular-orbitals and the ri as the electron variables, but the discussion that follows is quite general and it need not even be the case that the variables are those of identical particles. Now from what has been said before it can always be supposed that the orbitals are adapted to particular irreducible representations of the invariance group of the Hamiltonian and that one desires an approximate solution which is also adapted to a particular irreducible representation. The variable-space for the product function on the right-hand side of Eq. 116 is the direct sum of each of the variable-spaces for the particles, that is r1 ˚ r2 ˚ r3 ˚ :::: ˚ rN . The functions  r built on each of these variable-spaces individually constitute a

40

B. Sutcliffe and R.G. Woolley

separate function space for each variable and the total function space is, therefore, the direct product of all these function spaces. If, for the moment, we restrict each of these function spaces to sufficient functions to carry irreducible representations of the invariance group, then the direct product function space is just  .1/ ˝  .2/ ˝ :::: ˝  .N / ;

(117)

where the row matrix  (1) contains  1 and is a basis for the representation D(1) , of dimension n 1 , and so on, so that the total dimension of the space is n1  n2  ::::  nN D P . The symmetry operation R has an orthogonal representative R in each of the variable-spaces and it has a representative in the total variable-space which is the direct sum of N repeats of R. Thus the effect of OR on the product function (Eq. 117) is to produce  .1/ D.1/ .R/ ˝  .2/ D.2/ .R/:::: .N / D.N / .R/;

(118)

which, by the ordinary rules of manipulation for direct products, is  .1/ ˝  .2/ ˝ :::: ˝  .N / D.1/ .R/ ˝ D.2/ .R/:::: ˝ D.N / .R/:

(119)

The elements of the row matrix arising from the direct product of the orbitals can obviously be written (with suitable reordering if necessary) as a row matrix of functions like (Eq. 116) but involving such other partner orbitals as are necessary. Now if it happens that all the original orbitals belong to one-dimensional irreducible representations of the invariance group, then the matrix of direct products of representations in Eq. 119 is just a scalar, generally speaking ˙1, so that, in this case, ˚ obviously belongs to an irreducible representation of the invariance group of the problem. Otherwise the matrix of direct products of representations generally constitutes a reducible representation of the invariance group and in this case the product function (Eq. 116) and its partners do not, generally, carry an irreducible representation of the point group and must be suitably adapted to do so. There are clearly two possible ways in which this might be done. One could start off from ˆ and project out the required components, or one could recognize that there is a matrix relationship between the unadapted set and the adapted set and try to determine the matrix elements from group theoretical arguments. Both approaches have their uses but since the second approach has not yet been considered, it is appropriate to speak of it now. If, for the moment, we consider the direct product of just two irreducible representations, then it follows at once, from the character orthogonality theorem, that we can write down the number of times that any irreducible representation occurs in the reduction of the direct product representation. If we are thinking of representations  and in the direct product and  as the resulting representation, let us call this number (  ). It is zero or an integer and of course it is known once the character table for all the irreducible representations of a group are known. This

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

means that we can find a transformation U such that X .v / D./ ; U D./ ˝ D.v/ U D

41

(120)



where a direct sum is implied on the right-hand side, which runs over all the representations in the group. In Eq. 120 it is assumed that the representations D() , etc., are unitary so that from the definition of the direct product, D./ ˝ D. / is unitary and can, therefore, be reduced to irreducible form by means of a unitary transformation. Now if we denote by  () the row of n basis functions for D() , then the set of n D n n product functions  ./ ˝  . / are a basis for the direct product representations and clearly the set of product functions s./ ; s D 1; 2 : : : n defined by  ./ D  ./ ˝  . / U

(121)

are a basis for the reduced representation on the right-hand side of Eq. 120. It is usual to rewrite (Eq. 121) in the form s./ D

X

./ .v/

i j



ˇ  ˇ i; vj ˇ s ;

(122)

i;j

and refer to the elements of the unitary matrix in Eq. 122 as vector-coupling or Clebsch–Gordan coefficients. Care is needed here, because if (  ) is zero then clearly (Eq. 122) makes no sense, but this can be taken care of at the formal level by defining the vector-coupling coefficient as zero if (  ) is zero. There is also obviously some ambiguity if (  ) is greater than one and it will then be necessary to define some standard ordering. But nevertheless, the ideas are clear enough. The point is that if the irreducible representations of the groups are known then it is always possible to discover the vector coupling coefficients and hence to synthesize product functions that carry irreducible representations. This process is often called vector-coupling and obviously having coupled a pair of functions one can then couple another one to that pair and so on. This technique finds its principal use in dealing with the construction of eigenfunctions of the rotation group, but it is a quite general procedure. There are many different notations for the vector-coupling coefficients and great care must be exercised in determining just which convention a particular book is using, but it is perhaps appropriate to note that there is a common, pretty standard, notation in terms of the Wigner 3–j symbol defined, in our particular notation, so that (with the Wigner symbol as the object in brackets on the right-hand side) 

1=2

.i; j j  s/ D Œ



  i j s

 ;

(123)

42

B. Sutcliffe and R.G. Woolley

where [ ] is the dimensionality of the  th irreducible representation. There are, obviously, different sets of 3– j symbols for every group, but the ones most commonly tabulated are those for the three-dimensional rotation group; they are also available for the crystallographic point groups. Whether one chooses synthetic or projective methods to construct many-particle symmetry functions is a matter of choice in any given problem. Having considered the transformation properties of functions, consider how these properties may be used in the construction of matrix elements of operators and, from this, consider again how operators themselves transform. Consider the matrix elements of an arbitrary operator A between two functions. We symbolize the matrix element A12 by Z A12 D

1 .x/ A2 .x/ dx;

(124)

where the integral is a definite one over all space and over all coordinates and where dx symbolizes the appropriate volume element. Since the variables of integration are dummy we may also write Z A12 D

1 .y/ A0 2 .y/ dy;

(125)

where A0 symbolizes A made up with y just as A was made up with x. But now let it be supposed that y corresponds to a variable change induced by a transformation so that y D RT x. Then Z  A12 D  1 .x/ A  2 .x/ j R j dx; (126) where A is the transformed operator as in Eq. 98 and  i the transformed function as in Eq. 95. j R j is the Jacobian of the transformation and since the matrix R is orthogonal, j R j is ˙1. If it is 1 then it can be shown that the signs of the limits change also so that the integral is unchanged and thus jRj can always be treated as C1. Thus (Eq. 126) may be rewritten as Z A12 D



 1 .x/ A  2 .x/ dx:

(127)

Now assume for a moment that A is H then of course A is also H if R is a symmetry operator of the problem; thus it is the case that Z

But suppose it turned out that

1 H2 dx D

Z



 1 H 2 dx:

(128)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

 1  OR 1 D 1 ;

43

(129)

and  2  OR 2 D 2 ; then one would have proved that Z

1 H2 dx D 

Z

1 H2 dx;

(130)

and clearly the only way for this to happen is for the integral to vanish. This is a familiar result as is its generalization, namely, that if  1 is a basis function for a particular irreducible representation and  2 is one for another irreducible representation, then the matrix element vanishes unless the direct product of these two representations is reducible to a direct sum of representations which includes the totally symmetric (unit) representation. It should be remembered that although the integral certainly vanishes if the totally symmetric representation is absent, even if this representation is present, the integral could also vanish, for reasons other than the symmetry considered. Thus in the context of the selection rules based on the matrix elements one can say that a particular transition is forbidden by symmetry, one cannot say that it is allowed by symmetry.

Spin and Point Group Symmetry So far spin has not been considered explicitly. It is usual to ignore it in discussions of point group symmetry because the total spin operator is invariant under all point group transformations and the axis orientation is defined by the fixed frame choice so there is a fixed internal choice for the z-axis. Thus it is regarded as sufficient to require that the spatial part of any trial function has the correct point group symmetry and then to form properly antisymmetric functions from the spatial parts and the spin eigenfunctions in the manner outlined earlier. If orbitals are used to construct the spatial part, then it is usual simply to extend the symmetry orbitals to become symmetry spin-orbitals. Returning to the functional form used in Eq. 116 this would become ˆ .x1 ; x2 ; x3 ::::xN / D A1 .x1 / 2 .x2 / : : : N .xN / ;

(131)

where A is the projection operator for the antisymmetric representation of the symmetric group of the electrons. It is usually just called the antisymmetrizer: AD

1 X 2 pOp; NŠ p

44

B. Sutcliffe and R.G. Woolley

where OP is the permutation operator and OP is the parity of the permutation. OP operates on the electronic variables and should thus be associated with the inverse of a particular permutation. However since the symmetric group is ambivalent and the representation of interest is one-dimensional this is a distinction without a difference. In the case above where the initial function is an orbital product, the operator can equally well be treated as if it operated on the orbital index. The space parts of the spin-orbitals  i and  j can be the same if the spin parts are different. If any two spin-orbitals are the same the projected function simply vanishes. This vanishing is the basis of what is usually called the “Pauli exclusion principle.” The function (Eq. 131) is clearly a determinant of spinorbitals with the spin-orbital index designating a row (column) and the electron numbering designating a column (row). This was first recognized by Slater and so such determinants are called Slater determinants and often denoted by the shorthand M j1 .x1 / 2 .x2 / : : : N .xN /j ; where M is a normalizing factor. Such Slater determinants are not themselves always spin eigenfunctions but they are eigenfunctions of Sz . They can be either coupled or projected to yield spin eigenfunctions for a particular Ms value and the resulting functions can then be further projected by the step-up operator SC D Sx C i Sy to produce functions with Ms C 1 or the step-down operator S D Sx  i Sy to produce functions with Ms  1. The presence of spin in the trial function does not modify the previous point group symmetry discussion. However the requirements of spin and permutational symmetry may mean that a trial function is constrained so that it might not be possible to have a trial function with a particular point group symmetry and a specified spin symmetry. As an example consider a determinant of doubly occupied orbitals for four electrons symbolized as M j1 .r1 / ˛ .s1 / 1 .r2 / ˇ .s2 / 2 .r3 / ˛ .s3 / 2 .r4 / ˇ .s4 /j : This determinant is a spin eigenfunction with S D 0. If the orbitals are point group symmetry orbitals, the only possible symmetry of the many particle function is that generated by the direct product of each orbital symmetry with itself followed by the direct product of the resulting symmetries. If the orbitals belong to onedimensional irreps then the many particle function must belong to the totally symmetric representation of the point group. It is not possible to represent other symmetries with a function of this form.

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

45

The Construction of Approximate Eigenfunctions of the Clamped NucleiHamiltonian The basic mathematical tool used in the construction of approximate eigenfunctions is the variation theorem. This theorem asserts that for any square integrable function ˚ which is of a definite and allowed symmetry for the Hamiltonian, the quotient R  ˆ Hˆdx R D E  Ef ; ˆ ˆdx

(132)

where Ef is the lowest energy of the particular symmetry. The function ˆ is chosen to contain parameters which can be varied to minimize E to make it as close as possible to Ef . The function is usually scaled so that the denominator in the quotient is unity and the function with this scaling is said to be normalized to unity or just to be normalized. If ˆ is chosen as a linear combination of functions with the coefficients as the parameters then minimizing E leads to a secular problem the roots of which are upper bounds not only to the lowest states of the particular symmetry, but to excited states of increasing energy. At this stage the secular problem can be approximated and approximations to the eigenfunctions can be developed by perturbation theory. It is usual to construct approximate solutions to the electronic Schrödinger equation in terms of functions composed of spin-orbitals  r (x). In general it is supposed that we have available a set of m . N / orbitals (r) and to each of these we attach an ˛(s) or ˇ(s) spin factor function to form the spin-orbitals. When we write x as a variable designation we shall in future mean space and spin variables collectively. We can write formally .1 : : : 2m / D .˛ˇ/  .1 : : : m /   D   ;

(133)

where  denotes the standard direct (or Kronecker) product of matrices. The N-electron functions ˆk are composed of Slater determinants of N spinorbitals or perhaps as a fixed linear combination of a number of such determinants. The matrix elements of H between such functions can be written in terms of oneand two-electron integrals over the spin-orbitals thus Hkl  hˆk jHj ˆl i D

Xkl rs

1 Qrs hr jhj s i C

1 Xkl 1 Qrs;tu hr s jgj t u i ; 2 rstu (134)

where the sums go over all spin-orbitals and where kl Q1rs and the kl Q2rs,tu are simple coefficients sometimes called projective reduction coefficients or coupling coefficients or alternatively one- and two-particle density matrix elements. These coefficients will of course be zero if any of the spin-orbitals shown in their indices do not occur in the pair of functions considered. The notation for the one and twoelectron integrals is conventional,

46

B. Sutcliffe and R.G. Woolley

Z hr jhj s i D

r .r/ hs .r/ dr;

(135)

and Z hr s jgj t u i D

r .r1 / s .r2 / g.12/t .r1 / u .r2 / dr1 dr2 :

(136)

Similarly the overlap matrix elements can be written as D ˇ E D ˇ E Xkl ˇ ˇ 1 r ˇs : Mkl D ˆk ˇˆl D N 1 Qrs

(137)

rs

Now let it be supposed that from a given set of spin-orbitals all possible functions ˆk are constructed and that the resulting set has M members. The best wavefunctions that could be obtained from such a set would be a linear combination of these M terms with coefficients determined using the variation theorem. Use of the variation theorem on such a set leads to the familiar secular problem H  EM jD 0

(138)

whose solutions give M eigenvalues E1  E2  E3     EM each of which is an upper bound to the first, second, third, and so on exact solution of the electronic Schrödinger problem. In the special case where the spin-orbitals are orthonormal and the trial functions are Slater determinants the expressions for the projective reduction coefficients are both simple and limited, given by Slater’s rules to be discussed in detail in later chapters in this work. With such a choice there are Hamiltonian matrix elements between functions that differ from each other only in two or fewer orbitals and Mkl D ıkl . The expressions for these coefficients when the orbitals are not orthogonal involve the overlap integrals Sij between all the orbitals in the functions and there is no limitation on orbital differences between the functions and Mkl is not the unit matrix. Every electronic permutation must be considered in their evaluation. For non-orthogonal orbitals it is thus much more difficult to consider systems with more than a few electrons and, because atomic orbitals on different centers are not orthogonal, this difficulty has hindered the development of VB theory in a quantitative manner until very recently. An account of modern VB developments forms a later part of this handbook. Usually LCAO MOs are developed so as to be orthogonal so that given i D

m X pD1

p Cpi

i D 1; 2; 3 : : : ; m;

(139)

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

47

˛ ˝ to evaluate the two-electron integral i j jgj k l involves evaluating all the twoelectron integrals h r s jgj t u i and summing. To transform all the two-electron integrals would seem at first sight to involve m8 multiplications and additions and thus put it beyond computational possibility for anything other than a small number of orbitals. However it actually needs operations only of the order of m4 , which is bad enough, but can be dealt with. Although it proved possible to develop the LCAO method with Slater orbitals for diatomic molecules, the evaluation of two-electron integrals when the atomic orbitals were centered on three or four distinct nuclei proved very difficult and is not, even now, commonly undertaken. However it proved possible by the use of Gaussian functions, as pioneered by McWeeny (1950) and developed by Boys (1950) in the 1950s to evaluate the two-electron integrals swiftly and to high precision. Tests on atoms and diatoms showed that although many more Gaussian functions were required to obtain results comparable with a given set of Slater orbitals, the computational time involved was not much longer. Results using Gaussian functions are now regarded as standard for polyatomic systems. For ease of exposition we shall for the moment assume that the orbitals  r are orthonormal so that the spin-orbitals are also orthonormal. If the ˚ k are then chosen as normalized Slater determinants or proper linear combinations of them, they too may be chosen to be an orthonormal set so that Mkl D ıkl . The secular problem then simplifies to jH  E1M j D 0;

(140)

and this is the form that is typically taken in a CI calculation with orthonormal Mos  r and orthonormal configuration functions ˆk , while Eq. 138 is the form typically taken in a VB calculation where, in general, neither the orbitals nor the structure functions are orthonormal. Let us consider what happens if we subject the spin functions to the transformations  !  U.2/;

(141)

and the orbitals to the transformations  ! U.m/;

(142)

where U(n) is an n  n unitary matrix. In this case the spin-orbitals change  !  0 D  U.2/  U.m/ D .  / .U.2/  U.m// ; and it is easy to show that the  0 form an orthonormal set still.

(143)

48

B. Sutcliffe and R.G. Woolley 0

If we now make up the M functions ˆk but using the spin-orbitals  k rather than 0 the  k , it is easy to show that the  k continue to form an orthonormal set because 0 the  r are orthonormal. Since the set ˆk is complete, this means that the  k are at most a unitary transformation of the  k , that is: ˆ0k D

M X

ˆn Unk .M /;

(144)

nD1

where U(M) is an M  M unitary matrix. This in turn implies that the matrix H is transformed as H ! H0 D U .M /HU.M /:

(145)

The new secular problem is, therefore, ˇ 0 ˇ ˇH  E1M ˇ D 0;

(146)

ˇ ˇ  ˇU .M / .H  E1M / U .M /ˇ D 0;

(147)

which can be rewritten as

and using the property jABj D jAj jBj of determinants of square matrix products it follows at once that the secular problem (Eq. 147) is identical with the secular problem (Eq. 140). Thus the energies are completely unchanged by this unitary transformation. All possible unitary matrices of a given dimension, say n, form a group under matrix multiplication, a group that is usually written as U(n) and thus we can say that the secular problem is invariant under the operations that constitute the group U .2/  U .m/. It is wrong to think of this group as a transformation group in the same way as a point group. There is no coordinate transformation that corresponds to the unitary transformation on the orbital space. In this case the transformation on the function space is all that there is, the unitary matrices do not “correspond” to anything though of course, one can assert that each matrix formally represents a particular unitary operator if one wishes. If one does think in this way then we can regard the result that has just been shown as a very weak form of a fundamental result due to Wigner who showed that unitary invariance is a very deep invariance in any quantum mechanics based on square integrable eigenfunctions. At one level the result just shown is quite useless because for m reasonably large M is astronomical. One can refine the result a little to make M less big, for one

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

49

can show that if one starts off with each containing N˛˛-spin a set of determinants   orbitals and Nˇ ˇ-spin-orbitals N˛ C Nˇ D N so that Ms D N˛  Nˇ =2 for every determinant, then it is sufficient to include only transformed functions with the same Ms value. Further one can show that if one starts off with functions which are fixed linear combinations of determinants which are spin eigenfunctions (i.e., have a particular expectation value S say, of S2 ), then again one only need consider the transformed functions that preserve S. But to give some idea of how big the M can be, the following figures are relevant. For a system where the number of electrons, N, is equal to 10 (as in H2 O) and the number of orbitals, m, is equal to 30, the total number of determinants M is 7:54  1010 . If the process is restricted to those functions for which Ms D 0, M drops to 2:03  1010 and if the process is further restricted to those functions for which S D 0 then M becomes 4:04  109 . Clearly it is not possible to compute all the matrix elements and then to solve the secular problem. However it is possible to utilize the group theoretical results in the design of computational methods that yield tractable problems. The key to the connection between matrix element evaluation and the unitary group U(m) is because the Hamiltonian can be expressed in terms of the generators of this group. The spatial parts of the trial functions can be chosen to provide a basis for irreducible representations of U(m) corresponding to a particular choice of S and of Ms for spin symmetry by the use of Weyl tableaux. The required Young diagram for the particular choice of S and of Ms is constructed and then the conjugate diagram written down. Into this diagram the orbital indices are put in such a way that across any row the indices are nondecreasing and down any column they are strictly increasing. The resulting tableaux are the Weyl tableaux and each tableau labels a basis function for an irreducible representation of the group U(m) of dimension equal to the total number of tableaux. Thus for a three electron problem using four orbitals, choosing the Young diagram [2 1] would yield 20 distinct Weyl tableaux. Rather convenient rules may then be specified for matrix elements between such functions. This approach will be considered more fully later in this work. There is, however, one particularly simple and commonly useful case that is appropriate to consider here. That is the case where the trial wavefunction is simply a single Slater determinant of doubly occupied orbitals and it is a commonly useful form because many molecules have an even number of electrons and their electronic ground states are, in many cases, totally symmetric singlet states. The expected value of the Hamiltonian for such a trial function is X X .2 hr s jgj r s i  hr s jgj s r i/; ED2 (148) hr jhj r i C r

rs

where the sum is over orbitals, the spin having been integrated out leaving only the spatial integrations to be performed. Using the form (Eq. 139) for the orbitals E may be written as E D 2t rhR C t rGR;

50

B. Sutcliffe and R.G. Woolley

where ˛ ˝ hij D X i jhj j ; Rij D Cir Cjr; Gij D 2Jij  Kij ;

Jij D

X

r ˝ ˛ Rkl i k jgj j l ;

Kij D

kl

X

˝ ˛ Rkl i k jgj l j :

kl

It can then be shown that the elements of C that minimize the energy can be obtained by solving the generalized eigenvalue problem hF C D sC–; where hF D h C G;

D ˇ E ˇ Sij D i ˇ j ;

and – is a diagonal matrix with the orbital energies –r along the diagonal. Because the construction of hF involves a knowledge of C the construction of eigenvalues must start with a first guess at C too, obtain an improved guess, and be continued until no further improvement occurs. There are a number of ways in which this may be done but a popular way is simply to solve the generalized eigenvalue problem iteratively. This scheme has given the name self-consistent field (SCF) to the method generally. The contributions of Hartree and of Fock to its development are sometimes acknowledged by denoting it the HF SCF method. The SCF method is, at present, the most widely used method in computational chemistry and forms a central feature of all the commonly available computer packages for computational chemistry. Provided that the basis of AOs is full enough, the resulting functions yield energies that are often sufficiently good to enable meaningful distinctions to be made between molecules with different nuclear geometries and so to aid the interpretation of spectroscopic results and reaction mechanisms. However the basis of AOs cannot be made too large because of the number of electron repulsion integrals that must be computed and the difficulties of storing and manipulating these. It is seen, however, that the distinctive feature in the calculation of the electronic energy is the potential term V.i /, all the other terms being of precisely the same form whatever the geometry of a given molecule and between molecules. The potential depends only upon the coordinates of a single particle and thus can be realized in terms of one-particle integrals. This is usually expressed by saying that the potential depends only upon the one-particle density functional. It was shown by Hohenberg and Kohn (1964) that the electronic energy depended on the density functional in a unique way and that, if the density functional was known, then so was the energy. This result has led to the development of variants on the SCF method which involve approximations to the density functional form and require many fewer integral

The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

51

evaluations than does the direct use of an LCAO MO SCF method. Thus much larger molecules can be tackled in this approach than would otherwise be possible.

Conclusions It seems fair to say that if one treats the nuclei as distinguishable particles and takes the sum of the electronic energy obtained as an eigenvalue of the clamped nuclei Hamiltonian and the classical nuclear repulsion energy as a potential, the geometrical structure of the minimum in such a potential can be identified as the equilibrium molecular geometry. This can be done without appeal to the chemical bond, and without recognition of functional groups. Certain aspects of the transition state theory of chemical reactions can be rationalized in terms of structures anticipated using the potential at geometries away from the equilibrium one. If the occurrence of polar molecules is considered to be an aspect of molecular structure, polar molecules can be recognized from fixed nuclei electronic structure calculations by means of a dipole moment calculated as the sum of the electronic dipole and the classical nuclear dipole. And there are many other ways in which quantum mechanics formulated with the requirement that the nuclei may be treated as distinguishable clampable particles has been effectively used to illuminate chemical behavior. However, even if this approach is regarded as giving a satisfactory account of chemical structure, it still remains to justify by full quantum mechanical means the treatment of the nuclei that it involves. But at present such a justification still eludes us. It may be in the future that the multiple well approach to nuclear permutational symmetry will be shown to be properly founded and thus the eigenvalues of the molecular Hamiltonian10 to be just those anticipated from the previous approach; even so one will still be left with eigenfunctions which exhibit full permutation and rotation-inversion symmetry and it seems impossible to anticipate anything at all like classical chemical structure from these using the standard quantum mechanical machinery. An obvious objection to this discussion is that the full symmetries of the Coulomb Hamiltonian we have been discussing are manifested necessarily by its stationary states, whereas chemists are concerned with time-dependent states that may exhibit less symmetry. The time-dependent Schrödinger equation simply describes unitary time evolution of a prior state and will not change symmetries; any given initial state is the result of some previous time evolution so what is required is a quantum mechanical theory of initial states that are consistent with chemistry; about that nothing is known. The most that it seems possible to say is that chemical structure can be teased out in terms of a chosen ansatz (the imposition of fixed, distinguishable nuclei) if one has a good idea of what one is looking for. It seems

10

The Coulomb Hamiltonian for the electrons and nuclei specified by the molecular formula.

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B. Sutcliffe and R.G. Woolley

unlikely that one would ever guess that it was there in the full Coulomb Hamiltonian for a molecular formula unless one had decided on its presence in advance.

References Born, M., & Huang, K. (1955). Dynamical theory of crystal lattices. Oxford: Oxford University Press. Born, M., & Oppenheimer, J. R. (1927). Zur Quantentheorie der molekeln. Annalen der Physik, 84, 457. Boys, S. F. (1950). Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 200, 542. Cassam-Chenai, P. (2006). On non-adiabatic potential energy surfaces. Chemical Physics Letters, 420, 354. Collins, M. A., & Parsons, D. F. (1993). Implications of rotation-inversion-permutation invariance for analytic molecular potential energy surfaces. The Journal of Chemical Physics, 99, 6756. Combes, J. M., & Seiler, R. (1980). Spectral properties of atomic and molecular systems. In R. G. Woolley (Ed.), Quantum dynamics of molecules. NATO ASI B57 (p. 435). New York: Plenum. Czub, J., & Wolniewicz, L. (1978). On the non-adiabatic potentials in diatomic molecules. Molecular Physics, 36, 1301. Deshpande, V., & Mahanty, J. (1969). Born–Oppenheimer treatment of the hydrogen atom. American Journal of Physics, 37, 823. Fock, V. (1930). Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Zeitschrift für Physik, 61, 126. Frolov, A. M. (1999). Bound-state calculations of Coulomb three-body systems. Physical Review A, 59, 4270. Hagedorn, G., & Joye, A. (2007). Mathematical analysis of Born–Oppenheimer approximations. In F. Gesztesy, P. Deift, C. Galvez, P. Perry, & W. Schlag (Eds.), Spectral theory and mathematical physics: A festschrift in honor of Barry Simon’s 60th birthday (p. 203). London: Oxford University Press. Handy, N. C., & Lee, A. M. (1996). The adiabatic approximation. Chemical Physics Letters, 252, 425. Hartree, D. R. (1927). The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods. Mathematical Proceedings of the Cambridge Philosophical Society, 24, 89. Hartree, D. R., & Hartree, W. (1936). Self-consistent field, with exchange, for beryllium. II. The (2s)(2p) 3 P and 1 P excited states. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 154, 588. Heitler, W., & London, F. (1927). Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Zeitschrift für Physik, 44, 455. Herrin, J., & Howland, J. S. (1997). The Born–Oppenheimer approximation: Straight-up and with a twist. Reviews in Mathematical Physics, 9, 467. Hinze, J., Alijah, A., & Wolniewicz, L. (1998). Understanding the adiabatic approximation. The accurate data of H2 transferred to HC 3 . Polish Journal of Chemistry, 72, 1293. Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Physical Review, 136, 864. Hunter, G. (1975). Conditional probability amplitudes in wave mechanics. International Journal of Quantum Chemistry, 9, 237. Hunter, G. (1981). Nodeless wave functions and spiky potentials. International Journal of Quantum Chemistry, 19, 755. Kato, T. (1951). On the existence of solutions of the helium wave equation. Transactions of the American Mathematical Society, 70, 212.

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Klein, M., Martinez, A., Seiler, R., & Wang, X. P. (1992). On the Born-Oppenheimer expansion for polyatomic molecules. Communications in Mathematical Physics, 143, 607. Kołos, W., & Wolniewicz, L. (1963). Nonadiabatic theory for diatomic molecules and its application to the hydrogen molecule. Reviews of Modern Physics, 35, 473. Kutzelnigg, W. (2007). Which masses are vibrating or rotating in a molecule? Molecular Physics, 105, 2627. Longuet-Higgins, H. C. (1963). The symmetry groups of non-rigid molecules. Molecular Physics, 6, 445. McWeeny, R. (1950). Gaussian approximations to wave functions. Nature, 166, 21. Mohallem, J. R., & Tostes, J. G. (2002). The adiabatic approximation to exotic leptonic molecules: Further analysis and a nonlinear equation for conditional amplitudes. Journal of Molecular Structure: Theochem, 580, 27. Mulliken, R. S. (1931). Bonding power of electrons and theory of valence. Chemical Reviews, 9, 347. Nakai, H., Hoshino, M., Miyamato, K., & Hyodo, S. (2005). Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory. The Journal of Chemical Physics, 122, 164101. Pauling, L. (1939). The nature of the chemical bond. Ithaca: Cornell University Press. Roothaan, C. C. J. (1951). New developments in molecular orbital theory. Reviews of Modern Physics, 23, 69. Slater, J. C. (1930). Note on Hartree’s method. Physical Review, 35, 210. Sutcliffe, B. T. (2000). The decoupling of electronic and nuclear motions in the isolated molecule Schrödinger Hamiltonian. Advances in Chemical Physics, 114, 97. Sutcliffe, B. T. (2005). Comment on “Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory”. The Journal of Chemical Physics, 123, 237101. Sutcliffe, B. T. (2007). The separation of electronic and nuclear motion in the diatomic molecule. Theoretical Chemistry Accounts, 118, 563. Thirring, W. (1981). Quantum mechanics of atoms and molecules. A course in mathematical physics (Vol. 3, E. M. Harrell, Trans.). Berlin: Springer. Wilson, E. B. (1979). On the definition of molecular structure in quantum mechanics. International Journal of Quantum Chemistry, 13, 5. Woolley, R. G., & Sutcliffe, B. T. (1977). Molecular structure and the Born–Oppenheimer approximation. Chemical Physics Letters, 45, 393.

Remarks on Wave Function Theory and Methods D. Kedziera and A. Kaczmarek-Kedziera

Contents Introduction: What and Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Mechanics for Dummies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Way to Quantum Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational Principle: An Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation Calculus: The Art of Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Electron Approximation: Describe One and Say Something About All . . . . . . . . . . . . . . . . . . . Hartree–Fock Method: It Is Not That Sophisticated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Møller–Plesset Perturbation Theory: HF Is Just the Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond the HF Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled Cluster Approximation: The Operator Strikes Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 11 14 15 20 25 30 37 43 48 49

Abstract

Methods of computational chemistry seem to often be simply a melange of undecipherable acronyms. Frequently, the ability to characterize methods with respect to their quality and applied approximations or to ascribe the proper methodology to the physicochemical property of interest is sufficient to perform research. However, it is worth knowing the fundamental ideas underlying the computational techniques so that one may exploit the approximations intentionally and efficiently. This chapter is an introduction to quantum chemistry methods based on the wave function search in one-electron approximation.

D. Kedziera () • A. Kaczmarek-Kedziera Faculty of Chemistry, Nicolaus Copernicus University, Toru´n, Poland e-mail: [email protected]; [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_3-2

1

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D. Kedziera and A. Kaczmarek-Kedziera

Introduction: What and Why? Quantum chemistry is a branch of science originating from quantum mechanics that focuses on investigations of chemical systems. The mathematical roots of quantum chemistry allow it to be treated as a methodology for solving an eigenvalue equation for operators or – even simpler – finding solutions for some differential equations. We cannot totally escape this way of thinking, since this is how things really are. However, a chemist will comprehend quantum chemistry more as a helping tool in experimental work, supporting description of chemical reactions, a tool that plays a role similar to a spectrophotometer or a chromatographic column, a tool that can provide information about the system under consideration, and a powerful tool, the popularization of which was achieved thanks to the fast progress of computer power and the hard work of people who made the transformation from pure theory to computer programs possible. Their efforts were appreciated – in 1998, the Nobel Prize was awarded to Walter Kohn “for his development of the density functional theory” and John Pople “for his development of computational methods.” From the point of view of the experimentalist, the apparatus of quantum chemistry can be perceived similarly as the NMR spectrometer. One knows that the quality of the obtained NMR spectrum depends not only on the magnetic field of the magnet but also on the signal-processing capabilities. To successfully use NMR spectroscopy in experimental work, detailed knowledge about the technology of production and preparation of the magnets and electronic equipment is not a requisite. It is enough to keep in mind that with the given frequency one gets corresponding accuracy and information. All the rest is simply skill in sample preparation and expertise in spectrum interpretation. For effective usage of computational techniques of quantum chemistry, one must be aware of applied approximation to tune the accuracy of calculations and possess knowledge of the physicochemical phenomenon one wants to describe. The aim of the present chapter is to provide a gentle introduction to basic quantum chemistry methods – the methods of solving the electronic Schrödinger equation. The chapter is intended for people starting their adventure with computational chemistry and wanting it to become the tool, not the aim itself. When discussing quantum chemistry, we cannot totally avoid quantum mechanics. However, let us use another comparison: Traveling abroad it is good to know some basic expressions in the local language of the country you go to. It makes life easier and gives pleasure in interpersonal contacts. Still, no one expects a tourist to speak a language as fluently as a native speaker. Therefore, to efficiently apply computational techniques in experimental research, one has to learn some basic quantum mechanical terms that will help during the journey through the remainder of this chapter.

Remarks on Wave Function Theory and Methods

3

Quantum Mechanics for Dummies We will begin with the basic terms of quantum mechanics. In this section, they will be introduced in an intuitive manner, to enable understanding of the next sections’ content, even by beginners. We consider a system of N electrons in the field produced by the potentials arising from nuclei (the nuclei are not treated as particles consisting of nucleons but just as a point source of the electrostatic potential). We are interested only in one particular case: • The probability of finding the electrons of the considered system on the infinite distance from the nuclei is equal to zero. In other words, we want our system to constitute the whole entirety, not breaking into separate and independent parts (this would be the case of the interaction of two electrons with no attraction – two negative charges would repel each other to infinity). • The energies of this system constitute the discrete spectrum. • We want to know only the lowest value of energy (the wider approach can be found in the next volume of the present book). With such limitations, we do not need to consider all of the different general cases and can simply concentrate on the bound-state chemistry. A central notion in quantum chemistry is a wave function. This is a function characterizing a state of the system. Therefore, it depends on the variables that are adequate for the given system. This means that the wave function has to depend, at least, on spatial coordinates describing motions of the particles in the investigated system. Moreover, the wave function depends on so-called spin variables (spin is an additional degree of freedom included a posteriori in nonrelativistic quantum mechanics). This spin dependency can be built into the wave function by introducing a spin function. For instance, for the electron with a label 1, its wave function depends on the spatial coordinates x1 , y1 and z1 and is multiplied by the spin function ˛( 1 ) or ˇ( 1 ), where  1 is a spin variable. The spin functions must fulfill the following requirements: Z Z

˛  .1 / ˛ .1 / d 1 D ˛  .1 / ˇ .1 / d 1 D

Z Z

ˇ  .1 / ˇ .1 / d 1 D 1

(1)

ˇ  .1 / ˛ .1 / d 1 D 0;

(2)

where the integration is carried out over the spin variable, which can be treated as the integration variable only. Such a construction may seem to be somehow unnatural; however, it is a convenient way of ascribing spins to the electrons without dealing with its origins.

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D. Kedziera and A. Kaczmarek-Kedziera

In general, the wave function must depend on time to reproduce information about the time evolution of the system. However, since we are interested only in the ground state of the system, we can neglect the time dependency. Considering the bound state is equivalent to imposing the condition of the square integrability of the wave function. The integral over all variables in the full range must exist: “

Z :::

f  f d  D q;

(3)

where q is a finite real number. An asterisk under the integral denotes the complex conjugate; it comes from the fact that the wave function can be complex in general. The square-integrability condition ensures that the wave function vanishes for the infinite values of all spatial variables and, therefore, that our molecule is kept together. In the above expression, the integration intervals and the integration variables are not stated explicitly. For the investigated N-electron system, the wave function depends on the 3 N spatial variables (for each particle i, we have xi , yi , and zi coordinates) and additionally N spin variables ( i for the particle i): f D f .x1 ; y1 ; z1 ; 1 ; x2 ; y2 ; z2 ; 2 ; : : : ; xN ; yN ; zN ; N / :

(4)

The volume element in this 4 N-dimensional space is d  D dV  d ;

(5)

where the spatial part can be written as dV D dx1 dy1 d z1 dx2 dy2 d z2 : : : dxN dyN d zN ;

(6)

and the spin part is d  D d 1 d 2 : : : d N :

(7)

The spatial variables change from 1 to 1 and spin variables can take allowed values. One can see that writing all of the integrals, variables, and volume elements explicitly takes time and a lot of paper, even for relatively small systems. Therefore, one usually keeps them in mind, not writing them down. The wave function contains all of the information about the state of the system. In order to extract it, operators are applied. An operator can be understood by an analogy to a function. The function ascribes a number to a number and the operator ascribes a function to another function. In other words, the operator is a recipe for how to obtain one function from another: b Af D g:

(8)

Remarks on Wave Function Theory and Methods

5

We will denote operators by hats above the symbol to distinguish them from functions and numbers. One of the particularly interesting cases is when the function g is proportional to the function f, g D af;

(9)

where a is a number. Then Eq. 8 takes the form b Af D af:

(10)

Equation 10 is called an eigenvalue equation of the operator Â. The function f fulfilling this equation is called an eigenfunction and a is an eigenvalue of the operator Â. The Schrödinger equation b ‰ D E‰ H

(11)

is a typical eigenvalue equation in which the Hamilton operator Hˆ extracts the information about the energy E of the system from the wave function ‰. Like in the case of the wave function, we will not consider operators in general. Let us concentrate on the Hamilton operator and its properties to simplify our discussion. We need our operators ascribed to observables (Hamiltonian among others) to satisfy the following requirements: • Linearity – the operators must fulfill the condition b A .˛f C ˇg/ D ˛ b Af C ˇ b Ag;

(12)

where now ˛ and ˇ are numbers. This seems simple and obvious; however, it is not the property of all operators. For instance, the square root is not a linear operator, since the square root of the sum is not equal to the sum of the square roots. • Real eigenvalues – only real values can be measured in a laboratory. For these reasons, we will be interested in so-called Hermitian operators that can be defined by the relation “

Z :::

f1

“ Z     b Af2 d  D : : : f2 b Af1 d :

(13)

All the functions, variables, and integration intervals remain the same as in Eq. 3. Writing of all these things in the expressions was already troublesome enough, and things become even more complicated when operators appear. In order to make life easier, Dirac notation can be applied. In this notation Eq. 13 has the form ˇ E D ˇ E D ˇ ˇ Af2 D b f1 ˇb Af1 ˇf2 ;

(14)

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D. Kedziera and A. Kaczmarek-Kedziera

D ˇ ˇ E ˇ ˇ Aˇ f2 ; and the integral where the left-hand side can be equivalently written as f1 ˇb of Eq. 3 becomes simply D ˇ E ˇ f ˇf D q:

(15)

In this very convenient notation, it is also assumed that the integration intervals and variables flow from the context. Let us look at Hermitian operators more carefully, considering them in the example of Hamiltonian. It has already been mentioned that such operators have real eigenvalues. Furthermore, the eigenfunctions of the Hermitian operator that correspond to different eigenvalues are orthogonal. In other words, for b f1 D E1 f1 H

b f2 D E2 f2 ; and H

(16)

where .E1 ¤ E2 / ; one has D ˇ E D ˇ E ˇ ˇ f1 ˇf2 D f2 ˇf1 D 0:

(17)

This will be a very useful property, since it will cause various terms in complicated expressions to vanish. In the case of degeneration, or, in other words, when one of the eigenvalues corresponds to two or more eigenfunctions, the eigenfunctions f1 and f2 can be orthogonalized. It is worth considering the integral D

ˇ ˇ E ˇbˇ f1 ˇH ˇ f1 :

(18)

Since f1 is the eigenfunction of Hˆ with the eigenvalue E1 , it is obvious that D

ˇ ˇ E D ˇ E D ˇ E ˇbˇ ˇ ˇ f1 ˇH ˇ f1 D f1 ˇE1 f1 D E1 f1 ˇf1 :

(19)

It would be certainly more convenient if Dtheˇ result was a single number – the E ˇ eigenvalue E1 . This would be the case if f1 ˇf1 D 1 or if we say the function f1 was normalized to unity. It would be consistent with the interpretation of the D ˇ E ˇ integral f1 ˇf1 as the probability of finding the system in the whole space – it should be surely equal 1. This is a very handy requirement. Any function that does not possess this pproperty can be normalized by multiplying by the normalization factor N D 1= hf1 jf1 i. Then, the new function fQ1 will be given as fQ1 D N f1 :

(20)

Remarks on Wave Function Theory and Methods

7

This new function fQ1 isr also an eigenfunction of the Hamiltonian, since f1 was only D ˇ E ˇ f1 ˇf1 and Hamiltonian is linear: divided by the number b fQ1 D H b N f1 D N H b f1 D N Ef1 D EN f1 D E fQ1 : H

(21)

In the case of unnormalized functions, the expression for the eigenvalue E1 can be obtained from Eq. 19: ˇ ˇ E ˇbˇ f1 ˇH ˇ f1 E1 D D ˇ E : ˇ f1 ˇf1 D

(22)

However, many of the applied functions are not the eigenfunctions of the Hamiltonian. Therefore, let us investigate another interesting integral, D ˇ ˇ E ˇbˇ g ˇH ˇg ;

(23)

ˆ In order to calculate this integral, an alternate where g is not an eigenfunction of H. important property of the Hermitian operators needs to be exploited: the fact that their eigenfunctions constitute a complete basis set. Each function depending on the same variables as the eigenfunctions can be expressed as the linear combination of the basis functions. Now, this concept seems to be hard-core mathematics; however, anybody using computational techniques knows well that the two things one must input to the ab initio program are the method and the basis set. Hence, let us make a break from the general considerations of operators and abstract space functions and concentrate for a while on the basis set concept in the example of simple trigonometric functions. In a calculus course, one learns how to express a function using a set of other functions. For example, consider sinx function and expand it in the Taylor series around 0: 1 X x5 x3 .1/i 1 2i 1 x C  :::: sin x D Dx .2i  1/Š 3Š 5Š i D1

(24)

In Eq. 24, sinx function is expressed in the basis set of monomials: sin x D

1 X

ck x k ;

(25)

kD1

where ck are the expansion coefficients that need to be determined. In our case it is simple, since ck result directly from the Taylor expansion and are equal:

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D. Kedziera and A. Kaczmarek-Kedziera

Table 1 Taylor expansion of x function and standard deviation for various expansion lengths

n 1 3 5 7

(

.1/.k1/=2 kŠ

ck D

0

Fn x x  16 x 3 x  16 x 3 C x  16 x 3 C

1 x5 120 1 x5 120

for odd k; for even k:



1 x7 5;040

n 3.93  101 4.05  102 2.08  103 6.39  105

(26)

The summation in Eqs. 24 and 25 goes from 1 to 1. In practice, finite and possibly short expansions are applied: Fn .x/ D

n X

ci x i :

(27)

i D1

This truncation of the series introduces an approximation to our function. Let us analyze the x function in the range x 2 h 2 I 2 i : The standard deviation works well as the accuracy measure: sZ n

 2

 2

.sin x  Fn .x//2 dx:

(28)

Table 1 summarizes data for small n values. Increasing the number of expansion terms causes a decrease of the standard deviation values and more accurate representation of the original x function. Given the required accuracy of the calculation, the necessary value length of the expansion, n, can be found. The following question arises: Why use Taylor expansion instead of x function itself, if one needs to worry about the expansion accuracy? The answer is straightforward: simplifications and savings. It is much easier to operate on the polynomials R  i 2 than on the trigonometric functions (for instance, the integral x dx is much R 2 easier to handle than sin xdx). Moreover, the required accuracy can often be obtained with a relatively short expansion. Let us now make the considerations more general. As was stated before, the set of eigenfunctions of the Hermitian operator Hˆ is complete and orthonormal – functions are orthogonal and normalized: D ˇ E ˇ 8i;j fi ˇfi D ıij ; (29) where ı ij is a Kronecker symbol that takes value 1 for i D j and 0 otherwise. The basis set completeness means that each function depending on the same set of variables can be expressed by the basis functions:

Remarks on Wave Function Theory and Methods

gD

1 X

9

ci fi ;

(30)

i D1

where coefficients ci need to be found. Knowing the normalized g function makes this task simple, because of the orthonormality of the ffi g set, the coefficients will be equal: D ˇ E ˇ (31) ci D fi ˇg ; since 1 1 D ˇ E X D ˇ E X ˇ ˇ fi ˇg D ci fi ˇfi D cj ıij D ci : j D1

(32)

j D1

(Only one term for i D j, ci , remains; all other vanish for the Kronecker delta equals zero if i ¤ j .) Similarly, 1 D ˇ E X ˇ g ˇg D cj cj :

(33)

j D1

However, things are not that easy, since we usually apply the expansion (Eq. 30) when we do not know the g function. Thus, the integrals (Eqs. 31 and 33) should be perceived rather as the interpretation of the ci coefficients than the direct recipe for the calculations. From Eq. 30, the g function can be treated as the linear combination of the fi functions. Moreover (see Eq. 33), the probability that a system described by the function g is in the state fj is given by c*j cj . Now let us consider the expression D ˇ ˇ E D E ˇbˇ b ; g ˇH ˇg  H g

(34)

when g is not the Hamiltonian eigenfunction. Using the expansion Eq. 30 and the fact that fi are the Hamiltonian eigenfunctions (Eq. 16), one obtains 1 D ˇ ˇ E X ˇbˇ g D g ˇH cj cj Ej : ˇ

(35)

j D1

The above integral is called an average (expectation) value, and Eq. 35 for Hamiltonian carries the information about the average energy of the system in the state described by the g function. Looking closer at Eq. 35 shows that this average energy is simply a weighted average of all possible Ej energies of the system. The weights are determined by the c*j cj products – the probability of finding the system in the fj state. It should be noticed that we used the linearity of the Hamiltonian operator to achieve this result.

10

D. Kedziera and A. Kaczmarek-Kedziera

Conclusion? Very optimistic: We can say something about the sought energy value not knowing the eigenfunctions of the operator of interest, since for calculations of Eq. 35, we do not need fj functions. Strange? Not at all, if we recall some linear algebra: Using three basis vectors, we can describe each and every point in the 3D space. Likewise, the wave function can be perceived as a vector, D ˇ E ˇ the Hermitian operator as the symmetric transformation matrix, an integral f ˇg as the dot product, orthogonality of functions as orthogonality of vectors, and normalization as dividing the vector components by its length. Now, when the “linear algebra” term has already appeared, let us see how it is applied for solving the eigenequation. Almost all calculations are performed by applying the basis functions. This means that the unknown function ‰ describing the investigated system is expressed in the basis of known functions i (see Eq. 30): ‰

n X

ci i D ˆ;

(36)

i D1

where ‰ is the eigenfunction of the Hamiltonian corresponding to a given eigenvalue E (Eq. 11). The task is to find such ci coefficients that the function ˆ would be the best approximation to ‰. Since ˆ is an approximation to the wave function, the corresponding energy will also be only approximated. Let us call the approximation Eˆ . Basis functions i are not the Hamiltonian eigenfunctions; therefore, to estimate the energy, an average value must be calculated. Therefore, substituting Eq. 36 for Eq. 11 and multiplying by ‰  D h†i ci i j on both sides gives1 LHS D

DX i

ˇ ˇX E XX D ˇ ˇ E ˇbˇ ˇbˇ ci i ˇH ci j D ci cj i ˇH ˇ ˇ j ; j

RHS D Eˆ

i

XX i

(37)

j

E cj cj hi jj :

(38)

j

1

Here, different subscripts appear on the both sides of the integral. The sum does not depend on the name of the summation index; thus, any subscript can be applied. However, one should not apply the same index on both sides of the integral, since it can cause the erroneous omission of the off-diagonal terms. Compare the overlap integral ˇ + * ˇ D ˇ E X ˇX ˇ ˇ c i i ˇ cj j D hc1 1 C c2 2 C c3 3 C : : : jc1 1 C c2 2 C c3 3 C : : : i ‰ ˇ‰ D ˇ j i D c1 c1 h1 j1 i C c1 c2 h1 j2 i C c1 c3 h1 j3 i C : : : The explicit writing of all terms shows that not only the integrals hi ji i with the same function on both sides are present, but also the contributions hi jj i with i ¤ j . Therefore, the diversification of the subscripts prevents mistakes.

Remarks on Wave Function Theory and Methods

11

D ˇ ˇ E D ˇ E ˇbˇ ˇ In order to further simplify the notation, let us denote i ˇH ˇ j by Hij and i ˇj by Sij . Then, XX i

cj cj Hij D Eˆ

j

XX i

ci cj Sij :

(39)

j

Equivalently, in the matrix form Hc D ScEˆ ;

(40)

where H is the Hamiltonian matrix with the elements Hij , S is called the overlap matrix and is built of the overlap integrals Sij , and c denotes the vector of the ci coefficients. The basis sets applied in practice are usually non-orthogonal, which causes the off-diagonal terms in the S matrix to not vanish. Such a method of finding approximate eigenvalues and eigenvectors of the Hamiltonian is known as the Ritz method and is frequently applied in quantum chemistry. This simple introduction of basic terms of quantum mechanics is obviously far from complete. One can notice the lack of further discussion of the degeneration, the continuum spectrum, and many other topics. For these we encourage the reader to dive into the following excellent books on quantum mechanics and chemistry: Atkins and Friedman (2005), Griffiths (2004), Levine (2008), Lowe and Peterson (2005), McQuarrie and Simon (1997), Piela (2007), Ratner and Schatz (2000), and Szabo and Ostlund (1996).

On the Way to Quantum Chemistry For the sake of simplification, we assume that the energy of the ground state of our system differs from other energy values. This allows one to avoid embroilment in technical details that are unnecessary at this point. Our system is described by the wave function ‰ fulfilling the Schrödinger Eq. 11. It is important to notice that this eigenvalue equation can be solved exactly only for hydrogen atoms. Any more complicated system requires approximate techniques. In order to explain this complication, let us look into the Hamilton operator. For the system of N electrons and M nuclei, the full Hamiltonian is a sum of the following terms: • • • • •

be Kinetic energy of electrons, T bn Kinetic energy of nuclei, T bee Energy of interactions between electrons, V bnn Energy of interactions between nuclei, V bne Energy of interactions between a nucleus and an electron, V

12

D. Kedziera and A. Kaczmarek-Kedziera

In the atomic units, these terms have the following form: N  X 2 be D  1 T 2 i D1 ri

(41)

M X 1 2 bn D  T 2mi Ri i D1

(42)

bee D V

bnn D V

N N X X 1 r i D1 j >i ij

M M X X Zi Zj i D1 j >i

Rij

M N X X Zj bne D  ˇ ˇ V ˇr i  R j ˇ

(43)

(44)

(45)

i D1 j D1

charge, and rij where mi is the mass of the nucleus i, Zi stands for the nuclear ˇ ˇ ˇri  rj ˇ. Likewise, denotes the distance between the electrons i and j, r D ij ˇ ˇ Rij refers to the internuclear distance, Rij D ˇRi  Rj ˇ. The presence of the mutual distances between the particles causes a serious problem when solving the Schrodinger equation; it does not allow one to decouple the equations. Fortunately, from the chemist’s point of view, such a Hamiltonian is not very useful. The chemist is not interested in each and every bit of information one can get about any N-electron M-nuclei system; however, she or he is focused on the given molecule, its conformations, interactions with the environment, and properties (spectroscopic, magnetic, electric, and so on). What makes quantum mechanics a valuable tool for chemists is the Born–Oppenheimer approximation, discussed in detail in the previous chapter of this volume. Let us briefly summarize it to maintain consistent notation throughout the chapter. The chemist is concerned with the relative positions of the nuclei in the molecule and with the internal energy, but not with the motions of the molecule as a whole. This motion can be excluded from our consideration, for example, by elimination of the center-of-mass translation. Moreover, the intermolecular (electrostatic) forces acting on electrons and nuclei would be similar. This would cause much slower internal motion of the heavy nuclei in comparison to light electrons. For this reason, the approximate description of electron motion with parametric dependence on the static positions of nuclei is justified. Such reasoning leads to adiabatic approximation and finally to Born–Oppenheimer approximation. According to this approximation, the Hamiltonian can be written as b Db be C V bnn ; H Tn CH

(46)

Remarks on Wave Function Theory and Methods

13

where b T n now has a meaning of nuclear kinetic energy of the molecule for which the center of mass is stopped (however, there are still oscillations and rotations), and be D T be C V bee C V ben H

(47)

is called an electronic Hamiltonian, and it represents the energy of the system after omitting the nuclear kinetic energy term and nuclear repulsion. One can now focus on the solution of the equation of the form h

i bn .R/ C H b e .rI R/ C V bnn .R/ ‰ .r; R/ D E‰ .r; R/ : T

(48)

Here, the dependence on the electronic spatial variables r D .x1 ; y1 ; z1 ; : : : ; xN ; yN ; zN / and the nuclear spatial variables R D .X1 ; Y1 ; Z1 ; : : : ; XM ; YM ; ZM / is written explicitly. The semicolon sign in the Hˆ e term denotes the parametric dependence – for various R the various electronic equations are obtained. With such a Hamiltonian, it seems reliable to distinguish also the nuclear f (R) and electronic ‰ e (r; R) part in the wave function ‰ .r; R/  ‰e .rI R/ f .R/ ;

(49)

which leads to a significant reduction of the problem. Now Eq. 48 can be separated into three equations: b e .rI R/ ‰e .rI R/ D Ee .R/ ‰e .rI R/ H

(50)

  b e .rI R/ C V bnn .R/ ‰e .rI R/ D U .R/ ‰e .rI R/ H

(51)

  bn .R/ C U b .R/ f .R/ D Ef .R/ T

(52)

The first two describe electronic motion for a given position of nuclei. The difference between Ee (R) and U(R) is that in U(R) nuclear repulsion energy is taken into account. These equations are milestones in our considerations for two reasons. First, since we are now talking about “fixed positions of the nuclei,” finally we have got molecules instead of an unspecified system containing some electrons and some nuclei. The second is hidden in Eq. 52: Electronic energy and nuclear repulsion energy constitute the potential, in which nuclei are moving. That is why the proper description of electronic movement in a molecule is so important: The electrons glue the whole molecule together. Our attention in the rest of the chapter will be focused only on Eq. 50; hence, to simplify notation, all the subscripts denoting the electronic case will be omitted: be ! H b H

(53)

14

D. Kedziera and A. Kaczmarek-Kedziera

‰e ! ‰

(54)

b ‰ D E‰ b e ‰e .rI R/ D Ee ‰e .rI R/ ! H H

(55)

The electronic wave function ‰ satisfies all the requirements discussed in the previous sections, depends on the coordinates of N electrons, and additionally must be antisymmetric with respect to the exchange of coordinates of two electrons.2 It should be noted that the analytic solution of Eq. 50 is not known even for the smallest molecule, i.e., H2 . Therefore, the approximate techniques must be applied to extract the necessary information about molecules of interest. Quantum mechanics provides two tools: • Variational principle • Perturbation theory

Variational Principle: An Indicator The variational principle allows one to judge the quality of the obtained solutions. It can be formulated as follows: For the arbitrary trial function  that is squareintegrable, differentiable, and antisymmetric and depends on the same set of variables as a sought ground-state ‰ 0 function, we have D ˇ ˇ E ˇbˇ  ˇH ˇ (56) E0  D ˇ E ; ˇ ˇ where E0 is the ground-state energy corresponding to ‰ 0 (Eq. 50). The important consequence of the variational principle is that to estimate the energy of the system, one does not need to solve the eigenequation (this we already know; see Eq. 35), and moreover – what is crucial – the estimated energy value will always not be lower than the exact eigenvalue E0 . The proof of the inequality (Eq. 56) is straightforward and can be derived from Eqs. 30 and 35. The function  that satisfies the above requirements can be expanded on the basis of the Hamiltonian eigenfunctions: D

1 X

ci fi ;

(57)

i D0

2

In order to explain the antisymmetry requirement, we have to refer again to theory that is beyond the scope of the present chapter. Let us simply state here that wave functions must be antisymmetric without belaboring the point. This will mean that the exchange of the coordinates of the two electrons causes the wave function to change the sign: ‰ .1 ; 2 / D ‰ .2 ; 1 / :

Remarks on Wave Function Theory and Methods

15

b fi D Ei fi . Thus, where fi fulfill the eigenproblem H D ˇ ˇ E X1 X1 ˇbˇ  ˇH ˇ ci ci Ei c  ci E0 i D0 i D0 i D ˇ E D X  D E0 ; X 1 1 ˇ ci ci ci ci ˇ i D0

(58)

i D0

with the assumption that E0 is the lowest of all Hamiltonian eigenvalues (Atkins and Friedman 2005; Levine 2008; Lowe and Peterson 2005; McQuarrie and Simon 1997; Piela 2007; Szabo and Ostlund 1996).

Perturbation Calculus: The Art of Estimation Due to the variational principle that is satisfied for the electronic Hamiltonian, the group of methods of searching for parameters optimizing the energy value can be constructed. The way of verification of the given wave function is the corresponding energy value: The lower, the better. Besides this “quality control,” the variational principle does not give the prescription for the choice of the trial wave functions. Here comes the perturbation calculus – the method frequently applied in physics for the estimation of the functions or values on the basis of partial knowledge about the solutions of the investigated problem. We will consider here the Rayleigh– Schrödinger variant of the perturbation calculus (Atkins and Friedman 2005; Levine 2008; Lowe and Peterson 2005; McQuarrie and Simon 1997; Piela 2007; Ratner and Schatz 2000). Let us assume that the total electronic Hamiltonian of the investigated system can be divided into b DH b0 C H b 1; H

(59)

in such a fashion that we know the exact solutions of b 0 ‰ .0/ D E .0/ ‰ .0/ ; H k k k

(60)

where the subscript k enumerates the eigenvalues of the Hˆ 0 operator in such a way ˆ that E(0) 0 is the lowest energy. Now, one can say that the operator H describes the system for which Hˆ 0 is an unperturbed operator and Hˆ 1 denotes a perturbation. We can assume that if the change in the system represented by Hˆ 1 is minor, then ˆ0 the functions ‰ (0) k would be a good approximation to ‰ k . Considering H , one postulates its Hermicity and that its eigenvalues are not degenerate (in our case, for the ground state at least E(0) 0 must not be equal to any other eigenvalue). This condition will become clear in a moment. Knowing only the unperturbed solutions (Eq. 60), we would like to say something more about the ground-state energy of the investigated system. Nothing is easier – we can calculate the average value of the full electronic Hamiltonian with

16

D. Kedziera and A. Kaczmarek-Kedziera

the ‰ (0) 0 function. The variational principle states that the resulting energy will be not lower than the exact energy: ˇ ˇ E D .0/ ˇ b 0 b 1 ˇˇ ‰ .0/ D E .0/ C E .1/ : E 0  ‰ 0 ˇH CH 0 0 0

(61)

The term modifying E(0) 0 is simply ˇ ˇ E D .1/ .0/ ˇ b 1 ˇ .0/ E 0 D ‰ 0 ˇH ˇ ‰0 :

(62)

So far, the only new thing is the manner of partitioning the total energy into the energy of the unperturbed system and the corrections (where E(1) 0 is not the only term): .0/

.1/

.2/

E0 D E0 C E0 C E0 C ::::

(63)

Likewise, the wave function can be written as .0/

.1/

.2/

‰0 D ‰0 C ‰0 C ‰0 C : : : ;

(64)

(2) where ‰ (1) 0 , ‰ 0 and so forth are the corrections to the wave function of the unperturbed system ‰ (0) 0 . Now, the electronic Schrödinger Eq. 50 becomes

 0   b CH b 1 ‰ .0/ C ‰ .1/ C ‰ .2/ C : : : H 0 0 0    .0/ .1/ .2/ .0/ .1/ .2/ D E0 C E0 C E0 C : : : ‰ 0 C ‰ 0 C ‰ 0 C : : : :

(65)

Introducing the expansions (Eqs. 63 and 64) does not increase our knowledge about the energy or the wave function; it is only a different way of expressing the unknowns by other unknowns. However, now we have a starting point for further investigations. The comparison of the terms on the left- and right-hand side of the above expression is instructive. Let us regard as similar the terms with the same sum of the superscripts (so-called perturbation order, by analogy to the multiplication and ordering of polynomials). Simple multiplication in Eq. 65 and directing the terms of the same order to separate equations gives b 0 ‰ .0/ D E .0/ ‰ .0/ ; H 0 0 0

(66)

b 1 ‰ .0/ D E .0/ ‰ .1/ C E .1/ ‰ .0/ ; b 0 ‰ .1/ C H H 0 0 0 0 0 0

(67)

b 0 ‰ .2/ C H b 1 ‰ .1/ H 0 0

.0/

:: :

.2/

.1/

.1/

.2/

.0/

D E0 ‰ 0 C E0 ‰ 0 C E0 ‰ 0 :

(68)

Remarks on Wave Function Theory and Methods

17

These equations link the corrections to the wave function and to the energy. Before detailed investigation of the subsequent corrections, one more thing should be underlined. Up to now, the function ‰ 0 is not normalized; only ‰ (0) k are normalized. Until the corrections to ‰ 0 were found, we would not be able to normalize it. We can only write the normalization constant as N D ph‰1 j‰ i : However, it is not necessary 0 0 at this moment. Now the intermediate normalization condition is more useful: ˇ E D .0/ ˇ ‰0 ˇ‰0 D 1: (69) Such a concept is based on the information that the eigenfunctions of Hˆ 0 form an orthonormal complete set of functions (that is one of the reasons why the Hermicity of Hˆ 0 was required) and they can be applied to express any other function, for instance, ‰ 0 as ‰0 D

1 X

.0/

ck ‰k C

kD0

1 X

.0/

ck ‰k :

(70)

k¤0

In this linear combination, the function ‰ (0) 0 has a distinguished meaning .c0 D 1/; this is the approximation of the wave function of the considered system. Therefore, one can require that ‰ (0) 0 does not have a contribution to the higher corrections: ‰ 0 (1) , ‰ 0 (2) , and so on: 1 X

.0/

.1/

.2/

ck ‰k D ‰0 C ‰0 C ::::

(71)

k¤0

Here, the benefits from the intermediate normalization are obvious: The function ‰ (0) 0 is orthogonal to each of the corrections (or, in other words, the corrections are defined in such a way that they are orthogonal to ‰ (0) 0 ). Therefore, there is an additional set of equations to be satisfied: D

ˇ E .0/ ˇ .n/ ‰0 ˇ‰0 D ı0n ;

(72)

where the superscript n denotes the nth-order correction to the ground-state wave function ‰ 0 . Now we can go back to Eqs. 66, 67, and 68 and extract the corrections to energy. For this purpose, each of the equations must be multiplied from the lefthand side by ‰ (0) 0 and integrated: ˇ ˇ E D .0/ .0/ ˇ b 0 ˇ .0/ E 0 D ‰ 0 ˇH ˇ ‰0 ;

(73)

ˇ ˇ D E .1/ .0/ ˇ b 1 ˇ .0/ E 0 D ‰ 0 ˇH ˇ ‰0 ;

(74)

18

D. Kedziera and A. Kaczmarek-Kedziera

ˇ ˇ E D .0/ ˇ b ˇ .1/ D ‰ 0 ˇH ˇ ‰0 ;

.2/

E0

:: :

(75)

ˇ ˇ D E .0/ ˇ b 0 ˇ .n/ The integrals ‰0 ˇH ˇ ‰0 vanish, since D

ˇ ˇ ˇ ˇ E D 0 E D E .0/ ˇ b 0 ˇ .n/ b ‰ .0/ ˇˇ‰ .n/ D E .0/ ‰ .0/ ˇˇ‰ .n/ D 0: ‰ 0 ˇH ˇ ‰0 D H 0 0 0 0 0

(76)

Thus, obtaining the energy corrections of any order is straightforward. The general expression for the nth-order correction can be written as ˇ ˇ E D .n/ .0/ ˇ b 1 ˇ .n1/ E 0 D ‰ 0 ˇH ˇ ‰0

for n > 1:

(77)

The problem is that to obtain the corrections to the energy in the second or higher orders, the corrections to the wave function are necessary. Then, let us try to find ‰ (1) 0 . This function can be expressed as the linear combination of the functions from the orthonormal set f‰ (0) k g for k ¤ 0: .1/

‰0 D

1 X

.1/

.0/

ck ‰k ;

(78)

k¤0

where c(1) k are the expansion coefficients in the first-order correction. Again, the whole thing reduces to finding the coefficients ck . Substituting Eq. 78 to Eq. 67 gives 1  0 X   .1/ .0/ .1/ b  E .0/ b 1 ‰ .0/ : H ck ‰k D E0  H 0 0

(79)

k¤0

Integrating this equation with the ‰ (0) l function leads to * LHS D D

ˇ

.0/ ˇ b 0 ‰ l ˇH

1 X k¤0



1 ˇX

.0/ ˇ E0 ˇ

+ .0/ ck ‰k

k¤0

D

1 X

ˇ ˇ D E .0/ ˇ b 0 .0/ ˇ .0/ ck ‰l ˇH  E0 ˇ ‰ k

k¤0

1 ˇ  D E X   .0/ .0/ .0/ ˇ .0/ .0/ .0/ ‰l ˇ‰k D ck El  E0 ck El  E0 ılk

(80)

k¤0   .0/ .0/ D cl El  E0

and

ˇ ˇ ˇ ˇ 1ˇ E D E D E D .0/ ˇ .1/ b 1 ˇˇ ‰ .0/ D E .1/ ‰ .0/ ˇˇ ‰ .0/  ‰ .0/ ˇˇH b ˇˇ ‰ .0/ RHS D ‰l ˇE0  H 0 0 l ˇ0 ˇ l E 0 D .0/ ˇ b 1 ˇ .0/ D  ‰ ˇH ˇ‰ : l

0

(81)

Remarks on Wave Function Theory and Methods

19

Altogether, these allow one to write the coefficients of the expansion (Eq. 78) as D cl D

ˇ ˇ E .0/ ˇ b 1 ˇ .0/ ‰ l ˇH ˇ ‰0 .0/

.0/

E0  El

:

(82)

Hence, the first correction to the wave function is already known:

.1/

‰0 D

ˇ 1ˇ D E b ˇˇ ‰ .0/ 1 ‰ .0/ ˇˇH X 0 k k¤0

.0/ E0



.0/ Ek

.0/

‰k ;

(83)

and, thereby, the second-order correction to the energy can be calculated as

.2/

E0 D

ˇ 1ˇ D E b ˇˇ ‰ .0/ D 1 ‰ .0/ ˇˇH X 0 k .0/

k¤0

.0/

E0  Ek

ˇ ˇ E .0/ ˇ b 1 ˇ .0/ ‰ 0 ˇH ˇ ‰k :

(84)

This is also equivalently written as

.2/

E0 D

ˇD ˇ 1ˇ Eˇ2 b ˇˇ ‰ .0/ ˇˇ 1 ˇˇ ‰ .0/ ˇˇH X 0 k .0/

k¤0

.0/

E0  Ek

:

(85)

The energy difference in the denominator of the above expression cannot be equal to zero, and for this reason the non-degenerated ground state was assumed. The higher-order corrections are sought in a similar manner, which just requires more operations. One of the interesting issues is the problem of variationality of the perturbation calculus built upon the variational Hamiltonian. This is, however, a sophisticated problem for advanced readers and will not be discussed here. It should be added, in summary, that the manner of partitioning of the Hamilton operator was arbitrary. The only prerequisites were the Hermitian character of the operators (for the eigenfunctions to form the orthonormal set) and the non-degenerated ground-state eigenenergy and nothing more. One should also remember that, in practice, even solving the unperturbed problem cannot be performed exactly and the approximations must be applied. The consequence could be the loss of accuracy for higher-order corrections. Moreover, good convergence of the perturbation expansion can be expected when the consecutive corrections are small in comparison to the total estimated value. However, in such a case, the low orders of the series would reproduce the sought value with relatively good accuracy. Thus, application of the low orders of perturbation calculus is highly recommended.

20

D. Kedziera and A. Kaczmarek-Kedziera

One-Electron Approximation: Describe One and Say Something About All Equipped with general knowledge about the tools for the Schrödinger equation solution, one can move to many-electron systems. The electronic Hamiltonian for any many-electron system in atomic units has the following form (compare Eqs. 41, 42, 43, 44, and 45): M N N N X N X X X X Zj 1 b D 1 ˇ ˇC ˇ ˇ: H r i C ˇ ˇ ˇ 2 i D1 r  Rj r  rj ˇ i D1 j D1 i i D1 j Di C1 i

(86)

Let us look more closely. In the first term, we sum up over the number of electrons N; in the second term, the summations run over the number of electrons N and number of nuclei M; and the third term contains the double sum over the number of electrons N. Thus, one can simplify the notation of the first two terms:



N 1X

2 i D1

r i C

M N X X

Zj ˇ ˇD ˇr i  R j ˇ i D1 j D1

N X

0

M X

1

Zj @ 1 r i C ˇ ˇ A: ˇr i  R j ˇ 2 i D1 j D1

(87)

ˆ Now, denoting the term in parenthesis by h(i), M X Zj 1 b ˇ ˇ; h.i / D  ri C ˇ 2 r  Rj ˇ j D1 i

(88)

we get the part of the Hamiltonian depending only on the coordinates of one electron i (and nuclear coordinates, but it does not bother us). Hence, the total electronic Hamiltonian (Eq. 86) can be rewritten as the sum of one-electron and two-electron contributions: b D H

N X i D1

b h.i / C

N N X X

b g .i; j /;

(89)

i D1 j Di C1

where we introduced a symbol: 1 ˇ: b g .i; j / D ˇ ˇr i  r j ˇ

(90)

ˆ describes a It should be noticed that each of the one-electron Hamiltonians h(i) single electron in the field of some potentials. Therefore, the exact solutions of the eigenvalue problem for these one-electron operators are available. The problem lies

Remarks on Wave Function Theory and Methods

21

in the gˆ (i, j) operator that couples two electrons together: It is not possible to separate their coordinates exactly. All electronic Hamiltonians have the same general form; they differ only in the ˆ number of electrons N and the nuclear potential hidden in h(i). One can choose any possible chemical compounds and try to write the corresponding equations; however, one would soon note the similarity of all of them. Therefore, we will not invest time in describing the procedure for a polypeptide or a nanotube, but for simplicity we will start from the two-electron helium atom (the simplest manyelectron system) and later try to generalize the considerations. For the helium atom: • N D 2 two electrons • M D 1 one nucleus The generalization of the helium discussion into the larger (N-electron) systems should be straightforward: b h.1/ C b h.2/ D

2 X

b h.i / !

i D1

b g .1; 2/ D

2 2 X X

N X

b h.i /;

(91)

i D1

b g .i; j / !

N N X X

b g .i; j /;

(92)

X Zj 1 Z1 1 b ˇ ˇ: h.i / D  ri C !  r i C ˇ 2 2 jri  R1 j r  Rj ˇ j D1 i

(93)

i D1 j Di C1

i D1 j Di C1

and finally M

Recall that the exact solutions of the one-electron problem are known, and the task is to solve the full problem. The ideas of the perturbation theory were explained in the previous section. Now it is the time to apply the knowledge. The one-electron part can be treated as the unperturbed Hamiltonian and the rest as the perturbation: b0 D b H h.1/ C b h.2/;

b1 D b H g .1; 2/ ;

(94)

where the normalized solutions for the one-electron part are known: b h.1/i .1/ D –i i .1/;

(95)

b h.2/j .2/ D –j j .2/:

(96)

These functions require more attention. Although the electronic Hamiltonian – and thereby the one-electron operators – do not act on the spin variables, the wave

22

D. Kedziera and A. Kaczmarek-Kedziera

functions  k must carry on the spin dependence. Therefore, the function  k (l) is a product of the spatial part depending on the three spatial coordinates of the electron l and on the spin part and is called a spin-orbital. For simplicity, the set of coordinates  l is written as a label of the electron, i.e., l. Such convention will be applied from now on, with an exception where the  l labeling is really needed. If an operator can be written as a sum of contributions acting on different variables, its eigenfunction takes a form of the product of the eigenfunctions of the subsequent operators in the summation. In the case of the helium atom, where Hˆ 0 is ˆ and h(2), ˆ a sum of h(1) the wave function ‰(1, 2) can be denoted as the product of one-electron functions: ‰ .1; 2/ D i .1/j .2/:

(97)

The eigenproblem for such a function gives the eigenvalue that is simply the sum of the one-electron eigenvalues: h i b h.1/ C b h.2/ i .1/j .2/ D b h.1/i .1/j .2/ C b h.2/i .1/j .2/ h i h i D b h.1/i .1/ j .2/ C b h.2/j .2/ i .1/   D Œ–i i .1/ j .2/ C –j j .2/ i .1/   D –i C –j i .1/j .2/:

(98)

However, here the antisymmetry requirement should also be taken into account. The many-electron function must change the sign with respect to the exchange of labels of any two electrons: ‰ .1; 2/ D ‰ .2; 1/ :

(99)

The product (Eq. 97) is not antisymmetric; the interchange of the electron labels leads to ‰ .2; 1/ D i .2/j .1/ ¤ ‰ .1; 2/ ;

(100)

and the result is a function different from the original ‰(1, 2). But another function,   ‰ .1; 2/  i .1/j .2/  i .2/j .1/ ;

(101)

satisfies the antisymmetry condition. Moreover, the wave function ‰(1, 2) has to be normalized. This can be achieved by calculating the following (overlap) integral: ˇ D E D   ˇˇ  E ˇ ‰ .1; 2/ ˇ‰ .1; 2/ D N i .1/j .2/j .1/i .2/ ˇN i .1/j .2/j .1/i .2/ ˝ ˇ ˇ ˛ ˝ ˛ D N 2 i .1/j .2/ˇ i .1/j .2/  i .1/j .2/ˇ j .1/i .2/ ˇ ˇ ˛ ˝ ˛ ˝  j .1/i .2/ˇ i .1/j .2/ C j .1/i .2/ˇ j .1/i .2/ : (102)

Remarks on Wave Function Theory and Methods

23

The spin-orbitals of electrons 1 and 2 are mutually independent; thus, D

ˇ ˇ ˇ E D ED E ˇ ˇ ˇ i .1/j .2/ˇk .1/l .2/ D i .1/ˇk .1/ j .2/ˇl .2/ D ıi k ıj l ;

(103)

ˇ E D ˇ where i .1/ˇk .1/ D ıi k arises from the fact that the eigenfunctions of the ˆ form the orthonormal set. Hence, in Eq. 102 only the first Hermitian operator h(1) and last integral will be nonvanishing, and, finally, ˇ E D ˇ ‰ .1; 2/ ˇ‰ .1; 2/ D 2N 2 D 1;

(104)

p and, therefore, the normalization constant N must equal 1= 2. In order to fulfill the normalization and the antisymmetry request, the trial wave function can be written as ˇ ˇ  1  1 ˇ  .1/ i .2/ ˇˇ : (105) ‰ .1; 2/ D p i .1/j .2/  j .1/i .2/ D p ˇˇ i 2 2 j .1/ j .2/ ˇ Thereafter, the expectation value of the Hamiltonian can be calculated using the same tricks as in Eq. 1023 : D

E b .1; 2/ H

ˇ ˇ E D ˇ ˇb .1; 2/ˇ ‰ .1; 2/ D ‰ .1; 2/ ˇH ‰.1;2/ ˇ ˇ ˇ ˇ D E D E ˇ ˇ ˇ ˇ D i .1/ ˇb h.1/ˇ i .1/ C j .2/ ˇb h.2/ˇ j .2/ ˇ ˇ D E ˇ ˇ C i .1/j .2/ ˇ r112 ˇ i .1/j .2/ ˇ ˇ D E ˇ ˇ  i .1/j .2/ ˇ r112 ˇ j .1/i .2/ :

(106)

The only difference with respect to the overlap integral is that the integrals containing 1/r12 cannot be separated. In the spirit of perturbation calculus, the expectation value (Eq. 106) can be treated as the energy corrected up to the first order of perturbation expansion. According to the variational principle, this integral is an upper bound of the exact energy of the two-electron system under consideration. The recipe for correcting the trial function is given by the perturbation theory: Apply functions corresponding to the remaining states of the unperturbed system in the expansion. In other words, 3

Since the value of the integral does not depend on the name of the variable, the obvious equality Z b Z b f .x/dx D f .y/dy in the above case takes the form a

a

ˇ ˇ ˇ  ˇ  ˇ 1 ˇ ˇ ˇ 1 i .1/j .2/ ˇˇ ˇˇ j .1/i .2/ ˇˇ D i .2/j .1/ ˇˇ r r 12

12

ˇ ˇ ˇ j .2/i .1/ : ˇ

24

D. Kedziera and A. Kaczmarek-Kedziera

in order to improve the wave function, the expansion built from products of the remaining states of the systems should be used. Let us summarize. The many-electron function of the system can be approximately written as the antisymmetrized product of the one-electron functions being the solutions for the one-electron eigenvalue problem. This is the idea of the popular one-electron approximation (Atkins and Friedman 2005; Lowe and Peterson 2005; McQuarrie and Simon 1997; Ratner and Schatz 2000). In the N-electron case, one obtains the trial function as the antisymmetrized product of N one-electron functions that can be written in the form of the Slater determinant: ˇ ˇ ˇ 1 .1/ 1 .2/ : : : 1 .N / ˇ ˇ ˇ 1 ˇˇ 2 .1/ 2 .2/ : : : 2 .N / ˇˇ (107) ‰ .1; 2; : : : ; N / D p ˇ: ˇ: :: : : :: ˇ N Š ˇˇ :: :: : ˇ ˇ  .1/  .2/ : : :  .N / ˇ N N N p 1= N Š factor ensures the normalization of the wave function. Often, instead of writing the whole determinant, only the diagonal is written down: ˇ ˇ 1 .1/ 1 .2/ : : : ˇ ˇ 1 .2/ 2 .2/ : : : ˇ ˇ: :: :: ˇ :: : : ˇ ˇ  .1/  .2/ : : : N

N

ˇ ˇ ˇ ˇ ˇ ˇ D j1 .1/2 .2/ : : : N .N /j : ˇ ˇ N .N / ˇ

1 .N / 2 .2/ :: :

(108)

It is important to realize that the many-electron function in the form of the Slater determinant is only an approximation. Intuition tells us that describing the manyelectron system using only one-electron functions cannot be exact. One needs to be aware of the fact that such an approach causes the loss of some information included in the sought wave function. In particular, the one-electron function cannot “see” another electron; therefore, the terms coupling the mutual electron positions are missing in the one-electron approximation. For example, consider the two-electron function (Piela 2007)   F .1; 2/  e ar1 br2 cr12  e ar2 br1 cr12 ;

(109)

where r1 , r2 are the electron–nucleus distances, r12 denotes the distance between two electrons, and a, b, and c stand for the coefficients. F(1, 2) contains the factor c correlating the electron motions. In the one-electron approach, such a function would be approximated by the antisymmetrized function   f .1; 2/  e ar1 br2  e ar2 br1 ;

(110)

with total neglect of this correlation. From the practical point of view, this means that the trial function written as the single determinant does not allow reproduction of

Remarks on Wave Function Theory and Methods

25

the exact electronic energy, even when using the best possible one-electron functions for its construction. The question arises: Why use such an approach, knowing from the very beginning that it is bad? The answer is simple. First, including the electron correlation in the wave function is very expensive. For two electrons, one additional term appears; for three, three terms; for four, six. In general, for N electrons there are N(N  1)/2 terms (the triangle of the N N matrix without the diagonal elements). Therefore, the number of coefficients describing the electron correlation is much bigger than the number of one-electron terms. Calculating even the oneelectron coefficients is very time-consuming, and moreover, calculating the overlap integrals and Hamiltonian matrix elements becomes prohibitively complicated with the correlated functions. Second, perturbation theory provides ways of improving the results. Expansion built on a higher number of determinants will lead to better energy. Third, a chemist does not usually need exact data but only an appropriate accuracy (furnishing a house does not require caliper measurements, just a quick glance to estimate the size of the door and the furniture). Therefore, let us stick to the one-electron approximation. The next section will explain how to find the best possible spin-orbitals.

Hartree–Fock Method: It Is Not That Sophisticated Now we are prepared to concentrate on the methods for solving the electron equation. As you will see, they are only an extension of the already-discussed techniques. We start with the fundamental Hartree–Fock method. The main goal of this approximation is to find the spin-orbitals applied for construction of the Slater determinant that will best reproduce the exact wave function. We again begin our considerations with the two-electron system. The ˆ does not contain the part arising from the potential problem is that the operator h(i) of the second electron, and an operator responsible for this missing part must be found. What we do know is that such an interaction is included in the two-electron part of Eq. 106: 

1 r12

‰.1;2/

ˇ ˇ ˇ ˇ   ˇ 1 ˇ ˇ 1 ˇ D 1 .1/j .2/ ˇˇ ˇˇ i .1/j .2/  i .1/j .2/ ˇˇ ˇˇ j .1/i .2/ : r12 r12 (111)

Unfortunately, Eq. 111 contains the integrals that cannot be exactly separated into a product of the simpler integrals depending only on the coordinates of one electron. What we can propose is rewriting this equation in the following form: 

1 r12

‰.1;2/

˝ ˛ D hi .1/ jb .1/j i .1/i C j .2/ jb .2/j j .2/ C the rest;

(112)

26

D. Kedziera and A. Kaczmarek-Kedziera

where b .1/ and b .2/ are introduced to extract only the terms depending on the coordinates of a single electron from Eq. 111 and “the rest” is what remains from D E 1 r1 2 ‰ .1; 2/ after this extraction. If such extraction were possible, the operators b .1/ and b .2/ could be applied to improve our one-electron operators, which leads to the following equations: h i b h.1/ C b .1/ i .1/ D –i i .1/;

(113)

h i b h.2/ C b .2/ j .2/ D –j j .2/:

(114)

The solutions of such equations (spin-orbitals  i ,  j ) can be used to build up the Slater determinant. They ensure better approximation than Eqs. 95 and 96, since they somehow provide for the influence of the second electron. Therefore, our goal now is to utilize Eq. 106 (treating  i and  j ) as known functions) to find the best form of the operators b .1/ and b .2/. Adding zero, written as   1 1 0D  ; (115) r12 ‰.1;2/ r12 ‰.1;2/ to Eq. 111 allows one to ascribe, for instance, ˇ ˇ D E ˇ ˇ .1/j i .1/i D i .1/j .2/ ˇ r112 ˇ i .1/j .2/  j .1/i .2/ ; hi .1/ jb ˇ ˇ E ˝ ˛ D ˇ ˇ j .2/ jb .2/j j .2/ D i .1/j .2/ ˇ r112 ˇ i .1/j .2/  j .1/i .2/ ; ˇ ˇ D E ˇ ˇ the rest D  i .1/j .2/ ˇ r112 ˇ i .1/j .2/  j .1/i .2/ : Consider in more detail the integral containing b .1/. We want it to be expressed in such a way that only the coordinates of the electron labeled by 1 are explicitly written under the integral: ˇ ˇ D E ˇ ˇ .1/j i .1/i D i .1/j .2/ ˇ r112 ˇ i .1/j .2/  j .1/i .2/ hi .1/ jb RR  D  . /   . / 1  . /  . / d 1 d 2 R R i  1 j 2 r121 i 1 j 2  i .1 / j .2 / r12 j .1 / i .2 / d 1 d 2  R R  D i .1 / j .2 / r112 j .2 / d 2 i .1 / d 1 R  R  i .1 / j .2 / r112 i .2 / d 2 j .1 / d 1   R D i .1 / b J j .1/i .1 / d 1   R b j .1/i .1 / d 1  j .1 / K ˇ ˇ ˇ ˇ D E D E ˇ ˇ ˇb ˇ D i .1/ ˇb J j .1/ˇ i .1/  i .1/ ˇK j .1/ˇ i .1/ ˇ ˇ D E ˇ b j .1/ˇˇ i .1/ ; D i .1/ ˇb J j .1/  K

(116)

Remarks on Wave Function Theory and Methods

27

where two operators were defined:  1 j .2 / d 2 i .1/; r12 

Z 1  b K j .1/i .1/ D j .2 / i .2 / d 2 j .1/: r12 b J j .1/i .1/ D

Z

j .2 /

(117) (118)

Despite a slightly different way of defining, they are still ordinary operators. An b j .1/ act on the operator is a function of a function. The operators Jˆj (1) and K function  i (1), producing another function, as was written in Eq. 8. The operator Jbj .1/ acting on  i (1) transforms it into the same function: b J j .1/ i .1/ !

Z

j .2 /

 1 j .2 / d 2 i .1/; r12

(119)

 1 i .2 / d 2 j .1/: r12

(120)

b j .1/ produces  j (1): and the operator K b K j .1/ i .1/ !

Z

j .2 /

The expressions on the right-hand side of the arrows are some functions of a dependent variable  1 (the integration eliminates the dependence on  2 ; however, its result is not a number but a function depending on  1 ). The interpretation of the b j .1/ operators by ascribing them to observables is not straightforward. Jˆj (1) and K These operators appear in the equations when we try to write the interaction between two electrons as the average value of the one-electron operator calculated with the one-electron function. However, in fact, they do appear as the difference: b b b HF i .1/ D J j .1/  K j .1/:

(121)

And the physical sense should be sought in this difference. Here, b HF i .1/ is an operator of the average interaction of the electron labeled as 1, described by a spinorbital  i with the second electron characterized by  j (2). It should be noticed that the potential b HF i .1/ depends on  j (2), since this spin-orbital is necessary to define ˆJj (1) and K b j .1/ operators. Similarly, b b b HF j .2/ D J i .2/  K i .2/;

(122)

b j .1/ on the function  j (2) is defined by where the action of operators Jˆj (1) and K b J i .2/j .2/ D

Z

i .1 /

 1 i .1 / d 1 j .2/ r12

(123)

28

D. Kedziera and A. Kaczmarek-Kedziera

b i .2/j .2/ D K

Z

i .1 /

 1 j .1 / d 1 i .2/: r12

(124)

Here, one more important circumstance should be mentioned. The electronic Hamiltonian does not depend on spin. However, the electronic wave function is spin dependent. During the construction of the HF equations, this depenb since they depend on two different dence is introduced to the operators K spin-orbitals. Let us briefly review. We want to have a one-electron operator that includes the interaction between the electrons in some averaged way. Such an operator must depend on the function describing the motion of the second, adjacent electron. Therefore, for the two-electron case, two coupled equations must be solved (Eqs. 113 and 114). The word “coupled” that distinguishes this set of equations from Eqs. 95 and 96 is crucial. Denoting b .1/ D b h.1/ C b HF f i i .1/;

(125)

one can rewrite the above equations as b .1/k .1/ D –k k .1/; f k

for k D i; j:

(126)

b .1/ is called the Fock operator and Eq. 114 gives the Hartree–Fock equations. f i It should be noted that the label of electrons determines only the name of the integration variables, and the result of the integration does not depend on the name of the variable. Therefore, what is really important is the label of the spin-orbital. It will be even more pronounced in the N-electron case, when the electron labels are applied only to show that the operator acts on one or two electron coordinates. The form of the N-electron wave function depends only on the spin-orbitals and not on the electron labels; they are just the integration variables. The most popular way of solving Eq. 126 is the iterative procedure. It starts from the guessed or chosen spin-orbitals  i (1) and  j (2) applied to construct the potentials b HF HF i .1/ and b j .2/: Next, the obtained potentials are substituted to Eq. 125 and solutions of Eq. 126 give the improved form of the  i (1) and  j (2) spin-orbitals. These, on the other hand, are treated as the starting point again, and the whole procedure is repeated until the starting and final orbitals of the given iteration do not differ much. This technique is called self-consistent field (SCF). Often the abbreviations HF for Hartree–Fock method and SCF are used interchangeably. They can also be joined together as SCF–HF, denoting the self-consistent way of solving Hartree–Fock equations. Let us assume that the spin-orbitals are already known. Concentrate on the calculations of the average value of the Hamiltonian with the determinant build of these spin-orbitals using the operators defined previously. Writing down the twoelectron part:

Remarks on Wave Function Theory and Methods

D

E

1 r12 ‰.1;2/

29

ˇ ˇ D E ˇ b j .1/ˇˇ i .1/ D i .1/ ˇb J j .1/  K ˇ ˇ E D ˇ b i .2/ˇˇ j .2/ ; D j .2/ ˇb J i .2/  K

one obtains the integrals that can be denoted by ˇ ˇ D E ˇ ˇ D i .1/ ˇb J j .1/ˇ i .1/ ; ˇ ˇ D E ˇb ˇ Kij D i .1/ ˇK j .1/ˇ i .1/ :

Jij

With this notation, the expression (Eq. 106) takes the following form: D

E b .1; 2/ H

‰.1;2/

ˇ ˇ ˇ ˇ D E D E   ˇb ˇ ˇb ˇ D i .1/ ˇf .1/ˇ i .1/ C j .2/ ˇf .2/ˇ j .2/  Jij  Kij ; (127)

or if one wants to apply the spin-orbital energies – calculated earlier: D

E b .1; 2/ H

‰.1;2/

  D –i C –j  Jij  Kij :

(128)

Now it is time to generalize these considerations into the N-electron case. The recipe for this transformation was given previously (Eqs. 91, 92, and 93). For the system of N electrons, the set of N-coupled equations of the form b .1/i .1/ D –i i .1/; f i

for i D 1; : : : ; N

(129)

must be solved. Here, fi .1/ D h.1/ C

N   X b b j .1/ : J j .1/  K

(130)

j ¤i

The summation in the expression for the one-electron Fock operator arises from the fact that now the given electron described by the spin-orbital i interacts with (N  1) electrons in the states determined by the remaining spin-orbitals. In searching for the ground state energy, one is interested in the lowest possible energy. Therefore, the functions of choice are the spin-orbitals corresponding to the lowest values of –. Such a set of spin-orbitals, called occupied, is opposite to any other solutions of the Fock equations corresponding to higher energies. These are known as virtual (unoccupied) orbitals. Finally, the average value of the Hamiltonian can be written as D E b .1; 2; : : : ; N / H

‰ .0/ .N /

D

N X i D1

–i 

N N X X   Jij  Kij D E0HF i D1 j >i

30

D. Kedziera and A. Kaczmarek-Kedziera

It should be emphasized that the energy estimated in this manner is not the simple XN XN   Jij  Kij are neglected, sum of the orbital energies. If the terms i D1

j >i

the double summation of the electron–electron interaction would take place (Atkins and Friedman 2005; Cramer 2004; Jensen 2006; Levine 2008; Lowe and Peterson 2005; McQuarrie and Simon 1997; Piela 2007; Ratner and Schatz 2000; Roos and Widmark 2002; Szabo and Ostlund 1996).

Møller–Plesset Perturbation Theory: HF Is Just the Beginning From here forward, we will treat the Hartree–Fock function as the basis for the further investigations and denote it as ‰ (0) 0 , where the subscript 0 indicates the ground state and the superscript (0) is the reference function. We will also omit the explicit writing of the dependence of the Hamiltonian and the wave function on the coordinates of N electrons. As a consequence, D

ˇ ˇ E .0/ ˇ b ˇ .0/ ‰ 0 ˇH ˇ ‰0 D E0HF :

(131)

One of the possible ways of improving the HF results is the application of the perturbation theory (Atkins and Friedman 2005; Cramer 2004; Jensen 2006; Levine 2008; McQuarrie and Simon 1997; Piela 2007; Ratner and Schatz 2000; Szabo and Ostlund 1996). The Hamiltonian (Eq. 89) can be partitioned into the unperturbed part and the perturbation in the following manner: 0 1 N N N X X X 0 1 1 b.i /; H b D b D @ H b HF .i /A (132) f r ij j >i i D1 i D1 (compare Eq. 125). Up to the first order in the perturbation theory, the energy is ˇ ˇ E D E D .0/ .1/ .0/ ˇ b 0 b b 1 ˇˇ ‰ .0/ D H E0HF D E0 C E0 D ‰0 ˇH CH 0

.0/

‰0

:

(133)

The correction to the HF energy appears in the second order:

.2/

E0 D

1 X

ˇD ˇ ˇ Eˇ ˇ .0/ ˇ b 1 ˇ .0/ ˇ 2 ˇ ‰ 0 ˇH ˇ ‰ k ˇ .0/

k¤0

.0/

E0  Ek

:

(134)

During calculations, the expansion of the spin-orbitals in the finite basis set is applied. This allows identification of only the finite number of spin-orbitals. But the number of all possible ‰ (0) k functions can still be horrifyingly large. In practice, this is equivalent to the finite but long expansion in the above summation. In quantum chemistry we are interested in the best quality results with moderate expenses and are continuously searching for more economical methods.

Remarks on Wave Function Theory and Methods

31

Careful examination of the second-order energy correction E(2) 0 shows that a significant number of its terms do not contribute to the final result. Some work invested in the manipulation of expressions allows one not only to learn the basic computational apparatus but also to save a lot of effort. In order to proceed comfortably, the notation should again be simplified. Let us drop the superscript (0) denoting the zeroth-order functions (other functions will not appear in our considerations). Additionally, we omit the subscript k from the determinants containing the virtual spin-orbitals. In return, we explicitly specify the pattern of spin-orbital exchange. Using Eq. 108, the Slater determinant can be written as ˇ E ˇ (135) ˇ‰0 D j1 .1/2 .2/3 .3/4 .4/ : : : N .N /i : The numbers of the occupied spin-orbitals vary from 1 to N. Thus, the virtual spinorbitals will be labeled starting from N C 1 onward. Now consider the example of the determinant in which the occupied spin-orbital  3 is exchanged for the virtual one, N C8 . The new determinant can be written as ˇ E ˇ ˇ ˇ N C8 D ˇ1 .1/2 .2/N C8 .3/4 .4/ : : : N .N /i ; ˇ‰3

(136)

where the subscript in ‰3N C8 denotes the occupied orbital that is exchanged and the superscript denotes the virtual one that takes its place. To generalize, one could denote the occupied orbitals building the ‰ 0 function by first alphabet letters a, b, c, d : : : and the virtuals by p, q, r, s : : : . Therefore, ‰3N C8 can be written as ‰ pa , where a D 3 and p D N C 8. Likewise, any determinant can be represented. Thereafter, the influence of the function choice on the values of the integrals appearing in the electronic energy calculations can be investigated. Similarly, as in the case of Hamiltonian, the integrals can also be divided into two groups with one-electron operator b o1 D

N X

b o.i /;

(137)

i D1

b.i / will be used, and with two-electron ˆ (i) or f where in place of ô(i) operators h, operator, b2 D O

XX 1  : rij i j >i

(138)

To get a full picture, we should analyze the following types of integrals: o1 j ‰ap i h‰0 jb

D and

ˇ ˇ E ˇb ˇ p ‰ 0 ˇO 2 ˇ ‰a ;

(139)

32

D. Kedziera and A. Kaczmarek-Kedziera

˝ pq ˛ ‰0 jb o1 j ‰ab ˝

pqr ˛

‰0 jb o1 j ‰abc

D and D and

ˇ ˇ E ˇ b ˇ pq ‰ 0 ˇO 2 ˇ ‰ab ;

(140)

ˇ ˇ E ˇ b ˇ pqr ‰ 0 ˇO 2 ˇ ‰abc :

(141)

Any other will be equal to zero (Levine 2008; Szabo and Ostlund 1996). Recall that the determinant of the N N matrix can be represented as the sum of the N! products of the matrix elements. According to Laplace’s formula, a determinant can be expanded along a row or a column. Thus, calculation of the h‰ 0 jô1 j‰ pa i integrals, when the occupied orbital a in ‰ 0 has been exchanged with the virtual orbital p in ‰ pa , can be performed by expanding the determinant ‰ 0 along the a-th row and the determinant ‰ pa along the p-th row: N 1 X ‰0 D p a .i /Cai ; N Š i D1

(142)

N 1 X ‰ap D p p .i /Cpi : N Š i D1

(143)

The cofactors C can be perceived as the (N  1)-electron determinants that have been obtained from ‰ 0 and ‰ pa via the elimination of the a-th and p-th spin-orbitals, respectively. Thus, after elimination of what is different in the two determinants, we get both cofactors equal to each other. Hence, ˝

p˛

‰0 jb o 1 j ‰a D D

1 NŠ

1 NŠ

ˇ ˇ + ˇX ˇX ˇN ˇ N ˇ ˇ a .i /Cai ˇ b o.j /ˇ p .k/Cpi ˇj D1 ˇ kD1 i D1

*N X

N X ˝ ˛ ˛˝ a .i / jb o.i /j p .i / Cai j Cpi i D1

D

N X

˝

p .i / jb o.i /j p .i / i˝ D1 ˛ D a .1/ jb o.1/j p .1/ ; 1 N

(144)

˛

where D

N X

ˇ E ˇ Cai ˇCpi D .N  1/Š;

o.i /j a .i /i D N ha .1/ jb o.1/j a .1/i ha .i / jb

(145)

(146)

i D1

was applied. If the cofactors are not the same – this would be the case when the determinants differ from two or more spin-orbitals – the corresponding overlap

Remarks on Wave Function Theory and Methods

33

matrix is equal to zero according to the orthogonality condition. Thus, ˝ ˝ ˛ p˛ ‰ o ‰ D a .1/ jb o.1/j p .1/ ; ˝ 0 jb1 j apq ˛ o1 j ‰ab ˛ D 0; ˝‰0 jb pqr ‰0 jb o1 j ‰abc D 0:

(147)

Similar (but a little more time-consuming) considerations for the two-electron operators lead to the following expressions: ˇ ˇ N  ˇ ˇ E X ˇ 1 ˇ ˇb ˇ p ˇ ˇ ‰ 0 ˇO 2 ˇ ‰ a D a .1/i .2/ ˇ ˇ p .1/i .2/ r i D1 D ˇ ˇ12 E ˇ1ˇ  a .1/i .2/ ˇ r12 ˇ i .1/p .2/ ˇ ˇ D ˇ ˇ E D E ˇ b ˇ pq ˇ1ˇ ‰  ‰ 0 ˇO D  .1/ .2/ .1/ .2/ ˇ ˇ ˇ 2 a b p q ab r12 ˇ ˇ D E ˇ1ˇ  a .1/b .2/ ˇ r12 ˇ q .1/p .2/ D ˇ ˇ E ˇ b ˇ pqr ‰ ‰ 0 ˇO ˇ 2 abc D 0:

D

(148)

In the first contact with these equations, one can have the feeling that something is lost here. We start from the integrals with the N-electron functions and we finish with the integral of only the electrons labeled by 1 and 2. What happened to the rest? Again, it should be emphasized here that the electron labels symbolize only the integration variables. What matters is the functions of these variables, namely, spin-orbitals. Thus, in the integral with the one-electron operator, the electron label 1 means that the integration is performed only over the variables of one electron. Likewise, the two-electron operator integral depends on the variables of two electrons, which is symbolized by two labels: 1 and 2. In the integration of one-electron expressions, the electron label can be omitted without any harm: h¥a .1/ jb o1 .1/j ¥a .1/i. Likewise, in the two-electron case, we can declare that the spin-orbitals are written in the given order. So, ˇ ˇ ˇ ˇ   ˇ 1 ˇ ˇ 1 ˇ x .1/y .2/ ˇˇ ˇˇ v .1/z .2/ D x y ˇˇ ˇˇ v z : r12 r12

(149)

Moreover, now there is no reason to explicitly write the  symbol. Thereby, the next simplification of the notation is obvious: D

o1 j Ea i DD ha ˇjb o1 jˇai ;E ˇhaˇjb ˇ ˇ ˇ ˇ x y ˇ r112 ˇ v z D xy ˇ r112 ˇ vz ;

(150)

ˇ E D ˇ ˇ E D ˇ ˇ ˇ ˇ ˇ and finally, since the combination of the integrals xy ˇ1=r12 ˇvz  xy ˇ1=r12 ˇvz appears frequently, the following symbol is introduced:

34

D. Kedziera and A. Kaczmarek-Kedziera

 ˇ ˇ  ˇ ˇ 1 ˇ 1 ˇ hxy jj zvi D xy ˇˇ ˇˇ vz  xy ˇˇ r r 12

12

ˇ ˇ ˇ zv : ˇ

(151)

Now Eqs. 147 and 148 can be rewritten as ˝

p˛ ‰0˝jb oj pi ; o1 j ‰a D ha jb pq ˛ ‰0 jb o1 j ‰ab D 0; ˝ pqr ˛ ‰0 jb o1 j ‰abc D 0; N D ˇ ˇ E X ˇb ˇ p ‰ 0 ˇO 2 ˇ ‰ a D hai jj pi i; D ˇ ˇ E i D1 ˇ b ˇ pq ‰ 0 ˇO D hab jj pqi ; 2ˇ ‰ D ˇ abˇ E ˇ b ˇ pqr ‰0 ˇO 2 ˇ ‰abc D 0:

(152)

These simple equations, known as the Slater rules, allow for the following general remark: If the one-electron operator is integrated with functions that differ by more than one spin-orbital, the corresponding integral vanishes. Similarly, the result is zero for the integration of two-electron operators with functions differing by more than two spin-orbitals. Hitherto, only the integrals with ‰ 0 were considered. pq pqr However, it is easy to notice that functions ‰ ab and ‰ abc differ by only one exchange (spin-orbital c ! r ), etc. Therefore, the above considerations can be also applied in any other cases. In this abundance of equations, our main goal cannot be lost: All these derivations were necessary to limit the types of the ‰ k functions present in the MP2 energy expression. Now, with a recognition of the above Slater rules, one can safely neglect the integrals with the pairs of the ‰ k functions differing with more than two spinorbital exchanges. But this is not everything. Let us consider the integral D

ˇ ˇ E D ˇ ˇ E X  ˇ ˇ ˇbˇ p ‰ 0 ˇH aj hˇ p C ˇ ‰a D a ˇb j

Since

ˇ ˇ1 ˇ ˇr

ij

ˇ  ˇ ˇ jp : ˇ

(153)

ˇ D ˇ ˇ E ˇ ˇ pj D a ˇˇJbj ˇˇ p ; ˇ

(154)

ˇ ˇ1 ˇ ˇr

ˇ D ˇ ˇ E ˇ ˇ jp D a ˇˇK b j ˇˇ p ; ˇ

(155)

D ˇ ˇ E D ˇ 0ˇ E ˇb ˇ ˇbˇ p ‰ 0 ˇH ˇp ; ˇ ‰ a D a ˇH

(156)

 aj

ij

one gets

ij

ˇ  ˇ ˇ pj  aj ˇ

ˇ ˇ1 aj ˇˇ r



ij

and

ˇ ˇ1 ˇ ˇr

Remarks on Wave Function Theory and Methods

35

which has to vanish, because ˇ + * ˇN ˇX ˇ D ˇ 0ˇ E ˇ ˇb ˇ ˇ a ˇH ˇ p D a ˇ f .i /ˇ p D ha jf j pi D –p ıap D 0; ˇ ˇ

(157)

i D1

for it was assumed that a ¤ p. The obtained result, D

ˇ ˇ E ˇbˇ p ‰ 0 ˇH ˇ ‰a D 0;

(158)

is known as the Brillouin theorem. Applying it allows one also to show that ˇ 1ˇ E ˇb ˇ p ‰ 0 ˇH ˇ ‰a D 0:

(159)

b1 D H b H b 0; H

(160)

ˇ 1ˇ E D ˇ ˇ E D ˇ 0ˇ E ˇb ˇ p ˇbˇ p ˇb ˇ p ‰ 0 ˇH ˇ ‰ a  ‰ 0 ˇH ˇ ‰ a D ‰ 0 ˇH ˇ ‰a ! N D ˇ E X ˇ p –i ‰0 ˇ‰a D0

(161)

D

Indeed, using

one obtains D

i D1

D 0:

Now is the time for an important conclusion: In order to calculate the correction to the energy in MP2, the functions arising from ‰ 0 only by the exchange of precisely two spin-orbitals need to be applied, not more, not less. Life becomes easier when not calculating zero contributions in a complicated way. Let us exploit the above knowledge to transform Eq. 134. The integral on the pq right-hand side will be calculated with the functions ‰ ab . Assuming b > a and q > p allows avoidance of the integration with the same functions. The upper limit for the summations over a and b will be equal to N. For the remaining spinorbitals, it should be 1; however, in practice, the finite basis is applied and the upper limit will be determined by the basis set size. Let us look into X the numerator b.i /, one b of Eq. 134 carefully. Denoting the total Fock operator as F D f i gets D

ˇ 1ˇ E D ˇ ˇ E D ˇ ˇ E ˇ b ˇ pq ˇ b ˇ pq ˇ b ˇ pq ‰ 0 ˇH ˇ ‰ab D ‰0 ˇH ˇ ‰ab  ‰0 ˇF ˇ ‰ab D ˇ ˇ E D ˇ ˇ E ˇ ˇ ˇ ˇ D ab ˇ r1ij ˇ pq  ab ˇ r1ij ˇ pq ;

(162)

36

D. Kedziera and A. Kaczmarek-Kedziera

since D

ˇ ˇ E ˇ b ˇ pq ‰ 0 ˇF ˇ ‰ab D 0

(163)

(integration of the one-electron operator with the functions differing by two spinorbitals; see Eq. 152). Now consider the denominator of Eq. 134. The energy of the ground state is simply a sum of N lowest spin-orbital energies: .0/

E0 D

N X

–i :

(164)

i D1 pq

The energy of the zeroth order for the function ‰ ab is a similar sum with –a and –b replaced by –p and –q : E

.0/

.pq ab /

.0/

D E 0  –a  –b C –p C –q :

(165)

Thus, the final form of Eq. 134 is .2/

E0 D

XX b>a q>p

jhab jj pqij 2 : –p C –q  –a  –b

(166)

Why was it worth our hard work? Not only for satisfaction. These derivations are necessary to understand the mechanisms employed in computational methods of quantum chemistry. With the Slater rules, it is straightforward to recognize the vanishing integrals. The double exchanges appear to be most important, since these are the only exchanges that give rise to the energy corrections in the MP2 method. According to the perturbation calculus, the wave function corrected in the first order can be written as X ‰  ‰0 C ck ‰k ; (167) k¤0

where ck is given by D ck D

ˇ 1ˇ E ˇb ˇ ‰ 0 ˇH ˇ ‰k E0  Ek

:

(168)

From the Slater rules, it is easy to estimate that the only nonvanishing terms will arise from the double spin-orbital exchange in the wave function. The intuitive statement is that the low-order corrections should have the larger impact and the above considerations lead to the most important accomplishment of this section: The largest contribution to the corrections of the Hartree–Fock function arises from the functions with the doubly exchanged spin-orbitals.

Remarks on Wave Function Theory and Methods

37

Beyond the HF Wave Function Having done all this hard work, one can now sit comfortably in an armchair and think. The main goal of the quantum chemistry is to find the best possible description of the state of the system (the best possible wave function). The exact solutions of the Hamiltonian eigenproblem are unavailable and all we have are approximations. We have become used to approximations in everyday life. The important thing is to realize that the Hartree–Fock solutions can be improved. At the beginning of this chapter, various properties of operators were discussed. Among others, it was stated that the eigenfunctions of the Hermitian operators constitute the complete sets and any other function of the same variables can be represented by applying them. The one-electron spin-orbitals that are the eigenfunctions of the Fock operator are accessible. They form the complete set, but only for the one-electron functions. However, they can be applied to build up the N-electron determinants. The set of all possible determinants is also complete and, therefore, can be applied to express any N-electron function (Cramer 2004; Jensen 2006; Levine 2008; Lowe and Peterson 2005; Piela 2007; Ratner and Schatz 2000; Roos and Widmark 2002; Szabo and Ostlund 1996): ‰ D c0 ‰0 C

X a;p

cap ‰ap C

X

a;b;p;q

pq

pq

cab ‰ab C

X

pqr

pqr

cabc ‰abc C : : : :

(169)

a;b;c;p;q;r pq

pqr

For this purpose, only the coefficients c0 , cpa , cab , cabc , : : : need to be found. In the ideal case, all the summations would be infinite and the problem must be reduced. Still, instead of using the infinite expansions, the finite and relatively small number of terms can be sufficient. Moreover, solving the Hartree–Fock equations in the finite basis, one possesses only the finite number of orbitals that can be exchanged. Anyway, it is instructive to see how large the number of terms in Eq. 169 can be. Consider the methane molecule CH4 . Calculations with the minimal basis set (each orbital described by a single one-electron function; for carbon single functions for each of the orbitals, 1 s, 2 s, 2px , 2py , 2pz , and for hydrogen a single function for 1 s orbital) require 9 orbitals/18 spin-orbitals for the 10-electron system. From the probability theory, the number of combinations (K) of k elements from the nelement set can be calculated as

 nŠ n : (170) KD D k .n  k/ŠkŠ

 18 In the case of methane, 10 electrons can be placed in 18 spin-orbitals on D 10 43; 758 ways. This is equivalent to the 43,758 terms in the expansion (Eq. 169). Impressive. And one needs to remember that the minimal basis set gives relatively bad Hartree–Fock solutions and is not recommended in ab initio calculations. However, increasing the basis set size causes the number of expansion terms to grow dramatically. For instance, in the case of the so-called double- — basis set

38

D. Kedziera and A. Kaczmarek-Kedziera

(two functions per each orbital) for methane, one has 36 spin-orbitals, which makes 254,186,856 combinations! And double- — is still not much : : : . Therefore, it is necessary to find some way to reduce the size of the problem. The symmetry of the molecules can be applied here, and the fact that the chosen determinants (or their linear combinations) must be the given functions of the spin operators can be beneficial. Moreover, one would like to eliminate from the expansion the determinants that are not crucial for the quality of the wave function, and their neglect does not cause the deterioration of the description of the system (or causes only slight deterioration). In other words, only the determinants that have the significant contribution to the total energy must be chosen for the wave function construction. Let us begin with the classification of the determinants, taking into account the number of the spin-orbitals exchanged with respect to ‰ 0 . For this purpose, the averaged value of Hamiltonian calculated with ‰ will be useful. We can write the wave function expansion as ‰ D c0 ‰0 C ScS C DcD C TcT C QcQ C : : : :

(171)

The symbols’ meaning can be clearly deciphered by comparison with Eq. 169. S denotes a vector build of the determinants constructed from ‰ 0 by single exchanges, and cS is a vector of coefficients corresponding to the functions in S: ScS D

XX a

caq ‰aq :

(172)

q

In other words, S contains all the functions with the single exchanged spin-orbital. Similarly, D would be the combination of the functions with double exchanges, T with triple exchanges, and so forth. With such a notation, the function ‰ can be treated as the scalar product of the basis vectors ‰ 0 , S, D, : : : , and the coefficient vector 2 3 c0 6c 7 6 S7 7 ˇ E hˇ E ˇ E ˇ E ˇ E i 6 6 cD 7 ˇ ˇ ˇ ˇ ˇ 6 (173) ˇ‰ D ˇ‰0 ; ˇS ; ˇD ; ˇT ; Qi ; : : : : 6 c 7 7: 6 T7 6 cQ 7 4 5 :: : Using this notation, the Hamiltonian Hˆ of the system can be linked in an elegant way to a matrix H. Let us apply this form of the wave function for the calculation of the Hamiltonian average value. To simplify the expressions, let us limit ourselves to the truncated expansion: 2 3 c0 4 ‰SD D Œ‰0 SD cS 5 D c0 ‰0 C ScS C DcD : (174) cD

Remarks on Wave Function Theory and Methods

39

The average value of the Hamiltonian can now be written as

hH i‰SD

2 D ˇˇ 3 2 3 ‰0 ˇ ˇ ˇ E h ˇ E ˇ E ˇ Ei c0 ˇ ˇ ˇbˇ 6 ˇ 7 b ˇ‰0 ; ˇS ; ˇD 4 cS 5 : D ‰SD ˇH ˇ ‰SD D Œc0 cS cD  4 hS 5 H cD hD D

(175) The vector multiplication leads to the following expression: D ˇ ˇ E D ˇ ˇ E D ˇ ˇ E ˇbˇ ˇbˇ ˇbˇ hH i‰SD D c0 ‰0 ˇH ˇ ‰0 c0 C c0 ‰0 ˇH ˇ S cS C c0 ‰0 ˇH ˇ D cD D ˇ ˇ E D ˇ ˇ E D ˇ ˇ E ˇ ˇ ˇ ˇ ˇ b ˇ ‰0 c0 C cS S ˇH b ˇ S c S C c S S ˇH b ˇˇ D cD C c S S ˇH ˇ ˇ ˇ ˇ E D E D D ˇ ˇ E ˇbˇ ˇbˇ ˇbˇ C c D D ˇH ˇ ‰ 0 c 0 C c D D ˇH ˇ S c S C c D D ˇH ˇ D cD : (176) Such an equation is not very useful, since we still do not know the c0 , cS and cD coefficients determining the ‰ SD function. The only thing that can be said about them so far comes from the normalization requirement for ‰ SD : 1 D c0 c0 C cS C cS C cD cD :

(177)

This is not enough to uniquely determine the wave function. However, going back to Eq. 175 and multiplying only the inside vectors, one obtains ˇ3 2D ˇ ˇ E D ˇ ˇ E ˇ ˇbˇ ˇbˇ ‰ 0 ˇH h‰0 ˇ ˇ ‰ 0 ‰ 0 ˇH ˇS 6 ˇ 7 h ˇ E ˇ E ˇ Ei 6 D ˇ ˇ E D ˇ ˇ E ˇ 6 ˇ ˇbˇ 6 ˇ7b ˇ ˇbˇ S ˇH ˇ S D 6 hS ˇ 7 H ˇ‰0 ; ˇS ; ˇD D 6 S ˇH ˇ ‰0 ˇ5 4 D ˇ ˇ E D ˇ ˇ E 4 ˇbˇ ˇbˇ ˇ D ˇH ˇ ‰ 0 D ˇH hD ˇ ˇS 2

HSD

D ˇ ˇ E3 ˇbˇ ‰ 0 ˇH ˇD D ˇ ˇ E7 ˇbˇ 7 S ˇH ˇ D 7 : D ˇ ˇ E5 ˇbˇ D ˇH ˇD (178)

We can then associate finding the approximate Hamiltonian eigenvalues with its matrix in the ‰ SD basis:

hHi‰ SD

2D ˇ ˇ E ˇbˇ ‰ 0 ˇH ˇ ‰0 i 6 D ˇ ˇ E h 6 ˇbˇ



D c0 cS cD D 6 S ˇH ˇ ‰0 4 D ˇ ˇ E ˇbˇ D ˇH ˇ ‰0

D ˇ ˇ E ˇbˇ ‰ 0 ˇH ˇS D ˇ ˇ E ˇbˇ S ˇH ˇS D ˇ ˇ E ˇbˇ D ˇH ˇ S

D ˇ ˇ E3 ˇbˇ ‰ 0 ˇH ˇD 2 3 D ˇ ˇ E 7 c0 7 ˇbˇ S ˇH ˇ D 7 4 cS 5 : D ˇ ˇ E5 ˇbˇ cD D ˇH ˇD (179)

Because of the Hermitian character of the Hamilton operator, the HSD matrix is symmetric and real. Its diagonalization provides the set of the eigenvalues

40

D. Kedziera and A. Kaczmarek-Kedziera

corresponding to its eigenvectors. We are interested in the ground-state energy and thus, we need only the lowest eigenvalue of the HSD matrix and the respective normalized eigenvector ‰ SD . Knowing the procedure for the finite basis (only single- and double-orbital exchanges), we can see how it looks for the full (Eq. 171) expansion. The matrix notation leads to the average value of the Hamiltonian, written as 2D ˇ 3 ˇ ‰0 ˇ 2 3 ˇ7 6 c0 ˇ7 6 6 hS ˇ 7 6c 7 ˇ7 6 6 s7 ˇ7 6 6 7 6 cD 7  6 hD ˇ 7  ˇ 7 6 6 7 D b hH i‰ D c0 ; cS ; cD ; cT ; cQ 6 ˇ 7 H Œ‰0 i j Si ; j Di ; j Ti ; Qi ; : : :  6 cT 7 : 6 Tˇ 7 6 7 ˇ7 6 6 cQ 7 ˇ7 6 4 5 6 hQ ˇ 7 :: 5 4 : :: : (180) The vector multiplication permits one to perceive the average value as the eigenproblem of the Hamiltonian matrix: 2D ˇ ˇ E ˇbˇ ‰ 0 ˇH 0 ˇ ‰0 6 D ˇ ˇ E ˇbˇ 6 0 S ˇH ˇS 6 6 D ˇ ˇ ED ˇ ˇ E ˇbˇ ˇbˇ 6 6 D ˇH ˇ ‰ 0 D ˇ H ˇ S D ˇ ˇ E HD6 ˇbˇ 6 0 T ˇH ˇS 6 6 6 0 0 6 4 :: :: : :

D ˇ ˇ ˇbˇ ‰ 0 ˇH ˇ Di D ˇ ˇ E ˇbˇ S ˇH ˇD D ˇ ˇ E ˇbˇ D ˇH ˇ D D ˇ ˇ E ˇbˇ T ˇH ˇD D ˇ ˇ E ˇbˇ Q ˇH ˇ D :: :

3 0 D ˇ ˇ E ˇbˇ S ˇH ˇT D ˇ ˇ E ˇbˇ D ˇH ˇ T D ˇ ˇ E ˇbˇ T ˇH ˇT D ˇ ˇ E ˇbˇ Q ˇH ˇ T :: :

0 0 D ˇ ˇ E ˇbˇ D ˇH ˇ Q D ˇ ˇ E ˇbˇ T ˇH ˇQ D ˇ ˇ E ˇbˇ Q ˇH ˇ Q :: :

:::

7 7 :::7 7 7 :::7 7: 7 :::7 7 7 :::7 5 :: :

(181)

It can be clearly seen that some blocks in this matrix are equal to zero. This happens in two cases: • The integrals between ‰ 0 and functions of the S type (single exchange of spinorbitals) vanish due to the Brillouin theorem, as was shown in the previous section. • The integrals between the functions that differ by more than two exchanges, for instance, S and Q type, vanish due to the Slater rules. Even not knowing combinatorics, one can expect that the number of functions in a block will grow drastically with and increase in the number of exchanges (block S will contain less functions than D etc.). A bit of thinking in the beginning would help to save a lot of time by not calculating zero integrals. Let’s see: Only in the

Remarks on Wave Function Theory and Methods

41

D ˇ ˇ E D ˇ ˇ E ˇbˇ ˇbˇ case of ‰0 ˇH ˇ S blocks should all the elements be calculated. The ˇ D and S ˇH ˇ E D ˇ ˇ ˇ remaining matrices are sparse. For example, the Dˇnwidehat fH g ˇD block conˇ ˇ E D pq ˇ b ˇ pq tains the integrals of the following types: ‰ab ˇH ˇ ‰ab (the same function on both ˇ ˇ ˇ ˇ E D E D pq ˇ b ˇ pr pq ˇ b ˇ rs sides), ‰ab ˇH ˇ ‰ab (differing by one exchange), ‰ab ˇH ˇ ‰ab (differing by ˇ ˇ ˇ ˇ D E D E pq ˇ b ˇ pq pq ˇ b ˇ pq two exchanges), ‰ab ˇH ˇ ‰ac (differing by three exchanges), and ‰ab ˇH ˇ ‰dc (differing by four exchanges). Obviously, the two latter cases produce zeros. With the large number of exchanges, the size of the blocks grows abruptly, but most of the elements would be equal to zero. The simplification in this case would be the limitation of the H matrix size by the elimination of the functions including more than a given number of exchanges from the expansion. Let us leave only the single exchange block. Thus, the Hamiltonian matrix has the form 2D Hs D 4

ˇ ˇ E ˇbˇ ‰ 0 ˇH ˇ ‰0 0

3 0 D ˇ ˇ E5: ˇbˇ S ˇH ˇS

(182)

This is the block diagonal matrix. One of the properties of such matrices is that their eigenvalue set is a sum of the eigenvalues blocks. This means that E D ofˇ theˇ diagonal ˇbˇ the lowest possible eigenvalue is E0 D ‰0 ˇH ˇ ‰0 , and in consequence, there is no improvement in the ground-state energy with respect to the Hartree–Fock theory when taking only the single orbital exchanges. Hence, let us also include the functions of the D type. Now the Hamilton matrix can be written as D ˇ ˇ E3 2D ˇ ˇ E ˇbˇ ˇbˇ 0 ‰ 0 ˇH ‰ 0 ˇH ˇ ‰0 ˇD 6 D ˇ ˇ E D ˇ ˇ E7 ˇ ˇ ˇ ˇ 6 b ˇ S S ˇH bˇ D 7 (183) 0 S ˇH HSD D 6 7: ˇ E5 ˇ ˇ ˇ ˇ ˇ E D ED 4 D ˇbˇ ˇbˇ ˇbˇ D ˇ H ˇ ‰ 0 D ˇH ˇ S D ˇH ˇ D It is no longer a block diagonal matrix – all blocks contribute to its eigenvalues and one can count on some improvement. An interesting observation, however, is that here the functions exchange also have influence on the ˇ E ˇ Ethe single D ˇ spin-orbital D ˇ with ˇbˇ ˇbˇ energy via the S ˇH ˇ D and D ˇH ˇ S blocks. Next, subsequent groups of functions can be applied containing more than two spin-orbital exchanges. However, the calculations become prohibitively expensive, even for moderate size of the systems, and the consecutive corrections are smaller and smaller. The distinguished character of the double spin-orbital exchange was already discussed within the MP2 method. Now one can also expect that including double exchanges produces reasonable results with the moderate computational cost. Then, why not save more and diagonalize only

42

D. Kedziera and A. Kaczmarek-Kedziera

ˇ ˇ E ˇbˇ ‰ 0 ˇH ˇ ‰0 HD D 4 D ˇˇ ˇˇ E b ˇ ‰0 D ˇH 2D

D ˇ ˇ E3 ˇbˇ ‰ 0 ˇH ˇD D ˇ ˇ E5 ˇbˇ D ˇH ˇ D

(184)

instead of HSD ? This can be done; however, savings are not that great, since the number of S functions is significantly smaller than the number of D functions. Thus, if one can afford HD diagonalization, HSD diagonalization is probably also within easy reach. The above reasoning has led to the sequence of quantum chemistry methods. The best results can be obtained within full CI (FCI) by applying the full expansion (Eq. 171) within the given basis set. This is certainly the most expensive variant. Cheaper – but also worse – are, respectively, CISD based on the HSD matrix and CID neglecting single exchanges (Cramer 2004; Jensen 2006; Levine 2008; Lowe and Peterson 2005; Piela 2007; Ratner and Schatz 2000; Szabo and Ostlund 1996). So far, the reference function has been a single determinant. Such an approach is very limited. For instance, it does not allow one to describe a dissociation process. Correct description of dissociation requires at least one determinant for each subsystem. And, even in the cases when multi-determinant reference state description is not obligatory, such an elastic wave function will provide an improved description of the system of interest (Cramer 2004; Jensen 2006; Levine 2008; Piela 2007; Roos and Widmark 2002). The multi-determinant wave function ‰ depends both on the expansion coefficients and on the spin-orbitals building up the determinants. Both these sets of variables can be optimized simultaneously. The particular case of this procedure, when taking only the first expansion term, is the Hartree–Fock approximation (SCF– HF). Therefore, the optimization of the multi-determinant wave function is called the multiconfiguration (MC) SCF method. Even without a detailed study of the MC– SCF equations, an improvement in the results with respect to the HF energy can be expected. However, this approach is much more expensive, since the spin-orbitals are optimized several times. Again, a time saving is desired. Therefore, let us search for the spin-orbitals with the highest influence on the total energy value. It has been observed that not all doubly exchanged functions provide the same contribution to the energy. Some improve the result more and others less. This is due to the spin-orbital energy differences. The exchange of the spin-orbitals of significantly different energies does not contribute much to the total energy improvement. Therefore, it can be requested that the exchange is included in calculations only if the energy difference between the involved spin-orbitals is smaller than some given value. Hence, only some groups of spin-orbitals can be exchanged. Up to now, the spin-orbital notion was used. However, let us switch to the orbital language that is frequently used for MC–SCF considerations. For the N-electron system, the orbitals can be divided into three groups: • Core orbitals, which are not varied, since they have too low orbital energies, but are applied in the wave function expansion (doubly occupied orbitals) • Active orbitals, which are exchanged in the expansion (partially occupied orbitals)

Remarks on Wave Function Theory and Methods

43

• Virtual orbitals, which are not varied and not applied in the expansion (unoccupied orbitals) Instead of optimization of all the orbitals, only the active orbitals will be varied within the complete active space self-consistent field (CASSCF) approximation. In the acronym of this method, the number of active orbitals and active electrons is also provided for the given system. For instance, CASSCF (6,4) denotes the calculations with the expansion including all the possible exchanges of the four electrons within the six active orbitals. The CASSCF approach leads to all possible exchanges in the given active space, and for a moderately sized system, the size of the active spaces can quickly exceed the computational resources. In such a case, the solution can be the restricted active space self-consistent field (RASSCF) method, which supplies a way of limiting the size of the active space. Additionally, one needs to remember that for a powerful tool such as perturbation theory, there is no obstacle to applying the multi-determinant reference function as the unperturbed function in perturbation calculus. Thus, similar to the SCF–HF and MP2 approaches, CASPT2 would be the second-order perturbation theory complete active space method – the perturbationally corrected CASSCF.

Coupled Cluster Approximation: The Operator Strikes Back It would seem that all the straightforward ways to improve wave function in the oneelectron approximation have been exploited. However, we now next discuss one of the most accurate (and simultaneously most expensive) methods applied in quantum chemistry. The idea is simple. Consider again the expansion (Eq. 171). Introducing an operator bDC b0 C C b1 C C b2 C C b3 C C b4 C : : : ; C

(185)

b 0 j‰0 i D c0 j‰0 i ; C

(186)

b 1 j‰0 i D ScS ; C

(187)

b 2 j‰0 i D DcD ; C

(188)

b 3 j‰0 i D TcT ; C

(189)

b 4 j‰0 i D QcQ ; C

(190)

defined as

:: :

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D. Kedziera and A. Kaczmarek-Kedziera

allows one to write Eq. 171 in a very compact form: b ‰0 : ‰DC

(191)

Now the problem of finding the appropriate expansion can be replaced by the problem of finding the adequate operator. This is the essence of the coupled cluster (CC) method. Here the assumption is made that the wave function can be expressed by T ‰ D eb ‰0 ;

(192)

where ‰ 0 is a reference function (depending on the approach, this can be the onedeterminant HF function or the multi-determinant function arising from MC–SCF) and b T is a sought operator. Applying the expansion of the exponential function, it can be written that T b2 C 1 T b3 C : : : : bC 1T eb Db 1CT 2Š 3Š

(193)

b operators easier. Putting Such an expanded form makes the interpretation of the T bDb b2 C T b3 C b T T1 CT T4 C :::;

(194)

one can identify the subsequent Tbi operators as corresponding to i-tuple exchanges of the spin-orbitals in the reference function: b T 1 ‰0 D

X

tap ‰ap ;

(195)

a;p

b 2 ‰0 D T

X

pq

pq

tab ‰ab ;

(196)

a;b;p;q

and so forth. The coefficients t (called “amplitudes”) are in general not equivalent to the c coefficients in the CI expansion (see Eq. 171). In order to find their mutual relation, let us consider the approximate operator: bb b2 C T b3 C b T T1 CT T 4:

(197)

The operator (Eq. 193) takes the form T 1 Cb T 2 Cb T 3 Cb T4 eb Db 1 b b b CT1 CT2 CT3 Cb T4  2 1 b b b b4 C 2Š T 1 C T 2 C T 3 C T  3 b1 C T b2 C b b4 C 3Š1 T T3 CT  4 b1 C T b2 C T b3 C T b4 : C 4Š1 T

(198)

Remarks on Wave Function Theory and Methods

45

Limiting ourselves to the terms corresponding to not more than four spin-orbital exchanges and writing it in the ordered way according to the number of exchanges, one gets T 1 Cb T 2 Cb T 3 Cb T4 eb Db 1 Cb T1 b21 b2 C 1 T CT 2

(199)

b2 C 1 T b3 b3 C T b1 T CT 3 1 b3 C 1 T b22 C 1 T b21 T b2 C T b4 C b b41 : CT T 1T 2

2

Now the direct comparison can be made: b1 D b C T 1;

(200)

1 2 b2 D b C T ; T2 C b 2 1

(201)

b3 D T b31 ; b2 C 1 T b3 C T b1 T C 3

(202)

b22 C 1 T b21 T b3 C 1 T b2 C T b4 C b b41 : b4 D T T 1T C 2 2

(203)

bi operators, but still neither Cˆ i nor We have the relation between the Cˆ i and T b T i are known. Recall from the earlier sections that the double exchanges have a significant influence on the energy improvement with respect to the Hartree–Fock results. Taking double exchanges into account within the coupled cluster formalism means that the operators b T 1 and b T 2 need to be determined. However, as a side effect, they also allow inclusion of some not negligible contributions arising from the triple b1 and T b2 operators recover and higher exchanges. In the above comparison, the T ˆ ˆ two out of three terms in C3 and three out of five terms in C4 . This is the power of the CC method. Unfortunately, the strength of this method does not go together with ease b is ransomed with of calculations. Obtaining the expressions for the operators T compromises. Not only is the operator expansion (Eq. 194) truncated, but the basis set is finite. Moreover, the variational character of the method is sacrificed. In order to realize the complications, let us consider step-by-step the energy calculation within the CC formalism. We begin, as usual, with the electron Schrödinger Eq. 50. Substituting Eq. 192 gives T T b eb H ‰0 D Eeb ‰0 :

(204)

Taking into account that, due to (Eq. 193), D ˇ ˇ E ˇ Tˇ h ‰0 j ‰i D ‰0 ˇeb ˇ ‰0 D h‰0 j‰0 i D 1;

(205)

46

D. Kedziera and A. Kaczmarek-Kedziera

the energy can be calculated as ˇ E D ˇ ˇ E D ˇ ˇb b ˇ ˇbˇ E D ‰ 0 ˇH e T ˇ ‰ 0 D ‰ 0 ˇH ˇ‰ :

(206)

This is not the Hamiltonian average value expression. Additionally, the operator inside the bracket is not Hermitian. But, until we assume that (Eq. 192) is true, such an approach works. We can also construct an integral: D

ˇ ˇ E ˇ ˇ E D ˇ pq ˇ b b pq ˇ T ˇ ‰ab ˇH e T ˇ ‰0 D E ‰ab ˇeb ˇ ‰0 ;

(207)

which is the consequence of Eq. 204 and will be applied in the near future. We should now concentrate on the way of determining the form of amplitudes. To simplify the considerations, we can assume b b2 ; T T

(208)

which is equivalent to the CCD variant. We are interested in finding the amplitudes pq tab . The final result of the calculations will be the approximate energy: ˇ E D ˇ ˇb b ˇ e T 2 ˇ ‰0 : ECCD D ‰0 ˇH

(209)

pq

The information about the amplitude tab can be extracted from the integral ˇ ˇ E D pq pq ˇb ˇ tab D ‰ab ˇT 2 ˇ ‰0

(210)

(see Eq. 196). However, the amplitudes are still not known, since we do not know b2 operator. Therefore, one more equation is necessary to elicit the sought the T information. Let us begin with the approximated expression (Eq. 207): D

ˇ ˇ E ˇ ˇ E D ˇ pq ˇ b b pq ˇ T 2 ˇ e T 2 ˇ ‰0 D ECCD ‰ab ˇeb ‰ab ˇH ˇ‰ ˇ ED ˇ ˇ E 0 D ˇ ˇb b ˇ ˇ ˇ pq T2 D ‰0 ˇH e T 2 ˇ ‰0 ‰ab ˇeb ˇ ‰0 :

(211)

The expansion (Eq. 193) tailored to the present case, T2 b2 C : : : ; b2 C 1 T eb Db 1CT 2 2

(212)

and substituted to the left-hand side of Eq. 211 gives D

ˇ

ˇ ˇ ˇ E  1 b2 ˇˇ ˇ pq ˇ b b pq ˇ b b ‰ab ˇH e T ˇ ‰0 D ‰ab ˇˇH ‰0 : 1Cb T2 C T 2 2 ˇ

(213)

Remarks on Wave Function Theory and Methods

47

Further terms are not necessary; in such a case, the functions on both sides of the integral would differ with four and more spin-orbital exchanges (and Hamiltonian is still aDsumˇ of oneˇ and E two-electron operators). Similarly, the expansion in the ˇb b ˇ e T 2 ˇ ‰0 will also be truncated on the second term: integral ‰0 ˇH D

ˇ ˇ E D ˇ  ˇ E ˇb b ˇ ˇb b b ˇ e T 2 ˇ ‰ 0 D ‰ 0 ˇH 1 C T 2 ˇ ‰0 : ‰ 0 ˇH

(214)

D ˇ ˇ E ˇbˇ .0/ E 0 D ‰ 0 ˇH ˇ ‰0 ;

(215)

ˇ ˇ E ˇ E D ˇ ˇb b ˇ ˇbb ˇ .0/ e T 2 ˇ ‰ 0 D E 0 C ‰ 0 ˇH T 2 ˇ ‰0 : ‰ 0 ˇH

(216)

Remembering that

one gets D

ˇ ˇ E D pq ˇ T ˇ The last integral on the right-hand side of Eq. 211, ‰ab ˇeb ˇ ‰0 , can be nonvanishing only if the functions on the right and left side are the same. This is possible for ˇ ˇ E D ˇ ˇ E D ˇ pq ˇ T 2 ˇ pq ˇ T 2 ˇ ‰0 : ‰ab ˇeb (217) ˇ ‰0 D ‰ab ˇb This is the integral that can provide information about the desired amplitudes of Eq. 210. Putting all these together, one gets  pq ‰ab

ˇ

ˇ ˇH b b b2 C 1CT ˇ

ˇ  ˇ E D ˇ ˇ E D ˇ 1 b2 ˇˇ ˇbb ˇ pq ˇb ˇ T 2 ˇ ‰0 ‰ab ˇT T 2 ˇ ‰ 0 D E 0 C ‰ 0 ˇH 2 ˇ ‰0 : 2 (218)

ˇ ˇ E D pq pq ˇb ˇ Therefore, the amplitude tab D ‰ab ˇT 2 ˇ ‰0 can be expressed as

pq

tab

ˇ  ˇ E D pq ˇ b b b22 ˇˇ ‰0 b2 C 1 T ‰ab ˇH 1CT 2 ˇ E D ˇ D : ˇbb ˇ E 0 C ‰ 0 ˇH T 2 ˇ ‰ 0

(219)

Unluckily, this does not mean that the amplitudes are known. Still, the above pq b2 expression also contains the tab amplitudes on the right-hand side in the T operators. Moreover, all other amplitudes are also present on the right-hand side. The consequence of this aggravation is that the CC equations cannot be solved separately, one by one. All together, the complicated set of nonlinear equations must be handled. The number of equations is equal to the number of sought amplitudes. This is the main reason for the huge computational cost of the CC calculations, even

48

D. Kedziera and A. Kaczmarek-Kedziera

though the variationality of the method was abandoned (Atkins and Friedman 2005; Cramer 2004; Jensen 2006; Levine 2008; Piela 2007; Roos and Widmark 2002). It can be seen that solving the CC equations is quite complicated, even in the simplified case of the CCD approach. If one wanted to use the variational Hamiltonian and apply its average value, the following integrals would appear: ˇ D ˇ ˇ D ˇ ˇ E  ˇˇ t E ˇ ˇbˇ ˇbˇ b T b b T H e T ˇˇ ‰0 D eb ‰ ˇH ‰ 0 ˇH ˇ ‰ D ‰0 ˇˇeb ˇ e T ‰0 :

(220)

In order to calculate them, one needs to know the form of all the b T i operators, since not only the function on the right-hand side of the above integral will contain the exchanged spin-orbitals but also on ˇ ˇ E the left-hand side. Therefore, one D the function ˇbˇ b b needs to calculate terms like T 3 ‰0 ˇH ˇ T 2 ‰0 and many others. This causes the significant increase of the computational costs of the CC method. Like in the MP n case, the CC method is worth using for the short expansion of b operator. Thus, relatively good accuracy is obtained with a moderate price. the T

Conclusions We have finally reached the end of the zeroth iteration in the process of learning quantum chemistry methods. The beginner may feel saturated or even overwhelmed; however, we hope that this chapter arouses interest. Our aim was to show that simple ideas underlie quantum chemistry methods. The purpose is to put complicated things in a simpler and more convenient form. One of the most popular rules in computational chemistry is as follows: “If you cannot calculate something, divide it into parts in such a way that you can calculate some contribution while the other is too difficult.” For instance, nonrelativistic energy can be divided into HF energy and correlation energy. Correlation energy accounts for the contribution that we cannot calculate in practice, but methods such as MP n, CC, and CI allow one to find some part of it. It may happen (and it often does!) that what we can calculate will be enough. This chapter should be treated as the introduction to more advanced handbooks or as a guide through the symbols and concepts applied in the later parts of this book. Thus, some of the concepts are just touched upon, and many are omitted. If the reader noticed this and wants to know more, it means that this chapter has met its goal. Acknowledgments Authors are grateful to Dr Krzysztof Strasburger, Dr Andrej Antušek, Agnieszka Zawada, and Łukasz Mentel for helpful comments.

Remarks on Wave Function Theory and Methods

49

Bibliography Atkins, P., & Friedman, R. (2005). Molecular quantum mechanics. Oxford: Oxford University Press. Cramer, C. J. (2004). Essentials of computational chemistry: Theories and models. Chichester: Wiley. Griffiths, D. J. (2004). Introduction to quantum mechanics (2nd ed.). San Francisco: Benjamin Cummings. Jensen, F. (2006). Introduction to computational chemistry. Chichester: Wiley. Levine, I. N. (2008). Quantum chemistry (6th ed.). Englewood Cliffs: Prentice Hall. Lowe, J. P., & Peterson, K. (2005). Quantum chemistry (3rd ed.). Boston: Academic. McQuarrie, D. A., & Simon, J. D. (1997). Physical chemistry: A molecular approach. Sausalito: University Science. Piela, L. (2007). Ideas of quantum chemistry. Amsterdam: Elsevier. Ratner, M. A., & Schatz, G. C. (2000). Introduction to quantum mechanics in chemistry. Upper Saddle River: Prentice Hall. Roos, B. O., & Widmark, P.-O. (Eds.). (2002). European summerschool in quantum chemistry. Lund: Lund University. Szabo, A., & Ostlund, N. S. (1996). Modern quantum chemistry: Introduction to advanced electronic structure theory. Mineola: Dover.

Directions for Use of Density Functional Theory: A Short Instruction Manual for Chemists Heiko Jacobsen and Luigi Cavallo

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFT: A Paradigm Shift in Theoretical Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holes and Electron Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Climbing Jacob’s Ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical DFT and the Density Functional Zoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFT: Computational Chemistry in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Molecular and Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orbitals in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFTips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B3LYP Is No Synonym for DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choose Your DFT Method Carefully . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Read the Fine Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFT Does Not Hold the Universal Answer to All Chemical Problems . . . . . . . . . . . . . . . . . . . . . . . Make DFT an Integral Part of Your Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Follow Your Interests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Books on DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reviews and Overviews of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual Developments and Applications of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Developments and Applications of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 8 11 14 16 17 19 34 36 37 37 37 38 38 38 39 39 39 40 41

H. Jacobsen () KemKom, New Orleans, LA, USA e-mail: [email protected] L. Cavallo Physical Sciences and Engineering Division, Kaust Catalysis Center, King Abdullah University of Science and Technology (Kaust), Thuwal, Saudi Arabia e-mail: [email protected] © Springer Science+Business Media Dordrecht 2016 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_4-2

1

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H. Jacobsen and L. Cavallo

Abstract

Two aspects are quintessential if one seeks to successfully perform DFT calculations: a basic understanding of how the concepts and models underlying the various manifestations of DFT are built and an essential knowledge of what can be expected from DFT calculations and how to achieve the most appropriate results. This chapter expands on the development and philosophy of DFT and aims to illustrate the essentials of DFT in a manner that is intuitively accessible. An analysis of the performance and applicability of DFT focuses on a representative selection of chemical properties, including bond lengths, bond angles, vibrational frequencies, electron affinities and ionization potentials, atomization energies, heats of formation, energy barriers, bond energies, hydrogen bonding, weak interactions, spin states, and excited states.

Introduction Density functional theory (DFT) is an enticing subject. It appeals to chemists and physicists alike, and it is entrancing for those who like to work on mathematical physical aspects of problems, for those who relish computing observable properties from theory, and for those who most enjoy developing correct qualitative descriptions of phenomena. It is this combination of a qualitative model that at the same time furnishes quantitative reliable estimates that makes DFT particularly attractive for chemists. DFT is an alternative, and complementary, to wave function theory (WFT). Both approaches are variations of the basic theme of electronic structure theory, and both methods originated during the late years of the 1920s. Whereas WFT evolved rapidly and gained general popularity, DFT found itself in a state of shadowy existence. It was the appearance of the key papers by Hohenberg and Kohn (1964) and by Kohn and Sham (1965), generally perceived as the beginning of modern DFT, which changed the perception and level of acceptance of DFT. With the evolution of reliable computational technologies for DFT chemistry, and with the advent of the generalized gradient approximation (GGA) during the 1980s, DFT emerged as powerful tool in computational chemistry, and without exaggeration the 1990s can be called the decade of DFT in electronic structure theory. During this time period, despite the lack of a complete development, DFT was already competitive with the best WFT methods. Furthermore, the advancement of computational hardware as well as software has progressed to a state where DFT calculations of “real molecules” can be performed with high efficiency and without major technical hurdles. But at the end of the first decade of the new millennium, it appeared that DFT might have become a victim of its own success. DFT has transformed into an off-the-shelf technology and ready-to-crunch component and often was and still is used as such. Still and all, it became clear that the happy days of black-box DFT were over and that not all the promises of DFT came to fruition. DFT has its

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

3

own limitations and shortcomings, and the numbers obtained from DFT calculations began to lose some of their awe-inspiring admiration they enjoyed at the end of the first millennium. At the same time, DFT has matured into a standard research tool, used routinely by many experimental chemists to support their work. Despite the fact that DFT has evolved into a first-choice approach for a stunning potpourri of applications throughout chemistry and materials science, it is all too well known that DFT in its current form still has serious limitations; Cohen et al. (2012) identify and analyze current challenges to DFT. Moreover, it appears that the philosophical, theoretical, and computational framework of DFT might lose its prominent position in electronic structure theory. Fifty years after its formal inception, Becke (2014) reviewed the development and current status of Kohn– Sham density functional theory and raises the concern that increasing pressure to deliver higher and higher accuracy and to adapt to ever more challenging problems “may submerge the theory in the wave-function sea” (Becke 2014). But in whatever direction the path of extended development will guide users and developers of DFT, it seems evident that DFT will continue its role as research tool not only for computational but also for theoretical inclined chemists. We have composed our short instruction manual for chemists in view of the ontogeny of DFT. A chemist using DFT calculations should be aware of the fact that all approximations and simplifications of any general theory may lead to failures in computed data. Every principally correct theory, if not executed with specific care, may produce essentially wrong results and therefore erroneous predictions. Two aspects are quintessential if one seeks to successfully perform DFT calculations: a basic understanding of how the concepts and models underlying the various manifestations of DFT are built and an essential knowledge of what can be expected from DFT calculations and how to achieve the most appropriate results. Thus, we have divided the main body of our directions into two parts. In section DFT: A Paradigm Shift in Theoretical Chemistry, we expand on the development and philosophy of DFT. We do not present a course or textbook work on DFT; the interested reader will find a selection of references to the literature for more elaborate and detailed descriptions. Rather, we aim to illustrate the essentials of DFT in a manner that is intuitively accessible. For this reason, we will avoid mathematical equations as much as possible, incorporate formulas into the flow of the text, and only on occasion add the odd-numbered equation to the elaboration. As a consequence, we have to abandon the rigor that unambiguous and well-defined derivations require and introduce a certain degree of sloppiness. We hope that such a treatment will facilitate the flow of our arguments and emphasize that what we think are indispensable aspects of DFT. In section DFT: Computational Chemistry in Action, we present an analysis of the performance and applicability of DFT. We focus on a representative selection of chemical properties and system types and base our review on a selection of representing benchmarking studies, which encompass several well-established density functionals together with the most recent efforts in the field. Due to the multitude of papers that report DFT applications, our analysis is far from complete,

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H. Jacobsen and L. Cavallo

but we aim to present a representative snapshot of the current situation of DFT at the beginning of the second decade of the new millennium. We will close our work with two additional short sections DFTips and A Concise Guide to the Literature, where we present the reader with a few tips how to use DFT, and with a collection of selected references as a concise guide to the by now vast literature of DFT.

DFT: A Paradigm Shift in Theoretical Chemistry Density functional theory is primarily a theory of electronic ground-state structure, which is based on the electron density distribution ¡(r). In contrast to DFT, wave function theory is an approach to electronic structure, which is based on the manyelectron wave function ‰(rn ). In order to put the innovation of DFT into proper perspective, we begin with a brief overview of WFT, before we illustrate the essentials and growth of DFT. The objective of WFT is the exact solution of the time-independent Schrödinger equation (TISE), H‰ D E‰; for a system of interest. [We recall that in quantum mechanics, associated with each measurable parameter in a physical system is an operator, and the operator associated with the energy of a system is called the Hamiltonian H. The Hamiltonian contains the operations associated with the kinetic and potential energies of all particles that comprise a system. We further note that the terms function, operator, and functional are to be understood such that a function is a prescription which maps one or more numbers to another number, an operator is a prescription which maps one function to another function, and a functional takes a function and provides a number.] The solution to the TISE yields the wave function ‰ as well as the energy E for the system of interest. In a systematic, variational search, one looks for the wave function that produces the lowest energy and arrives at a description for the system in its ground state. If we consider a system of nuclei and N electrons, solving the TISE – within the Born–Oppenheimer separation between the apathetic ambulation of nuclei and the rapid ramble of electrons – yields the electronic molecular wave function ‰ el (r). This wave function depends explicitly on the 3 N coordinates of all N electrons, all of which might undergo positional permutation due to repulsive Coulomb interaction. A first approximation to the challenging task of solving the TISE slackens the interaction between electrons by any exchange and thus reduces the function  el (r) of 3 N variables to a product of N functions ¥ each depending only on three variables, ‰el .r/ D …iD1;N i .ri /. Atomic orbitals are conveniently chosen to represent the functions ¥. However, such a Hartree product of atomic orbitals violates the Pauli exclusion principle due to the fermion nature of electrons, and hence the appropriate form for a system of noninteracting electrons is a single determinant j1 .r1 / ; : : : n .rn /j ; known as Slater determinant. Such a wave function, from the mathematical properties of determinants, is antisymmetric with respect to exchange of two sets of electronic variables as it should be. One row of a Slater determinant carries contributions from all atomic orbitals  and is

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

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commonly referred to as molecular orbital (MO) §§MO . This approximate method for the determination of the ground-state wave function and ground-state energy is the well-known Hartree–Fock (HF) method. Although in the HF method the electrons obey exchange as required by the Pauli exclusion principle, the electrons are noninteracting, and the movement of one electron within the system is independent from the movement of all other electrons. However, as the presence of Coulomb repulsion between electrons would suggest, the electrons move in a correlated fashion. In order to allow for electron correlation, configuration interaction (CI) is introduced in that the wave function is constructed as a linear combination of several Slater determinants, obtained from a permutation of electron occupancies among all MOs available. Increasing the number of Slater determinants increases the accuracy of the calculations, although the added accuracy comes with the price of added computational cost that often becomes the limiting factor for WFT calculations. This brief exposition brings about two main differences between DFT and WFT. A WFT calculation in general, and increasing the accuracy of WFT calculations in particular, is computationally demanding. DFT seems to be more cost-efficient; after all, the simplest HF wave function ‰ el (r) depends on 3 N spatial coordinates, whereas the probability distribution of electrons in space ¡(r) depends only on three coordinates. But there exist strategies how the result of WFT can be systematically improved, whereas there is no methodical, standardized scheme to improve DFT calculations. In the following, we will explore reasons for the WFT–DFT differences. In WFT, atoms and molecules constitute the basic systems of interest. Since the distribution and redistribution of electrons within atoms and molecules are central to chemical properties and reactivity, we now limit ourselves to systems comprised of N electrons in motion, with some two-particle interaction. The Hamiltonian of such a system reads H D T C U C V, where T and U denote the operators for kinetic energy and for electron–electron interaction energy, respectively. Whereas U results in the internal potential energy of our system, the moving electrons might in addition interact with an external potential, and the operation V recovers the potential energy due to this extra interaction. For systems of electrons that move in a field of fixed nuclei, the external potential V is always just the nuclear field. Systems of electrons in combination with fields of fixed nuclei represent the essential building blocks of matter such as molecules or solids. For a chemist, it might appear counterintuitive that nuclei, being an essential part of a molecule, represent an external potential, but the nuclear potential – although internal to a molecule – is external to the density of the moving electrons. If we consider the external potential to be a uniformly distributed background positive charge, we arrive at the uniform or homogeneous electron gas (UEG or HEG), also known as jellium. At zero temperature, the properties of jellium depend solely upon the constant electron density distribution ¡ .r/ fD const:g. Such a treatment of electronic density, the Thomas–Fermi (TF) model, constitutes the origins of DFT (Parr and Yang 1989). The TF model is able to describe the kinetic energy of

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H. Jacobsen and L. Cavallo

the UEG as functional of the electronic density, and later on Dirac added a density functional for the exchange energy as a conclusion of the Pauli principle. The UEG formalism itself provides the basis for the local density approximation (LDA). Before we proceed, we take a small step back to WFT. At the beginning of the 1950s, Slater (1951) described the then current situation in WFT as follows: “The Hartree-Fock equations furnish the best set of one-electron wave functions for use in a self-consistent approximation to the problem of the motion of electrons in the field of atomic nuclei. However, they are so complicated to use that they have not been employed except in relatively simple cases.” Facing the decision “Do you want to calculate it, or do you want it to be accurate?,” Slater decided to replace the peculiar exchange term in the HF equations by something equivalent, yet easier to calculate. Slater used the free-electron approximation for the exchange potential, which, as Dirac has shown, could be expressed as a density functional. His new method, termed HFS, “was easy enough to apply so that we can look forward to using it even for heavy atoms” (Slater 1951), and in order to check its applicability, he performed calculations for the transition metal ion CuC . The exchange potential functional derived from the exchange energy functional contains one additional, scalable parameter ˛, which led to the development of the X˛.  method. This model enjoyed a significant amount of popularity among physicists and is still a topic of ongoing research activities (Zope and Dunlap 2006). The HFS – or the X˛  method – became the first practically used DFT method in chemistry. Two points that were fundamental for the progress of DFT were already anticipated within the advancements of the HFS – or X˛method: (1) Every density functional method to some degree contains one or more empirical parameters. Therefore, DFT has often been regarded as “Yet another Semi-Empirical Method” (YaSEM). The Hartree–Fock–Slater model, which can be regarded as ancestor of modern DFT, is such an example. But whereas the HFS method is intrinsically approximate, modern DFT is in principle exact (Kohn et al. 1996). (2) Transition metal chemistry has played and continues to play a major role in the progression of DFT. The work of Baerends and Ros (1978) is representative of the transition-metal– HFS era, and Ziegler’s contributions, summarized in his early review article (Ziegler 1991), have provided the impetus that changed the perception of calculations based on densities from YaSEM to DFT, “Das Future Tool.” A review article by Cramer and Truhlar (2009) is dedicated solely to developments and progress of DFT for transition metal chemistry and summarizes the state of affairs at the end of the first decade of the new millennium. At the heart of modern DFT is the rigorous, simple lemma of Hohenberg and Kohn (1964), which states that the specification of the ground-state density, ¡(r), determines the external potential V(r) uniquely (Eq. 1): ¡ .r/ ! V .r/ .unique/

(1)

This first theorem by Hohenberg and Kohn (HK-I) is not difficult to prove (Parr and Yang 1989), but for a chemist, the essentials of HK-I that given a density, only one external potential corresponds to that density, are intuitively clear. A

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

7

O O C3H6O ?

O

!

O

32 electrons

Propylene Oxide

OH

Fig. 1 Visualization of the first Hohenberg–Kohn theorem (density map drawn for a contour envelope of 0.01 a.u.)

pictorial representation of HK-I is shown in Fig. 1; we consider the density created by 32 electrons that move around the external potential created by one oxygen, three carbon, and six hydrogen nuclei. By inspection of a density map of a certain density value, it becomes obvious that of all the atomic constellations considered, only one seems to be consistent with the shape of the density map. Such a consideration reflects ideas developed in the context of “conceptual DFT” (Geerlings et al. 2003). HK-I also expresses the fact that there is a one-to-one mapping between the potential V(r), the particle density ¡(r), and the ground-state wave function ‰ 0 (Eq. 2): ¡.r/

! V .r/

! ‰0

(2)

This implies that all properties of a system are functionals of the groundstate density, since any property may be determined as the expectation value of the corresponding operator. With the help of this lemma, a minimal principle for the energy as functional of ¡(r) can be derived. The second Hohenberg– Kohn theorem (HK-II) provides the necessary guidelines to obtain the ground-state energy. Following HK-II, a variational principle is established, according to which the ground-state density of a system of interest can be determined. In order to put the promise of the HK theorems that all properties of a system can be obtained from its ground-state density, into reality, one would need a construction that is computationally accessible while maintaining the formal exactness of HK-I and HK-II. To this end, Kohn and Sham (1965) introduced a fictitious system of N noninteracting electrons that have for their overall ground-state density the same density as some real system of interest where the electrons do interact. Using some aspects of HF theory, the ground-state wave function ‰ 0 of such a noninteracting system is described by a single Slater determinant. The orbitals, which form this Slater determinant, known as Kohn–Sham (KS) orbitals ®KS , are solutions of

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H. Jacobsen and L. Cavallo

N single-particle equations. Following the variational principle, the ground-state energy and the ground-state density are determined from variations in ®KS . The essential contribution to the KS energy comes from the so-called exchangecorrelation energy EXC . It incorporates corrections to the kinetic energy due to the interacting nature of the electrons of the real system, all nonclassical corrections to the electron–electron repulsion, as well as electron self-interaction corrections. If EXC is ignored, the physical content of the theory becomes identical to that of the Hartree approximation. Thus, within the KS formalism, the electronic energy of the ground state of a system of N electrons moving within an external potential of nuclei is expressed – without approximations – as a functional of the ground-state density (Eq. 3): h E ¡ D Ts Œ ¡  C UŒ Œ¡.1/; ¡.2/  C Vne Œ ¡  C EXC Œ ¡

(3)

In Eq. 3, the first term represents the kinetic energy of the system of N noninteracting electrons, the second term corresponds to the Coulombic repulsions between the total charge distributions at two different positions within the system, and the third term accounts for nuclear–electron interactions, due to the presence of an external potential. It is the fourth term, the functional for the exchangecorrelation energy EXC , which is responsible for the power and magic of DFT. What makes current DFT applications approximate is the unknown analytic expression of EXC , for which an approximation is needed. The KS formalism is closely related to the HF formalism. What differentiates the KS operator from the HF operator is the exchange-correlation potential VXC . VXC in turn is a functional derivative of the exchange-correlation energy EXC . Furthermore, the Hamiltonian H operating on the wave function that is associated with the density of a fictitious system of N noninteracting electrons can be expressed as sum of oneelectron operators. Whereas increasing the accuracy of HF calculations is accompanied with a steep increase in computational cost, increasing the accuracy of DFT calculations apparently requires modifications in VXC , which – if at all – only lead to a moderate increase in computational cost. However, whereas it is well known how to systematically improve the accuracy and quality of HF calculations, no comparable strategy exists for DFT calculations within the confines of a KS approach. It appears that a detailed knowledge of the exchange-correlation energy EXC is essential for designing more accurate density functionals.

Holes and Electron Pairs The exchange-correlation energy EXC is a relatively small part of the total energy of a typical system, although it is by far the largest part of “nature’s glue” that binds atoms together (Kurth and Perdew 2000). It arises because the electrons do not move randomly through the density but avoid one another. Ziegler (1995) illustrates

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

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Fig. 2 Electrons in distress: While trying to maximize the attraction from the nuclei, an electron experiences enhanced repulsion from the other electron as it moves around in the molecular framework (top). To minimize the repulsion, each electron creates an exclusion area around itself, into which no other electron can penetrate (bottom) (Cartoon by Lauren Bertolino)

the situation as follows: An electron will try to maximize the attraction from the nuclei and minimize the repulsion from the other electrons, as it moves around in the molecular framework. To do so, it creates an exclusion area or “no-fly zone” around itself into which no other electron can penetrate, as pictorial exemplified in Fig. 2. The exclusion zone is referred to as the exchange and correlation (XC) hole, and it is the way in which the XC hole is modeled that distinguishes one electronic theory from another. Each density functional has its own characteristic XC fingerprint. The XC hole also determines to a large part EXC , which however contains three different contributions. The first is the potential energy of exchange, which also should include corrections for self-exchange or self-interaction. The second is the potential energy of correlation due to the effect of Coulomb repulsion. Both potential energies are negative and determined by the nature of the XC hole. The third contribution to EXC is a smaller positive kinetic energy of correlation due to the extra swerving motion of the electrons as they avoid one another (Perdew et al. 2009). The XC hole arises from an extension of the concept of the unconditional oneelectron probability density ¡(1) by considering pairs of electrons and a resulting conditional probability. When a reference electron is known to be at position 1, the conditional probability ¡cond (2, 1) of the other electron to be at position 2 can be written as the sum of the unconditional probability ¡(2) of the other electron and the XC-hole density ¡XChole .2; 1/. Thus, the hole ¡XChole .2; 1/ describes how the conditional density of the other electron deviates from its unconditional density ¡(2). It is instructive to have a closer look at hole profiles, and as simple example, we will consider the hydrogen molecule H2 with only two electrons or one electron pair. In Fig. 3, hole densities for H2 are shown; the two nuclei HA and HB are separated by 72 pm, and the reference electron is placed 15 pm to the left of nucleus HB . The XC hole can be split into contributions from the exchange or X hole, which arises from the fermion nature of an electron obeying the Pauli principle

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H. Jacobsen and L. Cavallo

HA

e

0.0 0.0 1.0 −0.1

2.0

3.0

HA

HB

4.0

D.V.

5.0

6.0

0.0 7.0 0.0 1.0 −0.1

2.0

e 3.0

X-hole

−0.3

HA

HB

4.0

D.V.

5.0

6.0

0.0 7.0 0.0 1.0 −0.1

2.0

e 3.0

4.0

HB 5.0

6.0

7.0

D.V.

−0.2

−0.2

−0.2 −0.3

0.1

0.1

0.1

C-hole

−0.3

XC-hole

Fig. 3 Hole densities in the hydrogen molecule: Only the full XC-hole ¡XChole .2; 1/ has physical meaning (Adapted from Baerends and Gritsenko (1997), with permission by the American Chemical Society)

and the correlation or C hole due to Coulomb repulsion within the pair of electrons. [The X hole and C hole are often referred to as Fermi hole and Coulomb hole, respectively.] The X hole puts an emphasis on the reference electron. It creates a taboo zone for the other electron with a negative ¡Xhole .2; 1/ probability density not only around the region in space where the reference electron currently is but also where it might be. Regions in the vicinity of both nuclei HA and HB are declared as “no-fly zone.” The C hole on the other hand puts an emphasis on the other electron. It excludes regions where the other electron might experience Coulomb repulsion with the reference electron, but it also maps out regions where the other electron can benefit from attractive Coulomb interactions with nuclei. We see a negative ¡Chole .2; 1/ probability density around HB but a positive ¡Chole .2; 1/ probability, a buildup of density, around HA far away from the reference electron. Whereas the X hole and C hole illustrate exchange and correlation, only the combined XC hole has physical meaning. Before we continue, a short remark on the use of some language is in order. Since the terms local and nonlocal are often recurred to in the context of DFT, and often with different meanings, we briefly define how the terms local and nonlocal are used in this work. An approximation is said to be local if its energy density and related properties at any position of interest depend only on the electron density neighborhood of the given position. Otherwise, an approximation is said to be fully nonlocal. [We note that some physicists separate local approximations into “strictly local” and “semilocal.”] From an inspection of Fig. 3, it appears that both the C hole and the X hole are inherently nonlocal (the same as the HF exchange energy). The XC hole, too, must therefore be nonlocal, but its dominant contributions arise from the region around its reference electron – the XC hole appears to be more local and less nonlocal than the X or C hole. This observation already anticipates that local density functionals might be able to produce approximate models for the nonlocal XC hole. Although it seems that there exists no systematic approach like in WFT to improve the accuracy of DFT, the advancement of density functionals depends

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

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Fig. 4 The Jacob’s ladder of density functional approximations to the exchange-correlation energy (Reprinted from Perdew et al. (2009), with permission by the American Chemical Society)

on more precise descriptions not only of the XC hole but also of the exchangecorrelation energy EXC . This task can be approached in a methodical manner.

Climbing Jacob’s Ladder Perdew and Schmidt (2001) compare the development of enhanced density functionals to a climb of Jacob’s ladder, leading the way from the Hartree world to the heaven of chemical accuracy, illustrated in Fig. 4. Each rung of the ladder adds a refinement to the approximation of the exchange-correlation energy.

First Rung: The Local Density Approximation The first rung employs only the local densities in the description of the exchangecorrelation energy. This method is known as the local density approximation (LDA). [Although vital to the fermion nature of electrons, so far we have treated spin rather novercal. But the issue of spin can be treated as well in density functional theory, and the local spin density approximation (LSD or LSDA) replaces the spin-averaged energy density with the energy density for a polarized homogeneous electron gas. LDA and LSDA are now often used synonymously.] LDA takes its densities from the uniform electron gas (UEG), and an analytical form of a density functional for the exchange energy of the UEG can be derived (the same exchange energy as used in the HFS method). No such expression exists for the correlation energy, but the UEG correlation energy can be calculated numerically and fit in various ways. One successful and popular parameterization comes from the work of Vosko, Wilk, and Nussair, referred to as VWN (Vosko et al. 1980). LDA performs surprisingly well in predicting molecular properties that are based on relative energy differences within a given density. Molecular geometries for

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H. Jacobsen and L. Cavallo

representative main group compounds could be reproduced in close agreement to the experiment (Versluis and Ziegler 1988). For transition metal complexes, metal–ligand separations are slightly underestimated, but still within acceptable conformity with X-ray data (Ziegler 1995). However, properties that are based on absolute energy differences between densities, such as bond energies, are not well described by LDA, where a clear overbinding tendency emerged. LDA is therefore a remarkably useful structural, though not thermochemical, tool. The disappointing performance of LDA in estimating thermochemical properties spawned the development of gradient-based methods, the second rung on Jacob’s ladder.

Second Rung: The Generalized Gradient Approximation The second rung or generalized gradient approximation (GGA) adds the gradients of the local densities to the exchange-correlation picture. It became clear that the homogeneous electron gas is only of limited use as a model of the inhomogeneous electron density within molecules, and approximations for exchange and correlation energy were augmented by density gradients. [In the older literature, GGAs are sometimes called nonlocal (LDA/NL), since the gradient implies a directional change within the density.] Two early models for correlation (Perdew 1986) and exchange (Becke 1988a) in combination resulted in the BP86 functional, the GGA that was most influential in the early developments of transition metal DFT (Ziegler 1991). Gradient corrections are essential for a quantitative estimate of bond energies as well as metal–ligand bond distances (Ziegler 1995). Third Rung: Meta-Functionals The third rung adds the kinetic energy density to the description of density functionals (Tao et al. 2003) and addresses the smaller third contribution to EXC . Such functionals are referred to as meta-functionals and, when built on second rung functionals, as meta-GGA (MGGA). The first three rungs of Jacob’s ladder all represent local functionals. They often work because of proper accuracy for a slowly varying density or because of error cancellation between exchange and correlation. Error cancellation can occur because the exact XC hole is usually more localized around its reference electron than the exact X hole (compare Fig. 3). Regions in which no error cancellation is expected are regions where exchange dominates correlation (Perdew et al. 2009). Climbing up the ladder, the approximations become more complicated, more sophisticated, and typically more accurate. Computation times increase modestly from the first to the third rungs and much more steeply after that. The added ingredients on each higher rung of the ladder can be used to satisfy more exact constraints or to achieve better agreement with experimental data (or both). These two strategies define the nonempirical and semiempirical approaches commonly used to improve density functionals. Beginning with the fourth rung, the nature of the density functionals changes from local to nonlocal.

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

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Fourth Rung: Hyper-Functionals The fourth rung, which also represents the first fully nonlocal rung, adds the exact exchange energy density. Such a functional is termed hyper-GGA (HGGA). After reaching the second rung, DFT progressed rapidly and took one giant step from the second to the fourth rung, omitting meta-GGAs. Following the idea of adiabatic connection, Becke derived a functional for exchange, which contained contributions from the exact HF exchange (1993a). He then designed an advanced functional for the exchange-correlation contribution containing three parameters for its various parts, including gradient corrections for correlation, gradient corrects for exchange, as well as an exact exchange contribution (1993b). These semiempirical coefficients have been determined by a linear least-square fit to 56 atomization energies, 42 ionization potentials, eight proton affinities, and ten first-row total atomic energies. Becke’s functional, combining HF theory and DFT with the use of three empirical coefficients, was the first example and initiated the evolution of so-called hybrid functionals. The hybrid functional B3LYP, based on Becke’s parameterization, was to a large part responsible for the meteoric ascent of DFT during the 1990s. While the first three rungs of Jacob’s ladder require no fitting of experimental data, empiricism seems unavoidable on the fourth rung. This has caused some skepticism, and it appeared that the success of “empirical DFT” would eventually be responsible for the death of “true DFT.” Gill humorously described the situation at the beginning of the new millennium in his obituary to DFT (Gill 2001). The Jacob’s letter metaphor puts the addition of exact exchange to density functionals into proper perspective. The step from the third rung to the fourth rung results in a new class of functionals, so-called HMGGAs. HMGGA functionals are currently a field of active development and appear to produce promising results. M06, for example, is a HMGGA with good accuracy for a variety of different chemical applications ranging from transition metals over main-group thermochemistry to barrier heights of chemical reactions. Thus, HMGGAs might be considered as a class of density functionals with broad applicability in chemistry (Zhao and Truhlar 2008a). Whether HMGGA is read as hybrid meta-GGA or hyper meta-GGA is a matter of taste; fortunately, both specifications result in the same acronym. Fifth Rung The fifth rung of Jacob’s ladder adds exact correlation as new ingredient. One might think of this as an expansion of the density space of a system by adding virtual densities into the picture. One approach to this problem is the use of the randomphase approximation (RPA). RPA in DFT in turn is closely related to time-dependent DFT (TD-DFT). The essence of RPA might be described as constructing the excited states of a system as a superposition of particle-hole excitations. When building a fifth-rung density functional for the exchange-correlation energy, the RPA utilizes full exact exchange and constructs the correlation with the help of the unoccupied Kohn–Sham orbitals. Like the first three rungs of Jacob’s

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ladder, the fifth rung requires no fitting. Although the essentials of RPA originated in the 1950s, fifth-rung methodologies were considered too computationally expensive for widespread use and application. Yet improvements in hardware and algorithms also entered the development of RPA technology, and it seems that “RPA has the potential to become a building block of future generations of electronic structure methods” (Furche 2008). In the early days of modern density functional theory, hazy clouds of ambiguity that enfolded the XC hole obscured the view of Jacob’s ladder. The existence of the third rung of Jacob’s ladder was recognized before the fourth rung entered the Jacob’s ladder picture (Becke and Roussel 1989), but at first it did not appear as a safe and secure stage for the ascent toward the heaven of chemical accuracy. Thus, although MGGAs predate HGGAs, the computational development of HGGAs predates that of MGGAs. Only after the clouds of ambiguity lifted, MGGAs became a recognized DFT approach in computational chemistry, and HMGGAs began to appear. The Jacob’s ladder scheme is not the only way to arrive at exact functionals. When leaving the confines of ordinary KS-DFT methods, and using ideas from WFT, one arrives at ab initio density functional theory, the seamless connection of DFT and WFT (Bartlett et al. 2005). While these methods have not yet positively established themselves as standard approaches in computational chemistry, the fundamental conception that guided the way to the fourth rung – capturing the best of both worlds – marks the beginning of ab initio DFT and led to the notion of double-hybrid theory (Grimme 2006a), in which DFT calculations are supported not only by exact exchange but also by HF correlation. Along the same lines, rangeseparated functionals (Leininger et al. 1997) base their decision on how to deal with electron–electron correlation – the DFT or WFT way – on the central variable that describes a pair of electrons. The idea also found its entry into the realm of exchange (Tsuneda and Hirao 2014), and at the time of writing, it appears that the offspring of parent functionals is ever more growing in number.

Practical DFT and the Density Functional Zoo A practical DFT-based calculation is in many ways similar to a traditional HF treatment in that the final outcome is a set of orbitals, the Kohn–Sham orbitals ®KS . The KS orbitals are often expanded in terms of a basis set as in the traditional linear combination of atomic orbital (LCAO) approach of traditional HF methods. Most often, Gaussian-type basis functions are used to construct atomic orbitals (GTO). A major exception is the Amsterdam Density Functional (ADF) suite of programs, where Slater-type basis functions (STO) are used. ADF constitutes one of the first programs developed essentially for applications of DFT. The evaluation of matrix elements of the Kohn–Sham exchange-correlation potential always requires at some step a 3-day numerical integration. Solutions to the problem of carrying out 3D numerical integration for polyatomic systems to arbitrary precision (Becke 1988b) provided a major thrust for computational DFT,

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

FUNCTIONAL (YEAR) B3LYP (1994) BLYP (1988) B3PW91 (1993) BP86 (1988) PBE (1996) BPW91 (1991) TPSS (2003)

TYPE

USAGE

HGGA

80 %

GGA

5%

15

91

PW

HGGA

4%

GGA

3%

GGA

2%

GGA

1%

MGGA

1%

B3

B3LYP

BLYP BP86

Fig. 5 Most popular density functionals at the end of the first decade of the new millennium (Data based on results of extensive literature searches (Sousa et al. 2007))

and proficient improvements were made in connection with developments of the ADF computer code (Boerrigter et al. 1988). The availability of economical numerical integration schemes made the choice of STOs over GTOs computationally compatible. The local nature of the effective potential in the one-electron Kohn–Sham equations affords efficient computational schemes. During the development of ADF, the remaining Coulomb problem, the two-electron-integral “bottleneck,” has been addressed by the introduction of auxiliary basis sets, the so-called density fitting (Baerends et al. 1973). Many of the pioneering improvements made during the development of the ADF suite of programs have become standard tools in density functional calculations, and as a result, DFT calculations perform compatible to, if not better than HF methods. We note that a density fit is not possible, when the chosen functional utilizes exact exchange. By now, a plethora of density functionals is available for electronic structure calculations. Toward the end of the first decade of the new millennium, Sousa and coworkers have presented an authoritative review, in which they evaluate the performance of over 50 different density functionals (Sousa et al. 2007). The authors also report the percentage of occurrences of the names of different functionals in journal titles and abstracts; we interpret these numbers as measure for usage and popularity of the corresponding functional. Although new functionals appear every year, the popularity ranking seems to possess some stability within a time interval of several years. Thus, in Fig. 5, we have compiled data for the seven most popular functionals, taken from the work of Sousa and coworkers, and include a popularity pie chart as well.

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H. Jacobsen and L. Cavallo

The key information conveyed in Fig. 5 is the fact that B3LYP is by far the most popular density functional in chemistry, representing 80 % of the total of occurrences of density functionals in the literature, in the period 1990–2006 (Sousa et al. 2007). Other popular density functionals such as BLYP, B3PW91, and BP86 acquire usage shares of 5 % and less and can only be considered as also-rans in the functional race. It seems advisable to briefly talk about how to decipher the density functional code. With the advent of GGAs at the end of the 1980s, the abbreviation for each GGA functional usually consisted of two parts: the first for exchange and the second for correlation. As an example, BP86 takes its exchange contribution from the work of Becke (1988a) and correlation from the work of Perdew (1986). Similarly, BLYP breaks down into B exchange and LYP correlation. Later, when gradient corrections for exchange and correlation were often taken from the same work, the functional is usually referred to by one combined code only. The GGA functional PBE takes its gradients for exchange as well as for correlation from the work of Perdew, Burke, and Ernzerhof. The same holds true for the MGGA TPSS. Strictly spoken, the PBE functional should be referred to as PBEPBE. It is also possible that the individual parts are combined with other functionals, for example, PBELYP or BPBE. HGGAs usually contain one number that indicates the degree of parameterization when building the hybrid: B3LYP refers to a three-parameter mixing of B exchange, LYP correlation, and exact exchange, whereas B1LYP refers to a one-parameter hybrid density functional. As density functionals get more elaborate and more complex, the XC coding is not always strictly followed. The HMGGA M06 and its variations (M06-L, M06-2X, M06-HF), for example, refer to a set of functionals developed at the University of Minnesota in 2006. The pie chart presented in Fig. 5 bears a striking resemblance to Pac-Man. Like the arcade game Pac-Man, often credited with being a landmark in video game history and virtually synonymous with video games, B3LYP has to be considered a landmark in electronic structure theory and is often used as a synonym for DFT. However, the same as there is more to video games than just Pac-Man, there is more to density functional theory than just B3LYP. In section DFT: Computational Chemistry in Action, we will explore the characteristics and capabilities of density functional in more detail. The reader, who would like to know more about the details and derivations of DFT, will find valuable information about the basics in the A Chemist’s Guide to Density Functional Theory (Koch & Holthausen 2002) and will learn more about advanced aspects in the A Primer in Density Functional Theory (Fiolhais et al. 2003).

DFT: Computational Chemistry in Action The breakthrough of DFT coincided with a rise of computational power at the end of the last millennium. CPU architectures advanced from CISC to RISC designs, supercomputers transformed from single vector processors to computing clusters.

Directions for Use of Density Functional Theory: A Short Instruction Manual. . .

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However, as the computational power increased, the problems too became more and more demanding, and the molecules that found their way into input files for density functional programs grew bigger and bigger. Computing time remains to be a crucial factor when assessing the performance of computational methods, and linear-scaling approaches are one of the great strengths of DFT (Yang 1991). However, these techniques fall out of the scope of our review and assessment, and we begin with a comparison of computational demands of representative density functionals, following standard approaches.

Computational Performance We start this section remarking that there is not something like “the best functional and basis set for all properties.” Rather, the specific methodological approach to be used depends from the specific problem at hand. Nevertheless, many functionals are robust enough to give rather reasonable results in a large series of chemical properties, and the scope of this section is to provide an overview of the performances of typical functionals and basis sets, trying to highlight which ones perform remarkably better or remarkably worse than the average, if this is known. Further, as a practical vademecum, the scope of this section is to give an overview of performances under “standard working conditions.” Thus, the focus will be on performances that can be expected when working with real-size systems (50–100 atoms including a transition metal), which requires a compromise between the computer resources available and the combination of functional and basis set used, rather than peak performances that can be reached with a sophisticated last-generation functional in combination with a very extended basis. On the other hand, there are several excellent reviews that provide an accurate and critical assessment of the various methods, with a particular focus on the best performances that can be achieved, independently from the computational cost (Sousa et al. 2007; Cramer and Truhlar 2009). Methods that currently are computationally too expensive might become the standard computational tools in the future. Finally, the number of possible functionals is very large, so that it is more confusing than enlightening to review all of them. In addition, it might well be that the best functional for a specific problem has not been tested in the several benchmarking studies published in the literature. As a general rule, before wasting huge amounts of computer power with the wrong functional and/or basis set, it is wiser to invest some time to read the literature to find which computational method works better (or acceptably well) for a given problem and possibly have a feeling of the accuracy through test calculations on small selected systems. To give an idea of the relative cost of the various functionals, the relative computational time required by some functionals in two standard applications such as the calculations of the energy and of the first derivatives of the energy with respect to the atomic coordinates (which must be calculated at each geometry optimization step) and the calculation of the second derivatives of the energy with respect to the atomic coordinates (which must be calculated for vibrational analysis) is reported

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Table 1 Relative performance of various functionals, as implemented in the Gaussian 09 package, in the calculation of energy and gradients, or second derivatives, for an organometallic complex of formula RuC31 H33 Cl2 N3 Method BP86 (density fit) BP86 PBE B3LYP PBE1PBE TPSS (density fit) TPSS TPSSh M06

GGA GGA GGA HGGA HGGA MGGA MGGA HMGGA HMGGA

E C gradients SVP 1.0 1.5 1.5 2.6 3.0 1.6 2.1 3.5 3.6

2nd derivatives 1.0 1.9 1.8 2.9 3.2 1.9 2.3 3.5 4.0

E C gradients TZVP 1.6 4.1 4.2 8.9 9.2 2.6 5.1 9.4 10.6

in Table 1. The system considered in these calculations is a 70 atom Ru complex whose brute formula is RuC31 H33 Cl2 N3 . The data reported in Table 1 clearly indicate that GGA calculations are computationally very effective. In addition, for DFT methods that do not rely on exact exchange, the performance can be further improved by using an auxiliary basis set to fit the electron density (usually called density fit or resolution of identity). Indeed, without this technical setup, the same GGA or MGGA functional – compare the BP86 (density fit) and the simple BP86 values in Table 1 – is roughly 50 % slower. The same consideration applies when MGGA functionals are considered; compare the TPSS (density fit) and the simple TPSS values in Table 1. We note that this technical acceleration is not possible when the Hartree–Fock exchange must be evaluated, and thus HGGA and HMGGA calculations cannot benefit from it. Generally speaking, there are marginal differences within a family of functionals, and HGGA functionals are roughly 2–3 times slower than GGA functionals. MGGA functionals, particularly when the resolution of identity technique is invoked, are roughly 50 % slower than GGA functionals and thus are quite faster than HGGA functionals. HMGGA functionals are roughly 3–4 times slower than GGA functionals. This relative speed between the different families of functionals is maintained when second derivatives are evaluated. Moving to the effect of the size of the basis set, calculations performed with a triple- plus one polarization function of main group atoms results in an increase of the required computational time by a factor of 2–3 roughly. Thus, on going from an accelerated GGA functional in combination with a split valence plus one polarization function basis set, to a HMGGA functional in combination with a triple- plus one polarization function, basis set result in an increase of the computational time by a factor of 10 roughly. In other words, a GGA/SVP calculation that would take one day would require more than one week, roughly, if performed at the HMGGA/TZVP level. This indicates that the selection of the most appropriate computational method (both functional and basis set) must be a trade between the accuracy needed and the computational time

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(or power) available. Of course, degradation of accuracy below the level required by the specific problem at hand is not possible.

Properties of Molecular and Electronic Structure To explore the capabilities of various density functionals, we have selected twelve representative properties of atoms and molecules. We begin with molecular structure (bond lengths, bond angles, vibrational frequencies) and basics of electronic structure (electron affinities and ionization potentials). We then proceed to the energetics of transformations of molecules (atomization energies, heats of formation, energy barriers), which will carry us to chemical bonding. The nature of the chemical bond remains a central theme in theoretical chemistry, and we discuss regular bonds as well as weak bonds, all being at the focus of ongoing research activities (bond energies, hydrogen bonding, weak interactions). The issue of spin in DFT deserves particular attention (spin states). Although DFT essentially is a groundstate theory, excited states too can be treated with density functional theory, and with our last property we briefly touch this topic (excited states). Time-dependent density functional theory (TD-DFT) is a topic in its own right, and an appropriate coverage of TD-DFT falls out of the scope of the present work. The reader will find an entry into this excited field when studying the articles compiled by Marques and coworkers (2006). The twelve topics we selected in no way exhaust the capabilities of DFT, and any property that can be treated with WFT is in principle accessible with DFT as well. As an example, we refer the reader to a most instructive review by Neese (2009) that illustrates the capabilities of DFT in the field of molecular spectroscopy. When evaluating the performance of computational methods, benchmarking is an essential procedure in which calculated properties are evaluated against accurate experimental data. By now, a large number of problem-specific databases have been established, which cover a wide variety of different physicochemical properties, such as proton affinities, atomization energies, barrier heights, reaction energies, and spectroscopic properties. However, these databases are not free from chemical biases and often narrowed by the structural space of chemical intuition. There is always the risk that when following established procedures, benchmark studies might lose some of their general appeal. As to avoid the dangers of casual benchmarking, Korth and Grimme have developed a “mindless” DFT benchmarking protocol. Here, the databases consist of randomly generated molecules that rely on systematic constraints (Korth and Grimme 2009) rather than on what is supposed to be chemical insight. In the following, we will make extensive reference to published benchmark studies. However, it might well be that a particular molecule of interest to the reader is not covered within one of the existing benchmark databases, and benchmark studies in general provide good starting points for calculations, but no guarantee for correctness.

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One: Bond Lengths It is well established that almost any DFT approach, beyond LDA, is able to reproduce correctly the geometry of molecular systems composed of main group atoms. With the increase of computer resources, it is becoming customary to test the performances of various methods through calculations on a rather large set of molecules and to report statistical values. In one of such comparative studies, 17 closed-shell molecules composed by first-row atoms, for which accurate experimental geometries determined in the gas phase were available, confirmed that the performance of commonly used GGA (BLYP, BPW91, and BP86) and HGGA (B3LYP, B3PW91, and B3P86) is quite accurate, although the mean unsigned error (MUE) on the bond length obtained with the GGA functionals, between 0.015 and 0.020 Å roughly, is slightly larger than that calculated with the HGGA functionals, usually below 0.010 Å (Wang and Wilson 2004). The convergence of the geometry was tested with respect to increasing basis set size from cc-pVDZ to aug-cc-pV5Z and was shown to occur quickly. Convergence is typically reached at the triple level, and beyond this level minor fluctuations, in the order of 0.002 Å, were observed. Thus, excellent performances require that at least a triple- basis set is used. Similar conclusions were reached in a different benchmark study on a dataset of 44 small molecules (Riley et al. 2007). Again, GGA functionals in combination with Pople basis sets of the 6-31G family result in MUE between 0.015 and 0.020 Å, while HGGA functional results in MUE below 0.010 Å. The MGGA functionals tested resulted in a minor improvement relative to GGA functionals, while the tested HMGGA functionals substantially reproduce the performance of HGGA functionals. This indicates that the advantage of meta-functionals certainly is not in bond distances. To give an idea of the performance of some popular functionals, and also to show the effect of the basis set, the dependence of the O–H bond length in water is reported in Table 2 as an exemplary case. The data indicate that reasonably accurate bond lengths (within 0.02 Å from the experimental value) can be achieved with computationally cheap GGA functionals and that HGGA performs slightly better with modest basis sets. As a general trend, the HGGA bond lengths are slightly shorter than the corresponding GGA value and independent of the computational approach; slightly shorter bond lengths are predicted with basis sets of increasing quality. It is noteworthy that with the extended aug-cc-pV5Z basis set, the GGA values slightly overestimate the experimental value, whereas the HGGA values slightly underestimate it. Importantly, rather good results can be achieved also with relatively small basis sets, which allow calculating geometries for fairly large systems with a reasonable accuracy. The very good performance of almost any functional to calculate accurately bond lengths of molecular systems composed by main group atoms is not replicated when bonds to transition metals are considered. Focusing on an extensive benchmark of 42 functionals on a database of 13 metal– ligand bond lengths, the MLBL13/05 database, all functionals provide rather good results, with MUE normally between 0 0.01 and 0.02 Å when a basis set of triple- quality is used (Schultz et al. 2005a).

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Table 2 Performance of selected functionals and basis sets in predicting the experimental value of the O–H bond length of water, 0.956 Å 6-31G SVP 6-31G(d,p) TZVP cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z

BP86 0.985 0.976 0.974 0.972 0.978 0.971 0.970 0.970 0.974 0.971 0.970 0.970

revPBE 0.984 0.974 0.973 0.971 0.977 0.970 0.968 0.968 0.974 0.970 0.968 0.968

B3LYP 0.976 0.967 0.965 0.962 0.969 0.961 0.960 0.960 0.965 0.962 0.961 0.960

TPSS 0.983 0.974 0.972 0.969 0.977 0.968 0.968 0.967 0.973 0.969 0.967 0.967

TPSSh 0.978 0.969 0.967 0.965 0.972 0.964 0.963 0.963 0.970 0.964 0.963 0.963

M06 0.970 0.963 0.960 0.958 0.964 0.958 0.956 0.956 0.961 0.958 0.956 0.956

Extending the benchmark to a database containing the bond length of eight metal–metal dimers, the TMBL8/05 databases, the situation deteriorates. GGA functionals, including the popular BP86, BLYP, and PBE functionals, still provide 0 rather good results, with MUE between 0.02 and 0.03 Å when a basis set of triple- quality is used, while reducing the quality of the basis set to double- deteriorates performances remarkably, with MUE between 0.06 and 0.09 Å (Schultz et al. 2005b). In the GGA family, the HTCH and OLYP functionals, with MUE 0 greater than 0.03 Å, should be avoided. The rather good performance of GGA functionals is not replicated by HGGA functionals, including the0 popular B3LYP and PBE1PBE functionals, with MUE between 0.08 and 0.09 Å. In the HGGA family, the BH&HLYP and MPW1K functionals, with MUE greater than 0.12 Å, should be avoided. Interestingly, MGGA functionals do not perform as or better than GGA functionals in predicting bond lengths, but rather worse. Indeed, including also the popular BB95 and TPSS functionals, they result in MUE greater than 0.06 0 Å. Introduction of HF exchange partially improves the performance of MGGA functionals, and the tested HMGGA functionals, including the B1B95 and the TPSSh functionals, result in MUE between 0.03 and 0.07 Å (Schultz et al. 2005a). Finally, the M06 functional performs particularly poor, with a MUE of 0.131 Å (Zhao and Truhlar 2008b). To give an idea of the performance of some popular functionals in the calculation of the M–ligand distances and of the effect of the metal on the bond distance of the ubiquitous CO ligand, analysis of these distances in three typical binary carbonyl complexes involving first-row transition metals is reported in Table 3. Basically, all the functionals reproduce the experimental M–CO distances well within 0.02 Å, but many of the functionals tested underestimate the difference in the axial and equatorial Fe–CO distances. In this respect, HGGA and HMGGA functionals seem to perform slightly better, although there is quite a debate on the exact assignment of the Fe–CO distances in Fe(CO)5 . Similar good behavior is shown by

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Table 3 Performance of selected functionals, in combination with the TZVP basis set on all the atoms, in predicting the experimental value of the M–C and C–O bond length (in Å) in three firstrow M(CO)n complexes

Exp. BP86 PW91 revPBE B3LYP PBE1PBE B98 TPSS mPWKCIS BB95 TPSSh M06

GGA GGA GGA HGGA HGGA HGGA MGGA MGGA MGGA HMGGA HMGGA

Ni(CO)4 M–C C–O 1.838 1.141 1.828 1.151 1.824 1.149 1.829 1.150 1.845 1.137 1.822 1.134 1.839 1.137 1.830 1.149 1.832 1.150 1.833 1.150 1.828 1.142 1.848 1.133

Fe(CO)5 M–Ceq , M–Cax 1.803, 1.811 1.809, 1.810 1.805, 1.806 1.809, 1.810 1.820, 1.828 1.796, 1.805 1.813, 1.820 1.813, 1.816 1.810, 1.811 1.811, 1.811 1.809, 1.815 1.820, 1.821

C–Oeq , C–Oax 1.133, 1.117 1.157, 1.153 1.154, 1.151 1.155, 1.153 1.142, 1.138 1.140, 1.136 1.142, 1.138 1.154, 1.151 1.156, 1.153 1.156, 1.153 1.147, 1.144 1.139, 1.135

Cr(CO)6 M–C C–O 1.918 1.141 1.910 1.155 1.906 1.153 1.911 1.154 1.926 1.141 1.900 1.138 1.915 1.141 1.918 1.152 1.911 1.154 1.912 1.154 1.915 1.146 1.920 1.138

all the functionals in the prediction of the CO distance when bonded to a transition metal, although the GGA and MGGA functionals tested yield systematically longer CO distances. In this respect, HGGA and HMGGA functionals do perform slightly better.

Two: Bond Angles The good performance of almost every functional to predict correctly bond lengths of molecular systems composed by main group atoms is confirmed in the case of bond angles. Again, a benchmark study on 17 closed-shell molecules composed by first-row atoms, for which accurate experimental geometries determined in the gas phase was available, confirmed that the performance of commonly used GGA (BLYP, BPW91, and BP86) and HGGA (B3LYP, B3PW91, and B3P86) functionals is quite accurate, with a MUE between 1.0ı and 1.5ı (Wang and Wilson 2004). Differently from bond lengths, GGA and HGGA methods perform rather similarly on bond angles. Also for bond angles, the convergence was tested with respect to increasing basis set size from cc-pVDZ to aug-cc-pV5Z and was shown to occur quickly, and again convergence is typically reached at the triple-— level. Similar conclusions were reached in a different benchmark study on a dataset of 44 small molecules (Riley et al. 2007). All the functionals considered resulted in MUE between 1.0ı and 1.5ı , independent of the functional used. To give an idea of the performance of some popular GGA and HGGA functionals, and also to show the effect of the basis set, the dependence of the H–O–H angle in water is reported in Table 4 as an exemplary case. Accurate bond angles (within 1.0ı from the experimental value) can be achieved with all functionals and moderate basis sets.

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Table 4 Performance of selected functionals and basis sets in predicting the experimental value of the H–O–H angle of water, 105.2ı 6-31G SVP 6-31G(d,p) TZVP cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z

BP86 107.2 102.2 103.1 104.1 101.7 103.6 103.9 104.2 103.8 104.2 104.2 104.2

revPBE 107.1 102.1 103.0 104.1 101.7 103.5 104.0 104.3 103.8 104.2 104.3 104.3

B3LYP 108.3 103.1 104.0 105.1 102.7 104.5 104.9 105.1 104.8 105.0 105.1 105.1

TPSS 107.1 102.3 103.3 104.3 104.9 103.7 104.0 104.3 103.8 104.3 104.3 104.3

TPSSh 107.7 102.7 103.7 104.6 102.3 104.0 104.3 104.5 104.1 104.6 104.6 104.6

M06 109.4 103.3 104.5 104.9 102.8 104.4 104.8 105.0 104.7 104.9 104.9 105.1

Three: Vibrational Frequencies Benchmarking various DFT methods to reproduce accurately vibrational frequencies of 35 molecular systems composed by main group atoms revealed that GGA methods, with a MUE of roughly 40 cm1 , are among the most accurate functionals (Riley et al. 2007). Indeed, the performance of several HGGA methods was at least 20 cm1 worse, with MUE between 60 and 80 cm1 , and meta-functionals are not an improvement. As for other geometrical properties, accurate performance requires that a triple- basis set be used. The GGA functionals also performed better than HGGA functionals in the prediction of the vibrational frequency in nine homonuclear 3d metal dimers, with a MUE around 100 cm1 for BLYP and BP86 and around 120 cm1 for B3LYP and B3P86. Nevertheless, both families of functionals resulted in a rather large deviation from accurate data (Barden et al. 2000). The performance of various functionals in the prediction of the CO stretching frequency in typical binary carbonyl complexes with first-row transition metals is exemplified in Table 5. Simple GGA and also MGGA functionals perform better and are able to capture the experimental value with an accuracy of roughly 50– 100 cm1 , while Hartree–Fock exchange seems to deteriorate results, since the HGGA and HMGGA functionals reproduce the experimental value with an accuracy of roughly 150–200 cm1 . In terms of percent, the GGA and MGGA functionals overestimate the experimental values by 3–4 %, while the HGGA and HMGGA functional by 6–10 %. While these results may seem quite accurate, almost all the functionals considered are unable differentiate too little between metals. In fact, the experimental value decreases by 58.1 cm1 on going from Ni(CO)4 to Cr(CO)6 , but the functionals examined are unable to capture this difference. The best performing are the B98 functional, with a difference of merely 15.0 cm1 , and the HGGA and HMGGA with differences slightly smaller than 10 cm1 .

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Table 5 Performance of selected functionals, in combination with the TZVP basis set on all the atoms, in predicting the symmetric frequency of the CO stretching mode in selected first-row transition metal binary carbonyl complexes. Next to each calculated frequency is reported the %error calculated as %err D 100*(Exp. /DFT ). The final column reports the difference between the Ni(CO)4 and the Cr(CO)6 frequencies Method Exp. BP86 PW91 revPBE B3LYP PBE1PBE B98 TPSS mPWKCIS BB95 TPSSh M06

GGA GGA GGA HGGA HGGA HGGA MGGA MGGA MGGA HMGGA HMGGA

Ni(CO)4 2061.3 2104.0 2116.0 2110.0 2194.0 2232.0 2212.0 2121.0 2102.0 2100.0 2169.0 2234.0

%err 2.1 2.7 2.4 6.5 8.3 7.3 2.9 2.0 1.9 5.2 8.4

Fe(CO)5 2038.1 2103.0 2115.0 2109.0 2184.0 2223.0 2202.0 2116.0 2102.0 2100.0 2162.0 2224.0

%err 3.2 3.8 3.5 7.2 9.1 8.0 3.8 3.1 3.0 6.1 9.1

Cr(CO)6 2003.0 %err 2102.0 4.9 2115.0 5.6 2109.0 5.3 2185.0 9.1 2223.0 11.0 2197.0 9.7 2117.0 5.7 2102.0 4.9 2100.0 4.8 2162.0 7.9 2226.0 11.1

Ni -Cr 58.1 2.0 1.0 1.0 9.0 9.0 15.0 4.0 0.0 0.0 7.0 8.0

NHC

N

N

N

N

N

N

NHC Cl

IMes

IAd

SIMes

Ir OC CO N

N

ICy

N

N

IPr

N

N

SIPr

Fig. 6 Iridium complexes that bear N-heterocyclic carbene ligands

On the other hand, the simple BP86 GGA functional has been also tested in the prediction of the CO stretching frequency in rather large organometallic complexes. For these large and computationally demanding systems, which are displayed in Fig. 6, computationally effective methods are needed. The data reported in Table 6 clearly show the very good performance of the BP86 functional, which is able to reproduce the higher frequency symmetric CO stretching with an error of roughly 20 cm1 only and the lower frequency asymmetric CO stretching with an error of roughly 30 cm1 . Further, despite the poor performances discussed above in the ability of GGA functionals to differentiate between different binary M(CO)n complexes, comparison of the saturated N-heterocyclic carbene complexes (SIPr and SIMes) with their unsaturated counterparts (IPr and IMes) indicates that the DFT values reproduce the experimental finding that both CO

Directions for Use of Density Functional Theory: A Short Instruction Manual. . . Table 6 Experimental and DFT calculated CO stretching frequencies, in cm1 , in several N-heterocyclic carbene complexes

Method (IAd)Ir(CO)2 Cl (ICy)Ir(CO)2 Cl (IPr)Ir(CO)2 Cl (SIPr)Ir(CO)2 Cl (IMes)Ir(CO)2 Cl (SIMes)Ir(CO)2 Cl

Experimental 2063.4 1979.8 2064.8 1981.2 2066.8 1981.0 2068.0 1981.8 2066.4 1979.8 2068.0 1981.8

25 BP86 2082 2083 2083 2084 2084 2085

2010 2015 2005 2007 2007 2008

Table 7 Calculated and experimental electron affinities, in kcal/mol, for neutral late transition metal M(CO)n complexes System Mn(CO) Fe(CO) Ni(CO) Fe(CO)2 Ni(CO)2 Ni(CO)3

Experimental 25.7 26.7 18.5 28.1 14.8 24.8

BP86/6-311G(d) 25.0 25.7 19.1 32.6 19.9 35.3

B3LYP/6-311G(d) 24.0 21.8 13.8 – 16.0 27.9

stretches are about 1 or 2 cm1 smaller in the complexes with the unsaturated N-heterocyclic carbene ligand (Kelly et al. 2008).

Four: Electron Affinities and Ionization Potentials A benchmark study of 24 molecules from the G2/97 dataset, with the addition of PO2 , indicated that, with some exceptions, DFT methods reproduce the electron affinity of molecular systems composed by main group atoms with a MUE close to 4 kcal/mol. Some hybrid functionals, such as the B98 functional, performs slightly better (MUE D 3.15 kcal/mol), while GGA functionals with the P86 correlation term usually perform rather poorly, with MUE around 6 kcal/mol (Riley et al. 2007). Finally, meta-functionals are not an improvement. As for any molecular system with a negative charge, the calculation of electron affinity requires that basis set containing diffuse functions are used. Moving to ionization potentials, the performance of the various methods on 36 molecules from the G2/97 dataset, with the addition of PO2 , substantially replicates that found for electron affinities, although the MUE, between 5 and 6 kcal/mol, is slightly larger (Wang and Wilson 2004). The B1B95 MHGGA functional provided the best performance, with a MUE of 4.25 kcal/mol. Moving to selected cases (see Table 7), the BP86 GGA functional seems quite more accurate than the HGGA B3LYP functional in predicting the electron affinity of highly unsaturated late transition metal M(CO)n complexes, but the HGGA functional seems to be more accurate when the unsaturation at the metal is reduced (Zhou et al. 2001). These results exemplify the difficulty to extract trends from benchmarks, if the specific case at hand has not been included in the testing dataset.

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Five: Atomization Energies A benchmark study on 17 first-row closed-shell molecules indicated that in the calculation of the atomization energy with the Dunning correlation-consistent basis sets, the HGGA functionals, with a MUE of roughly 2.2 kcal/mol, outperform GGA functionals, such as BLYP (MUE D 7.2 kcal/mol), with BP86 performing particularly bad (MUE D 16.2 kcal/mol) (Wang and Wilson 2004). In another benchmark study on the atomization energy of a dataset composed by 109 main groups of organic and inorganic molecules, calculated with the MG3S basis set, HGGA functionals were again confirmed to perform well, with MUE smaller than 1.0 kcal/mol, while GGA functionals, such as the PBE, with a MUE of 3.0 kcal/mol, again performed poorly. The peak performance of 0.40 kcal/mol was produced with the B1B95 functional, while the classic B3LYP resulted in a MUE of 0.91 kcal/mol. Finally, HMGGA functionals were shown to perform similarly to HGGA functionals (Zhao and Truhlar 2005a). A set of range-corrected functionals, including the CAM-B3LYP, the LC-¨PBE, and the ¨B97 family, also performed particularly well, with MUE between 0.51 and 0.89 kcal/mol (Peverati and Truhlar 2014). On the other hand, GGA functionals perform well when the atomization energy of 9 metal dimers in the TMAE9/05 database is calculated, with MUE in the range of 5–8 kcal/mol. HGGA functionals, instead, resulted in MUE around 15– 30 kcal/mol, with only the B97-1, B97-2, and B98 functionals with MUE below 10 kcal/mol. MGGA and HMGGA functionals substantially replicate the results obtained with non-meta-functionals, although the M05 and M06 functionals result in the very low MUE of 6.9 and 4.7 kcal/mol, respectively (Zhao and Truhlar 2008b). These results are another indication that hybrid functionals usually perform better for molecules composed by main group atoms, whereas GGA and MGGA functionals perform better for transition metal chemistry. Six: Heats of Formation The accurate calculation of this property, even for rather simple molecular systems composed by main group atoms, still represents a challenge for several functionals. Indeed, the MUE on the heat of formation in a benchmark study on 24 molecules from the G2/97 dataset, with the addition of PO2 , spans the rather broad range of 10– 30 kcal/mol, even if basis set of rather good quality, such as the 6-31CCG* basis set, are used. Functionals to be avoided are those containing the P98 correlation term among the GGA, the B1LYP and the B98 among the MGGA, the PBEKCIS among the MGGA, and the BB1K among the HMGGA. On the other side, good performances were obtained with the PBELYP, PW91LYP, MPWPW91, and MPWPBE GGA functionals, the PBE1PBE and B3PW91 HGGA functionals, and finally the TPSS and the TPSSKCIS MGGA functionals. With these functionals, the MUE on the heats of formation usually is below 10 kcal/mol (Riley et al. 2007). Augmented correlation-consistent Dunning-type basis sets usually lead to slightly better performances. Nevertheless, some computational results have cast a shadow on the common procedure to test heats of formation on small molecules. Indeed, it has been shown that almost all the functionals, including the popular

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B3LYP functional, are unable to predict correctly the heats of formation of n-alkanes (Curtiss et al. 1997, 2000). Due to the inability to describe properly long-range attractive dispersion interaction, practically all functionals introduce a systematic error in the calculation of the isodesmic stabilization energy (Eq. 4). This systematic error ranges between 0.90 kcal/mol for the HMGGA MPWB1K functional and 1.82 kcal/mol for the OLYP functional. The BP86 and PBE GGA functionals result in an error of 1.25 and 1.08 kcal/mol, respectively, while the HGGA functional B3LYP results in an error of 1.33 kcal/mol. While these errors may seem to be not dramatic, they are per CH2 unit, so that the errors are between 10 and 20 kcal/mol for a simple molecule such as n-decane (Wodrich et al. 2006). n  CH3  .CH2 /m  CH3 C m CH4 ! .m C 1/ C2 H6

(4)

Seven: Energy Barriers The performance of a series of functionals was tested to reproduce the barrier height for a series of reactions. Starting from hydrogen transfer reactions, the forward and backward barrier  for the following 3 reactions OH  C CH4 ! CH3  C H2 O; OH  C H  ! O 3 P C H2 ; and H  C H2 S ! HS  C H2 , which constitute the BH6 database (Lynch and Truhlar 2003), GGA functionals systematically underestimate the barriers, with a MUE between 7.8 and 9.4 kcal/mol. Better results were obtained with HGGA functionals, with MUE around 4.5–5.0 kcal/mol. The mPWPW91 MGGA functional is not a clear improvement, with a MUE of 8.5 kcal/mol, while HMGGA functionals such as the mPW1PW91 and the MPW1K perform better, with MUE of 3.9 and 1.4 kcal/mol, respectively (Zhao et al. 2004). Testing the functional on the larger BH42/04 database, consisting of 42 transition-state barrier heights of hydrogen transfer reactions in mostly openshell systems, gave substantially similar results (Zhao & Truhlar 2004). The HGGA functionals underestimate barriers by a MUE of roughly 4 kcal/mol, while HMGGA functionals give better results, with peak performances from the BB1K, XB1K, and MPWB1K resulting in MUE around 1.2–1.3 kcal/mol. Similar conclusions are achieved when the HTBH38 database, which contains 38 transition-state barrier heights for 19 hydrogen transfer reactions, 18 of which involve radicals as reactant and product, is considered. In this case, 10 examined HGGA functionals result in an average MUE of 3.7 kcal/mol, while 7 range-corrected functionals slightly decrease this value to 2.7 kcal/mol (Peverati and Truhlar 2014). Moving to 38 transition-state barrier heights for non-hydrogen transfer reactions constituting the NHTBH38/04 database, and comprising 12 barrier heights for heavy-atom transfer reactions, 16 barrier heights for nucleophilic substitution reactions, and 10 barrier heights for non nucleophilic substitution unimolecular and association reactions, GGA functionals, such as the PBE, still approximate barriers severely, with a MUE of 8.64 kcal/mol. HGGA functionals, such as the B3LYP and the PBE1PBE functionals, perform better, although the MUE, around 4 kcal/mol, still is quite large (Zhao et al. 2005). MGGA functionals perform similarly to the GGA functionals, while the performance of HMGGA is rather scattered. For example, the TPSSh and the MPW1KCIS functionals do not perform impressively, with MUE of 6.6 and

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Table 8 Selected mean errors for HTBH38 and NHTBH38 database Method PBE BLYP B3LYP PBE1PBE TPSS TPSSKCIS TPSSh BB1K

GGA GGA HGGA HGGA MGGA MGGA HMGGA HMGGA

Hydrogen transfer 9.32 7.52 4.23 4.22 7.71 4.69 5.97 1.16

Heavy atom transfer 14.93 14.66 8.49 6.62 14.65 9.26 11.51 1.58

Nucleophilic substitution 6.97 8.40 3.25 2.05 7.75 4.88 5.78 1.30

Unimolecular and association 3.35 3.51 2.02 2.16 4.04 2.12 3.23 1.44

4.6 kcal/mol, respectively, while the MPW1K and the BB1K are very accurate, with MUE of 1.3 and 1.2 kcal/mol, respectively (Zhao and Truhlar 2005a). As for hydrogen transfer reactions, comparison between 7 range-separated functionals and 10 HGGA functionals indicated that the range-separated functionals, with an average MUE of 2.9 kcal/mol, perform slightly better than the HGGA functional, with an average MUE of 3.7 kcal/mol. However, the peak performance of the MPW1K and B97-3 HGGA functionals, with MUE around 1.5 kcal/mol, is not reached by any of the range-separated functionals, with a minimum MUE of 2.41 kcal/mol achieved by the ¨B97 functional (Peverati and Truhlar 2014). The breakdown of these cumulative MUEs for selected functionals is reported in Table 8. Clearly, the most problematic cases are the transfer reaction of both hydrogen and heavy atoms. With the exception of the BB1K, all the other functionals fail to a large and embarrassing extent. For the HTBH38 dataset, the M06 and M062X HMGGA functionals, with MUE of 2.00 and 1.13 kcal/mol, also perform well (Zhao and Truhlar 2005a). Barrier heights of nucleophilic substitutions are predicted better, although GGA functionals still result in a too large MUE. Finally, the barrier heights of unimolecular and association reactions are predicted with reasonably accuracy by all functionals. Again, the performance of the BB1K functional is extremely good for the four classes of reactions considered. Similarly good and well-balanced performances are also obtained with other HMGGA functionals such as the PW6BK and the MPWB1K functionals. Similar extensive tests on another dataset, comprising the barrier height for 23 reactions of small systems with a radical transition state, also highlighted the good performances of the HMGGA BB1K functional, with a MUE of 1.05 kcal/mol, in predicting energy barriers. However, when the same functionals were tested on a dataset of 6 barrier heights of larger systems with a singlet transition state, the best performance was obtained with the simple HGGA B1LYP functional, with a MUE of 2.58 kcal/mol, and the performance of the B3LYP functional, with a MUE of 3.10 kcal/mol, was slightly worse, while all the HMGGA functionals tested, including the BB1K functional, resulted in MUE greater than 4 kcal/mol (Riley et al. 2007).

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Table 9 First metal–carbonyl dissociation energy, in kcal/mol, for selected first-row transition metal systems. DFT values, calculated with the TZVP basis set on the metal, and the SVP or TZVP basis sets on CO Exp. Method BP86 PW91 revPBE B3LYP PBE1PBE B98 TPSS mPWKCIS BB95 TPSSh M06

GGA GGA GGA HGGA HGGA HGGA MGGA MGGA MGGA HMGGA HMGGA

Ni(CO)4 24.9˙2a SVP 32.3 34.3 33.9 24.2 29.2 25.6 32.9 30.3 30.4 31.2 25.6

TZVP 29.4 31.4 30.9 21.3 26.4 22.8 30.5 27.3 27.4 28.6 22.5

Fe(CO)5 41.5˙2b SVP 50.1 52.8 52.1 42.1 50.1 44.7 50.8 48.4 48.8 50.0 44.2

TZVP 46.8 49.4 48.6 38.8 47.1 41.8 48.1 44.8 45.2 47.4 40.3

Cr(CO)6 36.8˙2b SVP 46.0 48.6 48.2 39.8 46.1 41.5 46.2 44.7 45.4 45.5 45.0

TZVP 42.8 45.5 44.9 36.5 43.4 38.6 43.8 41.3 42.4 43.0 41.7

Eight: Bond Energies The performance of several functionals to predict bond energies on a database of 21 metal–ligand bond energies, the MLBL21/05 database (Schultz et al. 2005a), indicated that GGA functionals predict metal–ligand bond energies with a MUE between 6 and 12 kcal/mol, with the BLYP, PBE, and BP86 among the worst, and that performances can be quite better at the HGGA level, with MUE normally in the 5–7 kcal/mol range. MGGA and HMGGA functionals do not offer an improvement, as exemplified by the MUE of the TPSS and TPSSh functionals, 7.9 and 5.5 kcal/mol, respectively, and by the M06 functional, with a MUE of 5.4 kcal/mol (Zhao and Truhlar 2008b). Taking again some selected binary carbonyl complexes of first-row transition metals as examples, the performance of various functionals in the prediction of the first bond dissociation energy is reported in Table 9. The first clean result is that the binding energy is strongly affected by the basis set quality, and a triple- basis set yields binding energies that are roughly 3–4 kcal/mol lower. Focusing on the TZVP results, the GGA functionals examined consistently overestimate the CO binding energy in the three complexes by roughly 3–6 kcal/mol. The PBE1PBE HGGA functional performs like the GGA functionals examined, whereas the B3LYP, and particularly the B98 functional, is among the best performing functionals. Metafunctionals only offer a marginal improvement. Nine: Hydrogen Bonding Benchmarking various DFT methods to reproduce the H–bond interaction energy of ten systems composed by main group atoms revealed that HGGA functionals generally perform better than GGA functionals, with MUE around 0.5 kcal/mol, and the B3LYP functional among the best. GGA functionals, instead, result in

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Table 10 H-bond interaction energy, in kcal/mol, of selected nucleic acid base pairs Base pair G:C Watson–Crick A:T Watson–Crick G:U Wobble U:U Calcutta

MP2/aug-cc-pVQZ 27.7 15.1 15.7 9.6

B3LYP/6-31G(d,p) 25.5 12.3 13.4 7.5

PW91/6-31G(d,p) 27.7 14.5 14.8 8.7

MUE between 0.5 and 2.0 kcal/mol, with the BLYP, MPWPW91, and MPWPBE among the best performing (Riley et al. 2007). MGGA and HMGGA functionals perform close to the B3LYP functional. In all cases, the quality of the basis sets had a strong impact on the quality of the results, and at least a 6-31G(d,p) basis set should be used. Further tests were performed on a database consisting of the binding energies of 6 hydrogen bonding dimers, the HB6/04 database (Zhao and Truhlar 2005b). Also these tests indicated that the performance of GGA functionals is generally not very impressive, with MUE normally between 0.5 and 4 kcal/mol, with the exception of the very well-performing PBE functional, with a MUE of 0.25 kcal/mol only. On the average HGGA performs better, with MUE between 0.3 and 1.0 kcal/mol, with the PBE1PBE functional, with a MUE of 0.27 kcal/mol, among the best, while the B3LYP functional, with a MUE of 0.87 kcal/mol, does not perform impressively. Meta-GGA and meta-HGGA functionals are not a great improvement. These results have been achieved with the rather large aug-cc-pVTZ basis set, using the 6-31CG(d,p) basis set leading to slightly lower performances. Extensive tests of the performance of the B3LYP and PW91 functionals in the case of H-bonded nucleic acid–base pairs indicated that the PW91 GGA functional replicates with better accuracy the MP2/aug-cc-pVQZ values than the B3LYP HGGA functional (Sponer et al. 2004). The B3LYP calculations underestimate the interaction energies by few kcal/mol with relative error of 2.2 kcal/mol. Representative values for Watson–Crick (G–C, A–T) purine–pyrimidine (G–U) and pyrimidine–pyrimidine (U–U) pairs are presented in Table 10.

Ten: Weak Interactions For a long time, one of the well-known failures of DFT was in the description of long-range dispersion forces, which are the glue that keeps together Van der Waals complexes and are at the basis of the well-known      stacking interactions. Members of the former family are the rare-gas dimers that almost invariably are predicted to be unbound (or bound by effect of the basis set superposition error) by many GGA and HGGA functionals. Indeed, an extensive benchmark on the WI4/04 datasets, composed by four rare-gas dimers (HeNe, NeNe, HeAr, and NeAr) indicated that the B3LYP functional simply does not predict a energy well for three out of the four dimers if the basis set superposition error is corrected by the counterpoise procedure and that even including the basis set superposition error the interaction energy is underestimated by 0.23 kcal/mol, and equilibrium distances 0 are overestimated by more than 1 Å. On the other hand, HMGGA functionals were shown to reproduce with good accuracy both the interaction energy and the

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equilibrium distance of these dimers, with MUE on the counterpoise-corrected binding energies below 0.05 kcal/mol by several functionals, such as the X1B95 and the MPWB1K, and with MUE on the equilibrium distances as low as 0.10 Å, which is a remarkable result for weakly interacting systems (Zhao and Truhlar 2004). Enlarging the dataset to include also organic molecules, such as the C6 H6 –Ne, and the (C2 H4 )2 and (CH4 )2 dimers, indicated that standard GGA functionals systematically underestimate the interaction energy by a MUE of roughly 0.4–1.0 kcal/mol, with the exception of the PBE functional, with a MUE of 0.30 kcal/mol. MGGA functionals do not perform much better, while a small improvement is shown by HGGA functionals, with the best results from B97-1 with a MUE of 0.20 kcal/mol. HMGGA functionals do not offer better performances, although the largest MUE is reduced to 0.64 kcal/mol (Zhao and Truhlar 2005b). Finally, better performances were shown by the M05-2X and M06-2X functionals, with a MUE of only 0.03 and 0.09 kcal/mol, respectively, when applied to a dataset formed by the four rare-gas dimers above, C6 H6  Ne; CH4  Ne; and the (CH4 )2 dimer. Surprisingly, on this reduced dataset, the well-performing B97-1 functional was not tested (Zhao and Truhlar 2005b). Moving to      stacking interactions, the performance of a series of functionals was tested on the PPS5/05 database, which consists of binding energies of five      stacking complexes, namely, (C2 H2 )2 , (C2 H4 )2 , and sandwich, T-shaped, and parallel-displaced (C6 H6 )2 (Zhao and Truhlar 2005a). Basically, all the functionals tested underestimated the binding energies in the PPS5/05 database, with MUE in the range of 1.0–3.8 kcal/mol. This is not a minor failure, since the average binding energy in the PPS5/05 dataset amounts to 2.02 kcal/mol only. The PBE GGA functional, with a MUE of 2.1 kcal/mol, performs better than the B3LYP HGGA functional, which results in a MUE of 3.2 kcal/mol. The best performances, slightly above 1 kcal/mol, are obtained with the PWB6K, PW6B95, and MPWB1K. A clear improvement is obtained with the M06-2X functional, with a MUE of 0.30 kcal/mol only. Reducing the amount of Hartree–Fock exchange deteriorates somewhat performances, as evidenced by the MUE of 0.60 kcal/mol yielded by the M06 functional Zhao and Truhlar 2008b). Focusing on      stacking interactions between nucleic acid bases, which are fundamental to describe properly the base–base stacking in nucleic acids, the popular B3LYP is simply unable to find a minimum corresponding to the AT as well as the GC base pairs in a stacked geometry. Of the six functionals tested, only the MPWB1K and the PWB6K functionals resulted in stable stacked base pairs, with an underestimation of the stacking energy of roughly 2–3 kcal/mol (Zhao & Truhlar 2005c). Range-separated functionals, which were developed with the particular intent of describing properly the exchange term at long distances, offer a clear improvement over the performance of classic HGGA functionals on weak interacting systems. For example, when the performance of different families of functionals is compared through the NCCE31/05 database, which comprises the various databases described above, seven range-separated functionals, including the CAMB3LYP, the LC-¨PBE, and the ¨B97 functionals, with an average MUE of

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0.61 kcal/mol, perform clearly better than 10 HGGA functionals, with a MUE of 0.92 kcal/mol. However, it is important to remark that on the same NCCE31/05 database, functionals fitted to reproduce non-covalent interactions, as the M06 functional, with a MUE of 0.41 kcal/mol, also perform remarkably well (Peverati and Truhlar 2014). As a final remark, we note that, based on an idea earlier proposed for Hartree– Fock calculations (Hepburn et al. 1975; Ahlrichs et al. 1977), the addition of an empirical C6 R6 dispersion term represents a very simple cure to the failure of standard density functionals to predict dispersion interactions (Grimme 2006b). This approach is treated extensively in Chapter 12, so we do not discuss it here in details. We only remark the very good performance of a range-separated functional that also include an empirical dispersion term, such as the wB97X-D functional, on the NCCE31/05 database, with a MUE of only 0.32 kcal/mol.

Eleven: Spin States The problem of a reliable prediction of the relative ordering of different spin states in transition metal complexes remains a tough challenge for DFT – not only for a quantitative judgment (energy separation between the different spin states) but even for a qualitatively assessment (correct prediction of the spin ground state). There is general consensus that GGA functionals overestimate the stability of low-spin states, whereas the inclusion of Hartree–Fock exchange in the HGGA functionals results in an overestimation of the stability of high spin states (Ghosh 2006; Swart 2008). This led to the development of the B3LYP* functional, in which the amount of the Hartree–Fock exchange is reduced from 20 % of the original B3LYP functional to 15 % (Reiher et al. 2001). Furthermore, it has been suggested that results obtained with Slater-type basis sets converge rapidly with the basis set size, while this convergence in case of Gaussian-type basis sets is much slower, and demanding basis sets like Dunning’s correlation-consistent basis are needed to achieve good results (Güell et al. 2008). Besides the above general comment, benchmark tests of the different functionals in this case are often hampered by the problem of accurate reference data and by the problem that functionals that seem to behave properly for a metal or system simply fail if the system changes (Ghosh 2006). Focusing on Fe complexes, a benchmark study versus CASPT2 values (see Table 11) indicated that the OPBE functional Table 11 Singlet–quintet splitting, Esinglet –Equintet in kcal/mol, for selected Fe complexes (Swart 2008) CASPT2 BP86 RPBE B3LYP PBE1PBE OPBE

Fe(H2 O)6 2C 46.6 28.4 34.3 33.1 46.0 49.3

MAD mean absolute deviation

Fe(NH3 )6 2C 20.3 5.1 6.3 14.1 24.7 19.0

Fe(bpy)3 2C 13.2 23.2 29.9 0.6 9.0 14.9

MAD 14.5 14.3 11.2 9.1 1.9

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performs definitely better than commonly used GGA and HGGA functionals. In a similar study, the performance of some DFT functionals to describe CASPT2 results for a series of five Fe complexes indicated that the OLYP functional performed remarkably well in these cases, whereas the success of the HGGA PBE1PBE, B3LYP, and B3LYP* functionals varied from case to case (Pierloot and Vancoillie 2008). Finally, another benchmark study, in which various properties of Fe2 , Fe2  , and FeOC , as obtained from a series of GGA, HGGA, MGGA, and HMGGA functionals, were calculated, indicated that no single functional was found to yield a satisfactory description of all characteristics for all states of these species (Sorkin et al. 2008). These results clearly indicate that the spin-state problem still is an open challenge for DFT.

Twelve: Excited States Testing the performance of several functionals in the TD-DFT prediction of 21 valence and 20 Rydberg excitation energies in N2 , O2 , HCOOH, and tetracene indicated that valence excitations are easier to predict than Rydberg excitation. Indeed, valence excitations were predicted by 15 functionals with a MUE of 0.36 eV, while Rydberg excitations were predicted with a MUE of 1.13 eV. Focusing on valence excitations, HMGGA functionals such as TPSSh, B98, and B97-3 are the best performers, with MUE as small as 0.25 eV. Nevertheless, GGA and HGGA functionals, with a MUE of 0.32 and 0.28 eV for the PBE and B3LYP functionals, respectively, also perform reasonably well. Differently, for the accurate prediction of Rydberg excitations, a high amount of Hartree–Fock exchange, as in the M062X and M06-HF functionals, with MUE lower than 0.4 eV, is beneficial. With the exception of the also well-performing BMK functional, almost all the other functionals tested results in MUE greater than 0.78 eV, with the PBE and B3LYP functionals resulting in a MUE of 1.95 and 1.07 eV, respectively. When Rydberg and valence excitations are combined into a single database, the best average performance is of the M06-2X and BMK functionals, with a MUE around 0.35 eV (Zhao and Truhlar 2008b). With regards to charge-transfer excitations, the performance of 16 functionals was tested in the prediction of three charge-transfer excitation energies in tetracene and in the NH3 F2 and C2 H4 C2 F4 complexes. The average MUE over the 16 functionals examined, 3.86 eV, is depressive. The only working functional is M06HF, with a MUE of 0.09 eV. However, the inclusion of Hartree–Fock exchange is not the only reason for this surprisingly good performance, since simple HF and the HFLYP functional results in MUE around 1 eV. With the exception of the M05-2X and M06-2X, with a MUE around 2.5 eV, all the other functionals tested resulted in a MUE greater than 3 eV (Zhao and Truhlar 2008b). As a final note, it would be interesting to test the performance of the M06-HF on a larger database. Moving to more complex organic molecules, the   !   transitions of more than 100 dyes from the major classes of chromophores have been investigated using a TD-DFT with the PBE, PBE1PBE, and long-range corrected hybrid functionals. The PBE1PBE and CAMB3LYP were shown to outperform all other approaches, with the latter functional especially adequate to treat molecules with delocalized

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excited states. The PBE1PBE functional resulted in a MUE of 22 nm (0.14 eV) with no deviation exceeding 100 nm (0.50 eV), thus delivering reasonable estimates of the color of most organic dyes of practical or industrial interest. Long-range functionals allowed a better description of the low-lying excited-state energies than HGGA functionals, and linearly corrected long-range approaches yield an average error of 10 nm (Jacquemin et al. 2008). An extended test of the performance of 41 functionals in the calculation of the electronic absorption spectra of Cu and Zn complexes by TD-DFT methods indicated that HGGA functionals outperform GGA functionals. In the case of the spin-unrestricted calculations on the CuII (thiosemicarbazonato) complex, the functional best performing in the reproduction of the experimental spectra and geometry was the B1LYP, while the B3LYP functional was ranked 8. This order was not replicated in case of the spin-restricted ZnII (thiosemicarbazonato) complex, where the best functional was PBE1PBE, with the B3LYP ranked 10. In both cases, HGGA functionals did not offer an improvement. In almost all the cases, the calculations underestimated the experimental excitation energies (Holland and Green 2010). Nevertheless, it is may be worth noting that the OPBE and the OBLYP functionals, which were shown to perform well in other cases, were not considered.

Orbitals in DFT Since chemists long have used and continue to use orbitals as natural language to explain and rationalize the complex reality of molecules that define the realms of inorganic and organic chemistry, we conclude with a few remarks on orbitals obtained from density functional calculations. Originally, chemists have built their understanding of orbitals on constructs that resulted from WFT analyses, and such an orbital is usually referred to as molecular orbital (MO). One important aspect of an MO analysis is the investigation of orbital overlap. A simplified wave function theory that emphasizes this particular feature of orbital interaction, the extended Hückel theory (EHT), has revolutionized the general perception of molecular structure and reaction mechanisms. Since the Kohn–Sham orbitals, introduced and used in DFT, serve a different purpose than creating a reasonable single determinantal wave function ‰, chemists were seeking answers to the question “what do the Kohn–Sham orbitals and eigenvalues mean?” as DFT moved into the spotlight of electronic structure theory. A simple answer was based on a comparison of orbitals of small molecules (H2 O, N2 , PdCl4 2 ) obtained from WFT (Hartree–Fock, EHT) and DFT calculations: The shape and symmetry properties of the KS orbitals are very similar to those calculated by HF and EHT methods. The energy order of the occupied orbitals is in most cases in agreement between WFT and DFT methods. Overall, the KS orbitals are a good basis for qualitative interpretation of molecular orbitals (Stowasser and Hoffmann 1999). This simple conception of the meaning and the use of KS orbitals by now has gained general acceptance, and chemists often use KS orbitals in ways that are familiar to them from MO analysis.

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Baerends and Gritsenko (1997) have provided an answer to the same question based on fundamental concepts of density functional theory. Among other aspects, the authors identify the following two important characteristics of KS orbitals: 1. The highest occupied Kohn–Sham orbital energy is equal to the exact first ionization energy. This is a property that is very desirable in qualitative MO theory in general and is often simply assumed in such theories. 2. The lowest unoccupied Kohn–Sham orbital energy and all other virtual orbital energies are solutions in exactly the same potential as the occupied orbitals. They are therefore not shifted toward higher energies in the same way as Hartree–Fock virtual orbitals are – HF orbital energy differences are not estimates of excitation energies. Further, it has been observed empirically for a long time that KS orbital energy differences are good approximations to excitation energies, and the KS orbital energy differences play a role as a first approximation to the excitation energy in the treatment of excitation energies using time-dependent DFT. The authors therefore recommend KS orbitals and one-electron energies as tools in the traditional qualitative MO considerations on which much of the rationalizations of contemporary chemistry are based. The Kohn–Sham one-electron model and KS orbitals provide an ideal MO theoretical context to apply concepts such as “charge control” and “orbital control.” KS orbitals usually follow the expected behavior, in terms of bonding and antibonding character in terms of geometrical distortions and in terms of interaction with other atoms or molecules. About 10 years later, Cramer and Truhlar (2009) presented a more conservative view of the use of KS orbitals. One should be careful not to stretch the interpretation of KS orbitals beyond its limits, since KS orbitals correspond to a fictitious noninteracting system with the same electron density as the correct many-body function. Since the density computed from KS orbitals is an approximation to the exact density, properties that depend on individual orbitals, with the exception of the energy of the highest occupied orbitals, should be interpreted with care. Nonetheless, many studies published in the literature do employ DFT molecular orbitals to interpret the electronic origins of chemical bonding and reactivity. The quality of the KS orbitals depends to a large part on the ability of a chosen density functional to correctly represent the ground-state density of a given molecule. In most cases, different density functionals produce qualitatively identical orbitals, which also agree with WFT orbitals. For molecules that might posses a spin-polarized ground-state density, different methods of electronic structure calculation not only produce quantitatively different results but also lead to qualitatively contrastive conclusions. One such case is illustrated in Fig. 7. Figure 7 Energies and shapes of the two highest occupied (red-blue) and the two lowest unoccupied (yellow-green) KS orbitals (contour envelope 0.05 a.u.) of Fe(S2 C2 H2 )2 extracted from an unpolarized ground-state density according to GGA and HGGA DFT calculations. The molecule was constructed according to D2h symmetry with a Fe–S bond length of 220 pm. Also shown for the sake of comparison are orbitals obtained from an extended Hückel calculation.

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E/e V –4.02 –4.38

–4.48

–5.50 –5.73 –5.79 –6.31

–7.19

EHT

GGA

DFT

HGGA

–9.63

–10.51

S

S HC

CH

–10.82 –11.03

Fe HC

CH S

S

Fig. 7 Energies and shapes of the two highest occupied (red-blue) and the two lowest unoccupied (yellow-green) KS-orbitals (contour envelope: 0.05 a.u.) of Fe(S2 C2 H2 )2 extracted from an unpolarized ground state density according to GGA and HGGA DFT calculations. The molecule was constructed according to D2h symmetry with a Fe-S bond length of 220 pm. Also shown for the sake of comparison are orbitals obtained from an extended Hückel calculation

But the essential understanding of chemical orbitals itself is open to interpretation; Autschbach (2012) elaborates on the use and misuse of orbitals in quantum theory and chemistry. Naturally, KS orbitals play an essential role in this challenging controversy. We leave it to the reader to decide whether or not chemically meaningful information can be extracted from the orbital picture as displayed in Fig. 7.

DFTips The advertent reader has noticed the many technical recommendations that we have given within the discussion of the twelve selected scenarios for properties of atoms and molecules. Here, we will close our instruction manual with a few pieces of general advice.

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B3LYP Is No Synonym for DFT Some researchers hold the opinion that given the fact that the B3LYP functional has been identified as the most successful DFT method in an overwhelming number of systematic investigations in very many areas of chemical research, there is no persuasive motivation to recommend its replacement by one of the other functionals. We do not agree with such a judgment. Although the B3LYP functional is largely responsible for DFT becoming one of the most popular tools in computational chemistry, it does have unsatisfactory performance issues, notably the unreliable results obtained for transition metal chemistry (Zhao and Truhlar 2008a). One should become aware of the capabilities and shortcomings of particular density functionals. It might well be that B3LYP is the proper approach to many chemical problems, but choosing a functional for its previous success while ignoring its potential failures cannot be the right strategy to approach DFT calculations.

Choose Your DFT Method Carefully We agree with Burke’s interpretation of “The good, the bad and the ugly”: It is good to choose one functional of each kind and stick with it. It is bad to run several functionals and pick the “best” answer. And it is ugly to design your own functional with 2300 parameters. In view of the many functionals available, it is likely that one will find a functional that fits one’s own particular problems. In view of possibilities offered by computational programs that make it fairly easy to create new sets of parameterized hybrid functionals, it is almost certain that for each chemical problem, the right functional can be designed. Following such an approach, density functional calculations lose their generality and meaning. One should aim for consistency within one’s calculations, rather than for the best agreement with experiment. If one choose a particular functional for its characteristics and capabilities, the results of DFT calculations gain a predictive quality.

Read the Fine Print During the early days of HGGA development, it became clear that B3LYP is not B3LYP (Hertwig and Koch 1997). Different programs based their implementation of the B3LYP functional on different models of the underlying LDA and produced slightly different results for the same chemical problem. The same considerations apply to the gradient corrections for correlation; P86 requires a different choice of LDA than LYP. Some computer programs automatically make the right choice, while other programs rely on correct input instructions given by the user.

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Familiarize yourself with the default values and basic implementations of your favorite DFT code. Default values are chosen with much consideration and, most of the time, are just adequate. However, for subtle problems, you might find it necessary, for example, to change the accuracy of the numerical integration routines.

DFT Does Not Hold the Universal Answer to All Chemical Problems You can expect that not all of your DFT calculations will produce satisfying results. An increasing number of problem molecules are currently identified for which standard density functionals fail to produce satisfactory results. Seemingly simple stereoelectronic effects in alkane isomers provided a serious challenge for many standard functionals, such as B3LYP, BLYP, or PBE (Grimme 2006c). However, such failures are not bad news for DFT but rather good news. The origin of the dissatisfying DFT performance has been carefully analyzed and led to the design of improved functionals (Zhao et al. 2006). Be honest with your DFT results and accept apparent failures – it might be just another small step climbing Jacob’s ladder.

Make DFT an Integral Part of Your Work Although in some areas of chemical research DFT is at the verge of becoming a standard research tool, necessary for the complete characterization of new molecules, not every chemical problem warrants a full-fledged DFT investigation. However, if you structure your computational approach beginning with an elaboration of qualitative aspects before addressing a problem in a quantitative way, a few quick DFT calculations might provide you with valuable insights how to further pursue your line of work.

Follow Your Interests As final advice we will leave you with the words of Nobel Prize laureate Harold Kroto: Do something which interests you or which you enjoy, and do it to the absolute best of your ability. If it interests you, however mundane it might seem on the surface, still explore it because something unexpected often turns up just when you least expect it. We hope that our work could satisfy some old interests as well as perk some new interests. We wish that our short instruction manual serves as a valuable guide for the perplexed and provides some food for thought for the enlightened, be it in agreement or in disagreement. If you follow your interests, keep an open mind, and maintain a broad perspective, good things will happen.

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Bibliography The literature on DFT is large and rich in excellent reviews and overviews. At the same time, while thousands of papers on DFT have been published, most of them will become out of date in the future, as the picture and perception of DFT, chemistry, and science in general is in a state of constant flux. It was necessary to make a conscious selection that of course is not unbiased – some references represent the most recent developments in DFT, while others have been part of our DFT folder for a long time. The reader will find references to books, review articles, and articles that illustrate developments and applications of DFT.

Books on DFT Fiolhais, C., Nogueira, F., & Marques, M. (Eds.). (2003). A primer in density functional theory (lecture notes in physics). Berlin/New York: Springer. Koch, W., & Holzhausen, M. C. (2002). A chemist’s guide to density functional theory (2nd ed.). Weinheim/New York: Wiley. Marques, M. A. L., Ullrich, C. A., Nogueira, F., Rubio, A., Burke, K., & Gross, E. K. U. (Eds.). (2006). Time-dependent density functional theory (lecture notes in physics). Berlin: Springer. Parr, R. G., & Yang, W. (1989). Density functional theory of atoms and molecules. New York: Oxford University Press.

Reviews and Overviews of DFT Baerends, E. J., & Gritsenko, O. V. (1997). A quantum chemical view of density functional theory. Journal of Physical Chemistry A, 101, 5383–5403. Becke, A. D. (2014). Perspective: Fifty years of density-functional theory in chemical physics. Journal of Chemical Physics, 140, art. 18A301. Cohen, A. J., Mori-Sánchez, P., & Yang, W. (2012). Challenges for density functional theory. Chemical Reviews, 112, 289–320. Cramer, C. J., & Truhlar, D. G. (2009). Density functional theory for transition metals and transition metal chemistry. Physical Chemistry Chemical Physics, 11, 10757–10816. Geerlings, P., De Proft, F., & Langenaeker, W. (2003). Conceptual density functional theory. Chemical Reviews, 103, 1793–1873. Kohn, W., Becke, A. D., & Parr, R. G. (1996). Density functional theory of electronic structure. Journal of Physical Chemistry, 100, 12974–12980. Neese, F. (2009). Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling. Coordination Chemistry Reviews, 253, 526–563. Perdew, J. P., Ruzsinszky, A., Constantin, L. A., Sun, J. W., & Csonka, G. I. (2009). Some fundamental issues in ground-state density functional theory: A guide for the perplexed. Journal of Chemical Theory and Computation, 5, 902–908. Sousa, S. F., Fernandes, P. A., & Ramos, M. J. (2007). General performance of density functionals. Journal of Physical Chemistry A, 111, 10439–10452. Ziegler, T. (1991). Approximate density functional theory as practical tool in molecular energetics and dynamics. Chemical Reviews, 91, 651–667. Ziegler, T. (1995). Density functional theory as practical tool in studies of organometallic energetics and kinetics. Beating the heavy metal blues with DFT. Canadian Journal of Chemistry, 73, 743–761.

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Zhao, Y., & Truhlar, D. G. (2008a). Density functionals with broad applicability in chemistry. Accounts of Chemical Research, 41, 157–167.

Conceptual Developments and Applications of DFT Ahlrichs, R., Penco, R., & Scoles, G. (1977). Intermolecular forces in simple systems. Chemical Physics, 19, 119–130. Baerends, E. J., Ellis, D. E., & Ros, P. (1973). Self-consistent molecular Hartree-Fock-Slater calculations – I. The computational procedure. Chemical Physics, 2, 41–47. Baerends, E. J., & Ros, P. (1978). Evaluation of the LCAO Hartree-Fock-Slater method – Applications to transition-metal complexes. International Journal of Quantum Chemistry, 12, 169–190. Bartlett, R. J., Lotrich,V. F., & Schweigert, I. V. (2005). Ab initio density functional theory: The best of both worlds? Journal of Chemical Physics, 123, art. 062205. Becke, A. D. (1988a). Density-functional exchange-energy Approximation with correct asymptotic behavior. Physical Review A, 38, 3098–3100. Becke, A. D. (1988b). A multicenter numerical-integration scheme for polyatomic molecules. Journal of Chemical Physics, 88, 2547–2553. Becke, A. D., & Roussel, M. R. (1989). Exchange holes in inhomogeneous systems – A coordinatespace model. Physical Review A, 39, 3761–3767. Becke, A. D. (1993a). A new mixing of Hartree-Fock and local density-functional theories. Journal of Chemical Physics, 98, 1372–1377. Becke, A. D. (1993b). Density-functional thermochemistry: 3. The role of exact exchange. Journal of Chemical Physics, 98, 5648–5652. Boerrigter, P. M., te Velde, G., & Baerends, E. J. (1988). 3-dimensional numerical-integration for electronic-structure calculations. International Journal of Quantum Chemistry, 33, 87–113. Furche, F. (2008). Developing the random phase approximation into a practical post-Kohn–Sham correlation model. Journal of Chemical Physics, 129, art. 114105. Gill, P. M. W. (2001). Obituary: Density functional theory (1927–1993). Australian Journal of Chemistry, 54, 661–662. Grimme, S. (2006a). Semiempirical hybrid density functional with perturbative second-order correlation. Journal of Chemical Physics, 124, art. 034108. Hepburn, J., Scoles, G., & Penco, R. (1975). Simple but reliable method for prediction of intermolecular potentials. Chemical Physics Letters, 36, 451–456. Hertwig, R. H., & Koch, W. (1997). On the parameterization of the local correlation functional. What is Becke-3-LYP? Chemical Physics Letters, 268, 345–351. Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Physics Reviews, 136, B646– B871. Kohn, W., & Sham, L. J. (1965). Self-consistent equations including exchange and correlation effects. Physics Reviews, 140, A1133–A1138. Kurth, S., & Perdew, J. P. (2000). Role of the exchange-correlation energy: Nature’s glue. International Journal of Quantum Chemistry, 77, 814–818. Leininger, T., Stoll, H., Werner, H. J., & Savin, A. (1997). Combining long-range configuration interaction with short-range density functionals. Chemical Physics Letters, 275, 151–160. Perdew, J. P. (1986). Density-functional approximation for the correlation-energy of the inhomogeneous electron gas. Physical Review B, 33, 8822–8824. Perdew, J. P., Burke, K., & Ernzerhof, M. (1996). Generalized gradient approximation made simple. Physical Review Letters, 77, 3865–3868. Perdew, J. P., & Schmidt, K. (2001). Jacob’s ladder of density functional approximations for the exchange-correlation energy. Density Functional Theory and Its Applications to Materials, 577, 1–20.

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Peverati, R., & Truhlar, D. G. (2014). The quest for a universal density functional: The accuracy of density functionals across a broad spectrum of databases in chemistry and physics. Philosophical Transactions of the Royal Society A, 372, art. 20120476. Slater, J. C. (1951). A simplification of the Hartree-Fock method. Physics Reviews, 81, 385–390. Tao, J. M., Perdew, J. P., Staroverov, V. N., & Scuseria, G. E. (2003). Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids, Physical Review Letters, 91, art. 146401. Tsuneda, T., & Hirao, K. (2014). Long-range correction for density functional theory. WIREs Computational Molecular Science, 4, 375–390. Versluis, L., & Ziegler, T. (1988). The determination of molecular structures by density functional theory: The evaluation of analytical energy gradients by numerical integration. Journal of Chemical Physics, 88, 322–328. Vosko, S. H., Wilk, L., & Nusair, M. (1980). Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Canadian Journal of Physics, 58, 1200–1211. Zope, R. R., & Dunlap, B. I. (2006). The limitations of Slater’s element-dependent exchange functional from analytic density-functional theory, Journal of Chemical Physics, 124, art. 044107.

Practical Developments and Applications of DFT Autschbach, J. (2012). Orbitals: Some fiction and some facts. Journal of Chemical Education, 89, 1032–1040. Barden, C. J., Rienstra-Kiracofe, J. C., & Schaefer, H. F. (2000). Homonuclear 3d transition-metal diatomics: A systematic density functional theory study. Journal of Chemical Physics, 113, 690–700. Curtiss, L. A., Raghavachari, K., Redfern, P. C., & Pople, J. A. (1997). Assessment of Gaussian2 and density functional theories for the computation of enthalpies of formation. Journal of Chemical Physics, 106, 1063–1079. Curtiss, L. A., Raghavachari, K., Redfern, P. C., & Pople, J. A. (2000). Assessment of Gaussian-3 and density functional theories for a larger experimental test set. Journal of Chemical Physics, 112, 7374–7383. Ghosh, A. (2006). Transition metal spin state energetics and noninnocent systems: Challenges for DFT in the bioinorganic arena. Journal of Biological Inorganic Chemistry, 11, 712–714. Grimme, S. (2006b). Semiempirical GGA-type density functional constructed with a long-range dispersion correction. Journal of Computational Chemistry, 27, 1787–1799. Grimme, S. (2006c). Seemingly simple stereoelectronic effects in alkane isomers and the implications for Kohn-Sham density functional theory. Angewandte Chemie International Edition, 45, 4460–4464. Güell, M., Luis, J. M., Solà, M., & Swart, M. (2008). Importance of the basis set for the spin-state energetics of iron complexes. Journal of Physical Chemistry A, 112, 6384–6391. Holland, J. P., & Green, J. C. (2010). Evaluation of exchange-correlation functionals for timedependent density functional theory calculations on metal complexes. Journal of Computational Chemistry, 31, 1008–1014. Jacquemin, D., Perpete, E. A., Scuseria, G. E., Ciofini, I., & Adamo, C. (2008). TD-DFT performance for the visible absorption spectra of organic dyes: Conventional versus long-range hybrids. Journal of Chemical Theory and Computation, 4, 123–135. Kelly, R. A., Clavier, H., Giudice, S., Scott, N. M., Stevens, E. D., Bordner, J., Samardjiev, I., Hoff, C. D., Cavallo, L., & Nolan, S. P. (2008). Determination of N-heterocyclic carbene (NHC) steric and electronic parameters using the [(NHC)Ir(CO)(2)Cl] system. Organometallics, 27, 202–210.

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Korth, M., & Grimme, S. (2009). “Mindless” DFT Benchmarking. Journal of Chemical Theory and Computation, 5, 993–1003. Lynch, B. J., & Truhlar, D. G. (2003). Small representative benchmarks for thermochemical calculations. Journal of Physical Chemistry A, 107, 8996–8999. Pierloot, K,. & Vancoillie, S. J. (2008). Relative energy of the high-(T-5(2g)) and low-((1)A(1g)) spin states of the ferrous complexes [Fe(L)(NHS4)]: CASPT2 versus density functional theory. Journal of Chemical Physics, 128, art. 034104. Reiher, M., Salomon, O., & Hess, B. A. (2001). Reparameterization of hybrid functionals based on energy differences of states of different multiplicity. Theoretical Chemistry Accounts, 107, 48–55. Riley, K. E., Op’t Holt, B. T., & Merz, K. M. (2007). Critical assessment of the performance of density functional methods for several atomic and molecular properties. Journal of Chemical Theory and Computation, 3, 407–433. Schultz, N. E., Zhao, Y., & Truhlar, D. G. (2005a). Density functionals for inorganometallic and organometallic chemistry. Journal of Physical Chemistry A, 109, 11127–11143. Schultz, N. E., Zhao, Y., & Truhlar, D. G. (2005b). Databases for transition element bonding: Metal-metal bond energies and bond lengths and their use to test hybrid, hybrid meta, and meta density functionals and generalized gradient approximations. Journal of Physical Chemistry A, 109, 4388–4403. Sorkin, A., Iron, M. A., & Truhlar, D. G. (2008). Density functional theory in transition-metal chemistry: Relative energies of low-lying states of iron compounds and the effect of spatial symmetry breaking. Journal of Chemical Theory and Computation, 4, 307–315. Sponer, J., Jurecka, P., & Hobza, P. (2004). Accurate interaction energies of hydrogen-bonded nucleic acid base pairs. Journal of the American Chemical Society, 126, 10142–10151. Stowasser, R., & Hoffmann, R. (1999). What do the Kohn-Sham orbitals and eigenvalues mean? Journal of the American Chemical Society, 121, 3414–3420. Swart, M. (2008). Accurate spin-state energies for iron complexes. Journal of Chemical Theory and Computation, 4, 2057–2066. Wang, N. X., & Wilson, A. K. (2004). The behavior of density functionals with respect to basis set. I. The correlation consistent basis sets. Journal of Chemical Physics, 121, 7632–7646. Wodrich, M. D., Corminboeuf, C., & Schleyer, P. v. R. (2006). Systematic errors in computed alkane energies using B3LYP and other popular DFT functionals. Organic Letters, 8, 3631– 3634. Yang, W. (1991). Direct calculation of electron density in density functional theory. Physical Review Letters, 66, 1438–1441. Zhao, Y., Pu, J., Lynch, B. J., & Truhlar, D. G. (2004). Tests of second-generation and thirdgeneration density functionals for thermochemical kinetics. Physical Chemistry Chemical Physics, 6, 673–676. Zhao, Y., & Truhlar, D. G. (2004). Hybrid meta density functional theory methods for thermochemistry, thermochemical kinetics, and noncovalent interactions: The MPW1B95 and MPWB1K models and comparative assessments for hydrogen bonding and van der Waals interactions J. Journal of Physical Chemistry A, 108, 6908–6918. Zhao, Y., & Truhlar, D. G. (2005a). Design of density functionals that are broadly accurate for thermochemistry, thermochemical kinetics, and nonbonded interactions. Journal of Physical Chemistry A, 109, 5656–5667. Zhao, Y., & Truhlar, D. G. (2005b). Benchmark databases for nonbonded interactions and their use to test density functional theory. Journal of Chemical Theory and Computation, 1, 415–432. Zhao, Y., Gonzalez-Garcia, N., & Truhlar, D. G. (2005). Benchmark database of barrier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions and its use to test theoretical methods. Journal of Physical Chemistry A, 109, 2012–2018. Zhao, Y., & Truhlar, D. G. (2005c). How well can new-generation density functional methods describe stacking interactions in biological systems? Physical Chemistry Chemical Physics, 7, 2701–2705.

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Zhao, Y., Schultz, N. E., & Truhlar, D. G. (2006). Design of density functionals by combining the method of constraint satisfaction with parametrization for thermochemistry, thermochemical kinetics, and noncovalent interactions. Journal of Chemical Theory and Computation, 2, 364– 382. Zhao, Y., & Truhlar, D. G. (2008b). The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theoretical Chemistry Accounts, 120, 215–241. Zhou, M., Andrews, L., & Bauschlicher, C. W. (2001). Spectroscopic and theoretical investigations of vibrational frequencies in binary unsaturated transition-metal carbonyl cations, neutrals, and anions. Chemical Reviews, 101, 1931–1961.

Introduction to Response Theory Thomas Bondo Pedersen

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadratic and Higher-Order Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gauge and Origin Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Nuclear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 7 13 15 17 21 24 25

Abstract

This chapter provides a concise introduction to quantum chemical response theory as implemented in a number of widely used electronic structure software packages. While avoiding technical derivations of response functions, the fundamental idea of response theory, namely, the calculation of field-induced molecular properties through changes in expectation values, is explained in a manner equally valid for approximate wave function and density functional theories. Contrasting response theory to textbook treatments of perturbation theory, key computational concepts such as iterative solution of response equations and the identification and calculation of electronic excitation energies are elucidated. The wealth of information that can be extracted from approximate linear, quadratic, and higher-order response functions is discussed on the basis of the corresponding exact response functions. Static response functions and

T.B. Pedersen () Department of Chemistry, Centre for Theoretical and Computational Chemistry, University of Oslo, Oslo, Norway e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_5-2

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their identification and numerical calculation as energy derivatives are discussed separately. Practical issues related to the lack of gauge and origin invariance in approximate calculations are discussed without going into too much theoretical detail regarding the sources of these problems. Finally, the effects of nuclear motion (molecular vibrations, in particular) and how to include them in computational studies are treated in some detail.

Introduction The ultimate vision of computational chemistry is to provide a virtual laboratory for chemical explorations. The goal is not to replace but rather complement the real lab. In the virtual lab, it is easy to change “experimental” conditions and to experiment with hypothetical what-if scenarios. Virtual synthesis of chemical compounds can be done with a few mouse clicks, whereas the virtual measurement of physical and chemical properties is the often burdensome task of a computational engine based on the laws of classical and/or quantum physics. Most research efforts, therefore, are aimed at improving the algorithms and approximations implemented in the computational engines. The main role of quantum chemical response theory, which is the subject of this chapter, is the virtual measurement of optical properties of molecules. The scope of this chapter is to provide a rudimentary understanding of response theory as implemented in a number of molecular electronic structure packages based on wave function models or density functional theory. Only the general structure of response theory and its computer implementation are discussed, leaving out the often complicated details of advanced wave function and density functional models. For these details the reader is referred to the literature mentioned in the last section and references therein. Although the discussion of this chapter is restricted to nonrelativistic theory, it is the same line of reasoning that is applied in relativistic response theory. In conjunction with the chapter on applications of response theory, the reader should become sufficiently familiar with the concepts and practices of response theory to allow educated use of quantum chemistry software packages. As the name indicates, response theory describes the response of a molecular system to external potentials such as electromagnetic fields. The meaning of external depends on the definition of the molecular system. For example, the magnetic field due to nuclear spins would typically be considered an external potential. The fundamental assumption is that the external potentials are weak compared to the internal potentials of the molecule. That is, the external potentials can be regarded as perturbations of the isolated molecule. This assumption can be justified by noting that the electric field strength in the hydrogen atom is on the order of 1011 Vm1 , which is well above the external field strengths applied in almost all experimental setups. Experimental observations are then rationalized by means of response functions, which may be expressed entirely in terms of states and energies of the unperturbed (isolated) molecule. Response functions are thus characteristic properties of a molecule. The objective of quantum chemical response theory is the computation of response functions from first principles.

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Response functions quantify the field-induced change in a given observable such as the electric dipole moment. It is therefore necessary to know the state of the molecule before the field is applied. This unperturbed state is normally taken to be the molecular ground state, which is usually assumed to be nondegenerate. Calculation of the molecular ground state is, however, a daunting task and approximations are invoked in practice. We will use the clamped-nucleus Born– Oppenheimer approximation to separate the electronic degrees of freedom from the nuclear ones. Focusing on the response of the electronic ground state, the coupling between the nuclei and the external field is neglected. This is justified for field frequencies much larger than vibrational and rotational frequencies. The electronic ground state j0i is computed as an approximate solution to the time-independent Schrödinger equation, that is, H0 j0i  E0 j0i

(1)

where H0 is the electronic Hamiltonian (see below) and E0 is the ground state energy in the absence of external fields. The approximate solution (energy and wave function or density) can be obtained using standard methods such as density functional theory (DFT), Hartree–Fock (HF) theory, multiconfigurational selfconsistent field (MCSCF) theory, configuration interaction (CI), or coupled cluster (CC) theory. For fixed nuclear positions (clamped-nucleus approximation), the electronic Hamiltonian is given by H0 D

Ne Nn Ne X Nn K1 X X X 1 X e 2 ZK e 2 ZK ZL .ps /2  C 2me sD1 4–0 jrs  RK j KD2LD1 4–0 jRK  RL j sD1 KD1 Ne X s1 X e2 C 4–0 jrs rt j

(2)

sD2 t D1

Here, Ne is the number of electrons, Nn is the number of nuclei, me is the mass of the electron, e is the elementary charge, "0 is the vacuum permittivity, rs and ps are the position and momentum operators of electron s, and RK and ZK are the position and atomic number of nucleus K. Starting from the Schrödinger equation (Eq. 1), relativistic effects as described by the Breit–Pauli Hamiltonian can be treated as perturbations on an equal footing with external fields. Effects of nuclear motion (vibrations and rotations) can be estimated once the electronic response functions have been calculated. Response functions are functions of the frequencies of the external fields, and although response theory is based on time-dependent perturbation theory, it is more appropriate to describe response theory as frequency dependent rather than time dependent. Nevertheless, response theory for approximate quantum chemical models is sometimes referred to as time dependent, for example, time-dependent DFT (TDDFT) and time-dependent HF (TDHF) theory. Related by Fourier transformation, the time and frequency domains are two sides of the same coin. In practice, however, response functions are computed for a limited number of perturbation frequencies without any explicit reference to the time evolution of the quantum state.

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Response Theory Approximate wave function and density functional theories provide information about the electronic structure of molecules in their electronic ground state. The information includes the electronic charge density, total energy, electric multipole moments (dipole, quadrupole, octupole, etc.), forces on the nuclei, and vibrational frequencies, which is sufficient to model a wide range of chemical phenomena. For example, equilibrium structures and transition states can be calculated from the forces, and vibrational frequencies are not only useful for the interpretation of vibrational spectra but also enable the calculation of thermochemical data from first principles. These theories are sufficient to model experimental conditions where only the electronic ground state is significantly populated. When a molecule is subjected to external fields, however, quantum mechanics tells us that the electronic ground state responds by becoming a superposition of many electronic states. Response theory takes this wave function change into account and thus facilitates calculations of molecular properties beyond the reach of the ground state theory. Computation of the ground state response allows extraction of information about transitions from the ground state to excited states induced by one or more photons and even allows us to calculate properties of the excited states without explicitly computing the excited state wave functions. Response theory describes the change in observable quantities such as electric and magnetic multipole moments due to external fields. The starting point of response theory therefore is the time evolution of the expectation value hAi of a Hermitian operator A representing the observable quantity (e.g., the electric or magnetic dipole moment). In order to monitor the change in an observable quantity, we need to know the state of the molecule before the external field is switched on. As mentioned above, the electronic ground state j0i is the proper choice under most experimental conditions. With this initial condition, the time evolution can be written as the perturbation expansion hAi D h0 jAj 0i Z 1 C hhAI V .!/ii! exp .i!t/ d! 1 Z 1Z 1 C 12 hh AIV .!/ ; V .! 0 / ii !;! 0 exp .i .! C ! 0 / t/ d!d! 0 C

(3)

1 1

where the operator V(¨) describes the interaction between the electrons and the external field of frequency ¨. It should be noted that the perturbation expansion cannot generally be guaranteed to converge. Moreover, the perturbation expansion does not make sense when the external potential is comparable to or larger than the internal potential of the molecule. In most cases of practical interest, including the interaction of molecular electrons with laser fields, the perturbation expansion may be used.

Introduction to Response Theory

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Frequency-dependent response functions can only be computed within approximate electronic structure models that allow definition of the time-dependent expectation value. Hence, frequency-dependent response functions are not defined for approximate methods that provide an energy but no wave function. Such methods include Møller–Plesset (MP) perturbation theory, multiconfigurational second-order perturbation theory (CASPT2), and coupled cluster singles and doubles with noniterative perturbative triples [CCSD(T)]. As we shall see later, it is possible to derive static response functions for such methods. Common methods that do allow calculation of frequency-dependent response functions include DFT, HF, MCSCF, and members of the CC hierarchy of wave functions (CCSD(T) is not a member of this hierarchy, as it does not provide a wave function). In these models, the wave function (or density in the case of DFT) is written in terms of time-dependent parameters which are determined from the time-dependent Schrödinger equation with the initial condition that the approximate ground state wave function (or density) is recovered in the limit .t ! 1 /, that is, before the external fields are switched on. The zeroth-order parameters are thus determined in the same way as in the ground state model, and response theory calculations are naturally performed on top of a standard ground state calculation. First-, second-, and higher-order parameters are determined from response equations that take into account the coupling with the external field in a manner consistent with the time-dependent Schrödinger equation. The response equations have the general matrix form (the detailed form of these matrices and vectors depends on the approximate theory used) ŒH  !S .n/ .!/ D V .n/ .!/

(4)

where (n) (!), the unknown quantity in this equation, is a vector containing the nth-order ( n >0) wave function parameters in a suitable order. While the matrices H and S only depend on the unperturbed ground state wave function, the vector on the right-hand side additionally depends on the lower-order wave function  parameters .1/ ; .2/ ; .3/ ; : : : ; .n1/ as well as on one or more perturbation operators V(¨). This implies that the solution of the first-order equations must be known before the second-order equations can be solved, and so on. The frequency ¨ represents the sum of the frequencies corresponding to the perturbations included in the vector on the right-hand side of Eq. 4. Solving the response equations (Eq. 4) is formally simple, namely, .n/ .!/ D ŒH  !S1 V .n/ .!/

(5)

where we assume that the matrix ŒH  !S is nonsingular. Those frequencies ¨ for which the matrix is singular have a special significance, as will be discussed below. Since the matrices H and S only depend on the ground state wave function or density, they may in principle be computed once and for all. For each frequency and right-hand side vector we may then calculate the inverse ŒH  !S1 followed by a simple matrix–vector multiplication to obtain the response parameters according to

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Eq. 5. Unfortunately, this simple solution strategy is too computationally demanding to be useful in practice. The large number of matrix elements in H and S makes it expensive to compute the matrices and to compute the inverse which, in addition, needs to be done for each frequency ¨. The number of matrix elements is given by the square of the number of ground state wave function parameters. For DFT or HF, for example, the number of ground state parameters is given by OV where O and V are the number of occupied and virtual (Kohn–Sham or HF) orbitals, respectively. For the more complicated coupled cluster singles and doubles (CCSD) wave function, the number of ground state wave function parameters is roughly O2 V2 /2. To get an impression of the orders of magnitude, consider a simple benzene molecule with the aug-cc-pVDZ basis set. In this case, there are approximately 3,600 DFT or HF ground state parameters and over six million CCSD parameters. The number of elements in the H and S matrices would therefore be approximately 13 million for DFT and HF and 36  1012 for CCSD. For two benzene molecules, these numbers would increase to approximately 207 million matrix elements for DFT and HF and 1016 for CCSD. Just storing the matrices in computer memory would quickly pose an insurmountable problem, and a different solution strategy is needed. Rather than straightforward application of Eq. 5, the response equations (Eq. 4) are solved in an iterative procedure in which the matrices and vectors are projected onto a small subspace. In this procedure, an initial guess (a trial vector) is generated according to Eq. 5 by neglecting all off-diagonal elements of H and S such that the inverse matrix ŒH  !S1 is readily computed. The iterations proceed by refining the trial vector until the norm of the residual vector R D .H  !S/ .n/ C V .n/

(6)

falls below a specified tolerance. Although quantum chemistry software packages provide reasonable default values for the tolerance, it may occasionally be necessary to lower the value to obtain sufficient accuracy in the computed response functions. From the perturbative corrections to the wave function parameters we obtain perturbative corrections to the expectation value of the operator A, that is, hAi D hAi.0/ C hAi.1/ C hAi.2/ C   

(7)

We may then identify response functions for each of the approximate electronic structure models by comparing with Eq. 3: hAi.0/ D h0 jAj 0i hAi.1/ D

Z

1 1

hhAI V .!/ii exp .i!t/ d! !

(8) (9)

Introduction to Response Theory

hAi

.2/

Z D :: :

1 2

1

Z

1

7

˝˝  ˛˛ AI V .!/ ; V ! 0

!;! 0

1 1

    exp i ! C ! 0 t d!d! 0

(10)

Response theory deviates in one crucial aspect from the formulation of timedependent perturbation theory in most textbooks on quantum mechanics: the response parameters (n) are not explicitly expressed in terms of the excited states. As a consequence, knowledge of the excited state wave functions is not needed in response theory. Instead, we must solve the response equations (Eq. 4) for each set of perturbation operators V(¨). This is a tremendous computational advantage as there are significantly fewer perturbation operators, and hence response equations to solve, than excited states in virtually all cases of practical interest. In order to illuminate the wealth of information that can be extracted from response functions for approximate wave functions, it is instructive to study the details of the exact response functions expressed in terms of electronic excited states.

The Linear Response Function The linear response function describing the first-order induced electric dipole moment due to an oscillating and spatially uniform electric field is related to the frequency-dependent electric dipole polarizability as ˛˛ ˝˝ ˛ij .!I !/ D  i I j !

(11)

The ith Cartesian component of the electronic electric dipole operator is given by i D e

X

ris

(12)

s

where the sum is over all electrons in the molecule. This identification is obtained by substituting A ! i and V .!/ !  F .!/ in Eq. 3. The uniform electric field amplitude vector at frequency ¨ is here represented by F(!). In an approximate calculation, the electric dipole polarizability is thus obtained by computing (minus) the linear response function from first-order wave function parameters determined by Eq. 4 with the three Cartesian components of the electric dipole operator as perturbation operators. Note that the response equations must be solved for each frequency. More information can be extracted from the linear response function, however. This is most easily seen by studying the exact linear response function. Unlike the approximate theories, the exact linear response function is expressed in terms of the ground and excited states satisfying the time-independent Schrödinger equation

8

T.B. Pedersen

H0 jki D Ek jki

(13)

where k 0 and H0 is the molecular electronic Hamiltonian in the clamped-nucleus Born–Oppenheimer approximation, Eq. 2. The exact linear response function is given by   1 X h0 jAj ki hk jBj 0i h0 jBj ki hk jAj 0i  hhAI Bii! D „ !  !k0 ! C !k0

(14)

k¤0

where „!k0 D Ek  E0 is the excitation energy for the transition from the ground state to the kth excited state. This formulation of the linear response function is often referred to as the sum-over-states expression or spectral resolution. The exact linear response function is singular at the molecular excitation energies, that is, it has poles at ! D ˙!k0 . This is exploited in approximate theories to identify excitation energies as those frequencies for which the approximate linear response function is singular. In principle, the search for excitation energies could be done by scanning over frequencies in a manner analogous to measurements of absorption spectra. In practice, however, it is straightforward to identify the poles of the approximate linear response function as the frequencies for which the response equations (Eq. 4) are singular. As discussed above, this occurs at frequencies where the matrix H  !S cannot be inverted. This, in turn, occurs when ¨ equals one of the eigenvalues wn of the generalized eigenvalue problem .H  wn S/ Vn D 0

(15)

where Vn is the eigenvector corresponding to the eigenvalue wn . The generalized eigenvalue problem does not depend on the perturbation, and it can therefore be solved to directly obtain the excitation energies, identifying !k0 D wk ; k D 1; 2; 3;    , without explicit calculation of neither the excited state wave functions nor their energies. The eigenvectors Vn are the closest we get to an explicit wave function representation of the excited states in response theory. In practice, therefore, analysis of these eigenvectors is used to extract the (approximate) orbital nature of an excitation, for example, as a    orbital transition. Since we know the ground state energy E0 from the prerequisite unperturbed calculation, the excited state energy may be deduced from the excitation energy as Ek D E0 C „!k0

(16)

Solving the eigenvalue problem, Eq. 15, does not necessarily yield all excited states. With a spin-adapted wave function, only excited states with the proper spin symmetry would be produced. For example, only spin-singlet excited states can be found with a spin-singlet wave function parameterization. Triplet states require a suitable spin-triplet adaptation of the wave function response. In unrestricted

Introduction to Response Theory

9

theories, the ground and excited states need not be spin eigenfunctions and the labeling of states as spin-singlet, spin-triplet, etc., is not well defined. It is important to realize, however, that the number of eigenvalues wn equals the number of excited states within a given approximate theory. Calculation of all eigenvalues is very computationally demanding. The number of excited states in an approximate theory equals the number of ground state wave function parameters, which grows quickly with the size of the molecule, as discussed above. One would most often be interested in just a few, typically on the order of 10, of the lowest excitation energies. Iterative techniques have been devised that allow the solution of Eq. 15 for a given number of eigenvalues and eigenvectors in ascending order. In order to calculate the excitation energy ! k0 , it is therefore necessary to compute all excitation energies below this one as well, that is, !10 ; !20 ; !30 ;    ; !k1;0 .1 The iterative solution of the generalized eigenvalue problem Eq. 15 is carried out by refining trial vectors and eigenvalues until the norm of the residuals Rn D .H  wn S/ Vn

(17)

falls below a given tolerance. Another important piece of information can be extracted from the pole structure of the linear response function. The residue of the linear response function at an excitation energy provides the one-photon transition strength: h0 jAj ki hk jBj 0i D lim „ .!  !k0 / hhAI Bii! !!!k0

(18)

This observation allows us to identify transition moments between the ground state and the kth excited state, h0 jAj ki, in approximate theories by computing the residue of the linear response function at the kth excitation energy. Again, this is achieved without explicitly computing the excited state wave function. In practice, the transition moment is expressed in terms of the kth eigenvector of Eq. 15 and the matrix representation of the operator A in the molecular orbital (or spin orbital) basis. It is therefore possible, in principle, to compute the linear response function for an approximate wave function using the sum-over-states expression with excitation energies calculated according to Eq. 15 and transition moments from Eq. 18. If all excitation energies and residues are included, the result is identical to the response function computed from the response equations (Eq. 4). From a computational point of view, of course, this approach is not advisable unless the combined number of perturbation operators and frequencies is larger than the number of excited states. This is practically never the case. Within approximate wave function or density functional theories, it thus becomes possible to calculate one-photon absorption intensities such as the electric dipole

1

Although techniques exist which solve an eigenvalue equation around a specific energy, these techniques are not used for quantum chemical calculations of excitation energies, as the energy range is rarely known in advance.

10

T.B. Pedersen

oscillator strength of a transition from the ground state to the kth excited state from the following residue of the frequency-dependent electric dipole polarizability: 2me !k0 h0 jj ki  hk jj 0i 3e 2 „ X 2me D 2 !k0 lim „ .!  !k0 / ˛i i .!I !/ !!!k0 3e „ i

fk0 D

(19)

where me is the mass of the electron. In approximate theories, the iterative solution of the response equations (Eq. 4) becomes difficult and may even fail to converge when the frequency is too close to one of the excitation energies. As a consequence, linear response functions should only be calculated in transparent spectral regions. In absorptive spectral regions (“close” to an excitation energy), so-called anomalous dispersion is observed experimentally. Anomalous dispersion can be understood theoretically by taking into account the finite lifetime of the excited electronic states, leading to damped response theory. The exact damped linear response function is given by hhAI Bii! D

  1 X h0 jAj ki hk jBj 0i h0 jBj ki hk jAj 0i  „ !  !k0 C i k =2 ! C !k0  i k =2

(20)

k¤0

The broadening  k is proportional to the probability of the excited state jki decaying into any of the other states, and it is related to the lifetime of the excited state as k D 1= k . For k D 0, the lifetime is infinite and Eq. 14 is recovered from Eq. 20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (Eq. 4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes cannot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, all decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions: (1) all broadening parameters are assumed to be identical, 1 D 2 D    D , and (2) the value of  is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in Eq. 4 with the substitution ! ! ! C i =2, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. Figure 1 shows the damped linear response function calculated from the real part of Eq. 20 with different values of the broadening. Only a section of the frequency spectrum around the lowest excitation energy is shown. With infinite lifetime . D 0/, the linear response function goes to ˙1 as the frequency approaches the

Introduction to Response Theory

11

1000 9.114 mHartree 4.557 mHartree 2.2785 mHartree 1.13925 mHartree 0 mHartree

Polarizability (arbitrary units)

500

0

–500

–1000 0.456

0.458

0.46

0.462

0.464

0.466

0.468

0.47

0.472

0.474

Frequency (Hartree)

Fig. 1 The damped linear response function computed from the real part of Eq. 20 with the indicated values of the broadening  (corresponding to 0, 250, 500, 1,000, and 2,000 cm1 ). Electric dipole transition moments and excitation energies were obtained using CCSD for the hydrogen molecule with the aug-cc-pVDZ basis set

excitation. This is avoided when the finite lifetime . > 0/ is taken into account. In the absorptive region, the behavior of the damped linear response function depends strongly on the value of the broadening. Since the broadening parameter can be chosen freely, the results obtained in this region must be treated with great care: changing the value of the broadening could give a very different result. Sufficiently far from the excitation energy, on the other hand, the effect of the broadening is small and can be neglected. In the transparent region, we may therefore safely use the undamped linear response function. A long list of molecular electromagnetic properties can be obtained by inserting suitable operators in the linear response function. The electric dipole polarizability is an important example. To further illustrate the procedure of obtaining molecular properties from linear response functions, we now consider the induced electric dipole moment due to higher-order multipole components of the electromagnetic field. The induced electric dipole moment due to a time-dependent uniform magnetic field is, to first order in the field, given by the electric dipole – magnetic dipole polarizability ˛˛ ˝˝ Gij0 .!I !/ D Im i I mj !

(21)

12

T.B. Pedersen

where the jth Cartesian component of the magnetic dipole operator is defined as mj D 

e X s L 2me s j

(22)

with the angular momentum operator of electron s given by Ls D rs  ps . This identification is obtained from Eq. 3 by substituting A ! i and V .!/ ! m B .!/, where B(!) is the magnetic field amplitude at frequency ¨. Using multipole expansions of the electric and magnetic fields, the Maxwell equations show that a time-dependent uniform magnetic field is always accompanied by a quadrupolar electric field, and the contribution to the induced dipole moment from the electric dipole – electric quadrupole polarizability ˛˛ ˝˝ ai;j k .!I !/ D  i I qj k !

(23)

P is of the same order of magnitude as G0 . Here, qj k D e s rjs rks is the j, k component of the Cartesian electric quadrupole moment operator. The electric dipole – magnetic dipole and electric dipole – electric quadrupole polarizabilities govern the optical rotation of chiral molecules. For samples of randomly oriented molecules, the contribution from the electric dipole – electric quadrupole polarizability averages to zero, and only the trace (the sum of the diagonal elements) of the electric dipole – magnetic dipole polarizability contributes to the optical rotation. The frequency-dependence of the linear response functions gives rise to optical rotation dispersion. The residues of the electric dipole – magnetic dipole polarizability are related to the rotatory strength X Rk0 D Im Œh0 jj ki  hk jmj 0i D  lim „ .!  !k0 / Gi0 i .!I !/ i

!!!k0

(24)

which determines the differential intensity of electronic circular dichroism spectra. If the molecules are oriented, the rotatory strength is a tensor (since the intensity will depend on the orientation of the molecule relative to the propagation direction of the photon) that also includes contributions from the residues of the electric dipole – electric quadrupole polarizability. Finally, we note that the linear response function satisfies the permutation and conjugation relations hhAI Bii! D hhBI Aii! ˛˛ ˝˝ hhAI Bii! D A I B  !

(25) (26)

where the asterisk denotes complex conjugation and A is the adjoint of the operator A.A D A for Hermitian operators, by definition). It follows from these relations that the electric dipole polarizability is an even function of frequency, ˛ij .!I !/ D ˛ij .!I !/, and a symmetric tensor, ˛ij .!I !/ D ˛j i .!I !/, which implies

Introduction to Response Theory

13

that only six of the nine Cartesian components of the polarizability need to be computed. The permutation and conjugation relations are satisfied by approximate linear response functions as well.

Quadratic and Higher-Order Response Functions The quadratic response function describing the second-order induced electric dipole moment due to a uniform time-dependent electric field is related to the frequencydependent electric dipole hyperpolarizability as   ˝˝ ˛˛ ˇij k !I !; ! 0 D  i I j ; k !;! 0

(27)

where ! D ! C ! 0 . With a procedure completely analogous to that of the linear response functions, this identification is obtained by substituting A ! i and V .!/ !  F .!/ in Eq. 3. A variety of optical processes are governed by the electric hyperpolarizability depending on the values of the frequencies, including Second Harmonic Generation through ˇij k .2!I !; !/ and Optical Rectification through ˇij k .0I !; !/. In an approximate calculation, the electric dipole hyperpolarizability can be obtained by computing (minus) the quadratic response function from first-order wave function parameters determined by Eq. 4 with the three Cartesian components of the electric dipole operator as perturbation operators at each set of frequencies ¨, ! 0 , and !. In accordance with Wigner’s 2 n C 1 rule, which states that the nth-order wave function parameters determine the response functions up to order 2 n C 1, second-order wave function parameters are not needed. In this context, the order of the response function corresponds to the number of operators it contains such that the ground state expectation value is of order 1, the linear response function is of order 2, the quadratic response function is of order 3, and so on. The 2 n C 1 rule is essential for efficient computer implementations of response theory. To learn more about the information that can be extracted from the quadratic response function in approximate calculations, we now study the exact one. Like the exact linear response function, the exact quadratic response function can be calculated from excitation energies and transition matrix elements of the unperturbed molecule: ( ˝ ˇ ˇ ˛ X h0 jAj ki k ˇB ˇ l hl jC j 0i 1 hhAI B; C ii!;! 0 D 2 PBC „ .! C ! 0  !k0 / .! 0  !l0 / k;l¤0 ˝ ˇ ˇ ˛ h0 jC j ki k ˇB ˇ l hl jAj 0i (28) C 0 .! C ! 0 C ˇ l0ˇ/ ˛.!  !k0 /) ˝ ! h0 jBj ki k ˇAˇ l hl jC j 0i  .! C !k0 / .! 0  !l0 /

14

T.B. Pedersen

The bar signals that the operator has been shifted to a zero-point defined by its ground state expectation value. For example, B D B  h0 jBj 0i I

(29)

where I is the identity operator. The operator PBC permutes the pairs (B, ¨) and (C, ! 0 ):       PBC f B; !; C; ! 0 D f B; !; C; ! 0 C f C; ! 0 ; B; !

(30)

giving a total of six distinct terms in the quadratic response function. The quadratic response function is singular at the molecular excitation energies, that is, when ! C ! 0 D ˙ !k0 or when ! D ˙ !k0 or when ! 0 D ˙ !k0 for some k >0. These singularities can be avoided by taking into account the finite lifetime of the excited molecular states, as discussed for the linear response function in the previous section. The quadratic response function, for exact as well as approximate states, satisfies the permutation and conjugation relations hhAI B; C ii!;! 0 D hhAI C; Bii! 0 ;! D hhBI A; C ii!! 0 ;! 0 D hhBI C; Aii! 0 ;!! 0

(31)

D hhC I A; Bii!! 0 ;! D hhC I B; Aii!;!! 0 ˛˛ ˝˝ hhAI B; C ii! D A I B  ; C  !;! 0

(32)

The assumption, known as Kleinman symmetry, that the quadratic (and higherorder) response function is symmetric under interchange of any pair of operators is only true at zero frequency, as is easily verified from Eq. 31. Note that the quadratic response function is symmetric when permuting the operators B and C at ! D ! 0 , which is the case for, for example, Second Harmonic Generation: ˇij k .2!I !; !/ D ˇi kj .2!I !; !/. Kleinman symmetry is often assumed in calculations of the electric dipole hyperpolarizability at low frequencies where it is approximately valid. This reduces the number of independent tensor elements and thus the computational effort. Owing to the more complicated dependence on field frequencies, the residues of the quadratic response function provide more information than those of the linear response function. One of the most important residues of the quadratic response function is lim „ .! 0  !m0 / hhAI B; C ii!;! 0 " # X  h0 jAj ki hk jBj mi ki mi h0 jBj hk jAj C D  „1 hm jC j 0i !k0  .!m0  !/ !k0  !

! 0 !!m0

k

(33)

Introduction to Response Theory

15

where the summation over k includes the ground state j0i. The quantity in square brackets is an element of the transition matrix for two-photon excitation from the ground state to the excited state m. Another useful (double) residue is 

 lim „ .! 0 C !m0 /

!!!n0

lim „ .! 0  !m0 / hhAI B; C ii!;! 0 0

! !!m0

D ımn h0 jAj 0i h0 jBj ni hn jC j 0i  h0 jBj ni hn jAj mi hm jC j 0i (34) From this expression it is possible to extract transition moments between excited states Œhn jAj mi. With the specific choice n D m it thus becomes possible to compute excited state properties such as dipole moments from the quadratic response of the ground state. Higher-order response functions (cubic, quartic, quintic, etc.) can also be written in terms of molecular excitation energies and eigenstates. These functions possess singularities at the excitation energies, and the associated residues (single, double, etc. residues) can be used to identify a wide range of molecular properties. For example, residues of the cubic response function provide two-photon transition strengths (the product of two-photon transition matrix elements) and three-photon transition matrix elements, while the three-photon transition strength is obtained from the quintic response function. Linear response functions for the excited states can be extracted from the cubic response function. By calculating ground state response functions we may thus extract all molecular properties.

Static Response Functions The interaction between a molecule and time-independent external fields is described by frequency-independent perturbation operators. Time-independent fields are nonoscillating, and interactions with molecular electrons are therefore characterized by perturbation operators at zero frequency. The limit ! ! 0 is often referred to as the static limit and response functions at zero frequency are called static response functions. Although static response functions can be calculated in the same way as dynamic (frequency-dependent) ones, they are special in the sense that they are related to changes in ground state energy. For example, the electronic ground state energy of a molecule in the presence of a static uniform electric field, F, can be expanded as E D E0  h0 jj 0i  F  F ˛ .0I 0/  F C   

(35)

where E0 is the energy in the absence of the field. The expectation value of the dipole moment operator thus determines the first-order energy due to the field, whereas the static polarizability determines the second-order energy. Higherorder energy corrections are determined by the static hyperpolarizability, second

16

T.B. Pedersen

hyperpolarizability, etc. This explains why ground state expectation values are often classified as first-order properties, linear response functions as second-order properties, and so on. The relation between static response functions and energy corrections implies that molecular properties can be identified as derivatives of the total energy at zero field strength. For example, @E @Fi

(36)

@2 E @Fi @Fj

(37)

h0 ji j 0i D  ˛ij .0I 0/ D 

where it is implicitly understood that the derivatives must be evaluated at zero field strength .F D 0/. Hence, the calculation of static response functions does not require a wave function and may therefore be computed with electronic structure theories that only provide the energy (e.g., MP2, CASPT2, and CCSD(T)). Even if a quantum chemical software package does not feature an implementation of a given static response property, it is often possible to evaluate the property as an energy derivative by finite difference. Most packages allow the addition of external static fields to the Hamiltonian, making it possible to compute the energy at a specified field strength, E(F). From the definition of the partial derivative, we may thus compute the dipole moment and polarizability from finite difference formulas such as E .ıi /  E .ıi / @E  @Fi 2ıi @2 E E .ıi /  2E.0/ C E .ıi /  @Fi2 ıi2         E ıi ; ıj C E ıi ; ıj  E ıi ; ıj  E ıi ; ıj @2 E  @Fi @Fj 4ıi ıj

(38)

(39)

(40)

where •i (•j ) is a small field strength along the ith (jth) Cartesian direction. The error in these numerical derivative formulas is quadratic in the field strength which should therefore be chosen as small as possible. Choosing too small field strengths, however, may lead to numerical instabilities when computing the difference between two large and almost identical numbers. It is therefore advisable to conduct a convergence study with respect to the field strength when computing numerical derivatives. A number of static perturbations arise from internal interactions or fields, which are neglected in the nonrelativistic Born–Oppenheimer electronic Hamiltonian. The relativistic correction terms of the Breit–Pauli Hamiltonian are considered as perturbations in nonrelativistic quantum chemistry, including Darwin corrections,

Introduction to Response Theory

17

the mass-velocity correction, and spin–orbit and spin–spin interactions. Some properties, such as nuclear magnetic resonance shielding tensors and shielding polarizabilities, are computed from perturbation operators that involve both internal and external fields.

Gauge and Origin Invariance As described above, molecular properties are identified as response functions by selecting the observable quantity whose change is to be monitored and the external field(s) inducing the change. The operator describing the coupling between the external field and the electrons is not unique, however. The electric field F(r, t) can be written in terms of the vector potential A(r, t) and the scalar potential (r, t) as F .r; t/ D 

@A .r; t/  r .r; t/ @t

(41)

while the magnetic field B(r, t) is given by B .r; t/ D r  A .r; t/

(42)

Since the electric and magnetic fields are obtained by differentiation, adding a constant to the vector and scalar potentials does not alter the physical electric and magnetic fields. More generally, adding the gradient of a continuous function (r, t) to the vector potential, that is, A ! A C r , while subtracting the time derivative of the same function from the scalar potential, that is, .r; t/ ! .r; t/  @ =@t will not change the electric and magnetic fields. This feature is known as gauge invariance and implies that there is a manifold of scalar and vector potentials related by gauge transformations that describe the same physical field. This leads to a manifold of different quantum mechanical operators describing the interaction between the electrons and the field. The time-dependent Schrödinger equation is invariant under such gauge transformations and the computed properties therefore also should be invariant. This is indeed the case for exact response functions. As an example, consider again the interaction between electrons and a uniform electric field. In section “The Linear Response Function,” the electric dipole polarizibality was identified using the length gauge perturbation operator V .!/ D  F .!/

(43)

An equally valid choice would have been the velocity gauge perturbation operator V .!/ D 

ie p A .!/ !me

(44)

18

T.B. Pedersen

where A(!) is the amplitude vector of the electromagnetic vector potential in the X electric dipole approximation and p D ps is the total electronic momentum s

operator. This operator leads to the following expression for the electric dipole polarizability: ˛ij .!I !/ D 

˛˛ i e ˝˝ i I pj ! !me

(45)

While Eq. 11 is called the length gauge or dipole-length gauge expression, Eq. 45 is often called the mixed gauge form since it involves both the electric dipole operator and the momentum operator. The length and mixed gauge polarizabilities are equivalent due to the equation of motion „!hhAI Bii! D hhŒA; H0  I Bii! C h0 jŒA; Bj 0i

(46)

which is satisfied by the exact linear response function, and due to the commutator relations 

i e„ j ; H0 D  pj me  i ; j D 0

(47) (48)

Such relations (equation of motion and commutators) guarantee that exact linear response functions are gauge invariant. In addition, transition moments deduced from residues of the exact linear response function are also gauge invariant. For approximate theories, however, different results are always obtained with different perturbation operators related by gauge transformations. The use of a finite basis set implies that operators are represented as finite matrices and, consequently, the commutator relations no longer hold. As the basis set quality is increased, the commutators converge to the exact values. Gauge invariance thus is recovered in the limit of a complete basis set, provided that the equation of motion Eq. 46, is fulfilled. While this is the case in theories with fully variational orbitals such as DFT, HF, and MCSCF, it is not the case in approximate theories with fixed (nonvariational) orbitals such as CI and CC. The exact quadratic and higher-order response functions satisfy similar equations of motion, for example, „ .! C ! 0 / hhAI B; C ii!;! 0 D hh ŒA; H0  I B; C ii!;! 0 C hh ŒA; B I C ii! 0 C hhŒA; C  I Bii!

(49)

ensuring gauge invariance of higher-order molecular properties. In analogy to the case of the linear response function, the higher-order equations of motion are

Introduction to Response Theory

19

satisfied in approximate theories with variational determination of the orbitals such as DFT, HF, MCSCF, but not in theories such as CI and CC. It is therefore necessary to specify which gauge (typically, length or velocity) is used when reporting computed response properties such as electric dipole (hyper-)polarizabilities and transition moments. With reasonably flexible basis sets, each of the DFT, HF, and MCSCF methods typically provides length and velocity gauge results that are quite close to each other, whereas CC results occasionally show significant differences between length and velocity gauge. For example, the specific optical rotation of (1 S,4 S)-norbornenone calculated at the CC singles and doubles (CCSD) level is 558 deg dm1 (g/ml)1 with the length gauge and 740 deg dm1 (g/ml)1 with the velocity gauge (corrected for an unphysical static limit, see below). The fundamental problem is that there is no general physical reason to trust one gauge formulation more than another. In some cases, however, there may be good reasons for favoring one gauge formulation over others in approximate calculations. One such example is optical rotation which is governed by the trace of the electric dipole – magnetic dipole polarizability, whose length gauge formulation is given by Eq. 21. Within the clamped-nucleus Born–Oppenheimer approximation, every electronic structure calculation is performed with fixed nuclear positions. The coordinate system used to specify the nuclear positions can be chosen arbitrarily. In particular, the results of the calculation should be independent of the choice of origin of the coordinate system. Suppose that we translate the origin of the coordinate system along a vector O. As a consequence, the positions of nuclei and electrons are shifted by the same vector, for example, rs ! rs  O, and the electric dipole – magnetic dipole polarizability changes according to G0 .!I !/ ! G0 .!I !/ C

e Im ΠhhI pii!  O 2me

(50)

which shows that the tensor is origin dependent. The culprit is the origin-dependence of the magnetic dipole operator: m! m

e pO 2me

(51)

This translation of the operator can also be achieved by a particular type of gauge transformation of the magnetic vector potential, explaining why the term gaugeorigin is often used for O in the context of magnetic properties. 0 It can be shown that the trace of the tensor G , and hence the computed optical rotation of a sample of randomly oriented chiral molecules, is independent of the origin provided that the linear response function satisfies Eq. 46 and that the commutator of Eq. 47 is fulfilled. Consequently, approximate linear response calculations of the length gauge optical rotation depend on the chosen coordinate origin. On the other hand, the trace of the velocity gauge formulation of the electric dipole – magnetic dipole polarizability

20

T.B. Pedersen

G0 .!I !/ D

e hhpI mii! !me

(52)

is unconditionally independent of the chosen origin, also in approximate theories. (Note that Eq. 52 is only valid for real wave functions.) It may thus be argued that the velocity gauge formulation of optical rotation should be preferred over the length gauge expression. It must be added, however, that the velocity gauge optical rotation calculated with an approximate theory suffers from a serious artifact: it predicts a nonvanishing optical rotation at zero frequency (at zero frequency, there is no light and the optical rotation must be zero). This can be traced back to the lack of gauge invariance and is rectified by the modified velocity gauge where the static limit simply is subtracted from the linear response function, that is, hhpi ; mi ii! is replaced by hhpi I mi ii!  hhpi I mi ii0 . The most widely used technique for ensuring origin invariance of magnetic properties, however, is based on so-called Gauge Including Atomic Orbitals (GIAOs, also known as London Atomic Orbitals [LAOs]). Belonging to the class of perturbation-dependent basis sets, GIAOs are obtained from any conventional atomic orbital basis set by multiplying each basis function with a complex phase factor that depends explicitly on an external uniform static magnetic field. In a finite GIAO basis set, the magnetic dipole moment operator depends on the chosen coordinate origin according to m!m

i Œ; H0   O 2„

(53)

which is identical to Eq. 51 in the limit of a complete basis set where the commutator of Eq. 47 is fulfilled. For fully variational approximate theories (DFT, HF, MCSCF), GIAOs thus remove the condition that the commutator must be fulfilled and the length gauge optical rotation becomes manifestly independent of origin. For approximate theories lacking variational orbital optimization (CI, CC), GIAOs are unable to remove the origin dependence of the length gauge optical rotation, since the equation of motion, Eq. 46, is not satisfied. The velocity gauge formulation with GIAOs is origin dependent in all approximate theories and, therefore, is never used. The greatest strength of GIAOs lies in origin invariant calculation of molecular properties involving static uniform magnetic fields, including static magnetic properties such as magnetizabilities and rotational g tensors, and nuclear magnetic shieldings as well as optical properties aided by a static magnetic field such as magneto-optical activity. The use of GIAOs makes it possible to calculate the total energy in the presence of a static uniform magnetic field in an origin-independent manner for variational as well as nonvariational approximate theories, implying that the molecular properties identified as energy derivatives are independent of origin as well. In addition, owing to their being atomic eigenfunctions of the angular momentum operator correct to first order in the applied magnetic field, GIAOs provide vastly improved basis set convergence of static magnetic properties.

Introduction to Response Theory

21

Effects of Nuclear Motion Within the clamped-nucleus Born–Oppenheimer approximation, electronic response functions are computed for fixed nuclear geometries. Just like the electronic ground state energy, the excitation energies and response functions thus become parametric functions of nuclear positions. For example, by solving the eigenvalue problem, Eq. 15, at a range of geometries, it becomes possible to compute excited state potential energy surfaces using Eq. 16 at each point. This, in turn, allows us to optimize excited state geometries using either a simple scan of the excited state potential energy surface or analytic gradients derived from Eq. 16. Since the ground state energy is required to compute the excited state energy in the response approach, calculations of excited state potential energy surfaces should only be carried out for geometries where the approximate wave function is valid. For single-reference methods such as HF and CC, one should only compute excitation energies (and thus excited state energies) for geometries not too far from the ground state equilibrium structure. As a consequence, these single-reference methods cannot be used to compute dissociative excited state surfaces, except at points close to the ground state equilibrium geometry. Multiconfigurational methods are required to describe such unbound excited states. For bound excited states, Eq. 16 also provides a viable path to excited state vibrational frequencies and wave functions. Along with those of the ground state, it then becomes possible to calculate 0–0 transition energies rather than the vertical excitation energies provided by the poles of the response functions. In addition, the vibrational structure of electronic bands becomes amenable to theoretical computations. Electronic ground state response functions are most often calculated at the (ground state) equilibrium geometry. The geometry optimization and response function evaluation may be performed at different levels of theory and using different basis sets. For example, it may be sufficient to optimize the geometry of a covalently bonded molecule at the DFT level of theory with a relatively small polarized basis set, while the response functions are evaluated at the CC level of theory (at the DFT equilibrium geometry) with a large polarized basis set augmented with diffuse functions to ensure the correct pole structure of the response functions. In other cases, it may be advantageous to use different DFT functionals and basis sets for the geometry optimization and property calculation. If high accuracy is required, vibrational effects must be taken into account. In a proper adiabatic Born–Oppenheimer treatment, the ground state wave function would be written as a product of an electronic and a vibrational wave function. The response of this wave function should then be computed and subsequently used to construct vibronic response functions. The sum-over-states expressions would include contributions from vibrational states in the electronic ground and excited states. Since each set of vibrational wave functions is tied to a specific electronic state within the adiabatic Born–Oppenheimer approximation, this approach is not feasible in practice. Hence, the electronic properties are considered as electronic ground state properties and therefore averaged in a vibrational state of the electronic ground state.

22

T.B. Pedersen

Computing the expectation value of the electronic property P(q), which may be a response function or a sum of response functions depending parametrically on the dimensionless normal coordinates q, in the vibrational state jv0 i of the electronic ground state, the vibrational correction becomes P D hv0 jP .q/j v0 i  P0

(54)

where P0 is the value of the electronic property at the equilibrium geometry. The analytic form of the dependence of P on normal coordinates normally is not known. Even if it were known, it would be too computationally demanding to evaluate the property on a grid of normal coordinates to fit the values to the analytic form. Instead, the property is expanded in a Taylor series. This implies that derivatives of the property with respect to the normal coordinates must be evaluated at the expansion point of the Taylor series. The property derivatives can themselves be expressed in terms of response functions (by considering the nuclear displacement terms of the Hamiltonian as perturbations), or they may be computed by numerical techniques analogous to the static properties of section “Static Response Functions.” The vibrational wave functions may be computed in the same manner as in the ground state theory. That is, they are computed on the basis of the harmonic approximation and often include anharmonic effects to first order in perturbation theory. In another approach, the vibrational problem is viewed as so-called mode dynamics and solved in a manner resembling electronic structure theory. This approach also makes it possible to compute the vibrational expectation value directly and only requires that an electronic structure program is available for computing energies and properties at specified nuclear geometries. The relatively small difference between vibrational energy levels implies that a range of vibrational states are occupied at finite temperature. Including anharmonic effects to first order, the temperature-dependent vibrational correction can be expressed as   1 X 1 @P X „!m knmm coth 4„ n !n @qn m 2kB T   X @2 P „! n coth C 14 2 @q 2k T B n n

P D 

(55)

where the property derivatives are to be evaluated at the equilibrium geometry, ¨n is the harmonic vibrational frequency of normal mode n, kB is the Boltzmann constant, T is the temperature, and the cubic (anharmonic) force constants are given by knmm D

@3 E0 2 @qn @qm

(56)

which is also evaluated at the equilibrium geometry (E0 is the electronic ground state energy surface, E0 D E0 .q//. The first term in Eq. 55 accounts for the

Introduction to Response Theory

23

anharmonicity of the potential energy surface, whereas the second term is purely harmonic and arises from the curvature of the property surface. If anharmonic effects are neglected, only the second term is included. Anharmonic effects can be as large as the harmonic contribution and should generally be included, in particular when the molecule contains low-frequency vibrational modes (such as torsional motions). Most quantum chemistry programs are able to compute the harmonic vibrational frequencies and normal coordinates, and some also provide cubic force constants (if not, these can be computed numerically). The property derivatives are most often computed by numerical differentiation. At zero temperature, only the vibrational ground state is populated and the zero-point vibrational correction (ZPVC) is obtained from Eq. 55 by taking the limit T ! 0 where the hyperbolic cotangent factors approach 1. Once the property and energy derivatives have been calculated, the temperature dependence of the vibrationally averaged property can be easily computed, including the ZPVC. Conformationally flexible molecules are characterized by multiple low-lying minima on the potential energy surface, each minimum defining a conformer of the molecule. All conformers contribute to the experimentally observed property and must be taken into account in theoretical calculations. In the simplest approach, provided that the barriers separating the minima are sufficiently large, the vibrationally corrected property can be computed along the lines discussed above for each conformer (letting the conformer structure play the role of equilibrium geometry in each case). The total (observed) property is then obtained as a Boltzmann average of the results for each conformer: P D

conformers X

hP ii Xi

(57)

i

Here, hPii and Xi are the vibrationally corrected property and mole fraction of conformer i, respectively. Note that the same formula can be applied to compute the average of the purely electronic contribution to the property (i.e., not vibrationally corrected). The mole fraction is computed according to exp .Gi =kB T / Xi D Pconformers   exp Gj =kB T j

(58)

where Gi is Gibbs free energy of conformer i, which, assuming the ideal gas approximation, is given by Gi D E0;i C 0;i C RT  T Si

(59)

where R is the universal gas constant. The electronic ground state energy of conformer i is denoted E0, i and "0, i is the zero-point vibrational energy of conformer i in the harmonic oscillator approximation. The entropy is computed from the electronic, vibrational, and rotational partition functions,

24

T.B. Pedersen

Si D kB T ln .Zel;i Zvib;i Zrot;i /

(60)

Zel;i D 1

(61)

vib gn;i exp . n;i =kB T /

(62)

rot gm;i exp . m;i =kB T /

(63)

Zvib;i D

X n

Zrot;i D

X m

The electronic partition function is unity, as only the electronic ground state is assumed populated. The vibrational energies of conformer i, –n,i , are computed in the harmonic oscillator approximation, which facilitates analytic summation over all vibrational levels. The rotational energy of conformer i, "m, i , is calculated within rot the rigid rotor approximation. The factors gvib n,i and gm,i are the degeneracies of the vibrational and rotational energy levels, respectively. Hence, the data needed to evaluate the Boltzmann average can be obtained from ground state calculations (electronic ground state energy at the equilibrium structure, harmonic vibrational energies, and rotational constants). The simple Boltzmann averaging relies on a range of approximations that tend to benefit from fortuitous cancellation of errors. In particular, the assumption that the conformers can be treated as independent with no couplings between their vibrational states is fragile, as barriers are often insufficient to warrant the assumption. A more correct description would be obtained by solving the vibrational problem for the entire ground state potential energy surface instead of using a firstorder anharmonic approximation in the vicinity of the minima. Calculation of the entire potential energy surface is, however, out of the question for all but the smallest systems. One way out might be to compute the nuclear motion using molecular dynamics and parametrized force fields, sampling the electronic property along the trajectory. Also this approach, however, may easily turn out to be too demanding in terms of the number of electronic structure calculations required.

Further Reading This section contains a short description of suggested references for further study of the concepts and techniques described above. The list of references is highly incomplete and should be considered as a mere starting point. The time-independent perturbation operators describing internal molecular interactions are taken from the Breit–Pauli Hamiltonian, which includes relativistic corrections to the molecular electronic Hamiltonian of Eq. 2. The Breit–Pauli Hamiltonian is discussed in more detail in the textbooks by Bethe and Salpeter (1957) and by Moss (1973). Appropriate operators for the interaction between charged particles and external electromagnetic fields are discussed in the paper

Introduction to Response Theory

25

by Fiutak (1963). The textbook by Craig and Thirunamachandran (1984) contains derivations of a long range of optical properties in terms of exact molecular states, while Barron’s book (1982) is highly recommended for its thorough discussion of optical activity phenomena. The expressions derived in these books may be translated into response functions amenable to calculation by means of approximate wave functions. Olsen and Jørgensen (1985, 1995) have derived and discussed response functions for exact, HF, and MCSCF wave functions in great detail, while Koch and Jørgensen (1990) presented a derivation for CC wave functions. The latter was modified by Pedersen and Koch (1997) to ensure proper symmetry of the response functions. Christiansen et al. (1998) have presented a derivation of dynamic response functions for variational as well as nonvariational wave functions that resembles the way in which static response functions are deduced from energy derivatives. Linear and higher-order response functions based on DFT have been presented by Sałek et al. (2002). Damped response theory has been discussed by Norman et al. (2001) in the context of HF and MCSCF response theory. Nonperturbative calculations of static magnetic properties at the HF level have been presented by Tellgren et al. (2008, 2009). Gauge invariance of HF and MCSCF response theory has been shown in terms of the equations of motion for the response functions by Olsen and Jørgensen (1985, 1995), whereas the lack of gauge invariance in CC response theory was demonstrated by Pedersen et al. (Pedersen and Koch 1997; Pedersen et al. 2001). The use of GIAOs to ensure origin invariance of static magnetic properties has been discussed by Helgaker et al. (Helgaker and Jørgensen 1991; Olsen et al. 1995) and for optical rotations by Bak et al. (1995). Different approaches to the calculation of vibrational corrections to response properties can be found in the work of Sauer and Pack (2000), Ruud et al. (2000), and Kongsted and Christiansen (2006). The Boltzmann averaging procedure for conformationally flexible molecules has been critically reviewed by Crawford and Allen (2009). Mort and Autschbach (2008) have proposed an approach based on a decoupling of hindered rotations from the remaining (high-frequency) vibrational modes, which allows for a separate calculation of the hindered rotations without invoking the harmonic approximation. Acknowledgments The author wishes to thank Profs. Michał Jaszu´nski, Antonio Rizzo, T. Daniel Crawford, and Trygve Helgaker for commenting on the manuscript.

Bibliography Bak, K. L., Hansen, A. E., Ruud, K., Helgaker, T., & Olsen, J. (1995). Ab initio calculation of electronic circular dichroism for trans – Cyclooctene using London atomic orbitals. Theoretica Chimica Acta, 90, 441–458. Barron, L. D. (1982). Molecular light scattering and optical activity. Cambridge, MA: Cambridge University Press.

26

T.B. Pedersen

Bethe, H. A., & Salpeter, E. E. (1957). Quantum mechanics of one- and two-electron atoms. New York: Academic. Christiansen, O., Jørgensen, P., & Hättig, C. (1998). Response functions from Fourier component variational perturbation theory applied to a time-averaged quasienergy. International Journal of Quantum Chemistry, 68, 1–52. Craig, D. P., & Thirunamachandran, T. (1984). Molecular quantum electrodynamics. London: Academic. Crawford, T. D., & Allen, W. D. (2009). Optical activity in conformationally flexible molecules: A theoretical study of large-amplitude vibrational averaging in (R)-3-chloro-1-butene. Molecular Physics, 107, 1041–1057. Fiutak, J. (1963). The multipole expansion in quantum theory. Canadian Journal of Physics, 41, 12–20. Helgaker, T., & Jørgensen, P. (1991). An electronic Hamiltonian for origin independent calculations of magnetic properties. Journal of Chemical Physics, 95, 2595–2601. Koch, H., & Jørgensen, P. (1990). Coupled cluster response functions. Journal of Chemical Physics, 93, 3333–3344. Kongsted, J., & Christiansen, O. (2006). Automatic generation of force fields and property surfaces for use in variational vibrational calculations of anharmonic vibrational energies and zero-point vibrational averaged properties. Journal of Chemical Physics, 125, 124108. Mort, B. C., & Autschbach, J. (2008). A pragmatic approach for the treatment of Hindered rotations in the vibrational averaging of molecular properties. European Journal of Chemical Physics and Physical Chemistry, 9, 159–170. Moss, R. E. (1973). Advanced molecular quantum mechanics. London: Chapman and Hall. Norman, P., Bishop, D. M., Jensen, H. J. A., & Oddershede, J. (2001). Near-resonant absorption in the time-dependent self-consistent field and multiconfigurational self-consistent field approximations. Journal of Chemical Physics, 115, 10323–10334. Olsen, J., & Jørgensen, P. (1985). Linear and nonlinear response functions for an exact state and for an MCSCF state. Journal of Chemical Physics, 82, 3235–3264. Olsen, J., & Jørgensen, P. (1995). Time-dependent response theory with applications to selfconsistent field and multiconfigurational self-consistent field wave functions. In D. R. Yarkony (Ed.), Modern electronic structure theory (Vol. 2, pp. 857–990). Singapore/River Edge: World Scientific. Olsen, J., Bak, K. L., Ruud, K., Helgaker, T., & Jørgensen, P. (1995). Orbital connections for perturbation-dependent basis sets. Theoretica Chimica Acta, 90, 421–439. Pedersen, T. B., & Koch, H. (1997). Coupled cluster response functions revisited. Journal of Chemical Physics, 106, 8059–8072. Pedersen, T. B., Fernández, B., & Koch, H. (2001). Gauge invariant coupled cluster response theory using optimized nonorthogonal orbitals. Journal of Chemical Physics, 114, 6983–6993. Ruud, K., Astrand, P.-O., & Taylor, P. R. (2000). An efficient approach for calculating vibrational wave functions and zero-point vibrational corrections to molecular properties of polyatomic molecules. Journal of Chemical Physics, 112, 2668–2683. Sałek, P., Vahtras, O., Helgaker, T., & Agren, H. (2002). Density-functional theory of linear and nonlinear time-dependent molecular properties. Journal of Chemical Physics, 117, 9630–9645. Sauer, S. P. A., & Packer, M. J. (2000). The Ab initio calculation of molecular properties other than the potential energy surface. In P. R. Bunker & P. Jensen (Eds.), Computational molecular spectroscopy (pp. 221–252). London: Wiley. Tellgren, E. I., Soncini, A., & Helgaker, T. (2008). Nonperturbative ab initio calculations in strong magnetic fields using London orbitals. Journal of Chemical Physics, 129, 154114. Tellgren, E. I., Helgaker, T., & Soncini, A. (2009). Non-perturbative magnetic phenomena in closed-shell paramagnetic molecules. Physical Chemistry Chemical Physics, 11, 5489–5498.

Intermolecular Interactions Alston J. Misquitta

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Two-Body Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Methods for the Two-Body Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supermolecular Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multipole Expansion for the Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Body Contributions to the Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Dispersion Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polarization in Organic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal Structure Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 8 8 9 16 20 21 22 23 23 27 28 28 30 32 33 33 34 35

Abstract

Van der Waals interactions determine a number of phenomena in the fields of physics, chemistry and biology. As we seek to increase our understanding of

A.J. Misquitta () TCM Group, Cambridge, UK School of Physics, Queen Mary, University of London, London, UK e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_6-2

1

2

A.J. Misquitta

physical systems and develop detailed and more predictive theoretical models, it becomes even more important to provide an accurate description of the underlying molecular interactions. The goal of this chapter is to describe recent developments in the theory of intermolecular interactions that have revolutionised the field due to their comparatively low computational costs and high accuracies. These are the symmetry-adapted perturbation theory based on density functional theory (SAPT(DFT)) for interaction energies and the Williams– Stone–Misquitta (WSM) method for molecular properties in distributed form. These theories are applicable to systems of small organic molecules containing as many as 30 atoms each and have demonstrated accuracies comparable to the best electronic structure methods. We also discuss the numerical aspects of these theories and recent applications which demonstrate the range of problems that can now be approached with these accurate ab initio methods.

Introduction Intermolecular, or van der Waals, interactions are responsible for a wide variety of phenomena in the fields of physics, chemistry, and biology. The thermodynamic properties of gases, liquids, and solids depend on these interactions. In fact, many liquids and many solids would not exist without van der Waals forces. Amongst other properties that depend strongly on the van der Waals interactions are microwave and infrared spectra of molecular complexes and bulk phases. Due to their long-range nature, van der Waals forces determine the entrance channels for chemical reactions. Some of the more exotic systems in which van der Waals interactions play an important role are Bose–Einstein condensates (BECs) and helium nanodroplets (Chang et al. 2003; Xu et al. 2003). In biological systems, van der Waals interactions are particularly important, subtle, and often hard to model. The stability of DNA and RNA arise, in part, from the stacking energy (Hobza and Sponer 2002), which is determined by this interaction. One of the important problems in which van der Waals interactions play an important role is the problem of protein folding (Lehninger 1970). This chapter is an introduction to the field of intermolecular interaction and to the modern ab initio electronic structure methods – primarily those based on perturbation theory – that have been developed to study them. We will be mainly concerned with applications to small organic molecules for which accuracies of the order of a kJ mol1 or less are sufficient. High-accuracy calculations on small dimers can be orders of magnitude more accurate, but these are the subject of a specialist review (see Szalewicz et al. 2005 for a review and references). Nor are we concerned with empirical methods. Our focus will be on first principles methods for the interactions of closed-shell systems in the non-relativistic limit. In the last decade, ab initio methods have been used to successfully model the structure of liquid water (Bukowski et al. 2007) studied the interactions between DNA base tetramers (Fiethen et al. 2008) and predicted the crystal structure of an organic

Intermolecular Interactions

3

molecule (Misquitta et al. 2008b). The goal of this chapter is to describe the main theoretical developments that have been responsible for these applications.

Definition of the Interaction Energy The interaction energy of a cluster of N interacting rigid molecules is defined to be Eint D EABC :::  EA  EB  EC     :

(1)

Here EABC : : : is the energy of the cluster and EX ; X D A; B; C; etc., is the energy of molecule X. Non-rigid molecules undergo a deformation with   in the  cluster,  an associated deformation energy cost: ıEX D EX xX  EX xX0 , where x0X is the geometry of monomer X in isolation and x*X is the geometry in the cluster. This deformation energy cost should be included as part of the interaction energy defined above, but since this chapter is mainly concerned with the interactions of rigid molecules, we will assume that the deformation energies ıEX are obtained in a suitable manner, and concern ourselves only with the interaction energy defined by Eq. 1. Eint can be evaluated directly, but for computational efficiency as well as physical interpretation it is worthwhile to partition the N-body interaction energy into contributions from dimers, trimers, and so on. This leads to an exact reformulation of Eint that is known as the many-body expansion: Eint .ABC    / D

X Xi jri  rj j

(7)

Here, MI and ZI denote the mass and atomic number of nucleus I ; me and e are the electronic mass and elementary charge, and 0 is the permittivity of vacuum. The nabla operators r I and r i act on the coordinates of nucleus I and electron i , respectively. The total wave function ˆ.r; RI t / simultaneously describes the motion of both electrons and nuclei.

6

C. Lorenz and N.L. Doltsinis

The Born–Oppenheimer approximation (Kołos 1970; Kutzelnigg 1997; Doltsinis and Marx 2002b) separates nuclear and electronic motion based on the assumption that the much faster electrons adjust their positions instantaneously to the comparatively slow changes in nuclear positions. The electronic problem is then reduced to the time-independent (electronic) Schrödinger equation for clamped nuclei, Hel .rI R/‰k .rI R/ D Ek .R/‰k .rI R/;

(8)

where Hel .rI R/ is the electronic Hamiltonian, Hel .r; R/ D T .r/ C Vnn .R/ C Vne .r; R/ C Vee .r/;

(9)

and ‰k .rI R/ is the electronic wave function of state k. Meanwhile, nuclear motion is described by ŒT .R/ C Ek .R/ k D i „

@ k @t

(10)

with the nuclear wave function k .R; t / evolving on the potential energy surface Ek .R/ of the electronic state k. The total wave function is then the direct product of the electronic and the nuclear wave function, ˆ.r; RI t / D ‰k .r; R/k .R; t /

(11)

In the classical limit (Doltsinis and Marx 2002b), the nuclear wave equation (10) is replaced by Newton’s equation of motion R I D r I Ek MI R

(12)

For a great number of physical situations the Born–Oppenheimer approximation can be safely applied. On the other hand, there are many important chemical phenomena such as charge transfer and photoisomerization reactions, whose very existence is due to the inseparability of electronic and nuclear motion. Inclusion of nonadiabatic effects is beyond the scope of this chapter and the reader is referred to the literature (e.g., Refs. Doltsinis and Marx 2002b; Doltsinis 2006) for more details. The above approximations form the basis of conventional molecular dynamics, Eqs. (22) together with (8) being the working equations. Thus, in principle, a classical trajectory calculation merely amounts to integrating Newton’s equations of motion (22). In practice, however, this deceptively simple task is complicated by the fact that the stationary Schrödinger equation (8) cannot be solved exactly for any many-electron system. The potential energy surface therefore has to be approximated using ab initio electronic structure methods or empirical interaction potentials (the so-called force-field molecular dynamics (Sutmann 2002; Allen and Tildesley 1987)). The former approach, usually referred to as ab initio molecular dynamics

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

7

(AIMD), will be the subject of section “Ab Initio Molecular Dynamics,” while the latter – force-field molecular dynamics – will be discussed in section “Classical Molecular Dynamics.”

Ab Initio Molecular Dynamics In the following, we shall focus on first-principles molecular dynamics methods. Due to the high computational cost associated with ab initio electronic structure calculations of large molecules, computation of the entire potential energy surface prior to the molecular dynamics simulation is best avoided. A more efficient alternative is the evaluation of electronic energy and nuclear forces “on the fly” at each step along the trajectory.

Born–Oppenheimer Molecular Dynamics In the so–called Born-Oppenheimer implementation of such a scheme (Marx and Hutter 2000), the nuclei are propagated by integration of Eq. (22), where the exact energy Ek is replaced with the eigenvalue, EQ k , of some approximate electronic Hamiltonian, HQ el , which is calculated at each time step. For the electronic ground state, i.e., k D 0, the use of Kohn–Sham (KS) density functional theory (Parr and Yang 1989; Dreizler and Gross 1990) has become the method of choice. Car–Parrinello Molecular Dynamics In order to further increase computational efficiency, Car and Parrinello have introduced a technique to bypass the need for wave function optimization at each molecular dynamics step (Car and Parrinello 1985; Marx and Hutter 2000). Instead, the molecular wave function is dynamically propagated along with the atomic nuclei according to the equations of motion R I D r I h‰k jHQ el j‰k i MI R

(13)

X ı i R i D  ? h‰k jHQ el j‰k i C ij ı i j

j;

(14)

where the KS one-electron orbitals i are kept orthonormal by the Lagrange multipliers ij . These are the Euler–Lagrange equations @L d @L D ; .q D RI ; dt @qP @q

? i /

(15)

for the Car–Parrinello Lagrangian (Car and Parrinello 1985) LD

X1 I

2

P2 C MI R I

X1 i

2

i h P i j P i i  h‰k jHQ el j‰k i C

X

ij .h

ij

ji

 ıij /

ij

(16)

8

C. Lorenz and N.L. Doltsinis

that is formulated here for an arbitrary electronic state ‰k , an arbitrary electronic Hamiltonian HQ el , and an arbitrary basis (i.e., without invoking the Hellmann– Feynman theorem).

Classical Molecular Dynamics While first-principles molecular dynamics simulations deal with the electrons in a system, this results in a large number of particles and inevitably requires the use of quantum mechanics. Therefore, the calculations become significantly timeconsuming. Classical molecular dynamics ignore electronic motions and calculate the energy of a system as a function of the nuclear positions only and therefore are used to simulate larger, less detailed systems for larger time scales. The successive configurations of the system are generated by solving the differential equations that constitute Newton’s second law (Eq. 22): d 2 XI FXI D dt 2 MI

(17)

This equation describes the motion of particle I of mass MI along one dimension (XI ), where FXI is the force on the particle in that dimension. The solution of these differential equations results in a trajectory that specifies how the positions and velocities of the particles in the system vary with time. In realistic models of intermolecular interactions, the force on particle I changes whenever particle I changes its position or whenever another atom with which particle I interacts changes its position. Therefore, the motions of all the particles are coupled together, which results in a many-body problem that cannot be solved analytically. Therefore, finite difference methods are used to integrate the equations of motion. Generally, the integration of Eq. (17) is broken into consecutive steps that are conducted at different times t that are separated by increments of ıt , which is generally referred to as the time step. First, the total force on each particle in the system at time t is calculated as the vector sum of its interactions with other particles. Then, assuming the force is constant over the course of the time step, the accelerations of the particles are calculated, which are then combined with positions and velocities of the particles at time t to determine the positions and velocities at time t C ıt . Finally, the forces on the particles in their new positions are determined, and then new accelerations, positions, and velocities are determined at t C 2ıt and so on. A common approach in the various finite difference methods used to integrate the equations of motions for classical molecular dynamics simulations is that it is assumed that the positions, velocities, and accelerations (as well as all other dynamic properties) can be approximated using Taylor series expansions:

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

9

1 1 1 R.t C ıt / D R.t / C ıt V.t / C ıt 2 A.t / C ıt 3 B.t / C ıt 4 C.t / C : : : 2 6 24

(18)

1 1 V.t C ıt / D V.t / C ıt A.t / C ıt 2 B.t / C ıt 3 C.t / C : : : 2 6

(19)

1 A.t C ıt / D A.t / C ıt B.t / C ıt 2 C.t / C : : : 2

(20)

P is the velocity, A (D R) R is the acceleration, and where R is the position, V (D R) B and C are the third and fourth derivatives of the positions with respect to time, respectively.

Verlet Algorithm One of the most widely used finite difference methods in classical molecular dynamics simulations is the Verlet algorithm (Verlet 1967). In the Verlet algorithm, the positions and accelerations at time t and the positions from the previous time step R.t  ıt / are used to calculate the updated positions R.t C ıt / using the equation: R.t C ıt / D 2R.t /  R.t  ıt / C ıt 2 A.t /:

(21)

The accelerations can be conveniently calculated using Newton’s equation (17), where the forces are obtained by differentiating the energy with respect to the nuclear positions, AD

F rE D M M

(22)

While the velocities do not explicitly appear in Eq. (21), they can be calculated from the difference in position over the entire time step: V.t / D jR.t C ıt /  R.t  ıt /j=2ıt

(23)

or the difference in position over a half time step (t C 12 ıt ): 1 V.t C ıt / D jR.t C ıt /  R.t /j=ıt 2

(24)

The fact that the velocities are not explicitly represented in the Verlet algorithm is one of the drawbacks to this method in that no velocities are available until the positions have been determined at the next time step. Also, in order to calculate the position of particles at t D ıt , it is necessary to determine the positions at t D ıt since the algorithm requires the position at time t  ıt to calculate the position at time t C ıt . Often, this drawback is overcome by using the Taylor series to calculate

10

C. Lorenz and N.L. Doltsinis

R.ıt / D R.0/ıt V.0/C 12 ıt 2 A.t /jC: : :. A final drawback of the Verlet algorithm is that there may be a loss of precision in the resulting trajectories that result from the fact that the positions are calculated by adding a small term (ıt 2 A.t / to the difference of two larger terms (2R.t / and R.t  ıt /) in equation (21).

“Leap-Frog” Algorithm In an attempt to improve upon the original Verlet algorithm, several variations have been developed. The leap-frog algorithm (Hockney 1970) is one of the variations that use the following equations to update the positions: 1 R.t C ıt / D R.t / C ıt V.t C ıt /; 2

(25)

1 1 V.t C ıt / D V.t  ıt / C ıt A.t /: 2 2

(26)

and the velocities:

In the leap-frog algorithm, the velocities V.t C 12 ıt / are first calculated from the velocities at time t  12 ıt and the accelerations at time t using Eq. (25). Then the positions R.t C ıt / are calculated from the velocities V.t C 12 ıt / and the positions R.t / using Eq. (26). The algorithm gets its name from the fact that the velocities are calculated in a manner such that they “leap-frog” over the positions to give their values t  12 ıt . Then the positions are calculated such that they “leap-frog” over the velocities, and then the algorithm continues. The “leap-frog” algorithm improves upon the standard Verlet algorithm in that the velocity is explicitly included in the calculations and also the “leap-frog” algorithm does not require the calculation of the differences of large numbers so the precision of the calculation should be improved. However, the fact that the calculated velocities and positions are not synchronized in time results in the fact that the kinetic energy contribution to the total energy cannot be calculated for the time at which the positions are defined. In response to this shortcoming in the “leapfrog” algorithm, a formalism to calculate the velocities at time t has been developed that follows V.t / D ŒV.t C

ıt ıt / C V.t  /=2 2 2

(27)

Velocity Verlet Algorithm The velocity Verlet method (Swope et al. 1982), which is a variation of the standard Verlet method, calculates the positions, velocities, and accelerations at the same time by using the following equations: 1 R.t C ıt / D R.t / C ıt V.t / C ıt 2 A.t / 2

(28)

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

1 V.t C ıt / D V.t / C ıt ŒA.t / C A.t C ıt /: 2

11

(29)

The velocity Verlet method is a three-stage algorithm because the calculation of the new velocities (Eq. 29) requires both the acceleration at time t and at time t C ıt . Therefore, first, the positions at t Cıt are calculated using Eq. (28) and the velocities and accelerations at time t . The velocities at time t C 12 ıt are then calculated using 1 1 V.t C ıt / D V.t / C ıt A.t /: 2 2

(30)

Then the forces are computed from the current positions, which results in being able to calculate A.t C ıt /. Then the final step consists of calculating the velocities at time t C ıt using: 1 1 V.t C ıt / D V.t C ıt / C ıt A.t C ıt /: 2 2

(31)

Therefore, the velocity Verlet allows for the velocities and positions to be calculated in a time-synchronized manner and thus allows for the kinetic energy contribution of the total energy. Also, the precision of the results will be improved upon those from the standard Verlet algorithm as there are no differences of large numbers within the formalism of the method. The selection of the best time integration method for a given problem and the size of the time step to use will be discussed in section “Setting the Timestep.”

Hybrid Quantum/Classical (QM/MM) Molecular Dynamics The ab initio and classical simulation techniques discussed in the previous sections can be viewed as complementary. While AIMD is capable of dealing with electronic processes such as chemical reactions, charge transfer, and electronic excitations, its applicability is limited to systems of modest size, precluding its use in complex, large-scale biochemical simulations. Classical MD, on the other hand, can describe much larger systems on longer time scales but misses any of the abovementioned electronic effects, e.g., bond breaking and formation. The basic idea of the QM/MM approach is to combine the strengths of the two methods treating a chemically active region at the quantum level and the environment using molecular mechanics (i.e., a force field). There are several excellent review articles on the QM/MM method in the literature (Senn and Thiel 2009; Thiel 2009; Groenhof 2013).

Partitioning Schemes The entire system, S, is partitioned into a chemically active inner region, I, and a chemically inert outer region, O. If the border between these regions cuts through chemical bonds, so-called link atoms, L, are usually introduced to cap the inner region (see section Bonds Across the QM/MM Boundary).

12

C. Lorenz and N.L. Doltsinis

Subtractive Scheme S In a subtractive scheme, the total energy, EQM=MM , of the entire system, I;L I;L S S EQM=MM D EMM C EQM  EMM

(32)

is calculated from three separate energy contributions: (i) the MM energy of the S entire system, EMM , (ii) the QM energy of the active region (including any link I;L I;L atoms), EQM , and (iii) the MM energy of the active region EMM . The role of the third term in Eq. (32) is to avoid double counting and to correct for any artifacts caused by the link atoms. For the latter to be effective, the force field has to reproduce the quantum-mechanical forces reasonably well in the link region. Additive Scheme In an additive scheme, the total energy of the system is given by I;L I;O S O EQM=MM D EMM C EQM C EQMMM

(33)

The difference to the subtractive scheme is that here a pure MM calculation is performed for only the outer region and the interaction between QM and MM regions is achieved by an explicit coupling term, I;O bond vdW el EQMMM D EQMMM C EQMMM C EQMMM

(34)

bond vdW el , EQMMM , and EQMMM are bonded, van der Waals, and electrowhere EQMMM static interaction energies, respectively. The simplest way to treat electrostatic interactions between the I and O subsystems is to assign fixed electric charges to all I atoms (mechanical embedding). In this case the QM problem is solved for the isolated subsystem I without taking into account the effects of the surrounding atomic charges in O. The majority of implementations use an electrostatic embedding scheme in which the MM point charges of region O are incorporated in the QM Hamiltonian through a QM-MM coupling term,

el HO QMMM D

n X X i

X X q˛ ZI q˛ C jr  R˛ j I 2ICL ˛2O jRI  R˛ j ˛2O i

(35)

where q˛ are the MM point charges at positions R˛ (all other symbols as defined in “section Born–Oppenheimer Approximation”). In this way, the electronic structure of the QM region adjusts to the moving MM charge distribution. A problem that arises when an MM point charge is in close proximity to the QM electron cloud is overpolarization of the latter, sometimes referred to as a “spill-out” effect. This can be avoided by modifying the Coulomb potential in the first term of Eq. (35) at short range (see, e.g., Laio et al. 2002).

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

13

At present, in all commonly used partitioning schemes, the partitions remain fixed over time, i.e., an MM atom cannot turn into a QM atom and vice versa. This can present a serious limitation, for instance, in the case of solvent diffusion through the chemically active region. A number of adaptive partitioning methods have been proposed to remedy this problem (Kerdcharoen et al. 1996; Hofer et al. 2005; Kerdcharoen and Morokuma 2002; Heyden et al. 2007; Bulo et al. 2009; Nielsen et al. 2010; Pezeshki and Lin 2011; Pezeshki et al. 2014; Waller et al. 2014); however, the computational overhead is significant and physical conservation laws are not strictly satisfied.

Bonds Across the QM/MM Boundary Partitioning the total system into QM and MM regions in such a way that cuts chemical bonds is best avoided. However, in many cases this is inevitable. Then one has to make sure that any atoms participating in chemical reactions are at least three bonds away from boundary. Furthermore, it is preferable to cut a bond that is unpolar and not part of a conjugated chain. Link Atoms Cutting a single covalent bond will create a dangling bond which must be capped by a so-called link atom, and in most applications a hydrogen atom is chosen. In the QM calculation, the atoms of region I together with the link atoms L are treated as an isolated molecule in the presence of the point charges of the environment O. The original QM–MM bond, cut by the partitioning, is only treated at the MM level. Boundary Atoms Boundary atom schemes have been developed to avoid the artifacts introduced by a link atom. The boundary atom appears as a normal MM atom in the MM calculation while carrying QM features to saturate the QM–MM bond and to mimic the electronic properties of the MM side. The QM interactions are achieved by placing a pseudopotential at the position of the boundary atom, parametrized to reproduce electronic properties of a certain chemical end group, e.g., a methyl group in the case of a cut C–C bond. Among the various flavors that have been proposed, the pseudobond method for first principles QM calculations (Zhang et al. 1999; Zhang 2005, 2006) and the pseudopotential approach for plane-wave DFT (Laio et al. 2002) are the most relevant in the present context. Frozen Localized Orbitals The basic idea behind the various frozen orbital methods (Warshel and Levitt 1976; Théry et al. 1994; Monard et al. 1996; Assfeld and Rivail 1996; Assfeld et al. 1998; Ferré et al. 2002; Fornili et al. 2003, 2006a, b; Sironi et al. 2007; Loos and Assfeld 2007; Philipp and Friesner 1999; Murphy et al. 2000; Gao et al. 1998; Amara et al. 2000; Garcia-Viloca and Gao 2004; Pu et al. 2004a, b, 2005; Jung et al. 2007; Jensen et al. 1994; Day et al. 1996; Kairys and Jensen 2000; Gordon et al. 2001; Nemukhin et al. 2002, 2003; Grigorenko et al. 2002) is to saturate the cut QM– MM bond by placing on either the MM or the QM atom at the boundary localized

14

C. Lorenz and N.L. Doltsinis

orbitals that have been determined in a prior quantum-mechanical SCF calculation on a model molecule containing the bond under consideration. To preserve the properties of the bond, the localized orbitals are then kept fixed in the subsequent QM/MM calculation. Different flavors are the local SCF (LSCF) method (Théry et al. 1994; Monard et al. 1996; Assfeld and Rivail 1996; Assfeld et al. 1998; Ferré et al. 2002), extremely localized molecular orbitals (ELMOs) (Fornili et al. 2003, 2006a; Sironi et al. 2007), frozen core orbitals (Fornili et al. 2006b), optimized LSCF (Loos and Assfeld 2007), frozen orbitals (Philipp and Friesner 1999; Murphy et al. 2000), generalized hybrid orbitals (Gao et al. 1998; Amara et al. 2000; GarciaViloca and Gao 2004; Pu et al. 2004a, b, 2005; Jung et al. 2007), and effective fragment potentials (EFP) (Jensen et al. 1994; Day et al. 1996; Kairys and Jensen 2000; Gordon et al. 2001; Nemukhin et al. 2002, 2003; Grigorenko et al. 2002). Of the three types of boundary treatment, the link atom method is the simplest both conceptually and in practice and is hence the most widely used. The boundary atom and in particular the frozen orbital methods can potentially achieve higher accuracy but require careful a priori parametrization and bear limitations on transferability (Senn and Thiel 2009).

Coarse-Grained Molecular Dynamics A large number of important problems in fields that are often studied using molecular dynamics simulations (i.e., soft condensed matter physics, structural biology, chemistry, and materials science) take place over a time span of microseconds to seconds and distances of few hundred nanometers to a few microns. However, these time and length scales are still unattainable via quantum or atomistic forcefield molecular dynamics methods despite significant computational hardware advances (Shirts and Pande 2000; Mervis 2001; Reed 2003) and the development of increasingly powerful software (MacKerell et al. 1998; Lindahl and van der Spoel 2001; Wang et al. 2004a; Phillips et al. 2005). Therefore, one approach that has been utilized in order to be able to study these complex problems is to reduce the computational demand of the simulation by reducing the number of atoms represented and therefore the degrees of freedom of the simulated system. This procedure of reducing the number of atoms represented in a system is done by grouping atoms together and representing them as a single interaction site and is generally referred to as “coarse graining” of the system. Figure 1 shows a comparison of the atomistic, united-atom, and coarse-grained representation. The “bead-spring” coarse-grained model of polymer chains that was created by Kremer and Grest in 1990 has served as the foundation for many of the coarse-grained models that have been developed for a wide range of phenomena (at the current date this paper has been cited over 1440 times) including various studies of polymers and biomolecules including DNA solutions. Many of the more recent coarse-grained models have been developed for biological macromolecules since there are many examples of interesting biophysical phenomena that occur at large length and time scales. The most widely used coarse-grained models for

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

All-atom representation

15

United-atom representation

coarse-grained representation Fig. 1 Atomistic, united-atom, and coarse-grained representations of organic molecules

biological systems include the generic model of Lipowsky et al. (Goetz et al. 1999; Shillcock and Lipowsky 2002), the solvent-free model of Deserno et al. (Cooke et al. 2005), and the specific models of the Klein group (Shelley et al. 2001), the Voth group (which is called the multi-scale coarse-grained model) (Izvekov and Voth 2005, 2006), the Essex group (called the ELBA force-field) (Orsi et al. 2011), and the Marrink group (called the MARTINI force-field) (Marrink et al. 2007). The above coarse-grained models have generally been developed for lipid membranes; however, there are also coarse-grained force fields for proteins (as reviewed in Tozzini (2005) and some more recent examples Betancourt and Omovie (2009), Bereau and Deserno (2009), de Jong et al. (2013), and Kar et al. (2013)), carbohydrates (Lopez et al. 2009), and DNA (Tepper and Voth 2005; Khalid et al. 2008). When developing a coarse-grained model for a system, there are two important decisions to be made: (1) how many atoms to combine (coarse grain) into a single interaction site and (2) how to parameterize the coarse-grained force field. In deciding the number of atoms to combine into a single interaction site, one must consider the obvious trade-off of how much detail are you able to sacrifice in order to simulate larger length and/or time scale phenomena and still be able to actually accurately model the phenomena of interest. The least amount of coarse graining that has been used is represented by what is called a “united-atom” representation of a molecule where all “heavy” atoms (generally all non-hydrogen elements in a molecule) are represented and the “light” (i.e., hydrogen) atoms are grouped with the heavy atom to which they are bonded into one interaction site. United-atom versions of many of the popular all-atom force fields listed are in section “Classical Force Fields” exist and have been successfully used in several studies. In addition to united-atom models, there are several existing coarse-graining methods that will combine a different number of atoms together into one interaction site.

16

C. Lorenz and N.L. Doltsinis

A mapping function C is required to develop a description of the desired system in terms of the coarse-grained interaction sites. This mapping function uses a set of n atomic coordinates in the atomistic system (rn D r1 ; : : : ; rn ) and maps them to a unique configuration of coordinates for N interaction sites (RN D R1 ; : : : ; RN ) in the coarse-grained system. Then the mapping function generally has the form:

R D Cr

(36)

where R and r are vectors of length 3N and 3n, respectively, and C is a matrix with dimensions 3N  3n and contains the elements cI i that are used to link the descriptions of each coarse-grained interaction site, I , to the atoms in the all-atom configuration it represents: RI D

n X

cI i ri :

(37)

iD1

The most common form of mapping is where the coarse-grained interaction sites are mapped to the centerPof mass of the atoms within the all-atom configuration, and therefore cI i D mi = niD1 mi . In general, coarse-grained systems are governed by similar potential terms as are found in atomistic models such as nonbonded terms (both pairwise interactions and electrostatic interactions) and bond- stretching terms, and then in more sophisticated models even angle and dihedral terms will be included as well. Generally, all specific models are parameterized based on comparison to atomistic simulations and/or detailed experimental data. The “top-down” approach generally parameterizes the interactions without explicit consideration of a more detailed model. The interactions are often determined using physicochemical intuition or generic physical principles or certain emergent structural or thermodynamic properties that are observed on even larger scales. The most popular example of a “top-down” coarse-grained model is the Martini model (Marrink et al. 2004; de Jong et al. 2013; Lopez et al. 2009). The Martini model provides transferable potentials that describe effects of hydrophobic, van der Waals, and electrostatic interactions between sites, which are characterized based on their polarity and charge. The Martini potentials have been optimized to reproduce the partitioning of model compounds between aqueous and hydrophobic environments (Marrink et al. 2004, 2007). For more information about developing Martini force fields for molecules which have as of yet to be modeled, please refer to this website (Martini http://md.chem.rug.nl/cgmartini/ index.php/parametrzining-new-molecule). Effective coarse-grained potentials have also been extracted from atomistic simulations (often referred to as “bottom-up” approaches) using a variety of different methods including, but not limited to, those that we briefly introduce in this chapter as they are currently among the most popular methods: inverse Iterative Boltzmann Inversion approaches (M uller-Plathe 2002; Reith et al. 2003), Monte

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

17

Carlo schemes (Lyubartsev 2005; Elezgaray and Laguerre 2006) and force matching (Ercolessi and Adams 1994; Izvekov and Voth 2005, 2006; Lu et al. 2013). Iterative Boltzmann Inversion (IBI) is one of the most commonly used bottomup approaches to parameterizing coarse-grained potentials because it utilizes a straightforward and generally robust algorithm. In general, this method is used to construct nonbonded pairwise additive potential functions between every type of interaction site in a given system. The potential functions are iteratively refined in order to match target radial distribution functions (see section “Spatial Distribution Functions”) that are determined from atomistic simulations, whose trajectories have been mapped into the coarse-grained representation of the systems. The general philosophy behind the IBI method is similar to the philosophy behind deriving atomistic pair potentials which have been used to reproduce structure factors determined from neutron scattering experiments (Schommers 1973; Soper 1996). In the iterative process of refining the pair potentials, the initial guesses for each of these interactions are calculated from the potential of mean force, UPMF , from the target radial distribution functions, g.r/: UPMF .r/ D kB T lnŒg.r/

(38)

where kB and T are the Boltzmann constant and temperature, respectively. The intramolecular potential energies are calculated using the observed distributions Pq of each type of interaction (i.e., bond lengths, bond angles, and dihedral angles) from the reference atomistic simulations using an analogous approach as used for the pair potentials in Eq. (38): U .q/ D kB T lnŒPq .q/

(39)

where q is the bond length, bond angle, or dihedral angle, for the various terms of the intramolecular potential energy. Then using the initial guess for the nonbonded pair potentials (Eq. (38)) and these intramolecular potential energies, a coarse-grained simulation is performed and radial distribution functions gCn G .r/ are calculated and compared to the target radial distribution functions gt .r/ to update the pair potential energy functions for the next iteration of the algorithm: U .nC1/ .r/ D U n .r/ C  U .n/ .r/ D U n .r/ C kB T ln

gCn G .r/ gt .r/

(40)

where  scales the potential update, and should be chosen in order to ensure stability of the algorithm. This iterative process is repeated until the measured radial distribution functions in the CG simulations are within a user-defined tolerance of the target radial distribution functions. This method has been automated in a software package called VOTCA (http://www.votca.org/home) that can be used in combination with the popular molecular dynamics simulation packages GROMACS (Spoel et al. 2005; http://www.gromacs.org/), LAMMPS (http://lammps.sandia. gov/; Plimpton 1995), and DL-POLY (Smith et al. 2002; http://www.cse.scitech. ac.uk/ccg/software/DL_POLY/).

18

C. Lorenz and N.L. Doltsinis

The Inverse Monte Carlo (IMC) algorithm is another iterative method that corrects a guessed interaction potential to reproduce a target radial distribution function. However, the IMC method applies a correction to the potential which is determined from statistical mechanical arguments that account for the fact that small changes in the potential energies at a given distance r can lead to variation in the radial distribution functions at all distances. Generally speaking, one can summarize this approach by stating that the Hamiltonian (potential energy) H of the system depends on a series of parameters 1 ; 2 ; : : : ; M , and the desire is to reproduce canonical averages of a given measurable < S1 >; < S2 >; : : : ; < SM > (e.g., radial distribution functions). In Inverse Monte Carlo, the desire is to calculate the values of the parameters used to calculate the potential energy M from the canonical averages < SM >. In order to do so, in the vicinity of an arbitrary point in the space of the Hamiltonians, it can be shown that: < S˛ >D

X ı < S˛ > 

ı



(41)

where        ıH ıH ı < S˛ > ıS˛ ˇ < S > :  D S  ı

ı

ı˛ ı˛

(42)

Using Eqs. (41) and (42), then this problem can be solved iteratively. The IBI and IMC methods are both designed to reproduce a target radial distribution function and therefore are commonly called structure-based coarsegraining methods. Another class of bottom-up coarse graining approaches are those that are designed to match the force distributions on the coarse-grained interaction sites to the force distributions measured within the atomistic trajectory on the atoms that are mapped into the various interaction sites. This force-matching approach was first used for obtaining atomistic potential energies by fitting forces and trajectories from ab initio calculations (Ercolessi and Adams 1994). It has since been extended to fitting coarse-grained potentials by Izvekov and Voth (Izvekov and Voth 2005, 2006; Lu et al. 2013), who call their method multi-scale coarse graining (MSCG). In this force-matching scheme, f D F, where f is a vector composed of the atomistic force data.  is a vector containing all of the coarse-grained force-field parameters, and F is a matrix that is related to the input atomistic configurations. Since this equation is a standard least squares matrix equation, it can be converted to G D b;

(43)

where G is the square matrix FT F and b is the vector FT f. Therefore, G  D ıb, which can be used to iterate the values of  in order to converge on the appropriate forces within the coarse-grained simulations. We have attempted to introduce some of the more popular approaches that are utilized to develop coarse-grained potentials for use in simulating biological,

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

19

polymeric, and colloidal systems in this section. However, by no means is this a complete summary of all of the approaches that have been used, so for more lengthy discussions of the various approaches, we would recommend some recent review articles including (Noid 2013; Saunders and Voth 2013; Brini et al. 2013; Riniker et al. 2012; de Pablo 2011). In general, the advantage of using a bottom-up approach is that the resulting force field will produce a higher level of accuracy and closer resemblance to atomistic simulations. However, these schemes produce force fields that are useful for a given state point and therefore are not transferable, whereas the advantages of the thermodynamic approach include that it produces a potential that has a broader range of applicability and also the thermodynamic approach does not require atomistic simulations to be done in the first place.

Interaction Potentials/Force Fields Classical Force Fields Classical, or empirical, force fields are generally used to calculate the energy of a system as a function of the nuclear positions of the particles within the system while ignoring the behavior of the individual electrons. As stated in section “Born–Oppenheimer Approximation,” the Born–Oppenheimer approximation makes it possible to write the energy as a function of the nuclear coordinates. Another approximation that is key to the implementation of classical Force fields is that it is possible to model the relatively complex motion of particles within the system with fairly simple analytical models of inter- and intramolecular interactions. Generally, an empirical force field consists of terms that model the nonbonded interactions (Enonbonded ), which include both the van der Waals and Coulombic interactions, the bonded interactions (Ebond ), the angle bending interactions (Eangle ), and the dihedral (bond rotations) interactions (Edihedral ): E.R/ D Enonbonded C Ebond C Eangle C Edihedral :

(44)

Figure 2 presents representative cartoons of the bond, angle, and dihedral interactions from a molecular perspective. The form that each of these individual terms takes is dependent on the Force field that you are using. There are several different force-field options available for various systems. The best way to find the most suitable force field for your specific problem is to conduct a literature and/or Internet search in order to find which force field has the capability to model the molecules you are interested in studying. However, if you are interested in modeling organic/biological molecules, there are several large force fields that may be a good place to start, including CHARMM (MacKerell et al. 1998), OPLS (Jorgensen and Swenson 1984), Amber (Cornell et al. 1995), and COMPASS (Sun et al. 1998). The CHARMM and AMBER force fields each have recently developed generalized versions of their force fields and tools to help researchers develop forcefield parameters for small molecules that they may be interested in modeling for

20 Fig. 2 Intramolecular terms of classical force fields: bond, angle, and dihedral interactions

C. Lorenz and N.L. Doltsinis lb Bond:

qa Angle:

f Dihedral:

which parameters do not currently exist. In the AMBER force field, this general force field is called GAFF (generalized AMBER force field) (Wang et al. 2004b; http://ambermd.org/antechamber/gaff.html). While in the CHARMM force field, CGENFF is the generalized force field (Vanommeslaeghe et al. 2010; Yu et al. 2012; CGENFF http://mackerell.umaryland.edu/~kenno/cgenff; ParamChem http:// cgenff.paramchem.org). Likewise, there are several well-known large force fields that can be used for solids, like the BKS potential (van Beest et al. 1990) for oxides and the embeddedatom method (EAM) (Daw and Baskes 1983, 1984; Finnis and Sinclair 1984) and modified embedded-atom method (MEAM) (Baskes 1992) force fields. In addition to defining the functional forms used for the various terms in the general potential formulation, a force field will also define the variables used in the potential which are derived from a combination of quantum simulation results and experimental observations. In the following sections, each of the terms in Eq. (44) will be discussed further and typical functional forms that are used in the previously mentioned force fields and others to represent each term will be shown. To this end, we limit the discussion to simple nonpolarizable force fields in which the individual atoms carry fixed charges. They capture many-body effects such as electronic polarization only in an effective way. More sophisticated, polarizable force fields have been developed over the past two decades (see, e.g., Ponder et al. 2010 and references therein); however, they are computationally substantially more demanding. Some of the most common polarizable force field methods will later be introduced at the end of section “Classical Polarizable Electrostatical Models”.

Nonbonded Interactions There are two general forms of nonbonded interactions that need to be accounted for by a classical force field: (1) the van der Waals (vdw) interactions and (2) the electrostatic interactions.

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

21

van der Waals Interactions In order to model the van der Waals interactions, we need a simple empirical expression that is not computationally intensive and that models both the dispersion and repulsive interactions that are known to act upon atoms and molecules. The most commonly used functional form of van der Waals energy (EvdW ) in classical force fields is the Lennard–Jones 12-6 function that has the form: "   #  X IJ 6 IJ 12 EvdW .R/ D 4IJ  ; (45) RIJ RIJ I >J where IJ is the collision diameter and IJ is the well depth of the interaction between atoms I and J . Both IJ and IJ are adjustable parameters that will have different values to describe the interactions between different pairs of particles (i.e., the values of  and  used to describe the interaction between two carbon atoms are different than the values of  and  used to describe the interaction between a carbon and an oxygen). Equation (45) models both the attractive part (the R6 term) and the repulsive part (the R12 term) of the nonbonded interaction. Other formulations of the Lennard–Jones nonbonded potential commonly have the same power law description of the attractive part of the potential but will have different power law dependence for the repulsive part of the interaction, such as the Lennard–Jones 9-6 function: "   #  X IJ 9 IJ 6 EvdW .R/ D 4IJ  : (46) RIJ RIJ I >J When the nonbonded interactions of a system that contains multiple particle types and multiple molecules are modeled using a Lennard–Jones-type nonbonded potential, it is necessary to be able to define the values of  and  that apply to the interaction between particles of type I and J . The parameters for these cross interactions are generally found using one of the two following mixing rules. One common mixing rule is the Lorentz–Berthelot rule where the value of IJ is found from the arithmetic mean of the two pure values and the value of IJ is the geometric mean of the two pure values: IJ D .I C J /=2 IJ D

p I J

(47) (48)

The other commonly used mixing rule is the one that defines both IJ and IJ as the geometric mean of the values for the pure species: IJ D

p

I J

(49)

22

C. Lorenz and N.L. Doltsinis

IJ D

p I J

(50)

Most force fields use the Lorentz–Berthelot mixing rule; however, the OPLS force field is one force fields that utilize the geometric mixing rule. In other nonbonded pairwise potentials, the repulsive portion of the interaction is modeled with an exponential term, which is in better agreement with the functional form of the repulsive term determined from quantum mechanics. One example of such a potential is the Buckingham potential (Buckingham 1938): EvdW .R/ D

X

AIJ exp.BIJ RIJ / 

I J

where there is a two-body term 2 :         IJ pIJ IJ IJ qIJ exp

2 .RIJ / D AIJ IJ BIJ  RIJ RIJ RIJ  aIJ IJ

(55)

and a three-body term 3 : 

2

3 .RIJ ; RIK ; IJK / D IJK IJK cos IJK  cos 0;IJK   exp

IJ IJ RIJ  aIJ IJ





IK IK :  exp RIK  aIK IK 

(56)

The Stillinger–Weber potential has generally been used for modeling crystalline silicon; however, more recently it has also been used for organic molecules as well. Another example of a three-body interatomic potential is the Tersoff potential (Tersoff 1988, 1989), which also was created initially in an attempt to accurately model silicon solids. Electrostatic Interactions Due to the fact that not all particles in a molecule have the same electronegativity, different particles will have stronger attractions to electrons than others. However, since classical force fields do not model the flow of electrons, the different particles within a molecule are assigned a partial charge that remains constant during the course of a simulation. Generally these partial charges qI are assigned to the nuclear centres of the particles. The electrostatic interaction between particles in different molecules or particles that are separated by at least two other atoms in a given molecule is calculated as the sum of the contributions between pairs of these partial charges using Coulomb’s law: Ecoul D

XX I

J

qI qJ 40 RIJ

(57)

where the charges of each particle are qI and qJ and 0 is the dielectric constant. In practice, an Ewald sum (Ewald 1921) is generally used to evaluate the electrostatic interactions within a classical MD simulation. However, this is a very computationally expensive algorithm to implement and it results in a computational

24

C. Lorenz and N.L. Doltsinis

cost of N 3=2 , where N is the number of particles in the system. In order to obtain better computational scaling, fast Fourier transforms (FFTs) have been used to calculate the reciprocal space summation required within the Ewald sum. By using the FFT algorithm, one can reduce the cost of the electrostatic algorithm to N log N . The most popular FFT algorithm that has been adopted for use in classical MD simulations is the particle–particle particle–mesh (pppm) approach (Hockney and Eastwood 1981; Luty et al. 1994, 1995). Classical Polarizable Electrostatical Models In traditional classical molecular dynamics simulations with nonpolarizable force fields, the charge distribution is mimicked by the fixed partial charges. As a result the electronic polarization effects are considered in a mean-field manner by optimizing the partial atomic charges such that they are representative of the condensed, typically aqueous, phase. In order to achieve a more accurate description of the response of the charge distribution to changes in the surrounding electrostatic field, models that explicitly account for electronic polarizability are essential. There are several different approaches to including explicit electronic polarizability into classical force fields. The most computationally achievable methods to mimic the quantum-mechanical response of the electronic distribution within molecules to changes in the surrounding electric field is to extend the potential energy function such that it is capable of capturing this physics. There are three different general approaches most commonly used to do so and we will briefly discuss below: the induced dipole model, the fluctuating charge model, and the classical Drude oscillator model. In the induced dipole model, generally an induced dipole, which is commonly calculated using a self-consistent field iterative procedure using isotropic atomic polarizabilities, is assigned to each atomic site (Warshel and Levitt 1976; Liu et al. 1998; Kaminski et al. 2002; Ren and Ponder 2002; Xie et al. 2007; Jorgensen et al. 2007; Shi et al. 2013). These induced dipoles are then used to calculate charge– dipole and dipole–dipole interactions, which are combined with the charge–charge interactions, to yield the electrostatic energy of the system. The induced dipole  by an external electric field E from these models is calculated using  D ˛E, where ˛ is the polarizability of the atomic site. The fluctuating charge model (Rick et al. 1994; Rick and Berne 1996; Stern et al. 2001; Patel and Brooks 2004; Patel et al. 2004) treats partial atomic charges as dynamical variables based on the electronegativity at each atomic site. Charges are propagated according to the principle of electronegativity equalization (Pauling and Yost 1932), and charge conservation constraints are assumed. Charges are generally allowed to only redistribute within molecules to avoid unphysical intermolecular charge transfer. During a molecular dynamics simulation, the charge variables used to model the fluctuating charges are usually propagated using an extended Lagrangian approach. In the fluctuating charge model, the induced dipole  by an external electric field E is calculated by  D .d 2 =/E, where d is the distance between centers of two atoms and  is hardness of charge flow between them.

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

25

The classical Drude oscillator model (Lamoureux et al. 2003; Lamoureux and Roux 2003) is also commonly referred to as the shell (van Maaren and van der Spoel 2001) or charge-on-spring (Kunz and van Gunsteren 2009) model. In these models, a Drude particle or oscillator, which has a charge assigned to it, is attached to the center of the atom with which it is associated via a harmonic spring, and this setup is designed to mimic nuclear-electron motion (Dick and Overhauser 1958). In the Drude model, the induced dipole by an external electric field is calculated from  D .qD =k 2 /E, where qD is the Drude charge and k is the spring constant of the harmonic spring that attaches the Drude particle to the center of its associated atom. Each of these different models has been applied in various classical force fields in order to create polarizable versions of the force field. CHARMM has adopted the Drude model (Chowdhary et al. 2013; He et al. 2013; Patel et al. 2015; Savelyev and MacKerell 2014; Lopes et al. 2013) and also has created a fluctuating charge model (Patel and Brooks 2004; Patel et al. 2004; Lucas et al. 2012; Zhong and Patel 2011). The induced dipole model has been used by the AMBER (Caldwell et al. 1990; Dang et al. 1991; Caldwell and Kollman 1995a, b) and AMOEBA (Ponder et al. 2010; Shi et al. 2013) force fields.

Bonded Interactions The bonded interactions are needed to model the energetic penalty that will result from two covalently bonded atoms moving too close or too far away from one another. The most common functional form that is used to model the bond bending interactions is that of a harmonic term: X .0/ Ebond D kb .`b  `b /2 (58) bonds

where kb is commonly referred to as the bond constant and is a measure of the bond .0/ stiffness and `b is the reference length or often referred to the equilibrium bond length. Each of these parameters will vary depending on the types of particles that the bond is joining.

Angle Bending Interactions The angle bending interactions are also modeled in order to determine the energetic penalties of angles containing three different particles compressing or overextending such that they distort the geometry of a portion of a molecule away from its desired structure. Again, the most common functional form to model the angle interactions is a harmonic expression: X Eangle D ka .a  a.0/ /2 (59) angles .0/

where ka is the angle constant and is a measure of the rigidity of the angle and a is the equilibrium or reference angle.

26

C. Lorenz and N.L. Doltsinis

Torsional Interactions The torsional interactions are generally modeled using some form of a cosine series. The OPLS force-field uses the following expression for its torsional term: Edihed D

X 1 .1/ 1 .2/ Kd Œ1 C cos. / C Kd Œ1  cos.2 / 2 2 dihedrals 1 .4/ 1 .3/ C Kd Œ1 C cos.3 / C Kd Œ1  cos.4 / 2 2

(60)

.i/

where Kd are the force constants for each cosine term and is the measured dihedral angle. The CHARMM force field uses the following expression: X

Edihed D

Kd Œ1 C cos.n  dd /;

(61)

dihedrals

where Kd is the force constant, n is the multiplicity of the dihedral angle , and dd is the shift of the cosine that allows one to more easily move the minimum of the dihedral energy.

First-Principles Electronic Structure Methods For the electronic ground state, i.e., k D 0, Kohn–Sham (KS) density functional theory is commonly used. In this case, the energy is given by E0  E KS Œ  D Ts Œ  C

Z d r vext .r/ .r/ C

1 2

Z d r vH .r/ .r/ C Exc Π

(62)

with the kinetic energy of noninteracting electrons, i.e., using a Slater determinant as a wave function ansatz,

‰ KS

ˇ ˇ ˇ 1 ˇˇ Dp ˇ nŠ ˇˇ ˇ

1 .x1 / 1 .x2 /

:: : 1 .xn / n

Ts Π D 

1X fi 2 i

ˇ

n .x1 / ˇˇ

2 .x1 /

 .x / 2 2  :: :: : : 2 .xn /   

Z dr

i .r/r

ˇ ˇ ˇ ˇ ˇ n .xn / 2

n .x2 / ˇ

(63)

i .r/

(64)

where fi is the number of electrons occupying orbital i , the external potential including nucleus–nucleus repulsion and electron–nucleus attraction, vext .r/ D

N 1 X N X I D1 J >I

N

X ZI ZI ZJ  jRI  RJ j I D1 jr  RI j

(65)

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

the Hartree potential (electron–electron interaction) Z .r0 / vH .r/ D d r0 jr  r0 j

27

(66)

the exchange–correlation energy, Exc , and the electron density .r/ D

n X

fi j

i .r/j

2

(67)

i

The orbitals which minimize the total, many-electron energy (62) are obtained by solving self-consistently the 1-electron Kohn–Sham equations,   1 ıExc Œ   r 2 C vext .r/ C vH .r/ C (68) i .r/ D i i .r/ 2 ı .r/ DFT is exact in principle, provided that Exc Œ  is known, in which case E KS (see Eq. (62)) is an exact representation of the ground state energy E0 (see Eq. (8)). In practice, however, Exc Œ  is not – and presumably never will be – known exactly; therefore, (semiempirical) approximations are used. The starting point for most density functionals is the local density approximation (LDA), which is based on the assumption that one deals with a homogeneous electron gas. Exc is split into an exchange term Ex and a correlation term Ec . Within the LDA the exchange functional is given exactly by Dirac (1930): Z LDA Ex Œ  D .r/xLDA . .r//d r (69) where xLDA . /

3 D 4

  13 3 1 .r/ 3 

(70)

The LDA correlation functional, on the other hand, can only be approximated. We give here the most commonly used expression by Vosko et al. (1980), derived from Quantum Monte Carlo calculations: Z LDA Ec Π D .r/cLDA . .r//d r (71) where     2 2b Q x C tan1 cLDA . / D A ln X Q 2x C b      .x  x0 /2 2.b2x0 / Q bx0 ln C (72)  tan1 X .x0 / X Q 2x C b

28

C. Lorenz and N.L. Doltsinis

q p p 3 with X D x 2 C bx C c, x D rs , rs D 3 4 .r/ , Q D 4c  b 2 , x0 D 0:104098, A D 0:0310907, b D 3:72744, c D 12:9352. This simplest approximation, LDA, is often too inaccurate for chemically relevant problems. A notable improvement is usually offered by so-called semilocal or gradient corrected functionals (generalized gradient approximation (GGA)), in which Ex and Ec are expressed as functionals of and the first variation of the density, r : ExGGA Π; r  D EcGGA Π; r 

Z Z

D

.r/xGGA . .r/; r /d r

(73)

.r/cGGA . .r/; r /d r

(74)

Popular examples are the BLYP (Becke 1988; Lee et al. 1988), BP (Becke 1988; Polák 1986), and BPW91 (Becke 1988; Perdew et al. 1992) functionals. The GGA expressions for x;c . .r/; r / are complex and shall not be discussed here. In many cases, accuracy can be further increased by using so-called hybrid functionals, which contain an admixture of Hartree–Fock exchange to KS exchange. Probably the most widely used hybrid functional is the three-parameter B3LYP functional (Becke 1993), B3LYP Exc D aExLDA C .1  a/ExHF C b ExB C .1  c/EcLDA C cEcLYP

(75)

where a D 0:80, b D 0:72, c D 0:81, and ExHF is the Hartree–Fock exchange energy evaluated using KS orbitals. New functionals are constantly proposed in search of better approximations to the exact Exc . Often functionals are designed to remedy a particular shortcoming of previous functionals. For systems in which van der Waals interactions are important, for instance, Grimme’s empirical D3 dispersion correction (Grimme et al. 2010) may be used in combination with a GGA or hybrid functional. The total energy is then simply the sum of the DFT energy E KS and the dispersion correction E disp , E DFTD3 D E KS C E disp

(76)

where E disp contains 2-body and 3-body terms, E disp D E .2/ C E .3/

(77)

Becke and Johnson proposed the following form for the 2-body contributions (Becke and Johnson 2005; Johnson and Becke 2005, 2006): E .2/ D 

X X I >J nD6;8

sn

n RIJ

CnIJ 0 n C f .RIJ /

(78)

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

29

with the damping function 0 0 f .RIJ / D a1 RIJ C a2

(79)

to avoid near singularities at short interatomic distances RIJ , and s 0 RIJ

D

C8IJ C6IJ

(80)

The functional-dependent scaling factors sn and coefficients a1 and a2 can be obtained from the website of the Grimme group (www.thch.uni-bonn.de/tc) together with the dispersion coefficients CnIJ . The reader is referred to a number of excellent textbooks (Parr and Yang 1989; Dreizler and Gross 1990; Koch and Holthausen 2000; Fiolhais et al. 2003) and online resources (Burke http://dft.uci.edu/doc/g1.pdf; Rappoport et al.) on DFT for further details on the theory and on how to choose the right functional for a specific application.

Building the System/Collecting the Ingredients Setting Up an AIMD Simulation Building a Molecule In many cases, the coordinates of a molecular structure are available for download on the Web, from crystallographic databases (Reciprocal Net www.reciprocalnet. org/edumodules/commonmolecules; Cambridge Crystallographic Data Centre www.ccdc.cam.ac.uk; RCSB Protein Data Bank www.rcsb.org/pdb; Inorganic Crystal Structure Database www.fiz-karlsruhe.de/icsd.html; Toth Information Systems www.tothcanada.com) or journal supplements. For relatively small molecules, an initial guess structure can be built using molecular graphics software packages such as Molden (www.cmbi.ru.nl/molden/molden.html) or Avogadro (Hanwell et al. 2012; www.avogadro.cc). Plane Waves and Pseudopotentials The most common form of AIMD simulation employs DFT (see section “First-Principles Electronic Structure Methods”) to calculate atomic forces, in conjunction with periodic boundary conditions and a plane-wave basis set. Using a plane-wave basis has two major advantages over atom-centered basis functions: (i) there is no basis set superposition error (Boys and Bernardi 1970; Marx and Hutter 2000) and (ii) the Pulay correction (Pulay 1969, 1987) to the Hellmann–Feynman force, due to basis set incompleteness, vanishes (Marx and Hutter 2000, 2009).

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C. Lorenz and N.L. Doltsinis

Plane Wave Basis Set As a consequence of Bloch’s theorem, in a periodic lattice, the Kohn–Sham orbitals (see Eq. 68) can be expanded in a set of plane waves (Ashcroft and Mermin 1976; Meyer 2006), k;j .r/

D

X

k;j

cG e i.kCG/r

(81)

G

where k is a wave vector within the Brillouin zone, satisfying Bloch’s theorem, .r C T/ D e ikT .r/

(82)

for any lattice vector T, T D N1 a1 C N2 a2 C N3 a3

(83)

N1 ; N2 ; N3 being integer numbers, and a1 ; a2 ; a3 the vectors defining the periodically repeated simulation box. In Eq. (81), the summation is over all reciprocal lattice vectors G which fulfill the condition G  T D 2M , M being an integer number. In practice, this plane-wave expansion of the Kohn–Sham orbitals is truncated such that the individual terms all yield kinetic energies lower than a specified cutoff value, Ecut , „2 jk C Gj2  Ecut 2m

(84)

The plane-wave basis set thus has the advantage over other basis sets that convergence can be controlled by a single parameter, namely, Ecut . Which cutoff value is the best compromise between accuracy and computational cost depends on the specific system studied, on the properties of interest, and, crucially, on the choice of pseudopotentials (see below). In this periodic setup, the electron density (see Eq. 67) can be approximated by a sum over a mesh of Nkpt k-points in the Brillouin zone (Chadi and Cohen 1973; Monkhorst and Pack 1976; Moreno and Soler 1992), .r/ 

1 X fk;j j Nkpt k

k;j .r/j

2

(85)

Since the volume of the Brillouin zone, VBZ D .2/3 =Vbox , decreases with increasing volume of the simulation supercell, Vbox , only a small number of kpoints need to be sampled for large supercells. For insulating materials (i.e., large band gap), a single k-point is often sufficient, typically taken to be k D 0 (-point approximation).

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

31

Pseudopotentials While plane waves are a good representation of delocalized Kohn-Sham orbitals in metals, a huge number of them would be required in the expansion (81) to obtain a good approximation of atomic orbitals, in particular near the nucleus where they oscillate rapidly. Therefore, in order to reduce the size of the basis set, only the valence electrons are treated explicitly, while the core electrons (i.e., the inner shells) are taken into account implicitly through pseudopotentials combining their effect on the valence electrons with the nuclear Coulomb potential. This frozen-core approximation is justified as typically only the valence electrons participate in chemical interactions. To minimize the number of basis functions, the pseudopotentials are constructed in such a way as to produce nodeless atomic valence wave functions. Beyond a specified cutoff distance from the nucleus, Rcut the nodeless pseudo-wave functions are required to be identical to the reference all-electron wave functions. Normconserving Pseudopotentials

are generated subject to the condition that the pseudo-wave function has the same norm as the all-electron wave function and thus gives rise to the same electron density. Although normconserving pseudopotentials have to fulfill a (small) number of mathematical conditions, there remains considerable freedom in how to create them. Hence, several different recipes exist (Hamann et al. 1979; Bachelet et al. 1982; Kerker 1980; Vanderbilt 1985; Troullier and Martins 1990, 1991; Goedecker et al. 1996; Hartwigsen et al. 1998). Since pseudopotentials are generated using atomic orbitals as a reference, it is not guaranteed that they are transferable to any chemical environment. Generally, transferability is the better the smaller the cutoff radius Rcut is chosen. However, the reduction in the number of plane waves required to represent a particular pseudowave function – i.e., the softness of the corresponding pseudopotential – increases as Rcut gets larger. So Rcut has to be chosen carefully and there is always a trade-off between transferability and softness. An upper limit for Rcut is given by the shortest interatomic distances in the molecule or crystal the pseudopotential will be used for: one needs to make sure that the sum of the two cutoff radii of any two neighboring atoms is smaller than their actual spatial separation. For each angular momentum l a separate pseudopotential VlPS .r/ is constructed. The total pseudopotential operator is written as PS VO PS D Vloc .r/ C

X

PS Vnl;l .r/POl

(86)

l

where the nonlocal part is defined as PS PS Vnl;l .r/ D VlPS .r/  Vloc .r/

(87)

PS and the local part Vloc .r/ is taken to be the pseudopotential VlPS .r/ for one specific value of l, typically the highest one for which a pseudopotential was created. The

32

C. Lorenz and N.L. Doltsinis

pseudopotential (86) is called semi-local, since the projector POl only acts on the l-th angular momentum component of the wave function, but not on the radius r. (Note: a pseudopotential is called nonlocal if it is l-dependent.) To achieve higher numerical efficiency, it is common practice to transform the semi-local pseudopotential (86) to a fully nonlocal form, PS VO PS D Vloc .r/ C

X

jˇi > Bij < ˇj j

(88)

ij

using the Kleinman-Bylander prescription (Kleinman and Bylander 1982). Vanderbilt Ultrasoft Pseudopotentials

An ultrasoft type of pseudopotential was introduced by Vanderbilt (1990; Laasonen et al. 1993) to deal with nodeless valence states which are strongly localized in the core region. In this scheme the normconserving condition is lifted and only a small portion of the electron density inside the cutoff radius is recovered by the pseudo-wave function, the remainder is added in the form of so-called augmentation charges. Complications arising from this scheme are the nonorthogonality of Kohn– Sham orbitals, the density dependence of the nonlocal pseudopotential, and the need to evaluate additional terms in atomic force calculations. How to Obtain Pseudopotentials?

There are extensive pseudopotential libraries available for download with the simulation packages CPMD (Hutter et al.), CP2K (Hutter et al.), QuantumEspresso (www.quantum-espresso.org), Abinit (www.abinit.org), or online (www.physics. rutgers.edu/~dhv/uspp/; http://cp2k.web.psi.ch/potentials; www.quantum-espresso. org; www.abinit.org). However, before applying any pseudopotentials, they should always be tested against all-electron calculations. Pseudopotentials used in conjunction with a particular density functional should have been generated using the same functional. In many cases, the required pseudopotential will not be available in any accessible library; in this case it may be generated using freely downloadable programs (Vanderbilt www.physics.rutgers.edu/~dhv/uspp/; Atomic Pseudopotentials Engine http://tddft.org/programs/APE; Hamann www.mat-simresearch.com/oncvpsp-2.1. 1.tar.gz; Opium – pseudopotential generation project http://opium.sourceforge.net; Martins www.abinit.org/downloads/psp-links/psp-links/MARTINS2ABINIT.tgz; Fuchs and Scheffler http://cpc.cs.qub.ac.uk/summaries/ADKA_v1_0.html; Dacapo pseudopotential library http://wiki.fysik.dtu.dk/dacapo/Pseudopotential_Library). How to Set the Plane-Wave Cutoff?

The number of plane-wave basis functions, i.e., the value of the cutoff energy Ecut , required for a sufficiently accurate DFT calculation sensitively depends on the choice of pseudopotential. For a calculation on H2 O with normconserving pseudopotentials of the Troullier–Martins type (Troullier and Martins 1990, 1991), for example, a cutoff energy of 70 Ry (D 35 a.u.) is typically used (see, e.g.,

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

33

Sprik et al. 1996), while 25 Ry is usually sufficient with ultrasoft pseudopotentials. Goedecker–Teter–Hutter normconserving pseudopotentials (Goedecker et al. 1996; Hartwigsen et al. 1998; Krack 2005) normally require a slightly higher cutoff than Troullier–Martins pseudopotentials (CPMD Manual www.cpmd.org:81/ manual/node73.html). Strictly speaking, the best cutoff energy should be determined for every new system and set of pseudopotentials by carrying out a series of single point calculations with different Ecut and analyzing the convergence of the total energy and the properties of interest.

Setting Up a Classical MD Simulation There are two general stages that make up the preparation to conduct force-field molecular dynamics simulations: (1) gathering preliminary information and (2) building the actual system.

Gathering Preliminary Information Gathering the preliminary information before conducting the simulation is mostly focused on making sure that the simulation is possible. First, it is important to identify the type and number of molecules that you wish to model. Then, it is necessary to find the force field that will allow you to most accurately model the molecules and physical system that you want to simulate. A brief synopsis of some of the larger classical force-field parameter sets is given in section “Classical Force Fields.” These force fields and references may be good starting points in searching for the correct classical force field to use for a given system, but the best way to find a specific force field is to just conduct a search for research articles that may have been conducted on the same system. If no force-field parameters exist for the system of interest, then you can use configurations and energies from quantum simulations to parameterize a given force field for your system. A methodology for how a force field was parameterized originally is presented in the relevant paper; however, this is a complicated exercise and is probably best left to the experts. Building the System After identifying that a force field exists for the system you wish to model, the next step is to build the initial configuration of the molecules within the system. The initial configuration will consist of initial spatial coordinates of each atom in each given molecule. When building a large system consisting of several molecules of various types, it is easiest to write a computer code that contains or reads in the molecular structure and coordinates of each molecule present in the system and then have the code replicate each molecule however many times is necessary in order to build the entire system. Alternatively, most of the molecular dynamics simulation packages previously mentioned have capabilities to build systems from a pdb file; however, these tools are often useful for only certain systems and force fields. There is unfortunately no one tool which can be used to build any system with any force field.

34

C. Lorenz and N.L. Doltsinis

These initial configurations can represent a minimum energy structure either from another simulation (i.e., a final structure from an energy minimization in a quantum or classical Monte Carlo simulation can be used as the starting state for classical simulations), from experimental observation (i.e., the pdb database for crystallographic structures of proteins) or building the initial coordinates based upon the equilibrium bond distances and bond angles from the force field. The placement of the molecules within the simulation box can be done in a number of different ways as well. The molecules can be placed on the vertices of a regular lattice or in any other regularly defined geometry that may be useful for conducting your simulation (i.e., in simulating the structural properties of micelles oftentimes the surfactant molecules will initially be placed on the vertices of a bucky ball such that they are in a spherical configuration). Also, molecules can be placed at random positions within the simulation box. The one advantage of placing molecules at regularly spaced positions is that it is easier to insure that there is no overlapping of molecules, whereas with the randomly placed molecules it can be quite difficult to ensure that a placed molecule does not overlap with another molecule in the box (particularly for large or highly branched molecules). For more complex geometries, there are several software packages and online resources which are very useful in generating initial configurations (most of these output xyz and/or pdb files and which are generally useable within the majority of molecular dynamics codes). These tools include Enhanced Monte Carlo (http://montecarlo. sourceforge.net/emc/Welcome.html), VMD TopoTools (http://sites.google.com/ site/akohlmeyer/software/topotools), Avogadro (www.avogadro.cc), Packmol (ch7-2:packmol-www), and Atomsk (http://pierrehirel.info/codes_atomsk.php? lang=eng), which are all freely available. Then if you are interested in carrying out simulations with the CHARMM or MARTINI force fields, you can use the online resource – CHARMM-GUI (http://www.charmm-gui.org/) – to create a variety of systems by choosing the molecules and arrangement of molecules (i.e., bilayer, micelle, . . . ) and it will output a pdb file and result force field files for you. In addition to containing the initial spatial coordinates of all of the molecules in the system, the initial configuration most also contain some additional information about the atoms and molecules in the systems. Each atom in the configuration must contain a label of what atomic species (i.e., carbon, nitrogen, etc.) it represents. This label will be different for each simulation code used, but all of them will have some type of label as it will inform the simulation code what force-field values to use to represent the interactions of that atom. A list of all of the covalent bonds, the bond angles, and the dihedrals in the system will also need to be included in the initial configuration. The lists of the bonds, angles, and dihedrals contain an identifier for each atom that make up the bond, angle, or dihedral and then an identifier for the type that informs the simulation package which parameters to use in calculating the energy of the bond, angle, or dihedral. The final component of the initial configuration of a classical simulation is a list of all of the various types of atoms, bonds, angles, and dihedrals in the system along with their corresponding force-field parameters (i.e.,  and  for atom types to describe their nonbonded interactions, force constants, and equilibrium values for bond, angle, and dihedral types).

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

35

Finally, after building the initial configuration, the simulation is about ready to be performed. The last step is to choose the simulation variables and set up the input to the simulation package in order to convey these selections. These options and the decision process behind choosing from the various options will be presented in the following sections.

Preparing an Input File Optimization Algorithms Optimization algorithms are often used to find stationary points on a potential energy surface, i.e., local and global minima and saddle points. It is often useful to carry out a geometry optimization before starting an MD simulation, in particular when the initial coordinates are likely to be far away from their realistic values. The only place where they directly enter MD is in the case of Born–Oppenheimer AIMD, in order to converge the SCF wave function for each MD step. It is immediately obvious that the choice of optimization algorithm crucially affects the speed of the simulation.

Steepest Descent The steepest descent method is the simplest optimization algorithm. The initial energy EŒ‰0  D E.c0 /, which depends on the plane-wave expansion coefficients c (see Eq. 81), is lowered by altering c in the direction of the negative gradient, dn D 

@E.cn /  gn @c

cnC1 D cn C n dn

(89) (90)

where n > 0 is a variable step size chosen such that the energy always decreases and n is the optimization step index. The steepest descent method is very robust; it is guaranteed to approach the minimum. However, the rate of convergence ever decreases as the energy gets closer to the minimum, making this algorithm rather slow.

Conjugate Gradient Methods The Conjugate Gradient method generally converges faster than the steepest descent method due to the fact that it avoids moving in a previous search direction. This is achieved by linearly combining the gradient vector and the last search vector, cnC1 D cn C n dn

(91)

36

C. Lorenz and N.L. Doltsinis

where

dn D gn C ˇn dn1

(92)

Different recipes exist to determine the coefficent ˇn (Jensen 2007) among which the Polak–Ribière formula usually performs best for non-quadratic functions, ˇn D

gn .gn  gn1 / gn1 gn1

(93)

In the case of a general non-quadratic function, such as the DFT energy, conjugacy is not strictly fulfilled and the optimizer may search in completely inefficient directions after a few steps. It is then recommended to restart the optimizer (setting ˇ D 0). Convergence can be improved by multiplying gn with a preconditioner matrix, e.g., an approximate inverse of the second derivatives matrix (Hessian in the case of Q The method is then called the preconditioned conjugate geometry optimization) H. Q is approximated by gradient (PCG). In the CPMD code, the matrix H

Q D H

for G Gcut

9 =

HGKScut Gcut for G < Gcut

;

8 < H KS 0 GG

:

(94)

KS where HGG 0 is the Kohn–Sham matrix is the plane-wave basis and Gcut is a cutoff value for the reciprocal lattice vector G (set to a default value of 0.5 a.u.).

Direct Inversion of the Iterative Subspace Having generated a sequence of optimization steps ci , the direct inversion of the iterative subspace (DIIS) method (Pulay 1980, 1982; Császár and Pulay 1984; Hutter et al. 1994) is designed to accelerate convergence by finding the best linear combination of stored ci vectors, cnC1 D

n X

ai ci

(95)

iD1

Ideally, of course, cnC1 is equal to the optimum vector copt . Defining the error vector ei for each iteration as ei D ci  copt

(96)

Equation (95) becomes n X iD1

ai copt C

n X iD1

ai ei D copt

(97)

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

37

Equation (97) is satisfied if n X

ai D 1

(98)

ai ei D 0

(99)

iD1

and n X iD1

Instead of the ideal case Eq. (99), in practice one minimizes the quantity h

n X

ai ei j

n X

aj ej i

(100)

j D1

iD1

subject to the constraint (98), which is equivalent to solving the system of linear equations 0

b11 B b21 B B :: B : B @ bn1

b12 b22 :: :

  :: :

b1n b2n :: :

10 1 0 a1 1 B a2 C B 1 C CB C B :: C B :: C D B B C B : C CB : C B A 1 @ an A @

bn2    bnn 1 1    1 0



0 0 :: :

1

C C C C C 0 A 1

(101)

where bij D hei jej i

(102)

and the error vectors are approximated by Q 1 .copt /gi ei D H

(103)

Q e.g., Eq. (94). using an approximate Hessian matrix H,

Controlling Temperature: Thermostats If understanding the behavior of the system as a function of temperature is the aim of your study, then it is important to be able to control the temperature of your system. The temperature of the system is related to the time average of the kinetic energy, which can be calculated by
N V T D 2 I 2

(104)

38

C. Lorenz and N.L. Doltsinis

if the system is in thermal equilibrium. The right-hand side of Eq. (104) can be approximated by 3N k T for a large number of particles. 2 B Below we introduce specific thermostating techniques for MD simulations at thermodynamic equilibrium, e.g., for calculating equilibrium spatial distribution and time-correlation functions. However, when MD simulations are performed on a system undergoing some nonequilibrium process involving exchange of energy between different parts of the system, e.g., when an energetic particle, such as an atom or a molecule, hits a crystal surface, or there is a temperature gradient across the system, one has to resort to specially developed techniques, see, for example, Kantorovich (2008), Kantorovich and Rompotis (2008), and Toton et al. (2010). In these methods, based on the so-called generalized Langevin equation, the actual system on which MD simulations are performed is considered in contact with one (or more) heat bath(s) kept at constant temperature(s), and the dynamics of the system of interest reflects the fact that there is an interaction and energy transfer between the system and the surrounding heat bath(s).

Rescale Thermostat One obvious way to control the temperature of a system is to rescale the velocities of the atoms p within the system (Woodcock 1971). The rescaling factor  is determined from  Ttarget =T0 , where Ttarget and T0 are the target and initial temperatures, respectively. Then, the velocity of each atom is rescaled such that Vf D Vi . In practice, the inputs generally required to use a rescale thermostat include: T0 – Initial temperature, Ttarget – Target temperature,  - Damping constant, (i.e., frequency with which to apply the thermostat), ıT – Maximum allowable temperature difference from Ttarget before thermostat is applied, • frescale – Fraction of temperature difference between current temperature and Ttarget is corrected during each application of thermostat. • • • •

If it is desired to have a strict thermostat (i.e., when first starting a simulation that might have particles very near one another), then ıT and  should have values of 0:01 Ttarget and 1 time step, respectively, and frescale should be near 1.0. However, if you wish to allow a more lenient thermostat, then the value of ıT should be of the same order of magnitude as Ttarget ,  should be 102  103 time steps, and frescale  0:01  0:1.

Berendsen Thermostat Another way to control the temperature is to couple the system to an external heat bath, which is fixed at a desired temperature. This is referred to as a Berendsen thermostat (Berendsen et al. 1984). In this thermostat, the heat bath acts as a reservoir of thermal energy that supplies or removes temperature as necessary. The velocities are rescaled each time step, where the rate of change in temperature

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

39

is proportional to the difference in the temperature in the system T .t / and the temperature of the external bath Tbath : 1 d T .t / D .Tbath  T .t // dt 

(105)

which when integrated results in the change in temperature each time step: T D

ıt .Tbath  T .t //: 

(106)

In Eqs. 105 and 106,  is the damping constant for the thermostat. In practice, the necessary inputs when using the Berendsen thermostat include: • Tbath – temperature of the external heat bath, •  – damping constant for the thermostat. Obviously the amount of control that the thermostat imposes on the simulation is controlled by the value of . If  is large, then the coupling will be weak and the temperature will fluctuate significantly during the course of the simulation, while if  is small, then the coupling will be strong and the thermal fluctuations will be small. If  D ıt , then the result will be the same as the rescale thermostat, in general.

Nosé–Hoover Thermostat While the Berendsen thermostat is efficient for achieving a target temperature within your system, the use of a Nosé–Hoover thermostat properly reproduces a canonical ensemble once the system has reached a thermal equilibrium. The extended system method, which was originally introduced by Nose (1984a, b) and then further developed by Hoover (1985), introduces additional degrees of freedom into the Hamiltonian that describes the system, from which equations of motion can be determined. The extended system method considers the external heat bath as an integral part of the system by including an additional degree of freedom in the Hamiltonian of the system that is represented by the variable s. As a result, the potential energy of the reservoir is Epot D .f C 1/kB T ln s;

(107)

where f is the number of degrees of freedom in the physical system and T is the target temperature. The kinetic energy of the reservoir is calculated by Ekin

  Q ds 2 D ; 2 dt

(108)

40

C. Lorenz and N.L. Doltsinis

where Q is a parameter with dimensions of energy  (time)2 and is generally referred to as the “virtual” mass of the extra degree of freedom s. The magnitude o Q determines the coupling between the heat bath and the real system, thus influencing the temperature fluctuations. Utilizing Eqs. 107 and 108, and substituting the real variables for the corresponding Nosé variables, the equations of motion are found to be as follows: P I; R I D FI  R R MI   f kB T  Ttarget ;

P D Q

(109)

(110)

where D ssP . quantity s NH D

Q f kB Ttarget

(111)

is an effective relaxation time or damping constant. In practice, the inputs that are necessary when utilizing the Nosé–Hoover thermostat during a molecular dynamics simulation include: • Ttarget – Target temperature, • Q - Fictitious mass of the additional degree of freedom s. The most significant variable in the above list is Q. Large values of Q may cause poor temperature control, with the infinite limit resulting in no energy exchange between the temperature bath and the real system, which is the case of conventional molecular dynamics simulations resulting in the microcanonical ensemble. However, if Q is too small, then the energy oscillates and the system will take longer in order to reach a thermal equilibrium. An in-depth discussion of the Nosé–Hoover thermostat can be found in Hunenberger (2005) and Tuckerman (2010), for example. Nosé–Hoover Chains The Nosé–Hoover thermostat discussed above has been demonstrated to run into ergodicity problems and does not generate the canonical ensemble in some pathological cases. To make the particle motion more “chaotic,” in the Nosé– Hoover chain thermostat, the simple Nosé–Hoover thermostat is itself coupled to another thermostat, which again can be coupled to a thermostat, and so forth. The equations of motion for a chain of m thermostats are given by (Martyna et al. 2002; Tuckerman 2010)

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

P1 D :: :

Pj D

41

R I D FI  1 R PI R MI

(112)

 f kB

T  Ttarget  1 2 Q1

(113)

i 1 h Qj 1 j21  kT  j j C1 Qj

(114)

 1

2 Qm1 m1  kT Qm

(115)

:: :

Pm D

The conserved quasi-Hamiltonian is HQ D H C

m X Qj j D1

2

j2

Z C 3N kT

1 dt C

m X

Z kT

j dt

(116)

j D2

Compared to the input parameters required for the simple Nosé–Hoover thermostat, the user now needs to also specify the length m of the chain (e.g., m D 4 is the default value in the CPMD program).

Langevin Thermostat The Langevin thermostat is a stochastic thermostat, in which the equations of motion of the particles in the system take the form: P I /: R I D FI C Ffriction .R MI R

(117)

Therefore, the force that a particle experiences is due to the gradient in the potential energy surface and a frictional force due to collisions of the particle with the particles in the thermal bath. This frictional force is defined as P I / D  I MI R P I C I .t / Ffriction .R

(118)

where I is a proportionality constant, which is often referred to as the “friction coefficient.” The term I .t / is a random force due to the stochastic collisions with the particles in the thermal bath, and it has a Gaussian probability distribution with P and an autocorrelation function no correlation to any past or present R or R < I .t /J .t 0 / >D 2 I kB Ttarget ıI;J ı.t  t 0 /;

(119)

42

C. Lorenz and N.L. Doltsinis

where kB is Boltzmann’s constant and Ttarget is the temperature of the thermal bath. In practice, Eq. (117) is solved using a difference equation over a small time step ıt , such that: q VI D AI .t /ıt  I VI .t /ıt C  2kB Ttarget I ıt =MI

(120)

where  is a random number with unit variance. When using a Langevin thermostat during a molecular dynamics simulation, the necessary inputs generally are: • Ttarget – Target (or bath) temperature, • – The friction coefficient. The most significant variable in the above list is . Choosing too large a value of will result in the system becoming overdamped due to the friction term, and choosing too small a value of may result in too weak of a coupling between the thermal bath and the system and therefore poor temperature control.

Controlling Pressure: Barostats It may be desired to study the behavior of the simulated system while the pressure is held constant (i.e., pressure-induced phase transitions). Many experimental measurements are made in conditions where the pressure and temperature are held constant and so it is of utmost importance to be able to accurately replicate these conditions in simulations. One thing of note is that the pressure often fluctuates more than other quantities such as the temperature in an N V T molecular dynamics simulation or the energy in an N VE molecular dynamics simulation. This is due to the fact that the pressure is related to the virial term, which is the product of the positions of the particles in the system and the derivative of the potential energy function. These fluctuations will be observed in the instantaneous values of the system pressure during the course of the simulation, but the average pressure should approach the desired pressure. Since generally the temperature and number of atoms will also be held constant during constant pressure simulations, and the volume of the system will be allowed to change in order to arrive at the desired pressure. Therefore, less compressible systems will show larger fluctuations in the pressure than the systems that are more easily compressed.

Berendsen Barostat Many of the approaches used for the controlling the pressure are similar to those that are used for controlling the temperature. One approach is to maintain constant pressure by coupling the system to a constant pressure reservoir as is done in the Berendsen barostat (Berendsen et al. 1984), which is analogous to the way

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

43

temperature is controlled in the Berendsen thermostat. The pressure change in the system is determined by: 1 dP .t / D .P0  P .t //; dt P

(121)

where P is time constant of the barostat, P0 is the desired pressure and P .t / is the system pressure at any time t . In order to accommodate this change in pressure, the volume of the box is scaled by a factor of 3 each time step; therefore, the coordinates of each particle in the system are scaled by a factor of  (i.e., RI .t C ıt / D 1=3 RI .t /, where 

ıt D 1 .P  P0 / P

 13 :

(122)

In practice, the inputs for the Berendsen barostat will include: • P0 – Desired pressure, • P – Time constant of the barostat, One other input that may be included in the use of the Berendsen barostat is to define which dimensions are coupled during the pressure relaxation. For example, you could define that the pressure is relaxed in a way that the changes in all three dimensions are coupled and therefore all of the dimensions change at the same rate. On the other hand, the pressure relaxation can be handled in an anisotropic manner, such that none of the dimensions are coupled and each dimension will have its own scaling factor that results from the individual pressure components.

Nosé–Hoover Barostat Similar to the Nosé–Hoover thermostat, the extended system method has been applied to create a barostat (Hoover 1986) that is coupled with a Nosé–Hoover thermostat. In this case, the extra degree of freedom  corresponds to a “piston,” and it is added to the Hamiltonian of the system, which results in the following equations of motion: d R.t / D V.t / C .t /.R.t /  RCOM /; dt F.t / d V.t / D  Œ.t / C .t /V.t /; dt M   1 T d .t / D 2 1 ; dt T T0

(123)

(124)

(125)

44

C. Lorenz and N.L. Doltsinis

1 d .t / D V .t /.P  P0 /; dt N kB T0 P2

(126)

d V .t / D 3.t /V .t / dt

(127)

where RCOM are the coordinates of the center of mass of the system,  is the thermostat extra degree of freedom and can be thought of as a friction coefficient, T is the thermostat time constant,  is barostat extra degree of freedom and is considered a volume scaling factor, and P is the barostat time constant. Equations 126 and 127 explicitly contain the volume of the simulation box, V .t /. Generally, this barostat is implemented using the approach described in Melchionna et al. (1993). In addition to the variables that are a part of the equations of motion, there is a variable Q that represents the “mass” of the “piston.” This is analogous to the “mass” variable in the Nosé–Hoover thermostat. In practice, the required input for the Nosé–Hoover barostat will include: • • • • •

P0 – Desired pressure, T0 – Desired temperature, P – Time constant of the barostat, T – Time constant of the thermostat, Q - The “mass” of the piston.

Like in the case of the Nosé–Hoover thermostat, care must be taken when selecting the value of the variable Q. A small value of Q is representative of a piston with small mass and thus will have rapid oscillations of the box size and pressure, whereas a large value of Q will have the opposite effect. The infinite limit of Q results in normal molecular dynamics behavior.

Setting the Timestep Born–Oppenheimer MD/Classical MD Since BOMD is a classical MD in the sense that the nuclei are classical particles, the same rules concerning the choice of timestep apply to both BOMD and atomistic or coarse-grained force-field MD. The largest possible timestep, ıt , is partially determined by the fastest oscillation in the system – in many molecules this would be a bond-stretching vibration involving hydrogen, e.g., CH, NH, or OH. It is immediately plausible that ıt must be smaller than the shortest vibrational period in order to resolve that motion and for the numerical integrator (see section “Classical Molecular Dynamics”) to be stable. Let us assume a particular molecule has an OH vibration at ! 0 D 3500 cm1 . Then corresponding to a period T can be calculated to be about 10 fs:

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

h  1 „! D D D hc hc c cT

!0 D

45

(128)

where we have used  D 1=T and c is the speed of light (remember to use c in cm/s if ! 0 is in cm1 ). Solving (128) for T , we get 1 c !0

T D

(129)

According to the Nyquist–Shannon sampling theorem, the lowest allowed sampling frequency is twice the maximum oscillation frequency of the simulated system, which would mean for the present example that the maximum timestep is 5 fs. Another important consideration is the stability limit of the numerical integrator used in the MD simulation (Sutmann 2006). For a one-dimensional harmonic oscillator, the propagation of nuclear position and momentum can be written in matrix form as     !RnC1 !Rn DM (130) VnC1 Vn In order for the propagation to be stable, the eigenvalues  of matrix M, which are obtained from the secular equation, det.M  1/ D 0

(131)

must satisfy the condition jj < 1. This then defines the maximum timestep. The velocity Verlet algorithm is usually implemented in the form ıt Fn 2M D Rn C ıt VnC 1

VnC 1 D Vn C 2

RnC1

2

VnC1 D VnC 1 C 2

ıt FnC1 2M

(132) (133) (134)

Using the force expression for harmonic oscillator, F D ! 2 MR

(135)

we obtain ! 2 ıt Rn 2 D Rn C ıt VnC 1

VnC 1 D Vn  2

RnC1

2

(136) (137)

46

C. Lorenz and N.L. Doltsinis

VnC1 D VnC 1  2

! 2 ıt RnC1 2

(138)

The total propagation matrix M (see Eq. 130) can be written as the product of the three propagation matrices corresponding to steps (136–138),  MD

10  ˛2 1



1˛ 01



10  ˛2 1



2

D

1  ˛2 ˛ 2 ˛2 ˛.1  4 / 1  ˛2

! (139)

where ˛ D ! ıt . The eigenvalues of M are then given by r ˛2 ˛2 ˙˛ 1 D1 2 4

(140)

The stability condition is satisfied if (Sutmann 2006) ! 2 ıt 2 < 2

(141)

Solving (141) for ıt , we obtain p ıt
D    dP dQA.P; QI 0/A.P; QI t /f .P; Q/ (166) where f .P; Q/ is the equilibrium phase space distribution function. From the velocity autocorrelation function, for example, one can calculate the diffusion coefficient as Z 1 1 DD < VI .0/VI .t / > dt (167) 3 0 where VI is the velocity of particle I . Alternatively, one can obtain the diffusion coefficient for long times from the associated Einstein relation, 6tD D< jRI .t /  RI .0/j2 >

(168)

In practice, D is then determined from a linear fit to the mean square displacement (rhs of Eq. 168) as one sixth of the slope. An example is shown in Fig. 4. Another common application of correlation functions is the calculation of IR absorption spectra. The line-shape function, I .!/, is given by the Fourier transform of the autocorrelation function of the electric dipole moment M,

Fig. 4 Mean square displacement of liquid water from CPMD simulations at 900 K and linear fit to determine the diffusion constant D using Eq. (168)

1 2

Z

1

< M.0/M.t / > e i!t dt

(169)

1

4 mean square displacement [Å2]

I .!/ D

simulation

3

y = 6*D*x + c

2

1

0

0

1 time [ps]

2

Molecular Dynamics Simulation: From “Ab Initio” to “Coarse Grained”

53

Fig. 5 A snapshot of a micelle formed from DDAO molecules and oil molecules formed using the VMD software package (http:// www.ks.uiuc.edu/Research/ vmd/)

Visualization Due to the nature of MD simulations, one of the most productive forms of analysis of a simulation is to be able to visualize the trajectory of the molecules of interest. This is particularly useful since experimental techniques are not able to produce visual pictures of atomistic interactions and therefore it is something that only simulations (at this point) are able to provide. In order to visualize a simulation trajectory, there are several different very powerful computer packages that are commonly used. These software packages include VMD (http://www.ks. uiuc.edu/Research/vmd/), PyMol (http://pymol.sourceforge.net/), RasMol (http:// rasmol.org/), and several others (http://www.umass.edu/microbio/rasmol/othersof. htm). Figure 5 shows an example of the type of pictures that can be made using the visualization software. Each of these codes will generally accept the trajectory in any number of standard inputs (i.e., pdb, xyz,. . . ) and then will generate snapshots which can be rendered individually or as a movie. In addition to providing the visualization, these codes have become progressively powerful analysis codes in their own right. They now have the ability to measure bond lengths, angles, and dihedrals as a function of time and determine the solvent accessible surface area, hydrogen bond network, and many other useful structure-related and dynamic properties of the system.

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Statistical Mechanics of Force-Induced Transitions of Biopolymers Sanjay Kumar

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymer and Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Enumeration Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3 7 8 9 9 11 12 13 18 21 21

Abstract

Single molecule force spectroscopy constitutes a robust method for probing the unfolding of biomolecules. Knowledge gained from statistical mechanics is helping to build our understanding about more complex structure and function of biopolymers. Here, we have review some of the models and techniques that have been employed to study force-induced transitions in biopolymers. We briefly describe the merit and limitation of these models and techniques. In this context, we discuss statistical models of polymer along with numerical techniques, which may provide enhanced insight in understanding the unfolding of biomolecules.

S. Kumar () Department of Physics, Banaras Hindu University, Varanasi, India e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_8-2

1

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S. Kumar

Introduction Recent technological developments of experimental techniques, for example, optical tweezers, atomic force microscope, etc., made it possible to apply a force of the order of pN to manipulate the single biomolecules (Kumar and Li 2010; Rief et al. 1999, 1997; Tskhovrebova et al. 1997). Many interesting results, for example, structural, functional, and elastic properties of biomolecules, information about the kinetics of biomolecular reactions, and detection of molecular intermediates have been obtained (Bustamante et al. 2000; Cecconi et al. 2005; Smith et al. 1992, 1996). Moreover, these experiments also provided a platform to verify theoretical predictions based on the models developed in the framework of statistical mechanics (Bhattacharjee 2000; Giri and Kumar 2006; Kumar 2009; Kumar and Giri 2007; Kumar et al. 2005; Lubensky and Nelson 2000; Marenduzzo et al. 2001a; Zhou et al. 2006). Many biological reactions involve large conformational changes which provide well-defined mechanical reaction coordinates, for example, the end-to-end distance of a polymer, that can be used to follow the progress of the reaction (Bustamante et al. 2004; Kumar et al. 2007). Such processes have been modeled by a simple two-state model (Bustamante et al. 2004). The applied force “tilts” the free energy surface along the reaction coordinate by an amount linearly dependent on the end-to-end distance. The kind of transitions induced by the applied force are the folding–unfolding transition of proteins, the stretching and unzipping transition of double-stranded DNA, or the ball–string transition of a polymer (Kumar and Li 2010). From thermodynamics point of view, the change in energy of system can be categorized into components related to the heat exchanged and the work done on or performed by the system. If the change in energy of the system is quite low then the system remains in quasi-static equilibrium. The aim of this chapter is to provide the concepts from statistical mechanics to describe the force-induced transitions of biopolymers. In section “Models”, we discuss some of the models of polymers, which have been used to study biopolymers in the presence of a force f. Section “Methods and Techniques” dwells with few techniques used in statistical mechanics for polymers. Among these techniques, we choose an exact enumeration technique to study force-induced desorption of polymer adsorbed on the surface in section “Applications”. The method developed for homo-polymers can also be extended to study the protein unfolding and DNA unzipping in quasi static equilibrium. The chapter ends with some results based on model studies which are interesting for future experiments in section “Conclusions”.

Models Macromolecules from living organisms are called biopolymers which vary in their size, shape, and function. Polymers including biopolymers are made up of long chains of “monomer” units (de Gennes 1979; des Cloizeaux and Jannink 1990;

Statistical Mechanics of Force-Induced Transitions of Biopolymers

3

Grosberg and Khokhlov 1994; Vanderzande 1998). The monomers can be of different natures. In DNA and RNA they are called nucleotides, and amino acids in proteins. Building blocks of simple artificial polymers can be group of just a few atoms, for example, CH2 in polyethylene or a complex structure in amino acids. The difference between polymers and biopolymers is in their structure. Biopolymers have generally well-defined structures. The most intensive theoretical study related to biomolecules can be performed with the help of all-atom simulations. All-atom models, which provide the most detailed description on the atomistic level, include the local interaction and interaction involved in non-bonded monomers. The later include the (6–12) Lennard-Jones potential, the electrostatic interaction, and the interaction with environment (Kumar and Li 2010). However, such study for a long chain is computationally demanding and is difficult to handle analytically. In order to get the key features, physicists attempt to simplify complex structures of biopolymers as much as possible. Such process is termed as coarse graining. They model polymer chains as threads or necklaces made of beads on a string which describes some essential properties of biopolymers (de Gennes 1979; des Cloizeaux and Jannink 1990; Grosberg and Khokhlov 1994; Vanderzande 1998) but not always. In the literature, coarse-grained models are divided into two broad categories: (1) Continuum models and (2) Lattice models, which are briefly discussed below.

Continuum Models We briefly describe three models, namely, the Gaussian Chain model, Freely Jointed Chain (FJC) model, and Worm Like Chain (WLC) model, which have been extensively used to describe the force-extension curves of biopolymers. The advantages of these models come from their simplicity and allowing one to derive analytical expressions in a simple form. In these models a polymer chain consists of N beads (monomers) of contour length L. A point in d-dimensional space represents each monomer and the distance between two consecutive monomers is Ri 1  Ri (see Fig. 1). The energy with force f (along x-direction) in Gaussian model is expressed as (Doi and Edwards 1986) 3kB T F D 2b

Z

LDN b 0



@r.s/ ds @s

2 f b x0

Z

Nb

ds 0

@r.s/ ; @s

(1)

where b is the effective bond length known as the Kuhn length and r(s) describes the local state at arc-length point s. For example, in the case of a flexible Gaussian chain, r is a three-dimensional position vector, whereas for WLC case it represents the unit tangent vector. kB is the Boltzmann constant and T the temperature of the system. Using the path integral technique (Doi and Edwards 1986; Kleinert 1990), the endto-end distance distribution function of a chain under the force can be written as

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S. Kumar

a

b

i+2 i+1

Arbitrary Angle

θ

Arbitrary Angle θ i i–1

Fig. 1 (a) Schematic of a freely jointed chain. (b) Freely rotating chain with fixed bond angle. These models do not incorporate excluded volume effects in their description

 PN .R; f / D

3 2N b 2

.3=2/

(

2 )  N b2f b x0 3 R exp  : 2N b 2 3kB T

(2)

The x-component of R can be expressed as Z

PN .Rx ; f / D dRy dRz PN .R; f / (  2 ) .3=2/  1 3 N b2f D Rx  exp  ; 2N b 2 2N b 2 3kB T

(3)

which gives the expression for the extension x in the presence of applied force f (Dai et al. 2003) x.f / D

Nb f b : 3 kB T

(4)

This is a linear force relation. The model describes the response of a single polymer chain under low force. The major limitation of the model is the property that the distance between two monomers can be extended without any limit. This shortcoming of Gaussian chain is removed in the FJC model, where the distance between two consecutive monomers (bond length) is kept fixed while the rotational angle occurs with equal probability (Fig. 1b). The free energy of the system can be written as N X F D f b x0 rn ; nD1

where rn are bond vectors with constant length jrn j D b.

(5)

Statistical Mechanics of Force-Induced Transitions of Biopolymers

5

The distribution function of end-to-end vector R in the presence of force f is defined as PN .R; f / D R

exp Œˇf b x 0  R PN .R; 0/ ; d R exp Œˇf b x 0  R PN .R; 0/

(6)

where ˇ D 1=kB T and PN (R, 0) is the end-to-end distance distribution function in the absence of force f, which is given as 1 PN .R; 0/ D N C1 .N  2/Šb 2 R 2  .N  2n  R=b/

Œ.N R=b/=2

X

N 2

.1/

nD0

n



N n

 (7)

:

The component of end-to-end distance along the x-direction may be obtained by integrating Eq. 6 with respect to Ry and Rz as (Dai et al. 2003) PN .Rx ; f / D

exp .ˇf Rx / Œ4 sinh .ˇf b/ =ˇf bN

PN .Rx ; 0/ ;

(8)

where the value of PN (Rx , 0) can be obtained from Eq. 7. The average extension along the force direction is   1 C coth .ˇf b/ : x.f / D bN  ˇf b

(9)

One of the important characteristics of biopolymers not described by the FJC model is the stiffness of the chain. A more realistic model for many systems is that of the Freely Rotating Chain, or Worm Like Chain (WLC), which includes stiffness in its description. This model builds an extension of the FJC model with the assumptions that the bond angles are fixed at certain angle, but are free to rotate. This gives rise to a uniform distribution of dihedral angles. The end-toend distance within the WLC model can be calculated by transfer matrix methods, numerical simulations, etc. One can find variational expressions for x as a function of f (Chatney et al. 2004; Marko and Siggia 1995; Rosa et al. 2003a). The one used extensively in the literature is (Marko and Siggia 1995) f D

kB T 2b



1 .1  x=L/2

1C

 4x : L

(10)

This equation also reduces to the Gaussian chain result in the small force regime i.e., f < kB T =b and for large force f > kB T =b, it acquires the form s x DL 1

kB T 2f b

! :

(11)

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S. Kumar

Fig. 2 Force-extension curves for the Gaussian chain, FJC, and WLC models

10

Force (pN)

8

WLC FJC Gaussian Chain

6

4

2

0

0.6

0.81 Relative extension

One can see from this equation that the difference between the contour length L and the extension x varies as f 0:5 . The force-extension curves for the Gaussian chain, FJC and WLC are shown in Fig. 2. The behavior of a biopolymer subjected to a large force as described by the WLC model is contrary to the behavior as described by the FJC model, whereas at low force all models show identical behavior. The Kuhn length of the WLC model corresponds to twice the persistence length. One of the important shortcomings is that all these models have only one free parameter, known as the persistence length or Kuhn length, and hence are not suitable when it comes to describing the entire force-extension curve involving many intermediates. It is important to recall that all these models ignore crucial excluded volume effect, i.e., the space occupied by a monomer is not available to other monomers (de Gennes 1979). Apart from these, in a polymer solution three types of interactions may be present (de Gennes 1979): (1) monomer–monomer interactions, (2) monomer–solvent interaction, (3) solvent–solvent interaction. If monomer–monomer (or solvent–solvent) interaction is more than the monomer–solvent interaction in the solution, such solvent is known as a poor solvent. However, if monomer–solvent interaction favors in the solution, the solvent is referred as a good solvent. It is possible by lowering the temperature, one can go from the expanded coil state to a compact globule state and such transition is known as a coil–globule transition and transition point is called ™-point. Generally, forceinduced transitions have been studied by these models where excluded volume effects and attractive interactions between chain segments have been ignored and is thus well suited only for modeling the stretching of polymers in a good solvent (Kumar and Li 2010).

Statistical Mechanics of Force-Induced Transitions of Biopolymers

7

Lattice Models Any model designed to represent a polymer chain over a full range of physical conditions must include important effects such as excluded volume and attractive interactions as a starting point. In a simplified assumption, a linear biopolymer chain in a solvent can be described by a walk on a lattice in which a step or vertex of the walk represents a monomer. If the walk is allowed to cross itself without restriction such walk is referred as Random Walk, which is same as that of the Gaussian chain (de Gennes 1979; Vanderzande 1998). The excluded volume effect has been incorporated in the Random Walk model of polymer with a constraint that a lattice site cannot be visited more than once. This kind of walk is known as Self-Avoiding Walks (SAWs), which simulates a linear polymer chain in a good solvent (Fig. 3a). A polymer chain in a poor solvent is modeled by Self-Attracting Self-Avoiding Walks (SASAWs) by including self-attraction among non-bonded monomers. This model exhibits collapse transition including the existence of ™-point (de Gennes 1979; Vanderzande 1998). Various kinds of underlying lattices have been proposed in the literature to study the conformational properties of linear polymer chains. The choice of the lattice depends on the mathematical convenience, nature of the system, and the interactions present in the system. As far as universality is concerned, the nature of underlying lattice and the detail

a

b

y

y ^ n

x

x

c

y

y1

y3

fx yb

a1

fy

a2 y2 y0

y4

yb−1

x

Fig. 3 Schematic diagrams of SASAW, DW, and PDSAW model. A force has been applied at one end (along x-axis) keeping other end fixed. In order to study the role of anisotropy, one may apply force along y-direction (Fig. 3c)

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S. Kumar

of interactions do not matter much and many important properties associated with polymers can be derived, which are in qualitative agreement with the experiments. Sometimes quantitative agreement has also been achieved particularly in determining the critical exponents (de Gennes 1979; Vanderzande 1998). Therefore, SAWs with suitable interactions on lattice has been studied extensively in describing the various properties of polymers and biopolymers. In addition to self-avoidance, one can also introduce additional constraint on the walks, for example, certain direction(s) is (are) not accessible to the walker, such walks are called Directed Walks (DWs) (Privman and Svrakic 1989; Vanderzande 1998). To define a directed walk, a preferred direction b n on the lattice is assigned. Walkers are allowed to take only those steps in the non-negative direction of b n. The directed walk can be seen as a model of polymer that is subject to some external force in the direction of b n, for example, flow in which the polymer is immersed or an electric field acting on electrically charged polymers. If the direction b n is assigned as shown in Fig. 3b and walks in the non-negative projection of b n are not allowed then such walks are called Fully Directed Walks (FDWs). If the direction b n is assigned, say, along the x-axis (Fig. 3c), then the walks are said to be the Partial Directed Walks (PDSAWs). For example in two dimensions, the walker can take step in ˙ y directions but only in C x direction. The major advantage of the directed walk model of polymer is that it can be solved analytically and many important results may be derived exactly. The drawback is that it is too far from the real chain.

Polymer and Critical Phenomena It is known that the certain quantities associated with polymers modeled by SAWs for, example, number of distinct conformation (CN ), number of closed polygons (PN ), and end-to-end distance of chain (Re ) of N monomers scale as (de Gennes 1979) CN  N N 1 ; PN  N N ˛3 ; Re  N  ;

(12)

where  is the connectivity constant giving the number of choices per step for an infinitely long walk. From the phase transition and critical phenomena, we also know that certain physical quantities like susceptibility (), specific heat (C), and correlation length () near the transition point TC scale as ˇ ˇ ˇ T  Tc ˇ ˇ ;   0 ˇˇ Tc ˇ ˇ ˇ ˇ T  Tc ˇ˛ ˇ ˇ ; (13) C  C0 ˇ Tc ˇ ˇ ˇ ˇ T  Tc ˇ ˇ ˇ :   0 ˇ Tc ˇ

Statistical Mechanics of Force-Induced Transitions of Biopolymers

9

A relation between polymer statistics and phase transition was established by de Gennes (1979) and des Cloiseaux (1974) showing a correspondence between the polymer chain modeled by SAWs and the n-vector spin model of magnetization in the limit n ! 0. Similarities between correlation length (Ÿ) and the end-to-end distance (Re ) can be noticed by comparing Eqs. 12 and 13. Correspondence between c c 1/N and T T is viewed as T T ! 0 and N ! 1. This equivalence allowed Tc Tc polymer science to benefit from the vast knowledge accumulated in the study of critical phenomena. For example, the following relations, which are quite non-trivial to derive directly, are also valid here (de Gennes 1979): ˛ C 2ˇ C  D 2; 2  ˛ D d ;  D  .2  / :

(14)

Here, ˛, ˇ,  , and are the critical exponents and d represents the dimensionality of the system. The above equivalence may serve an important role in describing the long range behavior of the polymers.

Methods and Techniques Methods and techniques used in critical phenomena can also be applied to polymers and biopolymers to gain further insight of the system. In the following, we discuss some of the techniques of statistical mechanics which have been used in analyzing the results of single molecule force spectroscopy.

Generating Function Technique The generating function technique is a very powerful technique, which may be adapted to study the conformational properties of biopolymers (Forgacs et al. 1995; Privman and Svrakic 1989). Here, we use a simple example to illustrate how the generating function approach can be applied and exact results can be derived. Let us consider a directed walk model of a polymer chain (Privman and Svrakic 1989; Vanderzande 1998). For simplicity, we consider partial directed walks (PDWs) in which the walker is allowed to move along x- and ˙ y-axes only (Fig. 3c). As discussed above biopolymers are in general semi-flexible. Stiffness is introduced into the polymer chain by putting an energy cost "b on every bend of the walk thus "  b giving rise to an associated Boltzmann weight, k D e kB T . For k D 1 ."b D 0/ the chain is said to be flexible, while for 0 < k < 1; .0 < "b < 1/ the chain is said to be semi-flexible. The grand canonical partition function of such a chain can be written as (Mishra et al. 2003; Privman and Svrakic 1989) Z .z; k/ D

1 X

X

N D0 al l walks

z N k Nb :

(15)

10

S. Kumar

y k x

x

x

k

x

y y k y

y

y

x

Fig. 4 The diagrammatic representations of the recursion relations Eqs. 16 and 17 for PDWs. The thick arrows X and Y denote all possible walks with the initial step (fugacity) along C x and ˙ y directions

Here, z is the fugacity of the walk and Nb is the number of bends in a given configuration. In two dimensions, the grand canonical partition function defined by Eq. 15 may be expressed as a sum of two components of PDWs. The recursion relations for the case of a semi-flexible polymer chain are X D z C z .X C 2kY / ;

(16)

Y D z C z .kX C Y / :

(17)

and

Schematic representations of the above equations (for 2D) have been shown in Fig. 4. It may be noted that the first term of Eqs. 16 and 17 is the fugacity of the walk and remain constant as z. Solving Eqs. 16 and 17 we get XD

z C .2k  1/ z2 ; 1  2z C z2  2z2 k 2

(18)

Y D

z C .k  1/ z2 : 1  2z C z2  2z2 k 2

(19)

The partition function of the system can, therefore, be written as Z .z; k/ D X C 2Y D

.4k  3/ z2 C 3z : 1  2z C z2  2z2 k 2

(20)

Statistical Mechanics of Force-Induced Transitions of Biopolymers

11

The critical point for polymerization of an infinite chain is found from the relation 1  2z C z2  2z2 k 2 D 0:

(21)

This leads to an expression for the critical value of the step fugacity as a function 1 p . In the limit k ! 1, it reduces to the well-known value for the of k, zc D .1C 2k / flexible polymer chain in 2D (Privman and Svrakic 1989). The approach has been successfully applied to study the DNA unzipping and unfolding of collapse polymer (Marenduzzo et al. 2001a, b; Rosa et al. 2003a, b).

Exact Enumeration Technique The conformational properties of a polymer chain and phase transition phenomena can be understood if one has the complete information about the partition function of the chain. In the lattice model, the canonical partition function of a polymer chain is calculated (Domb and Lebowitz 1989) by enumerating all possible walks of a given length. For example, we show total number of conformations (CN ) for N step walk in Table 1. The grand canonical partition function defined in Eq. 15 can be written as Z.z/ D

X

CN z N ;

(22)

N

where z is the fugacity associated with each step of the walk. As the singularity of the partition function is associated with critical phenomena, the partition function defined in Eq. 22 will follow Z.z/  .1  z/ in the thermodynamic limit. Table 1 Values of CN for different N for the square lattice. The value of CN can be obtained numerically or one may see the following webpage: http://www.ms. unimelb.edu.au/iwan/saw/ series/sqsaw.ser

N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CN 4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416596

N 16 17 18 19 20 ::: ::: ::: ::: ::: ::: ::: 53 54 55

CN 17245332 46466676 124658732 335116620 897697164 ::: ::: ::: ::: ::: ::: ::: 99121668912462180162908 263090298246050489804708 698501700277581954674604

12

S. Kumar 2.74 2.72 2.70 μ 2.68 2.66 2.64 2.62 0.00

0.02

0.04

0.06

0.08

0.10

1/N

Fig. 5 Variation of connectivity constant ( ) with 1/N for SAW on square lattice. The linear extrapolation gives D 2. 636 which is quite close to the best estimated value

In most of the cases, where model is not analytically solvable, one uses numerical techniques (Domb and Lebowitz 1989) to calculate the partition function. Once the partition function is known, other thermodynamic variables can be calculated. Since in a system of finite size the true phase transition cannot take place, therefore, one has to use suitable extrapolation scheme to calculate CN in the limit N ! 1. For this purpose suitable techniques, for example, ratio method, Pade approximants, differential approximation, etc. (Domb and Lebowitz 1989) can be used. In ratio method, the approximate value of for N ! 1 can be calculated by taking the ratios of consecutive terms of the series. These quantities should, for large N, be a linear function of 1/N log

CN C1 ' log  C .  1/ O .1=N / ; CN

(23)

where O .1=N / lower order correction. A simple fit then gives estimates for  and C  (Domb and Lebowitz 1989). However, there is an odd-even effect in CNNC1 . One q CN C2 can avoid this by using the square root of the successive ratios of CN as  D CN and extrapolate it to N ! 1. Linear extrapolation of  with

1 N

is shown in Fig. 5.

Monte Carlo Simulation The Monte Carlo (MC) technique is extremely simple in principle and has been greatly expanded and applied to a wide variety of problems (Binder 1995; Grassberger et al. 1999; Landau and Binder 2005). The typical goal of Monte Carlo sim-

Statistical Mechanics of Force-Induced Transitions of Biopolymers

13

ulations is first to find out the ground state of a given system (optimization problem) and then calculate ensemble averages through random sampling (equilibrium problem). The core of this approach is a set of predefined moves, which are traditionally on the lattice but may be defined in free space. At each time step a move is selected and may be accepted or rejected based on some criterion, for example, Metropolis method (Binder 1995), which has been used extensively, is defined as follows: 1. Start with some initial configuration I at a given temperature T 2. Define some rule to go from the present configuration to a new one 3. Compute the energy difference E D Enew  Eold , where Enew and Eold are energies of the system after and before the move 4. If E < 0, accept the move with new configuration. Otherwise, generate a random number r between 0 and 1 and accept the trial configuration if exp .ˇE/ > r. Such choice of acceptance of move is known as Metropolis method 5. Repeat Steps 2–4 till enough configurations have been sampled 6. Repeat Steps 1–5 for different temperatures The way in which temperature is decreased is known as the cooling process. Ideally one should try to devise an optimal way to find the annealing schedule, as the decrease rate and the number of MC steps per T can be varied during the numerical simulation. An important parameter to monitor during the annealing is the number of accepted moves that do not violate physical constrains. At high T, this number is very high but at low temperatures almost all moves may be rejected, and therefore, caution should be taken in sampling. One of the biggest advantages of the MC is that it can be very fast especially if the moves are selected carefully. The disadvantage is that it is not generally suitable for studying the dynamics since it is dependent on the move set, and method fails at low temperature. There are a number of good reviews and books available, and serious readers are advised to go through them (Binder 1997; Leckband and Israelachvili 2001; Muller et al. 2006; Muller-Plathe 1997).

Molecular Dynamics One of the important tools that has been employed in statistical mechanics is the molecular dynamics (MD) simulations (Allen and Tildesley 1987; Frenkel and Smit 2002). Quite frequently, this technique has been used to study the biomolecules (Kumar and Li 2010). Before studying the structural and dynamical properties of biomolecules, it is important to note that these molecules exhibit a wide range of time scales over which specific processes take place. For example, local motion, which involves atomic fluctuation, side chain motion, and loop motion occurs in the length scale of 0.01–5 Å and the time involved in such process is of the order of 10 15 to 109 s. The motion of a helix, protein domain, or subunit falls under the rigid body motion whose typical length scales are in between 1 Å and 10 Å and time involved in such motion is in between 109 and 1 s. Large-scale motion consists

14

S. Kumar

of helix–coil transition or folding–unfolding transition, which is more than 5 Å in length and, time involved is about 107 to 104 s. Hence the basic goal of molecular dynamics is to understand the role of different length and time scales involved in describing the physical phenomena. Since MD simulation provides information at microscopic level, it is desirable to use the basic concepts of statistical mechanics to derive the macroscopic observable like pressure, energy, heat capacity, etc. In MD one generates a sequence of points in phase space (pi , qi ) where pi is the momentum of the ith particle and qi is the position as a function of time. These points belong to the same ensemble and correspond to different conformations (Allen and Tildesley 1987; Frenkel and Smit 2002). The ergodic hypothesis states that the ensemble average is equivalent to time average. Hence if one allows system to evolve in time indefinitely then system will pass through all possible states and thus corresponds to the time average. Therefore, the goal is to generate enough representative point such that equality is satisfied. The method is based on Newton’s second law of motion, that is, Frenkel and Smit (2002), Allen and Tildesley (1987) Fi D m i

dEp d 2r D ; 2 dt dri

(24)

where F i is the applied force on the i-th particle of mass mi and Ep is the potential energy of the system. The potential energy, in general, is a complicated function and there is no analytic solution to the equation of motion and therefore, one has to solve these equations numerically. There are various algorithms/numeric schemes for integrating Newton’s equations of motion (Allen and Tildesley 1987; Frenkel and Smit 2002). They are generally derived from the Taylor expansion. The expansion for the positions (r), velocity (v), and acceleration (a) of the particles around some moment of time t are 1 r .t C t / D r.t / C v.t /t C a.t /t 2 C : : : ; 2

(25)

1 v .t C t / D v.t / C a.t /t C b.t /t 2 C : : : ; 2

(26)

a .t C t / D a.t / C b.t /t C : : : ;

(27)

where t is the time step used in the simulation. In the following, we will discuss some of the common schemes of integration algorithm, which have been extensively used in literature.

Statistical Mechanics of Force-Induced Transitions of Biopolymers

15

• Verlet algorithm: The Verlet algorithm uses positions and accelerations at time t and the positions from time t  t to calculate new positions at time t C t . In order to derive the Verlet algorithm one can write 1 r .t C t / D r.t / C v.t /t C a.t /t 2 ; 2

(28)

1 r .t  t / D r.t /  v.t /t C a.t /t 2 : 2

(29)

The sum of these equations gives r .t C t / D 2r.t /  r .t  t / C a.t /t 2 :

(30)

The algorithm does not use the explicit form of velocities. The usefulness of the algorithm are it is simple to implement, and does not require large memory. The algorithm is of moderate precision. • Leap Frog algorithm: In this algorithm, the velocities are first computed at time t C t and then used 2 to calculate the positions, r, at time t C t . The velocities leap over the positions and subsequently the positions leap over the velocities.   t t; r .t C t / D r.t / C v t C 2   t v .t C t / D v t  C a.t /t: 2

(31) (32)

The approximate form of velocities at time t is given by v.t / D

     t t 1 v tC Cv t  : 2 2 2

(33)

The major gain in this process is that the velocities are explicitly calculated, however, the shortcoming of the algorithm is that velocities are not calculated at the same time as the positions. • The Velocity Verlet algorithm: It may be noted that in the Leap Frog algorithm, velocities are not defined at the same time as the positions. As a consequence kinetic and potential energies are also not defined at the same time, and hence one cannot directly compute the total energy in the Leap Frog algorithm. It is, however, possible to implement in the Verlet algorithm a form that uses positions and velocities computed at same instant of times.

16

S. Kumar

1 r .t C t / D r.t / C v.t /t C a.t /t 2 ; 2 v .t C t / D v.t / C

1 Œa.t / C a .t C t / t: 2

(34) (35)

It has to be emphasized here that the Velocity Verlet algorithm is not memory consuming, because it is not required to keep track of the velocity at every time step during the simulation. Moreover, the long-term results of Velocity Verlet are equivalent to the Semi-implicit Euler method, and there is no compromise on precision. • Beeman’s algorithm: This algorithm is related to the Verlet algorithm and yields the same trajectories. However, it provides better estimate of the velocities and looks different than the Verlet algorithm. 2 1 r .t C t / D r.t / C v.t /t C a.t /t 2  a .t  t / t 2 ; 3 6

(36)

5 1 1 v .t C t / D v.t / C v.t /t C a.t /t C a.t /t  a .t  t / t: 3 6 6 (37) The advantage of this algorithm is that it gives accurate expression for the velocities and better energy conservation. The disadvantage is that because of the complex expressions computation is much more time-consuming. The procedure for a molecular dynamics simulation is subject to many userdefined variables. However, one should consider certain criteria in choosing these algorithms. It should conserve energy, momentum and computationally efficient so that long time step integration can be performed. Because the evaluation of the atomic positions is not performed on a continuous basis, but at intervals of a femtosecond (fs). Since this is the time-scale of stretches of the bonds with hydrogen atoms, these stretches should be constrained in order to permit time steps of 2 fs. The algorithm that permits this increase in simulation speed is called SHAKE (Frenkel and Smit 2002). Application of constraints on all bond lengths and bond angles would increase the permitted time step even more, without too much loss of information. Since most of the computational time is spent on evaluating the non-bonded interactions between atoms, the evaluation time step for non-bonded atom pairs can be increased to a small extent. Using a cutoff distance beyond which atoms are no longer considered to interact can significantly reduce the amount of non-bonded atom pairs. It may be noted that MD is a scheme for studying the time evolution of a classical system of N particles in volume V, where total energy is a constant of motion. As pointed out earlier, time averages are equivalent to ensemble averages, the observables, say A evolves according to constant energy simulation and hence,

Statistical Mechanics of Force-Induced Transitions of Biopolymers

hAiN;V;E D lim

Z

!1

17

A.t /dt:

(38)

0

It is possible to run many NVE simulations with different energies and result could be used in a Boltzmann-weighted average as hAiN V T D

X

hAiN VE e ˇE :

(39)

E

However, such process is not very practical, because simulations are quite expensive in terms of time. Moreover, we require a lot of E values to have enough statistics, and to conserve energy during simulation is a tough task. Thus, it is more appropriate to run simulation in constant temperature ensemble, that is, NVT. In the following, we will discuss some popular temperature control schemes (Allen and Tildesley 1987; Frenkel and Smit 2002). 1. Scaling velocities (Berendsen thermostat): Since in the beginning, the system is not equilibrated and hence remains in the incorrect state. The potential energy is converted to thermal energy or thermal energy is being consumed during equilibration. In order to stop the temperature from drifting, one of the possible ways is to constrain instantaneous velocities scaled by a factor at each step s v0i D

Td vi ; Ti

(40)

where Td is desired temperature and Ti is instantaneous temperature given by XN p 2 i Ti D 3N2kB after i-th step. This influences the dynamics severely i 2mi and does not give correct canonical ensemble because it does not generate fluctuations in temperature which are also present in the canonical ensemble. The method is good for initialization of phase. A weaker formulation of this approach is the Berendsen thermostat where to keep temperature constant, system is coupled to an external heat bath of temperature T0 . The velocities are scaled in such a way that the rate of change of temperature is proportional to the difference in temperature between system and bath, that is, T0  T .t / dT D ; dt c

(41)

where c is coupling parameter of the system with a bath. The scaling factor for velocities is given by

18

S. Kumar

s D

t 1C c



 T0 1 : T .t  t =2/

(42)

2. Adding stochastic forces and/or velocities (Langevin thermostat): This thermostat models the influence of a heat bath by adding to the velocity of each particle a small random (white) noise and a frictional force directly proportional to the velocity of that particle. These two factors are balanced to give a constant temperature. Since each particle is coupled to a local heat bath, the heat trapped in localized modes can be removed by using this model. ma D v C f .r/ C f 0 :

(43)

Here f (r) and f 0 are conservative force and random force respectively.  is the friction coefficient. The drawback of this thermostat is that momentum transfer is destroyed. So, it is not advisable to use Langevin thermostat in the simulations where one wishes to study the diffusion processes. 3. Nosé–Hoover thermostat: The Berendsen thermostat is quite efficient for relaxing a system to the target temperature. However, once your system has reached equilibrium, it is important to probe a correct canonical ensemble. The Nosé–Hoover thermostat is an extended-system method for controlling the temperature of simulated system (Huenenberger 2005; Thijssen 1999). It allows temperatures to fluctuate about an average value, and uses a friction factor to control particle velocities. This particular thermostat can oscillate when a system is not in equilibrium. Therefore, it is recommended to use a weak-coupling method for initial system preparation (e.g., Berendsen thermostat), followed by data collection under the Nosé–Hoover thermostat. This thermostat produces a correct kinetic ensemble. In this context, molecular dynamics simulation of nucleic acids have been considered as a bigger challenge because of the negative backbone charge and the poly-electrolytes behavior. In oligonucleotide dynamics simulations, particular attention should be paid to the atomic charges. The negatively charged phosphate groups may very well influence the trajectory. In contrast with molecular mechanics where a structural minimum is the end result, molecular dynamics offers so much information that is hard to quantify. Excellent reviews on molecular dynamics and its use in biochemistry and biophysics are numerous(see, e.g., Adcock and McCammon 2006 and references therein).

Applications FJC and WLC models have been used in analyzing the force-extension curve obtained from SMFS experiments. In this context, molecular dynamics and Monte Carlo simulations have been found to be quite useful in understanding the phenomena. Since there is a lot of good literature available in this subject, we are not going

Statistical Mechanics of Force-Induced Transitions of Biopolymers Fig. 6 Schematic diagram of the polymer chain adsorbed on the surface under the application of external force

19

f h

to discuss them here again. In this section, we shall illustrate that lattice model along with exact enumeration technique which may provide enhanced insight in the mechanism involved in the force-induced transitions because exact density of states are available and the entire phase diagram can be probed exactly for small chains. We consider SAWs that start from a point on an impenetrable surface and experience a force f in a direction perpendicular to the surface at the other end as shown in Fig. 6. The applied force, because of its direction, favors desorption and one expects a critical force (fc ) for the desorption. At a given temperature (T) when the applied force f is less than fc (T) the polymer will be adsorbed, while for f > fc .T / the polymer will be desorbed. The curve fc (T), therefore, gives the boundary that separates the desorbed phase from the adsorbed phase in the force– temperature (f, T) plane (Mishra et al. 2005). Here, CN (Ns , h) corresponds to the number of SAWs of N steps having Ns number of monomers on the surface and h, the height of the end-monomer away from the surface. In this case, the partition function may be defined as ZN .!; u/ D

X

CN .Ns ; h/ ! Ns uh ;

(44)

Ns ;h

where ! D e "s =kB T and u D e f =kB T are the Boltzmann weights for the surface interaction ("s < 0) and the applied force respectively. In the following, we set the Boltzmann constant kB D 1 and "s D 1. For a fixed force f, we locate the adsorption–desorption transition temperature ˝ ˛ from the maximum of fluctuation in number of adsorbed monomers (i.e., Ns2  hNs i2 ) (see Fig. 7), where hNs i and hN 2s i are defined as: hNs i D

X 1 Ns CN .Ns ; h/ ! Ns uh ; ZN .!; u/

(45)

X 1 Ns2 CN .Ns ; h/ ! Ns uh : ZN .!; u/

(46)

Ns ;h

and ˝ 2˛ Ns D

Ns ;h

20

S. Kumar

a

b

80 2D

−2

−2

60

40

20

f=0 f = 0.8 f = 1.1

10

20

0

3D

30

f=0 f = 0.5 f = 0.9

0

0.4

0.8

1.2

1.6

0

2

0

0.6

1.2

T

1.8

2.4

3

T

Fig. 7 The dependence of critical force fc (T) on T in (a) 2D and (b) 3D. The star corresponds to results obtained from the extrapolated values of the reduced free energy, and cross corresponds to the value obtained from finite size data of a step N D 20 (3D) and N D 31 (2D) respectively

a

b

2

2D

1

3D Desorbed 1.5

0.8 Desorbed f 0.6

f

Adsorbed

1

0.4

Adsorbed 0.5

0.2 0

0

0.5

1 T

0

0

1

2 T

Fig. 8 The dependence of critical force fc (T) on T in (a) 2D and (b) 3D. The star corresponds to results obtained from the extrapolated values of the reduced free energy, and cross corresponds to the value obtained from finite size data of a step N D 20 (3D) and N D 31 (2D) respectively

The force–temperature phase diagram is shown in Fig. 8. The occurrence of two maxima in fluctuation curve gives the signature of re-entrance in 3D, but absent in 2D (Mishra et al. 2005). Using the phenomenological argument and the probability distribution analysis, it was shown that the ground state entropy is responsible for the re-entrance which is absent in 2D (Mishra et al. 2005). It is possible to obtain better estimates of phase boundaries by extrapolating  for the large N. The reduced free energy per monomer in this case is defined as

Statistical Mechanics of Force-Induced Transitions of Biopolymers

G .!; u/ D lim

1

N !1 N

log ZN .!; u/ D log N .!; u/ :

21

(47)

N (¨, u) can be estimated from the partition functions found from the data of exact enumerations for finite N by extrapolating to large N. The fluctuation in terms of 2 reduced free energy is given as @ G 2 . @.log !/ The method described above has been applied to protein unfolding (Kumar and Giri 2005, 2007; Kumar et al. 2007), stretching of DNA (Kumar and Mishra 2008; Mishra et al. 2009), and unzipping of DNA (Giri and Kumar 2006; Kumar et al. 2005) and, many useful information about biomolecules have been derived which are in qualitative agreement with experiments.

Conclusions We have discussed some basic models of biopolymers and few techniques which have been used extensively in the past to understand the mechanism involved in force-induced transitions. In particular, we showed that lattice model along with exact enumeration technique may be used to interpret the results of SMFS. It is important to point out here that all the single molecule experiments have been performed to understand the structure and function of biopolymers in vivo by analyzing it in vitro. As a result, the effect of cellular environment has been ignored in all these studies. It is known that the interior of the cell contains different kinds of biomolecules like sugar, nucleic acids, lipids, etc. These macromolecules occupy about 40 % of the total volume with steric repulsion among themselves. This confined environment induces phenomena like “molecular confinement” and “molecular crowding” and has major thermodynamic and kinetic consequences on the cellular processes. Recently Singh et al. (2009a, b) have used exact enumeration technique and showed that the molecular crowding has significant impact on the force-induced transitions. In order to have better understanding of force-induced transitions, it is advisable to perform SMFS experiments in the environment similar to a cell. Acknowledgments We would like to thank D. Giri, A. R. Singh, and G. Mishra for many helpful discussions. Financial assistance from the Department of Science and Technology, New Delhi is gratefully acknowledged.

Bibliography Adcock, S. A., & McCammon, J. A. (2006). Molecular dynamics: Survey of methods for simulating the activity of proteins. Chemical Reviews, 106, 1589–1615. Allen, M. P., & Tildesley, D. J. (1987). Computer simulations of liquids. Oxford: Oxford Science. Bhattacharjee, S. M. (2000). Unzipping DNAs: Towards the first step of replication. Journal of Physics A, 33, L423–L428.

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Binder, K. (1995). Monte Carlo and molecular dynamics simulations in polymer science. New York: Oxford University Press. Binder, K. (1997). Applications of Monte Carlo methods to statistical physics. Reports on Progress in Physics, 60, 487–559. Bustamante, C., Smith, S. B., Liphardt, J., & Smith, D. (2000). Single-molecule studies of DNA mechanics. Current Opinion in Structural Biology, 10, 279–285. Bustamante, C., Chemla, Y. R., Forde, N. R., & Izhaky, D. (2004). Mechanical processes in biochemistry. The Annual Review of Biochemistry, 73, 705–748. Cecconi, C., Shank, E. A., Bustamante, C., & Marqusee, S. (2005). Direct observation of the threestate folding of a single protein molecule. Science, 309, 2057–2060. Chatney, D., Cocco, S., Monasson, R., & Thieffry, D. (2004). Multiple aspects of DNA and RNA: From biophysics to bioinformatics: Lecture notes of the Les Houches Summer School. Amsterdam: Elsevier. Cloizeaux, J. D. (1974). Langrangian theory for a self-avoiding random chain. Physical Review A, 10, 1665–1669. Dai, L., Liu, F., & Ou-Yang, Z.-C. (2003). Maximum-entropy calculation of the end-to-end distance distribution of force-stretched chains. Journal of Chemical Physics, 119, 8124. de Gennes, P. G. (1979). Scaling concepts in polymer physics. Ithaca/London: Cornell University Press. des Cloizeaux, J., & Jannink, G. (1990). Polymers in solution. Oxford: Clarendon. Doi, M., & Edwards, S. F. (1986). The theory of polymer dynamics. Oxford: Clardenden. Domb, C., & Lebowitz, J. L. (1989). Phase transition and critical phenomena (Vol. 13). New York: Academic. Forgacs, G., Lipowsky, R., & Nieuwenhuizen, T. M. (1995). The behaviour of interfaces in ordered and disordered systems (Vol. 14). Oxford: Clarendon. Frenkel, D., & Smit, B. (2002). Understanding molecular simulation. London: Academic. Giri, D., & Kumar, S. (2006). Effects of the eye phase in DNA unzipping. Physical Review E, 73, 050903(R). Grassberger, P., Nadler, W., & Barkema, G. T. (1999). The Monte Carlo approach to biopolymers and protein folding. Singapore: World Scientific. Grosberg, A. Y., & Khokhlov, A. R. (1994). Statistical physics of macromolecules. New York: American Institute of Physics. Huenenberger, P. (2005). Thermostat algorithms for molecular dynamics simulations. Advances in Polymer Science, 173, 105–149. Kleinert, H. (1990). Path integrals in quantum mechanics, satistics, and polymer physics. Singapore: World Scientific. Kumar, S. (2009). Can reentrance be observed in force induced transitions? Europhysics Letters, 85, 38003. Kumar, S., & Giri, D. (2005). Force-induced conformational transition in a system of interacting stiff polymers: Application to unfolding. Physical Review E, 72, 052901. Kumar, S., & Giri, D. (2007). Does changing the pulling direction give better insight into biomolecules? Physical Review Letters, 98, 048101. Kumar, S., Giri, D., & Bhattacharjee, S. M. (2005). Force induced tripple point for interacting polymers. Physical Review E, 71, 051804. Kumar, S., Jensen, I., Jaconsen, J. L., & Guttmann, A. J. (2007). Role of conformational entropy in force induced biopolymer unfolding. Physical Review Letters, 98, 128101–128104. Kumar, S., & Li, M. (2010). Biomolecules under mechanical force. Physics Reports, 486, 1–74. Kumar, S., & Mishra, G. (2008). Force-induced stretched state: Effects of temperature. Physical Review E, 78, 011907. Landau, D. P., & Binder, K. (2005). A guide to Monte Carlo simulations in statistical physics. New York: Cambridge University Press. Leckband, D., & Israelachvili, J. (2001). Intermolecular forces in biology. Quarterly Review of Biophysics, 34, 105–267.

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Lubensky, D. K., & Nelson, D. R. (2000). Pulling pinned polymers and unzipping DNA. Physical Review Letters, 85, 1572–1575. Marenduzzo, D., Bhattacharjee, S. M., Maritan, A., Orlandini, E., & Seno, F. (2001a). Dynamical scaling of the DNA unzipping transition. Physical Review Letters, 88, 028102. Marenduzzo, D., Trovato, A., & Maritan, A. (2001b). Phase diagram of force-induced DNA unzipping in exactly solvable models. Physical Review E, 64, 031901. Marko, J., & Siggia, E. (1995). Stretching DNA. Macromolecules, 28, 8759–8770. Mishra, G., Giri, D., & Kumar, S. (2009). Stretching of a single stranded DNA: Evidence for structural transition. Physical Review E, 79, 031930. Mishra, P. K., Kumar, S., & Singh, Y. (2003). A simple and exactly solvable model for a semi flexible polymer chain interacting with a surface. Physica A, 323, 453–465. Mishra, P. K., Kumar, S., & Singh, Y. (2005). Force-induced desorption of a linear polymer chain adsorbed on an attractive surface. Europhysics Letters, 69, 102–108. Muller, M., Katsov, K., & Schick, M. (2006). Biological and synthetic membranes: What can be learned from a coarse-grained description? Physics Reports, 434, 113–176. Muller-Plathe, F. (1997). Coarse-graining in polymer simulation: From the atomistictothemesoscopicscaleandback. ChemPhysChem, 3, 754–769. Privman, V., & Svrakic, N. M. (1989). Directed models of polymers, interfaces, and clusters. Berlin: Springer. Rief, M., Clausen-Schaumann, H., & Gaub, H. E. (1999). Sequence-dependent mechanics of single DNA molecules. Nature Structural Biology, 6, 346–349. Rief, M., Gautel, M., Oesterhelt, F., Fernandez, J. M., & Gaub, H. E. (1997). Reversible unfolding of individual titin immunoglobulin domains by AFM. Science, 276, 1109–1112. Rosa, A., Hoang, T. X., Marenduzzo, D., & Maritan, A. (2003a). Elasticity of semiflexible polymers with and without self-interactions. Macromolecules, 36, 10095–10102. Rosa, A., Marenduzzo, D., Maritan, A., & Seno, F. (2003b). Mechanical unfolding of directed polymers in a poor solvent: Critical exponents. Physical Review E, 67, 041802. Singh, A. R., Giri, D., & Kumar, S. (2009a). Force induced unfolding of bio-polymers in a cellular environment: A model study. Journal of Chemical Physics, 131, 065103. Singh, A. R., Giri, D., & Kumar, S. (2009b). Effects of molecular crowding on stretching of polymers in poor solvent. Physical Review E, 79, 051801. Smith, S. B., Finzi, L., & Bustamante, C. (1992). Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads. Science, 258, 1122–1126. Smith, S. B., Cui, Y., & Bustamante, C. (1996). Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules. Science, 271, 795–799. Thijssen, J. M. (1999). Computational physics. Cambridge: Cambridge University. Tskhovrebova, L., Trinick, K., Sleep, J. A., & Simons, R. M. (1997). Elasticity and unfolding of single molecules of the giant muscle protein titin. Nature, 387, 308–312. Vanderzande, C. (1998). Lattice models of polymers. Cambridge: Cambridge University Press. Zhou, H. J., Zhou, J., Ou-Yang, Z. C., & Kumar, S. (2006). Collapse transition of two-dimensional flexible and semiflexible polymers. Physical Review Letters, 97, 158302.

Molecular Mechanics: Principles, History, and Current Status Valeri Poltev

The ultimate justification for the many severe approximations and assumptions made in the present work comes from the fact that the agreement between the simple calculations and the available experimental data is as good as it is.N. L. Allinger, J. Amer. Chem. Soc., 81, 5727, 1959

Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Foundations and General Scheme of Molecular Mechanics. Atoms as Elementary Units of the Matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Expression and Terms of Molecular System Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intramolecular Contributions to Molecular System Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermolecular and Nonbonded Intramolecular Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Remarks on Molecular Mechanics, Its Accuracy, and Applicability. . . . . . . . . . . . . . . . A Bit of History: The “Precomputer” and Early Computer-Aided MM Calculations. . . . . . . . . . First MM Applications to Three-Dimensional Structure and Thermodynamics of Organic Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Molecular Crystal Study on the First Steps of Molecular Mechanics. . . . . . . . . . . Molecular Mechanics on the First Steps of Molecular Biology: Molecular Mechanics and Protein Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Mechanics on the First Steps of the Biophysics of Nucleic Acids. . . . . . . . . . . . . . . . The Problems and Doubts of Further Development of the MM Approach. . . . . . . . . . . . . . . . . . . . . . Approaches to Improvement of MM Formulae and Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Schemes of Water Molecule in MM Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modern Molecular Mechanics Force Fields: Description, Recent Development, and Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allinger’s Force Fields and Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merck Molecular Force Field (MMFF94). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Force Fields and Programs Designed by Scheraga and Coauthors. . . . . . . . . . . . . . . . . . . . . . . Force Fields and Programs Developed by Kollman and Coauthors. AMBER. . . . . . . . . . . . . . . . Other Popular Force Fields and MM Software. CHARMM, OPLS, GROMOS. . . . . . . . . . . . .

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V. Poltev () Autonomous University of Puebla, Puebla, Mexico e-mail: [email protected] [email protected]

© Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_9-2

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Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

A short survey of the general principles and selected applications of molecular mechanics (MM) is presented. The origin of molecular mechanics and its evolution is followed starting from “pre-computer” and the first computer-aided estimations of the structure and potential energy of simple molecular systems to the modern force fields and software for the computations of large biomolecules and their complexes. Analysis of the current state of physicochemical study of biological processes suggests that MM simulations based on empirical force fields have an ever-increasing impact on understanding the structure and functions of biomolecules. The problem of “classic mechanics” description of essentially quantum properties and processes is considered. Various approaches to a selection of force field mathematical expressions and parameters are reviewed. The relation between MM simplicity and “physical nature” of the properties and events is examined. Quantum chemistry contributions to MM description of complex molecular systems and MM contribution to quantum mechanics computations of such systems are considered.

Introduction Molecular mechanics (MM) deals with the classical (“mechanistic,” i.e., via Newtonian mechanics) description of molecular and supramolecular systems. The simplified assumptions and approximations enable one to use MM for wide applications to various systems, starting from simple low-molecular-weight molecules (such as numerous hydrocarbons) to large biomolecular complexes (such as those of proteins, nucleic acids, and membrane fragments) or material assemblies of many thousands of atoms. During a period a bit longer than a half of a century, the number of publications using this approach roused by many orders of magnitude. It was rather easy to follow nearly all the works related to MM calculations in the 1960s and in the early 1970s of the last century (and the author of this chapter tried to do so); today it is very difficult to follow all the papers related to the MM approach even in the rather restricted area of application (e.g., to biopolymer structure and function). Thus it is impossible to consider all the applications of MM in a short survey, and the choice of the material for this chapter is “a matter of taste” of the author who refers readers to other monographs and chapters for a more general description of the modern state and/or selected aspects of extended applications of MM; for general references see, e.g., monographs (Lewars 2011; Allinger 2010; Ramachandran et al. 2008; Cramer 2004; Leach 2001). We will follow the evolution of MM from the first “precomputer” and early computer-aided (i.e., before the era of personal computers) works to modern complex simulations impossible without supercomputers.

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The MM method is used now not only in theoretical and computational works but also as a part of experimental studies (e.g., many X-ray and NMR-derived structures of proteins, nucleic acids, and their complexes with other molecules deposited in Protein Data Bank (Berman et al. 2003) and Nucleic Acid Data Bank (Berman et al. 1992) are the results of MM refinements). The MM-like semiempirical terms are widely used now in some quantum mechanics-based studies. The systematic inclusion of such terms started from the addition of the dispersion term to the standard density functional theory computations, for creating the so-called DFTD method (see, e.g., Grimme 2004; Antony and Grimme 2006; Jurecka et al. 2007, and references herein for the first DFT-D results). Recent developments and applications of different DFT-D methods (DFT-D2, DFT-D3) demonstrated their predictive power for biomolecular systems (see, e.g., Grimme 2011; Brandenburg and Grimme 2014). Since the beginning of this millennium, MM-like terms are frequently included into so-called enhanced semiempirical quantum mechanics methods (see, e.g., Yilmazer and Korth (2015) for a recent review). Both dispersion (D) and hydrogen-bond (H) correction terms are used in such methods usually referred to as SQM-DH methods. The rapid extension of MM simulations is a result of development of new branches of natural science related to molecular biology with its numerous applications, to directed synthesis of new substances with desirable properties (including new drugs and new materials for the industry), and many other areas of socalled nanoscience. The MM simulations enable one to perform a preliminary selection of the compounds with desirable properties before experimental testing, and even before chemical synthesis itself. Sometimes it is impossible to rationalize completely experimental results without the construction of atom-level mechanistic model. In many cases, the model cannot be derived from experimental data only, e.g., the accuracy of X-ray diffraction patterns for biopolymer fragments is not sufficient for the extraction of precise atom coordinates due to a complexity of the system and inherent irregularities of atom positions (e.g., delocalization of water molecules). In other cases, the complex molecular system cannot be characterized by a single three-dimensional structure but by a set of conformations. MM simulations enable one to construct habitual atom-level molecular models of sufficiently complex systems favorable from energy viewpoint. These models can be visualized by computer graphics programs and are ready to use for further investigations and refinements (e.g., by quantum mechanics methods or by new experimental approaches).

Foundations and General Scheme of Molecular Mechanics. Atoms as Elementary Units of the Matter The modern MM can be considered as an extension of simple atom and bond representation of molecules and their complexes in classical chemistry. In the beginning of the 1950s, scaled paper images and hand-made primitive models of atoms and bonds from wood and wire enabled subtle scientists to construct the

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first successful models of proteins and DNA. The pioneering papers of Nobel Prize authors can be mentioned in this relation (Pauling et al. 1951; Watson and Crick 1953). Stick, ball-and-stick, and space-filling plastic models as well as their computer images are widely used until now for both teaching and investigation.

General Expression and Terms of Molecular System Energy The MM approach is a generalization and quantitative representation of these models via mathematical expressions. A set of such expressions together with numerical coefficients is usually referred to as the force field; the MM method itself is frequently referred to as the force field method. Rigid balls of computer graphics or plastic (metal, wood, etc.) models are replaced by “soft” ones, and hard sticks are replaced by springs. “Hard Sphere” approximation can be considered as simplified (in many cases oversimplified) MM approach. Nevertheless, very important results were obtained via this approximation in the middle of the last century; we will mention some of them below. The modern MM method and most of force fields mathematical expressions are the results of the efforts of many researchers for nearly half a century, trials and errors of both well-known and widely cited authors, as well as of those who contributed to specific parts of nonbonded energy computations for a selected class of molecules. In this section, we will describe general scheme and the main common assumptions of the MM method leaving any references to specific papers and contributions of the researchers for next sections. The main subject of this survey is rather simple MM scheme suggesting that the systems (molecules or their complexes) consist of atoms, each atom being simulated as a single point particle. Most computations for biologically important molecules use the so-called Class 1 model. More complicated schemes considering additional centers of interactions, additional terms, and more exact expressions (so-called Class 2 and Class 3 MM models) will be mentioned briefly. Molecular mechanics describes molecules in terms of “bonded atoms” (atoms in molecule), their positions are distorted from some idealized (equilibrium) geometry due to nonbonded interactions with the atoms of the same molecule and of other molecules. The MM approach implicitly uses the starting approximation for both classical and quantum computational chemistry schemes, namely, Born–Oppenheimer approximation. MM does not include electrons explicitly, but different configurations of electron shells of the same chemical element’s atoms are implicitly considered in force field parameters. Some authors consider not only atoms but lone pairs of electrons, chemical bonds, and other points in a molecule as centers of interactions. We will mention such “extension” of Class 1 models below. The first principal approximation of MM method is the additivity for several levels of calculations, namely, the additivity of energy terms responsible for the contributions of different physical nature, and the additivity of contributions of

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the atoms. The simple (“minimalist”) representation of the potential energy of a molecule or molecular complex as a function of atom coordinates, R, can be represented by a sum of four main terms each term being a sum of many contributions (Eq. 1). E.R/ D

X

Eb C

X

Ea C

X

Et C

X

Enb

(1)

The summing up in the terms is over all the chemical bonds (the first term), valence angles (the second term), torsion angles (the third term), and all pairs of atoms nonbonded to each other or to common third atom (the last term). The energy terms depend on mutual positions of atoms and on the adjustable constants (parameters); the parameters are suggested to be transferable between atoms and molecules of the same type. The transferability of the force field parameters is the second important assumption of MM, and the number of atom types of the same chemical element depends on particular force field.

Intramolecular Contributions to Molecular System Energy The first and the second terms (sums of Eq. 1) refer to the energy changes due to variations of bond lengths, stretching (Eq. 2), and bond angles, bending (Eq. 3). These terms are usually modeled as harmonic potentials centered on equilibrium values of bond lengths and bond angles, respectively, i.e., “simple Hooke’s law” dependences are used.  2 Ebi D kbi li  li 0

(2)

 2 Eai D kai ’i  ’i 0

(3)

In these equations, li and ’i , the current values of bond lengths and angles; parameters li 0 and ’i 0 , equilibrium values for bond lengths and angles; kbi and kai , stretch and bend force constants, respectively. These adjustable parameters depend on types of atoms forming the bond or the valence angle. Some force fields may also contain cubic and higher-order contributions to these terms, or sometimes more flexible Morse potential can be used instead, as well as “cross terms” can be included to account for correlations between stretch and bend components. In the latter case, the terms depending on both bond length and angle variations are added. The third sum of the expression (Eq. 1) refers to the changes of torsion energy; it is responsible for interactions of electron shells of two atoms A and D (and of two bonds A–B and C–D) which are connected through an intermediate chemical bond B–C. Eti D kti .1 C cos .ni i –•i //

(4)

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This periodic function (integer ni being the periodicity) contains the current value  i of the angle i of the rotation around B–C chemical bond (the angle between the two planes defined by atoms A, B, and C and by B, C, and D) and three parameters (kti , ni , and •i ; ni is multiplicity and •i is the phase angle) for each type of torsion (for each combination of four neighbor atoms of the molecule). These parameters can be estimated from experimental data on the structure and properties of the molecule considered and of related molecules, then they (as well as li 0 , ’i 0 , kbi , and kai of Eqs. 2 and 3) should be adjusted by trial computations. Many force fields include terms responsible for “improper” torsions or out-of-plane bending, i.e., terms related to four atoms not forming consecutive chemical bonds, which function as correction factors for out-of-plane deviations (for example, they can be used to keep aromatic rings planar). These terms can be expressed via harmonic potentials like those for bond stretching and valence angle bending. Cross terms depending on both torsion angle and bond length or valence angle are added in some force fields. Three terms mentioned above are positive for practically all states of all the molecular systems (or equal to zero when the term corresponds to equilibrium state for contributed bond, angle or torsion).

Intermolecular and Nonbonded Intramolecular Interactions The last sum of Eq. 1 refers to so-called nonbonded interactions, Enb, of all the atom pairs not bonding to each other or to the same third atom, Eq. 5. Each atomatom term is usually represented by a sum of electrostatic, Coulomb (the first term of Eq. 5), and van der Waals (the second and the third terms of Eq. 5) interactions.   Eij rij D Kqi qj =rij  Aij =rij 6 C Bij =rij 12

(5)

This equation contains rij , the current distance between i and j atoms; qi and qj , effective atom charges; Aij and Bij , adjustable parameters responsible for dispersion (London) attraction and short-range repulsion interactions respectively. The atomic charges are usually derived using calculations via various quantum chemistry methods, effective dielectric constant implicitly accounting for surrounding can be used (this value may be distant dependent). The Aij and Bij coefficient can be preliminarily estimated via equilibrium inter-atomic distance and energy values at equilibrium for neutral pairs of atoms (¡ij and ©ij respectively) and followed by the adjustment to reference experimental data. Most of the early force fields used for description of van der Waals interactions are Buckingham (6-exp) potential instead of Lennard-Jones (6-12) potential in Eq. 5. The total expression for nonbonded interaction term are usually referred to as (1-6-12) or (1-6-exp) potential relating to the dependency of the terms on the inter-atom distance. The Buckingham potential is more flexible (it has three adjustable parameters instead of two for 6-12 potentials for each atom pair type) and has more physics justifications for the range of experimentally observed distances (due to exponential dependence of electron wave functions on the distance from nuclei), but it is less convenient for computations. It

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has a maximum at short distance and then trend to negative infinite value. Majority of the modern force fields utilize 6-12 expressions for description of van der Waals interactions, the total atom-atom potential being referred to as 1-6-12 one. Some force fields substitute 6-12 potential by 10-12 one for the interactions of hydrogen atoms of hydrogen bonds in order to describe more sharp distance dependence in the most important area of energy minimum corresponding to H-bond formation (referred to as 1-10-12 potential). More complex expressions (including those dependent on the angles between two straight lines connecting three atoms of Hbond) were used for H-bond description in some early potential sets. The nonbonded terms of the intramolecular energy related to 1-4 interactions (i.e., the interactions between atoms in a molecule separated by three chemical bonds) are frequently accounted for with a coefficient less than 1 (1-4 scaling) as these interactions are already included into torsion term (Eq. 4). To reduce the number of adjustable Aij and Bij parameters of Lennard-Jones potential (and corresponding parameters of other potentials), the combination rules for equilibrium inter-atom distance, ¡, and minimal energy values, ©, for pairs of different atoms are usually applied.  1=2 ¡ij D ¡i C ¡j I ©ij D ©i ©j

(6)

Some force fields apply the combination rules directly to the coefficients of van der Waals terms. The calculations of potential energy via Eq. 1 are used to search for local energy minima (mutual atom positions corresponding to possible stable configurations), to construct and analyze multidimensional potential energy surfaces (PES), to follow trajectory of movement (in MD, molecular dynamics simulations), or to study averaged thermodynamic and geometry characteristics (via MC, Monte Carlo sampling) of the systems. Description of MD and MC methods is beyond the survey, while some results of these applications of MM method will be mentioned briefly.

General Remarks on Molecular Mechanics, Its Accuracy, and Applicability The first computer (and all “precomputer”) applications of mechanistic approach to molecule conformations and interactions ignored certain energy terms (e.g., stretching, bending, torsion, or electrostatic ones). Some modern works can ignore certain terms as well in order to reduce the number of variables of energy function, e.g., considering bond lengths as constants (their changes in many cases are very small and have no influence on energy and geometry of minimal energy structures). The simplest one of such approaches considers bond lengths and valence angles as constants, ignores torsion energy (the contributions of the first three terms of Eq. 1 being zeros), and utilizes “hard sphere approach” to nonbonded interactions. This approach is a mathematical representation of plastic (or wood, iron, etc.) space filling mechanic models or their computer images. The configurations are forbidden

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when any two nonbonded atoms are closer to each other than a sum of van der Waals radii (these configurations have infinite positive energy), all other being allowed (with zero energy). Already this oversimplified approach enables one to obtain some important results, e.g., to reject certain configurations and even a possibility to synthesize the molecule with inevitable too close positions of nonbonded atoms. The first “Ramachandran maps” for proteins (which will be discussed in the next section) have demonstrated allowed and forbidden regions on two-dimensional plots of the fragment of polypeptide chain. These maps were subsequently improved using more realistic MM functions or quantum mechanics calculations. Most modern MM computations include additional terms besides those already mentioned. These terms refer to direct imposition of experimental data (e.g., NMR-derived restrains on inter-atom distances or global characteristics of the macromolecule) and to description of specific quantum effects not accounting for by standard MM force field formulae. The complexity of mathematical expressions and the number of parameters depend on the systems considered. The problem of “which atoms pertain to the same type and which ones are of different types” is considered by the authors of specific force field and software depending on the tasks and computer resources. The atom type may depend not only on the chemical element and electron shell configuration but on neighbor atoms and on the structure of the whole molecular fragment (e.g., the carbons of six-member and five-member aromatic rings having the same three bonded atoms may be considered as pertaining to different atom types). The more broad the applications that are planned for the force field, the greater the number of atom types that should be involved and the more complex force field formulae that should be constructed. The first works which dealt with the tasks related to specific systems (e.g., the conformations of saturated hydrocarbons or peptide fragments) usually contained a few parameters; the modern force fields may contain thousands of parameters (in spite of use of combination rules mentioned above). The mathematical forms of the equations and numbers of atom types and adjustable parameters represent a compromise between the simplicity (for elaboration and for use) and the accuracy of the results obtained. Various physical considerations can be used for preliminary estimation of mathematical expressions and parameter values (rather simplified considerations were used in Eqs. 2, 3, 4, and 5). It is important to emphasize that neither dependences nor values of parameters can be “derived” (directly calculated) from universal principles or measured by any experimental method. The stretch and bend constants (of Eqs. 2 and 3) can be evaluated using infrared spectra; equilibrium bond lengths and valence angles can be estimated from X-ray data for crystals of simple molecules. The Aij coefficients of the attraction part of van der Waals interactions can be evaluated (and really were calculated and used without refinement in the first MM works) via approximate formulae for dispersion interactions; however, their exact values for a certain class of systems should be adjusted by comparison with experimental data or with the most exact quantum chemistry results after trial computations for reference set of related systems. The same is valid for other terms and their parameters. Some parameters have rather simple physical meaning

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and restricted areas of possible values (e.g., equilibrium distances between bonded and nonbonded atoms or barriers to rotation about the bonds); other parameters have only approximate relation to physical values (atom charges, Bij coefficients of Lennard-Jones potential). As all the parameters are adjustable ones, only the values of total energy and the equilibrium geometry of the molecular system can be compared with experimental data, and consequently have the strict physical meaning, not the individual contributions or the values of the individual parameters. As various force fields utilize different reference sets of data, the individual parameters are not transferable between different force fields even in cases where they use the same mathematical expressions. Different force fields may result in the nearly equal energy and geometry of local minima configurations but rather different values of the individual term contributions. Thus individual terms of the energy may have very approximate physical interpretation, although in some cases it is interesting to evaluate the certain energy contributions and to follow their changes for different molecular complexes and different configurations (and many researchers include these evaluations to their publications). It is worth mentioning that preliminary consideration of MM scheme has resulted already in some doubts and objections. Generally speaking, the classical description of the essentially quantum molecular systems cannot be exact and full. Most of the terms in Eqs. 2, 3, 4, and 5 refer to the first approximation or to the first term of expansion of the corresponding interaction energy. The atoms are not points, they have dipole and quadrupole moments (not only charges), and charge distribution in a molecule is continuous. The polarization or electron delocalization interactions are not considered in the classical “minimalist” MM approach, and the contributions of three-body and four-body interactions can be essential ones. Many attempts have been undertaken to overcome these inherent difficulties of the MM method as well as to justify the assumptions and simplifications; we will consider some of these attempts below. Few remarks for justification of the main principles of MM method are described here. The possibility of consideration of atoms as elementary subunits of the molecular systems is a consequence of Born-Oppenheimer or adiabatic approximation (“separation” of electron and nuclear movements); all the quantum chemistry approaches start from this assumption. The additivity (or linear combination) is a common approach to a construction of complex functions for a physics-based description of the systems of various levels of complexity (cf. orbital approximation, MO LCAO approximation, basis sets of wave functions, and some other approximations in quantum mechanics). The final justification of the method is good correlation of the results of its applications with the available experimental data and a possibility to predict the characteristics of molecular systems before experimental data become available. It can be achieved after careful parameter adjustment and proper use of the force field in the area of its validity. The contributions not considered explicitly in the force field formulae are included implicitly into parameter values of the energy terms considered.

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A Bit of History: The “Precomputer” and Early Computer-Aided MM Calculations The quantitative estimations of molecular properties via simple atom level mechanics representations originate from the communications of Hill (1946, 1948), Westheimer and Mayer (1946), and Barton (1948, 1950). All these papers refer to conformations of organic molecules. It is interesting to mention that mathematical expression of potential energy suggested in the pioneering work of Hill (1946) contains common for all the modern force fields stretch and bend components (Eqs. 2 and 3 of previous section) as well as the Lennard-Jones terms of nonbonded interaction energy. Westheimer and Mayer (1946) suggested the use of exponential term for description of steric repulsion. The first calculations for selected conformations of rather simple (“medium size”) molecules, such as diphenil derivatives in paper (Westheimer and Mayer 1946), cis-decalin and steroids in the papers of Barton (1948, 1950) were performed manually or using desk calculators. Some researchers constructed “hand-made” models of steel or wood (e.g., of cyclic saturated hydrocarbons in the papers of Allinger (1959)) for careful measurements of geometry parameters. The importance of quantitative estimations of nonbonded interactions for considerations of three-dimensional structure of organic molecules was emphasized starting from the first mechanical considerations, which was clearly shown by Bartell in 1960. He illustrated the preference of “soft sphere” over “hard sphere” approach to the analysis of hydrocarbon structure and suggested one of the first (6-12) parameters for hydrocarbons (Bartell 1960). As mentioned above, rather approximate calculations clearly demonstrated the utility of MM approach to the problems of organic chemistry as well as the need for further extensive computations and searching for more reliable parameters. We will refer to all these quantitative theoretical considerations of molecular properties as MM, not depending on the use of this term by the authors or on the method of estimation of different types of interactions. Rapid expansion of MM method starting from the 1960s was provoked by an introduction of computers into all the branches of natural science. In this section, we will briefly consider some examples of the first computer-aided applications of MM method to three research areas, namely, the physical organic chemistry (these works can be considered as a continuation of the “precomputer” papers mentioned above), the structure and properties of molecular crystals, and the interactions and conformations of biopolymers.

First MM Applications to Three-Dimensional Structure and Thermodynamics of Organic Molecules The first paper on computer study of organic molecule conformations was related to saturated hydrocarbons by Hendrickson (Hendrickson 1961). The angle bending, torsion, and (6-exp) van der Waals contributions to the conformation energy were

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taken into account, while constant values were assigned to bond lengths. The computer calculations of cyclo-alkanes containing 5, 6, and 7 carbon atoms enabled him to consider various conformers and to reproduce and rationalize the experimental data. During the 1960s and the beginning of the 1970s such computations were performed by several groups of investigators. Hendrickson (1962, 1973, and references herein), Allinger et al. (1971), and Allinger and Sprague (1973) extended MM approach to more complex hydrocarbons including those with delocalized electronic systems. Allinger’s computations took into account bond stretching in addition to terms used by Hendrickson. Wiberg (1965) was the first researcher who suggested use of the Cartesian coordinates of atoms (instead of “internal coordinates,” i.e., bond lengths, bond angles, torsions) for nonbonded (“strain”) energy minimization. Such approach enables researchers to use the same scheme and the same software to study completely different systems. Most modern MM programs use such an approach. The electrostatic term has not been included in the papers mentioned in this paragraph as hydrocarbons are nonpolar molecules; it was introduced later when more broad sets of molecules became to be considered. The most important results of early MM computations of organic compounds can be illustrated by Engler et al.’s paper (1973) titled “Critical Evaluation of Molecular Mechanics.” The calculations for various hydrocarbons have been performed using two rather different force fields, their own and that of Allinger et al. (1971). The two force fields have substantially different parameters as different sets of experimental characteristics were used for parameter adjustment. It results in significant difference of separate terms of the energy, which may vary by several times or even be positive for one force field and negative for another one. Nevertheless, total energies and relative values for various conformations calculated by two force fields are rather close to each other for most of the compounds considered. Two phrases of the paper’s abstract can be considered as characteristics of the whole situation with the MM approach for that time: “Most of the available data are reproduced with an accuracy rivaling that achieved by the experimental methods” and “The molecular mechanics method, in principle, must be considered to be competitive with experimental determination of the structures and enthalpies of molecules.” Since the abovementioned publications, the theoretical conformational analysis using MM force fields has become an inherent part of physical studies of organic molecules. It became clear that in spite of difficulties with the parameter choice, the MM calculations are not only a useful tool to rationalize experimental observations, to reproduce and to predict the structure and energy characteristics of “medium size” organic molecules with experimental accuracy, but they can also help in elaboration of pathways of chemical synthesis. Different mathematical expressions and parameters transform into force fields, i.e., to complete and verified sets of the formulae and constants and ready-to-use computer programs. One of the first such force fields and programs was MM1 program based on the paper by Allinger and Sprague (1973), which then was followed by MM2, MM3, and MM4 force fields and software for broad class of organic molecules. The inclusion of new atom types and new terms of energy as well as parameter refinement continued till last years,

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and we will discuss these program sets in the next sections together with other modern force fields and software.

The Role of Molecular Crystal Study on the First Steps of Molecular Mechanics The first publications on MM approach to molecule conformations mentioned above deal with intramolecular interactions. The first works on such approach to intermolecular interactions belong to one of the pioneers and founders of MM, A. I. Kitaigorodsky (other spellings, Kitaigorodskii, Kitajgorodskij, Kitaygorodsky), though he (and all the researchers at that time) did not use the term “Molecular Mechanics.” Nearly at the same period (the 1950s) as Hill and Westheimer, usually mentioned as initiators of MM approach, he suggested to consider mechanical models of molecular systems quantitatively via mathematical expressions. In the 1950s, he foresaw the applications of the method to various problems of physics and physical chemistry of organic and biological molecules, including the problems of crystal structure, adsorption, and conformational transitions. We refer to the most frequently cited works of Kitaigorodsky (1960, 1961, 1973). Kitaigorodsky used structure and thermodynamic data on the crystals of organic molecules for adjustment of parameters of atom-atom potential functions for calculations of nonbonded interaction energy. This term of the total energy is dominant for molecular crystals. The method of atom-atom potentials was suggested as a generalization of the “principle of close packing” of molecules in molecular crystals he discovered earlier; his book (Kitaigorodski 1959) describing this principle is still widely cited. Later he demonstrated that this principle is a consequence of more general atomatom approach to potential energy of molecular crystals. Kitaigorodsky was the first researcher who suggested to consider the interactions of nonbonded atoms in a molecule and of such atoms of different molecules via the same mathematical expressions and parameters (applications of MM approach to biopolymers would be impossible without this suggestion). The studies of molecular crystals via MM approach enabled one to derive potential functions for nonbonded interactions and to test the fidelity and accuracy of the approach itself using extensive sets of available quantitative data on structure and energy of molecular crystals. Some energy terms were ignored in the first MM works on conformations of organic molecules, but it was impossible to predict a priori what types of contributions to intramolecular interactions energy (sums in Eq. 1) can be reasonably neglected for specific types of molecules. Considering crystal structure of nearly rigid molecules (e.g., aromatic rings) it is possible to ignore (at least in the first approximation) all the terms but the last one in Eq. 1. This assumption is just a consequence of examination of the geometry of series of related molecules in crystals (e.g., aromatic hydrocarbons consisting six-atom rings); all of them have nearly equal values of bond lengths and angles not depending on molecular complexity and the type of crystal packing. For hydrocarbons, it is possible (again at least for the initial studies) to ignore electrostatic contributions as well (the

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molecules have no dipole moment, and all the estimations of atom charges result in small values, less than 0.1 of electron charge). Kitaigorodsky suggested stepby-step selection of potential functions starting from molecules of nearly neutral atoms of two types (hydrocarbons) and introducing next atom types one-by-one for selecting next parameters. His pioneering works on molecular crystals were followed by more extensive computations of other authors. Some of these works were inspired by his ideas and followed his methodology (e.g., first papers of Williams (1966, 1967)), other researchers performed computations using different expressions for nonbonded potentials and additional terms of the energy (e.g., Lifson and Warshel 1968; Warshel and Lifson 1970). The common set of (6exp) parameters for C : : : C, C : : : H, and H : : : H intermolecular interactions was obtained by Williams (1967) from energy and structure calculations for crystals of both aromatic and aliphatic hydrocarbons. Warshel and Lifson (1970) derived “consistent force field” parameters for description of both intermolecular and intramolecular interactions in crystals. This set contains both parameters for van der Waals interactions and other terms of energy mentioned in the previous section. Mason and coauthors (e.g., Craig et al. 1965; Rae and Mason 1968; Mason 1969) used combined approach to calculations of intermolecular interactions in crystals, namely, repulsive terms were calculated at atom-atom level, while attractive ones were considered on bond-bond level. Such approach and other changes of simple atom-atom scheme are not convenient for computations, and we would like to cite a phrase from Mason’s paper comparing his and Kitaigorodsky’s methods: “Although the representation of the attractive potential by spherically symmetric atom-atom interactions cannot be justified theoretically, it has been outstandingly successful in predicting the properties of crystals” (Mason 1969). The calculations for molecular crystals became a part of the selection and test of the parameters during elaboration of nearly all modern force fields. Later MM calculations for molecular crystals enable researchers to examine the approximations accepted in force field elaboration and to derive force field versions with additional terms of energy (including polarization and atom centers, e.g., Williams and Weller (1983)).

Molecular Mechanics on the First Steps of Molecular Biology: Molecular Mechanics and Protein Physics Development of molecular biology and biophysics in the 1960s required a quantitative consideration of the conformations and interactions of proteins, nucleic acids, and biomolecules in general. The problems of biological importance were to rationalize the structure and conformational properties of the proteins and nucleic acids, namely, to understand the contributions of the subunits and to construct the models of the most favorable and biologically important conformations of the fragments. We already mentioned in previous section that the first successful models of regular structures of proteins and DNA duplex have been constructed using “hand-made” fragments of paper, wood, wire in the beginning of the 1950s. For understanding, explanation, and prediction of molecular level mechanisms of

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biopolymer functions, it was necessary to work out the method of quantitative simulation of the three-dimensional structure and properties of biomolecules. The first molecular mechanics considerations of biopolymer fragments were performed in 1960s. We will follow briefly such studies for proteins in a few paragraphs below. The general problems on proteins which can be in principle solved via MM simulations are (1) the construction of three-dimensional structure of the macromolecule and prediction of the pathways of “protein folding” using restricted experimental data (ideally, the primary structure only); (2) the refinement of experimental structure (X-ray diffraction patterns usually do not supply us with information sufficient for precise atom coordinate assignment). The whole problem of the proteins functioning cannot be solved via MM only as chemical reactions are beyond the MM approach. Changes in electronic structure in processes that involve bond-breaking and bond-forming, charge transfer, and/or electronic excitation require quantum mechanics considerations. Nevertheless, MM approach is useful for the problems of enzyme-substrate complex formation and of molecular recognition which are crucial for the protein functions. First of all, two important results on protein structure obtained before classical MM computations should be mentioned. The atom-level structures of ’-helical and “-sheet fragments of polypeptide chains have been designed by Linus Pauling and Robert Corey in 1951 using “hand-made” models of wood (Pauling and Corey 1951; Pauling et al. 1951). Such regular structures are the intrinsic parts of the majority of the proteins. The success in the construction of these first models of the regular protein fragments (as well as of the DNA duplex) primarily depended on correct subunits geometry and a potential to predict the correct scheme of hydrogen-bond formation, all other contributions to intramolecular interactions being of secondary importance. The regular structures imply H-bond formation between N-H donors and CDO acceptors of the peptide groups of the same chain for ’-helix, and of other chain or of distant parts of the same chain for “-sheet. As it was pointed in more than half of century (Eisenberg 2003), “In major respects, the Pauling-CoreyBranson models were astoundingly correct, including bond lengths that were not surpassed in accuracy for >40 years.” The second important “precomputer” result refers to construction via hard-sphere models of two-dimensional map for possible (and impossible) conformations of the dipeptide unit, the so-called Ramachandran plot (Ramachandran et al. 1963). The polypeptide backbone consists of repeating peptide group connected via C’ atoms, the latter being connected to hydrogen atom and amino-acid residue (the first atom of the residue is designated as C“ atom). The peptide group is practically planar, and the only single bonds in the peptide chain are those between C’ atom and two neighbor peptide groups (C’ –N–H of one group, the torsion angle for rotation about this bond is designated as ˆ, and C’ –CDO of the next group, the torsion angle for rotation about this bond is designated as ‰). Using mathematical expressions for atom coordinates suggesting fixed bond lengths and bond angles, Ramachandran et al. (1963) have constructed the two-dimensional ˆ-‰ maps with two sets of van der Waals atom radii, “normally allowed” and “outer limit,” i.e., normal and shortened radii. The main part of the maps for all the amino-acid

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residues except glycyl corresponds to the conformations forbidden due to shorten atom-atom contacts, while ’-helix and “-sheet fall into the allowed regions. For glycyl residue, the allowed regions were considerably more extended. It took about half a year to construct these maps using a desk calculator (Ramachandran 1990); later such maps were computationally constructed using various force fields and quantum mechanics methods; superposition of ˆ-‰ combinations corresponding to experimentally or computationally constructed three-dimensional structure of proteins and peptides on the Ramachandran plot helped to check and rationalize the protein models nearly for half a century. The consideration of dependence of Ramachandran plot on amino acid residue via hard sphere approach enables Leach et al. (1966a) to evaluate semi-quantitatively the parts of two-dimensional map corresponding to allowed conformations for different amino-acid residues (from about 50 % of all conceivable conformations for glycyl to only 5 % for valyl). The evaluation of steric restrictions emphasizes their important role as a determinant in protein conformation; the consideration of ’-helices demonstrated that the preference of the right-handed ones in comparison to left-handed helices is due essentially to interactions of the C“ atom of the side chains with atoms in adjacent peptide units of the backbone (Leach et al. 1966b). The “hard sphere” approach was applied to search for allowed conformations of cyclic oligopeptides, for example (Némethy and Scheraga 1965). The main conclusion of the “hard sphere” works was that steric effects are one of the most important factors in determining polypeptide conformations. “Many conformations of a polypeptide can be classed as energetically unfavorable without consideration of other kinds of interactions; however, the method breaks down to the extent that it cannot discriminate between those conformations which are sterically allowed : : : The contribution that this method has made is that more than half of all the conceivable polypeptide conformations are now known to be ruled out by steric criteria alone” (Scott and Scheraga 1966a). Nearly at the same time (the middle of 1960s) as abovementioned hard sphere considerations of peptides, the first applications of MM formulae to protein structure were started using “soft atoms,” and the first parameters of potential functions suitable for peptide subunits calculations were suggested. The first such works were published by three groups of researchers, namely, those of De Santis and Liquori (De Santis et al. 1965), Flory (Brant and Flory 1965), and Scheraga (Scott and Scheraga 1966a). All these works were preceded by the publications of the authors related to synthetic polymers. The bond lengths, valence angles, and planar peptide group in all the early studies of polypeptides were fixed. De Santis et al. (1963) have carried out the calculations of van der Waals term of the energy using different potentials for regular conformations of linear polymers as functions of torsion angles of monomer units. The deepest minima of the conformational diagrams were found very near to the experimental structures, as obtained by X-ray fiber diffraction methods, for a series of polymers investigated including polyethylene, poly(tetrafluoroethylene), poly(oxy-methylene), and polyisobutylene. When the calculations were extended to the polypeptides (polyglycine, poly-Lalanine, and poly-L-proline), good agreement with the experiments was obtained

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as well (De Santis et al. 1965). After nearly 30 years De Santis wrote “While other contributions to the conformational energy are included in the calculations, the dominant role of the van der Waals interactions remains well established as the main determinant of the conformational stability of macromolecules” (De Santis 1992). Brant and Flory (1965) have carried out the calculations on peptide unit energy in which they included torsion potentials, van der Waals (6-exp) interactions, and dipole-dipole interactions between the permanent dipole moments of nearneighbor peptide groups. The conformational energies were used by Brant and Flory in statistical mechanical calculations of the mean-square unperturbed end-to-end distance of various polypeptide chains. They concluded that it was necessary to include the electrostatic interactions in the energy calculations to obtain agreement between the calculated and experimentally determined chain dimensions and thus that these interactions are important in determining polypeptide conformations. The first publication of Scheraga’s group related to “soft atoms” MM simulations related to proteins (on helical structures of polyglycine and poly-L-alanine (Scott and Scheraga 1966a)) included besides contributions considered in paper of Brant and Flory (1965) explicit accounting of hydrogen bonds and interactions of atom charges of peptide group. Rather complex expression for dipole-dipole energy of interactions between peptide groups used by Brant and Flory (1965) was substituted by Coulomb atom-atom terms, and (6-exp) expressions for van der Waals interactions were substituted in this paper and subsequent publications by (6-12) potentials, but additional complicated expression for description of hydrogen bond energy was added. The method and parameters for calculation of torsion and van der Waals energies were earlier developed by Scott and Scheraga (1966b) for normal hydrocarbons and polyethylene. It is interesting to note that both groups of authors (those of Flory and of Scheraga) applied without further refinement approximate Slater-Kirkwood equation for calculation of coefficient for attractive term of van der Waals energy (the coefficients were expressed via polarizabilities and effective numbers of electrons of the interacting atoms) and van der Waals radii for calculation of repulsion term parameters. Approximate character of such parameters is evident, but rather good agreement with experimental results was obtained in the both publications (Brant and Flory 1965; Scott and Scheraga 1966a). The existence of polyglycine and poly- L-alanine regular ’-helices was rationalized as a consequence of favorable both H-bond and van der Waals interactions, while helices of other possible types (¨ and 310 ) lie in relatively high-energy regions, and for isolated helices these structures have been excluded (Scott and Scheraga 1966a). The abovementioned calculations of Scheraga et al. on protein simulations were followed by a series of publications related to both improvements of the method (both parameter adjustments and new term additions) and of simplifications of computations. Such changes in the methods enable them to obtain preliminary results on peptides of up to 20 amino acid residues already in 1967 (Gibson and Scheraga 1967a, b). The improvements of the method refer to enlarging the equilibrium radii of atoms, assigning partial charges to each atom (but neglecting the electrostatic contribution when the atoms are not close together), and introduction of

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orientation-dependent term for H-bond description instead of complicated formulae of the previous paper (Scott and Scheraga 1966a). The simplified account for solvent contribution to the energy has been included considering the effect of removing the nearest neighbor to peptide water molecules (Gibson and Scheraga 1967a). The simplification of the model refers to introducing “extended atom,” i.e., CH, CH2 , and CH3 groups were considered as the single atom to reduce the number of atomatom interactions that had to be computed (this approach was used later in many force fields and such potential functions are commonly referred to as “united atom” potentials, we will discuss this simplification below). Scheraga’s group prolongs the improvement of the parameters and the method continuously till recent years. We will discuss these studies below, now we will mention some improvements performed in the first half of the 1970s. The most important improvements refer to description of H-bond potential by (10-12) expression (McGuire et al. 1972) (some other force fields adopted such dependence for description of interaction between hydrogen atom capable to participate in H-bond and atom-acceptor of H-bond), to calculation of partial charges via semiempirical CNDO/2 method (Yan et al. 1970), and to adjustment of repulsion part of van der Waals interactions to crystal data for extensive set of molecules including amino acid crystals (Momany et al. 1974, 1975). The results of the crystal calculations led to an internally self-consistent set of interatomic potential energies for interactions between all types of atoms found in polypeptides (ECEPP force field (Momany et al. 1975)). The bond lengths and bond angles were obtained from a survey of the crystal structures and were considered as fixed for each of amino acid residue (Scheraga’s force fields differ in this feature from most other widely used force fields). The preliminary set of potential functions they used to refine X-ray structures of some proteins, e.g., lysozyme (Warme and Scheraga 1974), ’-lactalbumin (Rasse et al. 1974), and rubredoxin (Warme et al. 1974). We will not consider details of these papers; instead we will mention the first publication on MM refinement of X-ray protein structure of Levitt and Lifson (1969). The refinements of protein structures (myoglobin and lysozyme) have been performed via two steps by minimization of the energy function containing all the MM terms and an additional term describing deviation of the atom coordinates from their values obtained from X-ray diffraction studies (penalty function). Three energy terms (corresponding bond stretching, bond angle bending, and torsion potentials) supplemented by penalty functions were used at the first step. At the second step nonbonded and hydrogen-bonded interactions were included, and penalty functions were omitted. The authors mention that energy function parameters are preliminary ones, intended for protein simulation in aqueous solution, differed significantly from those of Scheraga and subject to further improvement. The refine structures were close to X-ray diffraction ones and do not contain short atom-atom contacts or unusual bond lengths and valence angles. Starting from rough model coordinates, the method minimizes the assumed total potential energy of the protein molecule to give a refined conformation (Levitt and Lifson 1969).

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Molecular Mechanics on the First Steps of the Biophysics of Nucleic Acids The problems related to MM approach to nucleic acid structure and functions differ in some aspects from those for proteins. The main computational tasks for the 1960s in the area of nucleic acids were to rationalize the structure of native and modified DNA duplexes, t-RNAs, synthetic polynucleotides, i.e., to understand the contribution of the subunits (the bases, sugars, ionized phosphate groups, counterions, and surrounding water) to three-dimensional structure, and to evaluate the contributions of interactions of different physical nature to structure and functions of nucleic acids. The extensive applications of MM approach to nucleic acids were started a few years after those for proteins. The same is true for the adjustment of MM parameters for calculations of nucleic acids. Such a situation can be explained by “special role” of DNA in cells as a heredity material as well as by expectations of “specific” forces between the bases as conjugated molecules with delocalized electron system. Many researchers considered the electron exchange via H-bonds in base pairs and exchange interactions between stacked bases as the crucial contributions to nucleic acid functioning. It took some years before pioneers of quantum mechanics approach to biochemical problems would write: “The possible contributions of resonance energy stabilization through electronic delocalization across the hydrogen bond for horizontal interactions and of overlapping of their  -electronic cloud or of charge-transfer complexation for the vertical interactions appear to be of much smaller order the magnitude than the stabilization increments due to van der Waals-London forces” (Pullman and Pullman 1968). The author of this survey was the first researcher since the application of MM formulae in 1966 to consider the interactions of nucleic acid subunits. We will now briefly follow the route of investigations of nucleic acid interactions to “classic” MM approach. The first quantitative estimation of interactions of bases in DNA duplex via formulae of intermolecular interactions was performed by De Voe and Tinoco (1962). They used so-called molecule dipole approximation, i.e., permanent and induced electric point dipoles were placed into the center of each base. The energy of interaction between two paired or stacked bases was suggested to be a sum of the interactions of permanent dipoles (Coulomb electrostatic interactions), permanent dipole-induced dipole (polarization interactions), and of induced dipoles (London dispersion interaction). Such an approximation does not enable them to obtain even qualitatively correct results (e.g., the energy of interaction between the bases in Watson-Crick A:T pair was positive), but the computational problem of evaluation of interaction energy changes on the formation of the specific nucleic acid conformation has been defined in this chapter. That is why it is cited in hundreds of publications during nearly a half of century, many of them (including those of the author of this survey) were inspired by the paper of De Voe and Tinoco (1962). Shortly it became clear that such an approximation is invalid for this system not depending on dipole location as well as after replacement of point dipoles by “real” ones, i.e., by dipoles of definite size. Bradley et al. (1964)

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suggested that the main contribution into base-base interaction energy is of Coulomb electrostatic nature, which can be calculated in “monopole approximation.” The point charges were calculated via semiempirical methods of quantum chemistry and placed on each atom of the bases. This paper was followed by the papers of Nash and Bradley related to calculations of base-base interactions. One of their papers should be mentioned in relation to the general progress and development of MM approach. Starting from various mutual in-plane positions for all the combinations of RNA bases, the calculations and search for minima of electrostatic energy was performed (Nash and Bradley 1966). Van der Waals energy was taken into account in “hard sphere” approximation, i.e., any non-hydrogen atom pair should not be closer than sum of van der Waals radii. As a result of the calculations, 27 energy minima were obtained which have two or more short N-H   O or NH   N contacts (with N : : : N or N : : : O distances about 3 Å) corresponding to nearly linear or bifurcated hydrogen bonds. The results enable Nash and Bradley to rationalize the available up-to-date experimental data on base crystals data as well as base-base complex formation in solutions and polynucleotides. It appears that the observed base pairings with two H-bonds in crystals always correspond to one of the computed geometries of lowest energy (Nash and Bradley 1966). It is important to emphasize that Bradley et al. (1964) “monopole” approach to calculations of the base interaction was the first attempt to consider the electrostatic energy in the framework of the modern MM scheme. The abovementioned calculations of Scheraga’s group with inclusion of point charges of peptide group were started 2 years after the Bradley’s publication. The subsequent studies on the quantitative evaluation of the interactions and three-dimensional structure of nucleic acid fragments were continued via two routes. One of the approaches suggested construction of MM force field parametrized to nucleic acids interactions; another approach implied calculations using more rigorous physics concepts and molecular characteristics in a hope to understand physical nature of interactions. The first works on this way were performed by Pullman’s group and referred to interactions of the bases in fixed positions (e.g., Pullman et al. 1966). Considering the energy of interactions as a sum of electrostatic, polarization, and dispersion terms, two approximations were applied, “dipole” and “monopole”. The monopole approximation was applied to electrostatic energy only (following Bradley et al. 1964), while two other terms were really calculated in molecular dipole approximation. This approach was later augmented by inclusion of the short range repulsion term and representation of the molecule as a set of “many-centered multipole distributions obtained from ab initio SCF calculations (charges, dipoles, and quadrupoles located on the atoms and the middles of segments joining pairs of atoms)” (Langlet et al. 1981). We will not consider subsequent progress on this way of nucleic acid computations as new schemes (e.g., Gresh et al. 1986 and references herein) do not correspond to the MM approach. Nevertheless, it is worth to mention that the sophisticated schemes of base-base computations had no advantage for rationalization and prediction of experimental results in comparison to “standard” MM scheme reviewed in the next paragraph.

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In the middle of the 1960s, the author of this survey proposed the first atomatom scheme and numerical parameters for the calculation of interactions of nucleic acid bases (the terms “force field” and “molecular mechanics” were not widely used in those years). This scheme had three source points, namely, idea of De Voe and Tinoco (1962) on quantitative estimations of interactions of nucleic acid subunits, “monopole” approximation of Bradley et al. (1964) for electrostatic interactions of the bases, and Kitaygorodsky (1961) atom-atom approach to calculations of van der Waals interactions in molecular crystals. At that time, there was no “MMlike” calculation scheme suitable for quantitative considerations of nucleic acid interactions and structure. We refer here only to later papers summarizing some preliminary adjustments and applications of the approach (Poltev and Sukhorukov 1970; Poltev and Shulyupina 1986). When calculating the interactions of Pnucleic acid bases or other conjugated heterocyclic molecules the only term Enb of Eq. 1 was considered, corresponding to all the pair-wise interactions of atoms not pertaining to the same molecule. The bond lengths and valence angles were fixed and corresponded to averaged experimental geometry for each of the bases (like those for amino-acid residues and peptide backbone in Scheraga’s force field (Momany et al. 1975)). The atom-atom terms contained electrostatic, London attraction, and short-range repulsion contributions. This scheme was the first one assigning different parameters of van der Waals terms to the atoms of the same chemical element but different electron shell configuration (e.g., aromatic and aliphatic carbons, pyridine and pyrrol nitrogens, keto and enol oxygens). These parameters were preliminarily evaluated from approximate London formulae (for Aij coefficient of Eq. 5) and enlarged van der Waals radii of atoms (for Bij coefficient of Eq. 5 or corresponded values for the firstly used 6-exp potentials). The final values of the parameters were selected after step-by-step calculations and comparison with experimental data for crystals containing the atoms of some types only (starting from graphite, then aromatic hydrocarbons, pyrazine, benzoquinone, purine, and finally the bases). The effective charges of atoms were calculated via simple semiempirical quantum chemistry method (of Huckel and Del Re for  electrons and ¢-electrons, respectively) with the parameters adjusted to reproduce experimental values of dipole moments for related molecules. The primary scheme (Poltev and Sukhorukov 1970) contained polarization term as well, but it was shortly eliminated due to its small value for many cases and to avoid difficulties related to nonadditivity of atom-atom and molecule-molecule interactions. The parameter set was than amplified by consistent parameters for sugar-phosphate backbone (Zhurkin et al. 1980) and for water-DNA interactions (Poltev et al. 1984). This simple MM scheme specially adjusted to nucleic acid interactions enables us to rationalize extended set of experimental data and to forecast some nucleic acid properties before experimental evidence. Variability of DNA helix parameters (Khutorsky and Poltev 1976), dependence of mutual base-pair positions in energy minima on nucleotide sequence (Polozov et al. 1973), pathways for all the base-substitution errors (Poltev and Bruskov 1978), and DNA duplexes with mispairs (Chuprina and Poltev 1983) were predicted by such calculations before experimental data became available. The parameters of this simple scheme were refined later several times (like

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any other force field scheme discussed above and below) via adjustment to new experimental data and quantum mechanics considerations. We will mention here only one more series of early MM calculations related to nucleic acids performed by Indian scientists Renugopalakrishnan, Lakshminarayanan, and Sasisekharan (coauthor of Ramachandran et al. (1963) paper on ˆ-‰ plot for proteins). They suggested the first complete set of parameters for calculations of the conformations of nucleic acid fragments (Renugopalakrishnan et al. 1971, and references herein). The atom charges crucial for the base interactions have been calculated via the same procedure as in our earlier papers (e.g., Poltev and Sukhorukov 1970).

The Problems and Doubts of Further Development of the MM Approach As follows from the previous section, the usefulness of MM approach was already demonstrated in the first decade of its extensive applications and modifications. At the same time, the problems of justification and of pathways of future development of the approach had arisen. From the very beginning, it was clear that the method cannot provide us with exact description of the structures and processes due to its semiempirical nature. The interesting problem of the method is the relative extent of empiricism and physical meaning of the scheme and its parameters.

Approaches to Improvement of MM Formulae and Parameters Several ways of improvement of MM scheme were proposed during the 1960s and the 1970s. Two hypothetical approaches to refinement of MM representation and force fields can be considered. In the framework of the first approach, it is suggested to describe the physical nature of the system in the most exact way, e.g., to employ more exact equations for description of the interaction energy (e.g., not restricting expressions to first term in the expansion for bond stretching and angle bending, use of better grounded expressions for van der Waals interactions, etc.) and/or more detailed representation of the system considering not only point atoms but other centers of interactions, not restricting the scheme with scalar values but introducing vectors (such as dipole moments) or tensors (such as anisotropic polarizability). In the framework of the second approach, it would be possible to consider energy expressions as entirely empirical ones and to adjust the parameters to experimental data (or, later, to reliable quantum mechanics results) until the closest coincidence (e.g., via the least square method). Unsuccessful attempts of consistent and strict use of only one of the approaches had been undertaken during the first two decades of the MM computer simulations. The second approach mentioned above is practically impossible to realize; usually, there is no sufficient quantitative experimental data not only to derive the form of dependences of the energy on inter-atomic distances but to obtain the coefficients of already accepted potential form (e.g., of Lennard-Jones). Even in a

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case we have sufficient number of experimental values, the equations for parameter calculations have no single definite solution. It was demonstrated in the paper of Momany et al. (1968). The authors of the abovementioned paper tried to adjust the coefficient of (6-12) potentials for C : : : C, C : : : H, and H : : : H interactions for benzene molecule to the energy and structure data of benzene crystal. Using three sets of experimental data for different temperatures, they obtained entirely repulsive contribution from C-C interactions. It means that qualitative estimations (relative values of attraction terms for various atoms, of equilibrium radii, of atom charges in the same or similar molecules, etc.) should be taken into account in adjusting the parameters of the force field. Most of the modern force fields originate from previous sets of expressions and parameters derived from simple qualitative considerations (e.g., simplified formulae of dispersion term, van der Waals radii of atoms) and take into account these considerations automatically (e.g., using previous approximate parameters’ values as starting points). As for the first approach, the more complex expressions of the energy were suggested since the end of 1960s, and such expressions are used in some modern force fields, i.e., next terms in the energy expansions for stretching, bending, and van der Waals contributions. The inclusion of additional terms requires new parameters. Usually the number of reliable quantitative values is not sufficient, and such an approach becomes useful in case the area of applications of the force field is restricted to certain types of similar molecules (e.g., hydrocarbons). In any case, the expressions for each term remain the approximate ones, and more coefficients should be adjusted to target data. The more complex representation of the molecular system as compared to a set of point atoms was proposed by several researchers during the 1960s and the 1970s (including the author of this survey). Such a representation looks the most natural for atoms with lone pairs of electrons (inclusion of additional points for electrostatic interactions of pyridine nitrogen or keto oxygen located at the centers of lone-pair orbitals) or for atoms with  orbitals (additional points for aromatic heterocycles located above and below the ring planes to account for quadrupole moments of the molecules). Additional lone pair centers for electrostatic interactions are included in some modern force fields for better reproduction of electric field around the molecules and the directionality of hydrogen bonds. Although the introduction of such centers looks physically based, an inconsistency of such an approach can be detected rather easily. When assigning the negative charged interaction centers to some electron orbitals (and hence positively charged centers to nuclei locations), it looks natural to assign additional centers to chemical bonds (sites of the highest electron density between the nuclei). As a result, we have several times more centers (more computer resources are needed) and several times more adjustable parameters. Such an approach was consistently implemented into force field suggested by Scheraga and coauthors. The first version was titled “empirical potential using electrons and nuclei” (EPEN ; Shipman et al. 1975) and an improved version EPEN/2 was released in 3 years (Snir et al. 1978). In this model, the molecular interactions are modeled through distributed interaction sites, which account for the nuclei and electronic clouds. The positive charges are

Molecular Mechanics: Principles, History, and Current Status

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located at the atomic nuclei, and the negative charge centers are located off the nuclei. The bonding electrons are located along the bonds, while the lone-pair and  - electrons are located off the bonds. The energy of interaction is approximated by the sum of the coulomb interactions between all point charge centers, an exponential repulsion to represent electron–electron overlap repulsion, and an R6 (R D distance) attraction to simulate attractive energies between the nuclei. The distances between the charge locations are the method parameters; they are fixed in the molecule fragments. The parametrization was optimized by least square fits to spectroscopic, crystallographic, and thermodynamic data. This approach (as well as other approaches considering many in addition to atom centers of interactions) was not widely used in MM calculations. The situation can be accounted for by restrictions imposed by fixed geometry of the molecule fragments as well as by need for the parameter adjustment for molecules other than considered by the approach authors. A separate problem is the choice of the force field parameters for most exact reproduction of experimental and/or quantum mechanics data. All the experimental values are determined with a definite accuracy; all the quantum mechanics data refer to gas phase and do not include implicitly third body interactions. The use of quantum mechanics data only (not depending on level of theory) results in errors in reproduction of condensed phase properties. The adjustment of parameters to characteristics of condensed phase (most experimental data refer to crystals or solutions) leads to wrong values of characteristics for separate molecules or their simple complexes in gas phase. Hence, various experimental and quantum mechanics data should be considered with different weights in the process of the adjustment of force fields, the weights being dependent on areas of intended applications of the force field. Recently, a method and software package named ForceBalance for systematic force field optimization has been published (Wang et al. 2013a). The method enables researchers to parametrize a wide variety of functional forms using flexible combinations of target data. These data may come from experimental measurements or from quantum mechanics calculations. Various properties are included in a single objective function, which is then integrated over the entire configuration space. The method was used by the authors (Wang et al. 2013a) for developing a new polarizable water model with 27 parameters adjustable to quantum mechanics data for single water molecule and H-bonded water clusters. The water model contains five fluctuating charge sites including three virtual sites. The positions of virtual sites have been fully optimized; they do not coincide with lone pair centers. An interesting suggestion was made for description of van der Waals interactions. The authors propose specially modified (6-exp) potential. As it was mentioned in the first section, “classical” (6-exp) Buckingham function widely used on the first steps of MM-like calculations has unphysical dependence on interatomic distance at small distances. The authors of the model (Wang et al. 2013a) have chosen empirical van der Waals potential EvdW (r) with three parameters.

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   EvdW .r/ D 2© ¡6 = ¡6 C r6 = .1  3= .” C 3// Œ3 exp .” .1  r=¡// = .” C 3/  1 (7) This function depends on equilibrium interatomic distance, ¡; minimal energy values, ©; and a dimensionless constant, ”, describing the steepness of the repulsion. Like Buckingham function, it tends to an attractive r6 potential at large interatomic distances, tends to a repulsive exponential term at small distances, and has welldefined unique minimum. The authors note that their water model is quite accurate for some, but not all, properties of liquid water. We should note that this disadvantage is common for all the force fields derived from quantum mechanics results only, without adjustment to experimental data. Improvements of water model using ForceBalance software and both quantum mechanics and experimental data will be considered below.

Various Schemes of Water Molecule in MM Calculations In view of importance of water for life and for description of the biomolecular interactions as well as of “anomalous” physical properties of water in liquid and solid states, hundreds of models and their modifications for water molecule have been proposed. We will mention here only few of most frequently used and cited rigid models and some recent polarizable models. Simple atomic structure of the water molecule enables researchers to perform quantum mechanics calculations of rather high level unreachable for most organic molecules. The results of these computations for a single molecule and for H-bonded water clusters together with availability of quantitative experimental data for bulk water at various temperatures and pressures stimulate elaboration of new models and permanent improvement of existent ones. These models can be used for demonstration of possibilities and restrictions of the MM method in general, for comparison of various approaches to MM model construction. Most water models describe the molecule by the number of interaction centers not coinciding exactly with atom centers. This diversion from simple MM scheme can be understood as a consequence of small size of the molecule and the inability of any simple charge distribution on atom centers to adequately describe the molecular dipole and quadrupole of the monomer at the same time. Inclusion of additional centers or/and displacement of interaction centers from atoms is necessary to reproduce charge distribution of this simple molecule. (The author of this survey noticed this fact earlier for heterocyclic compounds, but fortunately in this case the contributions of permanent quadrupole interactions can be implicitly accounted for by other terms of interaction energy.) The first water molecule model (frequently referred to as BF model) has been proposed by Bernal and Fowler (1933). This model may be considered as the earliest molecular model suitable for MM calculations, and it was used for Monte Carlo calculations of liquid water performed in nearly a half of century for comparison with some later models (Jorgensen et al. 1983). The model contains four interaction sites: three charged sites, namely, two positive centers located at the hydrogen atoms and one negative center (so-called M-site) displaced by 0.15 Å from oxygen atom

Molecular Mechanics: Principles, History, and Current Status

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in the direction of H-O-H bisector, and one center for van der Waals interactions located on oxygen atom. It is interesting to note that Bernal and Fowler (1933) performed approximate evaluations of intermolecular interaction energy by the (16-12) potential functions. Jorgensen et al. (1983) used nearly the same position of centers and slightly modified charges in TIP4P four-site water model (one of their popular Transferable Intermolecular Potentials). Matsuoka et al. (1976) derived more complex four-site model from potential surface of water dimer constructed via ab initio quantum mechanics calculations with configuration interactions. The analytical expression for interaction potential function between two water molecules consists of pair-wise Coulomb interactions of three charges located similar to BF or TIP4P models and exponential interactions of all the three atoms of one molecule with the atoms of the other one. This model (the so-called MCY model) was widely used during the first decade of liquid water simulations. We will not consider many models including earliest ones which consist of three, five, or more centers but only briefly mention some of the rigid models cited till the present time and used in modern force fields. Stillinger and Rahman (1974) suggested the five-site model with two positive centers (hydrogen atoms), two negative centers (located at oxygen lone pairs), and one center (oxygen atom) for Lennard-Jones interactions. The model has been used successfully to study a wide variety of properties of liquid water. Mahoney and Jorgensen (2001), after extensive Monte Carlo simulations and parameters adjustment, suggested five-site model TIP5P which enables them to describe the density anomaly of liquid water better than previously existing models. Nada and Van der Eerden (2003) have designed six-site model of water molecule. Three of these sites are the three atoms of the molecule (they interact through a Lennard-Jones potential), and three other sites are point charges (O atom is electrically neutral; the two H atoms carry positive charges). The model can be considered as a combination of TIP4P and TIP5P models of Jorgensen. The Monte Carlo simulations of ice and water show that the six-site model reproduces well the real structural and thermodynamic properties of ice and water near the melting point. Two three-site rigid water models, namely, TIP3P of Jorgensen et al. (1983) and SPC of Berendsen et al. (1981) should be specially mentioned, as these models (or their modifications) are used in modern force fields for biomolecular simulations. These models have (in accord with general MM principles for class I models) three interaction sites centered on the nuclei. Each site has a partial charge for computing the Coulomb energy and only one site (oxygen atom) for Lennard-Jones interactions. The TIP3P model demonstrates better reproduction of experimental thermodynamic and structural data for bulk water (Jorgensen et al. 1983). Further progress in the study of water properties and its interactions with other molecules is related to new models introducing flexibility to the molecule (incorporating bond stretching and angle bending terms into interaction energy) and with the inclusion of electronic polarization. These effects are important for the quantitative description of water properties in liquid and solid states as well as of phase transitions. We mention here a few of recent water models beyond class I MM methods.

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The model elaborated by Ren and Ponder (2003, 2004), AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications), includes full intramolecular flexibility, permanent atomic monopole, dipole, and quadrupole moments placed on each atomic center and buffered 7-14 potential for pairwise atom-atom van der Waals interactions. Polarization effects are explicitly treated in the AMOEBA force field via mutual induction of dipoles at atomic centers. The functional forms for bond stretching and angle bending were taken from the Allinger’s MM3 force field and include anharmonicity through the use of higherorder deviations from ideal bond lengths and angles. An additional intramolecular term, the Urey-Bradley term, consists of a simple harmonic function. The force field does not contain off-atom sites. It was parametrized by hand to fit results from ab initio calculations on gas phase clusters up to hexamers (Ren and Ponder 2003) and temperature and pressure dependent bulk phase properties (Ren and Ponder 2004). The model enables authors to reproduce well both the high-level ab initio results for various gas phase clusters and bulk water thermodynamics data, including the second virial coefficients over the whole temperature range. The authors mention that AMOEBA water model is approximately eight times slower than the TIP3P one (for MD simulations of a periodic system of 216 water molecules); nevertheless, they consider it as “the first step in the derivation of a next-generation polarizable force field for biomolecular simulation.” Later, AMOEBA force field was extended to organic molecules (Ren et al. 2011) and to proteins (Shi et al. 2013). The iAMOEBA, i.e., “inexpensive AMOEBA” (Wang et al. 2013b) was developed using the ForceBalance automated procedure (Wang et al. 2013a). A direct approximation to describe electronic polarizability reduces the computational cost relative to a fully polarizable AMOEBA model, and the application of systematic optimization ForceBalance method results in raising the accuracy of iAMOEBA. Recently (Laury et al. 2015), a set of improved parameters for the AMOEBA water model was developed using the automated procedure, ForceBalance (Wang et al. 2013a). An adjustment of 21 force field parameters has been performed to both high-level ab initio results for gas phase clusters, containing 2–20 water molecules, and six condensed phase properties at a range of temperatures (249–373 K) and pressures (1 atm to 8 kilobar). The enthalpy of vaporization and density dependence on temperature were given a greater weight (as compared to the other properties) in the optimization. The model represents a significant refinement of the original AMOEBA parametrization, namely, it provides a substantial improvement in the ability to predict experimental data, particularly for a number of liquid properties across a range of temperatures. The authors (creators of original AMOEBA models are among them) refer to new model as AMOEBA14 and to the original model (Ren and Ponder 2003, 2004) as AMOEBA03. The AMOEBA14 model has a similar accuracy level as the AMOEBA03 model for the interaction energy of the gas phase cluster ranging from dimers to 20-mers. They promise compatibility of AMOEBA14 water model with the previously determined AMOEBA force fields for organic molecules (Ren et al. 2011) and for proteins (Shi et al. 2013). The authors of CHARMM force field and software package, widely used for biomolecular simulation (see description below), elaborate their own polarizable

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models of water based on classical Drude oscillator. Last model of their SWM (simple water model) series, six-site SWM6 one (Yu et al. 2013) was developed and optimized for liquid phase simulations under ambient conditions using CHARMM family of force fields. The atoms of the molecule kept its gas phase experimental geometry, and the positions of three other massless interaction sites correspond to the so-called M point located at a fixed distance from the oxygen atom along the bisector of the H–O–H angle, and two oxygen’s lone pairs (LP sites). The model is less computer time consuming as compared to more sophisticated AMOEBA; it has an improved balance between descriptions of microscopic and bulk phase properties compared to most additive and polarizable water models developed previously. Besides, the local hydrogen bonding structures in the liquid phase are better reproduced in the model as indicated by the O-O distance distribution function. However, the model does not reproduce correctly the experimental data on liquid water density maximum. Authors note that the model is not developed for routine use but for studies when the local hydrogen bonding is important and high accuracy is needed. We will not discuss many other recent sophisticated water models, including those developed using both experimental data and quantum mechanics computations (via both ab initio and DFT methods), such as Tröster’s polarizable molecular mechanics (PMM) model (Tröster et al. 2014) and Pinilla’s polarizable and dissociable force field (Pinilla et al. 2012). Selected old and recent MM water models, both simple and rather sophisticated, enable us to arrive at some conclusions on possibilities of molecular mechanics description of simple and complex molecular systems. Some of such conclusions are rather trivial, but additional evidence from the experience of hundreds of researchers on water molecule modeling can help to illustrate and reinforce them. Till recent time, complete and reliable MM description of even this very simple biologically important molecule is a challenge for computationally oriented researchers. More complex expressions for separate energy terms grounded on physical theories, increase of the number of adjustable parameters, improved algorithms for parameter adjustment to target data, and the use of more reliable quantum mechanics results, all together help to better reproduce and rationalize experimental data, but some results are not accurately reproduced. Each model of the molecule, each force field, reproduces (and predicts) some characteristics better as compared to other force fields. The more sophisticated model is not necessarily the better one for specific MM applications and specific tasks of the study. This is true not only because of restrictions of computer resources but due to approximate nature of the MM description. This conclusion is a result of many trials and errors in water simulations. Class I molecular mechanics representations, i.e., simple threesite rigid water models (TIP3P and its modifications) are successfully used in many biomolecular computations in spite of the development of new force fields. This success can be accounted for not by the importance of additional terms but by their implicit compensation by parameter choice for simple model (Class I force fields). More sophisticated force fields (Class II and Class III), together with quantum

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mechanics, both ab initio and semiempirical, computations are useful for detailed description of mutual positions of molecules in water solutions of biomolecules.

Modern Molecular Mechanics Force Fields: Description, Recent Development, and Applications The MM computations during the first two decades were mainly performed by the researchers using their own force fields and homemade computer programs. Most of the modern publications refer to standard, ready-to-use software with implemented (or sometimes slightly modified) force field parameters. The number of program and parameter packages is rather great and new or newly modernized software and force fields appear every year. We will present below a short overview of the selected program sets and force fields most frequently used for study of biomolecular systems. Some of them have been already mentioned in previous sections. During last decade, some software packages such as NAMD (Phillips et al. 2005) and TINKER (Ponder 2015) have appeared which give an opportunity to perform computations using various molecular mechanics force fields. Some popular quantum mechanics software packages such as GAUSSIAN and SCHRODINGER supply users with the possibility to use MM force fields as well.

Allinger’s Force Fields and Programs We started description of the modern force field with short characterization of Allinger’s parameters and programs as they were the first ones ready for use by any researcher. The first parameter set (MM1) was described in 1973 (Allinger and Sprague 1973), and the MM1 program was released through Quantum Chemistry Program Exchange in 1976. The force field and the program MM1 were followed by a series of improved and extended versions for various classes of organic molecules, MM2, MM3, MM4 (see Allinger et al. 1989, 1996; Lii et al. 2005 for references). These force fields have been parameterized based on experimental data, including the electron diffraction, vibrational spectra, heats of formation, and crystal structures. The calculation results were verified via comparison with high-level ab initio quantum chemistry computations, and the parameters were additionally adjusted. The last program versions include rather complex force field equations (including additional terms to harmonic potentials for stretching and bending and cross-terms; thus MM3 and MM4 force field are those of class II) and various calculation options (e.g., prediction of frequencies and intensities of vibrational spectra, calculating the entropy and heat of formation for a molecule at various temperatures). MM4’s molecular dynamics option creates a description of the vibrational and conformational motion of molecules as a function of time. The program examines the atoms of a molecule and their environment and decides which atom type is appropriate. The program includes more energy terms and more complicated expressions for some terms than most other popular MM force

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fields. The force field contains greater number of the constants which are adjusted to extended set of experimental data. The stretching and bending energy terms include higher order contributions, several cross-terms are inserted; the electrostatic contribution includes interactions of bond dipoles. The MM family of force fields designed by Allinger and coauthors is used till recent time for calculations on small and medium size organic molecules, e.g., steroidal-porphyrin aggregates (Lorecchio et al. 2014). New improvements to MM4 force field were suggested by Allinger (2011) recently. The classic of molecular mechanics, after some examples make the conclusion: “A good (well developed, Class 3) molecular mechanics force field will usually give molecular structures that are approximately of chemical accuracy. When it fails to do so, there is a good chance that nature knows something that we don’t” (Allinger 2011).

Merck Molecular Force Field (MMFF94) This force field of class II was developed for a broad range of “medium size” molecules primary of importance for drug design. It differs from Allinger’s and many other popular force fields in several aspects. The core portion of MMFF94 has primarily been derived from high-quality computational quantum chemistry data (up to MP4SDQ/TZP level of theory) for a wide variety of chemical systems of interest to organic and medical chemists (Halgren 1996, 1999b). Nearly all MMFF94 parameters have been determined in a mutually consistent fashion from the full set of available computational data. The force field reproduces well both computational and experimental data, including experimental bond lengths (0.014 Å rms), bond angles (1.2ı rms), vibrational frequencies, conformational energies, and rotational barriers. The mathematical expressions of the force field differ from many other force fields. The expressions for stretching and bending energies (like Allinger’s force fields) contain in addition to harmonic terms stretch-bend cross terms and outof-plane terms. The van der Waals energy is expressed by buffered (7-14) potentials (Eq. 8, the designations are the same as in Eqs. 2, 3, 4, 5, 6, and 7)  7       1:12¡ij 7 = rij 7 C 0:12 ¡ij 7  2 EvdW D ©ij 1:07 ¡ij = rij C 0:07 ¡ij

(8)

The electrostatic term contains buffering constant as well. The specially formulated combination rules for ©ij and ¡ij are used. The MMFF94 version was developed for molecular dynamics simulations, and MMFF94s version (Halgren 1999a) was modified for energy minimization studies (s in the title means static). Both versions provide users with nearly the same results for majority of molecules and complexes. Additional parameters of force field and careful adjustment to highlevel reference data result in good reproduction of extended set of experimental data on conformational energies of the molecules appeared after force field publication (Halgren 1999b). Nevertheless, Bordner et al. (2003) reveal that MMFF94 values of sublimation energy for a set of compounds are systematically 30–40 % lower

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than experimental data. It can be explained by “computational nature” of MMFF parameters and by the adjustment to quantum mechanics results without corrections for the experimental data, i.e., nonadditive contributions do not accounted for implicitly. This force field is used in many recent studies of biologically important molecules, which employ combined experimental and computational approaches. Anti-inflammatory drugs (Pawar et al. 2014; Sawant et al. 2014) and anticholinergic and antiasthmatic drugs (Guo et al. 2014) are among the compounds considered recently with MMFF.

The Force Fields and Programs Designed by Scheraga and Coauthors As we had already mentioned that the first MM parameter set for the simulation of proteins was described in 1975 by Scheraga and coauthors (Momany et al. 1975); their empirical conformational energy program for peptides (ECEPP) was released through Quantum Chemistry Program Exchange in the same year. This force field family was refined continuously till last years; we will briefly describe two versions of Scheraga’s force field used for studies of protein three-dimensional structure and protein folding up to recent time, ECEPP/3 (Némethy et al. 1992) and ECEPP-05 (Arnautova et al. 2006). The ECEPP/3 force field in turn is the modified version of ECEPP/2; it contains updated parameters for proline and oxyproline residues. The ECEPP force fields utilize fixed bond lengths and bond angles obtained from a survey of the crystal structures for each of amino acid residue. The peptide energy is calculated and minimized as a function of torsion angles only, and it is a significant advantage for minimization approaches as it drastically reduces the variable space that must be considered. The energy function of the ECEPP/3 force field consists of torsion and nonbonded (Coulomb and van der Waals) terms (Eqs. 4 and 5). The partial charges of atoms are calculated by the molecular orbital CNDO/2 method, the parameters of (6-12) term are adjusted by comparison with crystal data. The (6-12) terms are substituted by (10-12) ones for the interactions of the hydrogen atoms capable to participate in H bonds with H-bond acceptor atoms. In spite of appearance of new improved and augmented versions of ECEPP force field, the ECEPP/3 is used until recently for protein simulations (e.g., McAllister and Floudas 2010). The ECEPP-05 force field (Arnautova et al. 2006 and references herein) is adjusted to both new experimental data and quantum mechanics results. It has some distinctive features as compared to both ECEPP/3 version and many other popular force fields. The van der Waals term of the energy is modeled by using the “6-exp” potential function (potential of Buckingam)   Eij D Aij rij 6 C Bij exp Cij rij

(9)

where rij is the distance between atoms i and j; Aij , Bij , and Cij are parameters of the potential. The combination rules are applied directly to these parameters.

Molecular Mechanics: Principles, History, and Current Status

1=2 1=2     Aij D Aii Ajj I Bij D Bii Bjj I Cij D Cii C Cjj =2

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(10)

The parameters of (6-exp) potentials were derived by using so-called globaloptimization-based method consisting of two steps. An initial set of parameters is derived from quantum mechanics interaction energies (at MP2/6-31G* level of ab initio theory) of dimers of selected molecules; at the second step, the initial set is refined to satisfy the following criteria: the parameters should reproduce the observed crystal structures and sublimation enthalpies of related compounds, and the experimental crystal structure should correspond to the global minimum of the potential energy. The atomic charges were fitted to reproduce the molecular ab initio electrostatic potential, calculated at HF/6-31G* level; the fitting was carried out using the restrained electrostatic potential (RESP) method taken from the AMBER program. The method was applied to obtain a single set of charges using several conformations of a given molecule (the multiple-conformation-derived charges). An additional point charge (with zero nonbonded parameters) is used to model the lone pair electrons of sp2 nitrogen. The torsional parameters of the side chains of the 20 naturally occurring amino acids were computed by fitting to rotational energy profiles obtained from ab initio MP2/6-31G** calculations. The peptide backbone torsional parameters were obtained by fitting the energies to the ˆ-‰ energy maps of terminally blocked glycine and alanine amino acids constructed at MP2/6-31G**//HF/6-31G** level of theory. The performance of the force field was evaluated by simulating crystal structures of small peptides. As the important problems of the protein structure and functions as well as of the natural science as a whole refer to prediction of three-dimensional structure and pathways of folding of proteins, Scheraga and coauthors have modified and augmented the force fields mentioned above by some theoretical approaches and concepts. We will notice a few of them related directly to MM calculations. The all-atom ECEPP05 force field was coupled with an implicit solvent-accessible surface area model (ECEPP05/SA, Arnautova and Scheraga 2008). Recognizing the impossibility of searching enormous conformational space of real protein with an all-atom potential functions, a hierarchical procedure was developed. Its main features are the initial use of a united residue (UNRES) potential functions and an efficient procedure, conformational space annealing (CSA) (Scheraga et al. 2002, and references herein). The protein is first optimized with the low-resolution “coarse-grained” UNRES model (Liwo et al. 1999). In UNRES force field, the backbone is represented as a virtual-bond chain of C’ atoms, and the side chains are depicted as ellipsoids. The interaction sites are the united-atom side chains and the centers of the peptide groups between C’ atoms. The lowest-energy UNRES conformation, as well as a set of distinct low-energy conformations, is then converted to all-atom models. Finally, the all-atom models are refined with the EDMC method, with inclusion of continuum hydration model. The search for low-energy “native-like” structures includes several minimization and Monte Carlo procedures. The simplest of the MC methods is Monte Carlo with

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minimization, i.e., a Metropolis Monte Carlo (MCM) algorithm in which every trial state is first energy-minimized before the Boltzmann acceptance criterion is applied. The electrostatically driven Monte Carlo (EDMC) method employs a move set in which individual peptide groups are selected at random and rotated “in place” (i.e., the conformational change is localized to the peptide group as much as possible) so as to optimize the alignment of its dipole moment with the local electric field. The methodology assumes that a protein molecule is driven toward the native structure by the combined action of electrostatic interactions and stochastic conformational changes associated with thermal movements (Ripoll and Scheraga 1988). The use of the force fields and methods enables Scheraga and coauthors not only to rationalize but to predict some interesting features of protein three-dimensional structure and pathways of folding. “With increasing refinement of the computational procedures, the computed results are coming closer to experimental observations, providing an understanding as to how physics directs the folding process” (Scheraga 2008). Several recent successful applications of UNRES force to simulations of biological functions of proteins can be mentioned. The mechanism of growth of amyloid “-peptide fibrils (the main component of plaques formed in the human brain during Alzheimer disease) was suggested (Rojas et al. 2011) and supported by simulated 2D ultraviolet spectroscopy (Lam et al. 2013). The mechanism of HSP70 heat-shock protein action was proposed which was based on ATP-induced scissor-like motion of the two sub-domains of the nucleotide-binding domain (NBD) (Golas et al. 2012). The computationally obtained structure of this chaperon was confirmed then by experimental study. Some additional examples of UNRES applications are mentioned in very recently published review of Scheraga (2015). Similar approach was developed by Scheraga and coauthors for nucleic acids interactions. The first paper on coarse-grained models of nucleic acids represents NARES approach (Maciejczyk et al. 2010) to nucleic acid bases. Each base is simulated by a combination of three to five interaction centers of 6-12 van der Waals and permanent dipole interactions. Then, UNRES-2P model was developed (He et al. 2013) representing a polynucleotide chain by a sequence of virtual sugar atoms, located at the geometric center of the sugar ring, linked by virtual bonds with attached united base and united phosphate groups. This oversimplified model reproduces the double helix formation, some its thermodynamics properties (He et al. 2013), and was very recently applied to DNA-protein interactions (Yin et al. 2015).

Force Fields and Programs Developed by Kollman and Coauthors. AMBER The computer software and force fields referred to as “AMBER” (Assisted Model Building with Energy Refinement) are the most popular of those designed for wide class of biomolecules including proteins and nucleic acids. The first description of the computer program AMBER appeared in 1981 (Weiner and Kollman 1981), and

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the first detailed specification of force field was published 3 years later (Weiner et al. 1984). The force field contained usual “minimalist” set of energy terms (Eqs. 1, 2, 3, 4, 5, and 6 of this survey), the parameters being adjusted to both experimental and quantum mechanics data. The parameters of their previous publications and those of other authors were used as starting points for the adjustment. Atomic charges are obtained by fitting a partial charge model to electrostatic potentials calculated by ab initio quantum mechanics calculations (with minimal STO 3G basis set). The (10-12) potential was included for description of H-bonding (this term was deleted in new versions of AMBER force fields and software). The force field accounted for implicit solvent by using distance dependent dielectric constant (© D Rij ), united atom approach (i.e., several atoms, e.g., four atoms of a methyl group are considered as one particle) enabled one to perform calculations of sufficiently extended systems (such as fragments of DNA double helix of a few nucleotide pairs or peptides of dozens of amino acid residues). In spite of several deficiencies of the first AMBER force field (e.g., too short distances in stacking complexes due to short equilibrium radii for van der Waals interactions, defects in atomic charges due to very approximate quantum mechanics data), hundreds of papers were published with this force field and the program. In 2005, even after release of several new and improved AMBER force field versions, the paper (Weiner et al. 1984) was the 10th most-cited one in the history of the Journal of the American Chemical Society (Case et al. 2005). Since those publications new versions of both force fields and program sets were released. Cornell et al. (1995) published “second generation” AMBER force field for simulating the structures, conformational energies, and interaction energies of proteins, nucleic acids, and many related organic molecules in condensed phases. The charges were determined using the calculations with 6-31G* basis set and restrained electrostatic potential (RESP) fitting and had been shown to reproduce interaction energies, free energies of solvation, and conformational energies of small molecules to a good degree of accuracy. The new van der Waals parameters were derived to account for liquid properties. Retaining most of torsion parameters of “the previous generation” of force field, the improvement of potentials for peptide backbone was performed for ˆ and ‰ dihedrals as well as for the nucleoside ¦ dihedrals by adding extra Fourier terms. This “second generation” force field is usually referred to as ff94 (parameter set parm94). There are a number of AMBER parameter sets, with names beginning with “ff” and followed by a two-digit year number, such as ff99. We will briefly mention some new features of these force fields. Some versions differ from ff94 by minor (but important for some problems) modifications. The ff96 force field differs from ff94 only in the peptide ˆ-‰ torsional potential, and parm96 was developed because it was clear that parm94 overstabilized ’ helices relative to “ sheets. Thus, parm94 seems adequate for comparing structures with similar secondary structures, but parm96 seems to more accurately represent the relative stability of ’ and “ secondary structures. The improved version ff99 (Wang et al. 2000) was developed using the restrained electrostatic potential (RESP) approach to derive the partial charges. The adjustment of atom point charges and torsion constants for the set of 34

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molecules to the highest level ab initio model (GVB/LMP2) and experimental data enables authors to reproduce data for 55-molecule set with the better accuracy than MM3, MMFF, and early versions of CHARMM force fields. It was demonstrated that “a well-parameterized harmonic force field with a reliable charge method can describe the structure and intramolecular energies for organic systems very well” (Wang et al. 2000). The authors tried to use the adjusted parameters (parm99) in nonadditive models (which include polarization), and obtained “reasonable results using the same parameters derived for the additive model, although additional torsional parameterization is required to achieve the same high level accuracy as that found using the additive model” (Wang et al. 2000). This force field is widely used during more than 10 years. The other AMBER force field, “a thirdgeneration point-charge all-atom force field for proteins” (referred to as ff03) was developed by Duan et al. (2003). The main difference from previous force fields is that all quantum mechanics calculations were done in the condensed phase with continuum solvent models and an effective dielectric constant of © D 4. Initial tests on peptides demonstrated a high degree of similarity between the calculated and the statistically measured Ramanchandran maps for Glycine and Alanine peptide fragments. One more AMBER force field referred to as general Amber force field (GAFF) should be mentioned (Wang et al. 2004). GAFF have been designed as an extension of the AMBER force fields to be compatible with existing versions for proteins and nucleic acids and has parameters for most organic molecules of pharmaceutical importance that are composed of H, C, N, O, S, P, and halogens. The parameterization is based on more than 3000 MP2/6-31G* optimizations and 1260 MP4/6-311G(d,p) single-point calculations. Unlike most conventional force fields, parameters for all combinations of atom types are not contained in an exhaustive table but are determined algorithmically for each input molecule based on the bonding topology (which determines the atom types) and the geometry. A completely automated, table-driven procedure (called antechamber, a part of the late versions of AMBER program set) was developed to assign atom types, charges, and force field parameters to almost any organic molecule (Wang et al. 2006). New versions of AMBER software work with several force fields. The newest AMBER 11 version (Case et al. 2010) supports CHARMM fixed-charge force fields as well. The initial AMBER force fields and program packages (due to the state of the theory and computer facilities) were generally aimed at the search for energy minima of separate molecular systems in vacuum. The current AMBER versions are strongly aimed at simulations (molecular dynamics, evaluations of free energy changes) of biomolecules in water solutions. Both explicit solvent models (using TIP3P, TIP4P, TIP5P, and SPC water models as well as models of some organic solvents) and implicit ones are supported. By default, explicit solvation model involves electrostatic interactions handled by a particle-mesh Ewald (PME) procedure, and long-range Lennard-Jones attractions are treated by a continuum model. The PME is a modified form of Ewald summation, a method to efficiently calculate the infinite range Coulomb interaction under periodic boundary conditions, and PME is a modification to accelerate the Ewald reciprocal sum to near linear scaling using the three-dimensional fast Fourier transform (3DFFT).

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An accurate description of the aqueous environment is essential for atom-level biomolecular simulations but may become very expensive computationally. An implicit model replaces the discrete water molecules by an infinite continuum medium with some of the dielectric and “hydrophobic” properties of water. The continuum implicit solvent models have several advantages over the explicit water representation, especially in molecular dynamics simulations (e.g., they are often less expensive and generally scale better on parallel machines; they correspond to instantaneous solvent dielectric response; the continuum model corresponds to solvation in an infinite volume of solvent, there are no artifacts of periodic boundary conditions; estimating free energies of solvated structures is much more straightforward than with explicit water models). Despite the fact that the methodology represents an approximation at a fundamental level, it has in many cases been successful in calculating various macromolecular properties (Case et al. 2005). The total energy of a solvated molecule can be written as a sum of molecule’s energy in vacuum (gas phase) and the free energy of transferring the molecule from vacuum into solvent, that is, solvation free energy typically assumed as can be decomposed into the electrostatic and nonelectrostatic parts. The above decomposition is the basis of the widely used PB/SA (Poisson-Boltzman solvent accessible surface area) scheme. AMBER follows this way to compute the nonelectrostatic part of the energy. The most time-consuming part of solvation energy estimations is the computation of the electrostatic contribution. AMBER employs one of the most popular biomolecular applications the finite-difference method (FDPB) and analytic generalized Born (GB) method. In the GB model, a molecule in solution is represented as a set of spherical cavities of dielectric constant of 1 with charges at their centers embedded in a polarizable dielectric continuum solvent of a higher dielectric constant. The AMBER software and force fields are widely used for the simulation of various systems of biological and material science importance. Some researchers develop proper additions and corrections to “standard” parameters for better description of specific systems. We refer to only few examples. Perez et al. (2007) reparameterized the parm99 force field for nucleic acid simulations improving the representation of the ’/” conformational space. Zgarbova et al. (2011) refined description of glycosidic torsion angles of nucleic acids. Other work of Zgarbova et al. (2013) refers to © and — DNA torsions. Very recently, Ogata and Nakamura (2015) suggested the improvement of parameters of the force field for phospholipids. Song et al. (2008) adjusted the AMBER force field to reproduce conformational properties of an oxidative DNA lesion, 2,6-diamino-4-hydroxy-5formamidopyrimidine. New improved parameters for modified peptide chain were suggested by Grubiši´c et al. (2013) recently. Some of such improvements have been included into newer versions of AMBER package. AMBER force field can be used via other MM program packages, such as CHARMM, GROMACS, NAMD, and with widely used QM package GAUSSIAN. The latest version of AMBER program package is AMBER14 distributed with AmberTools 15 (Case et al. 2015). The new versions of AMBER software support the option of allowing part of the system to be described quantum mechanically

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in an approach known as a hybrid (or coupled potential) QM/MM simulation. Both semiempirical and the density functional theory-based tight-binding (DFTB) Hamiltonian can be used. The system may contain both two nonbonded parts and covalently bonded QM and MM subsystems (Case et al. 2010).

Other Popular Force Fields and MM Software. CHARMM, OPLS, GROMOS The force fields and program packages listed above are used for nearly the same type of simulations as those considered above. The main differences refer to the organization of the programs and to relative role of experimental data, and quantum mechanics results in force field parameter adjustment. The first versions of force fields and programs as well as the whole research programs were initiated by such pioneers of MM applications as Karplus (CHARMM, Chemistry at HARvard Molecular Mechanics), Jorgensen (OPLS, Optimized Potentials for Liquid Simulations), Gunsteren and Berendsen (GROMOS, GROningen MOlecular Simulation package), and GROMACS (GROningen MAchine for Chemical Simulation). The progress and improvement of the force fields and programs are connected to comparison with the approaches and results of other programs. The parametrization of second generation AMBER force field (Cornell et al. 1995) had been influenced by Jorgensen OPLS potentials for proteins (Jorgensen and Tirado-Rives 1988), whereas these OPLS united atom potentials adopted bond stretch, angle bend, and torsional terms from the AMBER united-atom force field (Weiner et al. 1984). The OPLS parameters for the 20 neutral peptide residues were obtained primarily via Monte Carlo simulations for the 36 organic liquids, and the parametrization for the charged protein residues was performed via comparisons with ab initio results for ion-molecule complexes (6-31G(d) basis set) and on Monte Carlo simulations for hydrated ions (Jorgensen and Tirado-Rives 1988). The OPLS-AA (all atom) force field (Jorgensen et al. 1996) retain most of bond stretch and angle bending parameters from AMBER all-atom force fields, but torsion and nonbonding constants are reparametrized utilizing both experimental and ab initio (RHF/6-31G* level) data. This force field was improved for peptides by means of refitting the key Fourier torsional coefficients using accurate ab initio data (LMP2/cc-pVTZ(f)//HF/6-31G** level) as the target (Kaminski et al. 2001). Most of popular program sets utilize various force fields, not only those developed in the research groups of the authors of the program with the same title. The software suite GROMACS was developed by Berendsen group at the University of Groningen, The Netherlands, starting in early 1990s. This fast program for molecular dynamics simulation does not have a force field of its own but is compatible with GROMOS, OPLS, and AMBER force fields. The GROMACS (van der Spoel et al. 2005) was developed and optimized especially for use on PC-clusters. It originated from the Fortran package GROMOS developed by van Gunsteren et al. (1987). The GROMOS force fields are not pure all-atom force fields; aliphatic CHn groups are treated as united atoms. The recent version, GRO-

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MOS05 (Christen et al. 2005), utilize new force field sets (53A5 and 53A6, the first number in the names of a parameter sets refer to the number of atom types) based on extensive reparameterization of the previous GROMOS force field (Oostenbrink et al. 2004). In contrast to the parameterization of most other biomolecular force fields, this parameterization is based primarily on reproducing the free enthalpies of hydration and apolar solvation for a range of organic compounds. Recently, 53A6 force field has been reparametrized by Pol-Fachin et al. (2014) by adjustment to QM data (HF/6-31G* level of theory) for applications to study Glycoproteins. Very popular for the computations of wide range of macromolecular systems are CHARMM (Chemistry at HARvard Macromolecular Mechanics) program and force field sets. The computations include energy minimization, molecular dynamics, and Monte Carlo simulations. CHARMM originated at Martin Karplus’s group at Harvard University, the first publication on CHARMM program and force field (refer to Brooks et al. 1983). Several versions of both software and force field were released since the first ones. The improvement of the programs and force fields are in progress till now. The mathematical expressions for calculation of CHARMM energy are nearly the same as “minimalist” ones for other force fields (MacKerell 2004). The minor differences with e.g., AMBER or OPLS force fields refer to inclusion of harmonic improper and Urey-Bradley terms. The parameters of various versions of force field are consistently adjusted with emphasis to specific molecules to be applied. The first versions of CHARMM force fields were aimed primarily to protein molecular dynamics simulations. Then, the special adjustments were performed for the simulations of the nucleic acids (Foloppe and MacKerell 2000) and lipids (Klauda et al. 2008). The improvement of parameter for specific molecules continues till recent times, while other parameters can be used from the previous versions. Raman et al. (2010) recently extended the CHARMM additive carbohydrate all-atom force field to enable modeling glycosidic linkages in carbohydrates involving furanoses. Like in most popular modern force fields, both the ab initio quantum mechanics computations and experimental data for solid and liquid states are used in adjustment of the parameters of CHARMM force fields. The distinct feature of this adjustment refers to charge adscription. Like OPLS force fields (Jorgensen et al. 1996), CHARMM utilize so-called supramolecular approach. The partial atomic charges are adjusted to reproduce ab initio (HF/631G* level) minimum interaction energies and geometries of model compounds with water or for model compound dimers (with energy and distance corrections for reproducing the correct experimental densities). On the other hand, the use of ab initio data on small clusters to optimize the van der Waals parameters leads to poor condensed phase properties. This requires that the optimization of these parameters be performed by empirical fitting to reproduce thermodynamics properties from condensed phase simulations, taking into account the relative values of the parameters obtained from high-level ab initio data on interactions of the model compounds with rare gases (Klauda et al. 2008). Very recently, Vanommeslaeghe and MacKerell (2015) published detailed description of the development of CHARMM method and software. We will mention here only some most important for biological and medical applications

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recent advancements of both all-atom additive force field (on the way to CHARMM36 version, modern class I model) and polarizable force fields (class II models). Best et al. (2012) presented a combination of refinements important with respect to the equilibrium between helical and extended conformations in protein folding simulations. Peptide backbone (®, §) potential has been refined against experimental solution NMR data. Side-chain ¦1 and ¦2 torsion parameters have been optimized by fitting to quantum-mechanical energy surfaces, followed by additional empirical optimization targeting NMR data. A comprehensive validation of the revised force field indicates that the revised CHARMM 36 parameters represent an improved model for modeling and simulation studies of proteins, including protein folding, assembly, and functionally relevant conformational changes (Best et al. 2012). The refinement of the CHARMM27 all-atom additive force field for nucleic acids via adjustment of backbone parameters to QM data (MP2/6-31 C G(d) level of theory) followed by trial MD simulations of DNA duplexes enabled Hart et al. (2012) to improve treatment of the BI/BII equilibrium of DNA conformations. Many other specific refinements of the CHARMM additive force fields for these and other classes of biologically important molecules have been mentioned in the recent paper of Vanommeslaeghe and MacKerell (2015). Along with comprehensive refinements and biomolecular applications of class I CHARMM force fields, polarizable CHARMM models are extensively developed during recent years for more accurate and physically based description of biologically important molecular systems. These force fields use the classical Drude oscillator (i.e., “charge-on-spring”) model. In the model, a virtual negatively charged particle (“Drude particle”) is connected to the core of each polarizable atom by means of a harmonic potential (“spring”). Description of this class II force field is out of scope of this survey, but we will mention some results of proteins and nucleic acid computations with such models. Very recently, Huang et al. (2014) summarized recent efforts to include the explicit treatment of induced electronic polarization in biomolecular force fields. They arrive with conclusion that the inclusion of explicit electronic polarizability leads to significant differences in the physical forces affecting the structure and dynamics of proteins (Huang et al. 2014). The polarizable force field more accurately models peptide-folding cooperativity (Huang and MacKerell 2014). Development of a CHARMM Drude polarizable model for nucleic acids (Baker et al. 2011) requires significant adjustments of the phosphodiester backbone and glycosidic bond dihedral parameters of basesodium interactions and of selected electrostatic parameters in the model (Savelyev and MacKerell 2014a). The modern polarizable model satisfactorily treats the equilibrium between the A, BI, and BII conformations of DNA (Savelyev and MacKerell 2014b). It is interesting to mention that multiple adjustments of different additive force fields mentioned above and other force fields result in an increase of similarity in both corresponding parameters values and the results of applications to complex molecular systems (such as protein and nucleic acid fragments), but many differences both in corresponding parameter values and computation results still remain.

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Conclusions The molecular mechanics computation approach to the study of the molecular systems arose a half of century ago and today has become a useful and powerful research tool for many branches of natural science. During this period the number of publications utilizing this approach increased by a few orders of magnitude. This increase continues up to present days, followed by new journals appearance and a flow of MM publications in the journals of rather wide area of coverage. The MM considers the molecular systems via classic Newtonian mechanics approach, i.e., using classical approximation to essentially quantum systems. The MM suggests representation of the system energy as a sum of terms responsible for interactions of various physical natures. The minimalist MM approach involves the simple expressions for chemical bond stretching, bond angle bending, torsions about the bonds, and nonbonded van der Waals and Coulomb contributions calculated via additive atom-atom scheme. The force fields (sets of mathematical formulae and numerical parameters) are derived from simple physical considerations followed by adjustment to the experimental data and precise quantum mechanics results. An additivity of the terms responsible for separate atom contributions and for interactions of different physical nature and transferability of the parameters between atoms and molecules of similar structure are the main assumptions of MM approach. Development of the approach results in elaboration of a variety of computer software and force fields for simulating different molecular systems of biological, medical, and industrial importance. Various force fields differ in the complexity of mathematical expressions and in the relative role of experimental and quantum mechanics results used for the parameter adjustment. The development of algorithms for the MM applications (including those for molecular dynamics and Monte Carlo techniques) and the increase of computer power enable researchers to approach such complex problems as predicting the pathways of the biopolymers three-dimensional structure, molecular recognition in various biological processes, assistance in creation of compounds with desirable properties (including drugs and industrial materials). Analysis of the current state of physicochemical study of biological processes suggests that MM simulations based on empirical force fields have an ever-increasing impact on understanding structure and functions of biomolecules. Rather close results obtained with different force fields can be accounted for by existing various solutions of the task of parameter adjustment even to the same target data (really, target data used when elaborating various force fields differ, slightly or considerably). Force fields are internally consistent. Their parameters are highly correlated within a force field. Several force fields have been elaborated recently going beyond the atomatom approach and the additivity principle. Some force fields utilize a hierarchical procedure for consideration of complex systems.

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Molecular Structure and Vibrational Spectra Jon Baker

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . He22C : An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Newton–Raphson Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hessian Matrix and Hessian Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition State Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Modified Newton–Raphson Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GDIIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry Optimization and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance for Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimization: An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition State Searches: An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Optimized Potential Scans in Transition State Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Experimental and Theoretical Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry Optimization of Molecular Clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry Optimization in the Presence of External Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,2-Dichloroethane: An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaled Quantum Mechanical Force Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,2-Dichloroethane: A Further Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density Functional Theory and Weight Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 6 8 10 11 13 16 18 18 21 29 36 39 41 43 44 49 56 58 65 69 71

J. Baker () Parallel Quantum Solutions, Fayetteville, AR, USA e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_10-2

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J. Baker

Abstract

This chapter deals with two very important aspects of modern ab initio computational chemistry: the determination of molecular structure and the calculation, and visualization, of vibrational spectra. It deals primarily with the practical aspects of determining molecular structure and vibrational spectra computationally. Both minima (i.e., stable molecules) and transition states are discussed, as well as infrared (IR), Raman, and vibrational circular dichroism (VCD) spectra, all of which can now be computed theoretically.

Introduction This chapter deals with two very important aspects of modern ab initio computational chemistry: the determination of molecular structure and the calculation, and visualization, of vibrational spectra. The two things are intimately related as, once a molecular geometry has been found (as a stationary point on a potential energy surface at whatever level of theory is being used) it has to be characterized, which usually means that it has to be confirmed that the structure is a genuine minimum. This of course is done by vibrational analysis, i.e., by computing the vibrational frequencies and checking that they are all real. A large percentage of the total expenditure in CPU cycles devoted to computational chemistry (variously estimated at between 60 % and 85 %) is spent optimizing geometries. In order to calculate various molecular properties, one first needs a reliable molecular structure so this is perhaps not surprising. Algorithms for geometry optimization are now highly advanced and usually very efficient and most of the quantum chemistry programs available for general use have solid and reliable geometry optimization modules. They also nearly all have analytical second derivatives, at least for the most common theoretical methods, which makes it relatively straightforward to compute vibrational frequencies once a structure has been found. In this chapter I deal primarily with the practical aspects of determining molecular structure and vibrational spectra computationally. I consider both minima (i.e., stable molecules) and transition states, as well as infrared (IR), Raman, and vibrational circular dichroism (VCD) spectra, all of which can now be computed theoretically. The program used to carry out the calculations presented here is the PQS package developed by Parallel Quantum Solutions (PQS 2010), although any modern general purpose package (e.g., Gaussian, Turbomole, GAMESS) would do just as well. As the name implies, all the major ab initio functionality of this package is fully parallel, including energies, gradients, and second derivatives. PQS was chosen because (a) it is the package I actually use in all my application work; and (b) I am one of its principal authors. A review of the capabilities and parallel efficiency of the PQS package was published recently (Baker et al. 2009). I have elected to use a standard level of ab initio theory for all of the examples presented in this chapter, namely, density functional theory (Hohenberg and Kohn

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1964; Kohn and Sham 1965) (DFT) using the B3LYP (Becke 1993) (see also Hertwig and Koch (1997)) functional and the 6-31G* (Ditchfield et al. 1971) basis set (B3LYP/6-31G* ). DFT is now the method of choice for routine chemical applications, and B3LYP – despite the large number of functionals developed since – is still one of the most popular. Many of the techniques and pitfalls in locating stable geometries are essentially independent of method, although DFT has its own issues as a result of the numerical quadrature required to handle many of the integrals; this will be discussed in some detail later. DFT is so popular that Hartree–Fock theory, which used to be the standard approach throughout the 1980s, has now almost disappeared other than as a precursor for higher level post-SCF calculations. At the time of writing, the latter are beginning to make a bit of a comeback as the limitations of DFT are being reached, in particular its inability to systematically improve the wave function (and hence the results). A good introduction to DFT for those, like me, whose background is in traditional quantum chemistry is the now classic 1993 paper from (Johnson et al. 1993). The Born–Oppenheimer approximation (Born and Oppenheimer 1927) (see also Wikipedia (2010)) is used throughout. This extremely important approximation, namely that electrons, being so much less massive, respond instantaneously to the motion of the nuclei, underpins the whole concept of the potential energy surface (PES). Stable arrangements of nuclei (stable or meta-stable molecules) correspond to local minima on the PES, first-order saddle points connecting two different minima correspond to transition states, and the “valleys” joining transition states to minima correspond to “reaction paths.” Electronic excitation is represented by a “jump” from one potential energy surface to another. We will be spending all of our time on the lowest energy (ground-state) PES.

Molecular Structure From a theoretical point of view the determination of a “molecular structure” means determining the geometry (i.e., the positions of atoms relative to one another) at a minimum (the bottom of a well) on the ground-state potential energy surface. Although a prerequisite, locating a minimum does not automatically mean that you have found the structure of a stable molecule; this depends on the barrier height, i.e., the energy required to get out of the well. The barrier height can be determined by locating the transition state, which for our purposes is defined as the highest energy point on the lowest energy path between reactants and products. (In a barrierless reaction, this will effectively be the energy of the products, assuming the reactant is taken to have the lower energy.) As a rough rule of thumb, if the barrier height is less than about 12–15 kcal mol1 , the system is kinetically unstable at room temperature. Determining the geometry of a stable molecule therefore requires knowledge of both the minimum itself and the barrier height (i.e., effectively locating the transition state) for all possible decomposition reactions. Of course, the structures of many stable molecules can be derived from “chemical intuition” based on preexisting knowledge, but for new structures in particular the transition state is

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equally important and the practicing theoretician must be able to find both. In this section we discuss the methods available to do so.

He22C : An Illustrative Example The simplest possible molecular orbital approach to the bonding in the hydrogen molecule (H2 ) would consider the overlap between two 1s atomic orbitals on each hydrogen producing two molecular orbitals (MOs), one of which is bonding (the in-phase combination) and the other antibonding (the out-of-phase combination). H2 is stable because its two electrons occupy the bonding orbital, giving an overall energy lowering compared to two separate H atoms. Similar considerations applied to helium would suggest, quite rightly, that He2 would not be stable, since both the bonding and antibonding MOs would be occupied, giving no net bonding. (In fact, as is well known, the antibonding MO is more antibonding than the bonding MO is bonding.) What happens if the electrons in the antibonding MO are removed? On the one hand, there should be a strong bonding interaction, while on the other the resultant positive charge should cause the two nuclei to repel one another. Figure 1 shows the ground-state PES of He2C 2 . This was obtained via a potential scan, varying the He–He distance from 0.6 to 4.0 Å in steps of 0.01 Å and computing the energy at each scanned distance. (An optimized potential scan – scanning one particular variable or combination of variables while optimizing all remaining degrees of freedom – is a useful tool for locating transition states, as will be seen later.) The upper curve shows the energy at the restricted RB3LYP/6-31G* level, while the lower curve is the unrestricted equivalent (UBLYP/6-31G* ). The two curves coalesce at He–He distances less than about 0.96 Å; at distances greater than this the restricted wave function is energetically unstable, i.e., a lower energy can be obtained by switching to the unrestricted formalism. During the restricted (upper) scan full symmetry .D1h / was maintained and the system is a pure singlet. For the unrestricted scan, symmetry was turned off and the highest occupied (HOMO) and lowest unoccupied (LUMO) MOs, both alpha and beta spin, were allowed to mix; this results – as can be seen – in an ultimately much lower energy, but the spin symmetry is lost and the system wave function is no longer a pure singlet. (Spin contamination gradually rises with increasing He–He distance until at 4.0 Å hS2 i is 1.000 and the multiplicity is 2.236, i.e., the system is somewhat “greater” than a doublet.) At B3LYP/6-31G* the energy of two separate HeC ions is 3.9863 Eh and so, despite the spin contamination, the unrestricted curve gives a far better description of the dissociation of He2C 2 than the restricted one. This is a general feature of the dissociation of essentially all diatomic molecules and is well known. The interesting feature from our point of view is that this simple, onedimensional PES contains both a (local) minimum and a transition state, and both types of stationary point can be examined and discussed in relation to the same

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−3.5 “J: \ he2-plots” −3.55 −3.6 −3.65 −3.7 −3.75 −3.8 −3.85 −3.9 0.5

1

1.5

2

2.5

3

3.5

4

* Fig. 1 Energy versus He–He distance (Å) for He2C 2 (RB3LYP/6-31G – upper curve; UB3LYP/6* 31G – lower curve)

potential energy curve. This is the only one-dimensional potential energy curve that I am aware of that has this feature. The gradient is zero at both turning points – minimum and transition state – on the potential energy curve. The key difference between them is of course the sign of the energy second derivative (the rate of change of the gradient) which is positive for a minimum and negative for a transition state. In this example, we have the complete energy curve between He–He distances of 0.5–4.0 Å available to us, but in general of course (in particular for anything more complicated than a diatomic molecule) this would not be the case, and if we wanted to search for, say, a local minimum we would have to provide an initial approximation for the geometry, i.e., a guess for the minimum structure. We would then calculate the energy, the gradient and either calculate, or provide a reliable estimate for, the second derivative and use this information to predict: (a) how far away we are on the PES from the nearest local minimum, and (b) the step to take (length and direction) in order to get there. Note that only in the one-dimensional case are the first and second derivatives single numbers; typically they would be vectors (with elements dE/dvi , where vi is the i-th variable) and matrices (with elements d2 E/dvi dvj ), respectively. Predicting the next step, i.e., how to change the current geometry to get to a new geometry that is closer to the desired stationary point than the old geometry was, is the job of the geometry optimization algorithm. For ab initio methods these algorithms can be very sophisticated as both the energy and the gradient are relatively expensive to compute (the second derivative is usually even more

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expensive), and so additional manipulations in the optimization algorithm in order to reduce the number of optimization cycles (i.e., the number of costly energy and gradient calculations) is more than justified. Not so in molecular mechanics where the energy and gradient cost very little and a highly efficient – in terms of CPU time – optimization algorithm is required. It is obvious that one of the factors that determines how quickly a given geometry optimization will converge is the starting geometry. In our He2C 2 example, in order to locate the metastable minimum, which occurs at a He–He distance of around 0.7 Å (shorter than in H2 , making this the shortest bond distance known), any initial starting bond length greater than ca. 1.2 Å – the distance in the transition state on the lower, unrestricted, curve – will almost inevitably lead to dissociation (see Fig. 1). If you want to find the transition state then the closer you start to a He–He distance of around 1.2 Å the better off you will be.

The Newton–Raphson Step Consider a starting He–He distance of about 1.0 Å, i.e., midway between the bound minimum and the transition state. This could probably go either way, toward the minimum or the transition state. Which way it does go is guided by the curvature of the surface around the local region, i.e., by the nature of the second derivative. If we do a standard Taylor series expansion about the current point, then we have E .x C h/ D E.x/ C h

h2 d 2 E dE h3 d 3 E C C C :::: 2 dx 2 dx 6 dx 3

(1)

In the region about a local minimum (or maximum) the potential energy curve is more-or-less parabolic, so assuming we are in or close to this region we can ignore all terms in the Taylor series expansion (Eq. 1) beyond the quadratic. Taking the derivative with respect to the displacement h gives dE d 2E dE .x C h/ D Ch 2 : dh dx dx At a stationary point

dE.xCh/ dh

(2)

D 0 and so the displacement h that achieves this is dE

h D  ddx 2E

or

dx 2

hD

g ; H

(3)

where g is of course the gradient and H is the energy second derivative. Equation 3 is the well-known Newton–Raphson step in one dimension. The multidimensional equivalent of this is h D H1 g;

(4)

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where h is now a vector (a set of displacements for all variables), g is the gradient vector, and H is the second derivative matrix, commonly known as the Hessian matrix. Going back to our He2C example we see that the Newton–Raphson step is 2 normally in the direction of minus the gradient (just like a steepest descent step) with a step length determined by the magnitude of the second derivative. If the second derivative is positive, the step will be downhill (opposite to the gradient) whereas if the second derivative is negative, the step will be uphill (in the same direction as the gradient). Thus at a He–He distance of 1.0 Å (see Fig. 1) the step taken will be uphill toward the transition state if the surface curvature is negative and downhill toward the minimum if it is positive. In the multidimensional case, we can always find a unitary transformation that diagonalizes the Hessian matrix (which is symmetric about its diagonal), i.e., D D UHUT ;

(5)

where D is a diagonal matrix and the superscript T represents a matrix transpose. The vectors in U are known as eigenvectors and the diagonal elements of D are the corresponding eigen values. Both the displacement vector h and the gradient g can be transformed into this new (diagonal) representation as g0 D UT gI

h0 D UT h:

(6)

Substituting Eq. 5 into Eq. 4 gives   h D  UD1 UT g:

(7)

Multiplying both sides by UT and rearranging gives     UT h D UT U D1 UT g ;

(8)

h0 D D1 g0 :

(9)

Diagonalizing the Hessian matrix results in a set of mutually orthogonal directions on the PES (the vectors in U) with the gradient along each direction given by the corresponding element in the vector g0 and the second derivative given by the corresponding diagonal matrix element in D. Thus we see that the Ndimensional Newton–Raphson step is equivalent to N separate one-dimensional Newton–Raphson steps along the orthogonal directions given by the vectors in U. If all the eigen values of the Hessian matrix (all the diagonal entries in D) are positive, then the step taken will attempt to minimize along all directions on the PES (all degrees of freedom) and will head downhill to the nearest minimum. If one (and only one) of the Hessian eigen values is negative, then the Newton–Raphson step will attempt to maximize (head uphill in energy) along that direction, while

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simultaneously minimizing along all the other directions, i.e., it will move toward a transition state. If the Hessian has two or more negative eigen values, then, left to its own devices, the optimization will head toward a second (or higher) order stationary point. As these are usually of much less interest than minima or transition states, the problem in these cases is normally how to take a decent optimization step when the Hessian signature (number of negative eigen values) is inappropriate for the stationary point being sought. I have spent a lot of time on the Newton–Raphson step as it, or a modification thereof, forms the heart of virtually all advanced geometry optimization algorithms. It is usually a good step to take, particularly in the immediate region around the stationary point, and from a practical point of view it utilizes all the information that is likely to be available at the current point on the PES, i.e., first and second derivatives; there are very few codes that routinely compute derivatives higher than the second, and they would be prohibitively expensive in any case.

The Hessian Matrix and Hessian Updates As I hope the above discussion has shown, the Hessian matrix plays a very important role in determining the step direction, which depends on the surface curvature in the local region of the PES. However, care must be taken with the step length. Hessian information is only local and only applies in the region immediately surrounding the current point. If the predicted step length is small, then you are probably near to a stationary point, in which case the step is best taken “as is”; however too large steps are usually not wise and you can easily get lost. Consequently it is a good idea to impose a limit on the step size – either an overall limit on the total step length or limits on the size of individual step components (or indeed both). Some algorithms attempt to define a “trust radius” which limits the step size dynamically depending on the local nature of the PES (usually determined by comparing the actual energy change from that predicted from the Hessian assuming quadratic behavior). In PQS a simple user-defined maximum step length is imposed (default 0.3 au). Calculation of the exact Hessian for ab initio wave functions is expensive and in many cases it can be replaced by an approximation with little effect on the optimization itself, i.e., on the number of optimization cycles required to reach convergence. In some cases having the exact Hessian is a definite disadvantage. If you are fairly close on the PES to the stationary point being sought then, although an exact Hessian will likely get you there faster, the reduction in the number of optimization cycles will probably not be enough to offset the extra computational time required to compute the Hessian in the first place. (The exception is if the surface if fairly flat – long, weak bonds, or, especially, floppy systems with flat torsional potentials – in which case an exact Hessian will be a tremendous help.) A good quality Hessian will usually help the most when you are not really close to a stationary point, but within the region where the surface curvature is still appropriate, i.e., the Hessian matrix still has the correct signature (all positive eigen

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values for a minimum and one, and only one, negative eigen value for a transition state). Computing an exact Hessian will almost certainly be a waste of time if you are a long way from where you finally end up; under these circumstances an exact Hessian will almost inevitably have more negative eigen values than required and the Newton–Raphson step alone will clearly be inappropriate. If you are searching for a minimum you will be far better off using an alternative for the Hessian matrix which has all positive eigen values (such a matrix is known as positive-definite) which should keep the energy going down until you get closer to the stationary point. If the gradient is large, then a few steepest descent steps, possibly with a line search, might be in order, simply to lower the energy and get to a more appropriate region of the PES. In fact, other than in exceptional circumstances, when searching for a minimum you do not normally need to compute a full Hessian matrix at all. In most cases reasonably good starting geometries can be derived simply from “chemical intuition,” for example, C–H bond lengths (except in the rare cases of bridging hydrogens) are likely to be of the order of 1 Å, so there is no point in using a starting geometry with C–H distances differing much from this value. Very reasonable estimates of stretching, bending and also torsional force constants (diagonal Hessian elements) are readily available – for example, stretching force constants via Badgers rule (Badger 1934, 1935) (see also Schlegel 1984) – which are perfectly adequate in the vast majority of cases. Consequently you can normally come up with an approximate estimate for at least the diagonal elements of the Hessian matrix at the starting geometry using some simple empirical rules. Once you have taken a step from one point on the PES to another (hopefully closer to the final stationary point), then you can compute an energy and a gradient at the new geometry. This information can be utilized, together with the energy, gradient, and anything else that might be to hand from the old point (or points), not only to work out a new step (the job of the geometry optimization algorithm) but also to improve (update) the original approximation for the Hessian matrix. You can think of this as similar to determining the surface curvature by finite difference on the gradient; you know the gradient where you started, you take a small step, calculate a new gradient, and determine the curvature by forward differences. Of course, updating a Hessian matrix on a multidimensional energy surface is a lot more approximate than this as typically the step taken is fairly large and you are attempting to get an update for the whole Hessian from a step in just one direction. There are a number of different Hessian updating formulae, e.g., Murtagh and Sargent (1970) and Powell (1971), Broyden-Fletcher-Goldfarb-Shannon (BFGS) (Broyden 1970; Fletcher 1970, 1980; Goldfarb 1970; Shanno 1970), which can be used where appropriate depending on the nature of the stationary point being sought. (The exact details are not important.) For minimization the BFGS update is a popular one as, if you start out with a positive-definite Hessian matrix (with all positive eigen values), as needed for a minimum search, you are more likely to retain this desirable feature with BFGS than with the other updates. (You may see statements to the effect than the BFGS update guarantees retention of a positivedefinite Hessian. This is not true. However, on those occasions – not too common,

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but they do occur – when the update fails in this regard, you can tell in advance that it is going to fail and either skip the update on that step or take a different one.)

Transition State Searches For a minimum search then, computing an exact Hessian matrix is rarely required. Things are unfortunately not so straightforward for transition state searches. First of all transition state geometries are not as obvious as are the geometries of stable minima and so it is often quite difficult to come up with good starting structures. This can be a major problem in itself. Secondly, the correct Hessian eigen value structure is much more important for a transition state than it is for a minimum. For a transition state search, the Hessian must have one, and only one, negative eigen value, and furthermore the direction of negative curvature must correspond to that for the structure being sought; to the reaction coordinate if you will. Difficult transition state searches can be plagued with additional small negative Hessian eigen values continually reappearing which correspond to rotation of, e.g., methyl groups, way off the reaction path, i.e., on some side chain or other nowhere near the reaction center. These additional negative eigen values often interfere with the search algorithm making the final structure difficult to locate. And thirdly, there are no simple empirical rules generally available for estimating a reliable starting Hessian for transition state searches as there are for minima. All these factors make transition state searches more difficult than minimizations and in all but the simplest cases a good estimate of the Hessian for at least the region round the active center is essentially mandatory. As is the case with minimization, once you are “on the right track,” it is not necessary to calculate fresh second derivative data at every optimization cycle, the Hessian matrix can be updated. Currently the best update for a transition state search is probably that proposed by Bofill (1994) which is a linear combination of the Powell (1971) and Murtagh and Sargent (1970) updates. This is a more flexible update than BFGS and allows the eigen value structure of the Hessian to change, which is obviously important if you get too close to a minimum. Unfortunately, unlike the case with minimization, there is no update that preserves the negative eigen value once you have it, and it is all to easy to find that the updated Hessian has either too few (zero) or too many (two or more) negative eigen values. There are transition state search algorithms that do not require an initial starting structure; instead they take as input the two minima that you wish to connect and attempt to derive some kind of “reaction path” connecting the two. Depending on the nature of the algorithm, the maximum energy structure on this path is either the transition state itself, or a pretty good estimate for it which is subsequently refined. The original algorithm of this type was the Linear Synchronous Transit (LST) approach of Halgren and Lipscomb (1977) which mapped each atom in the reactant to a corresponding atom in the product and derived the best linear path between them in a least-squares sense. The maximum energy structure along this path provided an estimate (unfortunately often not a particularly good one) of the

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transition state geometry which was subsequently used as input into a standard search algorithm. An improvement over this was Quadratic Synchronous Transit (QST) (Bell and Crighton 1984; Bell et al. 1981), which attempted to maximize along the line joining reactant and product while minimizing in all directions “orthogonal” (or conjugate) to this. One can easily see the rationale behind this approach: if the direction to be maximized could be reliably isolated from all other directions (imagine a Hessian matrix with what will become the direction of negative curvature having all zero off-diagonal matrix elements, i.e., Hii ; all Hij D 0) then it would be straightforward to maximize along the one direction while minimizing in the subspace of all the other variables. Unfortunately maintaining the orthogonality between the isolated direction and the rest of the space (i.e., keeping Hij D 0, j ¤ i) is rather difficult, and so the algorithm can often get lost. Nonetheless, there are several more modern algorithms based on these approaches, for example from Peng and Schlegel (1993) and Ionova and Carter (1993). If the “reaction coordinate” can be reliably confined to just a couple of (ideally one) variables, then a very good starting geometry for a transition state search can be obtained from an optimized potential scan. Here the variable in question is scanned over a range of values (hopefully including the value the variable has in the transition state) and at each scanned value all other degrees of freedom are minimized. In effect this involves a series of constrained optimizations over the scanned variable. Once again, the transition state is taken to be the geometry at the highest energy found during the scan. (If the highest energy occurs at either of the scan endpoints then either the scan range needs to be extended or there is no transition state in the region of the PES that is being explored.) An optimized potential scan is easy to set up with PQS and is normally a very reliable method, albeit potentially time consuming, of obtaining an excellent transition state starting geometry provided that the principal reaction coordinate can be successfully identified.

Choice of Coordinates In the discussion so far the choice of coordinates in which to carry out minimizations or transition state searches has not been specifically mentioned, although in certain cases – for example, when considering a potential scan – it is implicitly assumed that variables familiar to chemists, such as bond lengths and angles, have been employed. Virtually all quantities of interest that are actually computed in ab initio quantum chemistry, such as first (gradient) and second (Hessian) derivatives are calculated in terms of Cartesian coordinates and the molecular geometry, regardless of how it may be input, is stored and manipulated internally as a set of Cartesians. However, Cartesian coordinates are usually not the best choice when it comes to actually carrying out a geometry optimization, because they are generally too coupled. The only case where they are regularly used in the actual geometry optimization itself is in molecular mechanics.

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In the late 1960s and early 1970s the so-called Z-matrix was widely used in quantum chemistry, initially simply as a means of inputting the molecular geometry, but later as a means of defining a set of internal coordinates in which to carry out the optimization. With a Z-matrix, the geometry can be defined in terms familiar to a chemist using individual (primitive) bond lengths, bond angles, and torsions. The first atom defined in the Z-matrix is placed at the origin of a standard Cartesian axis system (i.e., at 0.0, 0.0, 0.0), the second is placed along the Z axis at a distance R1, say, and the third in the XZ plane at a distance, R2, from the second atom and making an angle, A1, with the first atom (or vice versa). Thereafter the position of all subsequent atoms is given in terms of a bond distance, a bond angle and, say, a torsion, relative to three previously defined atoms. In the days before graphical user interfaces (GUI) and model builders, the Z-matrix became a regular way of defining and reading a molecular geometry into the program doing the actual calculation. Dummy atoms could be included in the Z-matrix (and were needed to help define linear arrangements of three or more atoms for which the torsion was undefined) and, once geometry optimizations were regularly performed in Z-matrix coordinates, the writing of a successful Z-matrix became almost an art form, with the aim of defining the Z-matrix variables so that the value of any one of them could be changed without changing the values of any of the others, the idea being to reduce the coupling between the variables to a minimum. A well-defined set of internal coordinates usually performs better in a geometry optimization, i.e., converges in less optimization cycles, than the same optimization carried out in Cartesian coordinates. The playing field can be leveled, particularly for standard organic molecules, if the geometry is preoptimized in advance using a molecular mechanics force field (ensuring a reliable starting geometry) and if second derivative information – computed very cheaply during the mechanics minimization – is transferred to the ab initio optimization (Baker and Hehre 1991). However, as the system to be optimized gets larger and/or more complicated, Cartesian coordinates become less and less competitive with respect to a good set of internal coordinates. The Z-matrix is rarely used today. With the increasing popularity of GUIs and model builders it is no longer needed as a means of geometry input, and Z-matrix coordinates have been superseded as a coordinate set for optimization, first by natural internal coordinates (Fogarasi et al. 1992; Pulay et al. 1979) and then by delocalized internal coordinates (Baker et al. 1996). The latter, which are linear combinations of stretches, bends, and torsions (as opposed to the individual primitive internals in a Z-matrix), are readily generated from an underlying set of Cartesian coordinates and decouple the coordinate set (in a linearized sense) to the maximum extent possible (Baker et al. 1996). In large systems, reductions in the number of optimization cycles to reach convergence by an order of magnitude or more compared to Cartesians can be readily achieved. There is an additional computational cost is using delocalized internal coordinates however, as the transformations required to convert the Cartesian gradient into the gradient over delocalized internals, and – in particular – to convert the new geometry in internals back into Cartesians for the next optimization step (the back-transformation), are

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expensive and increasingly so with increasing system size. They scale formally as O(N3 ), where N is a rough measure of the system size, although this can readily be reduced to O(N2 ) (Baker et al. 1999). There have been attempts to reduce the scaling to O(N) (Farkas and Schlegel 1998; Paizs et al. 2000), but I think it is fair to say that these have not been entirely successful. It is principally because of the unfavorable scaling in the coordinate transformations that internal coordinates have not been adopted in molecular mechanics. The cost of the coordinate transformations and the overall complexity of the algorithm is not really a factor in ab initio methods and the majority of geometry optimizations in modern quantum chemistry packages are carried out in delocalized internals (Baker et al. 1996) or in some other set of redundant internal coordinates (Peng et al. 1996; Pulay and Fogarasi 1992).

The Modified Newton–Raphson Step We have already seen that the Newton–Raphson step is a pretty good one provided that the eigen value structure of the Hessian matrix is appropriate to the stationary point being sought. What should you do if it is not? An early effort to take remedial action under these circumstances was due to Poppinger (1975) who proposed that, when searching for a transition state, if the negative eigen value was lost then take a step uphill in the direction of the lowest positive eigenvector, while if an additional (unwanted) negative eigen value appeared, then take a step downhill in the direction of the least negative eigenvector. While this approach should lead you back into the right region of the PES, it has at least two drawbacks. Firstly, unlike the Newton–Raphson step which, when appropriate, minimizes/maximizes along all directions on the PES (all Hessian modes) simultaneously, here you are minimizing/maximizing essentially along one direction only. Secondly, successive such steps following a particular Hessian mode tend to become linearly dependent, which degrades many of the standard Hessian updating procedures. A much better step was subsequently proposed by Cerjan and Miller (1981) which involved modifying the standard Newton–Raphson step with a so-called shift parameter, œ. In terms of a diagonal Hessian representation, the Newton–Raphson step given by Eq. 4 can be written as h D †i 

Fi Ui ; di

(10)

where Ui is the i-th eigenvector, di is the corresponding eigen value, and Fi D UTi g is the component of the gradient along the local eigen mode Ui . The modified Newton–Raphson step is given by h D †i 

Fi Ui ; di  

(11)

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where œ can be regarded as a shift parameter on the Hessian eigen values. A suitable choice of œ can guide the step in a particular direction; for example if œ is close to a particular eigen value di , then the step taken will be predominantly in the direction of the corresponding eigenvector. In the early to mid-1980s, several groups were working on this topic, and several alternative transition state search algorithms were proposed differing only in the recipe used to determine the value of œ (Banerjee et al. 1985; Cerjan and Miller 1981; Simons et al. 1983). The principal optimization algorithm in PQS, both for minima and transition states, is the eigenvector following (EF) algorithm developed in 1986 (Baker 1986). It is based on the Rational Functional Optimization (RFO) approach of Simons and coworkers (Banerjee et al. 1985) which was found at the time to provide the best recipe for determining the shift parameter, œ. In this approach, the energy on the PES following a step h is written as (see Eq. 1): E .x C h/ D E.x/ C

gT h C 12 hT Hh : 1 C hT Sh

(12)

In the original formalism, the matrix S was a diagonal scaling matrix but in most practical implementations (certainly in the EF algorithm) it is taken as a unit matrix. Applying the stationary condition dE/dh D 0 to Eq. 12 gives the RFO eigen value equation (Banerjee et al. 1985): 

Hg gT 0

     h S0 h D : 1 01 1

(13)

The dimensionality of Eq. 13 is one more than the number of variables (n). As it is an eigen value equation, there are (n C 1) eigen values, œ; normally the lowest one, œ1 , is the one you need. Expanding out Eq. 13 gives .H  S/ h C g D 0;

(14)

gT h D :

(15)

In terms of a diagonal Hessian representation, and with S a unit matrix, Eq. 14 rearranges to Eq. 11, and substitution of Eq. 4 into the diagonal form of Eq. 15 gives †i 

Fi2 D ;   bi

(16)

which can be used to iteratively evaluate œ. The eigen values, œ, of Eq. 13 have the following properties (Banerjee et al. 1985):

Molecular Structure and Vibrational Spectra

15

1. The (n C 1) eigen values of Eq. 13 bracket the n eigen values of the Hessian: i  di  iC1 . 2. At convergence to a minimum both g and h are 0. The lowest eigen value, œ1 , of Eq. 13 is also zero and the other n eigen values are those of the Hessian at the minimum point, i.e., iC1 D di for all i. Thus, near a local minimum, 1 is negative and approaches zero at convergence. 3. For a saddle-point of order  (i.e., a stationary point with  negative eigen values) the zero eigen value of Eq. 13 at convergence separates the  negative and (n  ) positive Hessian eigen values. The separability of the positive and negative Hessian eigen values led Simons and coworkers to suggest that the problem be reformulated into  principal modes to be maximized and (n  ) modes to be minimized. There would be two shift parameters, œp and œn , one for modes along which the energy is to be maximized and the other for which it is minimized. The RFO eigen value equation (Eq. 11) could thus be partitioned into two smaller P-RFO equations, and each one solved separately. For a transition state, in terms of a diagonal Hessian representation, this gives rise to the two matrix equations: 

0 B B B @

d2 0 F2

d1 F 1 F1 0



0 :: : 

dn Fn

h1 1



 D p

h1 1

 ;

10 1 0 1 h2 h2 F1 B :: C :: C B :: C B C B C : C C B : C D n B : C : A @ A @ hn A Fn hn 0 1 1

(17)

(18)

œp is the highest eigen value of Eq. 17; it is always positive and approaches zero at convergence. œn is the lowest eigen value of Eq. 18; it is always negative and again approaches zero at convergence. The above discussion assumes that, for a transition state search, the energy is to be maximized along the lowest Hessian mode, d1 , and minimized along all the higher modes. However, as first pointed out by Cerjan and Miller (1981), it is possible to maximize along modes other than the lowest and in this way obtain transition states for alternative reactions and rearrangements from the same starting point. The ability of the algorithm to “walk” on the PES, maximizing along different Hessian eigen modes, is why the algorithm is known as Eigenvector Following. For maximization along the k-th mode, instead of the lowest, Eq. 17 would be replaced by

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J. Baker



dk F k Fk 0



hk 1



 D p

hk 1

 ;

(19)

and Eq. 18 would now exclude the k-th mode but include the lowest. Since what was originally the k-th mode is the mode along which the negative eigen value is required, if the search is to be successful this mode must eventually become the lowest mode. The mode that is actually followed during each optimization cycle is the mode which has the largest overlap with the mode followed on the previous cycle. In this way the same mode should hopefully be followed from cycle to cycle as it makes its way through the ranks from k-th position to first (lowest). Once the mode being followed has become the lowest, mode-following is switched off and it is always the lowest mode that is followed. Although it was developed primarily for transition states, the EF algorithm is also a very powerful minimizer. For minimization the standard RFO-step is taken (requiring a single shift parameter) using Eq. 13. Mode following for transition state searches is useful if you want to explore the PES in detail, but it is likely to be expensive and requires reliable Hessian information to be effective. There will almost certainly be more variables, and hence more Hessian modes, for a given system than there are different transition structures associated with that system; furthermore it is not always obvious which mode to follow at any given point in order to locate a transition state of a desired type. Beware of low energy modes which if followed will lead to rotational barriers for torsions in side chains rather than to the transition state for the main reaction being sought.

GDIIS As well as the EF algorithm, the PQS package has another algorithm for structure minimization: GDIIS (Csaszar and Pulay 1984). This is an extension of Pulay’s ubiquitous DIIS (Direct Inversion in the Iterative Subspace) procedure for accelerating SCF convergence (Pulay 1980, 1982), only applied instead to geometry optimization. The essence of the DIIS approach is that parameter vectors (in this case the geometry) generated in previous iterations x1 , x2 , : : : , xm are linearly combined to find the best geometry in the current iteration. We can express each parameter vector in terms of its deviation from the final solution, xf W xi D xf C ei . If the conditions †i ci ei D 0 and †i ci D 1 are satisfied, then it is also the case that †i ci xi D xf . The true error vectors, ei , are of course unknown. However, they can be approximated by ei D H1 gi (i.e., by the Newton–Raphson step). Minimization of the norm of the residuum vector †i ci ei taking into account the constraint equation †i ci D 1 leads to a system of m equations (Csaszar and Pulay 1984):

Molecular Structure and Vibrational Spectra

0

B11 B21 :: :

B B B B B @ Bm1 1

B12 B22 :: :

::: :::

Bm2 1

::: :::

B1m B2m :: : Bmm 1

17

1 1 :: :

10

CB CB CB CB CB 1 A@ 0

0 1 0 C B0C C B C C B :: C C D B :C; C B C cm A @ 0 A 1  c1 c2 :: :

1

(20)

Where Bij D hei jej i, the scalar product of the error vectors ei and ej , and œ is a Lagrange multiplier. The coefficients, ci , obtained from Eq. 20 are used to calculate an intermediate interpolated geometry: x0mC1 D †i ci xi ;

(21)

and a corresponding interpolated gradient vector g0mC1 D †i ci gi :

(22)

A new, independent geometry is generated from the interpolated geometry by acting on the interpolated gradient with the (approximate) Hessian, xmC1 D x0mC1  H1 g0mC1 :

(23)

Equations 20, 21, 22, and 23 constitute a complete cycle of the GDIIS method. Convergence can be tested for in the usual way (based on the gradient and the displacement from the previous step) or on the norm of the GDIIS residuum vector. The size of the GDIIS subspace, m, increases by one after each optimization cycle. Experience with the method suggests that m should not be allowed to grow too large; once a predefined maximum size has been reached (say 20, the current default in PQS), the earliest vectors are dropped out as new vectors are added. Intermediate quantities (geometries and error vectors) within the GDIIS subspace need to be stored, so limiting the subspace size also limits the storage requirements of the method. (This is not really an issue with GDIIS, but is more important with DIIS, where potentially large-dimension Fock and possibly density matrices are stored.) GDIIS was originally used in what is termed “static” mode, in which the original estimate for the Hessian matrix was used throughout the optimization procedure. In fact GDIIS performs very well with a simple unit matrix approximation for the Hessian. However, it certainly performs better if the Hessian matrix is updated. It is particularly important that the Hessian approximation remain positive definite, and so the BFGS update is strongly recommended with the update skipped if it is determined that this property is in danger of being lost.

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J. Baker

Geometry Optimization and Symmetry Many molecules have symmetry and it is definitely advantageous to utilize as much symmetry as possible during the calculation of the energy, the gradient and, if it is being computed, the Hessian. Various quantities, particularly the geometry and the gradient, can be symmetrized during the optimization step itself, thus removing numerical round-off error and resulting in a cleaner, and occasionally faster, optimization. Most programs can utilize at least Abelian point-group symmetry, resulting in savings in computational time by up to a factor of 8, depending on the molecular symmetry. If symmetry is utilized during a geometry optimization it obviously restricts the search on the PES to structures that retain the molecular symmetry, i.e., it excludes symmetry-breaking geometries. This can be particularly useful during a transition state search as, if it is known that the motion in the transition state breaks symmetry, then the transition state search can be replaced by a minimization within a symmetry-restricted subspace, usually a much simpler prospect. For example, consider ethane. The minimum, lowest energy structure has staggered C–H bonds and D3d symmetry. The transition state for rotation about the C–C bond has eclipsed C–H bonds and D3h symmetry. The latter can in fact be found by a simple minimization utilizing D3h point-group symmetry as the path to the D3d minimum is unavailable under the symmetry constraint. The minimum under D3h symmetry is actually a transition state once the symmetry constraint is removed. This illustrates another pitfall that occasionally catches out the novice: A structure that has been minimized under a symmetry constraint is not necessarily a true minimum and its identity should be confirmed by a full vibrational analysis. As a corollary to this, a transition state located under a symmetry constraint may not be a true transition state either; it could have one (or more) symmetry breaking modes with negative curvature.

Performance for Minimization We have had a lot of theory. How well, for example, does the standard eigenvector following (EF) algorithm perform in practice? This is illustrated in Table 1 for a selection of ten organic molecules, selected at random from the fragments library in the graphical user interface PQSMol available for use with PQS (2010). These systems were all optimized using a molecular mechanics force field (it is not important which one) and symmetrized prior to being stored. There have of course been a number of previous comparisons of the performance of various optimization algorithms. I mention two in particular, as I was involved in them myself. In 1991 Baker and Hehre compared the performance of the EF and GDIIS minimization algorithms, as implemented in the version of Spartan available at that time, on a test set of 20 organic molecules with a variety of structural motifs (Baker and Hehre 1991). The two algorithms performed similarly, but the main focus of this paper was to demonstrate that, provided reliable second derivative information was available, i.e., a good starting Hessian, then Cartesian coordinates

Molecular Structure and Vibrational Spectra

19

Table 1 Number of optimization cycles required to reach convergence for ten average-sized organic molecules under a range of starting conditionsa Molecule

Formula

Atoms Symm. Cartesians Unitb PM3c Alanine C3 H7 NO2 13 C1 28 11 Indole C8 H7 N 16 Cs 22 5 Cycloheptane C7 H14 21 C2 27 6 Bicyclo-2,2,2-octane C8 H14 22 D3 12e 10 Adamantane C10 H16 26 Td 12 3 cis-decalin C10 H18 28 C2 27 4 amp-5 C10 H14 N5 O 7 P 37 C1 193e 68 Porphine C20 H14 N4 38 D2h 21 7 Sucrose C12 H22 O11 45 C1 163 31 Cyclohexadecane C16 H32 48 S4 32 10

Delocalized internals Unitb Defaultd PM3c 20 10 8 11 5 5 23 5 6 29 8 8 11 4 4 26 4 4 85 29 35 11 6 7 67 17 21 27 6 9

a Standard EF algorithm used with a maximum allowed step size in the optimization space of 0.3 au and a BFGS Hessian update; convergence criteria were the standard PQS defaults – maximum component of the gradient vector less than 0.0003 au and either a maximum predicted displacement of 0.0003 au or an energy change from the previous cycle of less than106 Eh b Starting Hessian was a unit matrix c Starting Hessian was computed using PM3 by central-differences on the Cartesian gradient d Diagonal elements of the starting Hessian were estimated using simple empirical rules e Converged to higher energy local minimum

were just as efficient as Z-matrix coordinates for routine geometry optimization. A later paper Baker (1993) did the same thing for Cartesians versus natural internal coordinates on a larger test set of 30 mainly organic molecules. This latter test set became a kind of “standard” for checking improved optimization algorithms and several papers showing minor enhancements in performance (as would be expected with the passage of time) for these 30 molecules were published subsequently (Bakken and Helgaker 2002; Eckert et al. 1997; Lindh et al. 1995; Swart and Bickelhaupt 2006). A similar test set is also available for transition structures (Baker and Chan 1996). The conclusion in Baker (1993) was that with a reasonable starting geometry and a good initial Hessian, optimizations in Cartesian coordinates were as efficient as those employing natural internal coordinates. However, there were certainly signs that, as the system size increased, the use of natural internals became more and more advantageous even if a reliable starting Hessian was available, and it was further concluded that with no initial Hessian information, natural internals (for which read also delocalized internals which have more-or-less replaced them) were in general superior to both Cartesian and Z-matrix coordinates, and for unconstrained optimization were therefore the coordinates of first choice. One of the issues with the published studies (Baker 1993; Baker and Hehre 1991) is that the size of the systems in both test suites (all molecules had less than 30 atoms, and most much less), although typical for the time (the early 1990s), is pretty small by today’s standards. The molecules in Table 1 are about 50 % larger on the

20

J. Baker

average, four of them have more than 30 atoms, with the largest having 48. The level of theory used, B3LYP/6-31G* , is also more typical of today. They are still only average-sized molecules, but nonetheless are representative as far as performance is concerned. All the conclusions reached in the previous studies are fully supported by the results shown in Table 1. The use of a decent Hessian, even just an approximate one at the starting geometry (in this case computed by central differences using the semiempirical PM3 method (Stewart 1989)), improves the performance of minimizations in Cartesian coordinates substantially, by up to a factor of almost 7 (cis-decalin) and by an average over all 10 molecules of around 3.5. Despite this, Cartesian coordinates are not really competitive with delocalized internals under the same conditions, taking about 45 % more cycles on the average to reach convergence. The performance of Cartesians is notably worse for large, floppy molecules with no symmetry (amp-5 and sucrose). Perhaps surprisingly, the performance of delocalized internals using the standard default Hessian (estimating diagonal matrix elements only using simple empirical rules) is even better, with the number of optimization cycles being the same or less than with the PM3 starting Hessian for all systems except one (alanine). Of some surprise, at least to me, is the relatively poor performance of internal coordinates when no Hessian information (a unit matrix) is used. They are certainly better than Cartesian coordinates under the same conditions, which require about 75 % more cycles on the average to converge, but perform over three times worse than when the default diagonal Hessian is used, showing the advantages of even rudimentary information about the surface curvature. One of the molecules in Table 1, bicyclo-2,2,2-octane, took significantly less optimization cycles to converge in Cartesian coordinates with a unit Hessian (12) than with the same optimization in delocalized internal coordinates (29); the reverse is the case for all of the other systems. This apparent anomaly is due to the fact that the Cartesian unit Hessian optimization converged to a higher energy local minimum than the other optimizations. This was not the only occasion when this occurred: higher energy local minima were also found for amp-5 and to a lesser extent for sucrose and alanine (although in the latter case the difference was only 40 H and may well be simply due to premature convergence, a fairly common occurrence for unit Hessian Cartesian optimizations (Baker et al. 1996)). These results support another assertion made in Baker (1993), namely, that Cartesian optimizations tend to converge to the nearest local minimum whereas optimizations in natural (or delocalized) internals tend to converge more globally. This global tendency, and that is really all it is, with the internal coordinate optimizations is principally due to the fact that relatively small displacements in internal coordinates can lead to large changes when the geometry is converted back to Cartesian coordinates. For example a small change in a central bond angle can lead to a large Cartesian displacement at the end of a lengthy attached side chain; such large displacements enable internal coordinate optimizations to “jump over” local minima. (This phenomenon was first observed in the days of the Z-matrix.)

Molecular Structure and Vibrational Spectra

21

This is an appropriate point to emphasize that all the optimization algorithms discussed so far in this chapter are local methods, aimed at locating a single stationary point. Whether this is the lowest energy structure on the entire PES, i.e., the global minimum, is quite another matter. The number of (mainly conformational) minima a molecule has increases dramatically with system size and the problem of locating the one with the lowest energy quite quickly becomes almost insurmountable. Additionally, of course, unless the global minimum is noticeably lower in energy than other low-lying local minima, there will be a Boltzmann distribution at room temperature with several different minima having appreciable population; under these circumstances to talk about a well-defined structure does not make a lot of sense. Problems in locating global minima have only recently become an issue in ab initio calculations as the size of the systems that can be handled using these techniques has increased significantly in recent years. A number of global search algorithms are available, although none of them are bulletproof (i.e., absolutely guaranteed to locate the global minimum in a finite time). The problem of global minimization is outside the scope of this chapter and the interested reader is referred to the existing literature for more detail (Pardalos et al. 1995).

Minimization: An Example Figure 2 shows the optimization output at each cycle in the standard default EF minimization of indole. There is more printout than normal for the first and last cycles as I have used a higher print flag in order to show more detail. The output is of course specific to PQS but would be similar – at least in terms of the quantities printed – in other programs. At the beginning of each optimization cycle the current geometry is printed in Cartesian coordinates together with the point group (Cs ) and the number of degrees of freedom (29). Indole is planar and is oriented in the XY plane; consequently all Zcoordinates are zero. Next comes the SCF energy at that geometry and the Cartesian forces (the gradient – not normally printed). The latter should fully reflect the symmetry of the system, which it does in this case, and so all Z-gradient components are zero. This is followed by a symmetry analysis of the set of delocalized internal coordinates generated for indole which results in the elimination of all primitive torsions, all of which are fixed by symmetry. This analysis is typically done once only as thereafter the delocalized internal coordinates formed and used on the first cycle are retained unaltered throughout the optimization. One does not have to do this; several algorithms (e.g., Schlegel and coworkers (Peng et al. 1996)) effectively regenerate a set of active delocalized internals on each optimization cycle although in most cases this is not necessary and does not improve the overall performance. There is then a statement as to how many Hessian modes (eigenvectors) are used during the construction of the optimization step – this does not have to be all of them, but is usually most, and cannot be more than the total number of

22

Fig. 2 (continued)

J. Baker

Molecular Structure and Vibrational Spectra

Fig. 2 (continued)

23

24

Fig. 2 (continued)

J. Baker

Molecular Structure and Vibrational Spectra

Fig. 2 (continued)

25

26

Fig. 2 (continued)

J. Baker

Molecular Structure and Vibrational Spectra

Fig. 2 (continued)

27

28

J. Baker

Fig. 2 PQS output for the minimization of indole

degrees of freedom – followed by a listing of the corresponding Hessian eigen values (the curvature along each of the eigenvector directions). In this particular optimization, a diagonal Hessian in the underlying space of primitive internals (individual stretches, bends and torsions) is estimated using empirical rules, and is then transformed into the space of selected delocalized internal coordinates, U, via H D UT Hprim U (Baker et al. 1996). The simple RFO step is then computed and taken, appropriate for minimization, resulting in a single shift parameter, œ (see Eq. 16), with a value of  0.01172058. The resulting step length of 0.174879 is below the default maximum of 0.3 and so no scaling of the step is required. The parameter values, and their gradients and displacements are then printed out. As with the Cartesian forces, these are not normally printed, and are not particularly informative given that each coordinate is potentially a linear combination of all underlying primitive internals (minus the torsions in the case of indole). A persistently large gradient for a particular coordinate would suggest some problems with the overall coordinate set, suggesting that either the delocalized internals, the underlying primitives or likely both, should be regenerated. The convergence criteria – current value, tolerance, and whether or not convergence is satisfied – are then printed, together with details of the back-transformation

Molecular Structure and Vibrational Spectra

29

(conversion of the current geometry in internal coordinates back into Cartesians, again not normally printed). This constitutes a complete optimization cycle. On subsequent optimization cycles, assuming that the system is converging, the energy, the gradient components (both Cartesian and internal) and the step size should all decrease, with the latter two tending to zero. This is exactly what occurs for indole (see Fig. 2) which converges smoothly to the minimum energy structure. Another quantity that should tend to zero is the shift parameter, œ, which again clearly does so; its magnitude on the last cycle is 0.00000011. Note that, as previously stated (see the comments under Eq. 16) œ is always negative and so approaches zero from below. The closer œ is to zero, the closer is the step taken to the simple Newton–Raphson step. After the initial estimate, the Hessian matrix is updated at the beginning of each new optimization step. This is done using the BFGS update, as appropriate for a minimization. Indole is a planar, rigid molecule and so, not surprisingly, convergence is achieved quickly taking just five optimization cycles. At final convergence the displacement criteria is not fully satisfied; this is not an error as the default convergence criteria in PQS allow for just one of the displacement and energy change criteria to be satisfied (see the footnotes under Table 1); this is to prevent wasteful cycles near the end of the optimization for excessively floppy molecules for which relatively large displacements result in only small changes in energy. If it is desired that the displacement criterion be rigorously satisfied, then the energy criterion should be reduced so that it is almost impossible to satisfy (for example to 108 Eh); Convergence can then only be achieved by satisfying the displacement criterion.

Transition State Searches: An Example Figure 3 shows the optimization output at each cycle (truncated for the intermediate cycles) in the standard default EF transition state search for the reaction HCN$HNC. This is a very simple reaction but is still instructive. Both reactant and product are linear. Reaction can be considered to occur essentially by motion of the hydrogen atom in a semicircle centered about the midpoint of the C–N bond, with the †HCN angle going from 180ı in HCN to 0ı in HNC (CNH). Consider the starting geometry. A first approximation to the transition state (TS) geometry might be to position the hydrogen atom directly vertically above the midpoint of the C–N bond. (The H–C distance of course has to be decided.) This corresponds to †HCN a bit less than 90ı . However, knowing the relative energetics (HNC is less stable than HCN, the reaction as written is endothermic), the Hammond postulate (Hammond 1955) suggests that the TS is likely to resemble HNC more than HCN, in which case †HCN will very likely be noticeably less than 90ı , i.e., relative to HCN we have a late transition state. Now this is only a simple consideration, but the use of “chemical intuition” in this way can potentially save several optimization cycles. In this particular example, it does not save much because the system as a whole only has three atoms and the transition state is

30

J. Baker

fairly easy to locate, but starting with †HCN D 85ı instead of 70ı costs an extra optimization cycle, and starting with †HCN D 130ı takes three more cycles. (The other starting parameters were rC  N D 1.21 Å and rH  C D 1.5 Å.) As has already been mentioned, a good initial Hessian is almost mandatory for a TS search and a full Hessian at the starting geometry was computed at the semiempirical PM3 level (Stewart 1989) for the HCN$HNC reaction. The output for a transition state search is of course similar to that for a minimization. Quantities such as the Cartesian geometry, the SCF energy, the forces, parameter values, and displacements are common to both TS searches and minimizations and are printed out in the same way. The first difference occurs in the treatment of the starting Hessian: for the indole minimization (Fig. 2) a default Hessian, diagonal in the underlying space of primitive internals, is used whereas the TS search started with a full, albeit approximate, Cartesian Hessian which needs to be transformed into internal coordinates. This transformation is done once only, on the first optimization cycle, as thereafter the Hessian will be updated. Note the comment printed on cycle 1 that the Hessian transformation does not include the derivative of the B-matrix. The B-matrix (Wilson et al. 1955) formally relates a displacement in internal coordinates (q) to one in Cartesian coordinates (X),

Fig. 3 (continued)

Molecular Structure and Vibrational Spectra

Fig. 3 (continued)

31

32

Fig. 3 (continued)

J. Baker

Molecular Structure and Vibrational Spectra

Fig. 3 (continued)

33

34

Fig. 3 (continued)

J. Baker

Molecular Structure and Vibrational Spectra

35

Fig. 3 PQS output for the HCN$HNC Transition State Search

q D BX , and it turns out that a full transformation of the Hessian to internal coordinates requires the derivative of the B-matrix; the precise equation is Pulay (1977):   d B  inv T B Hint D Binv Hcart  gint ; dX

(24)

1  Binv D BBT B;

(25)

where

and a superscript T represents a matrix transpose. If you require an exact transformation of all quantities from Cartesian to internal coordinates, then the dB/dX term must be included; however it is usually advantageous to leave it out as its inclusion often changes the Hessian eigen value structure. In general, Cartesian Hessian matrices with a known eigen value structure will retain that structure if the dB/dX term is omitted; this is far from guaranteed when it is included. When starting off a geometry optimization (either a minimization or a TS search) using an optimized geometry and starting Hessian obtained from a previous calculation at a lower level of theory – a common occurrence – then the dB/dX term should almost certainly be omitted. Note that for a Hessian transformation at a stationary point, the gradient is zero in any case and so the dB/dX term will also be zero. For the HCN$HNC TS search the starting

36

J. Baker

Hessian has the correct eigen value structure, i.e., one negative eigen value, so our initial geometry was at least reasonable. The other major difference from the indole minimization is with the shift parameter(s). A TS search uses the P-RFO step, as opposed to the simple RFO step used for minimization (see Eqs. 17, 18, and 19), and so there are two shift parameters instead of just one. As explained previously, one shift parameter maximizes along the TS mode (the eigenvector with the negative eigen value), while the other minimizes along all the other Hessian modes. The first shift parameter is positive while the other is negative. On the first cycle the computed step length is initially greater than the default maximum of 0.3 and so the step is scaled down (by 0.888690). As with a minimization, on subsequent optimization cycles the gradient components and the step size should both decrease, tending to zero if the structure is converging. The two shift parameters, œ, also both tend to zero, one from above and the other from below. On the last cycle they are both zero to all eight decimal places printed out. The energy may increase or decrease, depending on whether the increase in energy from maximization along the TS mode is greater or smaller than the decrease in energy from minimization along all the other modes; the change in energy however should of course again tend to zero. (Overall a decrease in energy is the more likely, especially as the system gets larger, as the number of modes which are minimized becomes far greater than the single mode along which the energy is maximized.) The Hessian update, at the beginning of each new optimization step, is done using the Bofill formula (Bofill 1994), which is the default for a TS search. This is a linear combination of the Powell (1971) and Murtagh and Sargent (1970) updates, and the two mixing coefficients are printed out. They vary from cycle to cycle, as can be seen (Fig. 3). The final TS geometry is well converged (in six cycles), with the three convergence criteria all well below their respective tolerances.

Using Optimized Potential Scans in Transition State Searches We turn now to a somewhat more complicated example, namely, a search for the transition state for the parent Diels–Alder reaction between cis-1,3-dibutadiene and ethylene to give cyclohexene. Here we use an optimized potential scan to obtain an initial approximation to the TS geometry. The starting geometry for the potential scan is shown in Fig. 4. This was obtained by building cyclohexene using PQSMol, the graphical user interface for PQS (2010), and then stretching the appropriate C–C bonds (C10  C13 and C7  C16 as shown in Fig. 4) to separate off an ethylene fragment. The overall system symmetry (Cs ) was maintained using the symmetrization tools available in PQSMol. The †HCH angles in the ethylene fragment are likely to be too acute, but this will be rectified during the potential scan. The C10  C13 and C7  C16 distances in the starting geometry (see Fig. 4; the two distances are equivalent by symmetry) are 1.720 Å. What we are going to do

Molecular Structure and Vibrational Spectra

37

H4

H9 C7 H8

H11 F1 H3

C5

C 16

C 10

H 14 C 13

H 15

H6

H 12 H2

Fig. 4 Starting structure for potential scan for Diels–Alder TS search

is scan this distance over a range of (increasing) values, at each scanned value optimizing all the remaining degrees of freedom. If the scanned variable(s) is a good approximation to the reaction coordinate, and if the value of the variable at the transition state geometry lies within the scanned range, then what we should see is a maximum in the energy somewhere along the scan. The (optimized) geometry at this energy maximum should be a very good approximation to the transition state. An optimized potential scan is effectively a series of constrained geometry optimizations, with the constraint being the scanned variable. PQS has powerful algorithms for constrained optimization using the method of Lagrange multipliers (Baker 1992, 1997; Baker and Bergeron 1993). Because of the symmetry of the system, the scanned variable is not a single distance but rather a linear combination (with equal weights) of the C10  C13 and C7  C16 distances. The requested linear combination is normalized by the program and so the actual parameter scanned is p1 Œ.rC10  C13 / C .rC7  C16 /). The potential energy scan module in PQS 2 is capable of scanning any linear combination of primitive internal coordinates selected by the user. The optimized energies at C10  C13 and C7  C16 distances from 1.720 Å to about 2.65 Å are shown in Table 2. Note that, although the final level of theory used in this chapter for all examples given is B3LYP/6-31G* , the scan was actually carried out using the smaller 3-21G (Binkley et al. 1980) basis set. This is simply

38 Table 2 Optimized B3LYP/3-21G energies (Eh ) for the Diels–Alder Potential Scan at combined scan distances (in bohr) from 6.50 to 10.00 in steps of 0.25

J. Baker Current value: 6.5000 Current value: 6.7500 Current value: 7.0000 Current value: 7.2500 Current value: 7.5000 Current value: 7.7500 Current value: 8.0000 Current value: 8.2500 Current value: 8.5000 Current value: 8.7500 Current value: 9.0000 Current value: 9.2500 Current value: 9.5000 Current value: 9.7500 Current value: 10.0000

Energy is 233.352945649 Energy is 233.341556134 Energy is 233.328803153 Energy is 233.315693812 Energy is 233.303165136 Energy is 233.292127679 Energy is 233.283459373 Energy is 233.277798713 Energy is 233.275148592 Energy is 233.274846534 Energy is 233.276044729 Energy is 233.278051315 Energy is 233.280392073 Energy is 233.282767273 Energy is 233.285004883

to save computer time. I expect the B3LYP/3-21G energy surface to be very similar to the B3LYP/6-31G* one in this system, and since we are using the optimized scan to obtain a good, but still approximate, starting geometry for the proper TS search, there is no point using the larger basis set for the scan. As can be seen in Table 2, there is a clear energy maximum at a parameter value somewhere around 8.75. The parameter printed out is the sum of the two scanned C–C distances (unnormalized) given in atomic units (bohr); the maximum is equivalent to a C–C distance of 2.315 Å. During the scan, the first energy (from the starting geometry) took the most optimization cycles (12) to converge; thereafter each subsequent point typically converged in five to seven steps. Note that if the scanned range is insufficient to locate the maximum, it can simply be extended. For example, if the initial scan only went as far as 8.00 bohr (for the combined C–C distances), then it can be repeated for distances greater than this starting from the last optimized geometry found. That the scan needs to go to greater C–C distances is clear from the energy, which is increasing over the initial scan range; if it had been decreasing then one would need to go to shorter C–C distances. The full B3LYP/6-31G* TS search commenced using the optimized geometry at the scan maximum with a full Hessian matrix computed at this geometry using the 3-21G basis (the same basis as used in the scan). Not surprisingly this starting Hessian had the desired one negative eigen value, and the search converged in just four optimization cycles, indicating just how good the geometry resulting from the scan was. A full vibrational analysis at the B3LYP/6-31G* level on the converged transition state structure indicated that it was indeed a genuine TS. The C10  C13 and C7  C16 distances in the final TS were 2.272 Å, very close to those in the structure at the scan maximum.

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39

Comparison of Experimental and Theoretical Geometries Before turning to the second part of this chapter, the computational simulation of vibrational spectra, I want to make a few comments on the accuracy of theoretically computed geometries. Direct comparison with reliable experimental data shows that theoretical geometries are now really very accurate indeed, particularly for “normal” organic molecules. The level of theory used here, B3LYP/6-31G* , is typically accurate for, say, bond angles to within a few degrees and bond lengths to within 0.02 Å at worst, and if any experimental bond distance differs by much more than this from the theoretical value, it is more likely in my opinion for experiment to be in error rather than theory. One needs to bear in mind that none of the experimental methods commonly used in molecular structure determination, X-ray crystallography, microwave spectroscopy, nuclear magnetic resonance (NMR) etc., measure a bond distance directly; rather some kind of spectrum (or pattern) is obtained which has to be interpreted, and bond distances are extracted indirectly from this. It was fairly typical in the early 1970s (and even into the 1980s) for X-ray diffraction structures to be published, as a set of Cartesian coordinates, which resulted in quite ridiculous bond distances involving hydrogen. This was due principally of course to the difficulty in locating hydrogen atoms because of the much lower electron density surrounding hydrogen nuclei compared to heavier elements, and the fact that this density could easily be displaced in bonds involving more electronegative atoms. From the mid-1970s onward it became increasingly more common for theoretically computed bond distances involving hydrogen to be used in the fitting of X-ray data; this also resulted in much better agreement with heavy atom bond distances as well (Schäfer 1983). Another factor to be considered is that theory and experiment do not quite measure the same thing. Within the Born–Oppenheimer approximation (Born and Oppenheimer 1927; Wikipedia 2010), theory ideally determines the geometry at the bottom of the well, at the energy minimum, whereas this geometry is inaccessible to experiment due to the existence of zero-point vibrational energy. Such differences, however, are usually minor. Table 3 provides a comparison between optimized B3LYP/6-31G* and experimental geometrical parameters for a number of small, organic molecules. These results were taken from two papers published in the mid-1990s; a DFT study of some organic reactions (Baker et al. 1985) and a systematic DFT study of fluorination in methane, ethane, and ethylene (Muir and Baker 1996). The focus was mainly on the energetics, but many computed and experimental geometrical parameters were provided. The theoretical approaches did not include B3LYP, using instead a related functional (Becke’s original ACM (Becke 1993; see also Hertwig and Koch 1997), better known today as B3PW91), but I have reoptimized all the geometries at the B3LYP/6-31G* level for compatibility with this chapter. As can be seen from Table 3, the agreement between the computed and experimental geometrical parameters is excellent. The mean average deviation between the B3LYP/6-31G* and experimental bond lengths is just 0.006 Å, with

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Table 3 Optimized B3LYP/6-31G* and experimental geometrical parameters for a number of small organic molecules Parameter B3LYP Experiment Vinyl alcohol (CH2 D CHOH) rC  O 1.363 1.369 rC D C 1.334 1.335 rO  H 0.972 0.962 rC  H 1.086 1.080 †CCO 127.2 126.0 †COH 108.7 108.5 Acetaldehyde (CH2 CHO) rC D O 1.211 1.210 rC  C 1.508 1.515 rC  H 1.114 1.128 †CCO 124.7 124.1 trans-butadiene (CH2 D CHCH D CH2 ) rC D C 1.341 1.349 rC  C 1.458 1.467 †C D CH 119.4 120.9 †CCC 124.3 124.4 Ethylene (CH2 D CH2 ) rC D C 1.331 1.339 rC  H 1.087 1.087 †CCH 121.9 121.3 Cyclohexene (C6 H10 ) rC D C 1.337 1.334 rC  C 1.510 1.50 rC  C 1.537 1.52 CC 1.535 1.54 †CC D C 123.5 123.4 †CCC 112.0 112.0 †CCC 110.9 110.9 Tetrazine (C2 N4 H2 ) rN  N 1.326 1.321 rC  N 1.339 1.334 †NCH 126.6 127.4 Hydrogen cyanide (HCN) rC  N 1.157 1.135 rC  H 1.070 1.066 Nitrogen (N2 ) rN  N 1.105 1.098 rC D C 1.340 1.342 rC  C 1.519 1.517 rC  C 1.573 1.566 †CCC 86.5 85.5 †C D CC 94.4 94.2

Parameter B3LYP Experiment trans-1,1,2,2-tetrafluoroethane (CHF2 CHF2 ) rC  C 1.524 1.518 rC  F 1.360 1.350 rC  H 1.095 1.098 †CCF 108.8 108.2 †CCH 111.8 110.3 †FCF 108.8 107.3 Carbonyl fluoride (CF2 O) rC D O 1.180 1.17 rC  F 1.322 1.32 †FCF 107.8 112.5 Hydrogen fluoride (HF) rH  F 0.934 0.917 Hydrogen (H2 ) rH  H 0.743 0.741 Water (H2 O) rO  H 0.969 0.958 †HOH 103.6 104.5 Formaldehyde (CH2 O) rC D O 1.206 1.208 rC  H 1.110 1.116 †HCH 115.2 116.5 Acetylene (C2 H2 ) rC  C 1.205 1.203 rC  H 1.067 1.060 Ethane (CH3 CH3 ) rC  C 1.531 1.535 rC  H 1.096 1.094 †CCH 111.4 111.2 Fluoromethane (CH3 F) rC  F 1.383 1.382 rC  H 1.096 1.095 †HCH 109.3 110.4 Difluoromethane (CH2 F2 ) rC  F 1.361 1.357 rC  H 1.096 1.093 †HCH 112.2 113.7 †FCF 109.2 108.3 rC  F 1.342 1.333 rC  H 1.093 1.098 †FCF 108.6 108.8 Tetrafluoromethane (CF4 ) rC  F 1.329 1.323

Molecular Structure and Vibrational Spectra

41

a maximum error (for the C–N triple bond in HCN) of 0.022 Å. The mean average deviation for bond angles is only 0.8ı . Such results are typical. Included in Table 3 is the series of fluorine-substituted methanes: mono-, di-, tri-, and tetra-fluoromethane. These compounds exhibit a well-known substituent effect (the geminal effect) of decreasing C–F bond length (and increasing bond strength) with increasing fluorine substitution on the same carbon atom. This effect of fluorine (not seen with any of the other halogens) has been known for a long time (although there are still arguments about the explanation for it) and is very accurately reproduced by the theoretical calculations.

Geometry Optimization of Molecular Clusters The situation at the time of writing is that molecular structures for stable or metastable minima are readily located theoretically using powerful optimization algorithms usually carried out in natural, delocalized or redundant internal coordinates. The resulting structural parameters (computed using B3LYP or better with a decent basis set) are typically in excellent agreement with experimental results where these are available. The location of transition states can be more difficult, but perseverance usually brings its reward. Theoretical calculations are essentially the only means for providing structural data on transition states as they are extremely difficult to study experimentally. One area where standard natural or delocalized internal coordinates are not the best choice is for optimizing the geometry of molecular clusters. For larger clusters the problem of global versus local minimization becomes an issue, but even for relatively small clusters (containing half-a-dozen or so molecules) deriving a set of coordinates suitable for an efficient optimization from stretches, bends, and torsions alone usually results in poor optimization performance. One reason for this is that there are really two different types of forces involved in determining the structure of a molecular cluster: the normal, fairly strong intramolecular forces that determine the geometry of each individual molecule, and the weak intermolecular forces that determine the arrangement of each molecule relative to the others. The latter are best described in terms of scaled inverse distance coordinates (not standard distances) and an efficient algorithm which uses normal primitive internal coordinates to describe the geometry of each molecule in the cluster, and scaled inverse distances (with a cutoff) to describe the relative position of the cluster molecules with respect to each other (Baker and Pulay 2000) is available in PQS. The resulting set of coordinates, which are linear combinations of standard primitive internals and scaled inverse distances, are known as cluster coordinates. Examples of geometry optimization of some small molecular clusters are shown in Table 4. Optimization performance using cluster coordinates (1/R with a 5 Å cutoff) is compared with Cartesian coordinates. A modest level of theory was used (Hartree–Fock with the 3-21G basis set) and all optimizations were started with a unit Hessian matrix. In most cases the starting geometry was poor, and very far

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Table 4 Comparison of the performance of cluster coordinates and Cartesian coordinates for the optimization of some small molecular clusters Clusters 4  benzene 6  DMSO formamide C 6  water 6  methanol 5  pyridine 3  THF 8  water Average cycles

Cluster coordinates Cycles Energy 52 917.687877 124 3,292.233057 75 621.674713 39 686.489506 108 1,226.595255 65 689.117483 69 604.913202 76

Cartesian coordinates Cycles Energy 273 917.680357 660 3,292.192089 222 621.687307 301 686.492821 479 1,226.587376 348 689.113449 331 604.911548 373

Fig. 5 Starting and final structures for optimization of the (H2 O)8 cluster from Table 4: (a) starting geometry, (b) final geometry cluster coords, (c) final geometry Cartesians

from the final converged structure. No comparisons with natural or delocalized internal coordinates are included in Table 4; with such poor starting geometries this is usually a waste of time as the large changes that occur in the relative positions of molecules in the cluster often result in individual bond angles, as defined, exceeding 180ı causing problems with the various transformations, particularly the Cartesian backtransformation. This is yet another reason why standard internal coordinates are inappropriate for optimizing the geometry of molecular clusters. This is not a problem with inverse distance (or simple distance) coordinates as these are defined essentially from zero to infinity. As shown in Table 4, cluster coordinates typically converge much faster than Cartesians (on average for the seven clusters shown by a factor of almost 5) and usually – but not always – to lower energy structures. (Five out of the seven converged to lower energies than the corresponding Cartesian optimizations, two to higher.) The starting and final geometries are shown for the eight water molecule cluster in Fig. 5.

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43

Geometry Optimization in the Presence of External Forces I want to end this section by briefly mentioning a simple but relatively new technique which has proven to be very useful in deriving potentially new molecular structures. It was originally developed to study the geometrical changes inferred in atomic force microscopy and similar experiments (known collectively as mechano-chemistry) (Beyer and Clausen-Schaumann 2005), but can be applied as a general optimization tool which also has potential for the location of transition states. This is geometry optimization in the presence of external forces (Wolinski and Baker 2009) known as enforced geometry optimization (EGO). The force is applied along a straight line joining any two atoms in the molecule under study, either to push the two atoms together or to pull them apart. Whatever force is applied to one atom of the pair, an equal and opposite force is applied to the other. This ensures that there is no tendency for the molecule to translate or rotate; in essence it remains in place under simulated pressure. Forces can be applied between as many pairs of atoms as desired, although calculations to date have typically involved only a single pair of atoms. If you are applying an external force to, say, push two atoms together in a stable molecule there will be a tendency for a bond to form between the two atoms. If the applied force is insufficient to form a bond (or if a bond won’t form anyway), there will be some minor geometrical changes as the system endeavors to compensate for the excess force. If the applied force is sufficient, then what you should see is the energy rise as the two atoms are pushed together, then fall as the bond forms and additional molecular rearrangement occurs. The final “optimized” geometry will still involve a structural arrangement under considerable strain, but if this structure is allowed to relax (by reoptimizing with the additional force removed) it could well converge to a new, stable geometry, different from the initial structure. If it does so, then the maximum energy structure found as the two atoms were pushed together is often a reasonable estimate for the transition state joining the original and the new structure. The method as described above has been used to find seven previously unknown geometrical isomers of C10 N2 H8 , starting from the ground-state geometry of cisazobenzene, at the B3LYP/6-31G* level by pushing together various pairs of atoms, one from each of the two benzene rings (Wolinski and Baker 2009). All of the new structures were classified by vibrational analysis, and all were found to be minimal. Transition states between each of the new structures and cis-azobenzene were also located and characterized. Additional calculations were carried out at MP2/6-311G** to ensure that the energetics remained stable. The technique overall shows considerable promise for the discovery of new chemistry and new chemical structures.

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Molecular Vibrations A molecular system containing N atoms nominally has 3 N degrees of freedom. In Cartesian coordinates, one can consider each degree of freedom as separate X, Y, or Z displacements of each atom about its equilibrium position. This atomic motion can be induced for example by heat or by interaction with light or other electromagnetic radiation. Because the atoms in a molecule are connected by chemical bonds, the various atomic motions are coupled. In a classical treatment, it turns out that three of these motions involve overall translation of the system as a whole (along the X, Y, and Z directions of a standard Cartesian axis system), and three of them involve overall rotation of the system as a whole (about the same axes). These are known as rigid-body motions. The Cartesian axes pass through (they have their origin at) the center of mass of the system and the X, Y, and Z axes coincide with the principal axes of inertia of the undistorted molecule. Under these rigid-body motions the positions of the atoms in the molecule relative to each other remain unchanged, and so the potential energy of the system in a vacuum is also unchanged. The remaining 3 N  6 coupled motions, which do affect the relative atomic positions and hence the potential energy, are classified as molecular vibrations. (Note that a linear molecule has 3 N  5 vibrations, not 3 N  6, as one of the rotations – that about its own axis – cannot be observed, and is effectively ignored.) To a reasonable approximation, at least for small displacements, the vibrations in a polyatomic molecule can be described as a kind of simple harmonic motion. This is essentially equivalent to considering a chemical bond between two atoms as a weightless spring that obeys Hooke’s law (i.e., the force is proportional to the displacement). The simplest vibration is that in a diatomic molecule, and we can refer back to the potential energy curve for He2C 2 (Fig. 1) as an example. At the minimum, near the bottom of the well, the potential energy curve is indeed parabolic to a very good approximation. The force/displacement equation for a diatomic molecule obeying Hooke’s law is F D kq;

(26)

where F is the applied force, k is the so-called force constant, and q is the displacement from equilibrium. The negative sign indicates that the force exerted by the spring is in a direction opposite to that of the displacement. The potential energy stored in the spring (the integral of force over distance) is ED

1 2 kq : 2

(27)

By Newton’s second law of motion, the force can also be given by mass times acceleration

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F Dm

d 2q : dt 2

(28)

The two forces in Eqs. 26 and 28 are one and the same, and so the differential equation relating displacement, q, with time, t, for the harmonic oscillator is m

d 2q C kq D 0: dt 2

(29)

The solution to Eq. 29 is q.t / D A cos .2t / I

1 D 2

r

k ; m

(30)

where A is the amplitude of the vibrational coordinate q and  is the fundamental frequency of the vibration. The “mass,” m, is actually the so-called reduced mass, , which for a diatomic molecule A–B, with atomic masses mA and mB , respectively, is given by 1 1 1 D C : mA mB

(31)

Using the reduced mass ensures that the center of mass of A–B remains unchanged during the vibrational motion. From Eq. 27 the force constant, k, is given by kD

d 2E : dq 2

(32)

The force constant in a one-dimensional system (a system with a single degree of vibrational freedom) is thus directly related to the curvature of the potential energy surface, i.e., the shape of the well. From the force constant, one can directly determine the fundamental frequency (via Eq. 30). In a polyatomic molecule the single d2 E/dq2 term is replaced by the Hessian matrix, H (often called the force constant matrix). In Cartesian coordinates this is the same Hessian matrix introduced in Eq. 4 when discussing the Newton–Raphson step during a geometry optimization, and a classical treatment of molecular vibrations and the Newton–Raphson step both assume the same thing about the potential energy surface, namely that it is quadratic. The only major difference is that a vibrational analysis involves the atomic masses, whereas the potential energy surface is independent of mass. The fundamental vibrational frequencies for a polyatomic molecule can be extracted from the mass-weighted Hessian matrix, and the normal modes (the coupled motion of the atoms during the vibration) can be obtained by a suitable treatment of the Hessian eigenvectors. The precise steps involved are as follows:

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1. Mass-weight the Cartesian Hessian matrix. Hij HMW Dp : ij mi mj

(33)

2. Project out the translations and rotations from the mass-weighted Hessian. As discussed briefly above, the translational and rotational degrees of freedom can be separated out from the vibrations for a rigid body. This requires shifting the origin of the coordinate system to the center of mass, determining the principal moments and the rotation generators, setting up orthogonal coordinate vectors for translation and rotation about the principal axes of inertia, setting up a projection matrix P from these vectors and using it to “remove” the translations and rotations from the Hessian matrix, leaving only the vibrational modes. If the orthogonal set of vectors representing translation and rotation (there are six of them, each of length 3 N, where N is the number of atoms) are given by R, then the projection matrix P is formed by X X Pij .i ¤ j / D  Rik Rjk I Pii D 1  Rik Rik ; (34) k

k

and the projected Hessian is given by HP D PHMW PT :

(35)

3. Diagonalize the projected, mass-weighted Hessian matrix. The effect of the projection leaves the eigenvectors of the Hessian corresponding to translations and rotations with exactly zero eigen values (or at least zero within numerical round-off error), allowing the 3 N  6 nonzero vibrations to be separated out. These are converted to cm1 prior to print out, and the corresponding eigenvectors are mass-weighted and normalized to become the normal modes. The mass-weighting uses isotopically averaged atomic masses based on the average isotopic abundance for each atom. At first sight it might be surprising that the Hessian matrix, which after all in ab initio molecular orbital theory is inherently quantum mechanical, is amenable to a purely classical treatment. This is because the Born–Oppenheimer approximation (Born and Oppenheimer 1927; Wikipedia 2010) allows for a pretty good separation of the electronic and nuclear motion, allowing the latter to be treated classically. A quantum mechanical description of the simple harmonic oscillator leads to quantized energy levels given by   1 I En D h n C 2

1 D 2

r

k ; m

(36)

where the fundamental vibrational frequency, , is exactly the same as in a classical description (Eq. 30). The quantum number, n, takes integer values 0, 1, 2, : : : and so

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even in the vibrational ground-state (n D 0) there is a residual energy, 12 h, known as the zero-point energy. This is a fundamental quantum mechanical property of all oscillators at the absolute zero of temperature (Einstein and Stern 1913). A molecular vibration is excited when the system absorbs a quantum of energy, h, corresponding to the vibrational frequency, , of one of its normal modes. This is commonly accomplished under standard conditions by interaction with infrared (IR) radiation. An IR spectrum can be recorded by passing a beam of infrared light covering a range of frequencies through the sample. This can be done by using a monochromatic beam which changes in wavelength over time, or by using a Fourier Transform instrument which effectively measures all wavelengths at once. From this a transmission (or absorbance) spectrum can be produced showing at which frequencies/wavelengths the sample absorbs. Not all normal modes are IR active, i.e., result in the absorption of energy at the normal mode frequency. In order for a normal mode to be IR active, its motion must result in a change in the dipole moment of the vibrating molecule. For example, the totally symmetric C–H stretch in methane (in which all four C–H bonds are vibrating in the same way simultaneously) does not change the molecular dipole moment (which is zero in any case), and so this mode is IR inactive. Change one of the hydrogen atoms for a fluorine, however, and it is a different picture. In general, molecules with a permanent dipole moment in their ground-state geometries give rise to more intense signals in the IR spectrum (i.e., greater absorbance) than molecules in which a dipole moment is induced by the vibration itself. Vibrational excitation can occur, and usually does, in conjunction with rotational excitations, resulting in vibration-rotation spectra. For straightforward IR vibrational spectra, the simultaneous rotational excitations result in broardening of the vibrational bands. Bands are additionally broadened if two (or more) normal modes have frequencies close to one another, as well as by instrument resolution and other experimental artifacts. Additional bands, other than those caused by vibrational fundamentals, can arise in the spectrum by the absorption of two or more quanta of energy simultaneously, giving rise to combination and overtone bands, and by further excitations from already vibrationally excited states which give rise to what are called hot bands. It is worth noting at this point that although our analysis of vibrational motion derives from consideration of the simple harmonic oscillator, many common features observed in vibrational spectra are present precisely because the potential energy surface is not harmonic. For example, transitions between vibrational energy levels in the quantum harmonic oscillator formally allow the quantum number, n, to change only by 1 .n D ˙1/. As the vibrational energy levels are equally spaced (see Eq. 36), this means that light of only one frequency can be absorbed, that of the vibrational fundamental for each normal mode, and there are no overtone bands. The n D ˙1 selection rule does not apply to an anharmonic oscillator and the observation of overtones is only possible because vibrations are anharmonic. A further consequence of anharmoniticity is that the spacing between the vibrational energy levels decreases with increasing quantum number (eventually reaching a

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continuum allowing for vibrational dissociation) which results in the frequency of the first overtone being slightly less than twice the frequency of the vibrational fundamental. IR spectroscopy is a very important method in molecular structure determination. Even though the normal modes depend on the whole molecule, certain structural motifs have very similar motions regardless of how many atoms the molecule contains. For example, virtually all molecules containing a carbonyl group give rise to an intense (because of the bond polarity) signal in the IR spectrum around 1,800 cm1 due to excitation of the C D O stretch, thus allowing for the identification of this group. Theory can not only determine the fundamental frequencies and normal modes of a molecule, but by computing the dipole moment derivatives, it can reliably estimate the resulting band intensities as well, enabling the IR spectrum for the vibrational fundamentals to be predicted a priori. Another technique used in vibrational spectroscopy is Raman scattering (Raman and Krishnan 1928). This normally uses visible light, which is typically scattered by the molecule. Most of the light is scattered elastically at the same frequency as the initial radiation (Rayleigh scattering), but a small amount interacts with the system, excites a vibrational mode, and is scattered inelastically at a lower frequency. If the system is already in a vibrationally excited state, then the scattered radiation can have a higher frequency than the initial radiation. These are known as Stokes and anti-Stokes Raman scattering, respectively. Raman scattering constitutes only a very small component of the total scattered radiation (only about one part in 107 ) and a major difficulty in Raman spectroscopy is to separate out the weak, inelastic scattering from the intense Rayleigh scattered light. The frequency difference corresponds to, e.g., the frequency of one of the molecules normal modes. As with IR spectroscopy, not all of a molecules vibrations are active in the Raman spectrum. In order for a vibrational mode to be Raman active, the motion must result in a change in the polarizability of the electron charge cloud surrounding the molecule. The degree of change determines the intensity of the band. Whether or not a band is Raman, or indeed IR, active can be determined in advance by a symmetry analysis of the normal modes. In molecules with no symmetry, all bands are potentially both IR and Raman active, but if the molecule has symmetry then vibrations of certain symmetry types may be inactive in either the Raman or the IR spectrum, or both. For example, in a molecule with a center of symmetry (such as trans-1,2-dichloroethylene; point group C2h ), the symmetric modes (ag and bg for C2h ) are Raman active but IR inactive, and vice versa for the unsymmetric modes. The intensity of a Raman band can be determined theoretically by calculating the molecule’s polarizability derivatives. The polarization of Raman scattered light also contains useful structural information. This property can be measured by using plane polarized light and a polarization analyzer. Spectra recorded with the analyzer set both parallel and perpendicular to the excitation plane can be used to calculate the depolarization ratio of each vibrational mode. This provides insight into molecular orientation and the symmetry of the vibrational modes, as well as information about molecular shape. It is often

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used to determine macromolecular orientation in crystal lattices, liquid crystals, or polymer samples. Yet another spectroscopic method that can be used to help determine molecular structure is vibrational circular dichroism (VCD). This technique detects differences in attenuation of left and right circularly polarized light passing through the sample. VCD is sensitive to the mutual orientation of groups of atoms in a molecule and provides three-dimensional structural information. It is especially important in the study of chirality and molecular conformation. Only chiral molecules have a VCD spectrum. In particular, molecules that have either a plane of symmetry or a center of symmetry are VCD inactive. The differential absorption of polarized light in vibrational circular dichroism is proportional to the rotational strength, a quantity that depends on both the electric and magnetic dipole transition moments. For a vibrational transition from the ground (g) to an excited vibrational state (e), this is given by ˇ ˛  ˝ ˇ R .g ! e/ D Im hg j el j ei e ˇ mag ˇ g ;

(37)

where el and mag are the electric and magnetic dipole operators, respectively. These quantities can be calculated theoretically, enabling VCD intensities to be computed. For details see the review by Stephens and Lowe (1985). Modern computational approaches use so-called gauge-invariant atomic orbitals (GIAOs) for the magnetic component (Bak et al. 1995) (as does PQS) which are particularly efficient, and the computation of VCD rotational strengths as part of a vibrational analysis is now almost routine. IR, Raman, and VCD spectroscopy all excite the same vibrational fundamentals. The respective vibrational spectra are different because the mechanism by which light is absorbed is different in each case, the amount of absorption depending on changes in the dipole moment, the polarizability and the rotational strength, respectively. All these quantities are amenable to computation, and modern ab initio theory can reliably predict the frequencies of a molecule’s vibrational fundamentals as well as the intensity of the signal in IR, Raman, and VCD spectra. The latter in particular are greatly assisted by electronic structure calculations; comparison of experimental and theoretical VCD spectra enable absolute molecular conformations to be determined.

1,2-Dichloroethane: An Illustrative Example Figure 6 shows the PQS output for a full vibrational and thermodynamic analysis of trans (anti) 1,2-dichloroethane following a geometry optimization under C2h symmetry at the B3LYP/6-31G* level of theory. It includes IR and Raman intensities obtained from the dipole moment and polarizability derivatives, respectively. (Because of its symmetry the system is not VCD active.) In PQS, dipole moment derivatives are computed analytically along with the Hessian matrix and atomic axial tensors (for VCD rotational strengths) (Bak et al. 1995) are computed in the

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same module that calculates NMR chemical shifts. Polarizability derivatives are obtained numerically by finite difference on the force (gradient) in the presence of a series of external electric fields. The number of applied fields varies with the molecular symmetry but requires a maximum of 12 separate single-point energy plus gradient calculations in the worst case (when the system is C1 ). The first thing printed out is the name of the file containing the Hessian matrix. In PQS a vibrational and thermodynamic analysis is done independently of the Hessian computation enabling a full vibrational analysis of isotopomers to be carried out from the same Hessian matrix simply by changing the atomic masses. Analysis is done on any matrix read from the Hessian file, including partial and/or approximate Hessians (e.g., an updated Hessian left over from a geometry optimization), so care must be taken that the source of the Hessian matrix is known. A vibrational analysis of an approximate Hessian will of course, at best, be only approximate; in particular an analysis based on a geometry optimized at one level of theory and a Hessian matrix computed at another (lower) level of theory is, strictly speaking, invalid. Then comes a list of the vibrational frequencies in atomic units after the translations and rotations have been projected out. This is simply a printout of the eigen values of the projected mass-weighted Hessian (see Eq. 35); as can be seen, the six eigen values corresponding to the translations and rotations are all zero. The remaining eigen values are all positive, showing that the optimized structure of trans-1,2-dichloroethane is a genuine minimum. As any decent chemist should know, 1,2-dichloroethane has two stable conformations, trans and gauche; the former is the more stable by around 1.5 kcal mol1 (El Youssoufi et al. 1998a, b). This energy difference is sufficiently small that both conformers will be present, certainly at room temperature, and experimental vibrational spectra of 1,2dichloroethane, unless recorded at very low temperature, will contain bands from

Fig. 6 (continued)

Molecular Structure and Vibrational Spectra

Fig. 6 (continued)

51

52

Fig. 6 (continued)

J. Baker

Molecular Structure and Vibrational Spectra

Fig. 6 (continued)

53

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Fig. 6 PQS vibrational frequency output for trans-1,2-dichloroethane

the gauche as well as the more abundant trans conformer. This will be discussed in more detail later in this chapter. This is followed by a list of all symmetry types possible for the vibrational modes. This is simply a list of the irreducible representations present in the character table for the molecule’s symmetry point group; for trans-1,2-dichloroethane, having C2h symmetry, this is ag , bg (symmetric motions) and au , bu (antisymmetric). All 3 N  6 normal modes (Cartesian displacements for each atom) are then listed, together with as much information for each mode as is available, depending on which, if any, of the dipole derivatives, the polarizability derivatives or the rotational strengths have been computed for the molecule in question. For trans1,2-dichloroethane all these quantities are available except the latter, so listed for each mode is its symmetry type, its frequency in wavenumbers (cm1 ), whether or

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not the mode is IR active, the corresponding IR intensity, the X, Y and Z dipole derivatives with respect to the motion in each normal mode, whether the mode is Raman active, its corresponding Raman intensity and its Raman depolarization ratio. If the molecule were VCD active, its rotational strength (relative intensity in a VCD spectrum) would also be printed. As already noted, whether a mode is IR or Raman active can be determined in advance (although its intensity cannot) by group theoretical considerations and depends on its symmetry type. For trans-1,2-dichloroethane (because of its point group symmetry) each mode is either IR active or Raman active, but not both, and so IR and Raman spectroscopy excite completely different vibrational fundamentals. In the Raman spectrum, only bands of ag and bg symmetry will appear. These can be distinguished experimentally by measuring the depolarization ratio; according to theory (Wilson et al. 1955), the depolarization ratio of totally symmetric modes (ag under C2h ) is less than 0.75 (these are called polarized bands) whereas that of all other modes is 0.75 (depolarized bands). (See the depolarization ratios calculated for trans-1,2-dichloroethane in Fig. 5.) The vibrational analysis is followed by a standard, classical statistical thermodynamic analysis at 298.18 K (25 ı C) and 1 atm pressure. (For details, see McQuarrie (2000)). Computed quantities include the principal axes and moments of inertia, the rotational symmetry number and symmetry classification, and the translational, rotational, vibrational, and total enthalpy and entropy, respectively. Both the temperature and pressure can be altered from standard conditions and/or scanned across a requested range of values. The total zero-point energy at 0 K is P given by  12 h, summed over all real frequencies (converted to kcal mol1 ; see Eq. 36). The predicted IR and Raman spectra, based on the data printed out from the vibrational analysis in Fig. 6 are shown in Fig. 7. These are direct “screen captures” from PQSView, which is the job output and visualization component of the PQSMol GUI (PQS 2010). For each spectrum, the frequency range (horizontal axis), the intensity range (vertical axis), and the band half-width (fitted using a Lorentzian band profile) can be varied; it is also possible to “zoom in” on selected regions of the spectrum. So how good are the results? First of all it should be noted that we would not expect theory to be able to exactly reproduce experimental vibrational spectra as factors such as intensity and bandwidth depend strongly on experimental conditions (as to some extent do the frequencies themselves). Also, most importantly, the theoretical analysis assumes harmonic behavior, whereas the real spectrum is of course anharmonic. So we would expect the calculated vibrational fundamentals to deviate from experiment simply because they are harmonic, without even considering other sources of error (such as incomplete basis set and approximate treatment of electron correlation). For the level of theory used here, B3LYP/6-31G* , the average (mean) error between computed (harmonic) vibrational fundamentals and experimentally observed (anharmonic) values is around 50 cm1 . For example, in a study of 900 individual vibrational fundamentals from 111 different small molecules using B3LYP and a polarized-valence triple-zeta basis, Schlegel and

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coworkers (Halls et al. 2001) found an average absolute deviation between theory and experiment of 44.69 cm1 . Agreement with experiment can be much improved by scaling. This has been done since the early days of Hartree–Fock theory, when all computed HF/631G* frequencies were regularly scaled by a factor of around 0.89. Optimized Hartree–Fock bond lengths are typically too short compared with experiment; hence computed frequencies are almost uniformly too high. Single scale factors, applied directly to computed vibrational fundamentals, have since been derived for a range of other wave functions, including post-HF and DFT. A recent paper from Radom and coworkers derived least-squares fitted scale factors for more than 100 different levels of theory (varying both the methodology and the basis set) (Merrick et al. 2007). Scale factors for DFT-based methods are much closer to unity than scale factors derived using Hartree–Fock (which completely neglects electron correlation) showing that the raw (unscaled) results are simply better. The scale factor for B3LYP/6-31G* is 0.9614 (Merrick et al. 2007).

Scaled Quantum Mechanical Force Fields Although scaling of the computed vibrational frequencies using a single, global, scale factor is common, the first scaling methods applied to ab initio force constants used several different scale factors to correct for systematic errors in different types of molecular deformations, e.g., stretches, bends, or torsions. This procedure requires the transformation of the molecular force field (the Hessian matrix) into chemically meaningful internal coordinates and scales the resulting Hessian elements themselves. It cannot be applied directly to the calculated frequencies. It is thus less convenient than global scaling using a single scaling factor. Largely because of its simplicity, global scaling became popular, but using multiple scale

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factors yields much better results as was convincingly demonstrated by Blom and Altona in a series of papers starting in the mid-1970s (Blom and Altona 1976, 1977a, b; Blom et al. 1976). Their method forms the basis of the scaled quantum mechanical (SQM) procedure which has been in widespread use for over 25 years (Pulay et al. 1983). In the original SQM procedure, the molecular geometry was expressed in terms of a full set of nonredundant natural internal coordinates (Fogarasi et al. 1992; Pulay et al. 1979). On the basis of chemical intuition, the natural internal coordinates of all molecules under consideration (there can be more than one) are sorted into groups sharing a common scale factor, and factors for each group are determined by a least-squares fit to experimental vibrational frequencies. Force constants, originally calculated in Cartesian coordinates, are transformed into the internal coordinate representation, and scaling is applied to the elements of the internal force constant matrix (not to the individual vibrational frequencies) according to Fij .scaled/ D

p

si sj Fij ;

(38)

where si and sj are scaling factors for internal coordinates i and j, respectively. The accuracy obtained by selective scaling in this way is naturally greater than if just a single global scaling factor were used. Additionally, scaling the force constant matrix also affects the resultant normal modes, and hence the calculated intensities (which are unaffected if only the frequencies are scaled), leading to better agreement with experimental intensities. Furthermore the vibrational frequencies derive directly from the (scaled) Hessian matrix, and are thus fully consistent with it; not the case at all if the frequencies are scaled directly. The SQM procedure has been widely used in the interpretation of vibrational spectra. A further important role is the development of transferable scale factors which can be used to modify calculated force constants and so predict the vibrational spectrum a priori. The SQM module in PQS uses a modified scaling procedure involving the scaling of individual valence coordinates (Baker et al. 1998) (not the linear combinations present in natural internal coordinates). This has immediate advantages in terms of ease of use, as no natural internals need to be generated (a procedure which may fail for complicated molecular topologies), and it simplifies the classification and presorting of the coordinates. In addition, the extra flexibility involved in the scaling of individual primitive internals generally leads to an increase in accuracy and to more transferable scale factors. On typical organic molecules one can expect average differences between predicted SQM and experimental fundamentals of around 10 cm1 with maximum errors of about 30 cm1 or so (Baker and Pulay 1998). In my experience, if differences greatly exceed this maximum, it is more likely that the observed experimental peaks have been misassigned than the predicted SQM frequency is wrong.

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1,2-Dichloroethane: A Further Analysis We are going to apply the SQM method to the existing Hessian matrix for trans-1,2dichloroethane in order to improve the agreement with experiment; furthermore, we will also need to consider the gauche conformer which will almost certainly contribute to the experimental vibrational spectra at room temperature. The results of a vibrational analysis of gauche-1,2-dichloroethane, optimized at the B3LYP/6-31G* level, are shown in Fig. 8. This is direct printout from the corresponding PQS log file (not the output file). In addition to the full output, PQS produces a summary output – the log file – which contains only data that is of direct interest to the user and omits all intermediate printout (such as integral and timing data, intermediate steps in an optimization etc.). As can be seen, the log file reproduces the thermodynamic analysis that is printed in the full output file, but provides only a summary of the vibrational analysis and completely omits the normal modes. Gauche-1,2-dichloroethane has C2 symmetry and formally all of its vibrational fundamentals are both IR and Raman active. (Additionally, unlike the trans conformer, it is also potentially VCD active.) However, several modes have only relatively low intensity and may therefore be difficult to see in the experimental spectrum, especially given that the gauche is the higher energy conformer and is only expected to comprise about 13–25 % of the total at room temperature (El Youssoufi et al. 1998a, b). The computed B3LYP/6-31G* energy difference between the gauche and trans conformers is 1.5 kcal mol1 , including (minor) zero-point energy effects. In the paper that introduced the scaling of individual primitive internals into the SQM method (Baker et al. 1998), a set of 11 scale factors were derived from a test set of 30 molecules containing C, H, O, N and Cl at the B3LYP/6-31G* level. These scale factors will be used “as is” to scale the raw Hessian data for both trans and gauche-1,2-dichloroethane. The seven scale factors appropriate for use with 1,2dichloroethane as shown in Table 5. There have been several experimental studies of the vibrational spectra of 1,2dichloroethane. Of note are the 1975 IR and Raman measurements on the solid, liquid, and gaseous trans and gauche conformers from Mizushima and coworkers (1975), and the combined theoretical and FT-IR study of El Youssoufi et al. already referred to (El Youssoufi et al. 1998a, b). These results are summarized, together with the unscaled, scaled using a single scale factor (Merrick et al. 2007), and the SQM predicted fundamentals in Table 6. The first thing to note from Table 6 is that the order and the symmetry assignments of the fundamentals is predicted to be exactly the same theoretically as found experimentally. The actual assignments as to the type of motion for each vibrational mode can be readily checked by animating the mode and observing it visually. In this regard, theory is far easier to interpret than experiment. When you determine a vibrational fundamental computationally you automatically get its normal mode, its symmetry type, its IR and Raman intensities, its depolarization

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Fig. 8 PQS vibrational frequency summary from the log file for gauche-1,2-dichloroethane Table 5 Standard B3LYP/6-31G* SQM scale factors relevant to 1,2-dichloroethane

Type Stretch Stretch Stretch Bend Bend Bend Torsion

rC  C rC  H rC  Cl †CCCl †CCH; †ClCH †HCH all

Scale factor 0.9207 0.9164 1.0438 1.0144 0.9431 0.9016 0.9523

ratio, indeed the whole vibrational analysis, so there is absolutely no dispute as to the bands assignment. Additionally, it is possible to do an energy distribution analysis (Pulay and Torok 1966) of the normal modes which determines which primitive internals have more than a given weight (typically 3 %) to each mode, often enabling the principal motion in the mode (assuming there is one) to be found without having to animate it. The two sets of experimental fundamentals are for the most part in excellent agreement with each other and typically differ by 1–2 cm1 only. However, there are more serious discrepancies for the two highest frequencies. These are C–H stretches which, primarily due to tunneling, often show large differences from harmonic behavior, and both the position and the intensity of these bands in the experimental spectrum can deviate significantly from theoretical predictions. They are also often difficult to determine experimentally. In this case the highest frequency for the gauche conformer differs by 35 cm1 between the two sets of experimental values: 3,005 and 3,040 cm1 . Additionally, although the agreement in this region for the trans conformer looks fine, Mizushima and coworkers (1975) report a value for the au C–H antisymmetric stretch in the solid of 3,040 cm1 , compared to 3,009 cm1 in the gas phase, a substantial difference. Because of these experimental differences, we have omitted the two highest frequency C–H stretches from our error

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Table 6 Comparison of experimental and theoretical (B3LYP/6-31G* ) vibrational fundamentals for gas-phase 1,2-dichloroethane and their assignments Experiment Aa transconformer 122 (au ) 221 (bu ) 305 (ag ) 727 (bu ) 772 (ag ) 772 (au ) 996 (bg ) 1,057 (ag ) 1,124 (au ) 1,233 (bu ) 1,267 (bg ) 1,310 (ag ) 1,431 (ag ) 1,461 (bu ) 2,979 (ag ) 2,983 (bu ) 3,001 (bg ) 3,009 (au )c Mean errord (cm1 ) gaucheconformer 125 (a) 263 (a) 411 (b) 669 (a) 694 (b) 891 (b) 947 (a) 1,028 (a) 1,146 (b) 1,214 (a) 1,292 (b) 1,315 (a) 1,436 (b) 1,446 (a) 2,957 (b) 2,966 (a) 3,005 (a) 3,005 (b) Mean errord (cm1 ) a

Scaled (SQM) Scaled (0.9614) Unscaled

Assignment

117 (au ) 214 (bu ) 296 (ag ) 724 (bu ) 766 (ag ) 767 (au ) 999 (bg ) 1,027 (ag ) 1,128 (au ) 1,244 (bu ) 1,273 (bg ) 1,322 (ag ) 1,459 (ag ) 1,460 (bu ) 2,987 (ag ) 2,994 (bu ) 3,041 (bg ) 3,062 (au ) 9.3 (30)

Bb 123 220 728 773

1,122 1,232

1,460 2,983 3,011

111 (a) 260 (a) 408 (b) 668 663 (a) 694 689 (b) 891 889 (b) 948 938 (a) 1,027 1,021 (a) 1,146 1,146 (b) 1,214 1,211 (a) 1,292 1,303 (b) 1,313 1,325 (a) 1,437 1,438 (b) 1,443 (a) 2,957 2,971 (b) 2,963 2,971 (a) 3,025 (a) 3,040 3,037 (b) 5.4 (14) 16.1 (41)

114 (au ) 206 (bu ) 284 (ag ) 681 (bu ) 725 (ag ) 758 (au ) 985 (bg ) 1,025 (ag ) 1,115 (au ) 1,231 (bu ) 1,260 (bg ) 1,308 (ag ) 1,460 (ag ) 1,460 (bu ) 3,000 (ag ) 3,007 (bu ) 3,054 (bg ) 3,075 (au ) 17.9 (47)

119 (au ) 214 (bu ) 295 (ag ) 708 (bu ) 754 (ag ) 788 (au ) 1,025 (bg ) 1,066 (ag ) 1,160 (au ) 1,280 (bu ) 1,311 (bg ) 1,361 (ag ) 1,519 (ag ) 1,519 (bu ) 3,120 (ag ) 3,128 (bu ) 3,177 (bg ) 3,198 (au ) 38.8 (145)

Torsion CCCl def. CCCl def. C–Cl str. C–Cl str. CH2 rock CH2 rock C–C str. CH2 twist CH2 wag. CH2 twist CH2 wag. CH2 scissor CH2 scissor CH2 sym. str. CH2 sym. str. CH2 antisym. str. CH2 antisym. str.

109 (a) 251 (a) 393 (b) 629 (a) 653 (b) 869 (b) 924 (a) 1,014 (a) 1,133 (b) 1,199 (a) 1,290 (b) 1,312 (a) 1,438 (b) 1,442 (a) 2,976 (b) 2,983 (a) 3,038 (a) 3,051 (b) 40.9 (138)

113 (a) 261 (a) 409 (b) 654 (a) 679 (b) 904 (b) 961 (a) 1,055 (a) 1,179 (b) 1,247 (a) 1,342 (b) 1,365 (a) 1,496 (b) 1,500 (a) 3,095 (b) 3,103 (a) 3,160 (a) 3,173 (b)

torsion CCCl def. CCCl def. C–Cl str. C–Cl str. CH2 rock CH2 rock C–C str. CH2 twist CH2 twist CH2 wag. CH2 wag. CH2 sissor CH2 sissor CH2 sym. str. CH2 sym. str. CH2 antisym. str. CH2 antisym. str.

Ref. Mizushima et al. (1975) Ref. El Youssoufi et al. (1998a, b) c Reported as 3,040 cm1 in the solid d Omitting the two highest fundamentals, maximum deviation in parentheses (see text) b

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analysis, and the mean average (unsigned) and maximum errors between theory and experiment is reported over 16 vibrational fundamentals (not 18). The largest errors are of course found for the raw, unscaled frequencies, which show average differences from experiment for the two conformers of around 40 cm1 (and maximum errors of about 140 cm1 ). These are in line with the average errors reported by Schlegel (Halls et al. 2001) previously mentioned. Despite the large average differences, even the unscaled B3LYP/6-31G* frequencies are helpful in assigning the experimental spectrum as the order is the same as that found experimentally, even if the positions are off. Scaling even using just a single scaling factor significantly improves the comparison with experiment, reducing the average error by well over 50 %. However, the SQM results are by far the best and are genuinely predictive. Results for the trans conformer are typical for this approach (average error 9.3 cm1 , maximum error 30 cm1 ) and those for the gauche are even better (average error 5.4 cm1 , maximum error 14 cm1 ). In the latter case, they are three times better than using a single scaling factor, both on the average and for the maximum deviation. A recently published SQM reappraisal of the vibrational spectrum of toluene (Baker 2008) showed a similar average deviation from the 2005 experimental study of Keefe and coworkers (Bertie et al. 2005) of 5.3 cm1 (maximum error 17 cm1 ) and convincingly proposed that three observed bands either unassigned or assigned as combination bands were in fact fundamentals. In their 1998 SQM study of the vibrational spectra of several small fluorocarbons, Baker and Pulay (1998) proposed several reassignments to previously published experimental spectra and predicted the fundamental frequencies and IR intensities of nine other fluorocarbons for which there was, at that time, no experimental data. A subsequent jet-cooled FT-IR investigation by McNaughton and coworkers (Jiang et al. 2002) essentially confirmed all of the theoretical predictions. The final SQM-predicted IR and Raman Spectra for 1,2-dichloroethane are shown in Fig. 9 and 10, respectively. In each case individual spectra for the trans and gauche conformers are given followed by a combined spectrum assuming a room-temperature distribution of 85 % trans and 15 % gauche. Not surprisingly, because of its dominance at room temperature, the observed IR spectrum of 1,2-dichloroethane is predicted to look very much like that for the trans-conformer alone. However there are some peaks due to the gauche form that should be visible in room-temperature spectra and their intensity should increase at higher temperature. In particular, the CH2 wag. at  1,300 cm1 should have sufficient intensity to be seen in the IR spectrum and the CH2 twist at 1,211 cm1 should be visible in the Raman spectrum. In addition to their results for normal 1,2-dichloroethane, Mizushima et al. also reported the experimental vibrational fundamentals for the fully deuterated isotopomer (CD2 ClCD2 Cl) (Mizushima et al. 1975). These are compared with the SQM scaled frequencies (using the same scaling factors as for CH2 ClCH2 Cl) in Table 7. (As the Hessian matrix is independent of mass, the vibrational fundamentals for all possible isotopomers can be derived from a single Hessian matrix simply by changing the mass weighting.) The agreement is again excellent, with average

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Fig. 9 Predicted SQM IR spectra for 1,2-difluoroethane: (a) IR spectrum for trans-1,2dichloroethane, (b) IR spectrum for gauche-1,2-dichloroethane, (c) IR spectrum for 1,2dichloroethane at room temperature

errors (mean average deviation) of just over 7 cm1 and maximum errors around 20 cm1 for both deuterated conformers. There is one frequency for the gauche conformer where there is significant disagreement between theory and experiment (see Table 7); this is a b mode observed experimentally at 707 cm1 but predicted theoretically at 752 cm1 . In Table 3 in Mizushima et al. (1975) this frequency is in parentheses (the only one to be thus treated) which suggests that its value is uncertain. The trans conformer shows an intense band at precisely this frequency (707 cm1 ) which is predicted to be the most intense band in the entire IR spectrum of deuterated 1,2-dichloroethane, and it may well be that a gauche mode was assumed to lie hidden under this intense band. The prediction here is that this is not the case, and there is an unassigned feature in the IR spectrum around 752 cm1 which should be assigned as a gauche fundamental. Because of the possible experimental misinterpretation, this mode was omitted from the error analysis. As noted, gauche-1,2-dichloroethane is potentially VCD active. The two chiral forms of this molecule are shown in Fig. 11 together with their simulated VCD spectra. (They are denoted by gauche1 and gauche2, respectively; the rotational strengths given in Fig. 8 correspond to gauche1.) The VCD spectra for the two chiral conformers are precisely the opposite of each other; where one form has a positive rotational strength, the other has exactly the same magnitude but with a negative sign. In any real mixture of gauche-1,2-dichloroethane, however, both

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Fig. 10 Predicted SQM Raman spectra for 1,2-dichloroethane: (a) Raman spectrum for trans1,2-dichloroethane, (b) Raman spectrum for gauche-1,2-dichloroethane, (c) Raman spectrum for 1,2-dichloroethane at room temperature

forms will be present in equal amounts, i.e., it will be a racemic mixture, and so overall there will be no observable VCD spectrum as everything will cancel. However, this clearly shows the potential of VCD spectroscopy to determine the absolute molecular configuration of a pure, chiral conformer. Finally Table 8 compares optimized geometrical parameters for trans-1,2dichloroethane with experimental electron diffraction data from Kvesethi as reported in El Youssoufi et al. (1998a, b). I hope that the above has demonstrated just how useful modern theory can be for the prediction and interpretation of molecular vibrational spectra and the identification and assignment of vibrational fundamentals. I have attempted to show this with a fairly extensive theoretical study (carried out especially for this chapter) on one particular molecule, 1,2-dichloroethane, rather than merely summarizing the results of previous studies.

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Table 7 Comparison of experimental and theoretical (B3LYP/6-31G* ) vibrational fundamentals for fully deuterated 1,2-dichloroethane and their assignments Experimenta SQM trans conformer 128 (au ) 107 (au ) 217 (bu ) 201 (bu ) 299 (ag ) 292 (ag ) 568 (au ) 566 (au ) 707 (bu ) 701 (bu ) 708 (ag ) 709 (ag ) 768 (bg ) 770 (bg ) 817 (au ) 815 (au ) 909 (ag ) 895 (ag ) 944 (bu ) 946 (bu ) 975 (bg ) 977 (bg ) 1,036 (ag ) 1,046 (ag ) 1,085 (bu ) 1,078 (bu ) 1,160 (ag ) 1,146 (ag ) 2,179 (ag ) 2,180 (ag ) 2,185 (bu ) 2,180 (bu ) 2,264 (bg ) 2,271 (bg ) 2,271 (au ) 2,282 (au ) MAD (cm1 ) 7.3 (21) a b

Assignment torsion CCCl def. CCCl def. CD2 rock C–Cl str. C–Cl str. CD2 rock CD2 twist C–C str. CD2 wag. CD2 twist CD2 wag. CD2 sissor CD2 sissor CD2 sym. str. CD2 sym. str. CD2 antisym. str. CD2 antisym. str.

Experimenta SQM gauche conformer 120 (a) 106 (a) 229 (a) 224 (a) 365 (b) 359 (b) 624 (a) 616 (a) 643 (b) 620 (b) (707) (b) 752 (b) 759 (a) 766 (a) 828 (a) 822 (a) 828 (b) 826 (b) 941 (a) 935 (a) 1,022 (b) 1,022 (b) 1,052 (a) 1,053 (a) 1,067 (b) 1,060 (b) 1,153 (a) 1,135 (a) 2,171 (b) 2,159 (b) 2,171 (a) 2,168 (a) 2,265 (a) 2,256 (a) 2,265 (b) 2,261 (b) 7.7 (23)b

Assignment torsion CCCl def. CCCl def. C–Cl str. C–Cl str. CD2 rock CD2 rock CD2 twist C–C str. CD2 twist CD2 wag. CD2 wag. CD2 sissor CD2 sissor CD2 sym. str. CD2 sym. str. CD2 antisym. str. CD2 antisym. str.

From Ref. Mizushima et al. (1975); gas-phase where given, otherwise liquid Omitting mode in parentheses (707)

Density Functional Theory and Weight Derivatives Before concluding, I would like to comment on the use of weight derivatives in DFT calculations. As will be discussed, this has important ramifications for the reliable computation of DFT vibrational frequencies. Following Pople and coworkers (Johnson et al. 1993), we can write the DFT exchange-correlation energy as Z EXC D f . / dr; (39) where ¡ represents all the density dependence, of whatever form, in the density functional f (¡). This integral is too complicated to be evaluated analytically and numerical intgration has to be used. Virtually all DFT programs employ an atomic partitioning scheme pioneered by Becke (1988) to evaluate this integral, which separates the molecular integral into atomic contributions which may then be individually treated using single-center techniques. If this is done, we can replace Eq. 39 by

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Fig. 11 Predicted VCD spectra for the two chiral forms of gauche-1,2-dichloroethane: (a) gauche1, (b) gauche2, (c) simulated VCD spectrum of gauche1, (d) simulated VCD spectrum of gauche2 Table 8 Comparison of experimental and theoretical (B3LYP/6-31G* ) geometrical parameters for trans-1,2-dichloroethane

Parametera rC  C rC  Cl rC  H †CCCl †CCH £ClCCCl

B3LYP/6-31G* 1.518 1.815 1.090 109.2 111.6 180.0

Experimentb 1.528(6) 1.796(3) 1.120(10) 108.9(0.3) 113.0(1.3) 180.0

a

Distances in Å and angles in degrees From Ref. Kveseth (1974), standard deviation in parentheses

b

EXC D

X X A

i

!Ai f . I rAi / ;

(40)

where the first summation is over all atoms, A, and the second is over the numerical quadrature grid points, i, for the current atom. The ¨Ai are quadrature weights and the grid points, rAi are given by rAi D RA C ri , where RA is the position of nucleus A and ri represents a suitable quadrature grid centered on A.

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Because the grid points are atom-centered they are not fixed in space, but “move with” the atom. This means that the derivative of Eq. 40 (needed for the DFT contribution to the gradient) formally has two parts: OA EXC D

X X B

i

Œ!Bi OA f . I rBi / C .OA !Bi / f . I rBi / :

(41)

The first term in Eq. 41 looks like the numerical integral of the exchange-correlation contribution to the total energy gradient, and is the only term in a fixed grid, but the second term is formally needed and the implementation in Johnson et al. (1993) was the first time that this term – the weight derivative – was properly included. The error in neglecting the weight-derivative term can be reduced to virtually zero by taking a large enough number of grid points, but for smaller grids its omission is not justified (Johnson et al. 1993). I suspect that the hope was that if the second term in Eq. 41 were properly included, then small grids would no longer give unreliable results, but if this was indeed the hope, it was quickly dashed when it was shown that small grids which included weight derivatives often gave worse results than calculations in which they were omitted (Baker et al. 1994). Basically you have to have a decent quadrature grid to get reliable gradients; if the grid is not good enough using weight derivatives will not save you, if it is good enough then you probably do not need weight derivatives. At least this is typically the case for the gradient. One clear advantage of including weight derivatives is that the gradient zero coincides essentially exactly (for minimization) with the energy minimum. If the weight derivative term is omitted and grid quality is poor then there is a strong tendency for the energy to rise near the end of a geometry optimization as the gradient supposedly decreases. This was a common occurrence with many early DFT codes almost entirely due to sloppy numerical integration. The kind of errors in geometries seen when the DFT integration grid gets increasingly worse is shown in Table 9 which gives geometrical parameters for BLYP/6-31G* optimizations of hydrogen peroxide with a range of decreasing quality grids. The table is taken from Baker et al. (1994) for which see for more details. Results are shown with and without inclusion of the weight derivative term. As can be seen from Table 9, the HOOH torsion changes substantially with grid quality, as do some of the other parameters with the poorest quality grid. If anything, results with weight derivatives are worse than those without as grid quality worsens. The actual optimization for the poorest quality grid (grid 5) converged smoothly when weight derivatives were included, with the energy decreasing on every optimization cycle; it was only when examining the final geometry that one realizes that something is badly wrong. The situation as far as the first energy derivative (the gradient) is concerned is that for reliable results you need a good quality grid, but if you have that then you don’t really need to include weight derivatives (although there is no harm in doing so of course). For the second energy derivative (the Hessian matrix and hence vibrational frequencies) things are somewhat different. As convincingly

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Table 9 Optimized BLYP/6-31G* geometrical parameters for hydrogen peroxide using numerical quadrature grids of decreasing quality with and without weight derivativesa b

Grid quality Conv. 1 1 2 2 3 3 4 4 5 5

Weight derivs Yes Yes No Yes No Yes No Yes No Yes No

Geometrical parametersc rO  O rO  H 1.494 0.985 1.494 0.985 1.494 0.985 1.494 0.985 1.494 0.985 1.495 0.985 1.493 0.986 1.496 0.985 1.492 0.986 1.502 0.983 1.493 0.986

†HOO 98.4 98.4 98.4 98.4 98.4 98.4 98.6 98.3 98.7 97.7 98.4

£HOOH 121.9 122.0 121.9 122.0 121.9 122.8 118.3 126.2 113.9 142.8 114.3

a

From Ref. Baker et al. (1994) 1 D best grid, 5 D worst; Conv. represents converged results c In Å and degrees b

demonstrated by Malagoli and Baker (2003), weight derivatives are essentially de rigueur for the reliable computation of vibrational frequencies, particularly if heavier elements (beyond the first row) are involved. However, the reason for this is not what you might think. It turns out that there is a potential, major difficulty in integrating derivatives of basis functions that represent core electrons on their own atomic grid. Such basis functions have large exponents and their contribution to the electron density consequently dies off very rapidly with increasing distance from the nucleus. Exponents of the order of a million or more are not at all uncommon, especially if the core – as is often the case – is represented by a contracted function, and such exponents only increase in magnitude with increasing atomic number. In order to accurately integrate a Gaussian function that dies off so rapidly a high density of points are needed near the nucleus. The numerical inaccuracy does not matter very much at all for the exchange-correlation energy, but for the gradient and  especially for the second derivative, then a basis function exp !r 2 which   has a small value at some grid point near the nucleus, has a magnitude 2! exp !r 2   for its gradient and 4! 2 exp !r 2 for its second derivative. With ¨ of the order of 106 , then the contribution of this function to the second derivative has a magnitude four trillion r2 times its contribution to the energy at that particular grid point, i.e., it has changed from something fairly small to something potentially large, as well as having a greater radial extent. Consequently it requires a far more accurate numerical integration than it previously had. It is possible to ameliorate this inaccuracy by increasing the grid quality, but eventually with a large enough exponent, you will simply be swamped and it will not be possible to accurately integrate the function on any viable radial grid.

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The tremendous advantage of including the weight derivative term (as opposed to ignoring it and in effect assuming a fixed grid) is that, when considering any derivative term, because a grid “moves with” its atom there is no need to take the derivative of any basis function centered on that atom. Thus the highly inaccurate integration of basis functions with large exponents on their own atomic center no longer needs to be carried out. In other words, the inclusion of weight derivatives is so important for reliable frequencies with DFT, not because of anything inherent in the weight derivative term itself, but because, by including them, one can completely avoid a highly inaccurate numerical integration. If you are using DFT code which does not have weight derivatives, then consider carrying out any vibrational analysis with a Hessian matrix computed numerically. As noted above, the gradient is normally reliable even without weight derivatives provided the quadrature grid is of sufficient quality. The calculation will almost certainly take longer compared to computing a fully analytical Hessian, but at least it should give reasonable results. Table 10 shows the effects of not including weight derivatives in analytical Hessian matrices on the computed B3LYP/m6-31G* harmonic vibrational frequencies for Fe(CO)5 . (The m6-31G* basis (Mitin et al. 2003) is a modified version of the 6-31G* basis for first-row transition metals (Rassalov et al. 1998) that corrects deficiencies in the latter due to the lack of a sufficiently diffuse d-function in the 3 d-shell.) This Table, taken from Malagoli and Baker (2003), shows analytical frequencies computed both with and without weight derivatives using the standard grid in PQS (which is perfectly adequate for DFT energies and gradients) as well as a much better quality grid; additionally it gives numerical frequencies derived from a Hessian obtained via central differences on the (analytical) gradient using the standard grid. As can be seen, all computed frequencies are in excellent agreement with one another (all agreeing within 1 cm1 ) except for those computed with the standard grid without weight derivatives. These results are terrible. Apart from the high frequency C–O stretches (above 2,000 cm1 ), all the e0 modes show significant 00 errors, as do the a2 . The low frequencies are completely wrong, the order of the fundamentals is incorrect, and there are two imaginary frequencies even though the geometry was minimized. All this is solely due to integration error.

Conclusions Modern ab initio theory can compute a large number of molecular properties, particularly those for individual molecules – such as geometries and spectroscopic data – that are of direct relevance for experimental studies. This chapter has concentrated on molecular structure and vibrational spectroscopy where theory can be tremendously helpful to experimentalists. Theoretical data – for example, bond lengths involving hydrogen – are now regularly used as an aid when fitting X-ray and microwave data in order to extract geometrical parameters. IR, Raman and VCD spectra can readily be simulated theoretically, with reliable estimates of both band

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Table 10 Computed B3LYP/m6-31G* harmonic vibrational frequencies (cm1 ) for Fe(CO)5 a Vib. mode Symmetry e0 00 e e0 00 a2 0 a2 00 e 0 a1 0 e 0 a1 00 a2 e0 00 e 00 a2 e0 e0 0 a1 00 a2 0 a1 a

Standard grid No weight 150.09i 100.44 52.52 122.60i 349.78 360.98 419.73 417.52 441.88 440.85 463.61 561.95 579.54 627.12 2,089.44 2,115.53 2,116.03 2,187.58

Weight 47.44 98.96 102.00 109.13 350.44 361.49 419.44 428.61 441.57 471.81 484.05 562.26 620.21 658.12 2,089.55 2,115.43 2,116.09 2,187.40

Better grid No weight 46.55 98.46 101.72 108.75 349.78 360.89 419.74 428.40 441.85 471.77 484.22 561.97 620.05 658.07 2,089.59 2,115.69 2,116.20 2,187.59

Weight 47.38 98.81 101.79 108.93 350.33 361.25 419.41 428.50 441.57 471.81 484.01 662.14 620.22 658.07 2,089.40 2,115.43 2,116.20 2,187.31

Numerical Hessian 46.66 98.99 101.73 109.13 350.20 361.05 419.35 428.87 441.46 471.58 484.18 562.06 620.49 658.18 2,090.10 2,115.98 2,116.65 2,187.96

From Malagoliand Baker (2003)

positions and intensities, especially if the raw data is scaled via an SQM treatment. I have tried to illustrate the latter in particular by concentrating for the most part on an extensive study of a single system using a standard level of theory (B3LYP/6-31G* ) rather than attempting to summarize results on a large number of molecules using a variety of theoretical methods. Thirty years ago I was attending scientific meetings and conferences in which the theoretical chemists in the audience were often attempting to justify their existence to their experimental colleagues. At that time calculations were not especially helpful to the experimentalist. The standard level of theory for most applications was Hartree–Fock, with perhaps some higher level post-HF single-point energies to hopefully improve the energetics. Computed geometries were not particularly good, computed frequencies were typically much too high (indeed, the paper introducing one of the first useable analytical Hartree–Fock second derivative codes (Pople et al. 1979) had only just been published), reaction energetics were fairly poor and barrier heights were simply not reliable. All that has changed. Today the experimentalist who refuses to consider any input from theory is just making things harder for him/herself. This is especially the case in the field covered by this chapter.

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Molecular Electric, Magnetic, and Optical Properties ´ Michał Jaszunski, Antonio Rizzo, and Kenneth Ruud

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Molecular Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Molecular Breit–Pauli Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small Terms Due to the Scalar Potential in the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small Terms Due to the Vector Potential in the Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Response: Definitions, Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansions of Energy and Multipole Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric Multipole Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dipole Polarizability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Dipole Hyperpolarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second Dipole Hyperpolarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-Range Dispersion Interaction Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 5 8 9 10 15 15 18 19 20 25 27 29 31 32

We dedicate this work to the memory of our longtime friend Andrzej Sadlej. M. Jaszu´nski () Institute of Organic Chemistry, Polish Academy of Sciences, Warszawa, Poland e-mail: [email protected] A. Rizzo Istituto per i Processi Chimico Fisici del Consiglio Nazionale delle Ricerche, Area della Ricerca di Pisa, Pisa, Italy e-mail: [email protected] K. Ruud Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø, Tromsø, Norway e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_11-2

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´ et al. M. Jaszunski

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Electric Field Gradient at the Nucleus, Nuclear Quadrupole Coupling Constant . . . . . . . . . . . . One-Photon Absorption, Excitation Energies, and Transition Moments. . . . . . . . . . . . . . . . . . . . . . Two-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotational g Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Birefringences and Dichroisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Faraday Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypermagnetizabilities, Cotton–Mouton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Spin–Related Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NMR Effective Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Magnetic Shielding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Spin Rotation Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NMR Indirect Spin–Spin Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron Spin–Related Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Orbit Coupling Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phosphorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESR Effective Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESR Electronic g-Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero-Field Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESR Hyperfine Coupling Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 35 38 39 40 44 46 49 52 55 56 61 62 64 72 73 76 77 78 79 81 83 84 86 87

Abstract

The theory and applications of ab initio methods for the calculation of molecular properties are reviewed. A wide range of properties characterizing the interactions of molecules with external or internal sources of static or dynamic electromagnetic fields (including nonlinear properties and those related to nuclear and electron spins) is considered. Emphasis is put on the properties closely connected to the parameters used to describe experimentally observed spectra. We discuss the definitions of these properties, their relation to experiment, and give some remarks regarding various computational aspects. Theory provides a unified approach to the analysis of molecular properties in terms of average values, transition moments, and linear and nonlinear responses to the perturbing fields. Several literature examples are given, demonstrating that theoretical calculations are becoming easier, and showing that computed ab initio molecular properties are in many cases more accurate than those extracted from experimentally observed data.

Introduction The term “molecular properties” is generally used to denote properties characterizing the interactions of molecules with internal or external sources of static or dynamic electromagnetic fields. In the description of a complex

Molecular Electric, Magnetic, and Optical Properties

3

quantum-mechanical system consisting of a molecule and a field it is convenient, both in theory and in experiment, to introduce an approximation that separates the molecule, the field, and the molecule–field interaction. One can then introduce an additional approximation and analyze the effects of the interaction only for the molecule, assuming a passive role of the interaction-induced changes in the field. In this way we can describe the interaction between the field and the molecule in terms of what we can consider intrinsic properties of the molecule. Most spectroscopic methods assume that the experiment can be described in terms of these molecule–field interactions. There is, therefore, often a very close connection between the parameters used to describe the experimental spectra and the molecular properties that describe the molecule–field interactions probed by the experiment. The set of properties describing the influence of external electric and magnetic fields – static and time-dependent, spatially uniform or nonuniform – can be used to determine parameters that enter into the analysis of different experimental spectra. In particular, optical properties describe the interactions of a molecule with an external electromagnetic radiation field, and characterize both linear and nonlinear responses to the fields. Other sets of molecular properties describe the interactions of external magnetic fields and the magnetic fields due to the spins of all particles – that is, nuclear spins (specific for each nuclear isotope) and electron spin. These properties determine the parameters entering the analysis of Nuclear Magnetic Resonance (NMR) and Electron Spin Resonance (ESR) experiments. Additional fields can be created by the presence of neighboring molecules, and various effects due to weak long-range intermolecular interactions can also be described in terms of molecular properties. The treatment of molecular vibrational properties is in many cases based on the same principles as the formalism explored here; however, the calculation of such properties differs quite substantially from that of electronic properties due to the different quantum-mechanical treatments of the electronic and nuclear motions, and for this reason we will not consider vibrational properties in this chapter.

The Molecular Hamiltonian Our starting point for electronic structure calculations of molecules is the timeindependent, field-free nonrelativistic Schrödinger equation for the electronic degrees of freedom of a molecule HO .0/ ‰ D E‰;

(1)

where  is the wave function describing the electronic state of the molecule, E the energy of this state, and Hˆ (0) the Hamilton operator for the system, given by 1 X 2 e 2 X ZK e2 X 1 e 2 X ZK ZL HO .0/ D pOi  C C ; 2me i 4–0 iK riK 8–0 rij 8–0 RKL i¤j

K¤L

(2)

´ et al. M. Jaszunski

4

where the first term corresponds to the kinetic energy of the electrons, pi D i „ri is the linear momentum operator of particle i, the second term is the attractive potential between the electrons and the nuclei, the third term the electron–electron repulsion potential, and the last term the repulsive potential for the interaction between the nuclear point charges. In Eq. 2, e is the elementary charge, ZK e is the charge of nucleus K, 20 is the electric constant (electric permittivity of vacuum), and me the mass of the electron. We use ri and RK to denote the vectors giving the position of electron i and nucleus K with respect to a chosen origin of the molecular frame. If needed, for a specific frame of interest, we use the notation riO D ri  O

(3)

to emphasize that the positions are given relative to an origin O. We define vectors between two different points in space with the directional convention RKL D RK  RL ;

(4)

corresponding to a vector pointing from nucleus L to nucleus K. In the summations over electrons or nuclei, we always assume – unless explicitly stated otherwise – that the summation runs over all the particles of a given type in the molecule. Whenever needed, we shall use for vector and tensor coordinates the following subscripts: • Greek symbols for unspecified molecular frame coordinates; for these coordinates the Einstein summation convention is applied • x, y, z for specific molecular frame coordinates (for instance, determined by molecular symmetry) • X, Y, Z for laboratory frame coordinates • 1, 2, 3 when the tensor is specified using its own principal axes and values • a, b, c for coordinates defined by the moment of inertia I P where I D K MK RK RTK and MK is the mass of nucleus K. The Hamiltonian in Eq. 2 does not take into account the interactions with external electric or magnetic fields. Moreover, so far we have neglected numerous other interactions, related to the existence of the electron and nuclear spins (and the corresponding magnetic moments), their interactions with external fields, and relativistic effects. Thus, for the study of molecular properties which depend on these interactions we need to consider a more general approach, based for instance on the Breit–Pauli Hamiltonian (Dyall and Faegri 2007; Moss 1973). In many cases the contributions to molecular properties due to the additional terms appearing in the Breit–Pauli Hamiltonian can be estimated using the available solutions of the (nonrelativistic) Schrödinger equation and perturbation theory.

Molecular Electric, Magnetic, and Optical Properties

5

The Molecular Breit–Pauli Hamiltonian There are several ways to introduce and justify the use of the Breit–Pauli Hamiltonian. One can start from the Dirac equation which involves the relativistic Hamiltonian for the free electron, and extend it by applying various approximations to many electrons in the field generated by the nuclei. We will not discuss here the steps taken in the reduction of the Dirac Hamiltonian to the Breit–Pauli Hamiltonian (Dyall and Faegri 2007; Moss 1973). Instead, we only list the various contributions appearing in the Breit–Pauli Hamiltonian. The approximate form of the Breit– Pauli Hamiltonian we will discuss is accurate to order ˛ 2fs , where ˛ fs is the fine structure constant, and higher-order corrections are not considered at this stage (we will include later selected terms of the order ˛ 4fs needed for the calculation of response properties in the presence of magnetic fields). Finally, even though some of the operators appearing in the electronic Breit–Pauli Hamiltonian are of little interest from the point of view of molecular properties, they are included here for completeness. The purely electronic contributions to the Breit–Pauli Hamiltonian can be written as (Bethe and Salpeter 1957, p. 170; Moss 1973): X  2 i BP O Hel D  e 2m e i

(5)

mi  B

(6)

C

˛fs2 mi  .i  F  F  i / 4me

(7)

e„2 ˛fs2 .r F/ 8m2e

(8)

C



˛fs2 4  8m3e i

˛fs2 .mi  B/ i2 2m2e  1 X e2 1 C 4–0 2 rij C

(9)

(10)

(11)

j ¤i

  # i  rij rij  j i  j C rij rij3   e˛fs2 mi  rij  i C 2me rij3

e 2 ˛fs2  4m2e

"

(12)

(13)

´ et al. M. Jaszunski

6

  e˛fs2 mi  rij  j  me rij3 # "    3 mi  rij rij  mj ˛fs2 mi  mj 8   ı rij mi  mj   C 2 3 rij3 rij5  e 2 „2 ˛fs2    ; ı rij 2m2e

(14)

(15)

(16)

where ırij is the Dirac delta function   ı rij D 0

Z

ı

.rij /d 3 r D 1:

for rij ¤ 0I

(17)

V

In addition to these purely electronic contributions, we must also consider the interactions that involve simultaneously one of the electrons and one of the nuclei: BP HO eN D

 e2 1 X ZK  4–0 iK riK e˛fs2 mi  .riK  i / 3 2me riK e 2 „2 ˛fs2 C ı .riK / : 2m2e C

(18)

(19)

(20)

Finally, we have the contributions arising from the nuclei only BP HO nucl

D

X

" ZK eK  mK  B

(21)

K

C

X

L¤K

# e 2 ZK ZL : 8–0 RKL

(22)

We have omitted in these equations all the terms dependent on the (inverse) masses of the nuclei. In the purely electronic and purely nuclear parts we have not included the rest mass of the particles. In the above equations we have introduced the mechanical momentum i D pi  qi A .ri / ;

(23)

where A(ri ) is the magnetic vector potential experienced by electron i, and qi is the charge of the particle, which for an electron is  e. The corresponding magnetic field

Molecular Electric, Magnetic, and Optical Properties

7

induction is B (see below).  represents the external scalar electrostatic potential, and the electric field due to this external potential is F D r. We have also introduced the magnetic moment of the electron mi , which is related to the electron spin through mi D ge

e„ si D ge B si ; 2me

(24)

where ge is the electron g factor and the Bohr magneton B is the unit for magnetic moments. In a similar manner, mK is the nuclear magnetic moment, related to the nuclear spin IK as mK D „K IK D N gK IK ;

(25)

where  K is the nuclear magnetogyric ratio, N D e„=2mp is the nuclear magneton, mp the proton mass, and gK is the g factor of nucleus K, unique for each isotope. For a detailed description of the different terms appearing in the Breit–Pauli Hamiltonian we refer to Moss (1973). The Breit–Pauli Hamiltonian, Eqs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22, does not contain explicitly any coupling between the nuclear magnetic moments mK and the magnetic moments of the electrons mi . In sections “The Magnetic Vector Potential” and “Small Terms Due to the Vector Potential in the Hamiltonian,” we introduce these interactions by establishing a magnetic vector potential for the nuclear magnetic moments in a similar manner to the magnetic vector potential for the external magnetic field induction. These interactions, important for determining the observable NMR and ESR spectra, will thus appear through the couplings arising when we expand  2i (and the spin–Zeeman term leading to Eq. 6) using the full mechanical momentum operator given in Eq. 23. The Breit–Pauli Hamiltonian describes a number of contributions that can be considered small compared to the nonrelativistic Hamiltonian given in Eq. 2. These contributions are thus ideally suited to be treated by perturbation theory, at least as long as we do not consider the heaviest atoms of the periodic table, where the relativistic effects become substantial and the lack of a variational bound for the Breit–Pauli Hamiltonian makes any perturbation approach fail. For the energy of molecules consisting of light atoms, the relativistic effects can to a first approximation often be treated considering only the mass-velocity (Eq. 9) and oneelectron Darwin (Eq. 20) terms. Some of the terms included in the Breit–Pauli Hamiltonian also describe small interactions that can be probed experimentally by inducing suitable excitations in the electron or nuclear spin space, giving rise to important contributions to observable NMR and ESR parameters. In particular, for molecular properties for which there are interaction mechanisms involving the electron spin, also the spinorbit interaction (Eqs. 13 and 14) becomes important. The Breit–Pauli Hamiltonian in Eqs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22, however, only includes molecule–external field interactions through the presence of a scalar

´ et al. M. Jaszunski

8

electrostatic potential  (and the associated electric field F) and the appearance of the magnetic vector potential in the mechanical momentum operator (Eq. 23). In order to extract in more detail the interaction between the electronic structure of a molecule and an external electromagnetic field, we need to consider in more detail the form of the scalar and vector potentials.

Small Terms Due to the Scalar Potential in the Hamiltonian Let us begin with the simplest example, the interaction of a molecule with a static electric field. This interaction can be described by the Hamiltonian HO D HO .0/  e

X

 .ri / C e

X

i

ZK  .RK /;

(26)

k

where we have neglected the spin-orbit and relativistic terms given in Eqs. 7 and 8. We can expand the potential around a common origin  .ri / D .0/ C

@ 1 @2  ri˛ C ri˛ riˇ C    @ri˛ 2 @ri˛ @riˇ

(27)

with a similar expansion for the potential at the position of the nuclei, (RK ). Recalling that the electric field due to the external potential is F D r, we can rewrite Eq. 26 in terms of electric multipole operators HO D HO .0/ C Q.0/ C HO F  F C HO rF  rF C    ;

(28)

where Q is the total charge of the molecule. We note that the multipole operators are defined with respect to the chosen common origin. For a neutral system, the first two contributions describing the field effects are given by the: • External electric field perturbation: HO F D ;

(29)

where the electric dipole operator is O ˛ D e

X

rOi˛ C e

i

X

ZK RK˛ :

(30)

K

• External electric field gradient perturbation: 1 HO rF D  ‚; 3 where the electric quadrupole operator in its traceless form is

(31)

Molecular Electric, Magnetic, and Optical Properties

O ˛ˇ D  e ‚ 2

9

X  eX   2 3rOi˛ rOiˇ  rOi2 ı˛ˇ C ZK 3RK˛ RKˇ  RK ı˛ˇ : 2 K i

(32)

The interaction due to the external field gradient can also be expressed in terms of the second moments (traced form) qO ˛ˇ D e

X

rOi˛ rOiˇ C e

X

i

ZK RK˛ RKˇ :

(33)

K

Later we shall discuss also the effects due to the: • Electric field gradient at the nucleus K: K VO˛ˇ D

2 ı˛ˇ 1 X 3rOiK;˛ rOiK;ˇ  rOiK e 5 4–0 i riK X 2 ı 1 3R R RLK ˛ˇ  e ZL LK;˛ LK;ˇ ; 5 R LK 4–0

(34)

L¤K

since we will need it to describe the interaction with the nuclear quadrupole moments, QK , described by the interaction Hamiltonian 1 K HO V D  2

X

VK QK :

(35)

K

The Magnetic Vector Potential We consider a molecule in which the magnetic field arises from two primary sources, an external magnetic field induction and the permanent magnetic moments of nuclei possessing a spin. In the minimal coupling approximation, the mechanical momentum operator (Eq. 23) is given as i D pi C eAB .ri / C e

X

AmK .ri / ;

(36)

K

where the superscripts indicate the different sources of the magnetic vector potential. In Eq. 36 we have explicitly indicated that we are interested in the magnetic vector potential at the positions of the electrons. The magnetic vector potential is not uniquely defined, and we have to choose the gauge (Jackson 1998). In the Coulomb gauge, that is, where the magnetic vector potential is divergence free at all points in space, r A .r/ D 0;

(37)

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we may choose the vector potential describing the interaction with the external magnetic field induction and with the magnetic moments of the nuclei in the form A .ri / D

X mK  riK 1 B  riO C ˛fs2 : 3 2 riK K

(38)

It can easily be verified that the magnetic vector potential (Eq. 38) is divergence free. Since the magnetic field induced by the magnetic vector potential is given by B D r  AB .ri / ;

(39)

we can add any scalar differentiable potential rƒ to the magnetic vector potential in Eq. 38 without changing the description of the magnetic field induction experienced by the molecule (since r  .rƒ/ D 0). There are, therefore, multiple definitions of the magnetic vector potential that describe the same magnetic field induction, and thus the same physical situation. This invariance with respect to the choice of magnetic vector potential may be lost in approximate calculations. This in turn leads to what is known as the gauge dependence of computed magnetic properties, an undesirable feature of approximate calculations – the results depend on an arbitrary factor in the calculation. In practice, when interactions with external magnetic fields are considered, special computational methods are introduced to bypass this problem and to obtain reliable and unambiguous gauge-origin independent results. We will return to this point when we consider magnetic properties in section “Magnetic Properties.”

Small Terms Due to the Vector Potential in the Hamiltonian Having now established the form of the vector potential, we return to the molecular Hamiltonian. In the presence of magnetic fields, the kinetic energy is defined by the mechanical momentum operator, Eq. 23 (instead of the linear momentum operator), and in addition we need to consider the Zeeman term, Eq. 6. We can expand all the terms appearing in the kinetic energy, using the vector potential in the mechanical momentum, and introduce the magnetic field induction in the Zeeman term. This yields the following form of the Hamiltonian: e„ HO D HO 0  i me

X i

A .ri /  ri C

X e2 X 2 A .ri / C mi  .ri  A .ri // ; 2me i i (40)

where Hˆ 0 indicates the Hamiltonian in Eq. 2. In this equation we have taken advantage of the commutator relation ŒA .ri / ; ri  D 0 (this relation holds true for our choice of the Coulomb gauge, see Eq. 37). Inserting the magnetic vector potential, Eq. 38, into Eq. 40 we get

Molecular Electric, Magnetic, and Optical Properties

HO D HO 0 C HO B B C

X

11

HO K.PSO/ mK

(41)

K

X

1 X T O K;L.DSO/ mK H mL 2

(42)

X X X C mi HO s;B B C mi HO K.FC/ mK C mi HO K.SD/ mK :

(43)

CBT HO B;B B C

BT HO B;K mK C

K

i

K¤L

iK

iK

The summation over the nuclear magnetic moments in the last term of Eq. 42 should include contributions quadratic in mK , but since these contributions do not correspond to any measurable quantity, we have restricted the summation to different nuclei. We note also that these neglected terms are divergent, which shows the limitations of the applied theory. Let us analyze the explicit expressions for the different operators in Eqs. 41, 42, and 43, which are obtained by inserting the magnetic vector potential Eq. 38 into Eq. 40. The contributions in Eq. 41 arise from the unperturbed Hamiltonian and from the term linear in A(ri ). The operators involved are: • Magnetic dipole operator X e X B X .riO  ri / D m: HO B D 1iO D 1iO D i B 2me i „ i i

(44)

The magnetic dipole moment is proportional to the angular momentum, the coefficient of proportionality being the magnetogyric ratio ” e D e/2me . The orbital angular momentum of electron i is defined as liO D riO  pi D i „riO  ri :

(45)

The magnetic dipole operator describes the magnetic moment induced by the angular momentum of the electrons arising from the orbital motion, in contrast to the magnetic moment arising from the electron spin which we discuss when we consider the spin–Zeeman interaction. • Paramagnetic spin-orbit operator 2B HO K.PSO/ D ˛fs2 „

X 1iK i

3 riK

:

(46)

The paramagnetic spin-orbit interaction, often called the orbital hyperfine interaction, describes the interaction of the magnetic moment induced by the orbital motion of the electrons with the nuclear magnetic moments.

´ et al. M. Jaszunski

12

The contributions in Eq. 42 arise from the term quadratic in A(ri ) in Eq. 40. They involve the following operators: • Diamagnetic magnetizability operator  e2 X  2 HO B;B D riO 1  riO rTiO ; 8me i

(47)

where 1 is a 3  3 unit matrix. This operator is closely related to the second moment of the electronic charge, and thus reflects the size of the electron density distribution. The diamagnetic magnetizability tensor is a symmetric 3  3 tensor operator, and has, therefore, in general six independent tensor operator elements. • Diamagnetic shielding operator e 2 X .riO  riK / 1  riO rTiK HO B;K D ˛fs2 : 3 2me i riK

(48)

The diamagnetic shielding tensor describes the interaction of the magnetic field induction with the nuclear magnetic moment, mediated by the orbital motion of the electrons. We note that the diamagnetic shielding tensor is a 3  3 nonsymmetric tensor operator, and thus in general has nine independent elements. • Diamagnetic spin-orbit operator e 2 X .riK  riL / 1  riK rTiL HO K;L.DSO/ D ˛fs4 : 3 3 2me i riK riL

(49)

The diamagnetic spin-orbit operator describes the direct interaction between the two orbital magnetic moments induced in the electron density by the presence of the magnetic moments of nuclei K and L. As for the diamagnetic shielding operator, this is a 3  3 nonsymmetric tensor operator, with in general nine independent elements. Finally, the terms in Eq. 43 arise from the spin–Zeeman term, the last term in Eq. 40. • Spin–Zeeman operator HO s;B D 1:

(50)

The spin–Zeeman operator describes the direct interaction of an electron spin with the external magnetic field induction, and does not act on the spatial coordinates of the electrons, acting only in spin space (this contribution thus vanishes for closed-shell systems). We note that it is customary to describe this term in the Hamiltonian using the electron spin explicitly rather than the electron magnetic moment.

Molecular Electric, Magnetic, and Optical Properties

13

• Fermi contact (FC) operator 8 HO K.F C / D ˛fs2 3

X

ı .riK / 1:

(51)

i

The Fermi contact operator acts both in the spin and in the spatial space of the electronic coordinates, often leading to a very different behavior of the response properties than for instance the Hˆ K(PSO) operator. The Fermi contact operator represents the direct interaction of the electron spin-magnetic moment with the nuclear magnetic moment at the position of the nucleus, and can also be considered to be a measure of the spin-polarization at the nucleus. • Spin–dipole (SD) operator HO K.SD/ D ˛fs2

X 3riK rT  r 2 1 iK iK : 5 r iK i

(52)

The spin–dipole operator is in the form of a classical dipole–dipole interaction between the magnetic dipole of the nucleus and the magnetic dipole of the electron. For the contributions involving the electronic spin operators through the electron spin-magnetic moment mi – that is, Hˆ K(FC) and Hˆ K(SD) – we need to keep track of the electron spin index and of the nuclear spin index. Due to the nature of the Dirac delta function appearing in the operator for Hˆ K(FC) , this operator is diagonal in the two tensor indices, that is K.FC/ K.FC/ HO ˛ˇ D HO iso ı˛ˇ :

(53)

Both the Fermi contact and the spin–dipole operators can be derived considering the magnetic field induction created by the vector potential describing the nuclear magnetic moment iK r  AmK .ri / D ˛fs2 r  mKrr 3 iK X 3riK rT  r 2 1 iK iK D ˛fs2 C 5 r iK i

8 3

X

! ı .riK / 1 mK ;

(54)

i

which can be derived using standard vector product formulas. We shall in later sections also discuss the effects of the spin-orbit interaction. The total spin-orbit (SO) interaction operator (McWeeny 1992) can be defined combining the terms included in Eqs. 13, 14, and 19 as

´ et al. M. Jaszunski

14

HO SO

2 3   X X m  1 C 2m m  1 e i j ij i iK 5 D ˛2 4 ZK  3 8–0 me fs iK riK rij3

(55)

i¤j

 HO SO.1/ C HO SO.2/ : The corrections to the Hˆ SO operator, Eq. 55, which account for the presence of magnetic fields, arise through the angular momentum operators (Fukui et al. 1996): liK D riK  Œi „ri C eA .ri /

(56)

lij D rij  Œi „ri C eA .ri / :

(57)

The first part of the vector potential, related to the external magnetic field, causes the Hˆ SO operator to contain two contributions HO SO;B D HO SO;B.1/ C HO SO;B.2/ ;

(58)

# " T 2 X / .r 1  r  r r e iO iK iO iK HO SO;B.1/ D  ˛2 ZK mi  B ; 3 16–0 me fs iK riK 3 2   T 2 X r 1  r  r r  e iO ij iO ij HO SO;B.2/ D mi C 2mj  ˛2 4  B5 : 16–0 me fs rij3

(59)

(60)

i¤j

Similarly, the second part of the vector potential related to the magnetic moment of nucleus K leads to HO SO;K D HO SO;K.1/ C HO SO;K.2/ ; HO SO;K.1/ D 

(61)

" # X X .riK  riM / 1  riK rTiM e2 4 ˛ ZM mi   mK ; 3 3 8–0 me fs M riK riM i (62) 2

HO

SO;K.2/

3

  X  riK  rij 1  riK rTij e2 4 4 mi C 2mj  D ˛  mK 5 : 3 3 8–0 me fs riK rij i¤j

(63)

Molecular Electric, Magnetic, and Optical Properties

15

Response Theory Molecular Response: Definitions, Symbols The formalism we use to compare theory with experiment is based on the analysis of the expectation value ˇ ˛of some operator of interest. We denote the time-dependent reference state as ˇQ0.t / and consider hermitian (in general time-dependent) perturbing operators. The response functions correspond to the Fourier coefficients in the expansion of the time-dependent expectation value of an operator A (Olsen and Jørgensen 1995): Z 1 ˝ ˛ Q / jAj 0.t Q / D h0 jAj 0i C 0.t hhAI V !1 ii e i!1 t d !1 1 Z 1Z 1 C 12 hhAI V !1 ; V !2 ii e i.!1 C!2 /t d !1 d !2 1 1

C

1 6

Z

1Z 1Z 1

hhAI V !1 ; V !2 ; V !3 ii e i.!1 C!2 C!3 /t d !1 d !2 d !3 C    ;

1 1 1

(64) where each V !i represents a time-dependent perturbation operator. We have omitted the infinitesimal factors, which ensure that the perturbations are switched on adiabatically and vanish for t D  1. In the analysis of the different terms in Eq. 64 for the case of exact electronic states, the zeroth-order response function is the expectation value of the operator A in the reference state j0i ; h0 jAj 0i. The first-order, or linear, response function and its residue are given in the sum-over-states form by Olsen and Jørgensen (1995) and Sasagane et al. (1993): hhAI Bii!1 D  „1 D

X X h0 jAj ni hn jBj 0i

 „1

!n  !1 n P X  h0 jAj ni hn jBj 0i n

!n  !1

h0 jBj ni hn jAj 0i : C !n C !1

(65)

The poles correspond to the excitation energies of the system. In addition, from the residues: lim „ .!1  !e / hhAI Bii!1 D h0 j A j ei he j B j 0i ;

(66)

lim „ .!1 C !e / hhAI Bii!1 D  h0 j B j ei he j A j 0i;

(67)

!1 !!e

!1 !!e

´ et al. M. Jaszunski

16

we can extract information about the transition matrix elements. The second-order, or quadratic, response function is defined as hhAI B; C ii!1 ;!2 D 

1 X X h0 jAj mi hm jBj ni hn jC j 0i : .!m  ! / .!n  !2 / „2 P m;n

(68)

In Eq. 68 and expressions given below, ¨¢ equals the sum of the other

in the X  frequencies ! D !i and P permutes the operators and the associated i frequencies. For example, in Eq. 68, P permutes A(¨¢ ), B(¨1 ), C(¨2 ). We note that different expressions for nonlinear response functions, involving operators of the form B D B  h0 j B j 0i, are obtained when the state j0i is omitted from the summations (see, e.g., Sasagane et al. 1993). For the quadratic and cubic response functions there is a variety of residues, and we present only expressions for the residues that are of practical interest. A single residue of a quadratic response function is # "   1 XX h0 jAj mi hm jBj f i lim „ !2 !f hhAI B; C ii!1 ;!2 D  hf jC j 0i: !2 !!f .!m  ! / „ P m (69) This type of residue is used for instance to evaluate two-photon transition amplitudes, the B term in magnetic circular dichroism or the transition moments in phosphorescence. A double residue of the quadratic response function, given by   lim „ !1 C !f lim „ .!2  !e / hhAI B; C ii!1 ;!2

!1 !!f

!2 !!e

(70)

D  h0 j B j f i hf j .A  h0 j A j 0i/ j ei he j C j 0i ; allows us to determine (from a knowledge of the reference state j0i only) the transition moment between two other states, j ei and j f i. For the particular choice of j e iDj f i, we obtain lim „ .!1 C !e / lim „ .!2  !e / hhAI B; C ii!1 ;!2

!1 !!e

!2 !!e

D  h0 j B j ei .he j A j ei  h0 j A j 0i/ he j C j 0i ;

(71)

which allows us to extract the expectation value he j A j ei. In this way, using a double residue of the quadratic response function from the reference state function we can compute first-order properties of another state. Finally, the third-order, or cubic, response function can be expressed as

Molecular Electric, Magnetic, and Optical Properties

hhAI B; C; Dii!1 ;!2 ;!3 D 

17

1 X X h0 jAj mi hm jBj ni hn jC j qi hq jDj 0i :  „3 P m;n;q .!m  ! / .!n  !2  !3 / !q  !3 (72)

From the discussed residues, we can determine properties explicitly dependent on different electronic states, for instance all the transition moments coupling the reference state j0i and other states jei (from linear response residues) and all the one-photon transition moments coupling other pairs of states (from quadratic response double residues). In addition, from single residues of the second- and thirdorder response functions we can identify two- and three-photon transition matrix elements. The frequency(time)-independent properties are a special case of the response functions .!1 D !2    D 0/ and correspond to energy derivatives. We shall use a simplified notation omitting the subscripts indicating the (zero) frequencies for these properties. The response function, is therefore, in this case written as hhA; B, C, D : : : ii. For time-dependent oscillatory perturbations, the response functions can be related to the derivatives of the time-averaged quasienergy (Christiansen et al. 1998; Saue 2002). The derivation, based on a time-dependent variational principle and the time-dependent Hellman–Feynman theorem, shows explicitly the symmetry of the response functions with respect to the permutation of all the operators. We do not describe in this chapter the standard basis sets and wave function models in use in quantum chemistry, referring instead the interested reader to other chapters in this handbook and to a recent textbook (Helgaker et al. 2000). We only note that response theory has been formulated and corresponding programs have been implemented for most of the standard approximations, such as: • • • •

RHF, the restricted Hartree–Fock model MCSCF, the multiconfigurational self-consistent field model DFT, density-functional theory MP2, the Møller–Plesset second-order perturbation theory, which in the case of frequency-dependent perturbations has to be replaced by an iterative optimization of the perturbation amplitudes, as described by the CC2 model (Christiansen et al. 1995) • CCSD, the coupled-cluster singles-and-doubles model • CCSD(T), CCSD with a noniterative perturbative treatment of the triples correction, which in the case of frequency-dependent response functions has to be replaced by an iterative triples correction, as described by the CC3 model (Koch et al. 1997) • FCI, full configuration-interaction theory

´ et al. M. Jaszunski

18

There are also a couple of models designed primarily for the purpose of calculating response functions – that is, they cannot be derived in terms of energy or quasienergy derivatives of the energy expression for an electronic structure method. We note here the second-order polarization propagator approximation [SOPPA (Enevoldsen et al. 1998; Nielsen et al. 1980)], SOPPA using CCSD correlation amplitudes [SOPPA(CCSD) (Sauer 1997)], as well as the equation-of-motion CCSD model [EOM-CCSD (Perera et al. 1994a; Stanton and Bartlett 1993)]. In general, the accuracy of the computed molecular properties reflects the accuracy of the approach used for the unperturbed reference state. There is, however, one significant difference between variational and perturbative, nonvariational approximations which should be kept in mind in the calculation of properties. The response equations which hold for the exact wave function are in general not fulfilled for nonvariational wave function models, even in the limit of a complete basis set. For instance, in the MP2 and coupled-cluster approximations, the static first-order properties, which should be computed as the first energy derivatives, do not correspond to simple expectation values – corrections arising from orbital relaxation must in general be taken into account. Similarly, in the calculation of higher-order molecular properties when the basis set depends explicitly on the applied perturbations (such as geometrical distortions, or magnetic field perturbations described using London atomic orbitals), orbital relaxation corrections should be included (Gauss and Stanton 1995). We note, however, that as the excitation level is increased (triple excitations, etc.), the importance of orbital relaxation is reduced, since in the limit of the full CI wave function, which is exact, no orbital relaxation is necessary.

Expansions of Energy and Multipole Moments The static properties can be defined by an expansion of the total energy E D E .0/  ˛ F˛  12 ˛˛ˇ F˛ Fˇ  16 ˇ˛ˇ F˛ Fˇ F 

1  F F F F 24 ˛ˇı ˛ ˇ  ı

C 

 13 ‚˛ˇ F˛ˇ  13 A˛;ˇ F˛ Fˇ  16 C˛ˇ;ı F˛ˇ Fı C     12 ˛ˇ B˛ Bˇ  12 ;˛ˇ B˛ Bˇ F  14 ı;˛ˇ B˛ Bˇ F Fı C    C K;˛ˇ B˛ mK;ˇ C    : (73) For a molecule in a radiation field, the definitions of the frequency-dependent properties are related to the induced (in general also frequency-dependent) multipole moments. Let us consider a small harmonic perturbation of angular frequency ! from a plane-wave radiation field. We then have (Barron 2004):

Molecular Electric, Magnetic, and Optical Properties

19

• The induced electric dipole moment  ˛ D ˛˛ˇ Fˇ C

1 0 : 1 G˛ˇ B ˇ C A˛;ˇ Fˇ C    ! 3

: 1 0 1 0 : A˛;ˇ Fˇ C    ; Fˇ C G˛ˇ Bˇ C C ˛˛ˇ ! 3!

(74) (75)

• The induced electric quadrupole moment ‚ ‚˛ˇ D A;˛ˇ F 

: 1 0 D;˛ˇ B  C C˛ˇ;ı Fı C    !

: : 1 0 1 F ı C    ;  A0;˛ˇ F C D;˛ˇ B C C˛ˇ;ı ! !

(76) (77)

• The induced magnetic dipole moment m0 m0˛ D ˛ˇ Bˇ C

: 1 0 1 0 : D˛;ˇ Fˇ  Gˇ˛ Fˇ C  3! !

1 1 0 : B ˇ C D˛;ˇ Fˇ C Gˇ˛ Fˇ C    ; C ˛ˇ ! 3 :

(78) (79)

:

where F and B denote the time derivatives of the fields and we have considered only linear response properties. The properties we have included are the electric multipole polarizabilities – ’˛ˇ , dipole–dipole, A˛, ˇ , dipole–quadrupole; and C˛ˇ,ı , quadrupole–quadrupole – as well as the mixed polarizabilities – G˛ˇ , electric dipole-magnetic dipole, D,˛ˇ , magnetic dipole-electric quadrupole and the magnetizability (magnetic dipolemagnetic dipole polarizability) Ÿ˛ˇ . The frequency-dependent response properties labeled with a prime vanish for ¨ D 0, and the properties specified in the second rows of the equations (Eqs. 75, 77, 79) are zero for reference states described by real wave functions (in practice, closed-shell nondegenerate systems). We note that in the expansion of the induced dipole moment, higher-order corrections may also be important, in particular in connection with electric-field perturbations. For instance, considering only electric dipole contributions, it is customary to also include contributions to ˛ arising from frequency-dependent hyperpolarizabilities (see Eq. 84 below).

Electric Properties In this section we consider molecular properties which characterize the interactions with static and/or frequency-dependent electric fields. The electric properties of a molecule determine the electric properties of the bulk sample, such as the relative

´ et al. M. Jaszunski

20

permittivity (dielectric constant) and the refractive index. In addition, the electric properties can be used to describe intermolecular forces. Only static and dynamic molecular properties involving electric dipole and quadrupole operators will be discussed below. However, electric properties related to higher-order electric multipole operators can also be determined in a similar manner to the properties described here, in terms of expectation values, linear and nonlinear response functions. Nevertheless, it should be kept in mind that although the same formalism is applied in the calculation of response functions involving octupole, hexadecapole, and higher moments, in practice it may be difficult to obtain accurate results for these moments and related molecular properties. For instance, significantly larger basis sets with more polarization and diffuse functions may be required than for the properties involving lower-order electric moments. Most theoretical results are obtained in atomic units (a.u.), whereas experimental data often are given in SI or esu units. In Table 1 some useful conversion factors are given.

Electric Multipole Moments The permanent multipole moments characterize the charge distribution in a molecule. The zeroth-order (monopole) moment corresponds to the total charge, the first moment to the dipole, the second to the quadrupole, and so on. The multipole moments can be computed as expectation values of the multipole moment operators discussed in section “Small Terms Due to the Scalar Potential in the Hamiltonian.” As for the operators, we can use the primitive or traceless forms for the quadrupole and higher-order moments. The traceless multipole moments describe the deviation from an isotropic charge distribution. For instance, the traceless quadrupole moment vanishes for a spherical charge distribution when the center of the sphere coincides with the origin, since the transformation from the primitive (traced) quadrupole moment q˛ˇ to the traceless quadrupole moment ‚˛ˇ eliminates the isotropic Table 1 Unit conversion factors for selected electric properties F ‚ ’ “ ” VK

1 a.u. Eh e 1 a01 ea0 ea20 e 2 a02 Eh1 e 3 a03 Eh2 e 4 a04 Eh3 Eh e 1 a02

SI 5.142 206  1011 8.478 353  1030 4.486 551  1040 1.648 777  1041 3.206 362  1053 n 6.235 381  1065 9.717 365  1021

V m1 Cm C m2 C2 m2 J1 C3 m3 J2 C4 m4 J3 V m2

esu 1.715 3  107 2.541 7  1018 1.345 0  1026 1.481 8  1025 8.639 2  1033 5.036 7  1040 3.241 4  1015

statvolt cm1 statvolt cm2a statvolt cm3 cm3 statvolt1 cm4 statvolt2 cm5 statvolt cm2b

1Debye D 1018 statvolt cm2 The nuclear quadrupole moments, needed to compare the calculated NQCC with experiment (section “Electric Field Gradient at the Nucleus, Nuclear Quadrupole Coupling Constant”), are usually given in barn; 1b D 1028 m2 a

b

Molecular Electric, Magnetic, and Optical Properties

21

contribution. The traceless multipole moment of order n is defined by 2n C 1 components, whereas the traced tensor involves (n C 2)(n C 1)/2 independent quantities. A tabulation of nonzero symmetry-independent elements of multipole moments ( D dipole, ‚ D quadrupole, ˝ D octupole, and ˚ D hexadecapole) for different point groups is given by Kielich (1972). In general, when the multipole expansion of the potential is used to describe the influence of a static electric field, only the first nonzero moment is unambiguously defined. The next, higher-order multipole moments depend on the choice of origin, and when computed with respect to the origin of the coordinate system, they depend on the position of the molecule in space. This can be seen from Eq. 30, where a shift of the chosen common origin for the expansion of the potential by a vector D for a molecule with the total charge Q leads to a change of QD in the dipole moment; similarly, for a neutral molecule with a nonzero dipole moment, the quadrupole moment will change.

Dipole and Quadrupole Moments From the multipole expansion of the energy in the presence of a static electric field, Eq. 73, we find that the ’-component of the static permanent dipole moment is defined as the electric field derivative of E(F) ˇ @E .F/ ˇˇ ˛ D  : (80) @F˛ ˇFD0 For wave functions that fulfill the Hellman–Feynman theorem we can compute the dipole moment as the expectation value ˛ D h0 j O ˛ j 0i ;

(81)

where O ˛ is the ˛-component of the dipole moment operator defined in Eq. 30. Obviously, the expectation value expression is correct for the exact wave function. Furthermore, for most approximate wave functions it is much easier to compute the expectation value than to calculate the energy derivative. However, the results differ when the method used does not fulfill the Hellman–Feynman theorem. In this case, one can either derive the analytic expressions which reproduce the correct definition of the electric field derivative of E(F) (see for instance the discussion of the coupled cluster approach in Helgaker et al. 2000) or use the finite field method and numerical differentiation. The finite-field method appears to be very simple and has been often applied, but it has some drawbacks. First, the values of E(F) depend not only on the dipole moment, but also on the polarizability and hyperpolarizabilities, and high accuracy of the computed values of E(F) is required to perform the numerical differentiation accurately. Moreover, the presence of the finite electric field may lower the symmetry of the system, making the calculations much more timeconsuming.

´ et al. M. Jaszunski

22

MP2 / aug-cc-pVDZ

50

–0.1

50

0

0.1

–0.1

50

0

CCSD / aug-cc-pVDZ

–0.1

0

0.1

0

–0.1

0.1

CCSD / aug-cc-pVQZ

50

50

0

CCSD(T) / aug-cc-pVDZ

0.1

–0.1

0

0.1

CCSD(T) / aug-cc-pVTZ

50

–0.1

0.1

CCSD / aug-cc-pVTZ

50

–0.1

MP2 / aug-cc-pVQZ

MP2 / aug-cc-pVTZ

50

0

0.1

–0.1

0

0.1

Fig. 1 Correlation and basis set dependence of the dipole moment – normal distribution of errors (in Debye) (Reproduced by permission from Bak et al. (2000))

Figure 1, taken from Bak et al. (2000), illustrates the accuracy of equilibrium geometry dipole moments calculated for a series of small molecules. In Bak et al. (2000), the dipole moments of 11 molecules were studied at various levels of approximation using a sequence of basis sets, with the CCSD(T)/aug-cc-pVQZ results providing the benchmark values. The normal distribution of errors with respect to the reference values, shown in Fig. 1, demonstrates the systematic improvements of the computed results with increasing basis set and improved treatment of electron correlation effects. O that is, the traceless electric quadrupole operator as The expectation value of ‚, defined in Eq. 32 following Buckingham (1967), D E O ˛ˇ j 0 D 3 @E .rF/ ; ‚˛ˇ D 0 j ‚ @F˛ˇ

(82)

Molecular Electric, Magnetic, and Optical Properties

23

˝ ˇ ˇ ˛ can be related to the expectation values of second moments, q˛ˇ D 0 ˇqO ˛ˇ ˇ 0 , by a simple linear transformation, the same as that connecting the corresponding operators, Eqs. 32 and 33. Both ‚ and q are rank two, symmetric tensors. For molecules with an n-fold rotation axis with n  3: ‚  ‚zz D 2‚yy D 2‚xx ;

(83)

where z coincides with the rotation axis. In experiment, two diagonal components of ‚ are measured.

The Origin of the Frequency-Dependent Electric Dipole Polarizability and Hyperpolarizabilities The electric dipole moment Q in the presence of an applied spatially homogeneous non-monochromatic electric field †i F!i (where ¨i indicates the circular frequency) can be written as .! D 0/ C ˛˛ˇ .! I !1 / Fˇ .!1 / Q !˛  D ˛ ıX ˇ˛ˇ .! I !1 ; !2 / Fˇ .!1 / F .!2 / C 12 !1X ;!2 C 16 ˛ˇı .! I !1 ; !2 ; !3 / Fˇ .!1 / F .!2 /Fı .!3 / C    ;

(84)

!1 ;!2 ;!3

where ˛ is the unperturbed electric dipole moment. In this phenomenological expansion (which we use to describe elastic light scattering processes) the summations over optical frequencies will be restricted in order to preserve a constant ¨¢ equal to the sum of optical frequencies ! D

X

!i :

(85)

i

Equation 84 defines the components of the dynamic electric dipole polarizability tensor, ’ (¨¢ ; ¨1 ), those of the first electric dipole hyperpolarizability, ˇ  (¨¢ ; ¨1 , ¨2 ), and those of the second electric dipole hyperpolarizability   (¨¢ ; ¨1 , ¨2 , ¨3 ). The combination of frequencies and the presence or absence of static fields characterize various nonlinear processes. The calculations can be done for any combination of frequencies satisfying Eq. 85, but only selected examples of the most commonly studied optical phenomena will be discussed below (see, e.g., Willets et al. 1992 and Bogaard and Orr 1975 for more details). Figure 2 shows how some of these processes can be interpreted in terms of electronic transitions between ground and excited states. The excited states can be treated as virtual states, but if one of the excitation energies is close to the frequency of the light, the contribution of that state is often dominant. Simplified models that include in the analysis only a few excited states can be used when these computed states describe well the real, physical states of the molecule which are relevant for the property studied. However, it is important to emphasize that

´ et al. M. Jaszunski

24

ω3 ω2 ω

ω

(ω1+ω2+ω3)

ω1

ω2

ω2 ω3

(ω1+ω2)

ω1

ω1 (ω1+ω2–ω3)

ω1

ω2

(ω1–ω2) ω1

ω2

ω3 (ω1–ω2+ω3)

Fig. 2 Linear and nonlinear optical properties – a schematic illustration of the involved transitions for (left, top to bottom): ’(¨; ¨); “((¨1 C ¨2 ); ¨1 , ¨2 ); and “((¨1  ¨2 ); ¨1 , ¨2 ); (right, top to bottom) ”((¨1 C ¨2 C ¨3 ); ¨1 , ¨2 , ¨3 ); ”((¨1 C ¨2  ¨3 ); ¨1 , ¨2 , ¨3 ), and ”((¨1  ¨2 C ¨3 ); ¨1 , ¨2 , ¨3 )

even though these few-states models in many cases may give qualitative insight into the electronic processes, the convergence of the sum-over-states expansion for the (hyper)polarizabilities (see Eqs. 65, 68, and 72) with respect to the number of excited states included in the summation is in general very slow (Wiberg et al. 2006). We should also keep in mind that if an excitation energy equals a frequency, we have a very different situation in which we have to consider the finite lifetime of the excited state and also deal with the absorption process. In experiment, the molecule is often perturbed by a combination of a static and a dynamic electric field. We then have, assuming a monochromatic dynamic field, F˛ .t / D F˛0 C F˛! cos .!t /; and therefore the time dependence of the polarization can be described as

(86)

Molecular Electric, Magnetic, and Optical Properties 3! ˛ .t / D 0˛ C !˛ cos .!t / C 2! ˛ cos .2!t / C ˛ cos .3!t / C    ;

25

(87)

where n! ˛ are the Fourier amplitudes to be determined. In order to compare the measured polarization at a certain frequency to the theoretical expressions, we insert Eq. 86 into Eq. 84 and use the relations between cosn (!t) and cos(n!t). We obtain 0˛ D ˛ C ˛˛ˇ .0I 0/ Fˇ0 C 12 ˇ˛ˇ .0I 0; 0/ Fˇ0 F0 C 16 ˛ˇı .0I 0; 0; 0/ Fˇ0 F0 Fı0 C 14 ˇ˛ˇ .0I !; !/ Fˇ! F! C 14 ˛ˇı .0I 0; !; !/ Fˇ0 F! Fı! ; (88) !˛

2! ˛

D

˛˛ˇ .!I !/ Fˇ!

C

ˇ˛ˇ .!I !; 0/ Fˇ! F0

C 12 ˛ˇı .!I !; 0; 0/ Fˇ! F0 Fı0 C 18 ˛ˇı .!I !; !; !/ Fˇ! F! Fı! ; (89) 1 1 D ˇ˛ˇ .2!I !; !/ Fˇ! F! C ˛ˇı .2!I !; !; 0/ Fˇ! F! Fı0 ; (90) 4 4 1 ˛ˇı .3!I !; !; !/ Fˇ! F! Fı! ; (91) 3! ˛ D 24

where we have truncated the expansions at cos (3!t). Using the expansions given above and the expansion in Eq. 84 we can compare the experimentally measured polarization and the theoretical estimates.

Dipole Polarizability The frequency-dependent electric dipole polarizability is, next to the dipole moment, the most important effect characterizing the response of a molecule to electromagnetic radiation. It is a second-order symmetric tensor determined by the linear response function ˛˛ ˝˝ ˛˛ˇ .!I !/ D  O ˛ I O ˇ ! :

(92)

The frequency dependence of the polarizability may be described using the Cauchy moments (section “Cauchy Moments”). For the static dipole polarizability ’(0;0) we have ˛˛ˇ .0I 0/ D 

ˇ @2 E .F/ ˇˇ ; @F˛ @Fˇ ˇFD0

(93)

and the energy derivative can be evaluated using the finite-field approach (i.e., computing numerically the differences between E(F) and E(0)). Numerous molecular properties which describe nonlinear effects, such as the Kerr effect (section “Second Dipole Hyperpolarizability”) or magnetic circular

´ et al. M. Jaszunski

26

dichroism (section “Magnetic Circular Dichroism”), arising in the presence of radiation and additional electric or magnetic fields, are interpreted as derivatives of the dipole polarizability (Michl and Thulstrup 1995). They can be calculated as higherorder response functions. Similarly, relativistic corrections to the polarizabilities for heavy atoms can be estimated from higher-order response functions including the mass-velocity and Darwin operators, Eqs. 9 and 20, as additional perturbations (Kirpekar et al. 1995). In isotropic fluids, the experimentally measured quantity is usually the scalar component of the ’ tensor given by the isotropic average, defined as ˛iso D

1 ˛ : 3

(94)

The polarizability anisotropy may be defined as .˛anis /2 D

.3˛ ˛ ˛ ˛ / 2

;

(95)

which, using the components of the diagonalized tensor, leads to h i1=2 2  2 ˛anis D 21=2 ˛xx  ˛yy C ˛yy  ˛zz C .˛zz  ˛xx /2 :

(96)

For linear molecules, ˛ anis reduces to the difference between the parallel and perpendicular components, ˛anis D ˛k  ˛? :

(97)

The macroscopic property related to the molecular polarizability ’ is the dielectric constant ", defined by the ratio of the permittivity of the medium to the electric

p constant, "0 . It is also related to the refractive index n n D –; assuming the vacuum magnetic permeability to be equal to 1). The quantity measured in a dielectric constant experiment is  ˛Q D

Q FZ F

 D h˛II i C F !0

2z ; 3kT

(98)

where we use the notation < : : : > for the average over all the orientations, k is the Boltzmann constant, and T is the temperature. The temperature-dependent term is proportional to the square of the dipole moment, the temperature-independent term is the isotropic average of the polarizability (using Eq. 94 we can obtain a corresponding expression in terms of properties calculated in the molecular frame of reference). The anisotropy of the electric dipole polarizability can be determined in Rayleigh scattering experiments, where the observed depolarization of the incident light is a function of (˛ anis )2 (Bonin and Kresin 1997).

Molecular Electric, Magnetic, and Optical Properties

27

The static electric dipole polarizability of the He atom has been studied in more detail than any other response property of any other atomic or molecular system. Not only is the nonrelativistic value known accurately, and the relativistic corrections computed, but – in a recent benchmark calculation – the quantum-electrodynamic corrections have also been analyzed and numerous effects related to the nuclear mass have been taken into account (Łach et al. 2004). Standard energy-optimized basis sets are not suitable for accurate calculations of electric polarizabilities. The simplest solution – adding the necessary polarization and diffuse functions – makes the basis sets too large to enable efficient calculations for large molecules. Significantly smaller basis sets, designed considering the electric-field dependence of the orbitals (Benkova et al. 2005; Sadlej 1988), provide results of similar or better quality at lower computational cost. A practical problem that may be encountered in calculations using very extended basis sets is the appearance of linear dependencies in the basis set. This may make the response equations (as well as the equations determining the molecular energy/density) ill-conditioned and the calculations slowly convergent; in exceptional cases it may even be impossible to converge the equations. The problem is normally resolved by removing the linear dependencies in the basis set, or by removing manually the most diffuse basis functions from the basis set. Electron correlation effects on the isotropic polarizability are in general moderate, being at the most 5–10 % of the uncorrelated value and for nondipolar systems always leading to an increased polarizability. Correlation effects are usually larger on the individual tensor components than on the isotropic polarizability, and can, therefore, be expected to be more important for the polarizability anisotropies. In general, Hartree–Fock polarizability anisotropies cannot be considered accurate enough to allow for a quantitative comparison with experimental observations, or for the interpretation of the ellipticity arising in some birefringences, where the polarizability anisotropies often constitute an important (and dominant) orientational contribution. DFT in general reproduces quite well correlation effects in the polarizability of small molecular systems. In particular, polarizability anisotropies calculated using DFT will usually be in better agreement with experiment. However, due to the local nature of the exchange-correlation functionals and the lack of general implementations of current density functionals, DFT does not perform so well for extended conjugated systems (Champagne et al. 2000). We will return briefly to this point when discussing calculations of hyperpolarizabilities.

First Dipole Hyperpolarizability The first electric dipole hyperpolarizability, given by the quadratic response function ˛˛ ˝˝ ˇ˛ˇ .! I !1 ; !2 / D  O ˛ I O ˇ ; O  !1 ;!2 ; has physical meaning only when ¨¢ D ¨1 C ¨2 .

(99)

´ et al. M. Jaszunski

28

The static dipole hyperpolarizability “(0;0,0) can be obtained evaluating the third derivative of the energy. When we consider dynamic fields, the processes of particular interest are: Second Harmonic Generation (SHG) dc-Pockels Effect (dc-P) or Electro-Optic Pockels Effect (EOPE) Optical Rectification (OR)

˛˛ ˝˝ SHG .2!I ˇ O v;  O  !; !; !; !/ D   O I  ˛˛ ˝˝ EOPE .!I ˇ O v;  O  !;0 ; !; 0/ D   O I  ˛˛ ˝˝ OR .0I ˇ O ;  O  !;! : !; !/ D   O I 

According to Eqs. 88, 89, 90, and 91, in these processes we deal with a perturbation of frequency ¨ (and a static field in the Pockels effect) and we consider the resulting dipole moment oscillating with frequency 2¨, frequency ¨ or static, respectively. The general definition of the quadratic response function, Eq. 68, indicates its symmetry with respect to permutation of operators. Thus, for all the dipole hyperpolarizabilities we have ˇ .! I !1 ; !2 / D ˇ .! I !2 ; !1 / :

(100)

ˇ .! I !1 ; !2 / D ˇ .!1 I ! ; !2 /

(101)

Also the relation

OR EOPE .0I !; !/ D ˇ .!I !; 0/. We note that holds, and we have, for instance, ˇ this relation does not hold for macroscopic samples due to the differences in the local fields experienced by the molecule in these two different experimental setups. In the case of the first hyperpolarizability, the measured quantity is the vector component of ˇ in the direction of the permanent dipole moment , defining the molecular z axis. The relevant averages are given by:

 1 ˇz C ˇz C ˇz ; 5

(102)

 1 2ˇz  3ˇz C 2ˇz ; 5

(103)

ˇk D ˇ? D

where the same sequence of optical frequencies (not given explicitly) is used for the laboratory axes and the molecular axes. The number of independent nonzero tensor elements depends on the nonlinear optical process and on the symmetry of the molecule, see for instance Bogaard and Orr (1975). For example, “SHG is symmetric with respect to the permutation of the second and third indices (see Eq. 100) and this can be used to simplify the equations for the parallel and perpendicular components. For nonzero frequencies,

Molecular Electric, Magnetic, and Optical Properties

29

the number of independent tensor components to be computed decreases when Kleinman’s symmetry (Kleinman 1972) is assumed – that is, we assume that we can permute the indices of the incoming light without changing the corresponding frequencies, ˇ .! I !1 ; !2 /  ˇ .! I !1 ; !2 / :

(104)

Although for ¨1 ¤ ¨2 this is only an approximation and these two tensor components are not equal, Kleinman’s symmetry is often applied for low frequencies, where it is usually found to be a reasonable approximation. The basis set requirements for the calculation of first hyperpolarizabilities are much the same as for the linear polarizability. However, as the first hyperpolarizability probes even higher-order electric-field-perturbed densities of the molecule, care should be exercised to ensure that the basis set is sufficiently saturated with respect to diffuse polarizing functions. Special basis sets have been developed for the calculation of hyperpolarizabilities by Pluta and Sadlej (1998), though the same basis sets that can be used for polarizabilities in most cases give reliable estimates also for first hyperpolarizabilities. Due to the fact that the first hyperpolarizability involves multiple virtual excited states (Eq. 68), the property has proven to be more sensitive to electron correlation effects than the polarizability (Christiansen et al. 2006). Molecules with push–pull conjugated structures are often characterized by large first hyperpolarizabilities. Due to the problems facing many commonly used local exchange-correlation functionals, DFT has been shown to have difficulties describing correctly nonlinear optical properties such as the first hyperpolarizability for extended systems (Champagne et al. 2000). The recently introduced Coulomb-attenuated B3LYP functional (CAMB3LYP), see Yanai et al. (2004) has, in contrast, shown quite good performance also for rather extended systems (Paterson et al. 2006).

Second Dipole Hyperpolarizability The second dipole hyperpolarizability is described by the cubic response function ˝˝ ˛˛ ˛ˇı .! I !1 ; !2 ; !3 / D  O ˛ I O ˇ ; O  ; O ı !1 ;!2 ;!3 ;

(105)

where ¨¢ D ¨1 C ¨2 C ¨3 . Similarly to the lower-order static polarizabilities, the hyperpolarizability ”(0;0,0,0) corresponds to an energy derivative and can be computed using the finitefield approach. Among the possible frequency-dependent third-order processes, the most relevant are: The experimentally measured quantities in isotropic fluids are usually the scalar components of the tensor ” given by the isotropic average:

´ et al. M. Jaszunski

30 Electric Field – Induced Second Harmonic Generation (dc-SHG, ESHG, EFISH) ESHG .2!I !; !; 0/ 

˝˝ ˛˛ D  O I  O ;  O ;  O !;!;0 ;

Third Harmonic Generation (THG) THG .3!I !; !; !/ 

˛˛ ˝˝ D  O I  O ;  O ;  O !;!;! ;

Electro-optical or dc-Kerr effect (EOKE, dc-Kerr) dc-Kerr .!I !; 0; 0/ 

˝˝ ˛˛ D  O I  O ;  O ;  O !;0;0 ;

Optical or ac-Kerr effect (OKE, ac-Kerr) ac-Kerr .!1 I !1 ; !2 ; !2 / 

˝˝ ˛˛ D  O I  O ;  O ;  O !1 ;!2 ;!2 ;

Intensity-Dependent Refractive Index or Degenerate Four Wave Mixing (IDRI or DFWM) IDRI .!I !; !; !/ 

˝˝ ˛˛ D  O I  O ;  O ;  O !;!;! ;

dc Optical Rectification (dc-OR, EFIOR) dc-OR .0I !; !; 0/ : 

˛˛ ˝˝ D  O I  O ;  O ;  O !;!;0 :

k D

 1    C    C   ; 15

(106)

 1  2     ; 15

(107)

? D

where the same sequence of optical frequencies, not given explicitly above, is used for the laboratory axes and the molecular axes. As for the first hyperpolarizability, the number of independent nonzero tensor elements depends on the optical process and on the symmetry of the molecule. For example, ” THG is symmetric with respect to the second, third, and fourth indices, and this can be used to simplify Eqs. 106–107. For instance, the average value that should be compared with the experimental THG parallel component becomes kTHG D

1 THG  : 5 

(108)

Experimentally, if all fields have parallel polarization, one can measure the parallel components of the first and second hyperpolarizabilities which take into account the classical orientational averaging. In the case of ESHG, with the optical field polarized perpendicular to the static field, one measures the perpendicular components, and in the case of a dc-Kerr experiment, the differences between the parallel and perpendicular components. In the ESHG experiment a laser beam passes through the sample in a static electric field and a weak, collinear, frequency-doubled beam is detected. Absolute values for the hyperpolarizabilities cannot be extracted; the signal from the sample is compared to that of a known buffer gas (ultimately helium, for which there are accurate theoretical values (Bishop and Pipin 1989)) or a solid. In analogy with the derivation for ˛, Q the classical thermal averaging yields

Molecular Electric, Magnetic, and Optical Properties

 F Q ESHG D ˇQZZZ =F

F !0

D hZZZZ i C

31

z hˇZZZ i : 3kT

(109)

The molecular hyperpolarizabilities  ESHG and ˇ SHG can be obtained in this experiment. For non-centrosymmetric molecules a series of measurements over a range of temperatures has to be performed, whereas for centrosymmetric molecules ˇ D 0 and thus a single measurement at one temperature is sufficient. The majority of existing experimental data on hyperpolarizabilities are derived from ESHG and dc-Kerr measurements (Shelton and Rice 1994). The dc-Kerr effect differs from the other nonlinear optical processes as it allows for absolute measurements without the need for a reference measurement. The measured molar Kerr constant is (Shelton and Rice 1994): Adc-Kerr D



3 .hZZZZ i  hZXXZ i / C z .hˇZZZ i  hˇXXZ i / =kT 2   D E 3 1 2 0 ˛˛ˇ ˛˛ˇ z .˛zz  h˛ZZ i / ; C  h˛ZZ i ˛ZZ ı C  10kT kT (110) NA 81–0

where NA is Avogadro’s number, and it involves both dynamic hyperpolarizabilities (” dcKerr and ˇ EOPE ) as well as static (superscript 0) and dynamic polarizabilities. The computational requirements for the second hyperpolarizability are more or less the same as for the first hyperpolarizability. Although the number of studies that analyze the importance of electron correlation effects (not including here DFT) is rather limited, the available results confirm in general the findings made for the first hyperpolarizabilities (Christiansen et al. 2006).

Cauchy Moments Response functions for a system in its electronic ground state are analytic functions of the frequency arguments, except at the poles that occur when a frequency or a sum of frequencies is equal to an excitation energy. Thus, for frequencies below the first pole, the linear, quadratic, or cubic response functions can be expanded in power series (Hättig and Jørgensen 1998). Let us consider in more detail the simplest case when only electric dipole operators are involved and the frequency dependence of the (hyper)polarizabilities is of interest. The Cauchy series – that is, the power series expansion of the frequency-dependent polarizability – is usually written as: ˛ .!I !/ D

1 X

! 2k S .2k  2/ D S .2/ C ! 2 S .4/ C ! 4 S .6/ C    ;

kD0

(111)

´ et al. M. Jaszunski

32

where S(2) D ˛(0; 0). This expansion is valid also for purely imaginary frequency arguments. Similar expansions may be applied to describe the dispersion effects for hyperpolarizabilities. Often, the power series expansion in the frequency arguments may be truncated at second order leading to:     ˛iso .! I !1 / D ˛iso .0I 0/ 1 C A˛ !2 C !12 C O !i4 ;

(112)

    ˇk .! I !1 ; !2 / D ˇk .0I 0; 0/ 1 C Aˇ !2 C !12 C !22 C O !i4 ;

(113)

    k .! I !1 ; !2 ; !3 / D k .0I 0; 0; 0/ 1 C A !2 C !12 C !22 C !32 C O !i4 : (114) Each expansion contains only a single second-order dispersion coefficient A. Since these coefficients for different optical processes of a given order are equivalent (Bishop 1995), they express the dispersion of frequency-dependent properties in a way that is transferable between different optical processes, for instance Aˇ describes both the second-harmonic generation and the (dc-)electro-optic Pockels effect.

Long-Range Dispersion Interaction Coefficients For neutral, closed-shell weakly interacting systems, one of the dominant contributions to the interaction energy at large intermolecular distances is the dispersion energy. It can be analyzed using perturbation theory, and for two atoms or molecules A and B, we find (McWeeny 1992):

AB Edisp D

ˇD ˇ ˇ E ˇ2 A X B ˇˇ 0A 0B ˇˇVO AB ˇˇnA mB ˇˇ X 1 „

n¤0m¤0

A B !n0 C !m0

;

(115)

where VO AB denotes the intersystem interaction, j 0A i and j nA i represent the ground and excited states of system A, ! An0 is the excitation energy, and similarly for system B. We would like to express the dispersion energy in terms of response properties of molecules A and B since this allows for a physical interpretation of the dispersion interaction. Such an expression is obtained applying an integral transform to avoid the summation of the A and B excitation energies in the denominator of Eq. 115 and using the multipole expansion of the perturbing operator VO AB . For the second-order dispersion interaction between closed-shell atoms A and B we have:

Molecular Electric, Magnetic, and Optical Properties

AB Edisp .R/ D 

AB C6AB C8AB C10     ; R6 R8 R10

33

(116)

where the coefficient for the dipole–dipole term can be written as (Langhoff et al. 1971): C6AB

3 D 2  .4–0 / „

Z

1

˛ A .i !I i !/ ˛ B .i !I i !/ d !:

(117)

0

This equation is known as the Casimir–Polder formula. The polarizability at imaginary frequencies, ’(i¨; i¨), can be obtained for instance using Eq. 111 with an imaginary value of ¨. The CnAB ; n > 6 coefficients depend on similar higher-order multipole polarizabilities of one or both atoms. The same approach can be applied for atom–molecule and molecule–molecule interactions considering individual tensor components of the required polarizabilities. It can also be used for three-body interactions, and to describe the dispersion contributions to the pair polarizability function (Fowler et al. 1994). To determine the long-range dispersion coefficients we need the values of ’(i¨; i¨) for the whole range of imaginary frequencies, whereas the radius of convergence of the power series expansion of the polarizability (Eq. 111) is determined by the lowest excitation energy. However, using analytic continuation of the Cauchy expansion (Langhoff and Karplus 1970) we can obtain all the required values of ’(i¨; i¨) and thus we can determine accurate values of the dispersion interaction coefficient CAB 6 . The polarizabilities at imaginary frequencies can also be obtained directly, solving the complex linear response equations (Norman et al. 2003). Qualitatively correct estimates of the dispersion energy are obtained in the perturbational approach even using the dispersion coefficients Cn calculated at the DFT level. It has been shown that a successful description of intermolecular forces for large molecules may be obtained in this manner (Misquitta et al. 2005; Podeszwa and Szalewicz 2008).

Electric Field Gradient at the Nucleus, Nuclear Quadrupole Coupling Constant K The electric field gradient (EFG) tensor at the nucleus K  V˛ˇ – corresponds to the expectation value (Baker et al. 1989; Fowler et al. 1989) of the EFG operator VO K given by Eq. 34:

D E K K V˛ˇ D 0 j VO˛ˇ j0 ;

(118)

where the traceless form of the operator VO K is used. The components of the diagonalized EFG tensor, given in its own (EFG) principal axis system, fulfill therefore the equation

´ et al. M. Jaszunski

34 K K VzzK C Vyy C Vxx D 0:

(119)

They are by definition arranged so that K K j VzzK jj Vyy jj Vxx j:

(120)

For the asymmetry parameter K defined as K D

K K  Vyy Vxx

VzzK

;

(121)

we thus have 0  ˜K  1. On the other hand, for comparison with experimental data derived from the hyperfine structure of rotational spectra, tensor components defined in the principal axes of inertia system must be used. For quadrupolar nuclei – that is, nuclei for which the nuclear spin quantum number IK  1, the nuclear quadrupole coupling tensor is defined as eQK VK =„, where eQK is the electric quadrupole moment of the nucleus. There is an interaction of the nuclear quadrupole moment tensor with the EFG at the nucleus, mediated by the nuclear spin IK and described by the operator: HNq D

eQK 1X IT VK IK : 2 K IK .2IK  1/ K

(122)

The nuclear quadrupole coupling constant (NQCC) is defined as eQK VzzK =„. The EFG at the nucleus affects the NMR relaxation processes, thus it is relevant in the analysis of the linewidths of NMR signals. The linewidth is proportional to the inverse relaxation time (Abragam 1961), that is: 1=2 /

   2K eQK K 2 1 3 2IK C 3 V 1 C D c ; Tq 40 IK2 .2IK  1/ 3 „ zz

(123)

where  1/2 is the width of the peak at half of the maximum intensity and £c is the correlation time for the quadrupole coupling relaxation. From the formal 1/r3K dependence of the operator in Eq. 34, we realize that the basis used to calculate this property must be flexible in the core region and the outer-core/inner-valence region. This is an important requirement, as many modern basis sets are quite heavily contracted in these regions (since the orbitals do not change much in this region when going from the atomic to the molecular system). In general, much of the same basis set requirements as those arising in the calculation of nuclear shielding constants (and to some extent also the spin–spin coupling constants) apply, and we refer to section “Nuclear Spin–Related Properties” for further discussion.

Molecular Electric, Magnetic, and Optical Properties

35

An important application of theoretically calculated electric field gradients at the nucleus is in the determination of nuclear quadrupole moments. The interaction between the nuclear quadrupole moments and the electric field gradient at the nucleus can be measured with unprecedented accuracy (compared to other experimental approaches for determining nuclear quadrupole moments) in microwave spectroscopy. These interactions give rise to the fine structure in the rotational spectrum of the molecule. Highly accurate estimates for the electric field gradient at the nucleus can often be calculated theoretically (since this is a first-order molecular property). By combining these theoretical results with experimental observations, accurate values of the nuclear quadrupole moments can be obtained, and this approach has been used to revise previously estimated nuclear quadrupole moments (see, e.g., Kellö and Sadlej 1998).

One-Photon Absorption, Excitation Energies, and Transition Moments Calculations of one-photon absorption enable the determination of the extinction coefficient ", describing the attenuation of the intensity of an incoming light beam when passing through a sample of absorbing species. In the lowest-order perturbative expansion of the interaction between light and matter, " is related to the excitation energy and h0 j  j ni, the transition moment between the ground j 0i and excited j ni electronic states. The derived quantities are the dipole strength n

˝ ˛ D˛ˇ D h0 j O ˛ j ni n j O ˇ j 0 ;

(124)

and the oscillator strength. In the length gauge, the latter can be written as r f0n D

2me !0n 2me lim .!  !0n / !hhO ˛ I O ˛ ii! : h0 j O ˛ j ni hn j O ˛ j 0i D 2 3„e 3e 2 !!!0n (125)

Using the hypervirial relationship h0 j pO˛ j ni D i me !0n h0 j rO˛ j ni ;

(126)

we obtain in the velocity gauge v f0n D

2 2 lim .!  !0n / ! 1 hhpO˛ I pO˛ ii! ; h0 j pO˛ j ni hn j pO˛ j 0i D 3„me !0n 3me !!!0n (127)

and in the mixed length-velocity gauge

´ et al. M. Jaszunski

36

rv f0n D

2i 2i lim .!  !0n / hhO ˛ I pO˛ ii! ; h0 j O ˛ j ni hn j pO˛ j 0i D  3„e 3e !!!0n (128)

respectively (Hansen and Bouman 1980). To illustrate the differences between oscillator strengths computed in different 1 approaches, we present in Table 2 the results for the 11 †C g ! 1 …u singlet transition in the N2 molecule. The HF results and the results for the hierarchy of coupled-cluster approximations (CCS, CC2, CCSD and CC3) show the role of the wave function model. In addition, the role of the basis set is shown in the sequence of the results for the d-aug-cc-pV XZ correlation-consistent basis sets (Kendall et al. 1992), with X D D, T, Q, and 5, abbreviated in the table as daDZ, daTZ, daQZ, and da5Z, respectively. At the HF level the hypervirial relationship (Eq. 126) is fulfilled in complete basis sets and the differences between computed oscillator strengths are only due to the incompleteness of the basis set. We observe that all the HF results converge with X, and the da5Z values are almost identical. In the CC methods, in rv r addition to the differences arising due to basis set incompleteness, f 0n , f v0n , and f 0n may differ even for a complete basis set since Eq. 126 is not satisfied. This difference should be diminishing as we approach the full CI limit, and in fact for a given rv r one-electron basis the differences between f 0n , f v0n , and f 0n follow the sequence CCS > CC2  CCSD > CC3. Finally, we note that for evaluating the accuracy of the results the gauge invariance of a calculated oscillator strength is a necessary, but not sufficient requirement. Gauge invariance has been obtained for large basis sets both at the HF and at the CC3 level, whereas the correlated values differ significantly from the HF results, and only the CC3 results can be expected to be of high accuracy. The Thomas–Reiche–Kuhn (TRK) sum rule, which can be written in terms of the dipole oscillator strengths as X

f0n D Nel ;

(129)

n

where Nel is the total number of electrons, can be used as another test of the accuracy of the calculations. Of particular concern are Rydberg states, characterized by a very diffuse orbital occupied by the excited electron. DFT in general fails to properly describe Rydbergexcited states, as many of the common exchange-correlation functionals do not display the correct asymptotic behavior for the  1/r operator. This deficiency of modern exchange-correlation functionals can be partially rectified by introducing the correct asymptotic behavior for the exchange-correlation functional (Tozer and Handy 1998). It is worth noting that the time-dependent Hartree–Fock (TDHF) theory does not display the same problems in describing Rydberg states, due to use of the exact exchange operator. However, the lack of electron correlation effects in TDHF may limit the applicability of this approach for excited states in general.

Molecular Electric, Magnetic, and Optical Properties

37

Table 2 Oscillator strength and excitation energy (¨0n , in eV) for the singlet transition 11 †C g ! 11 …u of N2 ; results taken from Pawłowski et al. (2004) Model HF

CCS

CC2

CCSD

CC3

Basis set daDZ daTZ daQZ da5Z daDZ daTZ daQZ da5Z daDZ daTZ daQZ da5Z daDZ daTZ daQZ da5Z daDZ daTZ daQZ da5Z

f r0n 0.1669 0.1702 0.1709 0.1714 0.1629 0.1662 0.1670 0.1675 0.1202 0.1239 0.1248 0.1263 0.1671 0.1757 0.1794 0.1822 0.1844 0.2046 0.2141 0.2196

f v0n 0.1725 0.1724 0.1717 0.1715 0.1485 0.1504 0.1511 0.1512 0.1144 0.1211 0.1230 0.1238 0.1617 0.1729 0.1769 0.1788 0.1806 0.2045 0.2147 0.2194

f rv 0n 0.1697 0.1713 0.1713 0.1715 0.1556 0.1582 0.1589 0.1592 0.1172 0.1225 0.1239 0.1251 0.1644 0.1743 0.1782 0.1805 0.1825 0.2046 0.2144 0.2195

v r f0n  f0n 0.0055 0.0022 0.0008 0.0001 0.0144 0.0158 0.0158 0.0163 0.0058 0.0028 0.0018 0.0025 0.0055 0.0029 0.0025 0.0034 0.0037 0.0002 0.0006 0.0002

¨0n 13.26 13.22 13.22 13.22 13.27 13.24 13.23 13.23 12.42 12.58 12.65 12.69 12.80 13.00 13.08 13.11 12.71 12.87 12.92 12.94

To account for the broadening of the experimental spectrum (related not only to the finite lifetime of the electronic state, but also to the rovibrational structure, collisions, and other aspects of the interaction between light and matter), and in particular to investigate whether certain bands may be hidden in the experimental spectrum due to overlapping bands, simple Lorentzian line broadening is often added in the form L ./ 

 1 Q ;  .0  /2 C Q 2

(130)

where Q is related to the lifetime of the excited state, and 0 is the frequency of the electronic excitation. Assuming an isolated absorption band, Q is equal to half the width of the absorption band at half height (the value of Q is often adjusted in the calculations to fit the experimental data). It is common also to employ a Gaussian function to reproduce the line broadening L ./ D

2 2 2 p e 2.0 / = ;  2

(131)

´ et al. M. Jaszunski

38

and we have Q D 

r

 ln 2=2 .

Finally, we recall that the transition moment between two excited states hm j A j ni can be obtained using only the reference state wave function from a double residue of the quadratic response function Eq. 70 (and if j m iDj ni we may in this way determine the expectation value of A in the excited state).

Two-Photon Absorption In experimental studies of two- and multi-photon absorption processes, the multiphoton transition strength, a function of all the frequencies of all photons absorbed, is analyzed. For two laser sources of circular frequency ¨ 1 and ¨ 2 with associated wavelengths œ1 and œ2 , the two-photon transition strength •(¨1 , ¨2 ) for the transition between states j 0i and j ni in isotropic samples is given by, see e.g., Craig and Thirunamachandran (1984): on; 0n; TPA 0n 0n .!1 ; !2 / D FS˛˛ .!1 ; !2 / Sˇˇ .!1 ; !2 / C GS˛ˇ .!1 ; !2 / S˛ˇ .!1 ; !2 / C ı0n 0n; 0n C HS˛ˇ .!1 ; !2 / Sˇ˛ .!1 ; !2 / ; (132)

where F; G, and H are numbers depending on the polarization state of the two photons and on the geometrical setup (mutual direction of the laser beams) (Craig and Thirunamachandran 1984; McClain 1974) and, in the dipole approximation, 0n S˛ˇ

1X .!1 ; !2 / D „ m

(

  .O ˛ /0m O ˇ mn !m0  !1

 C

O ˇ

 0m

.O ˛ /mn

!m0  !2

) (133)

is the second-rank, two-photon tensor. In Eq. 133 the summation runs over the whole set of excited states, the energy conservation relation ¨1 C ¨2 D ¨n0 applies, and off-resonance conditions are implied – that is, the frequencies ¨1 and ¨2 are sufficiently far off the values at which the denominators vanish. The tensor is nonsymmetric in the exchange of the two frequencies except for ¨1 D ¨2 . For the special case of a one-color beam – that is, a monochromatic light source – the transition matrix is symmetric and (using ¨ D ¨1 D ¨2 ) 0n; TPA 0n 0n; 0n .!/ D S .!/ S .!/ C 2S .!/ S .!/ ı0n

(134)

for linear polarization of the incident light and 0n; TPA 0n 0n; 0n .!/ D S .!/ S .!/ C 3S .!/ S .!/ ı0n

for circular polarization, respectively.

(135)

Molecular Electric, Magnetic, and Optical Properties

39

It can be shown that the two-photon absorption transition rate (cross section) can be obtained from the single residue of the cubic response function (Hättig et al. 0n .!1 ; !2 /, Eq. 133, can 1998a). Two-photon absorption transition amplitudes S˛ˇ also be extracted from the single residue of a quadratic response function, see Eq. 69 above. The following relations hold for the matrix elements of the two-photon transition tensor 0n 0n .!/ D Sˇ˛ .!n0  !/ ; S˛ˇ

(136)

0n n0 .!/ D S˛ˇ .!/ ; S˛ˇ

(137)

For methods which do not fulfill Eq. 137 (such as the coupled cluster approach), one can instead use the two-photon transition strength (Hättig et al. 1998b): 0n .!/ D F˛ˇ;ı

o 1 n 0n 0n; 0n; n0 .!/ C Sı .!/ S˛ˇ .!/ ; S˛ˇ .!/ Sı 2

(138)

n0 where S0n ˛ˇ (!) and S˛ˇ (!) are the left and right transition moments. Three-photon absorption has been described through the single residue of the cubic response function (Cronstrand et al. 2003). Similarly to two-photon absorption, one can discuss the third-order (left and right) transition moments and strengths (Hättig et al. 1998a) and define the three-photon transition strengths in this manner for nonvariational wave functions. There has been a rather limited number of theoretical studies of three-photon absorption processes and these will therefore not be discussed any further here. In general, the computational requirements for the calculation of molecular properties from the residues of the response functions inherit the requirements from the response functions themselves; that is, the selection of the basis sets has to be done considering the operators appearing in the expression for the transition moments and excited-state properties. However, as the residues are connected to specific excited states, the nature of the probed excited state also needs to be considered.

Magnetic Properties Let us now consider perturbations involving external static magnetic fields only. In this case, the number of observable molecular properties that arise is rather limited. This is due in part to the fact that the strength of the magnetic field perturbation of the molecular energy is much smaller than the electric field perturbational strength, but also because only even-order interactions will survive for closedshell molecules. This is normally referred to as quenching of the induced magnetic moment, and can also be seen as an explanation why there are no permanent magnetic dipoles in such systems. The quenching of the magnetic moment in closed-

´ et al. M. Jaszunski

40 Table 3 Unit conversion factors for magnetizability Ÿ

1 a.u. .e„=me /2 Eh1

SI 7. 89104  1029

JT2

CGS 4.7521

ppm cgs/ppm cm3 mol1

shell molecules can easily be understood by considering that for any orbitally nondegenerate state, one can always choose the wave function to be real. For such a state, the expectation value of the magnetic moment is given as D

E D E D E 0 j HO ˛B j 0 D 0 j HO ˛B j 0 D  0 j HO ˛B j 0 D 0;

(139)

since the magnetic dipole operator is Hermitian and purely imaginary (see Eq. 44).

Magnetizability The magnetizability yields the first nonvanishing (second order in the external magnetic field) contribution to the energy of interaction between an external magnetic field and a closed-shell molecule 1 E .B/ D  B˛ ˛ˇ Bˇ ; 2

(140)

where Ÿ is a symmetric tensor of rank two with a total of six independent elements. For isotropic samples, the magnetizability is expressed in the principal axis system in which the tensor is diagonal, and in addition to the isotropic magnetizability defined in a standard manner as iso D

1 ˛˛ ; 3

(141)

two anisotropies: anis1 D 33  . 11 C 22 / =2;

(142)

anis2 D 11  . 22 C 33 / =2;

(143)

are usually defined, with j 33 j  j 22 j  j 11 j. However, results reported in the literature often do not follow these definitions, and care should be exercised when comparing with experiment to ensure that the same conventions are used for defining the anisotropies. For unit conversion factors, see Table 3. Two contributions to the magnetizability appear in the nonrelativistic electronic Hamiltonian in the presence of a magnetic vector potential Eq. 40. One arises as an expectation value of the diamagnetic magnetizability operator, see Eq. 47.

Molecular Electric, Magnetic, and Optical Properties

41

The second involves a linear response contribution arising from the interaction of the magnetic dipole operator Eq. 44 with itself. We can, therefore, calculate the magnetizability from the expression: ˇ ˇ E DD EE D ˇ @2 E .B/ ˇˇ ˇ B;B ˇ B OB O 0  H D  ˛ˇ D 2 0 ˇHO ˛ˇ I H ; ˇ ˛ ˇ @B˛ @Bˇ ˇBD0

(144)

where the last equality shows that the magnetizability can be obtained as an energy derivative and it holds because the magnetic fields discussed here are static. A negative magnetizability corresponds to an induced magnetic moment of the molecule opposing the applied magnetic field, and is experimentally manifested by the molecule trying to get away from the poles of a magnet and thus align itself across the applied magnetic field, an effect that is referred to as diamagnetism. The opposite effect, where the induced magnetic moment enforces the magnetic field, is referred to as paramagnetism, and in this case the molecule will seek to orient itself in the direction of the poles of the magnet and therefore along the magnetic field. The diagonal elements of the first contribution to the magnetizability in Eq. 144 are negative since ˇ E D ˇ ˇ ˛ e 2 ˝ ˇˇ 2 ˇ B;B ˇ dia 0 rOO ı˛ˇ  rOO;˛ rOO;ˇ ˇ 0 ; ˛ˇ D 2 0 ˇHO ˛ˇ ˇ0 D  4me

(145)

and this part is for this reason referred to as the diamagnetic contribution. The linear response function contribution to the magnetizability (including the negative sign, see Eq. 144) is always positive and is therefore referred to as the paramagnetic contribution. For closed-shell molecules, the expectation value almost always dominates, and these molecules are therefore diamagnetic. Exceptions occur for molecules for which there are near degeneracies in the electronic ground state. Then the denominator of the linear response function (65) may become small, leading to such an increase of the magnitude of the paramagnetic term that the overall magnetizability becomes positive. A well known and often studied example is the BH molecule (Ruud et al. 1995; Sauer et al. 1993). Although the expression for the magnetizability in Eq. 144 is conceptually simple and can in principle be used straightforwardly, the gauge origin problem turns out to be difficult. In order to ensure gauge-origin independent magnetizabilities, as well as significantly improve the basis set convergence, London atomic orbitals should be used when calculating magnetizabilities. A London orbital is obtained from a conventional basis function by multiplying it by an imaginary phase factor   i ! .B; RM / D exp  eAM  r  .RM / : „

(146)

 is here an ordinary basis function, such as a Gaussian or a Slater-type basis function. The magnetic vector potential appearing in the imaginary phase factor is

´ et al. M. Jaszunski

42

defined as AM D

1 1 B  ROM D B  .RM  O/ : 2 2

(147)

The effect of this complex phase factor is to move the global gauge origin O to the “best” gauge origin for the basis function located at center RM , namely, the nucleus M to which the basis function is attached. When London atomic orbitals (LAOs, known also as gauge-including atomic orbitals – GIAOs) are used, the definition of dia- and paramagnetic contributions to the magnetizability in Eq. 144 can no longer be used. We can instead define (Ruud and Helgaker 1997): ˇ E D ˇ ˇ B;B ˇ dia .O/ D 2 0 ˇHO ˛ˇ .O/ˇ 0 ; ˛ˇ para;Lon

˛ˇ

Lon dia .O/ D ˛ˇ .O/ :  ˛ˇ

(148) (149)

This partitioning leads to dia- and paramagnetic contributions that display the correct gauge origin dependence also for finite basis sets, since the diamagnetic contribution, dia ˛ˇ in Eq. 148, is an expectation value, and therefore it has the correct origin dependence. Furthermore, expectation values in general show fast basis set convergence. Therefore the significant improvement in convergence of both the total magnetizability and the diamagnetic magnetizability ensures fast basis set convergence also for the paramagnetic contribution, Eq. 149. To illustrate the importance of using London orbitals when calculating magnetizabilities, we consider in Table 4 the magnetizability (and rotational g tensor, vide infra) of the PF3 molecule. We note that the isotropic magnetizability calculated using the aug-cc-pVDZ basis set and without London atomic orbitals is wrong by a factor of almost three compared to the estimated Hartree–Fock limit. As shown in Table 4, even when the aug-cc-pV5Z basis containing more than 500 basis functions is used, the results obtained using Eq. 144 are still more then 10 % off the results obtained with London atomic orbitals, which can be expected to be very close to the Hartree–Fock limit considering the excellent basis set convergence of the London atomic orbital magnetizabilities. In contrast, all the calculations using London orbitals are within 3 % of the LAO aug-cc-pV5Z results. More importantly, the anisotropy is off by an order of magnitude at the aug-cc-pVDZ level, and even has the wrong sign at the aug-cc-pVQZ basis (containing 345 basis functions) if London orbitals are not used. Computational studies have shown that unless the ground electronic state is nearly degenerate with the lowest excited states, electron correlation effects on the isotropic magnetizability are in general negligible. All published results of correlated calculations of the magnetizability of closed-shell molecules without close-lying excited states have found electron correlation effects to be less than 3 % in most cases, much smaller than any uncertainty in experimentally determined

Molecular Electric, Magnetic, and Optical Properties

43

Table 4 Basis set convergence of the SCF magnetizability (in a.u.) and rotational g tensor of PF3 calculated without and with London atomic orbitals, Ruud and Helgaker (1997)

Ÿ iso Ÿ gk g?

aug-cc-pVDZ No-Lon Lon 16.55 6.80 3.66 0.25 0.0845 0.0618 0.0890 0.0378

aug-cc-pVTZ No-Lon Lon 11.50 6.67 0.77 0.32 0.0159 0.0636 0.0162 0.0380

aug-cc-pVQZ No-Lon Lon 9.06 6.61 0.48 0.33 0.0181 0.0643 0.0181 0.0383

aug-cc-pV5Z No-Lon Lon 7.66 6.57 0.28 0.34 0.0461 0.0649 0.0274 0.0385

magnetizabilities for such molecular systems. Results within 1–2 % of the Hartree– Fock limit can be obtained using the very modest-sized aug-cc-pVDZ basis set of Woon and Dunning (1994). The important point in the construction of a suitable basis set is the inclusion of diffuse polarizing functions in order to ensure highly accurate magnetizabilities. Commonly adopted exchange-correlation functionals (e.g., B3LYP, BLYP) are in general less efficient in recovering correlation effects on molecular magnetic properties than they are in recovering correlation effects on vibrational spectra or electric properties. Indeed, in many cases DFT may perform worse than Hartree– Fock in the calculation of isotropic magnetizabilities. A set of new functionals designed specifically for the calculation of magnetic properties has been developed by Keal and Tozer (2003). However, recent CCSD benchmark calculations indicate that the results obtained with one of these functionals deteriorate when vibrational corrections and solvent effects are taken into account (Lutnæs et al. 2009). The isotropic magnetizabilities are in general very little affected by nonelectronic contributions such as zero-point vibrational corrections, dielectric medium effects, or weak intermolecular forces. The situation is very different for the magnetizability anisotropy, which may display both fairly large electron correlation effects as well as sizeable contributions from nonelectronic contributions. Although highly accurate theoretical calculations using coupled-cluster wave functions have demonstrated that correlation effects are in general small (Gauss et al. 2007), and that nonelectronic contributions are small as noted above, agreement with experimentally determined magnetizabilities is in general poor. Part of the reason for this is the general lack of experimental gas-phase magnetizabilities. However, more important for the apparent disagreement between theory and experiment may be the fact that isotropic magnetizabilities are almost exclusively determined relative to the magnetizability of reference compounds. It has been suggested that the reference values used to determine experimental magnetizabilities are inaccurate (Ruud et al. 2000).

´ et al. M. Jaszunski

44

Rotational g Tensor The magnetizability describes the magnetic moment induced in a molecule by an external magnetic field induction, and how this induced magnetic moment in turn can interact with the external magnetic field and lead to an energy correction. A molecule also has its own source of a magnetic moment, arising from the rotational motion. Since the molecule consists of charged particles, the nuclei, and the electrons, the overall rotation of these particles will generate a small magnetic moment. In the case of the external-field-induced magnetic moment, the vector potential and the electron magnetogyric ratio determine the form of the magnetic dipole operator in Eq. 44. For the magnetic moment induced by the molecular rotation, the magnitude of the rotational angular momentum J is determined by the rotational quantum numbers of the molecule, giving rise to a rotationally induced magnetic moment mJa D

N X J gab Jb ; „

(150)

a;b

where N is the nuclear magneton and the rotational g tensor (also called the molecular g factor) is defined as g J D g J;nucl C g J;el D mp I1

X K

4mp me 1 para  I .RCM / : e2

 2  ZK RK;CM 1  RK;CM RTK;CM (151)

In Eq. 151 we have introduced the moment of inertia tensor I and we have used the definition of Ÿpara as given in Eq. 149, which is applicable both when using London orbitals and when using conventional basis sets. Both the nuclear positions and magnetic dipole operators are defined with respect to the center of mass of the molecule, as this is the point about which the molecule rotates. There are two contributions to the rotationally induced magnetic moment, one arising from the rotation of the nuclear framework, and one arising because of the rotation of the electron density, and these contributions are opposite in sign because the linear response function Ÿpara in Eq. 151 is positive. This rotationally induced magnetic moment is an example of a nonadiabatic effect. If the electrons were able to adjust their positions and momenta instantaneously to any change in the nuclear configuration, the magnetic moment created by the electrons and the nuclei would have the same magnitude but opposite sign, due to the different charges of the particles. However, this description, corresponding to the Born–Oppenheimer approximation, is not exact, and in fact a small permanent magnetic moment will arise from the decoupling of the electronic and nuclear rotation. The rotational magnetic moment can interact with either internal or external sources of magnetic moments or magnetic fields. If the rotational magnetic moment

Molecular Electric, Magnetic, and Optical Properties

45

interacts with the nuclear magnetic moment it gives rise to a contribution to the energy, discussed in more detail in section “Nuclear Spin Rotation Constants,” which involves the nuclear spin–rotation constant. If we apply an external magnetic field and consider the interaction of this field with the magnetic moment created by the molecular rotation, the interaction energy is (Flygare 1974; Gauss et al. 1996): E D 

N X J Ba gab Jb ; „

(152)

a;b

where Ja is the rotational quantum number along the principal rotational axis a. The rotational g tensor is a tensor of rank two with in principle nine independent elements. The rotational magnetic moment is only well defined in the principal rotational axis system, where the moment of inertia tensor is diagonal, and only the diagonal elements of gJ in this principal axes system contribute to the rotational Zeeman effect (Flygare 1974; Gauss et al. 1996). We also note that if we can determine both the magnetizability and the rotational g factor, we can indirectly determine the molecular quadrupole moment from the equation: qcc D

 e  J 2me J J . aa C bb  2 cc / C gaa Iaa C gbb Ibb  2gcc Icc : e 2mp

(153)

This relation shows that microwave Zeeman spectroscopy can be an important source of molecular quadrupole moments. From a theoretical point of view, this equation does not provide any additional information compared to calculating the molecular quadrupole moment directly as an expectation value, since the relationship (Eq. 153) is fulfilled exactly for any wave function. The electronic contribution to the rotational g factor is important in highresolution microwave spectroscopy even in the absence of an external magnetic field. In Eq. 150 we introduced the magnetic moment induced by the rotation of the molecular framework. Instead of letting this induced magnetic moment interact with an external magnetic field induction, we can let it interact with the rotational moment of the molecule. This gives a correction to the molecular energy which is quadratic in the rotational quantum number J. In ordinary microwave spectroscopy, the leading energy contribution is related to the moment of inertia tensor and is also quadratic in J. Therefore, the energy correction related to gJ can be treated as a correction to the moment of inertia tensor, and it is customary to introduce an effective inverse moment of inertia tensor defined as (Flygare 1974):   me J;el 1 1  : I1 D I g eff mp

(154)

When comparing highly accurate theoretical estimates of moments of inertia directly with the results observed in high-resolution microwave spectroscopy, it is essential to take into account these nonadiabatic corrections. It is particularly note-

´ et al. M. Jaszunski

46

worthy that the experimental determination of the equilibrium geometry involves the estimation of the effects of the rotational g tensor on the moment of inertia tensor, and that it is currently possible to calculate these effects with an accuracy much higher than obtainable from experimental data alone. Therefore, the use of highly accurate experimental data, for instance from rotational spectra, combined with highly accurate calculations of small corrections to these quantities, is currently the most accurate way to obtain experimental equilibrium geometries (Stanton et al. 2001) (for a related analysis of diatomic systems, see Sauer et al. 2005). As the rotational g tensor includes a linear response function involving two magnetic dipole operators, similar computational requirements apply for this property as for the magnetizability. We can therefore expect that the direct calculation of the rotational g tensor using Eq. 151 will display very slow basis set convergence. However, London orbitals are very efficient in getting fast basis set convergence, as illustrated for PF3 in Table 4. In microwave or molecular beam experiments, all of the three diagonal elements of the rotational g tensor are determined. As for the magnetizability, the individual tensor components show some dependence on electron correlation effects, and in order to allow for the calculated results to be within the very tight experimental errors bars for the rotational g tensor elements, electron correlation and the effects of zero-point vibrational corrections need to be included. This is illustrated for the water molecule in Table 5.

Birefringences and Dichroisms The consequences of the fact that an electromagnetic wave has both a frequencydependent electric field component and a frequency-dependent magnetic field component have so far not been discussed. The two components are perpendicular to each other, as well as perpendicular to the direction of the propagation of the light beam. Moreover, we have not discussed the implications of the possible polarization of the light beam. Light experienced in everyday life is mostly nonpolarized – that is, the tip of the electric or magnetic field vectors moves randomly in the Table 5 Electron correlation effects and zero-point vibrational corrections to the rotational g tensor components of H2 O. The molecule is located in the xz-plane with the dipole moment along the z-axis (gJ is dimensionless) Computational method

gJxx

gJyy

gJzz

Hartree–Fock (Ruud et al. 1998) RASSCF (Ruud et al. 1998) CCSD (Gauss et al. 2007) CCSD(T) (Gauss et al. 2007) RASSCF C ZPV (Ruud et al. 1998) Exp. (Molecular beam) (Verhoeven and Dymanus 1970) Exp. (Microwave Zeeman) (Kukolich 1969)

0.6635 0.651(1) 0.6545 0.6543 0.637(1) 0.645(6) 0.6465(20)

0.6814 0.667(3) 0.6707 0.6690 0.640(3) 0.657(1) 0.6650(20)

0.7342 0.728(3) 0.7304 0.7308 0.709(3) 0.718(7) 0.7145(20)

Molecular Electric, Magnetic, and Optical Properties

47

plane perpendicular to the propagation direction. With the use of appropriate tools, unpolarized light can be turned into a wide variety of polarization states, becoming linearly, elliptically, or circularly polarized. In these cases, the tip of the field vector(s) oscillates in the plane containing the direction of propagation (linear polarization) or rotates along a circle (circular polarization) or an ellipse (elliptical polarization) in the plane perpendicular to it. The polarization of light plays an important role in the interactions of light with a molecule. Birefringences and dichroisms are two aspects of this interaction which are closely related to the polarization status of light. They arise through a variety of different mechanisms and interactions between the electromagnetic field and the molecule, but they are all characterized by the simultaneous measurements of the refractive (birefringences) or absorptive (dichroisms) index of light along two directions, different with respect to some predefined laboratory or molecular frame. The difference in the dispersion or absorption in these two directions is described by a quantity of a specific sign and magnitude. All these properties give detailed information about the molecular structure, and in particular some of them can distinguish between a sample and its mirror image. Distinguishable mirror images can occur naturally, in chiral molecules where the mirror image of the molecule cannot be superimposed on the original molecule itself – that is, for so-called enantiomers – or they can occur by introducing an asymmetry in the experimental setup that makes the mirror image of the experimental design impossible to superimpose on the original setup. In the latter case, since the experimental design is itself chiral, the observed effect does not vanish even if the molecule is not chiral. We note that only molecules possessing no improper or rotation-reflection axes may be chiral. It is instructive to start by giving some examples of birefringences, before discussing the details of the most important birefringences and dichroisms. Let us consider first an isotropic liquid sample subject to a strong external static electric field. The refractive index nk experienced by the component of the polarization vector aligned parallel to the direction of the external field F in this case differs from the index of refraction n? for the component aligned perpendicular to the direction of the vector F (Kerr 1875a, b). As a consequence, the initially linearly polarized light beam will exit the region of the sample with an ellipticity. This is an example of a linear birefringence nlin D nk  n? ;

(155)

and the effect is known as electric-field-induced optical birefringence or more often as the Kerr (electro-optical) effect. It exhibits a quadratic dependence on the strength of the applied electric field. Its existence is due mainly to the fact that the external electric field, in a system possessing an anisotropic electric dipole polarizability tensor, tends to align the molecules preferentially in its direction. Other mechanisms in general also contribute to the emergence of an anisotropy, the most important being the effect of electronic rearrangements. They play a role through the higher-order polarizabilities of the system, in particular through the sec-

´ et al. M. Jaszunski

48

ond electric-dipole hyperpolarizability  and, in the presence of permanent electric dipoles, through the first electric-dipole hyperpolarizability ˇ (Kerr 1875a, b). Other examples of linear birefringences are the Cotton–Mouton and Buckingham effects, briefly discussed below, and the Jones and magneto-electric birefringences, which occur when linearly polarized light goes through an isotropic fluid perpendicularly to both static electric and magnetic fields. Circular birefringence is observed when the two circular components of a linearly polarized beam propagate with different circular velocities, and therefore an anisotropy arises between the two refractive indices nC and n : ncirc D nC  n :

(156)

The net result is a rotation of the plane of polarization. An example of a circular birefringence is optical activity, which is observed when a medium composed of chiral molecules (with an excess of one enantiomer) is subject to linearly polarized electromagnetic radiation. The dependence of the corresponding anisotropy on the wavelength is described by the optical rotatory dispersion (ORD). The Faraday effect, discussed below, is another well-known example of a circular birefringence. An axial birefringence occurs with unpolarized light, and an example is the socalled magnetochiral birefringence (Barron and Vrbancich 1984; Kalugin et al. 1999). It can be seen in isotropic samples composed of chiral molecules, and it is observed when a static magnetic induction field is switched on parallel to the direction of propagation of the unpolarized probe beam. The refractive index experienced by the beam propagating parallel to the external magnetic field, n"" , becomes different from that experienced by a beam propagating antiparallel to the field, n"# nax D n""  n"# :

(157)

The corresponding anisotropy has a different sign for the different enantiomers. An external field will have an orientational effect on the molecules of the sample, and this is a mechanism responsible for the emergence of birefringences. In the presence of external fields the sample becomes anisotropic, and the extent to which this happens depends on the temperature. Indeed, the general form of the optical anisotropy (for measurements taken at a fixed pressure) is usually of the form n / A0 C

A2 A1 C 2 C  ; T T

(158)

where the term A0 includes all contributions arising from the reorganization of the electrons due to the action of the external field(s), whereas the terms A1 , A2 , : : : are connected to different mechanisms of reorientation of the molecules, involving the interaction of the field(s) with permanent electric or magnetic multipole moments. The temperature-dependent terms vanish for systems of spherical symmetry; the contributions exhibiting a nonlinear dependence are usually connected to the pres-

Molecular Electric, Magnetic, and Optical Properties

49

ence of permanent electric or magnetic dipole moments, or higher-order processes involving more complicated interactions between fields and multipoles. When the inducing field is time dependent, the birefringence is said to be “optically induced.” In general, the information provided by birefringences induced by static fields can differ from that obtained observing the corresponding optically induced phenomena. For example, in the static Kerr effect of dipolar molecules, an important role is played by the permanent dipoles, which tend to be aligned by the static field. This contribution vanishes for dynamic inducing fields, where the birefringence arises only from the rearrangement of the anisotropy of the electric dipole polarizability. The contribution resulting from the interaction with permanent multipoles averages over time to zero.

Optical Rotation The first contribution to the induced electric dipole moment in Eq. 74 from the time dependence of the magnetic field, ! 1 G 0 , gives rise to two different observable properties. In the dispersive region, this property determines the optical rotatory power, or just optical rotation (OR) for short, and in the absorptive region it determines the rotational strength observed in electronic circular dichroism (ECD). There are no terms bilinear in the electric and magnetic fields in the Hamiltonian, therefore G0 , the mixed electric dipole–magnetic dipole polarizability (see also Eq. 74), can be expressed as a linear response function: 0 G˛ˇ

˝ ˇ ˇ ˛ ˛˛ ˝˝ O ˇˇ 0 h0 jO ˛ j ni n ˇm 2! X .!I !/ D  Im D Im O ˛ I m O ˇ !: „ !n2  ! 2

(159)

n¤0

We note that G0 vanishes if the electromagnetic field is static (! D 0), see the discussion in section “ Expansions of Energy and Multipole Moments.” For unit conversion factors, see Table 6. Measurements of optical rotation are almost exclusively carried out on liquid samples (though gas-phase measurements recently became possible (Müller et al. 2000)) and we will, therefore, be concerned primarily with the rotational average of the G0 tensor, which is described by the quantity ORD ˇ defined as ORD

ˇD

1 0 G : 3! ˛˛

Table 6 Unit conversion factors for electric dipole–magnetic dipole polarizability G0

1 a.u. e 2 a03 „1

SI 3. 60702  1035 C2 m 3 J 1 s 1

CGS 1. 08136  1027 Fr cm G1

(160)

´ et al. M. Jaszunski

50

The most common experimental setup involves the determination of the rotation of plane-polarized light as it passes through a sample in which there is an excess ORD of one enantiomer. The standard optical rotation [˛]25 ˇ, is D , proportional to reported for light with a frequency corresponding to the sodium D-line (589.3 nm) at a temperature of 25 ı C. Because the mixed electric dipole–magnetic dipole polarizability involves the magnetic dipole operator, in approximate calculations G0 carries an origin dependence. Indeed, the individual tensor elements of G0 are origin dependent. The trace of G0 must be origin independent, since the optical rotation is an experimental observable. In non-isotropic media, contributions to the optical rotation tensor arise from the mixed electric dipole–electric quadrupole polarizability A, EE DD O ˇ ; A˛;ˇ .!I !/ D  O ˛ I ‚ !

(161)

(see the last term in Eq. 74), and it is the combination of the G0 and A contributions that is gauge-origin independent for exact wave functions. The contribution from A vanishes in isotropic samples, since this mixed electric dipole–electric quadrupole polarizability is traceless. For approximate variational wave functions, the origin independence of the trace of G0 is only achieved in the limit of a complete basis set. One way to overcome this problem is to introduce local gauge origins by using London atomic orbitals (Helgaker et al. 1994). The dispersion of the optical rotation was for a long time also the focus of much experimental attention through optical rotatory dispersion measurements. Even after it became customary to restrict the optical rotation measurements to a single frequency, ORD served as an important tool for determining excitation energies in chiral molecules, although it has now been surpassed by electronic circular dichroism for these purposes (see section “Circular Dichroism”). At the sodium frequency, the optical rotation is rather small for most molecules. However, the individual diagonal elements of the mixed electric dipole–magnetic dipole polarizability may be fairly large in absolute value, often canceling each other in the trace. This is illustrated for a few selected molecules in Table 7. Consequently, the optical rotation is highly sensitive to numerical errors in the tensor components, because small residual errors in the individual tensor components, arising from the solution of the linear response equations, may lead to substantial errors in ORD ˇ. Another consequence of this cancellation is that ORD ˇ is very sensitive to the choice of molecular geometry, as well as zero-point vibrational effects (Ruud et al. 2001), since the small changes introduced by these effects in the molecular charge distribution, and thus on the different diagonal elements of G0 , can give rather large effects on ORD ˇ. The sensitivity of the optical rotation is perhaps most clearly illustrated in the case of conformationally flexible molecules – that is, molecules that have a significant population of multiple stable minima. As exemplified for paraconic acid (a substituted  -butyrolactone) in Table 8, different molecular conformations

Molecular Electric, Magnetic, and Optical Properties

51

Table 7 The diagonal elements of the mixed electric dipole–magnetic dipole polarizability and 3 1 1 the trace of the tensor for a few selected molecules (in a.u., [˛]25 g ) D in deg cm dm Molecule ı

H2 O2 (¥ D 120 ) (C)-methyloxirane (C)-camphor

0 ! 1 Gxx

0 ! 1 Gyy

! 1 Gzz0

ORD

1.201 0.820 0.240

0.387 0.812 4.891

0.960 1.587 5.847

0.049 0.015 0.239

[˛]25 D

ˇ

55.3 10.2 60.5

DFT/B3LYP results obtained with the Sadlej polarized basis sets, taken from Giorgio et al. (2004) 3 1 1 Table 8 The specific optical rotation ([˛]25 g ) for different conformers of D in deg cm dm (S)-()-paraconic acid calculated using different basis sets and at DFT/B3LYP and MP2 optimized geometriesa

Conformer A B C D E F [˛]25 D

DFT geometries 6-311CCG** 31.27 152.79 112.01 250.61 19.21 188.13 C14.78

aug-cc-pVDZ 42.35 b 144.53 b 95.73 b 245.03 b 10.78 b 182.86 b C6.72

aug-cc-pVDZ 47. 91 145.76 91.47 253. 88 0.36 182. 67 C4. 40

MP2 geometry aug-cc-pVDZ 10. 58 c

109.74 251. 71 13. 49 217. 33 85. 23

The bottom line of the table gives the total specific optical rotation [˛]25 D (all conformations weighted by the Boltzmann distribution). All the results taken from Marchesan et al. (2005) b Theoptical rotations were calculated for the B3LYP/6-311CCG** geometries c Anenergy minimum corresponding to a B-like conformer could not be found a

can have optical rotations that differ by orders of magnitude and even in sign. A thorough conformational search is therefore mandatory before the absolute sign of the optical rotation (or any birefringence or dichroism for that matter) is determined from theoretical calculations for conformationally flexible molecules. The optical rotation of the flexible molecule can then be determined by Boltzmann averaging over the dominant molecular conformations. In order to ensure reasonably well-converged results for the optical rotation, basis sets of polarized valence double-zeta quality are required, and sets such as the augcc-pVDZ of Woon and Dunning (1994), or the polarized triple-zeta basis of Sadlej (1988) have been shown to perform well for calculations of optical rotation. Most importantly, diffuse p functions in the outer regions of the electron density, which in most cases is described by the electron density of hydrogen atoms, are required to ensure qualitatively correct results (Zuber and Hug 2004). Since the optical rotation is very sensitive even to small changes in the electron density, electron correlation effects should also be taken into account in order to get accurate results. Due to the fact that chiral molecules in general have very low symmetry, if any symmetry at all, the only viable approach for calculating electron correlation effects in chiral molecules is currently density functional theory (Ruud et al. 2003; Stephens et al. 2001). Methods such as MP2 and CCSD could also be

´ et al. M. Jaszunski

52

used to describe electron correlation, but due to their nonvariational nature it is not ensured that gauge-origin independent results can be obtained in the conventional length-gauge formulation even in the limit of a complete basis set. In principle, the use of the dipole velocity gauge will ensure that the calculated results are independent of the gauge origin. For all but the smallest basis sets, the differences between the velocity and the length gauges are not very large, with the length gauge in general performing better. However, in the case of the optical rotation an additional complication arises: whereas G0 given by Eq. 159 will vanish in the limit of a static field, this is not the case for the dipole-velocity analogue of the optical rotation. It has been suggested that the much slower basis set convergence of the optical rotation compared to other properties calculated using the velocitygauge formulation can be improved by subtracting the static-limit value of the corresponding response function (Pedersen et al. 2004), such that EE EE DD DD G 0 / pO˛ I lOˇ  pO˛ I lOˇ : !

0

(162)

Since there are no empirical rules relating the stereochemistry of a molecule to the observed sign of the optical rotation, an important area of application for optical rotation calculations would be the combined theoretical and experimental determination of absolute configurations. However, this is a difficult task due in part to the large variations with geometry in the optical rotation of conformationally flexible molecules, and in part because of the small magnitude of ORD ˇ. These factors make the sign of the optical rotation hard to determine with confidence. One way to increase the predictive power of the calculations is to change the frequency of the incident light to shorter wavelengths. The magnitude of the optical rotation increases dramatically as the frequency approaches that of a resonance. Thus, if both theory and experiment could determine the optical rotation at frequencies closer to electronic excitation energies, theoretical calculations could provide a much more reliable proof of the absolute configuration of the molecule (Giorgio et al. 2004).

Circular Dichroism Natural (Electronic) Circular Dichroism Let us now turn to the lowest-order absorption process involving mixed electric and magnetic perturbations. This property, which is the absorptive analogue of the optical rotation, is known as electronic circular dichroism (ECD) or just CD for short. The differential absorption of circularly polarized light, corresponding to the difference between the absorptive index of the two circular components of linearly polarized light, is proportional to the rotational strength, which is normally calculated as the residue of the linear response mixed electric dipole–magnetic dipole polarizability:

Molecular Electric, Magnetic, and Optical Properties

˛˛ ˝ ˛ ˝˝ lim „ .!  !n0 / O ˛ I m O ˇ ! D h0 j O ˛ j ni n j m Oˇ j 0 :

!!!n0

53

(163)

Since this expression corresponds to the infinite lifetime approximation for the excited state, only a single number will be obtained at the frequency of the electronic excitation. In general n R, the rotatory strength for the transition j 0 i!j ni, includes an electric dipole–magnetic dipole contribution n

m R˛ˇ D

˛˝ ˛ ˝ ˛˝ ˛ 3i e 2 ˝ ı˛ˇ 0 j r j n n j lT j 0  0 j rOˇ j n n j lO˛ j 0 ; 4me

(164)

and an electric dipole–electric quadrupole contribution n

Q R˛ˇ D

˝ ˛˝ ˛ 3!n0 e 2 –˛ı 0 j rO j n n j qO ıˇ j 0 : 4

(165)

For randomly oriented molecules, the averaging leaves only the electric dipole– magnetic dipole contribution and the scalar rotatory strength is given by n

RD

˛˝ ˛ i e2 ˝ 0 j rT j n n j l j 0 ; 2me

(166)

corresponding to Eq. 163. These expressions are given in the length gauge. In the velocity gauge n

m R˛ˇ D

˛˝ ˛ ˝ ˛˝ ˛ 3e 2 ˝ ı˛ˇ 0 j p j n n j lT j 0  0 j pOˇ j n n j lO˛ j 0 ; 2 4me !n0

(167)

and n

Q R˛ˇ D

˝ ˛˝ ˛ 3e 2 C "˛ı 0 j pO j n n j TOıˇ j0; 2 4me !n0

(168)

where TC indicates the “velocity” form of the electric quadrupole operator TC D  .rp C pr/ :

(169)

This form has an advantage in comparison to the length form. Although with a translation of the reference frame the magnetic dipole and electric quadrupole components change, the total tensor in the velocity gauge is invariant to such a change of origin. For the length gauge, this invariance depends in addition on the fulfillment of the hypervirial relation, Eq. 126.

´ et al. M. Jaszunski

54

Two-Photon Circular Dichroism Two-photon circular dichroism (Tinoco 1975) arises in chiral systems due to the differential absorption of two photons, of which at least one is circularly polarized (De Boni et al. 2008). In this sense it can be seen as the nonlinear extension of ECD. The observable, the anisotropy of the two-photon transition strength, is proportional to the two-photon rotatory strength (Jansík et al. 2005; Rizzo et al. 2006; Tinoco 1975): n

RTPCD .!/ D b1 ŒB1 .!/0n  b2 ŒB2 .!/0n  b3 ŒB3 .!/0n ;

(170)

where b1 , b2 , and b3 are numbers, simple combinations of the analogous polarization and setup-related coefficients F; G, and H given for two-photon absorption in Eq. 132. The molecule-related parameters B 1 , B 2 , and B 3 take the form: ŒB1 .!/0n D

1 p;0n .!/ ; Mp;0n .!/ P ! 3 

(171)

ŒB2 .!/0n D

1 C;0n p;0n .!/ P .!/ ; T 2! 3 

(172)

ŒB3 .!/0n D

1 p;0n .!/ ; Mp;0n .!/ P ! 3 

(173)

and they are, therefore, appropriate contractions of generalized two-photon secondrank tensors, like the one given in Eq. 133. Indeed, these tensors are defined (for the general case of two photons of different frequency) as follows:

p;0n

P˛ˇ .!1 ; !2 / D

p;0n M˛ˇ

8 p p < O ˛ 0m O ˇ X 1 „

m

1X .!1 ; !2 / D „ m

: !m0  !1

C

(  p   O ˇ mn O ˛ 0m m

p

O ˇ

  0m

9

p = mn

O ˛

!m0  !2

; ;

   p ) m O ˇ 0m O ˛ mn

C ; !m0  !1 !m0  !2 8   p  p  C 9 C O˛ < =  O O  0m TO˛ T  X mn 1 C;0n 0m mn .!1 ; !2 / D "ˇ C T˛ˇ ; : !m0  !1 ; „ !m0  !2 m

(174)

(175)

(176)

where we have introduced the velocity form of the dipole operator, O p˛ D 

e X pOi˛ : me i

(177)

Within the formalism of response theory, the second-rank tensors of interest are obtained from the single residues of appropriate quadratic response functions, see

Molecular Electric, Magnetic, and Optical Properties

55

Eq. 69. The quadratic response functions of relevance for two-photon circular dichroism are: DD EE p p p;0n O ˛ I O ˇ ; V !n ) P˛ˇ .!1 ; !2 / ; (178) !1 ;!2

˝˝

p

O ˛ I m O ˇ ; V !n

˛˛

p;0n

!1 ;!2

DD EE p C "ˇ TO˛ I O  ; V !n

) M˛ˇ .!1 ; !2 / ;

!1 ;!2

C;0n .!1 ; !2 / ; ) T˛ˇ

(179) (180)

where V !n is an arbitrary operator (corresponding to the excitation vector to the state n). Single residues of quadratic response functions are efficiently and accurately computed nowadays with a number of wave function models. Nevertheless, DFT has been used almost exclusively in the few theoretical studies of two-photon circular dichroism that have been published (Jansík et al. 2007, 2008).

Faraday Effect In 1846, Faraday observed that when a static magnetic field was applied with its direction parallel to the direction of propagation of linearly polarized light traversing a transparent sample (glass, liquid, gas), the plane of polarization of the beam was rotated (Faraday 1846a, b). It was the first ever experimental evidence of a birefringence phenomenon, and in particular of a circular birefringence, and the effect is often referred to as the Faraday effect or magnetic field-induced optical rotation. The rotation is proportional to the optical anisotropy, see Eq. 156, and this depends linearly on the strength of the applied magnetic induction field. The mechanisms governing this effect involve the interaction of the static external magnetic induction field with the oscillating electric field of the light wave and with the electric multipoles of the sample. The term “magnetic optical rotatory dispersion” (MORD) is often used to denote the frequency dispersion of the Faraday effect, and thus of the corresponding circular birefringence. For a fluid sample, the rotation with respect to the molecular frame becomes (Barron 2004; Buckingham and Stephens 1966):   1 ˝ ˇˇ ˇˇ ˛ 0 B O  0 ˛˛ˇ Bz l; 0 m  D V .!/ Bz l / !–˛ˇ ˛ 0 ˛ˇ; C kT

(181)

where l is the path length, we have introduced the Verdet constant V(¨), and ˛ 0B denotes the magnetic field derivative of ’0 , the antisymmetric electric dipole polarizability. The second term in the˝ parentheses vanishes for closed-shell systems, ˇ ˇ ˛ O  ˇ 0 D 0. as the magnetic dipole is quenched, 0 ˇm The Verdet constant of a closed-shell system, free of orbital degeneracies both in the ground and in the excited state, is therefore obtained from the response function

´ et al. M. Jaszunski

56 Table 9 Unit conversion factors for the Verdet constant V (¨)

1 a.u. rad e a0 „1

SI 8. 03961  10 4 rad T1 m1

CGS 2. 76382  10 2 min G1 cm1

(Jaszu´nski et al. 1994; Parkinson and Oddershede 1997): ˝˝ ˛˛ V .!/ / i !"˛ˇ O ˛ I O ˇ ; m O  !;0 :

(182)

This contribution is sometimes referred to as the B term contribution to the Verdet constant. For molecules containing orbital degeneracies additional contributions arise (the A and C terms), in analogy with the closely related phenomenon – magnetic circular dichroism, discussed in section “Magnetic Circular Dichroism.” For unit conversion factors, see Table 9.

Hypermagnetizabilities, Cotton–Mouton Effect The Cotton–Mouton effect (CME) is a magnetic induction field-induced linear birefringence, observed in the presence of a strong magnetic induction field B with a component perpendicular to the direction of propagation of the beam. It is mainly due to the tendency of both the electric field associated with the light beam and the magnetic induction field to align the molecules exhibiting both an anisotropic electric dipole polarizability and an anisotropic magnetizability tensor (according to the Langevin–Born theory (Born 1918; Langevin 1910)). The rearrangement of the electron density again plays a part through the mixed electric and magnetic hyperpolarizabilities, or hypermagnetizabilities. The experimental quantity connected to the hypermagnetizability is the molar Cotton–Mouton constant m C. Semiclassically, for fluids composed of closed-shell systems (Buckingham and Pople 1956; Fowler and Buckingham 1989; Jamieson 1991): mC

D

2NA ΠC Q.T / ; 27

(183)

where ˜ is the hypermagnetizability anisotropy  D

 1  3˛ˇ;˛ˇ  ˛˛;ˇˇ ; 15

(184)

and Q.T / D

 1  3˛˛ˇ ˛ˇ  ˛˛˛ ˇˇ : 15kT

(185)

Molecular Electric, Magnetic, and Optical Properties

57

In Eq. 185, ’ is the (frequency-dependent) electric polarizability tensor, and Ÿ the magnetizability tensor. The two contributions to the molar Cotton–Mouton constant in Eq. 183 are, therefore, a temperature-independent term related to the hypermagnetizability ˜˛ˇ,ı and a temperature-dependent molecular orientational part – the “Langevin term.” When the magnetizability is expressed as a power series in the perturbing electric field: 1 ˛ˇ .F/ D ˛ˇ C ;˛ˇ F C ı;˛ˇ F Fı C : : : ; 2

(186)

the expansion coefficients —,˛ˇ and ˜ı,˛ˇ define the molecular first and second hypermagnetizability. The first indices of — and ˜ refer to the electric field and the last two to the magnetic field. In fluids, the experimentally measured property is related to the anisotropy of ˜. From the expansion above it can be seen that the static hypermagnetizability ˜ corresponds to the fourth derivative of the molecular energy: ˛ˇ;ı

ˇ @4 E .F; B/ ˇˇ D : @F˛ @Fˇ @B @Bı ˇF;BD0

(187)

It can also be expressed in terms of Ÿ(F) – the magnetizability dependent on F: ˛ˇ;ı D

ˇ @2 ı ˇˇ ; @F˛ @Fˇ ˇFD0

(188)

or ’ (B) – the polarizability dependent on B: ˇ @2 ˛.!I !/˛ˇ ˇˇ ˛ˇ;ı .!/ D ˇ @B @Bı ˇ

;

(189)

BD0

and we have indicated in the latter equation that it holds also for the frequencydependent polarizability. The hypermagnetizability is given as a sum of paramagnetic and diamagnetic contributions: para

˛ˇ;ı .!/ D ˛ˇ;ı .!I !; 0; 0/ C dia ˛ˇ;ı .!I !; 0/ ;

(190)

corresponding to a sum of a cubic response and a quadratic response function (Rizzo et al. 1997): ˛˛ ˝˝ para ˛ˇ;ı .!I !; 0; 0/ D  O ˛ I O ˇ ; m O ; m O ı !;0;0 ;

(191)

´ et al. M. Jaszunski

58

EE DD O B;B .!I dia I  O ; !; 0/ D  O ˛ ˇ ˛ˇ;ı ı

!;0

;

(192)

where (see for comparison Eq. 145) the “diamagnetic magnetizability operator” in the second term is B;B B;B D 2HO ˛ˇ : O˛ˇ

(193)

We recall here that the magnetic dipole moment and diamagnetic magnetizability operators, and thus the partitioning of the hypermagnetizability into para- and diamagnetic components, may depend on the chosen gauge origin. Finite field approaches corresponding to Eqs. 188 and 189 have been used (Rizzo et al. 1997). In these calculations ˛ˇ,ı is obtained combining the available analytic linear response methods with the numerical finite difference approach. The frequency-dependent electric dipole polarizability can be computed in the presence of a magnetic induction field, and the hypermagnetizability is obtained by numerical differentiation. This approach (Cybulski and Bishop 1994; Tellgren et al. 2008) in general requires complex algebra as the perturbation is purely imaginary, whereas most electronic structure codes use real wave functions. In the more often used finite field approach based on Eq. 188, one computes the magnetizability Ÿ with and without an external electric field perturbation. However, in this approach only the static value of the required hypermagnetizability is obtained. Analytic calculations of the quadratic and cubic responses Eqs. 192 and 191 also including London orbitals to ensure gauge origin independence can nowadays be carried out (Thorvaldsen et al. 2008). The effect of the London orbitals on improving the basis set convergence appears to be small. For atoms, choosing the origin of the gauge at the nucleus automatically makes the dependence on the gauge vanish, and there is no advantage in using LAOs. Moreover, the evaluation of ˜ may be reduced to the calculation of the Cauchy moments S(2n) and of the dipole–dipole–quadrupole hyperpolarizability frequency dispersion coefficients (Coriani et al. 1999a; Rizzo et al. 1997). For unit conversion factors, see Table 10.

Electric-Field-Gradient-Induced Birefringence As a magnetic induction field can give rise to an optical anisotropy through its interaction with the anisotropic magnetizability of a molecule, so can an electric field gradient, through its coupling with a permanent molecular quadrupole moment. The corresponding birefringence (Buckingham 1959; Buckingham and Disch 1963) is commonly called electric-fieldgradient-induced birefringence, more recently Table 10 Unit conversion factors for hypermagnetizability ˜

1 a.u.  2 2 ea0 .e„=me /2 Eh3

SI

CGS

2. 98425  1052 C2 m2 J1 T2

2. 68211  1044 cm3 G2

Molecular Electric, Magnetic, and Optical Properties

59

Buckingham birefringence, and it exhibits a linear dependence on the strength of the electric field gradient. Besides the orientational Langevin-Born-type effect, acting on quadrupolar molecules and due to both the electric field associated with the light wave and to the static external field gradient, it also involves the rearrangement of the electron density as a consequence of the electron interaction with the electric and magnetic wave vectors associated with the beam and with the externally applied electric field gradient. In this case, mixed electric-dipole, electric-quadrupole, and magnetic-dipole hyperpolarizabilities play their role in determining the strength of the effect. Buckingham birefringence has been often employed for the determination of molecular quadrupole moments (Buckingham 1959; Buckingham and Disch 1963). For an ideal gas at constant pressure, for light propagating along the Z axis n D nX  nY D

NA rF 3rF sD mQ .!; T / ; 15Vm ©0 2Vm

mQ .!; T / D

2NA s; 45©0

(194) (195)

Where Vm is the molar volume, rF is the external electric field gradient arranged so that rF D rFXX D rFY Y and rFZZ D 0. Equation 195 introduces the Buckingham constant m Q, which depends through the quantity s on the circular frequency of the light and on the temperature. For non-dipolar systems, where the quadrupole moment does not depend on the choice of origin, s D b .!/ C

1 ‚˛ˇ ˛˛ˇ .!I !/ ; kT

(196)

Involving the molecular quadrupole moment ‚˛ˇ and the frequency-dependent electric dipole polarizability ˛˛ˇ .!I !/. The temperature-independent contribution is b.!/ D B˛ˇ;˛ˇ .!I !; 0/  B˛;˛ˇ;ˇ .!I !; 0/ 

5 0 .!I !; 0/ ; "˛ˇ J˛;ˇ; ! (197)

And is a combination of three mixed hyperpolarizabilites: EE DD O ı B˛;ˇ;ı .!I !; 0/ D O ˛ I O ˇ ; ‚

;

(198)

EE DD O ˇ ; O ı B˛;ˇ;ı .!I !; 0/ D O ˛ I ‚

;

(199)

EE DD 0 .!I !; 0/ D i O ˛ I MO ˇ ; O  J˛;ˇ;

;

(200)

!;0

!;0

!;0

´ et al. M. Jaszunski

60

The expressions are somewhat more complex for dipolar fluids (Buckingham and Longuet-Higgins 1968), where the quadrupole moment becomes origin dependent. The formal expressions for s, Eq. 196, does not change provided that we refer to a frequency-dependent origin for the quadrupole operator, commonly labeled as the “effective quadrupole center” (Rizzo and Coriani 2005; Rizzo et al. 2007). For any other choice of the origin we have sDb.!/ C

1 kT



  5 0 .!I !/ : ‚˛ˇ ˛˛ˇ .!I !/ ˛ Aˇ;˛ˇ .!I !/ C "˛ˇ Gˇ ! (201)

The effective quadrupole center is defined as the point in space where the last term in Eq. 201, involving the molecular dispole moment ˛ and the mixed polarizabilities 0 .!I !/, vanishes. A˛;ˇ .!I !/ and G˛ˇ

Magnetic Circular Dichroism For a medium that is isotropic in the absence of magnetic fields, the Faraday A, B, and C terms determining the MCD of a transition from the electronic state to the electronic state jni are defined as (Buckingham and Stephens 1966; Schatz and McCaffery 1969; Stephens 1976): 1 Œhn jmj ni  h0 jmj 0i  Im .h0 jj ni  hn jj 0i/ ; 2 8 T4 > T1 > T2, associated with a relative population of the OH sites which is probably larger in the 12-ring channel for natural zeolites (Alberti et al. 1986). The T4 site has also been proposed as the most probable candidate for Al substitution by Demuth et al. based on the relative stability of Al embedded clusters with Al substituted in turn at the 4 possible sites (Demuth et al. 2001). Due to the low probability of Al in T2 and the high probability of more Al sites in the main channel (Alberti 1997), we have selected two models with the distributions T4, T3 and T4, T1, associated with both Na cations (NaMOR1, Na-MOR2) and protons (H-MOR1 and H-MOR2). The original models have been cut from a solid with the appropriate Al distribution, generated using the Cerius2 program (Cerius 2005) and terminated with hydrogens. In order to

Auxiliary Density Functional Theory: From Molecules to Nanostructures Na–MOR1

43

Na–MOR2

O4

O9

T3

O1 O6

O1

O7 T1 O3

O2 O10

T4

T4 b

O4

c H

O

Si

Al

Na a

Fig. 9 Structures, T and O atoms legend of the studied mordenite models. The orientation of the three crystallographic axis a, b, and c is also given

constrain the structure of the cluster to that of the solid, the coordinates of the terminal hydrogen atoms, positioned along the Si-O bonds, have been fixed during the subsequent optimizations, all other atomic coordinates being relaxed. Figure 9 illustrates the two models Na-MOR1 and Na-MOR2. In order to test the influence of the Al content on the mordenite geometrical parameters, we have also studied the same model without Al atoms, as will be discussed later. For these studies, we have used an ADFT-based approach, using a linear combination of atomic orbitals, as implemented in the deMon2k program (Köster et al. 2011a). Geometry optimizations and molecular electrostatic potential (MEP) calculations were performed at the local level using the correlation functional proposed by Vosko et al. (1980), whereas energetic properties (binding energies and proton affinities) have also been evaluated using the exchange-correlation functional of Perdew, Burke, and Ernzerhof (Perdew et al. 1996a). DFT-optimized double-zeta plus valence polarization (DZVP) basis sets were employed for all atoms (Godbout et al. 1992). For the fitting of the density, the A2 auxiliary function set was used (Godbout et al. 1992). The exchange-correlation potential was numerically integrated on an adaptive grid (Köster et al. 2004a). The grid accuracy was set to 105 a.u. in all calculations. The Coulomb energy was calculated by the

44

P. Calaminici et al.

variational fitting procedure proposed by Dunlap, Connolly, and Sabin (Dunlap et al. 1979; Mintmire and Dunlap 1982). A quasi-Newton method in internal redundant coordinates with analytical energy gradients was used for the structure optimization (Reveles and Köster 2004). The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 104 and 103 a.u., respectively. For the geometry optimization of the mordenite structures, the parallel version of deMon2k (Calaminici et al. 2006; Geudtner et al. 2006) was used. Each mordenite model possesses more than 400 atoms with around 6000 orbital basis functions and 11,000 auxiliary functions. The calculations were performed on 6 or 8 2.4 GHz Intel Xeon CPUs. These nodes were connected with a Myrinet switch. The geometry optimization of such models takes about 13 days on 8 CPUs. In this time, roughly 25 optimization steps are performed. The cation binding energies (BE) to the framework have been calculated using the formula: BE = E(neutral model) – E(cation) – E(anionic model). The O-H stretching frequencies of the H-MOR1 model have been calculated within the harmonic approximation. This normal mode being uncoupled with all the other vibrational modes of the model, its frequency has been calculated using a partial Hessian analysis, as proposed by Li and Jensen (2002).

Calculated Geometries and Properties General Structural Parameters of the Mordenite Models Our purpose here was to simulate averaged bond length and bond angle data that can be compared with those provided by X-ray experiments performed on natural and synthetic mordenites with various Si/Al ratios. Comparison with available theoretical data was also presented (Dominguez-Soria et al. 2007). The main geometrical parameters (T-O bonds and T-O-T bond angles) of the fully siliceous and sodium mordenite models have been compared in Table 1 of Dominguez-Soria et al. (2007) with experimental data. The calculated values have been averaged over Na-MOR1 and Na-MOR2, namely, considering different Al distributions. This procedure mimics the experimental averaging over the non- distinguishable Si and Al positions. Among several experimental structures of natural mordenites (Alberti et al. 1986; Alberti 1997; Schlenker et al. 1979; Ito and Saioto 1985; Simoncic and Armbruster 2004), the structure obtained by Alberti et al. (1986) has been chosen for comparison together with the experimental structure of a natural and a synthetic Na-mordenite (Simoncic and Armbruster 2004). As we have shown in Table 1 of Dominguez-Soria et al. (2007), the calculated bond lengths are very comparable with the experimental results, within 0.02 Å. We have noticed that this uncertainty is of the same order of magnitude as that obtained in previous theoretical work using the LDA level of theory and periodic boundary conditions (Demuth et al. 2001). The systematic lengthening of the T-O bonds in our calculations with respect to standard experimental bond distances could be interpreted in terms of a correction for thermal motions as given in Simoncic and Armbruster (2004). The average values of the T-O-T bond angles, presented in Table 2 of Dominguez-Soria et al. (2007), are in good agreement with experiment,

Auxiliary Density Functional Theory: From Molecules to Nanostructures

45

within 4 degrees, which is also the average difference of our calculated bond angles with respect to those reported previously by Demuth et al. (2001). It is worth noting that the isotropic temperature factors provided by X-ray refinements for the zeolite oxygens are all very large (2.4 to 4.6 Å) (Alberti et al. 1986; Simoncic and Armbruster 2004), leading to a substantial uncertainty on the oxygen positions. It was also shown that the averaged bond length and bond angle are rather insensitive to the Al content of the model. This allowed us to compare our models containing two Al sites with experimental mordenites which contain about 4 to 6 Al per unit cell. Al sites of Na-MOR Previous Data for Extra-Framework Cations Experimentally, the positions of the extra-framework cations are not easily determined. Moreover, there is a clear difference between natural and synthetic mordenites, due to the different natures of their extra-framework cations, as it was demonstrated in the literature (Simoncic and Armbruster 2004). Whereas Ca and K cations predominate in natural mordenites, synthetic zeolites are grown using Na cations, leading, presumably, to different locations of the (Al, cation) pairs; although T3 remains slightly more populated than the other sites, the distribution of Al among the possible T sites is much more balanced according to the pattern reported by Simoncic and Armbruster (2004) (for an Al content of 6, one can expect 1.3 Al at T1 and T4, 0.8 at T2 and 2 at T3). Consequently, the Na cations compensating the Al charges are found distributed within two regions, i.e., the side pocket and the main channel. Due to the presence of water, about 60 % of the cations cannot be precisely determined, presumably being located mainly in the main channel (Simoncic and Armbruster 2004). The remaining cations are found either bonded to three oxygens around T3 in the side pocket or pushed downward into the 8m-ring tube. Since our study concerns Na-MOR models, we focus on comparing our results with the experimental data provided for synthetic Na-mordenites. Previous theoretical work on Na-mordenite used periodic models involving 8 (Al, Na) pairs per unit cell, and results were compared with the natural zeolite data available (Demuth et al. 2001). The choice of occupying only T3 and T4 sites with equal populations was made, leading then to a strong concentration of cations in each cavity, related to the high symmetry of the solid. Therefore, it is hardly possible to make a direct comparison with this study, concerning the Na positions with respect to the framework oxygens. Analysis of the Properties of the Cationic Sites in Na-MOR1 and Na-MOR2 The substitution of one Si by Al in the framework generates one negative charge, which has to be compensated by the presence of a counterion. The framework negative charge is distributed among the framework oxygens, more particularly to the four oxygens adjacent to Al. The location of the cation thus depends on the local geometry around the T site for large Si/Al ratios, i.e., when the Al sites can be considered isolated. In these conditions, it is well established that smaller

46

P. Calaminici et al.

Al-O-Si angles are generally associated with larger electrostatic potentials, i.e., larger attractive interactions with an incoming positive charge (Goursot et al. 1998). In order to locate the most probable positions of the counterions, we have used the well-known strategy of approximating the cation position by searching the largest electrostatic interaction of the system density with a positive point charge. To do so, the molecular electrostatic potential (MEP) was calculated around the three AlO4 T4, T3, and T1 tetrahedra. The contour plots of these MEPs are illustrated in Figure 2a, 2b, and 2c of Dominguez-Soria et al. (2007), respectively. These MEP values have been calculated for the optimized structures of the corresponding T anions, at the T4 and T3 sites in Na-MOR1 and at the T4 and T1 sites in NaMOR2, respectively. As expected, the MEP wells in Na-MOR1 surround the two O2 and O10 in the main channel at T4 and the two O1 and O9 at T3 in the side pocket. The lowest MEP minimum at T4 is located between the two O2 and O10 (4:13 eV). The lowest minimum at T3 is located close to O9 (4:22 eV). In NaMOR2, the MEPs at T4 are identical to the ones calculated for T4 in Na-MOR1, showing that the two Al sites are independent. Interestingly, the T1 site of Na-MOR2 belongs to both the main channel and the side pocket. The O1 and O6 adjacent oxygens are in the side pocket, whereas O3 and O7 are located in the main channel (Fig. 9). The MEP well at T1 is essentially located within the side pocket, with the lowest minimum close to O6 (3:94 eV). It is worthwhile to underline that the MEP values at O7 and O3, i.e., in the main channel, are equal to 3:60 and 3:10 eV, respectively, showing that a compensating NaC or HC at those oxygens is unfavored with respect to O6. This result indicates that the counterion at T1 will be more stable within the side pocket than in the main channel, at least after dehydration. This conclusion is also valid for protons (Dominguez-Soria et al. 2007). Si substitution by Al leads, as expected, to lengthening the T-O bonds and decreasing the T-O-Si angles. The average elongation of the T-O bonds reaches 0.115 for T1, 0.124 for T3, and 0.123 Å for T4. The corresponding T-O-Si averaged decrease is found to be 3.3 degrees, 7.2 degrees, and 8.3 degrees for T1, T3, and T4, respectively. These changes are in a reasonable range, with a lower variation for the T1 site, due to its more rigid environment. The optimized structures of the NaC sites were reported in Table 3 of Dominguez-Soria et al. (2007), as well as the Na binding energies to the framework. From the reported Na-O distances, the cation can be considered as bi-coordinated at T4 and T1 and mono-coordinated at T3. This result correlates with the calculated positions of the MEP minima, as shown in Figure 2 of Dominguez-Soria et al. (2007). Interestingly we noticed that the shortest NaO distances correspond to the smallest Al-O-Si angles (smaller AlO-Si angles correlate with more electron density available for cations in the O lone pair due to better hybridization conditions) (Goursot et al. 1998). This result can be considered as the physical reason why the cation at T1 is located inside the side pocket. Moreover, this finding allows us to anticipate the preferred oxygen for protonation, i.e., O10 at T4, O9 at T3, and O6 at T1. The calculated Na binding energies follow the ordering T3 > T4 T1, which is very comparable with that indicated by the MEPs. This result indicates that electrostatics governs the cation – framework binding. Moreover, we noticed that

Auxiliary Density Functional Theory: From Molecules to Nanostructures

47

the use of the PBE GGA exchange-correlation functionals does not modify the trend established on the basis of LDA calculations. For sites T1, T3, and T4, it can be pointed out that the ordering of the Na binding energies reproduces the (Na, Al) population pattern, reported for synthetic Na-MOR. This result supports the suggestion of Simoncic et al. that there is a synergistic effect between cations and (Al, Si) order during the growth of the solid (Simoncic and Armbruster 2004). Finally, we remark that the binding properties found at T1 and T3 are different although both sites belong to the side pocket. This is related to a sum of different effects, the predominant one being most probably the global electrostatic potential felt by the cation. For the same reason, T1 and T4 present comparable binding properties, despite their different locations in the zeolite framework. It is important here to stress that these conclusions are made on the basis of the existence of two independent Al sites. If several (Al, Na) pairs are present within the side pockets, the distribution of the cations would most probably be governed by their mutual repulsion (Campana et al. 1997). For isolated sites, we have found that both T1 and T3 compensating cations are located in the side pocket area. In synthetic mordenites, at least two sites in side pockets are substituted by Al ( 2.5). Since the binding of NaC at T3 is favored with respect to that at T1, the three short Na-O bonds indicated by experiment (to O1, O10 and O9) can then be related to our calculated values at T3. In the presence of a second NaC to compensate Al in T1, one can expect that the mutual repulsion of the two cations will modify their position. Taking into account their relative stability, the experimental evidence that the other cation is pushed downward to the main cavity (Simoncic and Armbruster 2004) is supported by our results. Moreover we could argue further about Na-MOR with 6 Al per unit cell: this Al content would imply the presence of three (Na, Al) pairs in the side pockets (presumably two at T3 and one at T1). In this situation, one may expect that only two cations will remain inside the pocket, whereas the third one will be pushed outside due to cation-cation repulsion (this argument is verified by classical molecular mechanics simulations using Cerius2 associated with the cvff-aug-ionic force field (Demuth et al. 2001)). Al Sites of H-MOR As for the Na-MOR models, the protonated structures have been optimized, the original proton positions being those of the respective lowest MEP minima. At T4, the MEP well is situated at equivalent distances from the two O2 and O10. Whereas the NaC cation at T4 was found at equal distances from O10 and O2, the optimized position of the proton corresponds to a bonding to O2. In a previous theoretical study of the relative energies of various H-MOR models, the O2 protonation was also found to be more favorable than the O10 protonation (Li and Jensen 2002). Consistency with the mentioned relationship between Al-O-Si bond angle values and cation binding energies would lead one to expect comparable proton affinities at O2 and O10. At T1 and T3, the proton attaches to O6 and O9, respectively. Moreover, the O2-H and O6-H bonds are pointing toward the center of the 8m-ring which contains the T4 and T1 site, respectively, these bonds lying in the 8m-ring planes. The proton at T4 can thus be reached by a molecule being in the main

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channel as well as coming from the side pocket. In contrast, the O9-H bond at T3 is pointing toward the center of the side pocket. We could see that the Al-O bonds which do not connect to a proton are only little changed with respect to those of the Na-MOR models, with a maximum elongation of about 0.05 Å. As expected, the Al-O(H) bonds are elongated by about 0.12–0.16 Å with respect to the same bonds in the Na-MOR models. This effect is due to the covalent character of the OH bond, which induces the weakening of the Al-O bonds, whereas the NaC – oxygen interactions are weaker and essentially electrostatic. The presence of the sodium cation, bonded to two O (T4, T1) or one O (T3), had thus much less impact on the Si-O-Al framework than that of the proton. In contrast with the Na binding energies, we found that the proton affinity values are more sensitive to the inclusion of the GGA correction, leading to increased values with respect to LDA and to nonsignificant differences among the three acid sites (less than 0.08 eV) if one takes into account all methodological errors. From these values, we conclude that the sites belonging to the main channel and to the side pocket display similar proton affinities, showing thus that the strength of the OH bond is not dependent on the local structure of the studied sites. It is worth noting that proton affinities are not related to relative energies of models with different H sites, which are reported in previous theoretical work (Demuth et al. 2001), relative total energies being essentially related with relative Al site stabilities. In order to complete the description of the OH bond strength, the vibrational harmonic frequencies of O2-H and O9-H in the H-MOR1 model have been calculated. The values are 3667 cm1 for O9-H at T3 and 3659 cm1 for O2-H at T4, with IR intensities of 174 and 182 km/mol, respectively. These frequencies, not corrected by any scaling factor, can be considered similar within the theoretical error bar. This result is consistent with the corresponding proton affinity found for the proton at O9 (T3) with respect to O2 (T4). They are shifted toward higher energies by about 50 cm1 with respect to the reported infrared (IR) experimental asymmetric band at about 3610 cm1 . The comparison of our results with X-ray and IR experimental data allows us to suggest that the O9H stretching vibration (T3 site) is not responsible for the observed low-energy component at 3585 cm1 . Due to the similarity of the model and method used for both vibrations, it is most probable that the lower energy vibration is related to another Brønsted hydroxyl originating from: (i) Proton migration, after dehydration, to another oxygen, especially if both T1 and T3 sites are aluminated, leading to different local interactions; (ii) The presence of two T3 and one T1 aluminated sites in the side pocket (Simoncic and Armbruster 2004) will induce different MEP surfaces and different electrostatics; (iii) The presence of a (100) defect layer (Simoncic and Armbruster 2004) that induces geometry changes in the side pockets which are not reproduced in our models.

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We can thus argue that our models with isolated Al sites (Si/Al = 23) show that main channel and side pocket hydroxyls are energetically not different because of their framework environment. Their different energetical behaviors are most probably generated by the different content of framework Al at the 12-membered ring channels and in the side pockets obtained after crystallization.

Summary In this work, Na and protonated models of a mordenite zeolite, including Al in T1, T3, and T4 crystallographic sites, were studied using all-electron DFT calculations. From the analysis of the obtained results, the following conclusions can be drawn: (i) A good agreement with experimental bond length and bond angle data reported for synthetic Na-mordenites has been obtained; (ii) The binding energies of the Na cation follow the same ordering as the populations of Al T sites, as derived from X-ray measurements on synthetic Na-mordenites; this result shows the existence of a synergistic effect between cations and (Al, Si) order during the growth of the solid, as suggested experimentally; (iii) The compensating cation at the T1 site is found to be more stable in the side pocket than in the main channel, which questions the assignments of the T1associated proton to a main channel location; (iv) The calculated proton affinities at T1, T3, and T4 sites are equivalent, indicating that these Brønsted sites have similar acid strengths. One can thus infer that the different acidic behaviors at the different sites (Marie et al. 2000, 2004) do not originate from different acid strengths of these OH sites. This leads to the suggestion that the MOR acidic properties are more related to the electronic structure of the base than to effects of the solid framework. Thus, for the evaluation of differences in local acidity, the presence of the associated base (CO, CH3CN, NH3, etc. . . ) has to be taken into account. Further studies in this direction have been published by our laboratories (Dominguez-Soria et al. 2008, 2011).

Stability of Giant Fullerenes Fullerenes are carbon nanostructures formed by the closing of a graphitic sheet with the needed curvature supplied by intersecting, among a given number of graphitic hexagons, of 12 pentagons (Kadish and Ruoff 2007; Andreoni 2007). These carbon aggregates have been experimentally known for more than 20 years (Kroto et al. 1985) and, consequently, a large number of works, experimental as well as theoretical, focused on this subject (see, e.g., Seifert et al. 1996; Cioslowski 1995; Boltalina et al. 2000; Bühl and Hirsch 2001 and references therein). One main reason for the great interest in the study of fullerenes is certainly to be found in their particularly appealing geometrical form. The best-known fullerene

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is the so-called buckminsterfullerene that contains 60 carbon atoms (C60 ) and is composed of 12 pentagonal carbon rings located around the vertexes of an icosahedron and 20 hexagonal carbon rings at the centers of icosahedral faces (Kroto et al. 1985). Larger fullerenes that have an icosahedral symmetry can be constructed (Kroto and McKay 1988; Itoh et al. 1996) as well. These clusters, known as giant fullerenes, can be thought of as cutout pieces of graphene that are folded into their final shapes (icosahedrons). This kind of procedure generates 12 pentagonal carbon rings situated around vertices of an icosahedron, while all other carbon rings are hexagonal. Giant or large fullerenes have been the subject of different theoretical studies in the last years. We direct the interested reader to Dunlap et al. (1991), York et al. (1994), Bakowies et al. (1995), Scuseria (1995), Xu and Scuseria (1996), Scuseria (1996), Haddon et al. (1997), Bates and Scuseria (1998), Heggie et al. (1998), Geudtner et al. (2006), Dunlap and Zope (2006), Shao et al. (2006, 2007), Zope et al. (2008) and references therein. Most of these studies were focused either on understanding if the shape of these clusters is spherical or faceted (York et al. 1994; Bakowies et al. 1995; Scuseria 1995; Xu and Scuseria 1996; Scuseria 1996; Bates and Scuseria 1998; Geudtner et al. 2006), on calculating their response properties (Zope et al. 2008) or testing new algorithms developed for the investigation of large systems (Dunlap and Zope 2006; Geudtner et al. 2006). Most previous first-principles theoretical studies of large fullerenes have been performed either at the Hartree-Fock level of theory using symmetry restrictions and relatively small basis sets or employing analytic density functional theory (York et al. 1994; Bakowies et al. 1995; Scuseria 1995; Xu and Scuseria 1996; Scuseria 1996; Haddon et al. 1997; Bates and Scuseria 1998; Heggie et al. 1998; Dunlap and Zope 2006; Shao et al. 2006, 2007; Zope et al. 2008). In a recent study, we have performed state-of-the-art calculations on the large C180 , C240 , C320 and C540 fullerenes by employing the linear combination of Gaussian-type orbital density functional theory (LCGTO-DFT) approach (Calaminici et al. 2009). The structures of these clusters were fully optimized without any symmetry constraints. This work represents the first systematic study on large fullerenes based on non-symmetry adapted first-principles calculations, and it demonstrates the capability of ADFT for energy calculations and structure optimizations of large-scale structures without any symmetry constraint. In the next two sections, the computational details will be presented, and the most important results we have obtained in terms of structural changes, of the evolution of the bond lengths, and of the calculated binding energies will be reviewed.

Computational Details All calculations were performed using the DFT program deMon2k (Köster et al. 2011a). The exchange-correlation potential was numerically integrated on an adaptive grid (Köster et al. 2004a). The grid accuracy was set to 105 a.u. in all calculations. The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap, Connolly, and Sabin (Dunlap et al. 1979; Mintmire and Dunlap 1982). The calculation of the exchange-correlation energy was performed

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with the auxiliary function density (Köster et al. 2004b), i.e., auxiliary density functional theory was used. The structure optimizations were performed with the local density approximation (LDA) employing the Dirac exchange functional (Dirac 1930) in combination with the correlation functional from Vosko, Wilk, and Nusair (VWN) (Vosko et al. 1980). DFT-optimized double-zeta plus valence polarization (DZVP) all-electron basis sets optimized for local functionals (Godbout et al. 1992) were employed. For the structure optimization, a quasi-Newton method in internal redundant coordinates with analytic energy gradients was used (Reveles and Köster 2004). The geometry optimizations were performed using the parallel version of the deMon2k code (Calaminici et al. 2006; Geudtner et al. 2006). The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 104 and 103 a.u., respectively. The diamond and graphene calculations were performed in the same theoretical framework using the cyclic cluster model (CCM) (Janetzko et al. 2008). The obtained energies using the CCM are in the range of other calculated cohesive energies for graphene (Trickey et al. 1992; Dunlap and Boettger 1996). Because the fullerene and graphene calculations are performed within the same theoretical framework, the relative energy differences found here are reliable.

Results and Discussion The DFT-optimized singlet structures of C180 , C240 , C320 , and C540 are depicted in Fig. 10. These structures have been fully optimized at the all-electron level using DZVP basis sets in combination with the VWN functional. A long-standing discussion in the literature addresses the question of whether giant fullerenes prefer a faceted or a spherical shape. This question was raised considering pictures obtained by transmission electron microscopy (TEM) that have shown evidence of possible spheroidal structures in concentric carbon particles (Iijima 1980; Ugarte 1992, 1995). Using a divide-and-conquer method for density functional calculations, the structure and stability of C240 were studied, and the most stable structure was claimed to be highly spherical (York et al. 1994). However, this result was not confirmed by any successive theoretical work that have clearly shown evidence of a faceted shape for this fullerene (Dunlap et al. 1991; Bakowies et al. 1995; Scuseria 1995, 1996; Xu and Scuseria 1996; Haddon et al. 1997; Bates and Scuseria 1998). Depending on the viewing axis, simulated TEM of icosahedral fullerenes can provide either images with spherical or with faceted shapes (Scuseria 1995). In addition, an explanation of why experimental results showed rounder shapes for large fullerenes was also given (Scuseria 1995). As Fig. 10 shows, our first-principles-based structure optimizations predict that larger fullerenes, C240 , C320 , and C540 , prefer a faceted shape. Moreover, even for the smallest fullerene here studied, C180 , there is clear evidence that the faceted shape is preferred over a spherical shape if first-principles all-electron optimizations without any symmetry restriction are performed (see Fig. 10). Details about the timing of these calculations are reported in Geudtner et al. (2006) where it is shown that the optimization of such large fullerenes is feasible within a few days on a parallel architecture with 32 to 64 cores.

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Fig. 10 Optimized structure of C180 , C240 , C320 , and C540 fullerenes. The calculations have been performed with the VWN functional in combination with DZVP basis sets

We noticed that the obtained results are in agreement with most of the previous reported theoretical studies (Dunlap et al. 1991; Bakowies et al. 1995; Scuseria 1995, 1996; Xu and Scuseria 1996; Haddon et al. 1997; Bates and Scuseria 1998). In order to gain more insight into the structural changes of these systems as the number of carbon atoms increases, a detailed analysis of the bond length evolution was performed. In Fig. 11, the normalized number of bonds for C180 , C240 , C320 , and C540 are plotted versus the bond length. The dashed line at 1.419 Å represents the graphene bond length obtained from the periodic (CCM) deMon2k calculation. Most obvious from this figure is the difference of C320 to all other fullerenes. In fact, whereas usually a discrete distribution of bond lengths is found, in C320 , a wide, in some ranges almost continuous, bond length distribution is observed. This clearly indicates a break in the expected high symmetry of the system. Our studies show that the C320 fullerene possesses a ground-state potential energy surface (PES) of higher multiplicity, most likely either triplet or quintet. Further test calculations indicated that for this fullerene also in the cases of triplet and quintet PESs the continuous bond length distribution observed for the singlet PES persists. Of course, only non-symmetry adapted optimizations can lead to such a result. To the best of our knowledge, this symmetry breaking in larger fullerenes due to their electronic structure has never been observed in previous calculations. As Fig. 10 shows, the observed symmetry breaking does not alter the global shape of the giant fullerene.

Auxiliary Density Functional Theory: From Molecules to Nanostructures Fig. 11 Normalized number of bonds for C180 , C240 , C320 , and C540 versus the bond lengths (in Å)

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C540

normalized number of bonds

C320

C240

C180

1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 bond length [Å]

For the other systems, C180 , C240 , and C540 , the expected discrete bond length distribution is obtained, indicating that the symmetry of the electronic structures matches with the expected geometrical symmetry. In these systems, the number of different bond lengths increases with system size, and an accumulation of bond lengths around the graphene bond length is observed (Fig. 11). More surprising is the trend that the longest bond length in the cluster shortens with increasing cluster size. This indicates that delocalization increases with cluster size despite the global building pattern, i.e., the appearance of 12 pentagons. With the aim of guiding future desirable experiments on large fullerenes and to gain more information about their stability, we have also explored the behavior of the binding energy of the studied fullerenes with increasing fullerene size. The results of the uncorrected binding energy (in eV) per carbon atom obtained with the VWN functional have been illustrated in Figure 3 of Calaminici et al. (2009) showing that the binding energy increases monotonically with the increase in the number of carbon atoms. This indicates that the large fullerenes become more and more stable with increasing size. However, the increase turned out to be very moderate. We have also included the basis set superposition error (BSSE)

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in the calculation of the binding energies. The inclusion of the BSSE decreases the calculated binding energies of each studied fullerene by 0.02 eV but without altering the trend (Calaminici et al. 2009). From the comparison of the CCM calculations, we found that the calculated binding energy of C540 of 8.75 eV is very close to the cohesive energy of diamond (8.78 eV). This result indicates that C540 has a similar binding energy to diamond which fuels the hope that such giant fullerenes could indeed be prepared. However, the binding energy of even the largest studied fullerene, C540 , is still far away from the corresponding value in graphene which was calculated to be 8.91 eV (Janetzko et al. 2008).

Summary In this section, the results obtained from state-of-the-art density functional theory calculations performed on large fullerenes, such as C180 , C240 , C320 , and C540 , have been reviewed. The study was carried out with all- electron basis sets, and all structures were fully optimized without any symmetry restriction. The results obtained can be summarized as follows: (i) This work confirms that for all large fullerenes studied here, a faceted shape is preferred over the spherical shape; (ii) The analysis of the bond length evolution reveals for C320 a qualitatively different pattern than for the other fullerenes. The most likely explanation for this difference is a symmetry breaking in the electronic structure of this large fullerene; (iii) The shortening of the longest bond length with increasing cluster size indicates that delocalization increases with cluster size; (iv) The calculated binding energies are in the range of diamond but considerably below the graphene value. Thus, even giant fullerenes as those reviewed in this section are only metastable.

Transition State Search in Metal Clusters Generally, in cluster studies, the properties of interest are investigated considering the lowest minimum structure. However, the study of the full potential energy surface (PES) landscape is very important if one aims to understand how for a given finite system a chemical reaction may occur. A chemical reaction can be described in terms of an energy profile picture with which aspects of the mechanisms, structures, and energies of the reaction are explained. A way to characterize a PES is by the location of its relevant critical points (CPs). In general, stationary points of a function that depends on N variables are reached when the gradient of the given function vanishes. For a multidimensional function such as the PES, several different kinds of stationary points usually exist. These are minima, maxima, and saddle stationary points. These CPs are characterized by the eigenvalue spectrum of the corresponding Hessian matrix. A minimum CP possesses only positive eigenvalues

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that correspond to positive curvatures, in all principal directions. A maximum possesses only negative eigenvalues that correspond to negative curvatures in all principal directions. Saddle points of different order can be present on a PES, too. A transition state is a saddle point of first order, i.e., it possesses a negative curvature in one principle direction and positive curvatures in all other directions. From a chemical point of view, only minima and first-order saddle points are CPs of interest on a PES because they correspond to stable isomers and to transition states, respectively. For a molecular system, the reaction coordinate is defined by the minimum energy path on the PES that connects the energy minimum of the reactant with that of the product. For an elementary reaction step, this path has only one maximum which is a first-order saddle point and named the transition state. For metal clusters, the PES of even very simple system consisting of only a small number of atoms can possess a large number of local minimum structures. As a result, a complex network of rearrangement reactions exists in these systems. To unravel such networks, the systematic location of transition states is mandatory. We applied the hierarchical transition state search algorithm (Campo and Köster 2008) for the first time to rearrangement reactions of pure metal clusters, in particular selected sodium clusters. These systems are very challenging because neither the transition states nor the reaction coordinates are chemically intuitive. Therefore, it is usually not guaranteed that the two initial minima structures are indeed connected by an elementary rearrangement step free of intermediates. To this end, we performed intrinsic reaction coordinate (IRC) calculations starting from the located transition states. By the combination of the hierarchical transition state search algorithm with the IRC calculations, reaction intermediates can be found. In this way, complex rearrangement networks can be completely unraveled.

Computational Details All calculations were performed using the linear combination of Gaussian-type orbital density functional theory (LCGTO-DFT) deMon2k program (Köster et al. 2011a; Geudtner et al. 2011). The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap, Connolly, and Sabin (Dunlap et al. 1979; Mintmire and Dunlap 1982). The auxiliary density was expanded in primitive Hermite Gaussian functions using the GEN-A2 auxiliary function set (Calaminici et al. 2007a). The exchange-correlation functional was evaluated with this auxiliary density, i.e., the auxiliary density functional theory (ADFT) method was used (Köster et al. 2004b). The exchange-correlation energy and potential were numerically integrated on an adaptive grid (Krack and Köster 1998; Köster et al. 2004a). The grid accuracy was set to 105 a.u. in all calculations. The structure optimization and the frequency analysis of the clusters were performed employing the gradient corrected exchange-correlation functional proposed by Perdew, Burke, and Ernzerhof (PBE) (Perdew et al. 1996a) in combination with DFT-optimized double-zeta valence plus polarization (DZVP) basis sets (Godbout et al. 1992). In order to determine the geometry of the ground state and of corresponding low-lying states of the studied clusters, a very extensive search was performed

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and several dozens of initial configurations were studied. In this search, structures previously reported by other authors were included as well as many other initial structures which were extracted from BOMD trajectories recorded at a temperature of 250 K. The BOMD trajectories were generated in the canonical ensemble employing a Nosé-Hoover chain thermostat (Nosé 1984; Hoover 1985; Martyna et al. 1992; Gamboa et al. 2010) with seven thermostats in the chain and a coupling frequency of 200 cm1 . For all structures, a full geometry optimization without any symmetry restriction was performed using a quasi-Newton optimization method in delocalized internal coordinates (Reveles and Köster 2004). The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 104 and 103 a.u., respectively. The obtained minima were characterized by frequency analysis. The second derivatives were calculated by numerical differentiation (two-point finite difference) of the analytic energy gradients using a displacement of 0.001 a.u. from the optimized geometry for all 3N coordinates. The harmonic frequencies were obtained by diagonalizing the mass-weighted Cartesian force constant matrix. As already discussed, the transition states were located with the hierarchical transition state finder as implemented in the deMon2k code (Campo and Köster 2008; Geudtner et al. 2011). The only input information consisted of the minimum structures of the sodium clusters involved in the rearrangement. Starting from the obtained transition states, the IRC was calculated, using the approach from Gonzalez and Schlegel (Gonzalez and Schlegel 1989), either to find potential intermediates or to calculate the minimum energy path that connects the two starting minimum structures. Both the IRC and the uphill trust region optimization in the hierarchical transition state search were initialized by Hessian calculations in order to provide the correct eigenvalue spectrum.

Showcase Results In this section, the results of selected transition state searches in Na8 and Na10 clusters are discussed. The two lowest minimum structures of Na8 are depicted in Fig. 12 (top), as well as the lowest energy structure and a planar high-energy stable isomer for the Na10 cluster (bottom). All minimum structures are in their lowest multiplicity. The relative energies of these structures with respect to the ground-state structure are given in kcal/mol in this figure. The Na8 ground-state structure is a slightly distorted snub disphenoid formed by a square bipyramidal fragment with two extra Na atoms capping two triangular faces. The first low-lying minimum structure is formed by five slightly distorted tetrahedral fragments and lies 2.3 kcal/mol above the ground-state structure. The hierarchical transition state finder was applied with these two minimum structures as input. As result, a transition state characterized by an imaginary frequency of ! D 24i cm1 was found. Figure 13a illustrates some snapshots along the IRC for this cluster rearrangement reaction. The second structure in Fig. 13a is the transition state structure. The very small energy difference of only 0.2 kcal/mol between the high-energy isomer and the transition state challenges the transition state search algorithm. The hierarchical transition state finder worked flawlessly in the Na8 rearrangement

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Fig. 12 Lowest minima structures of the Na8 clusters (top) and lowest minimum structure with a planar high-energy structure of Na10 (bottom). The energy differences between the found ground-state structures (left) and the other structure (right) are in kcal/mol

investigated here. The above discussed rearrangement is, however, not the most common rearrangement in low-temperature Na8 molecular dynamics. Instead, rearrangements that transfer the minimum structure of Na8 into a mirror image are typical for the low-temperature dynamics of this cluster. Also this kind of rearrangement can be studied with the hierarchical transition state finder if the atoms of the reactant and product structures are correctly labeled (otherwise the initial alignment algorithm will superimpose the two identical structures). The corresponding IRC is depicted in Fig. 13b. At the starting point (first snapshot of Fig. 13b), the square bipyramidal fragment of the ground-state structure is at the right side of the structure. Following the black point on the energy path, we can see how the structure deforms going uphill to the transition state (second snapshot of Fig. 13). This transition state is characterized by an imaginary frequency of ! = 22i cm1 . The energy difference between this transition state and the groundstate structure is 0.9 kcal/mol. Once the transition state is formed, the structure deforms then downhill to reach again the ground-state structure configuration at the end point of the energy path (last snapshot of Fig. 13b). At this last point, the square bipyramidal fragment, characteristic of the ground-state structure, is now on the left side of the structure. The lowest minimum structure we found for the Na10 cluster is depicted at the bottom of Fig. 12 (left). We notice that as the cluster size increases, it becomes more complicated to locate the ground-state structure due to the increasing number of low-lying minima. The ground-state structure is built by two pentagonal bipyramidal moieties which are perpendicular to each other. Lying at 11.42 kcal/mol above the ground state, we found a completely planar minimum structure. The hierarchical transition state finder was applied considering this structure and the ground-state structure as input (see Fig. 12 bottom). Due to the shape of the two structures involved as reactant and product, this reaction mechanism is not chemically intuitive. Our result shows that also this rearrangement is free of intermediates. Snapshots along the corresponding IRC are illustrated in Fig. 13c. The activation energy is 1.5 kcal/mol and an imaginary frequency of ! D 26i cm1 characterizes the transition state.

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Fig. 13 Snapshots along the IRC for the Na8 rearrangement between the two lowest minimum structures of this cluster (a), between two mirrored images (b), and rearrangement between the lowest minimum and a planar structure for Na10 (c). The energy is plotted versus the IRC. The structure position is indicated by the black point on the energy path

Summary In this work, a hierarchical transition state algorithm which combines the socalled double-ended interpolation method with the uphill trust region method was applied to search for transition states of selected sodium clusters. These systems are prototypes of finite systems for which transition states are not chemically intuitive. Sodium clusters Na8 and Na10 have been investigated, and selected transition states have been analyzed. All found transition states have been confirmed by the intrinsic reaction coordinate calculations to connect the reactant and product of the corresponding isomerization reaction. The successful application of the hierarchical

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transition state finder to highly unsymmetrical reaction paths with small activation barriers underlines the robustness of this approach. A particular example is the Na8 cluster rearrangement. Again it is important to note that no other information than the minimum structures of reactants and products are used in these transition state optimizations. We also showed for Na8 that with the hierarchical transition state finder transition states between symmetry-equivalent minima can be found if reactant and product structures are correctly labeled. Similar studies for other metal clusters are currently underway in our laboratories.

Conclusion This review highlights recent developments in auxiliary density functional theory (ADFT) and their implementation in deMon2k. The simplifications associated with ADFT permit an efficient parallel code structure that is suitable for research applications in the nano-regime with chemical accuracy. The presented BornOppenheimer molecular dynamics simulation shows that simulation times on the nanosecond time scale can be reached with ADFT. As the applications presented here show, this opens new and exciting perspectives for computational chemistry and material simulations with first-principles methods. The development of ADFT in deMon2k is also a very educational example for the interplay of theoretical chemistry and computer science. The simplifications of the Kohn-Sham method by employing an atom-centered auxiliary density for the calculation of the Kohn-Sham potential yield simple and very efficient parallel algorithms. A typical example is the SCF acceleration by the MinMax procedure. Because only fitting vectors are involved, it can be applied to systems with hundreds of atoms like the zeolites and fullerenes discussed here. Another example is the formulation of the non-iterative auxiliary density perturbation theory (ADPT) that substitutes the computationally cumbersome CPKS method in ADFT. Similarly, the ADFT-GIAO methodology delivers magnetic properties that are indistinguishable from conventional DFT approaches but with a fraction of the computational cost. With the new variational fitting of exact exchange, almost all chemically relevant exchange-correlation functionals can now be used within the ADFT framework. The increased use of ADFT in the quantum chemical community over the last years has considerably enhanced our understanding of this fitting approach. While the present authors are responsible for errors or omissions in this small review, the credit for the advances in ADFT and ADPT is shared with a much wider group cited in the references here and elsewhere. Acknowledgements Financial support from CONACYT (U48775, 60117-F, CB-179409), ICYTDF (PIFUTP08-87), and CIAM (107310) is greatfully acknowledged. Parts of this review have been realized with the help of the bilateral CONACYT-CNRS project 16871.

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Zhao, Q., Morrison, R. C., & Parr, R. G. (1994). Physical Review A, 50, 2138. Zhao, Y., & Truhlar, D. G. (2006). Theoretical Chemistry Accounts, 120, 215. Zope, R. R., Baruah, T., Pederson, M. R., & Dunlap, B. I. (2008). Physical Review B, 77, 115452. Zuniga-Gutierrez, B., Geudtner, G., & Köster, A. M. (2011). Journal of Chemical Physics, 134, 124108. Zuniga-Gutierrez, B., Geudtner, G., & Köster, A. M. (2012). Journal of Chemical Physics, 137, 094113. Zuniga-Gutierrez, B., Camacho Gonzalez, M., Simon Bastida, P., Bendana Castillo, A., Calaminici, P., & Köster, A. M. (2015). Journal of Physical Chemistry A, 119, 1469. Zuniga-Gutierrez, B., Camacho Gonzalez, M., Simon Bastida, P., Bendana Castillo, A., Calaminici, P., & Köster, A. M. (2015). Journal of Chemical Physics, 143, 104103.

Guide to Programs for Nonrelativistic Quantum Chemistry Calculations Tao Zeng and Mariusz Klobukowski

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GAMESS-US . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Firefly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GAMESS-UK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dalton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NWChem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ORCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PSI3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACES II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACES III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFOUR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COLUMBUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commercial Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molcas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q-Chem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbomole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molpro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jaguar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PQS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HyperChem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ampac 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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T. Zeng () Department of Chemistry, Carleton University, Ottawa, ON, Canada e-mail: [email protected] M. Klobukowski Department of Chemistry, University of Alberta, Edmonton, AB, Canada e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_17-2

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Abstract

This chapter reviews most of the widely used nonrelativistic quantum chemistry program packages. Considering that information about availability and capabilities of the free quantum chemistry programs is more limited than that of the commercial ones, the authors concentrated on the free programs. More specifically, the reviewed programs are free for the academic community. Features of these programs are described in detail. The capabilities of each free program can generally be categorized into five fields: independent electron model; electron correlation treatment; excited state calculation; nuclear dynamics including gradient and hessian; and parallel computation. Examples of input files for the Møller–Plesset calculation of formaldehyde are presented for most of the free programs to illustrate how to create the input files. The main contributors of each free program and their institutions are also introduced, with a brief history of program development if available. All the key references of the cited algorithms and the hyperlinks of the home page of each program (both free and commercial) are given in this review for the interested readers. As the most important information of every cited free program’s documentation has been extracted here, it is appropriate to consider this chapter to be the manual of manuals.

Introduction Computational quantum chemistry has been experiencing recently a period of tremendous growth. As the capabilities of computer hardware increased, so did the appetites of researchers for modeling tools. Many computer packages became available, both commercially and as free programs, the latter usually requiring a simple registration procedure to obtain and use the code. The free access is usually restricted to researchers from academic institutions and the programs are sometimes unsupported; however, an associated users’ discussion forum often accompanies the program. In this chapter, we will review the software that is currently available, focusing on packages that are familiar to the reviewers. In addition to outlining the unique capabilities of each program, we added the URL links to the home pages to allow the readers to follow the latest developments of each code. The programs that use only the density functional theory and the programs for relativistic calculations at the level of Dirac theory and beyond are reviewed in other chapters. All the reviewed programs carry out fundamental tasks of a computational chemist or a computational molecular physicist: calculation of energy for various Hamiltonians; evaluation of gradients of energy (needed to locate stationary points on the potential energy surface); evaluation of the energy hessian (required to analyze the character of the located stationary point, identify local minima and saddle points, and perform vibrational frequency calculation); and evaluation of basic properties (population analysis, dipole moments). The components of the programs include basis set libraries and pseudopotentials.

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At the Hartree-Fock level (Hartree 1928; Fock 1930), the energies for closedshell systems are evaluated using the restricted Hartree-Fock (RHF) method (Hall and Lennard-Jones 1951; Roothaan 1951). For the open-shell molecules, there are several methods that are available in most programs: the unrestricted Hartree-Fock (UHF) method (Pople and Nesbet 1954), several variants of the restricted open-shell Hartree-Fock (ROHF) method (Hsu et al. 1976; McWeeny and Diercksen 1968), and the generalized valence bond (GVB) method (Bobrowicz and Schaefer 1977). The wavefunction based on a single-configuration Hartree-Fock method is often inadequate to describe bonding in a molecule, and in such a case several multiconfiguration methods are used; the approach is usually called multiconfiguration self-consistent field (MCSCF) (Olsen et al. 1983; Roos 1983, 1994; Schmidt and Gordon 1998; Shepard 1987; Werner 1987), and its practical implementations are the full orbital reaction space (FORS) (Ruedenberg et al. 1982a, b, c) and complete active space (CAS) (Roos 1987). Beyond Hartree-Fock, the energies (and wavefunctions) may be improved at several levels: various orders of the perturbation method at the Møller–Plesset level (1934) (MP2, MP3, MP4); for open-shell systems we could use either unrestricted MP2 (Pople et al. 1976) or one of several variants of the perturbation method based on the ROHF wavefunction: the Z-averaged perturbation theory (ZAPT) (Lee and Jayatilaka 1993; Lee et al. 1994) and RMP (Knowles et al. 1980; Lauderdale et al. 1991); configuration interaction (CI) method (Brooks and Schaefer 1979; Ivanic and Ruedenberg 2001). Very accurate energies may be obtained using the coupled-cluster theory (CC) (Paldus 2005; Shavitt and Bartlett 2009). Excited states may be studied using the general post-Hartree-Fock methods listed above, or some specialized techniques, such as configuration interaction with single substitutions (CIS) (Foresman et al. 1992), time-dependent density functional theory (TDDFT) (Dreuw and Head-Gordon 2005; Elliott et al. 2009), and equations-ofmotion coupled cluster (EOM-CC) (Kowalski and Piecuch 2004; Włoch et al. 2005). Solvent effects may be treated using several models: self-consistent reaction field (SCRF) (Karelson et al. 1986, 1993; Kirkwood 1934; Tapia and Goscinski 1975), polarizable continuum model (PCM) (Cammi and Tomasi 1995; Miertuš et al. 1981; Tomasi and Persico 1994; Tomasi et al. 2005), surface and simulation of volume polarization for electrostatics (SS(V)PE) (Chipman 1997, 2000, 2002), and conductor-like screening model (COSMO) (Baldridge and Klamt 1997; Klamt 1995; Klamt and Schüürmann 1993). Even though the programs described in this chapter are referred to as “nonrelativistic” (i.e., using single-component wavefunction), for molecules containing heavy atoms it is necessary to include at least scalar relativistic effects. A popular method of adding relativistic corrections is based on the formalism developed by Douglas and Kroll (1974) and by Hess (1986, 1989), and is usually referred to as DKH n (where n denotes the order of the method). Some scalar relativistic effects are included implicitly in calculations if pseudopotentials for heavy atoms are used to mimic the presence of core electrons; there are several families of pseudopotentials available: the effective core potentials

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(ECP) (Cundari and Stevens 1993; Hay and Wadt 1985; Kahn et al. 1976; Stevens et al. 1984), energy-adjusted pseudopotentials (Cao and Dolg 2006; Dolg 2000; Peterson 2003; Peterson et al. 2003), averaged relativistic effective potentials (AREP) (Hurley et al. 1986; LaJohn et al. 1987; Ross et al. 1990), model core potentials (MCP) (Klobukowski et al. 1999), and ab initio model potentials (AIMP) (Huzinaga et al. 1987). For very large molecular systems several programs offer semiempirical Hamiltonians (Stewart 1990), including Modified Neglect of Differential Overlap (MNDO) (Dewar and Thiel 1977), Semi-Ab initio Model 1 (SAM1) (Dewar et al. 1993), Austin Model 1 (AM1) (Dewar et al. 1985), Parametric Method 3 (PM3) (Stewart 1989), and Parametric Method 6 (PM6) (Stewart 2007) (the PM3 parameterization is available in the MOPAC2009 program which is freely available to academics, see http://openmopac.net/downloads.html). Zerner’s modification of the Intermediate Neglect of Differential Overlap approach (ZINDO) (Ridley and Zerner 1973) (see also Zerner 1991) is available in the ArgusLab program http://www.arguslab.com/ index.htm. The present Guide will focus on those programs that are freely available to academic community. In describing the capabilities of the programs we heavily borrowed from the available documentation. Commercial programs for ab initio computational chemistry (including several that employ semiempirical methods) will be mentioned only very briefly, guiding the reader to relevant home pages on the Internet. Several collections of references to quantum chemistry software may be found on the Internet http://en.wikipedia.org/wiki/Quantum_chemistry_computer_programs.

Free Software Free access to computer programs discussed in the present section is sometimes restricted to academic researchers. Please consult the Internet links for each program for specific restrictions.

GAMESS-US GAMESS-US (GAMESS D General Atomic and Molecular Electronic Structure System) is one of the early programs for the large-scale ab initio calculations in the public domain (Gordon and Schmidt 2005; Schmidt et al. 1993). The origins of GAMESS-US go back to the National Resources for Computations in Chemistry (NRCC), where the original GAMESS was assembled before 1981 (Dupuis et al. 1980). Since 1982 the development of GAMESS-US was carried out in the research group of Mark Gordon, first at North Dakota State and then at Iowa State University, with several research groups across the world making their contributions. The code is a good illustration of the care and effort that go into the development of any modern ab initio code: the October 2010 (Release 2) version has 769,161 lines of code and 221,147 lines of comments.

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The capabilities of GAMESS-US are listed below: Analytic gradients and hessians for RHF, ROHF, UHF, GVB, and MCSCF wavefunctions that are used to locate stationary points on the potential energy surface and identify their character (local minimum or transition state). MP2 energy and gradients for RHF, ROHF, and UHF wavefunctions. CI energy for very general CI excitation schemes (Ivanic 2003a, b; Ivanic and Ruedenberg 2001). Several methods of evaluating the CC energy, such as CC with single and double excitations (CCSD), with perturbative triple excitations (CCSD(T)), and including the latest completely renormalized (CR) CR-CC(2,3) method (Piecuch and Włoch 2005; Piecuch et al. 2002); numerical gradients for the CC methods are available. Excited states may be treated using the CIS, TDDFT, multiconfigurational quasidegenerate perturbation theory (MCQDPT) (Nakano 1993a, b), and equation-ofmotion coupled-cluster (EOM-CC) methodologies. There are many ubiquitous density functionals available. Extensive basis set libraries. Pseudopotentials: both effective core potentials and model core potentials are available. The intrinsic reaction coordinate method allows for checking whether the path from a transition state connects the two minima that correspond to the required reactant and product. Anharmonic vibrational analysis via the vibrational self-consistent field (VSCF) method (Chaban et al. 1999). Many molecular properties may be calculated (static polarizability and hyperpolarizability; frequency-dependent polarizability; electric moments; electric field, and electric field gradient). Solvent effects may be studied via continuum solvation methods using effective fragment potentials (EFP) (Adamovic et al. 2003; Day et al. 1996; Gordon et al. 2007; Jensen et al. 1984), polarizable continuum model (PCM), surface and simulation of volume polarization for electrostatics (SS(V)PE), conductor-like screening model (COSMO), and self-consistent reaction field (SCRF). A unique feature of the program is its ability to carry out all-electron calculations based on the Fragment Molecular Orbital (FMO) method (Fedorov and Kitaura 2007, 2009). Several possibilities for carrying out multireference perturbation calculations; spinorbit coupling may be treated as a perturbation. For spin-orbit coupling CI wave functions, two-component natural orbitals (natural spinors) may be formed and analyzed (Zeng et al. 2011a, b) Scalar relativistic effects may be included at the levels of DKH2, DKH3, normalized elimination of small components (NESC) (Dyall 2002), and relativistic elimination of small components (RESC) (Nakajima and Hirao 1999) methods. Semiempirical Hamiltonians (AM1, PM3, and MNDO) are available. The input style is illustrated below for the MP2 energy calculation for formaldehyde H2 CO at the experimental geometry r(CH) D 1. 116 Å, r(CO) D 1. 208 Å. The

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basis set is the correlation consistent cc-pVDZ basis (Dunning 1989), used in the form of spherical Gaussian functions. C2v point group symmetry is used. In this example, the geometric structure of formaldehyde is defined in terms of internal coordinates (bond lengths, bond angles, and dihedral angles) in the representation known as Z-matrix. For H2 CO, the first line in the Z-matrix definition specifies the starting atom, in this case oxygen O. The second line, C 1 1.208, defines the second atom, carbon, which is connected to the first atom, and the CO bond length equals 1.208 Å. The first hydrogen atom is defined on the third line, H 2 1.116 1 121.75; this H atom is connected to the second atom (carbon), the CH bond length equals 1.116 Å, and the bond angle H–C(2)–O(1) equals 121.75ı . Finally, the fourth line, H 2 1.116 1 121.75 3 180.0, defines the second hydrogen atom, which is connected to the second atom (C) with the CH bond length of 1.116 Å, the bond angle H–C(2)–O(1) of 121.75ı , and the dihedral angle H(4)– C(2)–O(1)–H(3) of 180.0ı . $contrl mplevl=2 runtyp=energy coord=zmt ispher=1 $end $basis gbasis=ccd $end $data H2CO at experimental geometry (CRC, p. 9-36) cnv 2 O C 1 1.208 H 2 1.116 H 2 1.116 $end

1 121.75 1 121.75

3 180.0

The home page of the program is http://www.msg.ameslab.gov/GAMESS/.

Firefly Formerly known as PC-GAMESS, the Firefly was developed on the basis of GAMESS-US, ending with the 1999 version. The program has been extensively modified, with large sections written to increase the efficiency of the program. The program is available as binaries for Windows-based computers as well as the Macintosh computers. The development of the code took place at the lab of A. A. Granovsky (Moscow State University). Some interesting features of the program are The Møller–Plesset energy corrections may be calculated up to the fourth order (MP2, MP3, and MP4). Compression of files allows for very large calculations to be carried out; the compression is accomplished via packing of integrals, matrix elements, and their indices reduce file size and input/output (I/O) time. Faster MCQDPT algorithm and availability of XMCQDPT implementation.

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Integral code allows for fast evaluation of generally contracted basis sets (Raffenetti 1973), such as atomic natural orbitals (ANO) (Almlöf 1987). Sample input file is illustrated below for the same MP2 energy calculation. At present, the program’s internal library does not have the cc-pVDZ basis set; in order to avoid the explicit (and thus rather lengthy) definition of this basis, the double-zeta basis set of Dunning and Hay (1977) is used instead. This input file also illustrates an alternative definition of the geometric structure of a molecule: in terms of Cartesian coordinates (according to Firefly’s defaults, in Ångströms), with only the symmetry-nonequivalent atoms defined. Each line defining atomic Cartesian coordinates specifies also the atom name and its nuclear charge. $contrl mplevl=2 runtyp=energy ds=1 $end $basis gbasis=DH $end $data H2CO at experimental geometry (CRC, p. 9~36) cnv 2 8.0 O 6.0 C 1.0 H $end

0.0000000000 0.0000000000 0.9489930831

0.0000000000 0.0000000000 0.0000000000

~0.6036077039 0.6043922961 1.1916470349

Home page http://classic.chem.msu.su/gran/gamess/index.html.

GAMESS-UK This code also originated from the NRCC code (Dupuis et al. 1980) and has been extensively modified by the Daresbury Laboratory (UK) group that included M. F. Guest, J. H. van Lenthe, J. Kendrick, K. Schöffel, P. Sherwood, and R. J. Harrison. A good summary of the code’s features has been published (Guest et al. 2005). A summary of the program’s capabilities is given below: Gradients and hessians for many energies (RHF, ROHF, UHF, MCSCF, CASSCF). Energies of the excited states and corresponding transition moments may be calculated using the multireference CI method (MR-DCI). For the studies of electronically excited states, direct random-phase approximation (RPA) and multiconfigurational linear response (MCLR) (Fuchs et al. 1993) excitation energies and oscillator strengths are also available. Ionization potentials may be calculated using the Green’s function methodology (Cederbaum and Domcke 1977) and the two-particle-hole Tamm-Dancoff method (Schirmer and Cederbaum 1978). MP2 (energies and gradients) and MP3 (energies). Multireference second-order perturbation method (CASPT2 method); multireference MP3 method (MR-MP3) is also available. CCSD and CCSD(T) energies.

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Many density functionals for the density functional theory calculations. Many properties may be evaluated, among them atomic charges (derived from Mulliken and Löwdin population analysis, natural population analysis (Read and Weinhold 1983; Read et al. 1985), and electrostatic potentials), polarizabilities, hyperpolarizabilities, and magnetizabilities. Semiempirical Hamiltonians (MNDO, MINDO/3, AM1, PM3, PM5, and MNDO-d) are available. Pseudopotentials (Hay and Wadt 1985; Roy et al. 2008) may be employed, with many pseudopotentials included in the library. The effects of the solvent may be modeled using the direct reaction field (DRF) method (de Vries et al. 1995; Duijnen and de Vries 1996). Relativistic effects may be treated using the zeroth-order regular approximation (ZORA) (Faas et al. 1995). Sample input file for the H2 CO calculation at the MP2/cc-pVDZ level of theory is shown below; notice that the symmetry of the molecule is determined by the program. title H2CO at experimental geometry (CRC, p. 9-36) zmatrix angstrom O C 1 1.208 H 2 1.116 1 121.75 H 2 1.116 1 121.75 3 180.0 end harmonic basis cc-pVDZ scftype mp2 enter

Home page is http://www.cfs.dl.ac.uk/gamess-uk/index.shtml.

Dalton To quote from the authors of the program, “the Dalton program system is designed to allow convenient, automated determination of a large number of molecular properties based on an HF, density functional theory (DFT), MP2, coupled cluster, or MCSCF reference wave function.” Given the great flexibility of Dalton’s computational capabilities, the authors describe it as directed at experts in ab initio calculations. The program has been developed at the University of Oslo. A very short list of Dalton’s capabilities is shown below; for those interested in the evaluation of molecular properties we suggest that Dalton’s manual be consulted (http://www.kjemi.uio.no/software/dalton/dalton.html).

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Complete Active Space (CAS) or Restricted Active Space (RAS) wave functions. Calculation of excited states and core hole states. Several coupled-cluster (CC) wave functions. MP2 and explicitly correlated R12-MP2 triples models. Dalton can calculate a large variety of molecular properties at many levels of theory: linear, quadratic, and cubic frequency-dependent response properties. Linear second-order polarization propagator approach (SOPPA) (Nielsen et al. 1980). NMR properties (magnetizabilities, nuclear shieldings, and all contributions to nuclear spin-spin coupling constants). EPR properties (electronic g-tensor, hyperfine coupling tensor, and zero-field splitting tensor). Circular dichroism properties (electronic circular dichroism, vibrational circular dichroism, and Raman optical activity). Gauge-origin independent magnetic properties. Solvent effects may be modeled via the self-consistent reaction field (SCRF) to arbitrary order in the multipole expansion. First-order properties for ground states: dipole and quadrupole moments, second moments of the electronic charge distribution and electric field gradients at the nuclei, as well as scalar-relativistic one-electron corrections (Darwin and mass velocity). Relativistic two-electron Darwin correction to the ground state. Excitation energies may be calculated for several levels of the coupled-cluster wavefunctions. Density functional theory. Effective core potentials. Input file for the calculation of the MP2/cc-pVDZ energy of H2 CO is shown below; the symmetry of the molecule is determined by the program. BASIS CC-pVDZ Z-matrix input 4 0 ZMAT O 1 8.0 C 2 1 1.208 H 3 2 1.116 H 4 2 1.116

6.0 1 121.75 1.0 1 121.75 3 180.0 0 1.0

**DALTON INPUT -RUN WAVE FUNCTION **WAVE FUNCTIONS

.HF .MP2 *MP2 INPUT .MP2 FROZEN 2 0 0 0 **END OF INPUT

Home page http://www.kjemi.uio.no/software/dalton/dalton.html.

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NWChem The program has been developed in the Molecular Sciences Software Group at the Pacific Northwest National Laboratory (Richland, WA, USA). The authors of the program state that “NWChem is a computational chemistry package designed to run on high-performance parallel supercomputers. Code capabilities include the calculation of molecular electronic energies and analytic gradients using HartreeFock self-consistent field (SCF) theory, Gaussian density function theory (DFT), and second-order perturbation theory. For all methods, geometry optimization is available to determine energy minima and transition states.” Furthermore, they add that “Classical molecular dynamics capabilities provide for the simulation of macromolecules and solutions, including the computation of free energies using a variety of force fields.” Details of the NWChem program system were described (Kendall et al. 2000). Features of the program include Spin-orbit density functional theory. Electron correlation energies may be calculated using many methods, including linearized coupled-cluster doubles (LCCD), coupled-cluster doubles (CCD), linearized coupled-cluster singles and doubles (LCCSD), coupled-cluster singles and doubles (CCSD), coupled-cluster singles, doubles, and triples (CCSDT), coupled-cluster singles, doubles, and active triples (CCSDTA), coupled-cluster singles, doubles, triples, and quadruples (CCSDTQ), CCSD and perturbative connected triples (CCSD(T)), CCSD and perturbative connected triples (CCSD[T]), completely renormalized CCSD[T] method (CR-CCSD[T]), completely renormalized CCSD(T) method (CR-CCSD(T)), CCSD and perturbative locally renormalized CCSD(T) correction (LR-CCSD(T)), quadratic configuration interaction singles and doubles (QCISD), configuration interaction singles and doubles (CISD), configuration interaction singles, doubles, and triples (CISDT), configuration interaction singles, doubles, triples, and quadruples (CISDTQ), second-, third-, and fourth-order Møller–Plesset perturbation theory (MP2, MP3, and MP4). Multiconfiguration SCF wavefunctions. Selected configuration interaction with perturbation correction. Effective core potentials that allow to include both the scalar relativistic effects and spin-orbit effects (Pacios and Christiansen 1985). Relativistic corrections may be added at the levels of DKH2 and ZORA. Solvent effects may be described using COSMO (Baldridge and Klamt 1997; Klamt 1995; Klamt and Schüürmann 1993). For the studies of excited states, the CIS, time-dependent Hartree-Fock (TDHF), and TDDFT methods are available. Anharmonic vibrational analysis may be carried out using the VSCF method (Chaban et al. 1999).

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Sample input file for the H2 CO calculation at the MP2/cc-pVDZ level of theory is shown below; the symmetry of the molecule is determined by the program. The Cartesian coordinates of all atoms must be defined, and the atomic symbol must be used as atom name, eliminating the need to define nuclear charges. echo start h2co geometry units angstrom O 0.00000 C 0.00000 H 0.94899 H –0.94899 end basis O library cc-pvdz C library cc-pvdz H library cc-pvdz end task mp2 energy mp2 freeze core 2 end

0.00000 0.00000 0.00000 0.00000

–0.60361 0.60439 1.19165 1.19165

Home page http://www.emsl.pnl.gov/capabilities/computing/nwchem/.

ORCA The principal developer of the code has been Frank Neese, first at the Max Planck Institute for Bioinorganic Chemistry in Mülheim and then at the University of Bonn. In the words of the program’s author, “ORCA is a flexible, efficient and easy-to-use general purpose tool for quantum chemistry with specific emphasis on spectroscopic properties of open-shell molecules. It features a wide variety of standard quantum chemical methods ranging from semiempirical methods to DFT to single- and multireference correlated ab initio methods. It can also treat environmental and relativistic effects.” An abbreviated list of program’s features includes Calculation of MP2 and RI-MP2 energies and gradients. Coupled cluster methods. CASSCF wavefunction. Scalar relativistic corrections at the levels of Douglas-Kroll, ZORA, and the infiniteorder relativistic approximation (IORA) (Dyall and van Lenthe 1999). Excited states energies may be calculated using the CIS, CIS(D), and TDDFT methods.

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Multireference CI and perturbation theory. Many properties may be calculated, including spin-orbit coupling, hyperfine and quadrupole couplings, the EPR g-tensor, and zero-field splitting. Sample input file for calculation of the MP2/cc-pVDZ energy of H2 CO is shown below (the symmetry of the molecule is determined by the program). ! MP2 RHF cc-pVDZ * int 0 1 C(1) 0 0 0 0.00 0.0 O(2) 1 0 0 1.208 0.0 H(3) 1 2 0 1.116 121.75 H(3) 1 2 3 1.116 121.75 *

0.00 0.00 0.00 180.00

Home page http://www.thch.uni-bonn.de/tc/orca/.

PSI3 In the description of the designers of the code, C.D. Sherrill, T.D. Crawford, and E.F. Valeev (from Georgia Institute of Technology and Virginia Tech), “the PSI3 suite of quantum chemical programs is designed for efficient, high-accuracy calculations of properties of small to medium-sized molecules. The package’s current capabilities include a variety of Hartree-Fock, coupled cluster, complete-activespace self-consistent-field, and multi-reference configuration interaction models.” PSI3, written in the programming languages C and CCC (rather than in the Fortran language, which has been traditionally used in computational quantum chemistry), was recently reviewed (Crawford et al. 2007). Some interesting features of the program are Arbitrarily high angular momentum levels in integrals and derivative integrals (up to m-type functions have been tested). Coupled cluster methods including CC2, CCSD, CCSD(T), and CC3 with RHF, ROHF, UHF, and Brueckner orbitals. Determinant-based CI including CASSCF, RAS-CI, and full CI. MP2 and MP2-R12 methods. Methods for excited state: CIS, CIS(D), random phase approximation (RPA), EOMCCSD, and CC3. Analytic energy gradients for CCSD based on RHF, ROHF, and UHF orbitals. Coupled cluster linear response methods for static and dynamic polarizabilities and optical rotation. Diagonal Born–Oppenheimer correction (DBOC) for RHF, ROHF, UHF, and CI wave functions.

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Sample input file for the H2 CO calculation at the MP2/cc-pVDZ level of theory is shown below (symmetry of the molecule is determined by the program). psi:( label = “H2CO at experimental geometry (CRC, p. 9-36)” wfn = mp2 reference = rhf basis = “cc-pvDZ” zmat = ( O C 1 1.208 H 2 1.116 1 121.75 H 2 1.116 1 121.75

3 180.0

) )

Home page http://www.psicode.org/.

ACES II The program ACES II (Advanced Concepts in Electronic Structure) was developed in the research group of R.J Bartlett at the Quantum Theory Project (QTP) of the University of Florida in Gainesville. The program’s backbone was written in the early 1990s and has undergone extensive developments since that time. ACES II focused on the coupled-cluster theory (Paldus 2005) and its applications (Bartlett 2005; Bartlett and Musiał 2007). Single point energy calculations may be carried out for the following cases: Independent particle models RHF, UHF, and ROHF. Correlation methods utilizing RHF and UHF reference determinants, including MBPT(2), MBPT(3), SDQ-MBPT(4), MBPT(4), CCD, CCSD, CCSD(T), CCSD C TQ*(CCSD), CCSD(TQ), CCSDT-1 CCSDT-2, CCSDT-3, QCISD, QCISD(T), QCISD(TQ), UCCS(4), UCCSD(4), CID, and CISD. Correlation methods that can use ROHF reference determinants include MBPT(2), CCSD, CCSDT, CCSD(T), CCSDT-1, CCSDT-2, and CCSDT-3. Correlation methods that can use quasi-restricted Hartree-Fock (QRHF) or Brueckner orbital reference determinants include CCSD, CCSDT, CCSD(T), CCSDT-1, CCSDT-2, and CCSDT-3. Two-determinant CCSD calculations for open-shell singlet state. Equation-of-motion CCSD calculation of dynamic polarizabilities and of NMR spin-spin coupling constants. Equation-of-motion CCSD calculations of excitation energies. Kohn-Sham DFT density functional methods combined with a wide selection of density functionals.

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Analytical gradients may be calculated for Independent particle models RHF, UHF, and ROHF. Correlation methods utilizing RHF and UHF reference determinants, including MBPT(2), MBPT(3), SDQ-MBPT(4), MBPT(4), CCD, CCSD, CCSD C T(CCSD), CCSD(T), CCSDT-1, CCSDT-2, CCSDT-3, QCISD, QCISD(T), UCC(4), UCCSD(4), CID, and CISD. Correlation methods that can utilize ROHF reference determinants, including MBPT(2), CCSD, and CCSD(T). Correlation methods that can utilize QRHF reference determinants, including CCSD. Two-determinant CCSD calculations for open-shell singlet state based on QRHF orbitals. EOM-CCSD for excited states. TD-CCSD analytical derivatives. Analytical hessians may be carried out for the independent particle models RHF, UHF, and ROHF. Home page http://www.qtp.ufl.edu/ACES/.

ACES III The program represents a significant advance in parallel processing and resulted from completely rewriting ACES II for efficiency in parallel calculations. The parallel technology based on the super instruction assembly language (SIAL) that marked the progress from ACES II to ACES III was discussed by Bartlett and coworkers (Lotrich et al. 2008). Home page http://www.qtp.ufl.edu/ACES/.

CFOUR The four Cs in the name of the program stand for Coupled-Cluster techniques for Computational Chemistry. The predecessor of this program was known as the Mainz-Austin-Budapest version of ACES II which was replaced by CFOUR in April 2009 (see http://www.aces2.de/). At present, the development of the program is continued at the University of Mainz (J. Gauss), at the University of Texas at Austin (J.F. Stanton and M.E. Harding), and at Eötvös Loránd University (P.G. Szalay). Some of the features of the program are NMR chemical shifts. Many properties, including nuclear spin-rotation constants, magnetizabilities, rotational g-tensors, indirect spin-spin coupling constants.

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Electronically excited states may be calculated via CIC, CIS(D), and several EOM-CC methods; the ionized and electron-attached states may be described via EOM-CC methods. Anharmonic force-field calculations may be carried out. Vibrationally averaged properties. Relativistic corrections. Diagonal Born–Oppenheimer corrections. Automated addivity and basis-set extrapolation schemes. Basis sets up to i-type Gaussian functions. Effective core potentials. Home page http://www.cfour.de/..

COLUMBUS COLUMBUS originated in 1980 at the Ohio State University and was developed by I. Shavitt, H. Lischka (University of Vienna), and R. Shepard (Battelle Columbus Laboratories). The program was designed to carry out a variety of multireference calculations on ground and excited states of molecules (Lischka et al. 2006). Specific features include Geometry optimization to find minima and saddle points. Searches for minima on the crossing seam (conical intersections). General MCSCF with quadratic convergence including state averaging. Direct MR-CISD (single and double excitations), using an arbitrary set of reference configurations. Multireference averaged coupled-pair-functional (MR-ACPF) and multi-reference average quadratic coupled-cluster (MR-AQCC) with size-extensivity corrections. Spin-orbit CI (based on graphical unitary group approach) (Yabushita et al. 1999). Search for conical intersections may be done at the MRCI level. For excited states, MCSCF, MR-CISD, and MR-AQCC-LRT transition moments may be calculated. Finite field method for second-order properties (polarizabilities). Home page http://www.univie.ac.at/columbus/.

MPQC The Massively Parallel Quantum Chemistry (MPQC) program is an object-oriented code that was written in CCC. The work on the code has been initiated by C.L. Janssen at the Sandia National Laboratories (USA). The capabilities of MPQC include

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Energies and gradients are available for closed shell, unrestricted and general restricted open shell Hartree-Fock and density functional methods. Energies may be calculated for the second-order open-shell perturbation theory. Energies and gradients may be evaluated for the second-order closed-shell Møller– Plesset perturbation theory (MP2). MP2 energies of closed-shell systems, including an R12 correlation factor and using an auxilary basis set, are supported. Home page http://www.mpqc.org/index.php.

Commercial Software For the details of commercial software the reader is referred to the links listed below for the most up-to-date information about each program. For completeness, both ab initio and semiempirical programs are included.

Gaussian One of the first programs for large-scale calculations in the commercial domain, dating its origins to 1970, is currently in its 2009 edition.1 Please see home page http://www.gaussian.com/.

Molcas Please see home page http://www.teokem.lu.se/molcas/introduction.html

Q-Chem Please see home page http://www.q-chem.com/

1 The original 13,370 lines of Gaussian 70 code were released to general public via the now defunct Quantum Chemistry Program Exchange (QCPE). Historic information about QCPE may be found on http://www.ccl.net/ccl/qcpe/QCPE_removed/. QCPE offered as the first ab initio program Polyatom (Version 1 with 3,275 lines of code) was made available by Csizmadia et al. in 1964. It is worth mentioning in passing that the fees that QCPE charged for the programs were very modest by today’s standards: $175 for codes greater than 10,000 lines plus $35 for media and handling. The programs grew as new capabilities were added: in 1974 Polyatom (Version II for IBM 360) grew to 20,000 lines, while the 1980 release of Gaussian (IBM Version II) contained about 60,000 lines of code. The current status of QCPE was explained in a brief note saved in Computational Chemistry List http://ccl.net/chemistry/resources/messages/2009/06/04.001-dir/ index.html.

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Turbomole Please see home page http://www.turbomole.com/

Molpro Please see home page http://www.molpro.net/

Jaguar Please see home page https://www.schrodinger.com/products/14/7/.

PQS Parallel Quantum Solutions. Please see home page http://www.pqs-chem.com/ software.shtml

Spartan Please see home page http://www.wavefun.com/

HyperChem Please see home page http://www.hyper.com/

Ampac 9 Please see home page http://www.semichem.com/ Acknowledgments The work was partly supported by the Research Grant No. G12121041441 to MK from the Natural Sciences and Engineering Research Council of Canada.

Bibliography Adamovic, I., Freitag, M. A., & Gordon, M. S. (2003). Density functional theory based effective fragment potential method. The Journal of Chemical Physics, 118, 6725–6732. Almlöf, J. (1987). General contraction of Gaussian basis sets. I. Atomic natural orbitals for firstand second-row atoms. The Journal of Chemical Physics, 86, 4070–4077.

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Baldridge, K., & Klamt, A. (1997). First principles implementation of solvent effects without outlying charge error. The Journal of Chemical Physics, 106, 6622–6633. Bartlett, R. J. (2005). How and why coupled-cluster theory became the pre-eminent method in an ab initio quantum chemistry. In C. E. Dykstra, G. Frenking, K. S. Kim, & G. E. Scuseria (Eds.), Theory and applications of computational chemistry: The first forty years (pp. 1191–1221). Amsterdam/Boston: Elsevier. Bartlett, R. J., & Musiał, M. (2007). Coupled-cluster theory in quantum chemistry. Reviews of Modern Physics, 79, 291. Bobrowicz, F. W., & Schaefer, H. F., III. (1977). The self-consistent field equations for generalized valence bond and open-shell Hartree-Fock wave functions. In H. F. Schaefer III (Ed.), Methods of electronic structure theory (modern theoretical chemistry) (Vol. 3, pp. 79–127). New York/London: Plenum. Brooks, B. R., & Schaefer, H. F., III. (1979). The graphical unitary group approach to the electron correlation problem. Methods and preliminary applications. The Journal of Chemical Physics, 70, 5092–5106. Cammi, R., & Tomasi, J. (1995). Remarks on the use of the apparent surface charges (ASC) methods in solvation problems: Iterative versus matrix-inversion procedures and the renormalization of the apparent charges. Journal of Computational Chemistry, 16, 1449–1458. Cao, X., & Dolg, M. (2006). Relativistic energy-consistent ab initio pseudopotentials as tools for quantum chemical investigations of actinide systems. Coordination Chemistry Reviews, 250, 900–910. Cederbaum, L. S., & Domcke, W. (1977). Theoretical aspects of ionization potentials and photoelectron spectroscopy: A Green’s function approach. Advances in Chemical Physics, 36, 205–344. Chaban, G. M., Jung, J. O., & Gerber, R. B. (1999). Ab initio calculation of anharmonic vibrational states of polyatomic systems: Electronic structure combined with vibrational self-consistent field. The Journal of Chemical Physics, 111, 1823–1829. Chipman, D. M. (1997). Charge penetration in dielectric models of solvation. The Journal of Chemical Physics, 106, 10194–10206. Chipman, D. M. (2000). Reaction field treatment of charge penetration. The Journal of Chemical Physics, 112, 5558–5565. Chipman, D. M. (2002). Comparison of solvent reaction field representations. Theoretical Chemistry Accounts, 107, 80–89. Crawford, T. D., Sherrill, C. D., Valeev, E. F., Fermann, J. T., King, R. A., Leininger, M. L., Brown, S. T., Janssen, C. L., Seidl, E. T., Kenny, J. P., & Allen, W. D. (2007). PSI3: An open-source ab initio electronic structure package. Journal of Computational Chemistry, 28, 1610–1616. Csizmadia, I. G., Harrison, M. C., Moskowitz, J. W., Seung, S., Sutcliffe, B. T., & Barrett, M. P. (1964). POLYATOM. Quantum Chemistry Program Exchange, 11, 47. Cundari, T. R., & Stevens, W. J. (1993). Effective core potential methods for the lanthanides. The Journal of Chemical Physics, 98, 5555–5565. Day, P. N., Jensen, J. H., Gordon, M. S., Webb, S. P., Stevens, W. J., Krauss, M., Garmer, D., Basch, H., & Cohen, D. (1996). An effective fragment method for modeling solvent effects in quantum mechanical calculations. The Journal of Chemical Physics, 105, 1968–1986. de Vries, A. H., van Duijnen, P. T., Juffer, A. H., Rullmann, J. A. C., Dijkman, J. P., Merenga, H., & Thole, B. T. (1995). Implementation of reaction field methods in quantum chemistry computer codes. Journal of Computational Chemistry, 16, 37–55. Dewar, M. J. S., & Thiel, W. (1977). Ground states of molecules. 38. The MNDO method. Approximations and parameters. Journal of the American Chemical Society, 99, 4899–4906. Dewar, M. J. S., Zoebisch, E. G., Healy, E. F., & Stewart, J. J. P. (1985). AM1: A new general purpose quantum mechanical molecular model. Journal of the American Chemical Society, 107, 3902–3909. Dewar, M. J. S., Jie, C., & Yu, J. (1993). SAM1; the first of a new series of general purpose quantum mechanical molecular models. Tetrahedron, 49, 5003–5038.

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Dolg, M. (2000). Effective core potentials. In J. Grotendorst (Ed.), Modern methods and algorithms of quantum chemistry (Vol. 1, pp. 479–508). Jülich: John von Neumann Institute for Computing. Douglas, M., & Kroll, N. M. (1974). Quantum electrodynamical corrections to the fine structure of helium. Annals of Physics, 82, 89–155. Dreuw, A., & Head-Gordon, M. (2005). Single-reference ab initio methods for the calculation of excited states of large molecules. Chemical Reviews, 105, 4009–4037. Duijnen, P. T. V., & de Vries, A. H. (1996). Direct reaction field force field: A consistent way to connect and combine quantum-chemical and classical descriptions of molecules. International Journal of Quantum Chemistry, 60, 1111–1132. Dunning, T. H. (1989). Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. The Journal of Chemical Physics, 90, 1007–1023. Dunning, T. H., & Hay, P. J. (1977). Gaussian basis sets for molecular calculations. In H. F. Schaefer III (Ed.), Methods of electronic structure theory (modern theoretical chemistry) (Vol. 3, pp. 1–27). New York/London: Plenum. Dupuis, M., Spangler, D., & Wendoloski, J. (1980). NRCC software catalog (Vol. 1, Program No. QG01 GAMESS Tech. rep.) Berkeley: National Resource for Computations in Chemistry, University of California. Dyall, K. G. (2002). A systematic sequence of relativistic approximations. Journal of Computational Chemistry, 23, 786–793. Dyall, K. G., & van Lenthe, E. (1999). Relativistic regular approximations revisited: An infiniteorder relativistic approximation. The Journal of Chemical Physics, 111, 1366–1372. Elliott, P., Furche, F., & Burke, K. (2009). Excited states from time-dependent density functional theory. Reviews in Computational Chemistry, 26, 91–166. Faas, S., Snijders, J. G., van Lenthe, J. H., van Lenthe, E., & Baerends, E. J. (1995). The ZORA formalism applied to the Dirac-Fock equation. Chemical Physics Letters, 246, 632–640. Fedorov, D. G., & Kitaura, K. (2007). Extending the power of quantum chemistry to large systems with the fragment molecular orbital method. The Journal of Physical Chemistry. A, 111, 6904– 6914. Fedorov, D. G., & Kitaura, K. (2009). The fragment molecular orbital method: Practical applications to large molecular systems. Boca Raton: CRC Press. Fock, V. A. (1930). Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Zeitschrift für Physik, 61, 126–148. Foresman, J. B., Head-Gordon, M., Pople, J. A., & Frisch, M. J. (1992). Toward a systematic molecular orbital theory for excited states. The Journal of Physical Chemistry, 96, 135–149. Fuchs, C., Bonaˇci´c-Koutecký, V., & Koutecký, J. (1993). Compact formulation of multiconfigurational response theory. Applications to small alkali metal clusters. The Journal of Chemical Physics, 98, 3121–3140. Gordon, M. S., & Schmidt, M. W. (2005). Advances in electronic structure theory: GAMESS a decade later. In C. E. Dykstra, G. Frenking, K. S. Kim, & G. E. Scuseria (Eds.), Theory and applications of computational chemistry: The first forty years (pp. 1167–1189). Amsterdam: Elsevier. Gordon, M. S., Slipchenko, L. V., Li, H., & Jensen, J. H. (2007). The effective fragment potential: A general method for predicting intermolecular interactions. In D. Spellmeyer & R. Wheeler (Eds.), Annual reports in computational chemistry (Vol. 3, pp. 177–193). Amsterdam: Elsevier. Guest, M. F., Bush, I. J., van Dam, H. J. J., Sherwood, P., Thomas, J. M. H., van Lenthe, J. H., Havenith, R. W. A., & Kendrick, J. (2005). The GAMESS-UK electronic structure package: Algorithms, developments and applications. Molecular Physics, 103, 719–747. Hall, G. G., & Lennard-Jones, J. (1951). The molecular orbital theory of chemical valency. III. Properties of molecular orbitals. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 202, 155–165. Hartree, D. R. (1928). The wave mechanics of an atom with a non-coulomb central field. Part I. Theory and methods. Mathematical Proceedings of the Cambridge Philosophical Society, 24, 89–110.

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Hay, P. J., & Wadt, W. R. (1985). Ab initio effective core potentials for molecular calculations. Potentials for K to Au including the outermost core orbitals. The Journal of Chemical Physics, 82, 299–310. Hess, B. A. (1986). Relativistic electronic-structure calculations employing a two-component nopair formalism with external-field projection operators. Physical Review A, 33, 3742–3748. Hess, B. A. (1989). Revision of the Douglas-Kroll transformation. Physical Review A, 39, 6016–6017. Hsu, H., Davidson, E. R., & Pitzer, R. M. (1976). An SCF method for hole states. The Journal of Chemical Physics, 65, 609–613. Hurley, M. M., Pacios, L. F., Christiansen, P. A., Ross, R. B., & Ermler, W. C. (1986). Ab initio relativistic effective core potentials with spin-orbit operators. II. K through Kr. The Journal of Chemical Physics, 84, 6840–6853. Huzinaga, S., Seijo, L., Barandiarán, Z., & Klobukowski, M. (1987). The ab initio model potential method. Main group elements. The Journal of Chemical Physics, 86, 2132–2145. Ivanic, J. (2003a). Direct configuration interaction and multiconfigurational self-consistent-field method for multiple active spaces with variable occupations. I. Method. The Journal of Chemical Physics, 119, 9364–9376. Ivanic, J. (2003b). Direct configuration interaction and multiconfigurational self-consistent-field method for multiple active spaces with variable occupations. II. Application to oxoMn(salen) and N2 O4 . The Journal of Chemical Physics, 119, 9377–9385. Ivanic, J., & Ruedenberg, K. (2001). Identification of deadwood in configuration spaces through general direct configuration interaction. Theoretical Chemistry Accounts, 106, 339–351. Jensen, J. H., Day, P. N., Gordon, M. S., Basch, H., Cohen, D., Garmer, D. R., Krauss, M., & Stevens, W. J. (1984). An effective fragment method for modeling intermolecular hydrogen bonding-effects on quantum mechanical calculations. In D. A. Smith (Ed.), Modeling the hydrogen bond (ACS symposium, Vol. 569, pp. 139–151). New York: ACS. Kahn, L. R., Baybutt, P., & Truhlar, D. G. (1976). Ab initio effective core potentials: Reduction of all-electron molecular structure calculations to calculations involving only valence electrons. The Journal of Chemical Physics, 65, 3826–3853. Karelson, M. M., Katritzky, A. R., & Zerner, M. C. (1986). Reaction field effects on the electron distribution and chemical reactivity of molecules. International Journal of Quantum Chemistry, 30, 521–527. Karelson, M., Tamm, T., & Zerner, M. C. (1993). Multicavity reaction field method for the solvent effect description in flexible molecular systems. The Journal of Physical Chemistry, 97, 11901– 11907. Kendall, R. A., Aprà, E., Bernholdt, D. E., Bylaska, E. J., Dupuis, M., Fann, G. I., Harrison, R. J., Ju, J., Nichols, J. A., Nieplocha, J., Straatsma, T. P., Windus, T. L., & Wong, A. T. (2000). High performance computational chemistry: An overview of NWChem a distributed parallel application. Computer Physics Communications, 128, 260–283. Kirkwood, J. G. (1934). Theory of solutions of molecules containing widely separated charges with special application to zwitterions. The Journal of Chemical Physics, 2, 351–361. Klamt, A. (1995). Conductor-like screening model for real solvents: A new approach to the quantitative calculation of solvation phenomena. The Journal of Physical Chemistry, 99, 2224– 2235. Klamt, A., & Schüürmann, G. (1993). COSMO: A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient. Journal of the Chemical Society, Perkin Transactions, 2, 799–805. Klobukowski, M., Huzinaga, S., & Sakai, Y. (1999). Model core potentials: Theory and applications. In J. Leszczynski (Ed.), Computational chemistry: Reviews of current trends (Vol. 3, pp. 49–74). Singapore: World Scientific. Knowles, P. J., Andrews, J. S., Amos, R. D., Handy, N. C., & Pople, J. A. (1980). Restricted Møller-Plesset theory for open-shell molecules. Chemical Physics Letters, 186, 130–136.

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Kowalski, K., & Piecuch, P. (2004). New coupled-cluster methods with singles, doubles, and noniterative triples for high accuracy calculations of excited electronic states. The Journal of Chemical Physics, 120, 1715–1738. LaJohn, L. A., Christiansen, P. A., Ross, R. B., Atashroo, T., & Ermler, W. C. (1987). Ab initio relativistic effective core potentials with spin-orbit operators. III. Rb through Xe. The Journal of Chemical Physics, 87, 2812–2824. Lauderdale, W. J., Stanton, J. F., Gauss, J., Watts, J. D., & Bartlett, R. J. (1991). Many-body perturbation theory with a restricted open-shell Hartree-Fock reference. Chemical Physics Letters, 187, 21–28. Lee, T. J., & Jayatilaka, D. (1993). An open-shell restricted Hartree-Fock perturbation theory based on symmetric spin orbitals. Chemical Physics Letters, 201, 1–10. Lee, T. J., Rendell, A. P., Dyall, K. G., & Jayatilaka, D. (1994). Open-shell restricted Hartree-Fock perturbation theory: Some considerations and comparisons. The Journal of Chemical Physics, 100, 7400–7409. Lischka, H., Shepard, R., Shavitt, I., Pitzer, R. M., Dallos, M., Müller, Th., Szalay, P. G., Brown, F. B., Ahlrichs, R., Böhm, H. J., Chang, A., Comeau, D. C., Gdanitz, R., Dachsel, H., Ehrhardt, C., Ernzerhof, M., Höchtl, P., Irle, S., Kedziora, G., Kovar, T., Parasuk, V., Pepper, M. J. M., Scharf, P., Schiffer, H., Schindler, M., Schüler, M., Seth, M., Stahlberg, E. A., Zhao, J.-G., Yabushita, S., Zhang, Z., Barbatti, M., Matsika, S., Schuurmann, M., Yarkony, D. R., Brozell, S. R., Beck, E. V., & Blaudeau, J.-P. (2006). COLUMBUS, an ab initio electronic structure program, release 5.9.1. Lotrich, V., Flocke, N., Ponton, M., Yau, A. D., Perera, A., Deumens, E., & Bartlett, R. J. (2008). Parallel implementation of electronic structure energy, gradient, and hessian calculations. The Journal of Chemical Physics, 128, 194104/1–15. McWeeny, R., & Diercksen, G. H. F. (1968). Self-consistent perturbation theory. II. Extension to open shells. The Journal of Chemical Physics, 49, 4852–4856. Miertuš, S., Scrocco, E., & Tomasi, J. (1981). Electrostatic interaction of a solute with a continuum. A direct utilization of ab initio molecular potentials for the prevision of solvent effects. Chemical Physics, 55, 117–129. Møller, Ch., & Plesset, M. S. (1934). Note on an approximation treatment for many-electron systems. Physical Review, 46, 618–622. Nakajima, T., & Hirao, K. (1999). A new relativistic theory: A relativistic scheme by eliminating small components (RESC). Chemical Physics Letters, 302, 383–391. Nakano, H. (1993a). MCSCF reference quasidegenerate perturbation theory with Epstein-Nesbet partitioning. The Journal of Chemical Physics, 99, 7983–7992. Nakano, H. (1993b). Quasidegenerate perturbation theory with multiconfigurational selfconsistent-field reference functions. Chemical Physics Letters, 207, 372–378. Nielsen, E. S., Jørgensen, P., & Oddershede, J. (1980). Transition moments and dynamic polarizabilities in a second order polarization propagator approach. The Journal of Chemical Physics, 73, 6238–6246. Olsen, J., Yeager, D. L., & Jørgensen, P. (1983). Optimization and characterization of a multiconfigurational self-consistent field (MCSCF) state. Advances in Chemical Physics, 54, 1–176. Pacios, L. F., & Christiansen, P. A. (1985). Ab initio relativistic effective potentials with spin-orbit operators. I. Li through Ar. The Journal of Chemical Physics, 82, 2664–2671. Paldus, J. (2005). The beginnings of coupled-cluster theory: An eyewitness account. In C. E. Dykstra, G. Frenking, K. S. Kim, & G. E. Scuseria (Eds.), Theory and applications of computational chemistry: The first forty years (pp. 115–147). Amsterdam: Elsevier. Peterson, K. A. (2003). Systematically convergent basis sets with relativistic pseudopotentials. I. Correlation consistent basis sets for the post-d group 13–15 elements. The Journal of Chemical Physics, 119, 11099–11112. Peterson, K. A., Figgen, D., Goll, E., Stoll, H., & Dolg, M. (2003). Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16–18 elements. The Journal of Chemical Physics, 119, 11113– 11123.

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Piecuch, P., & Włoch, M. (2005). Renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed Hamiltonian. The Journal of Chemical Physics, 123, 224105/1–10. Piecuch, P., Kucharski, S. A., Kowalski, K., & Musiał, M. (2002). Efficient computer implementation of the renormalized coupled-cluster methods: The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches. Computer Physics Communications, 149, 7196. Pople, J. A., & Nesbet, R. K. (1954). Self-consistent orbitals for radicals. The Journal of Chemical Physics, 22, 571–572. Pople, J. A., Binkley, J. S., & Seeger, R. (1976). Theoretical models incorporating electron correlation. International Journal of Quantum Chemistry, S10, 1–19. Raffenetti, R. C. (1973). General contraction of Gaussian atomic orbitals: Core, valence, polarization, and diffuse basis sets; Molecular integral evaluation. The Journal of Chemical Physics, 58, 4452–4458. Read, A. E., & Weinhold, F. (1983). Natural bond orbital analysis of near-Hartree-Fock water dimer. The Journal of Chemical Physics, 78, 4066–4073. Read, A. E., Weinstock, R. B., & Weinhold, F. (1985). Natural population analysis. The Journal of Chemical Physics, 83, 735–746. Ridley, J. E., & Zerner, M. C. (1973). Intermediate neglect of differential overlap techniques for spectroscopy: Pyrrole and the azines. Theoretical Chemistry Accounts: Theory, Computation, and Modeling, 32, 111–134. Roos, B. O. (1983). The multiconfiguration SCF method. In G. H. F. Diercksen & S. Wilson (Eds.), Methods in computational molecular physics (pp. 161–187). Dordrecht: Reidel. Roos, B. O. (1987). The CASSCF method and its application in electronic structure calculations. Advances in Chemical Physics, 69, 339–445. Roos, B. O. (1994). The multiconfiguration SCF theory. In B. O. Roos (Ed.), Lecture notes in quantum chemistry (Vol. 58, pp. 177–254). Berlin: Springer. Roothaan, C. C. J. (1951). New developments in molecular orbital theory. Reviews of Modern Physics, 23, 69–89. Ross, R. B., Powers, J. M., Atashroo, T., Ermler, W. C., LaJohn, L. A., & Christiansen, P. A. (1990). Ab initio relativistic effective core potentials with spin-orbit operators. IV. Cs through Rn. The Journal of Chemical Physics, 93, 6654–6670. Roy, L. E., Hay, P. J., & Martin, R. L. (2008). Revised basis sets for the LANL effective core potentials. Journal of Chemical Theory and Computation, 4, 1029–1031. Ruedenberg, K., Schmidt, M. W., Gilbert, M. M., & Elbert, S. T. (1982a). Are atoms intrinsic to molecular electronic wavefunctions? I. The FORS model. Chemical Physics, 71, 41–49. Ruedenberg, K., Schmidt, M. W., Gilbert, M. M., & Elbert, S. T. (1982b). Are atoms intrinsic to molecular electronic wavefunctions? II. Analysis of FORS orbitals. Chemical Physics, 71, 51–64. Ruedenberg, K., Schmidt, M. W., Gilbert, M. M., & Elbert, S. T. (1982c). Are atoms intrinsic to molecular electronic wavefunctions? III. Analysis of FORS configurations. Chemical Physics, 71, 65–78. Schirmer, J., & Cederbaum, L. S. (1978). The two-particle-hole Tamm-Dancoff approximation (2ph-TDA) equations for closed-shell atoms and molecules. Journal of Physics B: Atomic and Molecular Physics, 11, 1889–1900. Schmidt, M. W., & Gordon, M. S. (1998). The construction and interpretation of MCSCF wavefunctions. Annual Review of Physical Chemistry, 49, 233–266. Schmidt, M. W., Baldridge, K. K., Boatz, J. A., Elbert, S. T., Gordon, M. S., Jensen, J. H., Koseki, S., Matsunaga, N., Nguyen, K. A., Su, S., Windus, T. L., Dupuis, M., & Montgomery, J. J. A. (1993). General atomic and molecular electronic structure system. Journal of Computational Chemistry, 14, 1347–1363. Shavitt, I., & Bartlett, R. J. (2009). Many-body methods in chemistry and physics: MBPT and coupled-cluster theory. Cambridge, UK: Cambridge University Press. Shepard, R. (1987). The MCSCF method. Advances in Chemical Physics, 69, 63–200.

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Stevens, W. J., Basch, H., & Krauss, M. (1984). Compact effective potentials and efficient sharedexponent basis sets for the first- and second-row atoms. The Journal of Chemical Physics, 81, 6026–6033. Stewart, J. J. P. (1989). Optimization of parameters for semiempirical methods. I. Method. Journal of Computational Chemistry, 10, 209–220. Stewart, J. J. P. (1990). MOPAC: A semiempirical molecular orbital program. Journal of ComputerAided Molecular Design, 4, 1–103. Stewart, J. J. P. (2007). Optimization of parameters for semiempirical methods V: Modification of NDDO approximations and application to 70 elements. Journal of Molecular Modeling, 13, 1173–1213. Tapia, O., & Goscinski, O. (1975). Self-consistent reaction field-theory of solvent effects. Molecular Physics, 29, 1653. Tomasi, J., & Persico, M. (1994). Molecular interactions in solution: An overview of methods based on continuous distributions of the solvent. Chemical Reviews, 94, 2027–2094. Tomasi, J., Mennucci, B., & Cammi, R. (2005). Quantum mechanical continuum solvation models. Chemical Reviews, 105, 2999–3093. Werner, H. (1987). Matrix formulated direct MCSCF and multiconfiguration reference CI methods. Advances in Chemical Physics, 69, 1–62. Włoch, M., Gour, J. R., Kowalski, K., & Piecuch, P. (2005). Extension of renormalized coupledcluster methods including triple excitations to excited electronic states of open-shell molecules. The Journal of Chemical Physics, 122, 214107/1–15. Yabushita, S., Zhang, Z., & Pitzer, R. M. (1999). Spin-orbit configuration interaction using the graphical unitary group approach and relativistic core potential and spin-orbit operators. The Journal of Physical Chemistry, 103, 5791–5800. Zeng, T., Fedorov, D. G., Schmidt, M. W., & Klobukowski, M. (2011a). Two-component natural spinors for two-step spin-orbit coupled wave functions. The Journal of Chemical Physics, 134, 214107-1–214107-9. Zeng, T., Fedorov, D. G., Schmidt, M. W., & Klobukowski, M. (2011b). Effects of spin-orbit coupling on covalent bonding and the Jahn-Teller effect are revealted with the natural language of spinors. The Journal of Chemical Theory and Computation, 7, 2864–2875. Zerner, M. C. (1991). Semiempirical molecular orbitals methods. In K. B. Lipkowitz & D. B. Boyd (Eds.), Reviews in computational chemistry (Vol. 2, pp. 313–365). New York: VCH Publishers.

Photoactive Semiconducting Oxides for Energy and Environment: Experimental and Theoretical Insights Malgorzata Makowska-Janusik and Abdel-Hadi Kassiba

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Insights on Materials and Related Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Titanium Dioxide (TiO2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BiVO4 Nanostructures for Photocatalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Crystalline Structures: Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFT Methodology and Related Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cluster Approach to Model Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Features of TiO2 , NiTiO3 , and BiVO4 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Properties of NiTiO3 and BiVO4 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Properties of BiVO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Properties of NiTiO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural and Electronic Properties of TiO2 , NiTiO3 , and BiVO4 Nanosized Systems . . . . . BiVO4 Nanostructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 5 11 17 17 19 23 24 25 26 26 27 29 31 39 41 42

M. Makowska-Janusik () Institute of Physics, Faculty of Mathematics and Natural Science, Jan Dlugosz University in Czestochowa, Czestochowa, Poland e-mail: [email protected] A.-H. Kassiba Institute of Molecules and Materials of Le Mans – UMR-CNRS 6283, Université du Maine, Le Mans, France e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_18-2

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Abstract

This chapter reports experimental investigations and theoretical approaches devoted to analyze the electronic, optical, and vibrational properties occurring in semiconducting photoactive materials in their bulk form or as nanosized objects. An original cluster approach was developed to describe the physical features of nanoparticles and the impact of their surface on photoinduced charge transfer phenomena. Photovoltaic and photocatalysis are two major applications for both clean environment and sustainable sources of energy as well. In this context, semiconducting nanocrystals from defined oxide families have attracted increasing interest during the last decade for their promising potential in renewable energy applications because of their versatile and coupled optical and electronic properties. More specifically, the semiconductor oxides as titanium oxide and bismuth vanadate-based materials can be tailored in nanosized and mesoporous structures. The electronic and optical features may be modulated in large extents when dye molecules are used as sensitizing vectors enhancing the efficiency of solar cells, energy storage, and photocatalyst. These applications depend on the reactive surface area, morphology and nanostructuration, doping and sensitizing agents, as well as controlled vacancy rates acting on the charge transfer peculiarities. In parallel to experimental investigations dedicated to selected forms of photoactive semiconducting oxides, original theoretical approaches were developed to analyze the key features of the considered functional systems. Using an integrated approach associating theoretical models and numerical simulations, the influence of the size and morphology of nanoparticles on their electronic and optical properties was pointed out. The cluster approach methodology was developed to simulate the electronic properties of semiconducting nanocrystalline materials as isolated objects or functionalized by organic dye molecules. The construction of the system proceeds by the crystal structure frozen in the cluster core while the surface is modified according to the environmental interactions. Theoretically, it was proved that nanostructures exhibit patchwork properties coming from the bulk material including core crystal structure and from surface effects caused by environment. On the other hand, the role of doping of the considered structures by metallic elements was investigated on the photoactivity mechanism involved in nanoparticles. The nature of vacancies located close to the dopants plays a crucial role on the electronic and optical features in the photoactive materials. Thus, the theoretical approaches and the carried out numerical simulations contribute to draw quantitative insights of the physical properties of functional semiconducting oxides in agreement with relevant experimental analyses.

Photoactive Semiconducting Oxides for Energy and Environment:. . .

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Introduction Photoactive semiconducting oxides are subject of intensive investigations during the last two decades with a major aim to realize active media for photovoltaic and photocatalytic processes. The applications concern important societal challenges, such as green energies based on alternative and nonfossil resources and environmental problems related to water and atmosphere purification and preservation. Particularly, hydrogen production from water dissociation and depollution of environment from contaminants are major concerns of the twenty-first-century society. Increased industrial activity due to the emergence of new economic powers along with important demographic pressure contributes to high energy demand. Average estimates indicate energy resources exploited nowadays consummate around 80 % from nonrenewable energy origin even if the part of other energy sources (solar, wind) increase continuously in industrial countries during the last 20 years. Similarly, the improvement of living conditions in emerging countries through technological developments requires an increasing energy demand and water consumption. It is obvious that the factors listed above (water, energy, industry, demography, development) generate an impact on the environment. The increasing air and water pollution is a crucial health and societal problem which requires control and regulation and then the societal issues must find upstream solutions to prevent crises. Beyond the precautionary principles and good practices from the societies, scientific approaches at large scale should be conducted to support changes in the demand for energy and environmental preservation required for suitable evolution of the human life standards. In this context, the scientific activity related to photocatalytic processes has been exploited intensively for more than two decades in the active research area dealing with energy and environment. In particular, harvesting efficiently the solar spectrum in the visible light-driven photocatalysis is subject of important developments to create photoactive materials with an implementation of experimental protocols devoted to increase the efficiency of the photocatalysts. These mechanisms may be applied to the production of clean energy sources such as hydrogen production obtained by water splitting. They also apply to the degradation of organic pollutants or bacterial organisms that can affect water reserves or some critical atmospheres as required in medical environments. Other large-scale applications of photocatalytic reaction principles were tested on road surfaces containing photocatalysts to destroy polluting gases such as oxynitrides or hydrocarbons. These examples demonstrate the wealth of applications that rely on photocatalysis from the point of view of the active material designs as well as for the optimization of photocatalytic reactions. Among the wide classes of active materials, two families of semiconducting oxides are worthy of interest and were subject of important developments. They are

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related, on the one hand, to titanium dioxide (TiO2 ) materials as pure or associated with organic or inorganic components following the searched applications from photovoltaics or photocatalysis. On the other hand, bismuth vanadate (BiVO4 ) and related structures are emerging smart materials for photocatalysis. The experimental developments on such classes of photoactive materials are supported by theoretical and numerical models able to predict and analyze quantitatively the experimental properties of bulk and nanosized materials from charge transfer peculiarities to optical absorption and structural organization. Original approaches were developed on clusters with different size giving rise to behaviors related to molecular scales, nanosized clusters, or to macroscopic media. For photovoltaic or photocatalytic processes activated by light irradiation in semiconducting materials, the phenomena of charge transfer and the optical absorption features are of primary importance, and the simulation methods can contribute to shed light on the strength of the phenomena and orient the ways of their optimization. For such approaches, quantum chemistry codes and software packages based on quantum mechanics are active and hot topics for the simulations in material sciences. The density functional theory (DFT) method is an efficient tool to predict the physical properties of materials. The built of realistic geometries of the materials, the optimization of the structures and atomic position, as well as the evaluation of the related properties by DFT approach are currently applied to wide classes of functional materials including semiconducting photoactive media. Several models were explored on selective classes of photoactive materials such as TiO2 , BiVO4 , and NiTiO3 dedicated to photocatalysis and photovoltaic applications. For titanium dioxide structures, original theoretical studies have been realized to model anatase structure of nanocrystals with sizes ranging from 0.7 nm up to 2.0 nm organized as (TiO2 )n clusters. Also, the systems like BiVO4 or close ones as NiTiO3 were modeled by applying the cluster approach developed in our previous work (Makowska-Janusik and Kassiba 2012). For the nanostructures, the influence of surface and environment at their electronic properties was studied. Similar theoretical and numerical modeling studies were applied to predict and describe quantitatively the key physical properties of BiVO4 and NiTiO3 . Numerical approaches based on cluster model and quantum-chemical calculations were implemented to explain the electronic properties of nanoparticles and to elucidate the influence of surface effects on photocatalytic-driven mechanism such as charge transfer. The developed approaches are emanating from former works developed on bulk BiVO4 structures leading to their electronic properties (Walsh et al. 2009; Yin et al. 2011; Cooper et al 2014; Ma and Wang 2014). The implemented theoretical approach applies to model the features of mesoporous structure offering high specific surfaces which enhance photocatalytic and photovoltaic properties. The experimental investigations ensuring the synthesis of such structures and the evaluation of related vibrational, electronic, and optical features are confronted to the predicted properties obtained by numerical simulations.

Photoactive Semiconducting Oxides for Energy and Environment:. . .

5

Experimental Insights on Materials and Related Features Titanium Dioxide (TiO2 ) Photoactivity of Titanium Dioxide Among the efficient materials for their photocatalytic applications, titanium dioxide (TiO) is undeniably the most known and investigated. The green energy such as hydrogen production and the new generation of solar cells as well as the photocatalysis dedicated to the preservation of the environment have been massively exploiting active materials based on TiO2 . Particularly, photovoltaic applications were investigated using TiO2 in its pure or doped forms by metallic or nonmetallic ions or alternatively associated with other components including metallic clusters or organic groups. Various morphologies were considered such as micrometric or nanometric powders (Ishigaki et al. 2003), mesoporous structures or photoactive gels (Pattier et al. 2010), and thin films (Nam et al. 2004) or nanocomposites. The morphology and structural organization of titanium dioxide plays key roles on its electronic and optical features and was concerned by exhaustive investigations by a large scientific community. Thus, nanostructured titanium oxide as thin films, nanoparticles (Reyes-Coronado et al 2008), or mesoporous organizations (Kassiba et al. 2012) were realized with the relevant parameters leading to the requested structural organization (rutile, anatase, amorphous), morphologies (nanoparticles, nanodots, nanotubes), as well as self-assembled media with controlled porosity. The functionalization of titanium dioxide surface by active vectors was realized in order to improve the photovoltaic performances of devices based on such class of composites (Melhem et al. 2013). However, such elaboration processes are relevant to graft other functional groups on high specific surface materials. This is the case of hybrid networks composed by inorganic nanocrystals – polymer blend for new generation of hybrid solar cells (Lin et al. 2009). Other structures based on titanium oxides were also considered to improve the photo-conversion efficiency by high specific surfaces between the active components of solar cells (Hwang et al. 2009). Thus, sol–gel batches based on titanium oxide gels (Pattier et al. 2010) were realized, and their photoactivity under UV was investigated in the aim to realize new generation of hybrid solar cells. Nevertheless, despite the major limitations of TiO2 due to a wide band gap (3.2 eV), large-scale applications have been conducted and extensively used as appropriate structures for photovoltaic and photocatalysis. For applications, the photoactivity enhancement can be achieved by mechanisms as doping by metallic or nonmetallic ions able to modulate the electronic structure. The combination of TiO2 in composites with semiconductor structures such as ZnO (Chandiran et al. 2014) and CdS (Biswas et al. 2008) has been made, and the photoinduced effects were compared to those involved in pure materials. However, even with an increase in the efficiency by such experimental modifications, the UV radiation required for the photoactivity is costly, and it

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M. Makowska-Janusik and A. Kassiba

is necessary to develop alternative media with visible driven photoactivity or for high-efficient solar cells. Also, the well-known chemical procedure based on dyes sensitized solar cells (DSSC) has created the most investigated systems based on TiO2 and offers the best efficiency compared to heterostructure-based devices. Other hybrid nanostructured architectures were realized by using titanium dioxide and graphene to ensure challenging energy sources in new-generation of Li-ion batteries (Wang et al. 2014). Recently other approaches have created oxygen-deficient titanium dioxide associated with conjugated polymer. The realized functional media shows a noticeable improvement of efficiency of the based solar cell and the optoelectronic device such as OLEDS. In addition to these structural or composition of the functional materials based on titanium dioxide, the doping process is worthy of interest to modulate the electronic and optical responses of the materials. Also, several strategies were used for doping with metallic (Fe, Ag, Mo, Eu, etc.) or nonmetallic elements (N, C, etc.). Exhaustive investigations of the doping effects were carried out to analyze the electronic, optical, and magnetic features of the substitutional ions and to identify their local environments and the quality of their distribution in the host titanium dioxide structures. Such studies have been made on mesoporous architectures where several chemical reactions were also involved and catalyzed by oxygen being adsorbed in voids or incorporated in the bulk as anti-sites. This contributes to create new electronic states allowed in the band gap of the doped material and were discussed in several works as illustrated in the forthcoming part. However, instead of the doping effect, electronic and optical features of photoactive gels composed by titanium oxide can be also modified by UV irradiation leading to photoinduced electrons localized as Ti3C centers (Pattier et al. 2010). The mechanism of the Ti3C formation is well described by a series of reactions starting by the creation of a photoinduced electron–hole pairs. This process makes titanium oxide as efficient as dye sensitized one and can be also as crucial as doping agents may do on   h the electronic behavior. Such photoactive media Ti4C ! Ti3C was tested for new generation of solar cells (Pattier et al. 2010). The experimental investigations need quantitative analysis by using theoretical and modeling approaches in order to improve the photovoltaic or photocatalytic activities based on doped titanium dioxide-based materials (Makowska-Janusik et al. 2014). Several theoretical works have focused on the study of the electronic and optical features of titanium oxide possessing different stoichiometry and applying different computational approaches. The performed numerical simulations deal mainly with the electronic properties of the infinite crystal (Stausholm-Møller et al. 2010). Other performed approaches consider titanium oxide clusters with sizes up to 1–2 nm possessing crystalline core structure emanating from anatase or rutile (Makowska-Janusik et al. 2014). In all cases, the surface structure, morphology, and organization are inherent problem in numerical simulations on nano-objects with regard to their key role on the predicted properties.

Photoactive Semiconducting Oxides for Energy and Environment:. . .

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Synthesis of Mesoporous TiO2 Material and Selected Investigations The synthesis of mesoporous TiO is dedicated to create high specific surface. Titanium oxide-based mesoporous structures were synthesized by using the sol– gel method (Kassiba et al. 2012; Mei et al. 2010) with precursors as titanium tetraisopropoxide and thiourea for nitrogen doping. Surfactants as pluronic P123 are usually used to realize porous (structured) titanium dioxide with or without nitrogen doping. The materials can be obtained in the form of mesoporous powders or thin films. The as-formed systems were submitted to annealing treatments at temperatures up to 500 ı C in order to improve the crystalline structure of the titanium dioxide with a stable polytype such as anatase. The experimental investigations were carried out to identify the crystalline structures involved at defined synthesis conditions and after treatments by using Xray diffraction and particularly wide-angle X-ray scattering (WAXS) technique well adapted for mesoporous structures. The major information concern also the porosity features and in the case of doped structures the doping efficiency through the location of substitutional ions in the crystalline sites of the host structure. In this aim, electron paramagnetic spectroscopy (EPR) is a relevant technique to characterize the active electronic centers in mesoporous structures. Beyond to identify the nature of the active centers, EPR informs on the local environments of doping agents and their electronic, optical, and magnetic properties. The microstructure of the samples plays a key role in the stability of the paramagnetic centers and their incorporation in the network (gel, mesoporous) (Pattier et al. 2010; Kassiba et al. 2012). Thus, comparative investigations performed on mesoporous materials inform on the efficiency of the doping process and on the concentration and nature of the involved oxygen-based radicals and vacancies. The large specific surfaces of these nanostructured systems enhance the concentration of oxygen- or nitroxide-based radicals and contribute to lower the doping efficiency. X-ray diffraction measurements were conducted on TiO2 powders with porous morphologies (structured) or without pores (nonstructured) with or without doping. Figure 1 (top) shows WAXS patterns recorded on nonstructured samples with and without nitrogen doping (N-TiO2 -400-4.5 and TiO2 -400-4.5); the samples were annealed at 400 ı C during 4.5 h under air (Mei et al. 2010). The analysis of the diffraction line features by using the MAUD software program (Ferrari and Lutterotti 1994) evaluates the involved crystalline polytypes and their respective fractions. The structural investigations reveal the role of the doping and the morphology (structured or nonstructured) on the stabilized crystalline polytypes. Thus, the appropriate annealing conditions have been defined at 500 ı C during 12 h to realize pure anatase structure with suitable substitutional doping in the crystal sites. In this case, the same WAXS patterns show similar crystalline features whatever the samples such as structured or not and doped or free from any doping agents. Sol–gel method and spin-coating technique were used to realize mesoporous thin films (Pattier et al. 2010; Pattier 2010). The structural analysis can be made by X-ray reflectivity which informs on the ordering of pore lattices as a function of the

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M. Makowska-Janusik and A. Kassiba

Scattering Intenisty (arb.units)

*

N-TiO2

4000

TiO2

3000 2000

+ ¤ 20

Scattering Intenisty (arb.units)

*

*

1000 +

* *

30

1400

*

40

50

60 2θ (°)

70

* 80

Meso-N-TiO2

+

1100

*

*

* +

+

800 + 500 * 200 0

20

30

40

+

*

+

*

* *

* 50

60 2θ (°)

70

80

Fig. 1 WAXS patterns showing the involved crystalline structures in nonstructured TiO2 samples (top) with nitrogen doping or without (*:anatase, C: rutile, ¤: brookite) and in mesoporous sample (bottom) doped with nitrogen and annealed for 4.5 h at 400 ı C

thermal treatment of the films. Thus, characterizations by X-ray reflectivity were performed on mesoporous samples as a function of their annealing temperature and the time duration of this process. The existence of ordered porous structure is demonstrated as well as the evolution of the pore sizes with the annealing conditions (Fig. 2). Evaluations of the pore sizes were obtained by adjusting the X-ray reflectivity curves. The pore size was estimated to 10.9 nm in the as-formed

AbsoluteX-ray Reflectivity(arb.units)

Photoactive Semiconducting Oxides for Energy and Environment:. . .

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Meso-N-TiO2

10–1

10–3 As-formed

10–5

80°C 400°C

10–7 0.00

0.10

0.20

0.30

Wavevector Qz(Å–1)

Fig. 2 X-ray reflectivity measurements performed on thin films of mesoporous TiO2 with nitrogen doping. The organization of the pores is uniform at defined annealing temperature and time duration. The period of the pore on the layer changes from 10.9 nm for the as-formed sample to 5.4 nm in mesoporous film annealed at 400 ı C during 30 min (Pattier et al. 2010)

sample and change to 5.4 nm when the annealing is performed at 400 ı C during 4.5 h. This behavior is well understood by the contraction of the porous phase after extraction of the surfactant by annealing (Pattier et al. 2010; Kassiba et al. 2012). When doping is performed, the location of the substitutional ions in defined crystal sites or alternatively the existence of clusters or aggregates made from doping species can be identified by EPR. Such experiments were carried out to analyze the nitrogen doping from thiourea precursors as well as all active electronic centers (oxygen radicals, titanium ions, nitro-oxide radicals, etc.) which can contribute to the conductivity or act as traps for charge carriers. For the considered mesoporous structures, the existence of high specific surface favors the location of doping elements at the surface. Thus, Fig. 3 shows the EPR spectra recorded on the mesoporous samples and nonporous samples, annealed under air or argon at 400 ı C during 30 min. These experiments show that the atmosphere has a drastic effect on the nature of involved paramagnetic centers. Instead of the observation of nitrogen radicals, a variety of paramagnetic species, e.g., O2  , O3  , NO, Ti3C , was demonstrated. However, the nature of paramagnetic radicals and the role of the annealing temperatures and its duration were clarified on the considered N-TiO2 -doped samples. The nitrogen doping is effective but without to be incorporated in oxygen sites. Resolved signals implying nitrogen was observed for mesoporous samples annealed at 500 ı C during 12 h with the formation of NO radicals located on the surface of the porous structure. For nonporous samples, the paramagnetic species such as oxygen radicals were suppressed for the same treatment (500 ı C, 12 h) with the formation of nitrogen radicals (N*). The developed EPR signal which appears after annealing is characterized by narrow lines (Fig. 3) which indicate well-crystallized sites around the nitrogen ions. Thus, only in the

10

a EPR spectrum Intensity (arb.units)

Fig. 3 EPR spectra of N-TiO2 samples annealed at 500 ı C. Nitrogen ions are substituted in the crystalline sites of TiO2 only for samples without porous structures (a-experiment, b-simulation). For the porous one, only surface (NO)* radicals dominate the paramagnetic species behind the observed EPR signal (c-experiment, d-simulation)

M. Makowska-Janusik and A. Kassiba

N

radicals

b

c

(NO) radicals d

3300 3500 3400 Magnetic field (Gauss)

case of nonstructured (nonporous) TiO2 , nitrogen ions are indeed incorporated in the bulk structures as substitutional species. This contrast with the observations on structured (mesoporous) samples where nitrogen remains associated with oxygen as NO radicals located on the pore surfaces. As a matter of fact, doping with nitrogen is inefficient in mesoporous structures due to the tendency of doping ions to migrate at the outermost pore surfaces. The applications of titanium dioxide-based structures in photovoltaic devices or for visible light-driven photocatalytic reactions require appropriate electronic structure and band gap leading to maximum absorption of the solar spectrum. Intrinsically, TiO2 possesses wide band gap (3.1–3.2 eV) and only UV radiation can contribute to create photoinduced charge transfers from valance to conduction band. The electronic structure of TiO2 crystalline structure is characterized by O-2p orbitals which form the valence band, while Ti3C (3d1 ) states form the conduction band. In nitrogen-doped samples, it was shown from former work (Melhem et al. 2013) that reduction of the band gap occurs and a spectral extension of the absorption of solar radiation. Quantitative analysis is outlined in the theoretical approach developed below and supported by numerical simulations.

Photoactive Semiconducting Oxides for Energy and Environment:. . .

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BiVO4 Nanostructures for Photocatalysis The semiconducting oxide BiVO is currently a subject of intensive investigations devoted to create several morphologies such as nanoparticles, thin films, or composites associating metallic clusters. As thin films, BiVO4 represents an ideal photo-anode with a high quantum efficiency in the decomposition of water and hydrogen production (Kim and Choi 2014). The application as photocatalyst is also a very active research area, for example, in water depollution from organic contaminants (Wang et al. 2012). The realization of nanostructures with large specific surfaces by using suitable deposition conditions of rf -sputtered films or by the growth of mesoporous architectures or nanoparticles are well-improved approaches. The synthesis methods and annealing modify in a large extent the structural and morphological characteristics of the BiVO4 (Venkatesan et al. 2012). The physical properties can be modulated by several procedures to enhance the efficiency of the reaction mechanisms in visible light-driven photocatalytic processes or for photovoltaic conversion based on the photoinduced charge transfers (Zhang et al. 2007; Yu and Kudo 2006; Ng et al. 2010). Indeed, the efficiency of the phenomena depends on the charge densities involved in the semiconducting structure, on the optical band gap, as well as on the lifetimes of photogenerated excitons. However, the intrinsic properties of BiVO4 give rise to limited performances as, for example, in the photodegradation of organic molecules under visible light. A challenging strategy consists of doping the considered structures by suitable elements such as metallic ions incorporated in the structure or clusters contributing to realize composites with improved photocatalytic properties. Suitable doping was realized by using Cu, Mo, or other ions (Merupo et al. 2015a, b) as substitutional ions or using silver nanoparticles associated with BiVO4 to enhance visible light photocatalytic degradation of organic dyes in aqueous solutions. Other complex BiVO4 -based materials were also realized to enhance the photoactivity by increasing the rates of photogenerated carriers as well as their lifetimes required for photocatalysis. Thus, nanostructured-conducting media such as graphene layers have been associated with BiVO4 to limit the rate of recombination of photogenerated electron–hole pairs in water dissociation (Ng et al. 2010) or for degradation of organic dyes (Gawande and Thakare 2012). The association of TiO2 and BiVO4 with particular bridging agents was also investigated leading to remarkable photoactivity (Xie et al. 2015). In similar approaches, nanocomposites were also synthesized based on other oxides (Bajaj et al. 2013; Madhusudan et al. 2011). The relevant methodology ensuring the best composition and features of the efficient photocatalysts based on BiVO4 associates exhaustive experimental contribution for the synthesis and characterizations and theoretical approaches with numerical simulations to validate and analyze quantitatively the phenomena. The developed numerical approaches were partially conducted in our former works devoted to functional semiconducting systems (Makowska-Janusik and Kassiba 2012). The methodology proceeds by modeling at macroscale the structural, vibrational, and electronic properties of bulk crystal. Prediction and quantitative

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M. Makowska-Janusik and A. Kassiba

Fig. 4 FE-SEM image of BiVO4 powders obtained by ball milling

description are achieved for the key physical properties of BiVO4 . In the second step, numerical approaches based on cluster model and quantum-chemical calculations were implemented to explain the electronic properties including quantum confinement effects and related optical changes of nanoparticles and elucidate the influence of surface effects on photocatalytic-driven mechanism such as charge transfers.

Structural Properties of BiVO4 BiVO crystallizes in three main polytypes such as tetragonal zircon, monoclinic scheelite, and tetragonal scheelite structures. For applications in photocatalysis, it was shown that the monoclinic structure offers the best photocatalytic efficiency (Merupo et al. 2015b; Venkatesan 2014). On the other hand, the specific surface of nanomaterials is an important parameter which acts on the reaction area at the interfaces between the photocatalysts and its reactive environment. Suitable methods for the synthesis of BiVO4 were developed including ball milling or hydrothermal for micro- and nanosized particles as well as rf -sputtering for thin films with nanostructured surface profiles. For ball milling technique, mechanochemical processes were carried out with initial pure reactants Bi2 O3 and V2 O5 . Suitable operating conditions were chosen (time of reactions, speed of rotating reaction chambers, nature of the used grinding balls) to obtain homogeneous powders with defined stoichiometry and stable crystalline structure (Merupo et al. 2015b). An example of powder obtained by mechanochemical process during short milling time (6 h) is shown in Fig. 4 by field effect scanning electron microscopy (FE-SEM) (Venkatesan 2014). The microstructure exhibits individual aggregates made from small nanoparticles with sizes in the order of 20 nm. Structural investigations were carried out to identify the involved crystalline phase as a monoclinic scheelite in agreement with lattice parameters a D 5.1950 Å, b D 11.701 Å, and c D 5.092 Å (JCPDS no. 14–0688) (Fig. 5). The vibrational properties give complementary information due to their sensitivity to local distortions of chemical bonding involved in defined environments of the materials.

Fig. 5 XRD pattern of ball-milled BiVO4 , its refinement by Fullprof software is in agreement with the monoclinic sheelite structure according to the diffraction data reference (JCPDS no. 14–0688)

Scattering intensity (arb.units)

Photoactive Semiconducting Oxides for Energy and Environment:. . .

13

Monoclinic BiVO4

20

30

40

Fig. 6 Raman spectrum of BiVO4 obtained by mechanochemical method

Raman scattering intensity (arb.units)

2.q (°)

Monoclinic BiVO4

200

400 600 800 Raman shift (cm–1)

1000

The investigations by Raman spectrometry determine the vibrational modes allowed by the symmetry of the crystal structure. The analysis of the corresponding Raman bands including their full width at half maximum (FWHM) or the shift of their wave number position are related to the quality of the crystalline order of the atomic environments submitted to stretching or bending. Thus, Raman active modes (Fig. 6) of monoclinic clinobisvanite bismuth vanadate are involved at wave number positions at 211, 327, 369, 710, and 831 cm1 (Venkatesan et al. 2012). These active modes are associated with stretching and bending modes of VO4 groups involved in the unit cell of the monoclinic BiVO4 structure. The intense band at 831 and the weak shoulder at 710 cm1 are assigned to the symmetric Ag and asymmetric stretching modes of VO4 groups. The modes at 369 and 327 cm1 correspond to the symmetric and antisymmetric bending modes for the vanadium oxide tetrahedrons.

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M. Makowska-Janusik and A. Kassiba

Such modes are sensitive to bond angles and length distortions within the VO4 groups. The low wave number mode at 211 cm1 is assigned to an external mode associated with Bi-VO4 .

Fig. 7 EPR spectra (experiment and simulation) related to isolated V4C ions due to charge compensation from oxygen vacancies and from clustering of vanadium ions on the surfaces of BiVO4 particles (Venkatesan et al. 2013)

EPR spectrum intensity (arb.units)

Electronic Active Centers of BiVO4 The existence of high specific surfaces in the ball-milled synthesized powders and crystalline defects such as oxygen vacancies favor the appearance of active electronic centers. F-centers from electrons trapped in the vacancy sites or valence change of vanadium ions as V4C (3d1 ) play a crucial role on the photoinduced charge transfer process. Both centers are characterized by an effective electron spin S D 1/2 leading to an EPR signal which can be used to evaluate the nature of the center, its local environment, thermal stability, and behavior under light irradiation. For the ball-milled sample, the characteristic experimental EPR spectrum is reported in Fig. 7. The simulation of the experimental spectrum is made by superimposition of an EPR signal from isolated V4C ions in orthorhombic local environments, while a second signal is related to vanadium clustering. These results are inferred from the magnetic interactions between close V4C ions which alter the EPR line widths (FWHM) and average the anisotropy of the environments. Vanadium pairs or clustering of V4C ions accounts for the EPR signal with the features of a broad background absorption line. The structured details observed on the EPR signal such as several sharp lines are emanating from isolated ions with anisotropic local environments (Venkatesan et al. 2013). In one hand, magnetic interactions are characterized by g-tensor reflecting an orthorhombic local symmetry around V4C . This is understood by the occurrence of similar electronic structure and local environment of V4C in the sample BiVO4 irrespective to the sample microstructures. The location of V4C within crystal sites of the particle cores accounts for the experimental observations. On the other hand, the clustering of V4C is probably involved at the outermost particle surfaces. The exhaustive analysis of the magnetic interactions has been carried out in the former report (Venkatesan et al. 2013) and points out the high probability of

2500

BiVO4:V 4+

experimental

simulated

3500 Magnetic field (Gauss)

4500

Photoactive Semiconducting Oxides for Energy and Environment:. . .

15

occupation of excited orbitals of vanadium ions. The photoinduced charge transfer under light irradiation can be then favored and contribute to the efficiency of photocatalytic reactions which exploit BiVO4 with such active electronic centers. Additional investigations have been carried out in order to characterize the evolution of the electrical conductivity in BiVO4 samples with different microstructures as function of V4C concentration which is found in the range 1016 -1017 ion/gram. As the molecular bonding V5C -O2 accounts for the charge carrier motions within the crystalline structure, the presence of high or low concentration of V4C did not alter the effective conductivity. This fact excludes the possibility for such electronic active centers to act as trap for charge carriers. The photoinduced processes are expected to be enhanced in samples with higher concentration of V4C ions.

Optical Behavior of BiVO4 The optical absorption spectrum informs on the electronic structure and can be used to evaluate the band gap of materials. Previous theoretical investigations were developed on BiVO4 which possesses the features of direct band gap with the extreme of the valence (VB) and conduction band (CB) away from the center of the Brillouin zone (Walsh et al. 2009). Indeed, electronic band structure calculations demonstrate the occurrence of direct transitions between VB and CB at the Brillouin zone boarder, i.e., at the Z point associated with (0.000, 0.500, 0.500) point. The electronic states which compose the valence band (VB) and the conduction band (CB) were identified. Thus, the contributing orbitals to VB consist in interacting of Bi (6 s) and O (2p) states. For the CB, the main contribution occurs from V (3d) orbitals which can interact with Bi (6p) and O (2p). The optical band gap is then defined by transitions between (VB) and (CB) with finite probability at the Z point. The experimental investigations are traduced in Fig. 8 which reports the UV– vis absorption spectrum of BiVO4 versus the photon energy. The photon absorption range indicates the possibility of investigated structure to act as visible light-driven photocatalyst. In order to evaluate precisely the electronic band gap,   the so-called Tauc plot is usually exploited through the equation .˛:h/2 D C: h  Eg where ˛, C, and Eg represent, respectively, the absorption coefficient, a proportionality constant, and the direct band gap energy. The asymptotic extrapolation of the plot with the photon energy (h) axis gives the direct band Eg with a good approximation. By this representation we evaluate the band gap of the considered ball-milled powder with the monoclinic sheelite structure of BiVO4 to Eg D 2.40(5) eV. It is worth to notice that the optical behavior depends on the microstructure of the material, on its crystalline order, and on the eventual electronic active centers as discussed above notably for V4C . Annealing treatment of the powders at temperatures up to 500 ı C is relevant. As an example, as-formed milled powder and that obtained after annealing show different absorption edges of the BiVO4 (Fig. 9) with a net tendency to lower the band gap for annealed samples. These experimental results will be discussed based on the theoretical calculations developed for the mentioned materials.

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M. Makowska-Janusik and A. Kassiba

a Absorption coefficient a (arb.units)

Fig. 8 (Top) Experimental UV–vis optical absorption spectrum of ball-milled powders of BiVO4 . (Bottom) Tauc plot allows determining precisely the direct band gap of the sample

100

Monoclinic BiVO4

50

0 350

550 Wavelength (nm)

a . hn (arb.units)

b 100

750

Monoclinic BiVO4

50

Eg = 2.4 eV

0 2

2.5 Photon energy hν (eV)

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Absorption coefficient a (arb.units)

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0 450

525 Wavelength (nm)

600

Fig. 9 Optical absorbance spectra of ball-milled BiVO4 with different milling times modifying the microstructure (Venkatesan 2014) and after post-annealing treatment (A). Annealing contributes to widen the absorption band to higher wavelengths (lower band gap)

Photoactive Semiconducting Oxides for Energy and Environment:. . .

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Theoretical Background Modeling of Crystalline Structures: Preamble Theoretical models and numerical simulation programs were extensively developed in the last decades with the aim to describe the physical properties of semiconducting materials and to offer qualitative and quantitative supports for experiments. These approaches are used to predict physical properties (electronic, optic, magnetic, etc.) in a wide class of functional materials. In this context, electronic band structures have been evaluated, and several models were developed to ensure the calculations within reasonable time duration of the required computational operations. In all situations, the starting point deals with the Hamiltonian describing mutual interactions between electrons and nuclei and summarized as follows: X p2 X p2 X 2 j 1 i ˇe ˇ H D ˇ ˇ 2mi C 2mj C 2 4"0 ˇˇri r ˇˇ i j ’ i i;i ’ (1) X Z e2 X Zj Z e2 j 1 ˇ j’ ˇ  C ˇ ˇ 2 4"0 jri Rj j 4"0 ˇˇR Rj ˇˇ i;j j’ j;j ’ where ri and ri’ indicate the position of the i-th and i’-th electron, Rj and Rj’ are the positions of the nucleus, Zj denotes the atomic number of the nucleus, and pi , pj are, respectively, the momentum operators of the electrons and nuclei. However, as written, the eigenstates and eigenvalues of such Hamiltonian cannot be found simply and same approximations must be introduced to solve the problem. One of the approaches consists of separation of the electrons into the core and valence parts with the possibility to frozen the first one. The frozen-core approximation is thus fundamental in the pseudopotential approach. In this frame, the introduced pseudopotential reproduces the true potential outside the core region (valence electrons) while smoothing of such potential for the core electrons inside some defined radius and contributing then to eliminate the oscillations of the electron wave function inside the core region. According to normconserving condition, the pseudopotential Vi (r) can be composed by a long-range local part (VLR ) and a short-range semi-local one (VSR ) as follows: V NC PP D V LR C V SR ;

(2)

  where V LR D  Z e 2 =r is a radial function, and the semi-local part is the deviation from the all-electron potential inside the core region. In the same context, the ultrasoft pseudopotential approximation (Vanderbilt 1990) uses more than one projector for each momentum and further smooth the electron wave function. Within the Born–Oppenheimer approximation and for crystalline structure, the mean-field approximation should be implemented to solve the Schrodinger equation. In such approach, each electron feels the same average potential V(r) and its Schrodinger equation takes the form:

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M. Makowska-Janusik and A. Kassiba



 p2 C V .r/ ˆN .r/ D EN ˆN .r/ ; 2m

(3)

2

p C V .r/ denotes one-electron Hamiltonian (He1 ) identical for all where 2m valence electrons. V(r) posseses the symmetry of the lattice and contains both electron–electron and core–electron interactions. Additionally, the potential V .r/ D V .r C Rn / has periodicity of the lattice, where R is the lattice vector. Bloch theorem states that the eigenstates of a periodic Hamiltonian can be written as a product of periodic function with a plane wave of momentum k restricted to the first Brillouin zone (BZ). The quantum-chemical calculations of the crystal structure are based on a supercell method. The wave function in periodic potential named as Bloch function is defined as (Payne et al. 1992):

ˆk .r/ D exp .ik  r/ uk .r/ ;

(4)

where uk .r/ D uk .r C R/ is a periodic function with the translational period equal to R as it was mentioned for the potential V(r). As defined, the plane wave character of the wave function is seen with an amplitude modulated by the periodic function uk (r). Therefore, the wave function satisfies: ˆk .r C R/ D exp .ik  R/ ˆk .r/

(5)

Since the uk (r) has the periodicity of the lattice, it can be expanded in Fourier series: uk .r/ D

X j

  Aj .j / exp iGj  r ;

(6)

where Aj is the minimum translation vector and Gj is the reciprocal lattice vector. When translation parameter TR acts at the Bloch function: TR ˆk .r/ D ˆk .r C R/ D exp .ikR/ ˆk .x/

(7)

This is an eigenfunction of the TR with its eigenvalue equal to exp(ikR). According to the commutator rules, the eigenfunction of the one-electron Hamiltonian (Eq. 3) may be defined as the sum of Bloch functions: ˆ.x/ D

X k

Ak ˆk .x/ D

X k

Ak exp.i kx/uk .x/

(8)

Thus the one-electron wave functions can be indexed by k, which are the wave vectors of the plane waves forming the “backbone” of the Bloch function. A plot of the electron energies in (Eq. 3) versus k is known as the electronic band structure of the crystal.

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19

The electronic states for the crystal structure are calculated over the numbers of k-points. The symmetry considerations suggest that only k-points within the irreducible segment of the Brillouin zone should be taken into account (Srivastava and Weaire 1987; Payne et al. 1992). Using the mentioned method, it can be obtained an accurate approximation of the electronic potential and the total energy of an insulator by calculating electronic states at restricted number of k-points. The calculations for metallic systems require a more dense set of k-points to determine the Fermi level accurately. One of the most popular schemes for generating k-points proposed by Monkhorst and Pack (1976, 1977) produces a uniform grid along the three axes in reciprocal space. They defined a uniform set of special k-points as: k D u p b1 C u r b2 C u s b3 ;

(9)

where b1 , b2 , and b3 are the reciprocal lattice vectors, and up , ur , and us are numbers from the sequence: ur D .2r  q  1/ =2q r D 1; 2; 3; : : : ; q

(10)

The total number of k-points is q3 , but the number of the irreducible k-points can be drastically decreased by symmetry.

DFT Methodology and Related Functionals Another approach widely developed to compute the electronic structure of materials consists of the density functional theory (DFT) used in wide variety of approaches (Bermudez 2010; Na-Phattalung et al. 2006; Le et al. 2005; Chretien and Metiu 2011; Martsinovich et al. 2010; Persson et al. 2003; Albaret et al. 2000; Walsh et al. 1999; Qu and Kroes 2006, 2007; Gały´nska and Persson 2013; Sahoo et al. 2011; Barnard and Zapol 2004a, b; Blagojevic et al. 2009; Hamad et al. 2005; Albuquerque et al. 2012; Zhang et al. 2005 ; Labat et al. 2007). Initially, DFT was developed by Hohenberg and Kohn (Hohenberg and Kohn 1964) and then by Levy (1979). They define the charge density  as the key parameter in energy functions instead of the electron wave functions. Thus, the total energy is composed of the kinetic energy T[] of a system of noninteracting particles superimposed to the electrostatic part due to Coulomb interactions U[] and to the manybody interaction Exc [] called as the exchange and correlation energy. To achieve computational tasks within reasonable time delays, the exchange-correlation energy requires some approximation. In this aim, DFT methodology was implemented within two different approximations, namely, local (LDA) and gradient-corrected (GGA) one. The LDA approximation is based on the exchange-correlation energy of uniform electron gas (Hedin and Lundqvist 1971; Ceperley and Alder 1980). The local

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M. Makowska-Janusik and A. Kassiba

density approximation assumes that the charge density varies slowly on an atomic scale. The total exchange-correlation energy can be obtained by integrating over the uniform electron gas: Z LDA EXC Œ Š

.r/"XC Œ.r/ dr;

(11)

where "xc [] represents the exchange-correlation energy per particle in the considered uniform electron gas. Slater proposed that the exchange part of the mentioned potential may be defined as equal to "XC D1=3 (Slater 1951) where in fact the correlation is not included. Generally, the exchange-correlation energy is described by the approximation implemented by Vosko, Wilk, and Nusair (VWN) (Vosko et al. 1980) or by Perdew and Zunger (PW) (Perdew and Zunger 1981). To investigate the character of the non-homogeneity of the electron gas, the LDA approximation is not sufficient. This phenomena is taken into account by the density-gradient expansion referred to the nonlocal spin-density approximation (NLSD) based on the GGA functional and depends on the derivative d/dr as well as on . This provides a considerable increase in the accuracy of predicted energies and structures, but with an additional computational time delay. The mentioned methodology was implemented by Perdew and Wang in 1991 (Perdew et al. 1992) with the energy defined as: Z GGA EXC Œ Š

.r/"XC Œ.r/; r.r/ dr

(12)

One of the GGA functionals is the Perdew–Burke–Ernzerhof (PBE) one (Perdew et al. 1996), which is used in several problems related to material sciences. However the BLYP functional (Becke 1988) is exploited to evaluate accurately the energy parameters of organic groups leading to more relevant results than the PBE model. The situation is quite different for the correlation energy related to metallic systems where the BLYP potential gives unsatisfactory results. So far, DFT methodology contributes to precise insights on the electronic properties of different class of materials (Hohenberg and Kohn 1964). This seems related to the development of new and more accurate approaches through the used functionals as well as the versatility of numerical methods with regard to different classes of computed systems. As a major drawback of all functionals, the underestimation of the calculated electronic band gap (Hybertsen and Louie 1989) for semiconductors and insulators is frequently encountered due to the large uncertainty in the calculations of the electron self-interaction energies. The pure LDA (Kohn and Sham 1965) or the GGA approximation (Langreth and Perdew 1980) or even the hybrid functional (Stephenset et al. 1994; Perdew et al. 1996; Heyd et al. 2003) may suffer from lack of precision for the so called “strongly correlated” systems. Hybrid functionals were introduced by Becke (1993) replacing a fraction of GGA exchange with HF exchange. Such an approach represents a judicious compromise

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between Hartree–Fock exchange and DFT exchange and correlation part. The hybrid version of PBE and BLYP was proposed as PBE0 (Perdew et al. 1996), HSE (Krukau et al. 2006), and B3LYP (Becke 1993). B3LYP is currently the most popular functional used for organic groups. The formulation is based on Becke 88 exchange functional and LYP correlation functional (Becke, three parameter, Lee– Yang–Parr):       B3LYP LDA EXC D EXC C a0 EXHF EXLDA C ax EXB88  EXLDA C aC ECLYP ECV W N ; (13) where a0 D 0.20, ax D 0.72, and ac D 0.81. The PBE0 functional can be traduced as follows: PBE0 EXC D 0:25EXHF C 0:75ExPBE C ECPBE

(14)

and HSE uses an error function screened Coulomb potential to calculate the exchange part of the energy in order to improve the computational evaluation, especially for metallic systems: HSE EXC D 0:25EXSR ./ C 0:75EXPBE;SR ./ C EXPBE;LR ./ C ECPBE :

(15)

The separation of the electron–electron interaction into a short- and long-ranged part, labeled SR and LR respectively, is realized only in the exchange interactions. It is generally known that classical DFT potentials do not reproduce the correct far-nucleus asymptotic behavior (Almbladh and Pedroza 1984; Levy et al. 1984) and underestimate the excitation energies notably for charge transfer (Dreuw et al. 2003) as also noticed for calculation based on time-dependent DFT (TDDFT) functional (Appel et al. 2003; Dreuw et al. 2003). It is presumed that classical functionals overestimate local contributions and underestimate nonlocal ones. The long-range electron–electron exchange interaction is neglected in pure functionals and is supposed that the appropriate inclusion of the mentioned interaction into DFT calculations improves the obtained results (Tawada et al. 2004). Therefore, the potential solution leads to the partitioning of the total exchange energy into short-range and long-range contributions: EX D EXsr C EXlr

(16)

Such methodology is called long-range correction (LC) scheme (Iikura et al. 2001). It uses the standard error function to modify and improve the exchange functional for evaluating the long-range electron–electron interaction by the HF exchange integral. In LC approach, the repulsion electron operator is also divided into shortrange and long-range part and can be defined as:      1  ˛ C ˇerf rij ˛ C ˇerf rij 1 D C ; (17) rij rij rij

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M. Makowska-Janusik and A. Kassiba

ˇ ˇ where rij D ˇri  rj ˇ is the distance between two electrons located in position ri and rj . The parameter  represents the weighting factor which controls the separation between short-range exchange functional and the long-range part of HF exchange integral. The optimal  factor corresponds to the atomic bond length and hence is expected to be dissimilar for different atoms. When  is equal to zero, the LC DFT calculation corresponds to pure DFT methodology, and conversely  D 1 corresponds to the standard HF calculation. The ˛ and ˇ parameters define the exact exchange percentage between short- and long-range exchange functionals. Since ˛ C ˇ D 1; the exchange potential in LC functionals has the exact asymptotic behavior. Also, the parameter ˛ determines the contribution of the HF exchange at the short-range region. At a distance of about 2/ the short-range interactions become negligible. The LC-BLYP method uses ˛ D 0 and ˇ D 1: The CAM-B3LYP functional (Yanai et al. 2004) (Coulomb-attenuating method applied to B3LYP) also eliminates the long-range self-interaction error. In this case, the parameters are assigned to ˛ D 0:19 and ˇ D 0:46; and hence the exact asymptote of the exchange potential is lost. To sum up, the parameter ˛ allows incorporating the HF exchange contribution over the whole range of distance. Similarly, the parameter ˇ defines the DFT counterpart over the whole range by a factor of 1  .˛ C ˇ/ : Another major problem of standard LDAs and GGAs is that they could not describe the system when the electrons tend to be localized and strongly interacting, such as for transition metal oxides or rare earth elements and compounds. One of the successful models to describe correlated electrons in solids is the Hubbard model (Anisimov et al. 1991) based on extended LDA approach to what is termed LDACU. It decreases the electron self-interaction error by selectively adding an energy correction to localized electron states as for d or f orbitals, where the selfinteraction is particularly large. The additional interactions is usually considered only for high localized atomic-like orbitals on the same site, as the “U” interactions in Hubbard models: EU D

1 X U ni nj: i ¤j 2

(18)

In the multiband Hubbard model, the effective LDACU energy functional is written as: E LDACU D E LDA C E U C E dc ;

(19)

where ELDA denotes standard LDA energy functional and the E dc D UN .N  1/ =2 is called the double-counting term, to remove the energy contribution X of these orbitals which are twice introduced by the Hubbard term (N D ni ). i In the LDA C U approach, the Kohn–Sham equation is supplemented by the nonlocal potential. From the other side, the DFT/Hubbard method fails to compute the correct energy difference in systems with localized/correlated and delocal-

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ized/uncorrelated electronic states (Jain et al. 2011). Also, the LDA/GGA C U methodology was successfully applied to compute the electronic properties of different ternary oxides such as CuAlO2 (Laskowski et al. 2009), CuAl2 O4 (Liu and Liu 2011), Pr2 Ti2 O7 or Ce2 Ti2 O7 (Sayede et al. 2013), and MnFe2 O4 (Huang and Cheng 2013). Based on the outlined theoretical framework and as discussed below, the influence of the Hubbard parameter U on the electronic properties of NiTiO3 was investigated on the crystalline structure in bulk material or in nanosized clusters taking into account the strong correlation of the d-orbital electrons.

Cluster Approach to Model Nanostructures The study of nanosized systems is of great interest with regard to the critical role of the specific active surface in physical phenomena. Particularly, interfaces between the active material and the surrounding media contribute critically to the efficiency of heterogeneous catalysis. In this context, theoretical modeling and numerical simulations to predict electronic, optical, structural, vibrational, and related properties of nanoparticles are challenging tasks. Among the numerical approaches for prediction of the physical properties of nanoscale system, the cluster approach is worthy of interest with respect to combined aspects from molecular physics methodology as well as from that applied to macroscopic systems. Such approach requires DFT methodology supported by the semiempirical PM6 and PM7 parameterized approach due to the cluster size. Some of quantum-chemical methodologies use bulk crystal approach to predict the electronic properties of nanosized materials by building various surfaces and by reducing the periodicity of the crystal lattice (Martsinovich et al. 2010). Other methods proceed by full optimization of atomic positions of nano-clusters without defined crystal structure (Qu and Kroes 2006; Gały´nska and Persson 2013). In the present work, the cluster approach is developed within a methodology based on crystalline structure inside the core of the investigated nanoparticles surrounded by a reconstructed surface according to surface energy requirement and interactions with the environment. The nanostructures are built from their native bulk crystals with specified atomic positions as involved in defined space groups. Modeling of bulk systems is of primary importance to settle the methodology for the structural, vibrational, and electronic properties of nanostructures. In order to simulate realistic structures able to traduce experimental data, the clusters should be constructed with the appropriate size (Makowska-Janusik and Kassiba 2012). However, experimental observations are usually made on nanoparticles with diameters in the range of a few nanometers, but numerical modeling based on quantum-chemical methods are highly time consuming and cannot be applied on such large systems. In this context, semiempirical approaches are more relevant, and the parameterized PM6 or PM7 methods are preferred because their bases include parameterized data for frequently studied atoms. The following section reports

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M. Makowska-Janusik and A. Kassiba

selected semiconducting nanostructures which were modeled based on the above cited approach.

Structural Features of TiO2 , NiTiO3 , and BiVO4 Crystals The crystal structures of investigated semiconducting oxides were computed for two different configurations of the material. One of them was considered like bulk monocrystal and the second one was created in nanosized crystal future. In this case the TiO2 , NiTiO, and BiVO4 crystals were chosen. To build all structures, the Materials Studio program package was used. The structural parameters of all investigated crystals are presented in Table 1. These data are in agreement with those from experimental and other theoretical works (Burdett et al. 1987; Sleight et al. 1979; Zhao et al. 2011). The example of semiconducting NiTiO3 crystal unit cell and its nanostructure are shown in Fig. 10. Table 1 Structural parameters of investigated crystals: TiO2 in anatase phase, NiTiO3 in ilmenite phase, and the BiVO4 in monoclinic scheelite structure TiO2 anatase phase (Burdett et al. 1987) Space group I41 /amd, local symmetry Cell parameters a D b D 3.782 Å, c D 9.502 Å, ” D 90ı Atom position Atom name x/a y/b z/c Ti1 0 0 0 Ti2 0 0.5 0.25 O1 0 0 0.2066 O2 0 0 0.2066 O3 0 0.5 0.4566 O4 0 0.5 0.0434 NiTiO3 – ilmenite phase Space group (No. 148) Cell parameters a D b D 5.029 Å c D 13.795 Å ’ D “ D 90ı ” D 120ı Atom position Atom name x/a y/b z/c Ni 0 0 0.3499 Ti 0 0 0.1441 O 0.3263 0.0214 0.2430 BiVO4 – monoclinic scheelite phase (Sleight et al. 1979; Zhao et al. 2011) Space group C2/c space group and its point group is Cell parameters a D 7.247 Å b D 11.697 Å c D 5.090 Å ” D 134.226ı Atom position Atom name x/a y/b z/c Bi 0 0.134 0.750 V 0 0.370 0.250 O1 0.261 0.051 0.380 O2 0.354 0.208 0.861

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Fig. 10 Unit cell of NiTiO3 crystal (left) and the unpassivated nanocrystal made by atomic group (NiTiO3 )82 (right) (Ruiz Preciado et al. 2015)

Computational Parameters The electronic properties of all investigated single crystal were calculated using DFT method. The quantum-chemical calculations were performed using Cambridge Serial Total Energy Package (CASTEP), the module of Materials Studio program. The CASTEP is based on total energy plane wave pseudopotential method. In the first step, the geometries of the investigated structures were optimized with respect to the total energy minimization within the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (Pfrommer et al. 1997). The symmetries of all structures were frozen during the geometry optimization procedure, but in some cases the size of the unit cell was allowed to be modified. The convergence parameters were chosen for the energy as low as 2  105 eV/atom, the force on the atom was less than 0.01 eV/Å, the stress on the atom limited below 0.02 GPa, and the atomic displacement less than 5  104 Å. The electronic exchange-correlation energy was treated within the frame of the generalized gradient approximation (GGA) using Perdew–Burke–Ernzerhof (PBE) (Perdew et al. 1996) potential. The calculations were performed with the cutoff energy of the plane wave basis set equal to 350 eV for BiVO4 and 500 eV for TiO2 and NiTiO3 crystals. To accelerate the computational runs, the functional such as ultrasoft pseudopotential was used for the calculations related to the following electronic configuration: Ti 3 s2 3p6 3d2 4 s2 , Ni 3d8 4 s2 , Bi 6 s2 6p3 , V 3 s2 3p6 3d3 4 s2 , and O 2 s2 2p4 . The numerical sampling integration over the Brillouin zone was carried out using the Monkhorst–Pack method with a 8  8  8 and 5  5  7 special k-point mesh for (TiO2 ,NiTiO3 ) and BiVO4 , respectively. The total energy was assumed to converge when the self-consistent field (SCF) tolerance is equal to 105 eV/atom. The electronic properties were computed using the same parameters as used during the geometry optimization procedure. The calculations were performed in spin restricted as well as spin unrestricted procedures applying GGA/PBE potential.

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For the TiO2 and NiTiO3 crystals, the Kohn–Sham equation was also solved using the GGA/PBE functional with the Hubbard parameters. As for bulk systems, Materials Studio software was also used to build the nanostructures. The clusters possess spherical shape and were realized by limiting defined volume within the bulk crystal. According to stoichiometry, the clusters were built as n-multiple of the (TiO2 ), (NiTiO3 ), or (BiVO4 ) units with sizes ranging from 0.7 nm up to 2.6 nm. For NiTiO3 , the clusters consist of (NiTiO3 )2 up to (NiTiO3 )183 units. The atomic positions within the cluster were frozen to the ones related to bulk crystal. All clusters were modeled without any external saturation. Only the (TiO2 )n clusters were saturated by using hydrogen atoms or by hydroxyl (OH) groups. The electronic properties of clusters were computed by using the parameterized PM6 and PM7 method (Stewart 2007). This procedure allows pertinent evaluation of physical features of clusters with large sizes and with involvement of heavy atoms. The numerical simulations were then performed using MOPAC (Stewart 2012) and GAMESS (Schmidt et al. 1993) packages. All the calculations were carried out under restricted Hartree–Fock (RHF) regime with the SCF convergence criterion sets at 105 Hartree and no more than 500 iterations. The eigenvalues were calculated for the frozen atomic positions in the clusters with the molecular point group C1. On the other hand, the DFT calculations were performed by using DMol3 software (Delley 1990, 2000) with implemented PBE and B3LYP functionals applying DND basis set. No symmetry rules were imposed during the electronic structure analysis and the SCF density convergence criterion was fixed to 105 eV. In order to reduce the required computational time, all core electrons were represented by effective core pseudopotentials (ECP) (Dolg et al. 1987; Bergner et al. 1993). The electronic properties of the (TiO2 )n clusters were also calculated applying DFT methodology implemented in GAMESS program package with the 6-31G basis set and the effective core potential named Stevens–Basch–Krauss–Jasien–Cundari (SBKJC) (Stevens et al. 1984; Cundari and Stevens1993) required to limit the number of contributing electron wave functions. The calculations were performed with different exchange-correlation (XC) potentials in generalized gradient approximation. The BLYP and B3LYP potentials have been chosen, and the results were compared to different long-correlated hybrid and CAM-B3LYP functional.

Physical Properties of NiTiO3 and BiVO4 Crystals Structural Aspects Quantum-chemical calculations of all investigated crystals proceed first by optimizing their geometries using the GGA/PBE functional. The evaluated atomic distances were relaxed, but their changes remain limited below 5 % compared to experimental results reported elsewhere (Rajalingam et al. 2012). The small discrepancies could be attributed to the fact that the present calculations were made

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Fig. 11 Primitive unit cell of NiTiO3 single crystal (left panel) and of the BiVO4 single crystal (right panel)

at T D 0 K, whereas the experimental measurements are generally recorded at finite temperature. However, the small deviations between theoretical and experimental values indicate that GGA/PBE is satisfactory as a computational method to describe the structures of all investigated crystals. Following the geometry optimization, the investigated ilmenite NiTiO3 structure exhibits layered organization (see Fig. 10) with the Ti and Ni atoms forming the layers separated by oxygen atoms. Figure 11 shows that for the optimized BiVO4 crystal structure, the Bi–O and V–O bonds are more symmetric compared to the starting geometry. All Bi–O and V–O bonds have tendency to align at the same values. The geometry of monoclinic scheelite structure tends to the tetragonal geometry during the total energy minimization procedure. The possible reason of such isotropic bond lengths for Bi–O may be due to strong correlation between the Bi (6 s) and O (2p) orbitals (Ma and Wang 2014).

Electronic Properties of BiVO4 The electronic parameters of the geometry optimized structure of BiVO4 were calculated using DFT/PBE potential. The coordinates of the special points of the BZ are (in units of the reciprocal lattice vectors) L (1/2, 0, 1/2); M (1/2, 1/2, 1/2), A (1/2, 0, 0),  (0, 0, 0), M (0, 1/2, 1/2), and V (0, 0, 1/2). The features of the obtained energy dispersion curves are similar to former results of Zhao et al. (Zhao et al. 2011), but the value of the band gap is significantly different. The minimum band gap obtained for such structure is equal to 2.25 eV compared to 2.17 eV reported in Zhao et al. study. The energy band calculations were also performed for BiVO4 structure without any geometry optimization. The energy gap calculated for the frozen structure shows a departure of 0.16 eV higher than the band gap obtained

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M. Makowska-Janusik and A. Kassiba

4

E [eV]

2

0

–2

–4 L

M

A

G

Z

V

Fig. 12 Band structures calculated for frozen BiVO4 monoclinic structure

by the same methodology for the relaxed geometry. Figure 12 reports the energy band structure calculated for the frozen BiVO4 monoclinic structure where the band gap value is equal to 2.41 eV. The obtained energy band structure has the same shape for both relaxed and frozen structures. The valence band maximum (VBM) is located at the A point of BZ, while the minimum of conduction band (CBM) is realized at the Z point. The evaluated indirect band gap agrees with experimental values predicted by Rajalingam et al. (2012) and contrasts with former evaluation performed by Walsh et al. (2009) where they characterize the BiVO4 crystal as direct semiconducting material. The main contributions to the valence band states for the frozen structure are from O (2p) orbitals slightly hybridized with Bi (6 s) and V (3d) states (see Fig. 13). Metal oxide semiconductors such as ZnO or TiO2 usually have wide band gaps because the valence band has mainly O 2p character and is located at a positive potential (Kudo et al. 1999). The hybridization between the Bi (6 s) state and the O (2p) state at the top of the VB is indeed responsible for the relatively smaller band gap of BiVO4 monoclinic scheelite structure (Walsh et al. 2009). In contrary to the frozen structure, the partial DOS of optimized monoclinic BiVO4 underlines that the VBM possesses contribution mainly from V (3p) states hybridized with O (2p) electrons. The mentioned oxygen–vanadium interaction decreases the calculated energy gap. In both frozen and optimized structure, the bismuth and oxygen contributions to the VBM are similar. The conduction band of the frozen BiVO4 crystal is dominated by V (3d) states with small contribution from Bi (6p) and O (2p) states. The same data were also obtained for relaxed crystal. These results are in agreement with the data measured

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Fig. 13 Partial density of states of monoclinic BiVO4 frozen structure

by XPS (Payne et al. 2011). The oxygen and bismuth contribute to antibonding orbitals forming the CB states. The energy band structure obtained for frozen BiVO4 single crystal gives better results compared to the experimental data. This indicates that the no-relaxed structure gives sound analysis for the electronic features. The distortions of Bi–O and V–O bond lengths enhance the impact of the Bi (6 s) states and contribute to more dispersion in the valence band.

Electronic Properties of NiTiO3 The same calculations performed for the frozen and relaxed NiTiO single crystal give unrealistic energy band gap values. Therefore, the electronic properties of the mentioned crystal were calculated at the spin-polarized approach for a primitive unit

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Fig. 14 Electron density plotted in a selected plane with the constitutive elements Ti, O, and Ni atoms (Ruiz Preciado et al. 2015) Table 2 Hubbard U parameter values chosen for the Ti d and Ni d orbitals with the calculated energy gap value by using PBE C U methodology

Ti d 0 2.5 3.5 4.5 4.5

Ni d 0 2.5 4.5 3.5 4.5

Eg [eV] 0.77 1.75 2.33 1.94 2.46

cell applying Hubbard parameters. These calculations were performed in k space within the BZ directions where the coordinates of the special points are (in units of the reciprocal lattice vectors) F (1/2, 1/2, 0),  (0, 0, 0), K (1/4, 1/4, 1/4), and Z (1/2, 1/2, 1/2). The DFT/PBE method with or without Hubbard approximation was taken into account. Such functional when used in the spin-polarized regime underestimates the electronic energy band gap. The obtained value equal to 0.77 eV is far from experimental data which is in the range 2.12–2.18 eV (Singh et al. 1995; Wang et al. 2013). The reason of such large discrepancy is probably due to the presence of titanium and nickel atoms which possess strongly correlated electrons. An improvement of the predicted results requires the use of Hubbard approximation. Figure 14 reports the electron density calculated by DFT/PBE potential and shows the covalent character of the O–Ti, O–Ni, and Ti–Ni bonds. This behavior makes us confident in the relevance of the statement about the studied material characterized by strongly correlated electrons. The Hubbard-modified Hamiltonian is required to obtain band energy gaps in agreement with experimental data. The different Hubbard parameter (U) values were chosen for the valence Ti (3d) and Ni (3d) electrons and the energy gap values were calculated. The chosen U values summarized in Table 2. show that the close agreement between estimated band gap and experimental results are obtained for the Hubbard parameters equal to 3.5 eV and 4.5 eV for Ti (3d) and Ni (3d) electrons, respectively. These results underline that the PBE C U method provides a satisfactory qualitative electronic structure calculations with the Hubbard parameters UTi and UNi in a reasonable range. The electronic band structure computed with UTi D3.5 eV and UNi D4.5 eV is presented in Fig. 15. and indicates a direct semiconducting nature of NiTiO3 crystal

Photoactive Semiconducting Oxides for Energy and Environment:. . .

31

4 3

E [eV]

2

Eg 1 0 –1 –2

F

G

K

Z

Fig. 15 Electron band structure calculated by PBE/LDACU methodology with the energy levels created by alpha electron – black and energy levels created by beta electron – red (Ruiz Preciado et al. 2015)

with an energy gap at 2.33 eV. The top of valence band is build by alpha electrons and the bottom of conduction band is composed by beta ones. The alpha electrons are involved in hybridized orbitals based on the Ni (3d) and O (2p) (see Fig. 16). The bottom of conduction band is also composed by the hybridization of the Ti (3d) and Ni (3d) states in agreement with former report by Salvador et al. (1982). In both investigated materials, NiTiO3 and BiVO4 , the valence band is created by the oxygen O (2p) electron possessing anion character, while the conduction band contains mostly V (3d) and Bi (6p) of the monoclinic BiVO4 and Ti (3d) and Ni (3d) cations in the ilmenite NiTiO3 crystal. Similarly, as discussed below, the valence band of TiO2 is dominated by the O (2p) electrons, while the conduction band is populated mainly by the unoccupied Ti (3d) electron states.

Structural and Electronic Properties of TiO2 , NiTiO3 , and BiVO4 Nanosized Systems TiO2 Nanostructures The built of nanocrystalline structures with defined geometries was described in Chapter, “Cluster approach to model the nanocrystals and nanostructures”. The computational procedure considers (TiO2 ) clusters with size ranging from 0.6 up to 2.0 nm possessing indivisible number of TiO2 units as n D 2140. All the clusters were spherical in agreement with the work of Kshirsagar et al. (Kshirsagar and Kumbhojkar 2008) establishing that small crystallites exhibit a spherical

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5 4

Ni s p d

3 2 1 0 –1 –2 –3 –4 5

Ti

DOS [electrons/eV]

4

s p d

3 2 1 0 –1 –2 –3 –4

O s p

2

0

–2

–4 –6

–4

–2

0

2

4

E [eV] Fig. 16 Electron density of state (DOS) calculated for NiTiO3 crystal using PBECU method (Ruiz Preciado et al. 2015)

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Fig. 17 HOMO–LUMO energy gap splitting versus size obtained by using parametrized PM6 semiempirical method for (TiO2 )n applying different passivation elements (Makowska-Janusik et al. 2014)

shape. The HOMO–LUMO energy gap splitting ( EHOMO-LUMO) was calculated for (TiO2 )n clusters using the semiempirical PM6 parameterized method. The electronic parameters were evaluated versus the cluster size and the different surface passivation nature such as hydrogen, oxygen, or –TiOH groups. The obtained results point out the occurrence of quantum size confinement effect for all clusters. The size effect and the nature of surface passivation show correlated impact on the EHOMO-LUMO values for all clusters (see Fig. 17). Indeed, the most pronounced size effects were obtained with different features for unpassivated and oxygen-passivated clusters. Passivation by oxygen contributes to drastic shift of EHOMO-LUMO with a particular strength for the clusters with sizes less than 0.8 nm. These results are different than those from the work of Sahoo et al. (2011) probably due to our approach in the present work taking into account the cluster surface reconstruction which was implemented in accordance to the energy minimization procedure and environmental impact. According to our former analysis (Makowska-Janusik et al. 2014), in the surface reconstruction process, atoms at (TiO2 )n outermost cluster shell were moved according to the conjugated gradient algorithm with RMS gradient convergence equal to 0.01 kcal Å1 mol1 . These calculations indicate the important role of saturating atoms and surface reconstruction for small clusters, but for the clusters with diameter larger than 1.0 nm, the saturation effects can be omitted. This is convenient from the point of view of computer simulations because choosing realistic physical passivation is

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Table 3 HOMO–LUMO energy gap splitting calculated by using different DFT functionals for unpassivated (TiO2 )n anatase clusters versus its size variation Number of Cluster (TiO2 )N units n size [nm] BLYP B3LYP LC-BLYP D 0.33  EHOMO-LUMO [eV] 2 0.6 0.97 1.04 4.63 9 0.8 0.77 0.83 3.62 18 1.0 0.47 0.50 2.94 40 1.2 0.42 0.45 2.91

CAMB3LYP LC-BLYP D 0.80 2.92 2.28 1.85 1.46

4.76 3.97 3.03 3.03

always problematic, even if it can bring satisfactory results (Kassiba et al. 2002). Therefore, for the detailed studies, only the unpassivated (TiO2 )n clusters were taken into account. An evaluation of the EHOMO-LUMO of (TiO2 )n clusters was also calculated for structures with various sizes and different DFT functionals. The calculations were performed by using pure BLYP and hybrid B3LYP functional, as well as the different LC and CAM-B3LYP potentials. The obtained data are reported in Table 3. The HOMO–LUMO energy gap values obtained by BLYP and B3LYP functionals are quite small, but their evolution as a function of cluster size reproduces the quantum confinement effect. It is known that the band gap in semiconductors is underestimated by more than 30–50 % within the LDA approximation (Burke 2012). The calculations show also that the evaluated energy gap value can be substantially smaller than the real gap for semiconducting solids (Grüning et al. 2006). To overcome these discrepancies, a possibility consists of treating the exchange term of KS theory as orbital dependent by using hybrid functional (Seidl et al. 1996). However, in bulk crystal approach, the B3LYP potential does not match with the prediction of electronic features of the (TiO2 )n clusters. For representative clusters, the use of LC-BLYP, LC-BOP, LC-BECK, or LCBVWN functionals with a range separation parameter  D 0:33; gives similar values of the HOMO–LUMO energy gap splitting. In Table 3 only data obtained for LC-BLYP potential are presented. The LC and CAM functionals give better predictions of the electronic features than the pure BLYP or B3LYP hybrid functionals. However, LC-BLYP and CAM-B3LYP underestimate the HOMO– LUMO energy gap splitting compared to the band gap energy of experimental bulk anatase which is about 3.02 eV (Amtout and Leonelli 1995). The more relevant results are obtained by using the LC-BLYP functional, leading to a HOMO–LUMO gap of 2.94 eV and 2.91 eV for clusters with diameters of 1.0 nm [(TiO2 )18 ] and 1.2 nm [(TiO2 )40 ], respectively. Thus, the LC functional smoothly replaces the DFT exchange by the HF exchange at long inter-electron distances. Furthermore, it is possible to obtain larger HOMO–LUMO energy gap values by using different range separation parameter . Several works on the LC scheme suggested that the separation parameter should be larger than the standard one equal to 0.33 in order to reproduce atomization energies or barrier height energies, for example (Gerber and Ángyán 2005). In this case, the HOMO–LUMO energy gap splitting was calculated

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Table 4 HOMO–LUMO energy gap splitting calculated for unpassivated (TiO2 )18 anatase cluster with diameter equal to 1.0 nm) using LC-BLYP functional versus range separation parameter

DFT functional LC-BLYP

 D 0.10  D 0.33  EHOMO-LUMO [eV] 2.88 2.94

 D 0.50

 D 0.80

2.99

3.03

for (TiO2 )18 clusters of 1 nm diameter using different range separation parameters, within the LC-BLYP potential (see Table 4). The EHOMO-LUMO value increases with the separation parameter . A relevant evaluation of the HOMO–LUMO energy gap is obtained for a  parameter equal to 0.8 (see Table 3 last column). Higher values of  from 0.9 to 1.5 were also examined, but they did not influence significantly the band gap. These results are consistent with former works of Iikura et al. (Iikura et al. 2001). It is important to notice that an increase of  leads to a higher contribution of long-range exchange interactions from the HF exchange potential limiting also the self-exchange and self-correlation effect for the interacting electrons at the long inter-charge distance. The performed calculations are consistent with data obtained for strongly correlated systems with the electron density distributed close to the nuclei, as it is observed for ionic bonds. In the considered TiO2 system, Ti atoms possess octahedral coordination, and the strong ionic character of the bonds results in localized charges around the O2 anions. The HOMO orbital is dominated by the O (2p) electrons, while the LUMO is populated mainly by the unoccupied Ti (3d) electron states. The same results were obtained for the valence and conduction bands of TiO2 crystal. It was found that LC functionals in DFT method quantitatively reproduces energies of frontier orbitals including HOMO and LUMO. It is shown that for the strongly correlated systems, the LC functional with appropriate long-range separation factor may accurately reproduce the orbital energies in agreement with the prediction of Singh and Tsuneda (Singh and Tsuneda 2013). Within the developed approach, the performed calculations indicate that the choice of the separation coefficient  modulates critically and in large extent the electronic properties of the considered system.

NiTiO3 Nanostructures The implementation of LC functional to calculate the electronic parameters of NiTiO clusters did not evaluate coherently the HOMO–LUMO energy gap splitting. Reliable results were obtained by applying the semiempirical methodology. In Fig. 18, the evaluation of the energy difference between HOMO and LUMO calculated for the NiTiO3 clusters is presented and the size quantum confinement effect is clearly seen. Thus, the results obtained by using PM6 methodology reproduce quantitatively the energy gap of nanosized NiTiO3 . In the work of Vijayalakshmi and Rajendran (Vijayalakshmi and Rajendran 2012), it was specified that the size reduction of the NiTiO3 nanoparticle gives blue shift in the absorption spectra compared to the bulk material.

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5.0

ΔEHOMO-LUMO [eV]

4.5 4.0 3.5 3.0 2.5 2.0

0

10

20

30

40

50

60

70

80

90

100 110

n Fig. 18 Evaluation of the EHOMO-LUMO energy splitting versus number n of NiTiO3 units calculated by PM6 methodology for the different (NiTiO3 )n clusters (Ruiz Preciado et al. 2015)

To evaluate the electronic properties of NiTiO3 nanostructures versus their size, UV–vis absorption spectra were calculated by PM6 methodology. Three different clusters were used, namely, (NiTiO3 )2 , (NiTiO3 )8 , and (NiTiO3 )17 . Largesized clusters were not evaluated because of the computer memory problem. The calculated spectra are presented in Fig. 19. These results for the cluster (NiTiO3 )17 agree with the experimental data obtained on NiTiO3 powder (Fig. 20 top panel). The increase of the cluster size shifts the spectral A peak position to red range (see Fig. 19). The peak B which is well pronounced for the smallest cluster is also shifted to red side but in less marked rate for larger particles. The additional broad peak (C) at the position of 750–900 nm appears for the cluster (NiTiO3 )17 . The two peaks B and C are not observed in the spectrum calculated for the bulk NiTiO3 single crystal (Fig. 20 down panel). Probably this behavior results from surface influence on the electronic properties of nanocrystal material. In our previous work (Ruiz Preciado et al. 2015), it was shown that the HOMO and LUMO orbitals observed for the investigated NiTiO3 clusters are located at the separated atoms. The valence band minimum and conduction band maximum are diffused through the different atoms as it was shown in part concerning crystal electronic properties evaluation. The mentioned separation of HOMO and LUMO orbitals is not seen for the bulk materials where band structure exhibits different behavior. Probably, this explains why the B and C peaks are not visible in UV–vis absorption spectra of the single crystal. The comparison between the theoretical UV–vis absorption spectrum calculated for the cluster (NiTiO3 )17 and the experimental one (see Fig. 20, top panel) assigns

Photoactive Semiconducting Oxides for Energy and Environment:. . . Fig. 19 The UV–vis absorption spectra calculated by the PM6 methodology for different clusters

200

300

400

500

600

37

700

900 1000

(NiTiO3)17

A

C

B Absorption [arb. units]

800

(NiTiO3)8

A B B A

200

300

B

400

500

(NiTiO3)2

600 700 l[nm]

800

900 1000

quantitatively the origin of the observed features. The broad absorption edge situated at 410 nm is attributed to O2 ! Ti4C charge transfer interaction. The lower energetic shoulder is associated with the crystal field splitting of NiTiO3 , associated with the Ni2C ! Ti4C charge transfer (Lopes et al. 2009). However, in contrast to experimental features, additional peaks are obtained on the theoretical UV–vis absorption spectra. These discrepancies are explained by the absence of temperature influence which causes structural relaxation of the nanoparticles. Indeed, taking into account such temperature influence, the electron–phonon interaction and the Franck–Condon rule lead to the calculated absorption peaks with more broad shapes. Taking into account appropriate passivation of clusters and their reconstructed surfaces, the theoretical modeling of the nanosized structures reproduces their main physical features. This is illustrated on the vibrational properties through the calculated Raman spectra for selected clusters. Therefore, ilmenite crystal nanostructure with frozen atomic positions and nanoparticle with optimized geometry according to the total energy minimization was chosen. The studied structures possess common starting geometry indexed as (NiTiO3 )17 cluster. The calculated and experimental Raman spectra are presented in Fig. 21 for the cluster (NiTiO3 )17 with ilmenite

M. Makowska-Janusik and A. Kassiba

Absorbance [a.u.]

38

400

500

600

700

800

900

1000

Absorption [arb. units]

l [nm]

100

A

200

300

400

500

600

l [nm]

700

800

900

1000

Fig. 20 The UV–vis absorption spectra measured for NiTiO3 powder samples (top panel) and the spectrum calculated by PBE/LDA C U functional (top right panel) and the experimental data (bottom right panel)

structure. One may note that only one peak at position C is obtained. The origin of such vibrational band around 700 cm1 is attributed to the Ti–O–Ti stretching mode which can be slightly shifted in real crystal structure up to 720 cm1 (Su and Balmer 2000; Llabrés i Xamena et al 2003; Baraton et al. 1994). The peaks A, B, and D observed for the optimized nanostructure indicate the relaxed distance between Ni, Ti, and O bonds. Figure 21 reports also the convolution of the spectrum obtained for ilmenite nanocrystal and optimized structure with ratio 9/1. The strength of the developed theoretical and numerical modeling is clearly illustrated from the close correspondence between predicted Raman signatures and the experimental ones of nanopowder systems. The obtained results such as vibrational or optical absorption support the fact that the nanocrystalline oxides give rise to two defined contributions from the core of particles and the external and outermost surface with amorphous or reconstructed atomic positions. They traduce the properties of bulk nature and those traducing the surface broken periodicity.

Photoactive Semiconducting Oxides for Energy and Environment:. . .

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ilmenite NiTiO3 amorphous NiTiO3

Intensity [arb. units]

90% of ilmenite NiTiO3 + 10 % of amorphous NiTiO3 experimental data

D A

B

C

500

600

700

800

900

1000

Raman Shift [cm–1] Fig. 21 Raman spectra of (NiTiO3 )17 cluster calculated by the parametrized PM6 method and compared to experimental spectrum (Ruiz Preciado et al. 2015)

BiVO4 Nanostructures Beyond NiTiO3 , similar behavior was also noticed from the size quantum confinement effect involved in BiVO nanostructures. Using the semiempirical PM6 parameterized method, the electronic structures of isolated (BiVO4 )n clusters were calculated. The number n of structural units was changed from n D 1 up to n D 56 leading to cluster diameter up to 2.0 nm. The calculated EHOMO-LUMO energy gap splitting is presented in Fig. 22. The predicted values of the HOMO–LUMO difference are divided into two parts as marked by rectangles and triangles. The first class of mentioned data can be fitted by an exponential curve. Qualitatively, this behavior is in agreement with quantum confinement size effect. The second class of data (triangles) represents the clusters with frozen structure as for bulk system. These data fluctuate with increasing the cluster size but have tendency to be aligned by one asymptotic value close to 2.4 eV. Such behavior is in agreement with the experimental value of the energy gap defined for bulk BiVO4 . The area marked by dashed line contains cluster size range where the frozen bulk-like and surfacerelaxed structures defining the property of nanoparticle compete one with the other. This area corresponds to cluster sizes ranging from 1.1 up to 1.8 nm where the surface and volume atoms contribute differently to the electronic band structure of the cluster. The origin of the calculated EHOMO-LUMO is the effect of emulation between two competing mechanisms. One is believed to come from the crystalline cores of the nanometer-sized BiVO4 and the other one is attributed to the surface layer. Similar mechanism was used to explain the photoexcitation effect occurring in the crystalline Si cores (Wilcoxon et al. 1999).

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ΔEHOMO-LUMO [eV]

5

nanocluster representation bulk representation

4

3

2

1

0

20

40

60

number of (BiVO4)n units Fig. 22 The EHOMO-LUMO energy gap splitting versus size of BiVO4 cluster represented by the number of (BiVO4 )n units. Dashed line indicates the intermediate area between bulk frozen atoms and nanoparticle behavior with contributions from core and surface atoms

The electronic properties of the BiVO4 nano-clusters computed by PM6 semiempirical method confirm that the HOMO orbital is built by the oxygen states, while LUMO orbital include vanadium states. It confirms that the limitation of the cluster size did not change drastically its electronic configuration and properties. The obtained results show that the electronic structure of nanoparticle is the consequence of bulk crystal material. The HOMO and LUMO orbitals agree with the VBM and CBM of frozen BiVO4 single crystal, and performed calculations are in agreement with the works of Huang et al. (Huang et al. 2014) and Jo et al. (Jo et al. 2012). The theoretical calculations are consistent with the experimental features reported in the first part of the presented chapter. This fact makes us confident in the developed numerical approach to analyze the electronic and optical effects of this class of semiconducting oxides. The quantum confinement effect is well demonstrated theoretically on the optical spectrum of nanosized cluster. However, our experimental synthesis did not allow so small nanocrystal sizes, and only the features 20–30 nm particles where indeed demonstrated experimentally. Dealing with the photoinduced charge transfer required for visible-driven photocatalysis, the experimental results point out the visible light absorption in a wide spectral range. The nanoparticle sizes 20–50 nm contribute to enhance the specific surface which is critical in heterogeneous photocatalysis. Indeed, as the contacts at the interfaces increase between photocatalysts and surrounding media, an improvement occurs on the efficiency of photocatalysis reactions. The benefit from nanosized clusters is thought to be related to drastic enhancement of specific surfaces. However, as it was shown from the calculations (Fig. 23), the spectral range of absorbed sun light is shifted to higher energies limiting then the absorption quantum efficiency. Thus,

Photoactive Semiconducting Oxides for Energy and Environment:. . .

41

Absorption [arb. units]

(a) Nano

Bulk

1.5

2.0

2.5

3.0

hn [eV] Fig. 23 Calculated optical absorption spectra of BiVO4 as nanocrystalline object and infinite bulk system. The blue shift due to quantum confinement effect is traduced on the nanocrystal absorption curve

a balance must be achieved between the size reduction (specific surface) and the induced widening of the band gap.

Conclusions Exhaustive experimental investigations and theoretical approaches supported by numerical simulations were dedicated to selected classes of semiconducting oxides such as TiO2 , BiVO4 , and NiTiO3 characterized by promising visible-driven photoactive responses. The ambition consists of tailoring original smart materials from chemical composition, structural and morphological features for pohotovoltaic and photocatalysis applications. Original synthesis strategies were developed to obtain these materials as thin films, mesoporous structures, as well as nanoparticles. To create visible-driven photoactive media with higher efficiency, suitable doping was realized to modulate the electronic and optical properties. Structural and vibrational investigations were carried out and inform on the stabilized crystalline polytypes and their evolution with the synthesis and treatment conditions. The physical properties including electronics and optics, as well as the role of electronic active centers realized by doping or intrinsically induced by oxygen vacancies or form the stoichiometry departure, were also investigated, and their interpretation was made quantitatively with the support of exhaustive computational work. Quantum theory approaches based on DFT and semiempirical models were applied to predict the physical peculiarities of photoactive semiconducting oxides. Original approaches were tested and the appropriate functionals (potentials) were defined coherently with the chemical nature of the materials. Computer simulations were performed

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for single crystals and implemented successfully for nanosized structures. For all chosen nanoparticles as TiO2 , NiTiO3 , and BIVO4 , the appearance of the size quantum confinement effect was demonstrated on the electronic and optical features. The HOMO orbitals of all investigated nanocrystals are composed of the O (2p) states and the LUMO depends, respectively, on the V (3d) or Ti (3d) atoms. The hybridization of the O (2p) orbitals in the area of occupied orbitals with the other orbitals as Bi (6 s) decreases the energy gap for BiVO4 . For structural aspects, surfaces of nanostructured objects are reconstructed due to the environmental influence inducing changes on the interatomic distances. Orbital hybridization contributing to extremes of valance and conduction bands is modified by such surface effects with particular strength for low-sized clusters as traduced from the carried out numerical simulations. The close correlation between the experimental investigations on such semiconducting oxide systems and the theoretical approaches supported by numerical simulation paves the way to understand quantitatively the physical peculiarities of such photoactive systems. Charge transfer effects, role of doping, and structural arrangements as function for the material morphology are understood by such combined experimental investigations and theoretical models. Our ambition is to contribute by such methodology to design the relevant structures with improved photoactive properties able to be implemented in new technologies required for societal challenges like clean energy and safe environment.

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Structures and Stability of Fullerenes, Metallofullerenes, and Their Derivatives Alexey A. Popov

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structures and Stability of Empty Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Fullerenes and Enumeration of Their Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Isolated Pentagon Rule and Steric Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Isomers of IPR Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bonding, Structures, and Stability of Endohedral Metallofullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metal–Cage Bonding in Endohedral Metallofullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isomerism in Endohedral Metallofullerenes: Stability of the Charged Carbon Cages . . . . . . . Isomerism in Endohedral Metallofullerenes: Cage Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Violation of the Isolated Pentagon Rule in Endohedral Metallofullerenes . . . . . . . . . . . . . . . . . . . Structures and Stability of Fullerene Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addition of X2 to C60 : Isomers of C60 X2 and General Considerations . . . . . . . . . . . . . . . . . . . . . . . . Addition of H and F to C60 : Contiguous Addition, Benzene Rings, and Failures of AM1 . . Addition of Bulky Groups to C60 : Bromination and Perfluoroalkylation . . . . . . . . . . . . . . . . . . . . . Addition to C70 and Higher Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 4 6 12 21 21 26 31 35 36 36 38 43 48 52

Abstract

This chapter describes general principles in the stability and bonding of empty fullerenes, endohedral fullerenes, and exohedral derivatives of empty fullerenes. First, an overview of the structural properties of empty fullerenes is given. The problem of isomers’ enumeration is described and the origin of the intrinsic steric strain of the fullerenes is discussed in terms of POAV ( -orbital vector analysis) leading to the isolated pentagon rule (IPR). Finally, theoretical studies of the isomers of fullerenes are discussed. In the second part of the chapter,

A.A. Popov () Leibniz-Institute for Solid State and Materials Research (IFW Dresden), Dresden, Germany e-mail: [email protected] © Springer Science+Business Media Dordrecht 2016 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_19-2

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bonding phenomena and molecular structures of endohedral metallofullerenes (EMFs) are reviewed. First, the bonding situation in EMFs is discussed in terms of ionic/covalent dichotomy. Then, the factors determining isomers of EMFs, including those favoring formation of non-IPR cage isomers, are reviewed. In the third part, general principles governing addition of atomic addends and trifluoromethyl radicals to fullerenes are analyzed.

Introduction Carbon cluster research, originally the field of scientists interested in Cn species that might exist in deep space, “landed” on Earth with two discoveries that eventually changed the face of carbon science in particular and the nanoscience/nanotechnology in general. The first discovery was reported in 1985, when Kroto, Curl, Smalley, and coworkers made the Nobel-prize-winning suggestion that the high-mass peaks in the mass spectra of laser-evaporated graphite belonged to closed-cage structures now known as fullerenes (Kroto et al. 1985). These molecules remained the playthings of mass spectrometrists, the only scientists who could “observe” fullerenes by producing them in situ in infinitesimal amounts, until 1990 when Krätschmer and coworkers discovered that this “new form of carbon” can be produced in bulk (i.e., macroscopic) amounts by the arc burning of graphite (Krätschmer et al. 1990). C60 followed by C70 is the major fullerenes produced by arc burning; however, starting from C72 , classical fullerene cages are possible for each even number, and many of such fullerenes (known as higher fullerenes) are produced as well. Studies of higher fullerenes are complicated by the increasing number of possible isomers, even if the strict isolated pentagon rule (IPR; see below for more details) is applied. The fullerenes with the number of atoms less than 60 are also available, but still remain exotic objects. Thus, the world of fullerenes themselves is very rich, but they also provide uncountable possibilities for further chemical modifications. One of the attractive properties of fullerenes intrinsic to their closed-cage structure is the possibility of using them as robust containers for other species. The first evidence that metal atoms can be put inside fullerenes was reported in 1985 and was based on mass spectrometry (Heath et al. 1985), while the first bulk samples of fullerenes with metal atoms inside (endohedral metallofullerenes; hereinafter EMFs) were obtained in 1991 by laser evaporation (Chai et al. 1991) or arc discharge (Alvarez et al. 1991) of the graphite rods mixed with lanthanum oxide. In the first decade after the discovery of macroscopic fullerene production, the field of EMFs remained very “hot.” Many metals were put inside fullerenes during this time, and many new molecules were reported (Shinohara 2000; Popov et al. 2013), but the yields of EMFs were very low (usually on the order of 1 % of the empty fullerenes in the arc-discharge soot). Until 1999, EMFs were mostly molecules with one to three metal atoms encapsulated in the carbon cage. In 1999 it was found that the presence of nitrogen gas in the arc-burning reactor resulted in metal nitride cluster fullerenes (NCFs) with the composition M3 N@C2n (M D Sc, Y, Gd–Lu; 2n D 68–96; the symbol “@” in the formula denotes how the EMF

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molecule is divided into the endohedral cluster, the composition of which is given before this symbol, and the carbon cage with optional exohedrally attached groups, which follow after this symbol) to be produced (Dunsch and Yang 2007; Stevenson et al. 1999). In 2001 it was discovered that some “conventional” EMFs actually had C2 carbide units inside the cage. For example, “Sc3 @C82 ” and “Sc2 @C86 ” were shown to be Sc3 C2 @C80 and Sc2 C2 @C84 , respectively (Iiduka et al. 2005; Wang et al. 2001). Judicious choice of the nitrogen source led the group in Dresden to the invention of the reactive atmosphere method, in which the use of ammonia in the arc-burning reactor resulted in NCFs as major fullerene products (Dunsch et al. 2003, 2004). The same method with the use of CH4 resulted in Sc3 CH@C 80 (Krause et al. 2007). Oxides Sc4 O2,3 @C80 were synthesized using copper nitrate as the source of oxygen (Stevenson et al. 2008), while the use of guanidinium thiocyanate afforded formation of sulfide clusterfullerenes M2 S@C82 (M D Sc, Y, Dy, Lu) (Dunsch et al. 2010). The chemical route to open the empty fullerene cage, insert the small molecule inside (e.g., H2 ), and then close the carbon cage was also recently reported (Komatsu et al. 2005). Finally, it is also possible to introduce one or two atoms of inert gases into the fullerene by high-pressure–high-temperature treatment (Saunders et al. 1993). An extended review of the field was published in 2013 (Popov et al. 2013). Another important property of fullerenes from the chemist’s point of view is a very rich  -system based on the formal sp2 hybridization state of all carbon atoms. In chemical reactions fullerenes behave as polyalkenes (original expectations of the “superaromatic” properties were not confirmed) and thus exhibit very rich addition chemistry. That is, numerous cycloaddition reactions and addition of the groups forming one single bond to the fullerene core (such as atoms, e.g., halogens or hydrogen, or radicals, e.g., alkyl groups) are reported (Hirsch and Brettreich 2005). Development of fullerene science was always accompanied – and sometimes preceded – by theoretical studies. The principles governing stability of fullerenes were revealed in the late 1980s and early 1990s with the use of computational approaches, and such important data as relative energies of the fullerene isomers are still available only from the results of quantum-chemical calculations. Likewise, the most mysterious question about the fullerenes – why and how they are formed – is successfully addressed by theoreticians (Irle et al. 2003, 2006). The rise of the fullerene era to a large extent coincided with a dramatic increase in the capabilities of computational chemistry – either from advances in hardware, software, or theory itself. If studies of fullerenes in the mid-1980s, even by semiempirical approaches, were possible only in special laboratories, presently (at 2015), accurate DFT calculations can be done by virtually any scientist using a standard office computer. Yet, the large size of fullerenes and their derivatives still imposes serious limitations: even now it is barely possible to perform post-Hartree–Fock ab initio calculations. Thus, DFT is the main method of choice in fullerene chemistry and will probably remain so in years to come. Although MP2 calculations are becoming more feasible (Darzynkiewicz and Scuseria 1997; Haser et al. 1991; Krapp and Frenking 2007), a significant increase of computational demands in many cases is not justified by the reliability of the results when compared to DFT (the modeling of noncovalent

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interaction, the weak point of DFT, is, however, an important exception). Routine use of methods with higher scaling than MP2 is still very difficult at present. In this chapter we will show how quantum-chemical calculations can assist in studies of fullerenes. In particular, we will discuss prediction and elucidation of the molecular structures of the empty fullerenes, endohedral metallofullerenes, and selected fullerene derivatives. Taking into the account the huge amount of literature published on this subject, any review of the field is necessarily incomplete. We will not try to cover the whole field; instead, we will focus on several representative examples.

Structures and Stability of Empty Fullerenes Definition of Fullerenes and Enumeration of Their Isomers IUPAC defines a fullerene as a “compound composed solely of an even number of carbon atoms which form a cage-like fused-ring polycyclic system with twelve fivemembered rings and the rest six-membered rings.” In practice, all other closed-cage structures built from three-coordinate carbon atoms are also called fullerenes. The definition of fullerenes is based on the Euler theorem, which states that for a given polyhedron, the numbers of vertices (n), edges (e), and faces (f ) are related by: nCf DeC2

(1)

Each carbon atom in a fullerene is bonded to three other atoms; therefore, for a fullerene Cn , the number of edges is e D 3n/2, which yields the number of faces f D n/2 C 2. Since all arc-discharge produced fullerenes have only five- and sixmembered faces (it is possible to obtain fullerenes with four- or seven-membered rings only by chemical modification), one can calculate the number of vertices and faces through the number of pentagons (p) and hexagons (h): .5p C 6h/ =3 D n p C h D n=2 C 2

(2)

Solution of Eq. 2 yields p D 12 (and hence the number in the definition given above) and h D n/2 – 10. Obviously, the smallest structure with 12 pentagons is C20 (it has no hexagons), and starting from n D 24 at least one fullerene structure is possible for each even n (note that it is not possible to build a fullerene for C22 ). The structural consequences from Euler’s theorem applied to fullerenes with pentagons, hexagons, and heptagons were discussed in detail by Fowler and coworkers (Fowler et al. 1996). The difference between the number of pentagonal and heptagonal faces is always 12. Therefore, addition of a heptagon increases the number of pentagons by one, while decreasing the number of hexagons by two.

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Fig. 1 Unwinding C60 and C70 into spirals (thick black line)

The number of the isomers is rapidly increasing with n, raising the problem of systematic enumeration of all possible isomers. The problem was solved for all practical fullerene sizes by the invention of the spiral algorithm (Manolopoulos et al. 1991; Fowler and Manolopoulos 1995). The algorithm is based on the spiral conjecture, which states that The surface of a fullerene polyhedron may be unwound in a continuous spiral strip of edge-sharing pentagons and hexagons such that each new face in the spiral after the second share an edge with both (a) its immediate predecessor in the spiral and (b) the first face in the preceding spiral that still has an open edge (Fowler and Manolopoulos 1995). Figure 1 shows one of the ways in which C60 and C70 can be unwound into a spiral. The spiral can be coded by a sequence of “5” and “6” digits corresponding to the positions of the pentagons and hexagons. For instance, the spiral of the fullerene C60 shown in Fig. 1 can be written as: 56666656565656566565656565666665

(3)

Enumeration of all fullerene isomers is now straightforward: one should generate all possible spiral codes consistent with the given number of atoms (i.e., for Cn fullerene the length of the code is f D n/2 C 10, and positions of the pentagons can be used as running indices) and then check for each spiral if it can be wound up into a fullerene. The task, however, is not complete at this point since the fullerene molecule can be unwound into up to 6 n spirals, and hence each fullerene will be generated many times. For instance, if C60 is unwound into a spiral starting from a hexagon, the following codes can be also obtained: 65656566656656656566566565656566 66565656566566565665665666565656

(4)

There is a simple way to solve the problem of checking the uniqueness of the given fullerene. One can consider the spiral codes given in Eqs. 3 and 4 as numbers, which can then simply be compared to each other. For the spirals of C60 under discussion,

6

A.A. Popov

(Eq. 3) < (Eq. 4) < (Eq. 4). The spiral with the smallest number is called a canonical spiral; all noncanonical spirals should be sorted out. Thus, in addition to checking whether a given spiral can be wound up into a fullerene, one should also check if it is canonical. This problem can be also solved straightforwardly: if it is found that the spiral forms a fullerene, this fullerene should then be unwound into all possible spirals and the canonical one should be found among them and compared to the testing spiral. The algorithm outlined above has been proved to be very successful, and its use in enumeration of the fullerene isomers is now standard. Importantly, the sequential number of the given fullerene isomer (i.e., the number of the isomer in the list of all isomers generated for the given fullerene by the spiral algorithm) is its sufficient and unique identifier. Usually, in designation of the fullerene isomer, one also adds its symmetry (to be precise, the highest symmetry possible for this topology of atoms; the ground state structure of the molecule can have lower symmetry), and we will use this notation in this chapter. To conclude this section, we note that the spiral codes in Eqs. 3 and 4 can be further simplified, taking into account that each fullerene has only 12 pentagons, and hence the spiral code can be uniquely defined by the numbers of the pentagon positions. In this notation, the spiral of C60 in Eq. 3 is written as: 1 7 9 11 13 15 18 20 22 24 26 32

(5)

This notation is usually used in the literature, including the famous “An Atlas of Fullerenes” (Fowler and Manolopoulos 1995).

The Isolated Pentagon Rule and Steric Strain The importance of exhaustive systematic enumeration of all isomers for a given fullerene becomes clear when the problem of structure elucidation of fullerenes is raised. Highly symmetric structures of C60 and C70 were proposed as a result of playing with the models. However, the success of this approach was to a large extent fortuitous, and this is hardly possible for other fullerenes. Of course, an unambiguous elucidation of the molecular structure might be possible through single-crystal X-ray diffraction studies. However, the quasi-spherical shape of fullerene molecules results in strong disorder, which in most cases precludes reliable elucidations of the isomeric structures. This problem can be solved by chemical functionalization of fullerenes (in particular, chlorination and perfluoroalkylation of higher fullerenes have been extensively studied during the last decade) and X-ray studies of their derivatives (Kareev et al. 2008a; Shustova et al. 2007; Simeonov et al. 2007; Troyanov and Tamm 2009; Troyanov and Kemnitz 2012; Boltalina et al. 2015), but this promising approach is not always applicable. Thus, since early 1990s and up to now, 13 C NMR spectroscopy has been the most important method in structural studies of newly isolated fullerenes. However, through the number of lines and their relative intensities, 13 C NMR spectrum gives at best only the symmetry of

Structures and Stability of Fullerenes, Metallofullerenes, and Their Derivatives

7

Table 1 The total number of isomer of C60 –C102 fullerenes and the number of fullerene C90 isomers of different symmetry types compared to the number of corresponding IPR isomers Cn 60 70 72 74 76 78 80 82 84

Total 1812 8149 11190 14246 19151 24109 31924 39718 51592

IPR 1 1 1 1 2 5 7 9 24

Cn 86 88 90 92 94 96 98 100 102

Total 63761 81738 99918 126409 153493 191839 231017 285914 445092

IPR 19 35 46 86 134 187 259 450 616

C90 -symmetry group C90 -C1 C90 -C2 C90 -Cs C90 -C3 C90 -C2v C90 -D3 C90 -D5 C90 -D3h C90 -D5h

Total 97936 1266 655 3 50 4 1 1 2

IPR 16 16 6 0 7 0 0 0 1

the fullerene molecule, but not the exact isomeric pattern. Therefore, it is necessary that all isomers with NMR-determined symmetry are considered as possible structural guesses. Quantum-chemical calculations and studies by other spectroscopic techniques can be then used to determine the most appropriate structure. Table 1 lists the total amount of possible isomers of fullerenes Cn for different n. These numbers clearly show that consideration of all isomers is hardly feasible for the low-symmetric structures. For instance, 99918 isomers are possible for C90 , and the information that an experimentally studied isomer has C2 symmetry allows one to reduce the structural guesses to 1266 isomers, which is still not very feasible. On the other hand, the number of isomers formed in arc-discharge synthesis is usually rather low. Therefore, it seems possible and highly desirable to develop some simple rules to reduce the number of isomers that should be considered. The most successful rule of this sort is the isolated pentagon rule (IPR) proposed by H. Kroto back in 1987 on the basis of general considerations (Kroto 1987) and by Schmalz and coworkers in 1988 on the basis of more careful considerations of the sources of strain in carbon clusters and Hückel calculations (Schmalz et al. 1988). The IPR rule states that the fullerene isomers with adjacent pentagons are less stable than the isomers in which pentagons are surrounded only by hexagons (i.e., pentagons are isolated). In subsequent works numerous isomers of fullerenes were studied and the number of the pentagon–pentagon edges was found to be a parameter exhibiting very good correlation to the stability of the fullerenes. Figure 2a shows the correlation between the number of pentagon–pentagon edges and the relative energies of all 1812 isomers of C60 calculated at the QCFF/PI level of theory (Austin et al. 1995) (QCFF/PI is an inexpensive semiempirical computational approach combining classical mechanics for ¢-bonds and the Pople–Pariser–Parr method for  -system). The correlation is close to the linear form, and the energy penalty for one pentagon–pentagon edge, 111 kJ/mol for C60 , can be estimated from the slope of the fitted line. In a detailed study of the C30 –C60 fullerene isomers at the QCFF/PI level (Campbell et al. 1996), it was found that the penalty is increasing with the cage size (Fig. 2b), from 72 kJ/mol for C30 to 105 kJ/mol for C50 . The study of 55 C76 isomers

8

A.A. Popov

b E

120

/ kJ mol–1

2000

110

1500

100

1000

90

500

80 pp

0 0

10

c

20

E, kJ/mol

a 2500

n in fullerene Cn 70 28 32 36 40 44 48 52 56 60

800

600

ΔE

400

200

0 22.75

H 23.00

23.25

23.50

23.75

Fig. 2 (a) The relative energy of C60 isomers computed at the QCFF/PI level as a function of the number of pentagon–pentagon edges (Reproduced with permission from (Austin et al. 1995), © (1995) American Chemical Society); (b) energy penalty for the pentagon adjacency as a function of the fullerene size (Based on the data from Campbell et al. 1996); (c) relative energy of the IPR isomers of C116 computed at the QCFF/PI level as a function of the steric strain parameter H (Reproduced with permission from (Achiba et al. 1998a), © (1998) American Chemical Society)

of D2 and higher symmetry at the QCFF/PI level (Austin et al. 1994) has shown that the penalty of the pentagon adjacency can be estimated as 150 kJ/mol (but this value is probably overestimated since the non-IPR isomers with lower symmetry can be more stable). The study of all 40 isomers of C40 at various levels of theory, including DFT (B3LYP, BLYP, and LDA), the Hartree–Fock method, semiempirical methods (AM1, PM3, MNDO, QCFF/PI), the density-functional based tight-binding method (DFTB), and molecular mechanics (MM), has shown that for all methods except for MM, the penalty for C40 isomers falls in the narrow range of 82–100 kJ/mol (Albertazzi et al. 1999). In particular, B3LYP/STO-3G and QCFF/PI values are 92 and 91 kJ/mol, respectively, which confirms that the predictions on the broader range of isomers obtained at the QCFF/PI levels are also reliable. It should be noted,

Structures and Stability of Fullerenes, Metallofullerenes, and Their Derivatives

9

however, that the increase of the basis set to 6-31G resulted in a 13 % increase in the penalty at the BLYP level, showing that the basis set effects can be rather important. The energy penalty for the pentagon adjacency is sufficiently large to suggest that all non-IPR isomers can be sorted out for higher fullerenes (relative energies of the isolable fullerene isomers are usually less than 100 kJ/mol; see below). Increase of the energy penalty with the cage size is another factor favoring the application of the IPR. Indeed, the rule is strictly followed by all experimentally available empty fullerenes (the notable exclusion is C72 , which is discussed below). For the experimentally accessible fullerenes, the IPR reduces the number of possible isomers to be considered from many thousands to tens and hundreds (Table 1). It is, however, possible to stabilize pentagon adjacencies by exohedral derivatization or by coordination to the metal atoms in EMFs (Tan et al. 2009), and hence larger sets of isomers have to be considered in these cases. High energy penalty is the result of the combination of at least two factors. First, a pair of adjacent pentagons (pentalene) has an 8-electron  -system and is anti-aromatic. Another factor, which is probably more important, is the steric strain induced by adjacent pentagons. The formal hybridization state of carbon in fullerenes is sp2 . Thus, carbon atoms should be planar (by this, we mean that it has three neighboring carbon atoms in the same plane). From the geometrical point of view, planarity is achieved when a carbon atom is located on the fusion of three hexagons. In this case,  -orbitals (pz ) of the neighboring atoms are exactly parallel and the optimal overlap of these orbitals is achieved. However, one cannot build fullerene only from hexagons – pentagons are necessary to make closed-cage structures. The carbon atoms in pentagons are not planar anymore (the small angle of 108ı vs. 120ı in hexagons has to be compensated by the out-of-plane shift of the central atom), which results in rehybridization, the effect easily explainable in terms of  -orbital axis vector (POAV) analysis. POAV is defined as the vector that forms equal angles (POAV angles  ¢  ) to all three ¢-bonds (it is assumed that ¢-orbitals are aligned toward the neighboring atoms). Admixing of s-orbitals into the hybrid  -orbital (the essence of rehybridization) can be quantified from geometrical point of view as: mD

2sin2 .  =2/ 1  3sin2 .  =2/

(6)

where m determines admixture of s-orbital to the hybrid  -orbital (  D sm p; for perfect sp2 hybridization,  ¢  D 90ı and m D 0). When m is not equal to zero,  -orbitals of the neighboring atoms are not parallel anymore. Because of this misalignment, the  -orbital overlap is reduced, and therefore the  -bonding is weakened, the effect being the stronger the larger the value of m. Figure 3b shows possible junctions of pentagon and hexagons and hybrid  -orbital of the central atom computed for idealized polygonal bond angles. Misalignment of  -orbitals increases with the increase of the number of fused pentagons, and the IPR is a manifestation of this tendency.

10

A.A. Popov

Fig. 3 (a) Definition of the  ¢  angle in the POAV approach; (b) the component of the hybrid  orbital of the central atom in different junctions of pentagons and hexagons computed for idealized polygonal bond angles (Based on the data from Fowler and Manolopoulos 1995)

It is obvious from Fig. 3 that all fullerenes are inherently strained. For instance, it was estimated that the steric strain constitutes approximately 80 % of the excess energy of C60 with respect to graphite (Haddon 1993). To quantify the steric strain, Raghavachari introduced the concept of hexagonal indices (Raghavachari 1992). The neighbor index k of a given hexagon is the number of its edges which are shared with other hexagons, and every fullerene isomer can be characterized by a set of indices hk (k D 0–6), where hk is the number of hexagons with neighbor index k. In the aforementioned study of 1812 isomers of C60 , it was shown that the parameter H D

X k

k 2 hk

(7)

also gives good linear correlation with the relative energies of the isomers. The concept of hexagonal indices was in fact invented to explain relative stabilities of the IPR isomers. As each hexagon in the IPR isomer is adjacent to at least three other hexagons, h0 , h1 , and h2 indices are equal to 0, and the combination of four indices (h3 , h4 , h5 , h6 ) is sufficient to characterize hexagon adjacencies in a given fullerene isomer (Fowler and Manolopoulos 1995; Raghavachari 1992). Raghavachari suggested that to minimize the steric strain, the indices of all hexagons should be as close to each other as possible. In other words, the lowest strain energy can be achieved when the non-planarity is uniformly distributed over the whole carbon cage. Hence, the lowest strain is expected for those structures, in which all hexagons have the same neighborhood (all indices are equal), the condition fulfilled only for highly symmetric C60 -Ih (1), C80 -Ih (7), and C80 -D5h (6) in the whole range of IPR fullerenes C2n with 2n < 120 (Fowler and Manolopoulos 1995). More complex conditions were derived for other IPR fullerenes, namely:

Structures and Stability of Fullerenes, Metallofullerenes, and Their Derivatives

11

Table 2 Optimal hexagon indices, the IPR isomers with optimal hexagon indices, and the lowest energy IPR C2n and C2n 6 isomers (2n D 76–90) C2n

C76 C78 C80 C82 C84 C86 C88 C90

(h3 ,h4 ,h5 ,h6 )

The least-strain IPR isomersa

(4,24,0,0) (2,27,0,0) (0,30,0,0) (0,29,2,0) (0,28,4,0) (0,27,6,0) (0,26,8,0) (0,25,10,0)

Td (2) D3h (5) D5h (6), Ih (7) C2v (9) D2 (21), D2 (22), D2d (23) D3 (19) D2 (35) C2 (40), C2 (41), C2 (43),C2 (44), C2 (45)

The lowest energyC2n IPR isomersb D2 (1) C2v (3) D5d (1), D2 (2) C2 (3) D2 (22), D2d (23) C2 (17) Cs (17) C2 (45)

The lowest energyC2n 6 IPR isomersc Td (2) D3h (5) D5h (6), Ih (7) C2v (9) D2 (21) D3 (19) D2 (35) C2 (43)

a

Based on the data from (Fowler and Manolopoulos 1995) See Table 3 for more details c Based on the data from (Popov and Dunsch 2007) b

8 < .80  n; 3n=2  90; 0; 0/ I ! 60  n  80 .h3 ; h4 ; h4 ; h6 / D .0; 70  n=2; n  80; 0/ I ! 80  n  140 : .0; 0; 60; n=2  70/ I ! 140  n

(8)

Table 2 lists optimal combinations of hexagonal indices as well as fullerene isomers in the range of C70 -C90 with such sets of indices. Importantly, C72 is the only fullerene that has no isomers satisfying these conditions. As a numerical parameter to quantify the steric strain in IPR fullerenes, one can use the standard deviation of the index distribution (Fowler and Manolopoulos 1995): q h D

hk 2 i  hki2

(9)

where kD6 X ˝ i ˛ˇ k ˇi D1;2 D k i hk kD3

! 

kD6 X

!1 hk

(10)

kD3

It was also shown that  h correlates well with the standard deviation of the square of POAV pyramidalization angle, ( p 2 ) ( p D  ¢  – 90), as can be seen in Fig. 4a for a set of 24 IPR isomers of C84 (Boltalina et al. 2009). Therefore,  h and ( p 2 ) can be used interchangeably to quantify the steric strain, the latter parameter being more universal (for instance, it can be equally used for fullerene derivatives, in which some carbon atoms are in sp3 hybridization state). Figure 4b also plots the relative energies (E) of C84 computed at the B3LYP/6-31G level of theory versus ( p 2 ).

12

A.A. Popov

Fig. 4 (a) Correlation between  ( p 2 ) and  h parameters for 24 IPR isomer of C84 (Based on the data from Boltalina et al. 2009); for a fitted line, R2 D 0.97; (b) Correlation between  ( p 2 ) and B3LYP/6-31G*-computed relative energies of 24 IPR isomer of C84 (Based on the data from Sun and Kertesz 2001b)

The lack of correlations between E and ( p 2 ) for the isomers with medium values of these parameters indicates that for IPR fullerenes, the other factors (e.g., electronic) are at least equally important as the steric strain. As a result, the lowest energy isomers for C76 –C90 are usually not the isomers with the lowest steric strain (see Table 2), and steric strain analysis is not widely used now in the studies of higher fullerenes of medium size. If a larger set of isomers is considered (e.g., 6,063 IPR isomers C116 ), then the steric strain indicators (such as H in Eq. 7) indeed show correlation with the relative energies, at least at the QCFF/PI level (Fig. 2c; Achiba et al. 1998a). However, the standard deviation of 75 kJ/mol is still too large, and such data can be of help only to sort out some high-energy isomer. Below we will show that more precise schemes have been developed based on the combination of topological motifs and their increments (Cioslowski et al. 2000).

The Isomers of IPR Fullerenes With the advent of the IPR, the number of fullerene isomers to be considered was drastically reduced, and detailed computations of reasonable (i.e., IPR) isomers and comparison to the experimental results became feasible. In this section we will review the results of the search of the most stable isomers of fullerenes beyond C70 (starting from C70 , at least one IPR isomer is possible for each even number of atoms). It is necessary to point out that the term stability comprises some ambivalence. It can be used in terms of thermodynamic stability, which in the studies of isomers means low relative energy, or in terms of kinetic stability, which implies reactivity of the compound in the given experimental conditions (in the studies of fullerenes, it is usually determined by the HOMO–LUMO gap).

Structures and Stability of Fullerenes, Metallofullerenes, and Their Derivatives

13

When predicting the possible isomeric structure of fullerenes, it is necessary to consider a combination of both thermodynamic and kinetic stability factors, the former determining the possibility of the formation of the given fullerene and the latter showing whether a given fullerene can be extracted from the soot and further processed by standard fullerene separation techniques. In fact, conclusions based on thermodynamic stability can be valid only when the reaction is close to the equilibrium (which is hard to prove for fullerenes). Moreover, for the equilibrium, Gibbs energies (rather than relative energies) should be taken into account, and it was shown by Slanina and coworkers that the equilibrium composition can be significantly altered at high temperatures (Slanina et al. 2004b). On the other hand, there are some examples (see below) showing that if thermodynamically stable fullerene has small HOMO–LUMO gap, it is possibly produced but remains in the insoluble part of the soot after extraction (presumably because of its polymerization). In this respect, a combination of the relative energy and the HOMO–LUMO gap still seems to be convenient parameters for the basic characterization of the fullerene isomers. Table 3 lists the data on the relative energies of some higher fullerene isomers obtained using B3LYP and PBE0 (also known as PBE1PBE) hybrid DFT functionals. B3LYP results are compiled from several works by different groups and are obtained mainly with the use of 6-31G or 6-31G basis sets (Chen et al. 2001; Slanina et al. 2000; Sun and Kertesz 2000, 2001b, 2002; Sun 2003a, b; Zhao et al. 2003), while PBE0 values are results of single-point calculations with 6-311G basis set and DFTB-optimized atomic coordinates (Shao et al. 2006, 2007). Table 3 also lists PBE0-obtained HOMO–LUMO gaps. Figure 5 shows the structures of the major isomers of C60 and C70 –C84 . The first fullerene after C70 , C72 , is a “missing” fullerene since it has never been obtained in considerable amounts. The sole IPR isomers of C72 with D6d molecular symmetry are predicted to have large HOMO–LUMO gap (1.42 eV at the PBE/TZ2P level), however, DFT calculations (Kobayashi et al. 1997; Kobayashi and Nagase 2002b; Slanina et al. 2004a) supported also by the measurements of electron affinity of C72 (Boltalina et al. 2000a) show that trace amounts of C72 present in the fullerene extract are most probably based on the non-IPR C2v (11188) isomer, which is 46–48 kJ/mol more stable than the IPR D6d (1) isomer at the DFT level of theory (Table 3). Thus, C72 is the only empty fullerene Cn (n > 70) which violates IPR. C74 represents to some extent the opposite situation to C72 : according to DFT calculations, the sole IPR isomer of C74 , D3h (1), has small HOMO–LUMO gap and probably has a triplet ground state (Kovalenko and Khamatgalimov 2003). This fullerene was also thought to be “missing” for a long time, until it was found that it is insoluble in common fullerene solvents (Karataev 1998), most probably because of polymerization. C74 as well as some other “insoluble” fullerenes can be solubilized by either electrochemical (Diener and Alford 1998) or chemical modification (Goryunkov et al. 2004b; Shustova et al. 2006, 2007). For instance, trifluoromethylation of “insoluble” fullerene mixture afforded C74 (CF3 )12 as a major product and D3h (1) carbon cage in this compound was unambiguously confirmed by single-crystal X-ray diffraction (Shustova et al. 2007).

14

A.A. Popov

Table 3 Relative energies (kJ/mol) and HOMO–LUMO gap (eV) in selected isomers of C72 –C94 fullerenes as computed at the B3LYP and PBE0//DFTB levels of theory Cn C72 C74 C76 C78

C80

C82

C84

a

Sym. D6d (1) C2v D3h (1) D2 (1) T d (2) C2v (3) D3h (5) C2v (2) D3 (1) D3h (4) D2 (2) D5d (1) C2v (3) C2v (5) D3 (4) C2 (3) Cs (4) Cs (2) C2 (1) C2 (5) Cs (6) D2d (23) D2 (22) D6h (24) Cs (16) C2 (11) D3d (19) Cs (15) C1 (12) D2d (4) Cs (14) C2v (18) D2 (5)

B3LYP 0.0 48.1 0.0 0.0 0.0 18.8 27.6 41.4 102.9 0.0 9.2 11.7 27.7 37.2 0.0 16.2 27.4 32.1 34.1 51.1 0.0 0.4 29.7 33.7 35.2 42.3 47.0 51.1 61.3 62.7 65.4 66.4

PBE0 0.0 46.1 0.0 0.0 91.2 0.0 22.7 26.3 41.3

Gap 2.64 1.63 0.74 2.10 0.71 1.71 1.55 2.15 1.71

Ref.a Cn [b,c] C86

[b,c]

C88

0.0 1.1 17.1

1.39 1.15 0.95

[b,c]

C90

0.0 17.8 27.3 33.2 38.5 57.0 0.0 2.5 30.3 34.5 36.4

1.69 1.63 1.69 1.28 1.33 1.14 2.15 2.07 2.46 1.96 1.70

[b,d]

50.6

1.60

[b,c] [b,c]

C92

[h,d]

C94

Sym. C2 (17) Cs (16) C1 (12) C1 (11) C3 (18) Cs (17) C2 (33) C2 (7) C1 (11) C2 (20) C2 (45) C2v (46) Cs (35) C1 (30) C2 (28) C2 (40) C1 (32) C2 (18) D3 (28) D2 (84) C2 (26) C1 (38) D2 (82) D2 (81) C2 (43) Cs (42) Cs (44) C2 (133) C2 (61) C1 (37) C1 (34) C1 (15) C1 (91)

B3LYP 0.0 25.9 43.1 43.5 47.3 0.0 9.6 10.9 43.5 47.7 0.0 14.2 15.1 28.5 34.3 36.8 38.5 46.9 51.4 0.0 102.6 51.1 84.9 134.7 15.5 15.1 35.1 0.0 24.7 64.0 25.9 33.9

PBE0 0.0 23.6 43.5 45.9 46.7 0.0 8.9 5.1

Gap 1.58 1.97 1.24 1.17 1.27 1.60 1.80 1.63

45.3 0.0 12.8 12.0 27.1 32.4 39.6

1.40 1.74 1.87 1.91 1.81 1.75 1.45

43.6 0.0 20.5 22.2 22.2 24.1 40.5 0.0 8.3 25.2 29.4 33.4 33.8 35.0 43.6 44.7

1.52 2.20 1.98 1.73 1.89 1.76 1.61 1.91 1.99 1.74 1.66 1.63 1.71 1.55 1.44 1.50

Ref.a [b,d]

[e,d]

[f,d]

[g,d]

[i,d]

In each pair of references, the first one is for B3LYP values, the second one is for PBE0/6311G //DFTB values; experimentally characterized fullerene isomers are listed in bold b B3LYP/6-31G (Chen et al. 2001) c Shao et al. 2007 d Shao et al. 2006 e B3LYP/6-31G(Sun 2003a) f B3LYP/6-31G(Sun 2003b) g B3LYP/6-31G //SAM1(Slanina et al. 2000) h B3LYP/6-31G (Sun and Kertesz 2001b) i B3LYP/6-31G //SAM1(Zhao et al. 2003)

Structures and Stability of Fullerenes, Metallofullerenes, and Their Derivatives

15

Fig. 5 Molecular structures of C60 , C70 , and selected higher fullerenes. The atoms in adjacent pentagon pair in C72 are highlighted

Identity of C76 as D2 (1) isomer was unambiguously confirmed by 13 C NMR spectroscopy in the very early studies of the higher fullerenes (Ettl et al. 1991). According to DFT calculations, D2 (1) is considerably more stable than the second IPR isomer, Td (2). Moreover, the latter is subject to Jahn–Teller distortion (the actual symmetry is D2 ) and has a small HOMO–LUMO gap. Recently, it was shown that C76 -Td (2) is probably also formed in the fullerene synthesis but remains insoluble; however, it can be solubilized and characterized in the form of trifluoromethyl derivative (C76 -Td (2))(CF3 )12 (Shustova et al. 2006). C78 has five IPR isomers; four of them are quite stable (their E covers the range of 41 kJ/mol) and are proved to be formed in the arc-discharge synthesis. Three isomers, D3 (1), C2v (2), and C2v (3), are obtained in approximately comparable amounts (the actual ratio depends on the pressure used in the synthesis) and can be easily extracted from the soot as they have comparably high HOMO–LUMO gaps (Kikuchi et al. 1992; Wakabayashi et al. 1994). The isomer D3h (5), although it is the second in the stability order (Table 3), can be found in the fullerene extract in much smaller amounts (Simeonov et al. 2008). A possible reason is the low HOMO–LUMO gap, which results in the poor solubility. Indeed, this isomer was found among the “insoluble” fullerenes, its structure being unambiguously confirmed by the single-crystal X-ray structure of the trifluoromethyl derivative, (C78 -D3h (5))(CF3 )12 (Shustova et al. 2007). From seven IPR isomers of C80 , only the two most stable structures (Table 3), D2 (1) and D5d (2), are obtained in the form of empty fullerenes (Hennrich et al.

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1996; Wang et al. 2000b). DFT predicts that they are almost isoenergetic (the D2 (1) isomer is somewhat more stable), but D5d (2) isomer has considerably smaller HOMO–LUMO gap. Besides, the higher symmetry of D5d (2) isomer also reduces its content if entropy factors at higher temperatures are taken into account. As a result, the yield of D2 (1) is about 30 times higher (Wang et al. 2000b). Isolation of (C80 -C2v (5))(CF3 )12 after trifluoromethylation of the “insoluble” fullerene mixture has shown that C80 -C2v (5) is also produced in the arc-discharge synthesis (Shustova et al. 2006). C2v (5) isomer is rather stable but has small HOMO–LUMO gap (E D 25 kJ/mol, gap D 0.71 eV at the B3LYP/6-31G level). Importantly, icosahedral C80 -Ih (7) isomer, which should have the least steric strain among all IPR isomers of C80 , is in fact the least stable isomer (E D 114 kJ/mol at the B3LYP/631G level of theory (Slanina et al. 2008). The reason is its electronic instability – in the ideal icosahedral symmetry, its HOMO is fourfold degenerate and is occupied by two electrons. BP/TZVP studies have shown that it has D2h symmetry in the ground state, and its singlet and triplet states are isoenergetic (Furche and Ahlrichs 2001). However, this isomer becomes the most stable C80 isomer in the hexaanionic state, when its HOMO is fully occupied, which makes it a key structure for the endohedral metallofullerenes (see section “Bonding, Structures, and Stability of Endohedral Metallofullerenes” of this chapter). C82 has seven IPR isomers, but only one of them, C2 (3), was isolated in pure form (Kikuchi et al. 1992; Zalibera et al. 2007). C2 (3) is the most stable isomer of C82 and has large HOMO–LUMO gap (Sun and Kertesz 2001a), so its high abundance is not surprising. Meanwhile, Cs (4) isomer is also rather stable and has similar gap, but this isomer has never been isolated in a pure form. However, its presence in the fullerene soot was unambiguously confirmed by the characterization of its chloro-derivative (C82 -Cs (4))Cl20 (Yang et al. 2013). The studies of the “insoluble” fullerene fraction have shown that C2 (5) isomer is also formed in the arc-discharge synthesis, as verified by the isolation and characterization of its CF3 derivative, (C82 C2 (5))(CF3 )12 (Shustova et al. 2006). Starting from C84 , the increase in the number of possible structures becomes quite significant (see Table 1), which seriously complicates the studies, both experimentally and theoretically. This fact also increases the value of theoretical studies, since they can help in sorting out unrealistic isomers. The question of high importance in this situation is how to handle the rapidly increasing number of isomers. The obvious strategy of choice is to use several (usually two) computational approaches, first the “cheap” one (for instance, semiempirical method) for the whole set of isomers and then the more expensive but more reliable approach (usually DFT) for a reduced set of structures. It is important that the “cheap” method should be also sufficiently reliable so that the possible errors in the relative energies should be not very high, and a reasonable cutoff in the relative energies might be applied for the subsequent higher-level computations. Twenty-four IPR isomers of C84 form a convenient training set, since it is still feasible to perform a full DFT study for all of them. Figure 6 compares relative energies of 24 isomers obtained at the B3LYP/631G level (Sun and Kertesz 2001b) to those obtained with three semiempirical approaches widely used in the fullerene chemistry, QCFF/PI, MNDO, and DFTB.

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Fig. 6 Correlations between relative energies of 24 IPR isomer of C84 computed at the B3LYP/631G level and those computed using QCFF/PI (a), PM3 (b), and DFTB (c), all values are in kJ/mol. R2 values for the fitted lines are 0.925, 0.912, and 0.925, respectively

One can see that in all cases the correlation is quite reasonable (R2  0.92), and the standard deviations (¢ E ) are less than  20 kJ/mol. Roughly estimating the energy cutoff for the isomers computed by means of “cheap” method as 3¢ E , one can consider then only the isomers with 4E < 3 E for the subsequent DFT study (the cutoff should be appropriately increased, of course, if one is interested in the whole set of isomers within the given energy window). Assessment study of three semiempirical approaches, MNDO, AM1, and PM3, in predicting relative energies of fullerene isomers in comparison to B3LYP/6-31G was reported for a set of fullerenes including C30 –C50 , all IPR isomers of C72 –C86 , and selected isomers of C88 –C102 (Chen and Thiel 2003). Quite good correlations were found between the results of DFT and semiempirical calculations for higher fullerenes (R2 D 0.92, 0.88, and 0.86, for MNDO, AM1, and PM3, respectively; standard deviations are 20–25 kJ/mol), while for small fullerenes with a large number of pentagon adjacencies, performance of semiempirical methods was significantly worse. In another work, AM1 and PM3 methods were compared to DFTB and B3LYP/631G for all IPR isomers of C72 –C86 fullerenes and selected isomers of C28 –C36 (Zheng et al. 2005). At variance with the results in Chen and Thiel (2003), the authors have found that AM1 and PM3 performed rather bad (R2 D 0.77 and 0.73,

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A.A. Popov

respectively, for C72 –C86 ), while DFTB-predicted values were close to B3LYP/631G (R2 D 0.88 and 0.93 for NCC and SCC-variants of DFTB, respectively). The significant difference in performance of AM1 and PM3 methods reported by two groups is quite remarkable. Close comparison of the relative energies from the two works shows that the difference in relative energies predicted for the same isomers by the same method can be as high as 50 kJ/mol and in one case (C3v (8) isomer of C82 ) reaches 75 kJ/mol. At the same time, both works report the same values for the isomer of C84 and agree in that semiempirical methods give reliable predictions for this fullerene. C84 is also a special case among the higher fullerenes since, on the one hand, it is the most abundant empty fullerene after C60 and C70 (at least, if all C84 isomers are taken together); on the other hand, it provides the broadest range of the isomers characterized for one fullerene. The lowest energy is predicted by many approaches for D2 (22) and D2d (23) isomers, separated from the other isomers by the gap of ca 30 kJ/mol (Table 3). They are essentially isoenergetic (E varies within few kJ/mol in favor of one or another isomer depending on the method) and have large HOMO– LUMO gaps. Indeed, these are the isomers produced in the largest amounts (the D2 to D2d ratio is ca 2:1 and varies depending on the conditions of synthesis) (Kikuchi et al. 1992). C84 also serves as a good example to emphasize that with the increase of the number of isomers, the density of their distribution in the energy scale is also increasing. The energies of at least 12 isomers fall in the range of E D 30 – 71 kJ/mol at the B3LYP/6-31G level (Sun and Kertesz 2001b), and many of them can be isolated as “minor” isomers, including (in the order of their yield) C2 (11), Cs (16), Cs (14), D2d (4), and D2 (5) (Dennis et al. 1999). Isolation of D6h (24) and D3d (19) isomers from the arc-discharge soot of Gd-doped composite rods was also reported (note that these isomers were not formed in noticeable amounts with pure graphite rods) (Tagmatarchis et al. 1999). Perfluoroalkylation of the higher fullerene mixture resulted in the isolation and single-crystal X-ray characterization of the derivatives of several C84 isomers, including (C84 –C2v (18))(C2 F5 )12 , the carbon cage isomer of which has not been found in the previous studies (Tamm et al. 2009b). For C86 , two isomers, with C2 and Cs symmetry, could be isolated from the arc-discharge soot and characterized by 13 C NMR spectroscopy (Miyake et al. 2000). From 19 IPR isomer of C86 , 6 have C2 symmetry and 3 have Cs symmetry. DFT calculations show that isomers C2 (17) and Cs (16) have the lowest energies and large HOMO–LUMO gaps and hence can be assigned to the experimentally observed structures (Sun and Kertesz 2002). The assignment was also confirmed by comparison of B3LYP/6-31G computed and experimentally measured 13 C NMR chemical shifts (Sun and Kertesz 2002) and by single-crystal X-ray diffraction study of the mixture of two C86 isomers (Wang et al. 2010). In the most recent experimental study of C88 , three isomers of this fullerene were isolated and assigned by C NMR spectroscopy to one Cs and two C2 -symmetric structures (Miyake et al. 2000). Among 35 IPR isomers of C88 , 11 and 7 structures have Cs and C2 symmetry, respectively. DFT calculations show that Cs (17) has the lowest energy, followed by C2 (7) and C2 (33) isomers, all within the range of 10 kJ/mol (Table 3). These three structures have large HOMO–LUMO gaps and are

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19

separated from the less-stable isomers by a gap of at least 30 kJ/mol. Therefore, they can be readily assigned to the experimentally observed structures, which are also confirmed by the comparison of the B3LYP/6-31G-computed 13 C NMR spectra to the experimental data (Sun 2003b). The presence of C2 (7), Cs (17), and C2 (33) isomers in the arc-discharge produced fullerene mixture was confirmed by the isolation and single-crystal X-ray diffraction characterization of their chloro- and trifluoromethyl derivatives (Troyanov and Tamm 2009; Yang et al. 2012a; Wang et al. 2016). Detailed experimental study of C90 has shown that at least five isomers are formed in the arc-discharge synthesis and that the major fraction is a mixture of C2 and C2v -symmetric isomers (Achiba et al. 1995). The B3LYP/STO-3G calculations of all 46 IPR isomers with subsequent B3LYP/6-31G calculations of the isomers with E < 100 kJ/mol have shown that the lowest energy structures are C2 (45) and C2v (46) (Sun 2003b). Analogous result was also obtained in the DFTB and PBE0/6-311G //DFTB study (Shao et al. 2006). Therefore, C2 (45) and C2v (46) are the most probable candidates for the main major isomers of C90 . Chlorination of the mixture of higher fullerenes (Kemnitz and Troyanov 2009) and several C90 fractions (Troyanov et al. 2011) resulted in a series of C90 Cl2n structures with six C90 isomers: C2v (46), Cs (35), C1 (30), C2 (28), C1 (32), and Cs (34) cage isomers (E D 14.2, 15.1, 28.5, 34.3, 38.5, and 92.0 kJ/mol, respectively, at the B3LYP/6-31G level). Likewise, trifluoromethylated derivatives were structurally characterized for C2 (45) (E D 0.0 kJ/mol), Cs (35), C1 (30), and C1 (32) isomers (Kareev et al. 2008b; Tamm and Troyanov 2014; Tamm and Troyanov 2015a). D5h (1) (E D 76.2 kJ/mol), C1 (30), and C1 (32) isomers were also isolated from the soot produced by vaporization of Sm2 O3 /graphite rods and characterized by single-crystal X-ray diffraction (Yang et al. 2010, 2011). Isolation and 13 C NMR characterization of five major isomers of C92 , of which two have C2 symmetry and three have D2 symmetry, have been reported (Achiba et al. 1998b); however, further spectroscopic details were not given. In another work, one C2 -symmetric isomer and inseparable mixture of at least three other isomers were obtained from dysprosium arc-burned soot (Tagmatarchis et al. 2002). C92 (CF3 )16 with either D2 (81) or D2 (82) carbon cage was identified among the other products of the trifluoromethylation of the higher fullerene mixture (Troyanov and Tamm 2009), whereas chlorination and trifluoromethylation of C92 fraction gave derivatives of the C1 (38) isomer (Tamm and Troyanov 2015b). Results of DFT studies are more controversial for C92 than for smaller fullerenes discussed above (Table 3). Two detailed studies were reported, one based on the screening of the whole set of IPR isomers at the semiempirical SAM1 level with subsequent B3LYP/6-31G //SAM1 calculations (Slanina et al. 2000), while another study included screening at the DFTB level followed by PBE0/6-311G //DFTB calculations of the most stable isomers (Shao et al. 2006). As can be seen in Table 3, the two approaches resulted in substantially different relative energies and stability order of the isomers. While B3LYP//SAM1 approach favors D2 (84) isomer as the most stable one, PBE0//DFTB predicts that D3 (28) is 20 kJ/mol lower in energy than D2 (84). Full optimization of D3 (28) and D2 (84) isomers at the B3LYP/6-311G

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and PBE0/6-311G levels of theory in Shao et al. (2006) showed that the former is indeed more stable by 18.2 and 20.8 kJ/mol, respectively, in very close agreement with the results of PBE0//DFTB calculations. It appears thus that PBE0//DFTB values are more reliable than the results of B3LYP//AM1 calculations. Experimental structural information on the empty fullerenes beyond C92 is very scarce and usually limited to the mixtures of isomers. X-ray crystallographic studies of perfluoroalkylated and chlorinated higher fullerene mixtures or partially separated fractions characterized a series of isomer of C94 , C96 , C100 , and C104 (Tamm et al. 2009a, 2015; Yang et al. 2012b, 2014; Fritz et al. 2014), but these data are inconclusive since the complete isomeric composition of the initial fullerene mixture remains unknown. Yet a lot of calculations were performed for larger fullerenes, and systematic information on the lowest energy isomers is available up to C180 . The aforementioned PBE0/6-311G //DFTB study was performed for all IPR fullerenes in the C82 –C120 range as well as for all IPR and non-IPR isomers of C38 –C80 fullerenes (Shao et al. 2006, 2007). For the most stable isomer, 13 C NMR spectra were also predicted for comparison with experiment. In another series of works, the most stable isomers of C122 to C180 were identified (Xu et al. 2006, 2008). To screen the huge number of the isomers, the authors used the empirical scheme of Cioslowski and coworkers (Cioslowski et al. 2000). According to the original scheme, formation enthalpies of fullerenes were factorized into contributions of 25 independent structural motifs, namely, six-member rings with their first and second neighbors, and the contributions of the motifs were fitted using the energies of 115 IPR fullerenes in the C60 – C180 size range computed at the B3LYP/6-31G level. In 2008, parameters were refitted using the energies of 536 fullerene molecules in the C60 –C160 range (Xu et al. 2008). The serious advantage of this empirical method is that it is very inexpensive, and the energies can be estimated with high accuracy without quantumchemical calculations. B3LYP/6-31G calculations were then used by the authors to determine the lowest energy isomers (Xu et al. 2006, 2008). The same authors also performed benchmarking computations of C116 , C118 , and C120 isomers to analyze performance of QCFF/PI, AM1, PM3, MNDO, and tight-binding (TB) methods in comparison to B3LYP/6-31G //B3LYP/3-21G (Xu et al. 2007). It was found that, in contrast to the relative energies of the medium-size higher fullerenes such as C84 , for large fullerenes semiempirical methods performed badly and did not show any reasonable correlations with DFT-predicted relative energies. However, it should be noted that DFT is not a panacea itself. DFT has difficulties with description of the extended  -systems (Reimers et al. 2003), and hence it is possible that DFT is also not reliable for large fullerenes. Unfortunately, results of computations for larger fullerenes cannot be verified by experimental information at this moment. In conclusion, numerous studies have shown that the most stable isomers of the middle-size higher fullerenes predicted by DFT are usually the major isomers in the experimentally produced fullerene mixtures. At the same time, it is also clear that experimental results cannot be fully described by considering only thermodynamic control (equilibrium), since some other unknown factors are also important in

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determining the isomeric composition. For instance, it is not clear yet why some stable isomers are not formed in the arc-discharge synthesis; besides, the influence of the metals on the distribution of the arc-discharge produced fullerene isomers is not understood.

Bonding, Structures, and Stability of Endohedral Metallofullerenes Metal–Cage Bonding in Endohedral Metallofullerenes Since La@C82 was first isolated in 1991, the nature of the metal–fullerene bonding was of interest to researchers. Because of the radical nature of La@C82 as well as other MIII @C82 (M D Y, Sc), the ESR spectroscopy appeared useful in revealing the main features of the electronic structure of EMFs. Small hyperfine splitting constant of the metal observed in the spectra indicated that the spin density in M@C82 is presumably localized on the fullerene cage. This, along with the high electron affinity of fullerenes, resulted in the concept of the endohedral metallofullerenes as non-dissociative salts (e.g., La3C @C82 3 ), with the metal cation encapsulated in the negatively charged carbon cage. Likewise, ionic model is often used to describe electronic structure of endohedral fullerenes with more complex encapsulated species, such as dimetallofullerenes (e.g., (La3C )2 @C80 6 ), nitride clusterfullerenes (e.g., (Sc3 N)6C @C80 6 ), or carbide clusterfullerenes (e.g., (Sc3C )2 C2 2 @C82 4 ). The concept of the ionic metal–fullerene bonding was further developed by Kobayashi et al. (Kobayashi and Nagase 1998; Nagase et al. 1997). The authors analyzed the spatial distribution of electrostatic potential inside fullerene and showed that it has large negative values inside negatively charged carbon cages. This results in the strong stabilization of the metal cations if they are placed inside fullerene anions thus favoring formation of EMFs. Importantly, the minimum of the electrostatic potential in C82 3 was found in the position where metal resides in M@C82 as predicted by calculations (Kobayashi and Nagase 1998; Lu et al. 2000) or shown by experimental X-ray diffraction studies (Takata et al. 2003). At the same time, it was also found that electrostatic potential inside C80 6 has no distinguishable minima, and hence there are no distinct bonding sites for metal atoms or clusters in C80 -Ih (7). This is indeed confirmed by several computational studies of La2 @C80 -Ih (7) (Kobayashi et al. 1995; Shimotani et al. 2004; Zhang et al. 2007) and Sc3 N@C80 -Ih (7) (Campanera et al. 2002; Gan and Yuan 2006; Kobayashi et al. 2001; Popov and Dunsch 2008; Yanov et al. 2006). For La2 @C80 it is shown that there are two almost isoenergetic minima for positions of La atoms, with D2h and D3d molecular symmetries, and each minimum is multiplied by the group symmetry operations of Ih group. As a result, La2 unit exhibits almost free rotation inside C80 cage, and the charge density of La atoms forms a pentagonal dodecahedron as revealed by synchrotron radiation powder diffraction study (Nishibori et al. 2001). Another important argument in favor of the ionic model is the carbon cage isomerism of EMFs, which shows a good correlation with

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the relative stability of the appropriately charged empty cages, as will be discussed in detail below. Although the ionic model explains some spectroscopic and structural properties of EMFs, there are numerous studies which indicate that this model is oversimplified. While the analysis of MO energy levels in the EMFs and the corresponding empty cages can be indeed interpreted as an electron transfer to the cage from the metal, experimental (Alvarez et al. 2002; Kessler et al. 1997) and theoretical (Campanera et al. 2002; Lu et al. 2000; Muthukumar and Larsson 2008; Nagase et al. 1997; Yang et al. 2008) studies clearly show a substantially non-zero population of nd-levels of the endohedral metal atoms. The mixing of the cluster and fullerene MOs as well as the corresponding change of the orbital energies in EMF when compared to the MOs of the empty cage is the most apparent for Sc3 N@C78 (Fig. 7; Campanera et al. 2002; Krause et al. 2006). In many other EMFs, the metal contribution to individual  -MO orbitals of the cage is very small, yet the significant metal–cage interaction can be also revealed when the electron density distribution is analyzed (Liu et al. 2006; Popov and Dunsch 2008; Wu and Hagelberg 2008; Yang et al. 2008; Popov 2009). This can be best done with the use of promolecule deformation density approach, i.e., by visualizing the changes in the electron density distribution ¡ D [¡(Mol)  ¡(Ref)], where ¡(Mol) is the electron density of the molecule under study, while ¡(Ref) is the electron density of the reference system. A natural choice of the reference system in the studies of EMFs can be either noninteracting metal atoms or clusters (e.g., Sc3 N) and corresponding C2n molecules or metal or cluster cations (e.g., Sc3 N6C ) and appropriately charged

Fig. 7 (a) Frontier Kohn–Sham MO levels in Sc3 N@C78 -D3h (5) and C78 -D3h (5), the dash arrows show correspondence between the orbitals of the empty cage and Sc3 N@C78 ; (b) LUMO (A2 ’) MO of C78 and corresponding occupied orbital in Sc3 N@C78 ; (c) LUMO C 1 (E0 , only one component is shown) in C78 and corresponding occupied orbital in Sc3 N@C78 (Based on the data from Krause et al. 2006 and Popov 2009)

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C2n anions (e.g., C2n 6 ). In particular, when two sixfold charged ions were taken as a reference, a considerable concentration of the density at Sc atoms was found in all Sc3 N@C2n (2n D 68, 78, 80) EMFs (Popov and Dunsch 2008; Popov 2009), the effect referred to by some researchers as “back donation” (Liu et al. 2006; Wu and Hagelberg 2008). The ¡ plots have also revealed that metal–cage bonds are not localized (such as, e.g., carbon–carbon bonds) but can be better described by “coordination” bonds such as those found in transition metal complexes (CortésGuzmán and Bader 2005). The cluster–cage bonding in Sc3 N@C78 and Sc3 N@C80 was also analyzed using the energy decomposition method (Campanera et al. 2002). The authors have shown that after the Sc3 N6 C cation is encapsulated inside C78,80 6 cages, a strong orbital mixing and an electronic reorganization take place. Computed atomic charges are also much smaller than the values expected for the purely ionic bonding. For instance, net Bader, Mulliken, and NBO charges of the Sc3 N cluster in Sc3 N@C80 are C3.50, C2.88, and C2.99, respectively, at the B3LYP/6-311G level of theory (Popov and Dunsch 2008); another group reported the net Sc3 N charges of C1.28 (Mulliken), C1.10 (Hirshfeld), and C0.94 (Voronoi) (Valencia et al. 2008) at the BP/TZVP level of theory with ZORA relativistic corrections. Note that the charges strongly depend both on the method of theory used to compute wavefunction and on the electron density partitioning, but all reported values are significantly smaller than C6 expected for the ionic Sc3 N6C @C60 6 . In summary, from the molecular orbital point of view, the mechanism of the metal(cluster)–cage interactions in EMFs can be described by the formal transfer of the appropriate number of electrons from the endohedral species to the carbon cage with subsequent coordination of the metal cations by the cage as a “ligand” and reoccupation of the metal nd orbitals (see also (Liu et al. 2006) for a more detailed discussion). Note that in most cases there are no special localized orbitals that could be responsible for the cage-to-cluster electron transfer. Instead, this kind of interaction occurs through the overlap of many cage  -orbitals with nd orbitals of the metal. One of the consequences of this type of bonding is a spin-charge separation found in the radical anions of the EMFs with the metal-localized LUMO (such as Sc3 N@C80 , La2 @C72,78,80 , Ti2 C2 @C78 ). In these anions, the spin density is mostly localized on the metal atoms; however, the changes of the metal charges compared to the neutral state of the EMF molecule are rather small; instead, the surplus charge is mostly delocalized over the carbon cage (Popov and Dunsch 2008). Since the effect of the metal–cage bonding in EMFs is best revealed in the analysis of the electron density as a whole, rather than by studying the individual MOs, it is natural to use the method based on the analysis of the electron density for the quantitative description of the metal–cage interactions in EMFs. The quantum theory of atoms in molecules (QTAIM) developed by Bader is probably the most refined and well-established method for the analysis of the topology of the electron density frequently used for the revealing and quantifying of the bonding situation between the atoms (Bader 1990; Matta and Boyd 2007). Kobayashi and Nagase analyzed a topology of the electron density distribution in Sc2 @C84 , Ca@C72 , and

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Sc2 @C66 using QTAIM (Kobayashi and Nagase 1999, 2002a). Their studies have shown that although the bond critical points (BCPs: critical points of the electron density, in which the density has a minimum along one axis and has a maximum in the plane perpendicular to this direction) could be found between Sc or Ca and certain carbon atoms of the cage, the values of the electron density at BCPs (¡bcp ) were very small (0.05 a.u. for Sc-C and 0.01–0.02 a.u. for Ca-C), and the Laplacian of the density (r 2 bcp ) at metal–carbon BCPs was always positive. The authors concluded that such values for these descriptors are signatures of the highly ionic character of the metal–cage bonding. However, care should be taken when the bonding in transition metal compound is analyzed with QTAIM (Cortés-Guzmán and Bader 2005). It was revealed that for transition metals r 2 bcp values are usually positive, while bcp is small because of the diffuse character of electron distribution (Macchi and Sironi 2003), and hence these values alone cannot be proper descriptors of the bonding situation. Therefore, more specific descriptors should be applied in the analysis of the bonds involving transition metals, and the analysis of the energy density appears more useful than the analysis of the electron density alone. While kinetic energy density, G, is positive everywhere, potential energy density, V, is negative everywhere, and their sum, the total energy density H, defines which of the energy components is dominating. Cremer and Kraka suggested that bonding between the atoms can be considered as covalent if the total energy density at BCP, Hbcp , is negative (Cremer and Kraka 1984). The in-depth study of the metal–cage and intra-cluster bonding by DFT and QTAIM was recently reported for the four major classes of endohedral metallofullerenes, including monometallofullerenes, dimetallofullerenes, metal nitride clusterfullerenes, metal carbide clusterfullerenes, as well as Sc3 CH@C80 and Sc4 Ox @C80 (x D 2, 3) (Popov and Dunsch 2009). The analysis of the bond critical point indicators showed that both the intra-cluster and the metal–cage interactions in EMFs were characterized by the negative total energy density, which means that bonding in EMFs exhibits a high degree of covalency. Furthermore, from the point of the electron density distribution, the interior of the EMF molecules was found to be a complex topological object, with many critical points, unpredictable number of the bond paths, and large bond ellipticities, similar to the bonding situation in the complexes of transition metals with  -carbocyclic ligands (Farrugia et al. 2009). In this respect, the analysis of the metal–cage bonding based only on the properties of the bond paths was found to be insufficient, and the metal–cage delocalization indices, •(M, C), were also analyzed (by definition, •(M, C) is number of the electron pairs shared by the atoms M and C, which is very similar to the “bond order” in Lewis definition). It was found that in the majority of EMFs, •(M, C) values do not exceed 0.25; however, when summed over all metal–cage interactions, values of •(M,cage) close to 2–3 were obtained. While QTAIM atomic charges of the metal atoms were found to be approximately two times smaller than their formal oxidation states, the total number of the electron pairs shared by the metal atoms with the EMF molecule, (M), was found to be close to the typical valence of the given element (Table 4).

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Table 4 QTAIMatomic charges (q) of the metal atoms and the clusters and the metal–cage and metal–cluster electron-pair sharing in selected EMFs a EMF C2 -Ca@C72 Cs -Ca@C82 Cs -Sc@C82 D3 -Sc3 N@C68 C3 -Sc3 N@C80 D2 -Sc2 C2 @C84 Cs -Sc3 C2 @C80

q(M) QTAIM 1.50 1.52 1.60 1.74 1.76 1.72 1.67, 1.67

q(cluster) QTAIM

3.45 3.50 2.01 3.30

q(cluster) formal 2 2 3 6 6 4 6

C3 -Sc3 CH@C80 C1 -Sc4 O2 @C80

1.72 1.41, 1.75

3.48 3.66

6 6

C1 -Sc4 O3 @C80

1.87, 1.82

3.47

6

C2 -Ti2 C2 @C78 Cs -Y@C82 Cs -Y2 @C82 Cs -Y2 @C79 N C2 -Y3 N@C78 D3 -Y3 N@C80 D2 -Y3 N@C88 C1 -Y2 C2 @C82 C2 -La@C72 C2v -La@C82 D2h -La2 @C80 C2 -La3 N@C88 C2 -La3 N@C96

1.67 1.80 1.35 1.52 1.90 1.90 1.93 1.89 1.91 1.82 1.65 1.88 1.89

2.17

6 3 4 5 6 6 6 4 3 3 6 6 6

2.69 3.04 3.90 3.87 3.95 2.40

3.29 3.92 3.99

ı(M, cluster)

0.81 0.82 0.63 0.86, 0.63 0.82 1.21, 1.32 1.43, 1.11 0.82 0.63 0.37 0.83 0.85 0.79 0.66

0.08 0.95 0.90

ı(M, cage) 1.34 1.21 2.71 1.89 1.77 2.00 1.87, 2.13 1.82 1.74, 1.41 1.07, 1.48 3.29 2.54 2.25 2.37 1.76 1.77 1.65 1.90 2.72 2.69 2.95 1.98 1.89

(M) 1.34 1.22 2.71 2.70 2.59 2.63 2.73, 2.76 2.64 2.97, 2.65 2.52, 2.60 4.11 2.54 2.88 2.74 2.61 2.62 2.44 2.56 2.72 2.69 3.04 2.93 2.80

•(M, cluster) is the number of the electron pairs shared by the metal with the cluster; •(M,cage) is the number of the electron pairs shared by the metal with the carbon cage; (M) is the total number of the electron pairs shared by M with all other atoms of the molecule; when EMF has inequivalent metal atoms, corresponding values are averaged, unless they are significantly different; in the latter cases, different values are listed in the table separated by comma. Based on the QTAIM-B3LYP/6311G calculations from (Popov and Dunsch 2009) a

To conclude the section on the metal–cage bonding in EMFs, it appears useful to distinguish two ways to determine the charge on the atom in endohedral fullerene: (1) a “formal” charge and (2) the “actual” charge. The formal charge is integer and implies that the metal (cluster)–cage bonding is purely ionic. As such, formal charges do not describe the actual electron distribution in the endohedral fullerenes, but still appear to be useful in understanding the spectroscopic and structural properties of endohedral fullerenes. For instance, compounds with the same formal cage and metal charges exhibit very similar absorption and vibrational spectra. In

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the next section, we will show that the formal charge of the cage largely determines the isomeric structure of a given endohedral fullerene. The “actual” charge should represent the electron density distribution in the endohedral fullerenes, but unfortunately there are no unique ways to partition the electron density between the atoms. Depending on the definition and the method of theory used to evaluate the charges, they can vary in a large range in the same molecule as already noted for Sc3 N@C80 . However, whatever method is used, computed charges are always considerably smaller than the formal charges, which is a clear manifestation of the covalent contribution to the metal–cage interactions. At the same time, the charges are still considerably large to indicate that ionic contribution to the cluster–cage bonding is also important.

Isomerism in Endohedral Metallofullerenes: Stability of the Charged Carbon Cages One of the specific features of the endohedral metallofullerenes is that their carbon cage isomers are usually different from those of the isolated empty fullerenes. This fact can be understood by taking into account the electron transfer from the encapsulated species as discussed in the previous section. Using the semiempirical QCFF/PI method, Fowler and Zerbetto have shown that charging dramatically changes relative stabilities of the fullerene isomers (Fowler and Zerbetto 1995). For instance, Ih (7) isomer of C80 , being the least stable structure for (C2) and (0) charge states, is the lowest energy structure for (4) and (6). In agreement with this finding, it was shown that the most stable isomer of La2 @C80 has Ih (7) cage, as opposed to the D2 -symmetric empty C80 (Kobayashi et al. 1995). Variation of the relative energies of C82 isomers in the negatively charged states was analyzed by Kobayashi and Nagase at the HF/3-21G//AM1 level to rationalize experimentally observed isomers of M@C82 (Kobayashi and Nagase 1997, 1998). The authors have found that while C2 (3) is the lowest energy isomer of the neutral C82 , the C2v (9) cage is the most stable for both C82 2 and C82 3 as well as for M@C82 (M D Ca, Sc, Y, La, etc.) (Kobayashi and Nagase 1997, 1998). This finding is confirmed by experimental 13 C NMR studies of La@C82 and Ce@C82 anions (Akasaka et al. 2000; Tsuchiya et al. 2005; Wakahara et al. 2004). More recently, DFT calculations of C82 4 isomers were reported (Valencia et al. 2008) and the C3v (8) isomer was shown to be the most stable tetraanion, in line with the finding of this cage in the most abundant isomers of Sc2 C2 @C82 (Iiduka et al. 2006; Nishibori et al. 2006a) and Y2 C2 @C82 (Inoue et al. 2004; Nishibori et al. 2006b). The data on the relative energies of all IPR C82 isomers in (0)–(6) charge states obtained at the PBE/TZ2P level (Popov 2009) are summarized in Fig. 8a. DFT calculations show that C2v (9) is the most stable isomer for all charge states starting from (2), however, there is a change in favor of C3v (8) for (4) (Valencia et al. 2008). Note also that starting from (3) charge state, the IPR isomers C2v (9), C3v (8), and Cs (6) are considerably more stable than the other IPR isomers; as a result, only these isomers are found for MIII @C82 and M2 C2 @C82 .

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27

Fig. 8 (a) Relative energies IPR isomers of C82 computed at the PBE/TZ2P level in different charge states (Based on the data from Popov 2009); (b) PBE/TZ2P computed energies of the most stable C2n 6 hexaanions on the per-atomic basis plotted as a function of the cage size (Based on the data from Popov and Dunsch 2007)

The data on representative fullerenes discussed above show that (1) the relative energies of the fullerene isomers are dramatically affected by the charge and (2) the stability of the isomers of appropriately charged empty fullerenes provides a good estimation to the relative stability of the cage isomers of endohedral fullerenes. Thus, though ionic model is not realistic for the description of the electron density

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distribution in endohedral metallofullerenes, it can be useful for understanding the carbon cage isomerism of endohedral fullerenes. In the last decade, the higher yields, increased stability, and broad variety of the formed structures attracted special attention to the family of nitride clusterfullerenes. Their electronic structure may be conceived as a result of a formal sixfold electron transfer from the M3 III N cluster to the fullerene. Predicting of the lowest energy isomers of M3 N@C2n endohedral fullerenes is a difficult task since, in addition to the carbon cage isomerism, different orientation of the M3 N cluster inside the cage can give structures with substantially different relative energies; besides, DFT calculations of the molecules with transition metals are more complex than those of the empty fullerenes. In this respect, attempts were made to predict the possible structures of M3 N@C2n molecules based on the calculation for the empty fullerenes. The simple use of this formal sixfold electron transfer in M3 N@C2n to predict the most suitable cage isomers capable of encapsulating a nitride cluster was proposed by Poblet and coworkers. It was supposed that only the fullerenes with a considerable gap between LUMO C 2 and LUMO C 3 (which become HOMO and LUMO, respectively, in the C2n 6 hexaanion and presumably in M3 N@C2n ) should be considered as suitable hosts for nitride clusters (Campanera et al. 2005). With the application of this criterion, only C60 , C78 -D3h (5), C80 -Ih (7), and C80 -D5h (6) were found to be suitable cage isomers among all IPR fullerenes in the C60 –C84 range. Indeed, besides C60 , which is too small to host a Sc3 N cluster, only these and no other IPR cage isomers were found for Sc3 N@C2n clusterfullerenes. Later this work was expanded by considering also all IPR isomers in the C86 –C100 range (Valencia et al. 2007), and the isomers with the largest orbital gaps found for C86 and C88 hexaanions (D3 (19) and D2 (35), respectively) were those proved to exist in Tb3 N@C86 and Tb3 N@C88 by single-crystal X-ray diffraction studies (Zuo et al. 2007). A different approach to the search of the suitable cage isomers for nitride clusterfullerenes is based on the stability of the hexaanions (Popov and Dunsch 2007; Popov et al. 2007c; Yang et al. 2007a, b). In the view of the correlation found between the relative energies of the empty fullerenes in appropriate charge state and the relative energies of endohedral fullerenes, it was suggested that the suitable cage isomers of M3 N@C2n should be found among the most stable isomers of C2n 6 . The growing number of endohedral fullerenes violating IPR shows that non-IPR isomers should also be included in the search. Importantly, good correlation was found between the relative energies of the C2n 6 isomers computed at the AM1 and DFT levels of theory (Yang et al. 2007b). Since optimization of the structure of the fullerene at AM1 levels normally takes only about 1–2 min with the standard office computer, this fact dramatically facilitates the search of the stable isomers. That is, screening of all possible isomers can be done at AM1 level, and only few lowest energy C2n 6 isomers (and corresponding and M3 N@C2n isomers) can then be studied at the higher – and more computationally demanding – level of theory. By screening through the large number of the C70 6 , C76 6 , and C78 6 isomers, the molecular structures of Sc3 N@C70 (Yang et al. 2007b), M3 N@C78 (M D Dy,

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29

Tm) (Popov et al. 2007c), and DySc2 N@C76 (Yang et al. 2007a) were proposed, all violating the IPR. The structural guesses were then confirmed by comparison of the experimental and DFT-computed vibrational spectra. The full search of the most favorable cage isomers of Sc3 N@C2n and Y3 N@C2n (2n D 68–98) at the PBE/TZ2P level was reported in 2007 (Popov and Dunsch 2007). For C68  C88 , both IPR and non-IPR isomers were considered, which however resulted in the huge number of possible structures (for instance, there are totally 81,738 for C88 ). Taking into account that metal atoms in all known non-IPR fullerenes are coordinated to the adjacent pentagon pairs, the authors suggested that only the isomers with such pairs should be considered (while the structures with three or more adjacent pentagons could be omitted). This assumption considerably reduced the number of the isomers (e.g., 16,717 isomers were studied for C88 ), making the computations more feasible. Table 5 lists the lowest energy isomers of C2n 6 found in that work with their relative energies and HOMO–LUMO gaps computed at the PBE/TZ2P level; experimentally available structurally characterized EMFs are listed in the last column. Importantly, the isomers shown to exist by single-crystal X-ray studies can be found among the lowest energy isomers of C2n 6 . C78 -C2 (22,010) and C82 -Cs (39,663) isomers were predicted as possible structures of M3 N@C78 and M3 N@C82 (Popov and Dunsch 2007; Popov et al. 2007c) before their structures were confirmed by single-crystal X-ray studies of Gd3 N@C78 and Gd3 N@C82 (Beavers et al. 2009; Mercado et al. 2008). Even better correlation with experimental data is found if the relative energies of Sc3 N@C2n and Y3 N@C2n isomers are considered (Popov and Dunsch 2007). It is remarkable that C2n 6 isomers with high thermodynamic stability (i.e., low relative energy) quite often exhibit high kinetic stability (i.e., large HOMO–LUMO gap). As a result, the IPR isomers proposed by the group of Poblet (Campanera et al. 2005; Valencia et al. 2007) in most cases coincide with predictions based on the relative stability of the hexaanions (Popov and Dunsch 2007). The broad range of the fullerene sizes studied (Popov and Dunsch 2007) enabled the authors to follow the general trends in their stabilities. To compare the energies of the fullerenes of different size, the absolute energies were normalized to the number of atoms in the given fullerene. Figure 8b plots the normalized energies of the most stable C2n 6 isomers versus the number of atoms. The smooth decrease of the energy with the increase of the fullerene size is the result of a combination of two factors: (1) the decrease of the curvature of the cage, which results in smaller curvature-induced strain, and (2) the decrease of the on-site Coulomb repulsions of six surplus electrons in C2n 6 with the increase of the cage size. There is, however, a significant deviation from the smooth curve found for C80 -Ih (7) and C80 -D5h (6) isomers. These isomers are 185 and 98 kJ/mol more stable than they might be if they were like all other fullerenes (i.e., like those which obey the smooth decay in the normalized energy). The enhanced stability of the two C80 6 isomers explains the increased yield of M3 N@C80 compared to all other cage sizes. Besides, it explains why the C80 -Ih (7) isomer is always the most preferable structure in other

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Table 5 The lowest energy C2n 6 (2n D 68–88) isomers with their relative energies (kJ/mol) and HOMO–LUMO gaps (eV) computed at the PBE/TZ2P level of theory and the experimentally available EMFs with corresponding carbon cages (Based on the data from Popov and Dunsch 2007) Isomer D3 (6140) C2v (7854) D2 (10611) C2 (13295) Cs (17490) D3h (5)

Non-IPR Non-IPR Non-IPR Non-IPR Non-IPR IPR

4E 0.0 0.0 0.0 0.0 0.0 0.0

Gap 1.20 1.24 1.12 1.22 1.12 1.21

C2 (22010)

Non-IPR

59.1

1.60

Ih (7)

IPR

0.0

1.83

D5h (6)

IPR

88.2

1.51

Cs (hept)a C2v (9)b C2v (39705)b Cs (39663)b

Heptagon IPR Non-IPR Non-IPR

171.0 0.0 30.1 61.2

1.27 0.79 1.32 1.42

C84

D2 (21) Cs (51365)

IPR Non-IPR

0.0 1.8

0.80 1.34

C86

D3 (19)

IPR

0.0

1.51

C88

D2 (35)

IPR

0.0

0.97

C2n C68 C70 C72 C74 C76 C78

C80

C82

Exp. EMFs Sc3 N@C68 Sc3 N@C70 M2 @C72 (M D La, Ce) Not known DySc2 N@C76 , La2 @C76 M2 @C78 (M D La, Ce), Ti2 C2 @C78 , Sc3 N@C78 M3 N@C78 (M D Y, Dy, Tm, Gd), Sc3 NC@C78 M2 @C80 (M D La, Ce), M3 N@C80 (M D Sc, Y, Gd–Lu), Sc3,4 C2 @C80 , Sc4 O2,3 @C80 Ce2 @C80 , M3 N@C80 (M D Sc, Y, RE) LaSc2 N@C80 Sc3 N@C82 Not known M3 N@C82 (M D Y, Gd, Tm, Dy, Tb) Not known M3 N@C84 (M D Y, Gd, Tm, Dy, Tb) M3 N@C86 (M D Y, Gd, Tm, Dy, Tb) M3 N@C88 (M D Y, Ce, Nd, Gd, Tm, Dy, Tb)

a

For LaSc2 N@C80 , the relative energies for isomers Ih (7), D5h (6), and heptagon-containing Cs (hept) are 0, 61, and 27 kJ/mol, respectively, at the PBE/TZ2P level (Zhang et al. 2015) b For Y3 N@C82 , the relative energies for isomers C2v (9), C2v (39,705),and Cs (39,663) are 29.6, 0.0, and 32.6 kJ/mol, respectively, at the PBE/TZ2P level

clusterfullerenes with formal sixfold electron transfer to the cage, such as La2 @C80 , Sc4 O2,3 @C80 (Stevenson et al. 2008), and Sc4 C2 @C80 (Wang et al. 2009). The exclusive stability of the Ih (7) and D5h (6) isomers of C80 6 can be rationalized using the concept of hexagon indices (Raghavachari 1992) discussed in section “Structures and Stability of Empty Fullerenes.” The lowest strain condition (all indices are equal) is fulfilled only for C60 -Ih (1), C80 -Ih (7), and C80 -D5h (6) in the whole range fullerenes C2n with 2n < 120 (Fowler and Manolopoulos 1995). Thus, the exceptional stability of two C80 6 isomers can be explained by the favorable distribution of the pentagons, which leads to the least steric strain. Moreover, the

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list of the least-strained IPR isomers (Table 2) perfectly corresponds to the lowest energy IPR isomers of C76 6 –C90 6 (Popov and Dunsch 2007). For larger cages these conditions are less instructive, because many of the IPR isomers satisfy them. Interestingly, the relative energies of the IPR C2n 6 isomers follow the rationalization of the stability based on the steric strain much better than the relative energies of the uncharged IPR fullerenes, which usually violate the requirement of the minimized strain (the only exclusions are C84 and C90 ; see Table 2). Very recently, this phenomenon was clarified by Poblet and coworkers (Rodriguez-Fortea et al. 2010). The authors have suggested that the relative stabilities of the isomers of multiply-charged fullerene anions are largely determined by the on-site Coulomb repulsion. Since the negative charge in anions is mostly localized on the pentagons because of their non-planarity, the lowest energy structures should be those in which the maximal separation (and hence the most uniform distribution) of the pentagons is achieved. The authors have constructed the inverse pentagon separation index to show that its correlation with the relative stabilities of the isomers is improving with the increase of the charge. Thus, minimization of the on-site Coulomb repulsion and minimization of the steric strain are achieved by fulfillment of the same condition, namely, the most uniform distribution of the pentagons. If for the non-charged fullerenes, the steric strain factor can be outweighed by other factors such as stability of the  -system (and hence the lowest strain condition is usually violated), with the increase of the negative charge on the cage, the Coulomb repulsion sooner or later outweighs all other factors. Therefore, to predict the lowest energy isomers of the EMFs with high formal charge of the cage (such as M3 N@C2n ), one can apply the same arguments as were developed in early 1990s to find the isomers of empty fullerenes with the lowest steric strain. Another group has also shown that the most suitable host cages have the highest aromaticity, and so aromaticity criterion can be used to explain preference of certain isomers (Garcia-Borràs et al. 2013).

Isomerism in Endohedral Metallofullerenes: Cage Form Factor The carbon cage stability is very important in determining the molecular structure of endohedral fullerenes; however, the isolated structures are not always based on the lowest energy isomers of the hollow anions. In addition to the high stability, the fullerene cage should also provide a suitable shape for enclosed metal atoms or clusters; that is, there should be enough inner space in the cage for the endohedral species. If then a fullerene has adjacent pentagon pairs, they should be located in such a way that their coordination by the endohedral metal atoms is possible. This factor can be especially important for the nitride clusterfullerenes since the shape of the cluster itself imposes significant limitations on the possible location of the binding sites. Figure 9a plots the relative energies of the 20 lowest energy isomers of C72 6 and corresponding Sc3 N@C72 isomers calculated at the PBE/TZ2P level. The significant difference between the two sets of data is observed. While the most stable isomer of C72 6 is D2 (10,611) (Slanina et al. 2006; Popov and Dunsch 2007), this cage

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Fig. 9 (a) Relative energies of C72 6 and corresponding Sc3 N@C72 isomer computed at the PBE/TZ2P level (Based on the data from Popov and Dunsch 2007); (b) relative energies of C78 6 and corresponding Sc3 N@C78 and Y3 N@C78 isomer computed at the PBE/TZ2P level (Based on the data from Popov et al. 2007c)

isomer is destabilized for Sc3 N@C72 , and the lowest energy is found for Sc3 N@C72 with Cs (10,528) cage. The explanation of this phenomenon becomes clear when the structures of the cage isomers are analyzed. C72 -D2 (10,611) has elongated shape with two adjacent pentagon pairs on the opposite poles of the cage (Fig. 10). It is impossible to coordinate both pentagon pairs at once with the Sc atoms of the triangular Sc3 N cluster, and hence one pentagon pair in Sc3 N@C72 -D2 (10,611) is not stabilized by the metal, which explains the high relative energy found for

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Fig. 10 Molecular structures of selected non-IPR endohedral metallofullerenes. Carbon cages are shown in gray except for the adjacent pentagon pairs, which are shown in black. Note that in Sc2 @C66 -C2v (4059) has two fragments with three fused pentagons in each. In M3 N@C78 , and M3 N@C84 , M can be Y, Gd, Dy, Tm, Tb, and possibly some other lanthanides

this isomer by theory. The lowest energy C72 6 isomer with favorable location of adjacent pentagon pairs is Cs (10,528), and it corresponds to the most stable isomer of Sc3 N@C72 . However, this isomer is based on the relatively unstable carbon cage and, besides, it has a small HOMO–LUMO gap (0.39 eV at the PBE/TZ2P level). As a result, the Sc3 N@C72 nitride clusterfullerene has never been observed experimentally. Similar situation was also found for C74 6 and Sc3 N@C74 , which also has never been observed. Since the bimetallic cluster is more flexible, the steric factor is less important for dimetallofullerenes, and La2 @C72 is based on the lowest energy D2 (10,611) cage isomer (Kato et al. 2003; Lu et al. 2008). A different situation was found for the clusterfullerenes based on C78 (Popov et al. 2007c). Figure 9b compares relative energies of the lowest energy isomers of

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C78 6 and M3 N@C78 (M D Sc, Y). The most stable C78 6 isomer is IPR D3h (5), which is followed by the non-IPR C2 (22,010) isomer with two adjacent pentagon pairs. The relative energies of Sc3 N@C78 isomers closely follow those of C78 6 , which agrees with the formation of the Sc3 N@C78 -D3h (5) isomer (Olmstead et al. 2001). C78 -D3h (5) was also proved for M2 @C78 (M D La, Ce) and Ti2 C2 @C78 (Cao et al. 2004, 2008; Tan and Lu 2005; Sato et al. 2006; Yumura et al. 2005; Yamada et al. 2008). However, with the increase of the cluster size from Sc3 N to Y3 N, the relative energies of the M3 N@C78 isomers are changed dramatically. The isomer based on the D3h (5) cage is significantly destabilized, and the lowest energy is now found for the Y3 N@C78 -C2 (22,010) (84 kJ/mol more stable than Y3 N@C78 D3h (5)). The dramatic change in the stability of the isomers can be explained by insufficient size of the C78 -D3h (5) cage for encapsulation of the clusters larger than Sc3 N. As a result, Y3 N cluster is forced to be pyramidal in Y3 N@C78 -D3h (5) (nitrogen atoms is displaced from the plane of the metal atoms by 0.554 Å), which is an indication of the strong strain exerted by the cage. On the contrary, the inner space in C78 -C2 (22,010) isomer is sufficient to retain the planar shape of the Y3 N cluster (Fig. 10). Results of these calculations were supported by experimental spectroscopic studies of Tm3 N@C78 and the major isomer of Dy3 N@C78 (Popov et al. 2007c) and by single-crystal X-ray diffraction studies of Gd3 N@C78 (Beavers et al. 2009). M3 N@C80 -Ih (7) represents another important example of the cluster size influence on the stability of the clusterfullerenes (Popov and Dunsch 2007). The lowest energy isomers for C80 6 and Sc3 N@C80 are Ih (7) and D5h (6), and there is a gap of approximately 80 kJ/mol between these two isomers and the gap of approximately 120 kJ/mol between D5h (6) and all other, less-stable isomers. Stability order of C80 6 and Sc3 N@C80 isomers is rather close, but relative energies of unstable Sc3 N@C80 isomers are 20–40 kJ/mol smaller than those in C80 6 . These data agree with formation of Sc3 N@C80 -Ih (7) as the most abundant product of the Sc3 N@C2n clusterfullerene synthesis, followed by Sc3 N@C80 -D5h (6) as the second most abundant structure. With the increase of the cluster size to the Y3 N, the Ih (7) isomer is still the most stable one, and the energy difference between Ih (7) and D5h (6) isomers is almost the same as in the Sc3 N@C80 series, but all other isomers of Y3 N@C80 are significantly stabilized with respect to the Ih (7) isomer. For instance, relative energies of the D5h (6) and non-IPR C1 (28,325) isomers are quite close (70.2 and 90.0 kJ/mol, respectively). Similar to the case of M3 N@C78 , this can be explained by the relatively small size of almost spherical C80 -Ih (7) and C80 -D5h (6) cages, which leads to the increase of the strain when large M3 N clusters are encapsulated. As a result, M3 N@C80 -Ih (7) is still the most abundant nitride clusterfullerene for the clusters up to the size of Gd3 N, but overall yields are suppressed compared to the clusterfullerenes with smaller clusters (Krause and Dunsch 2005). For the larger cluster size (M D Nd, Pr, Ce, La). Echegoyen and coworkers have shown that the larger cages (in particular, C88 ) are produced in higher yield than that of M3 N@C80 (Chaur et al. 2008).

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Violation of the Isolated Pentagon Rule in Endohedral Metallofullerenes The specific feature of endohedral fullerenes is that the isolated pentagon rule, which imposes a strict limitation on the possible isomers of empty fullerenes, is often violated when a metal or a cluster is encapsulated inside the cage. Such possibility was first suggested by Kobayashi et al. in theoretical studies of Ca@C72 isomers (Kobayashi et al. 1997; Nagase et al. 1999), and the first experimental evidence was provided in 2000 when Sc2 @C66 -C2v (4,059) (Wang et al. 2000a; Yamada et al. 2014) and Sc3 N@C68 -D3 (6,140) (Stevenson et al. 2000) were isolated. Since that time, a lot of non-IPR endohedral fullerenes have been reported (Fig. 10), including Sc3 N@C70 -C2v (7,854) (Yang et al. 2007b), Sc2 S@C70 -C2 (7,892) (Chen et al. 2013), Sc2 S@C72 -Cs (10,528) (Chen et al. 2012), Sc2 C2 @C72 -Cs (10,518) (Feng et al. 2013), La@C72 -C2 (10,612) (Wakahara et al. 2006), La2 @C72 -D2 (10,611) (Kato et al. 2003; Lu et al. 2008), DySc2 N@C76 -Cs (17,490) (Yang et al. 2007a), La2 @C76 -Cs (17,490) (Suzuki et al. 2013), Sm@C76 -C2v (19,138) (Hao et al. 2015), M3 N@C78 -C2 (22,010) (M D Dy, Tm, Gd) (Beavers et al. 2009; Popov et al. 2007c), Gd3 N@C82 -Cs (39,663) (Mercado et al. 2008), and Y2 C2 @C84 -C1 (51,383) (Zhang et al. 2013), and M3 N@C84 -Cs (51,365) (M D Tb, Gd, Tm) (Beavers et al. 2006; Zuo et al. 2008). The state of the art in the studies of non-IPR fullerenes has been reviewed (Tan et al. 2009). Recently, LaSc2 N@C80 -Cs (hept) with one heptagon and 13 pentagons (two pentalene units) has been synthesized (Zhang et al. 2015). The tendency to violate the IPR in endohedral fullerenes can be explained by the changes of the relative energies of the fullerene isomers with the increase of the cage charge. It was shown already in 1995 that for C60 , the non-IPR C2v (1,809) isomer with two pairs of adjacent pentagons is substantially less stable than the IPR isomer in the neutrally charge state, but with the increase of the negative charge on the cage, the difference in the relative energies is diminishing, reaching almost zero for (4) and inversing for (6) states (Fowler and Zerbetto 1995). Semiempirical and DFT calculations for the whole set of 6,332 isomers of C68 in 0, (2), (4), and (6) charge states were reported (Chen et al. 2008). This study has shown that the lowest energy isomers of C68 4 and C68 6 are C2v (6,073) and D3 (6,140), in agreement with isolation of these isomers for Sc2 C2 @C68 (Shi et al. 2006) and Sc3 N@C68 (Olmstead et al. 2003; Stevenson et al. 2000), respectively. By plotting the relative energies of the isomers of the same charge versus the number of pentagon adjacencies, the authors have found that the penalty for a pentagon adjacency is decreasing from 92 kJ/mol in C68 to 70 kJ/mol in C68 6 at the AM1 level, confirming that the increase of the charge stabilizes pentagon adjacencies. An explanation of the stabilization of the pentagon adjacencies in metallofullerenes was proposed by Slanina et al. (2006). The authors noted that the pentalene (a unit of two fused pentagons) is an 8  anti-aromatic system in the neutrally charged state, but becomes 10  aromatic in the dianionic state (Zywietz et al. 1998). In a similar fashion, when the metal in the non-IPR endohedral fullerenes is coordinated to the pentagon pair, it donates two electrons to the

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pentalene unit to make it aromatic. This reasoning agrees with the fact that the maximum number of pentagon pairs found in endohedral fullerenes is equal to the half of the formal charge of the cage: three pairs for (6) charge as in Sc3 N@C68 -D3 (6,140) or Sc3 N@C70 -C2v (7,854), two pairs for (4) state as in Sc2 S@C72 -Cs (10,528) (Chen et al. 2012) or Sc2 C2 @C72 -Cs (10,518) (Feng et al. 2013), and one pair for (3) and (2) states as in La@C72 -C2 (10,612) or Sm@C76 C2v (19,138). A correlation of the total number of adjacent pentagon pairs in the lowest energy isomers of C2n 6 (2n D 68–88) with the cage size was revealed in Popov and Dunsch (2007). The authors found a pronounced tendency to decrease the number of pentagon pairs in the molecules with the increase of the cage size: while three pairs are common for C68 6 and C70 6 isomers, two and one pairs are preferable for C72 6 –C78 6 and C80 6 –C86 6 , respectively. At the same time, relative energies of the non-IPR isomers are increasing with respect to the IPR isomers, and starting from C86 formation of the non-IPR fullerenes is not expected (at least, for the cages with the formal charge of 6 and smaller). Indeed, the largest cage reported to date for any non-IPR fullerene is C84 –Cs (51,365) (Beavers et al. 2006; Zuo et al. 2008). This tendency agrees with the increase of energy penalty for the adjacent pentagon pair in the empty fullerenes with increase of the cage size as discussed in the section “The Isolated Pentagon Rule and Steric Strain.” With the increase of the cage size, a more uniform distribution of the pentagon-induced strain over the fullerene is possible, and hence, localization of such a strain in pentagon adjacencies should become more unfavorable than for the smaller cages. Besides, for the charged cages, the influence of the charging of the fullerene should be diminished with the growth of the cage size, and hence its stabilizing role for the pentagon adjacencies is leveled down for larger fullerenes.

Structures and Stability of Fullerene Derivatives Addition of X2 to C60 : Isomers of C60 X2 and General Considerations The C60 molecule has two types of C–C bonds, hexagon/hexagon (hex/hex) edges, and pentagon/hexagon (pent/hex) edges. Determination of their bond lengths by different experimental and theoretical methods gives the values of ca 1.40 and 1.45 Å, respectively. As might be expected, these values fall in the range of the bond lengths typical for conjugated systems; yet, they also show that there are formal double and single bonds in C60 , and it is more appropriate to call it polyalkene rather than superaromatic molecule. The chemical properties of fullerenes are to a large extent determined by the presence of the conjugated double bonds, and vast majority of fullerene reactions are addition reactions. We will not cover here the cycloaddition, mainly because quantum-chemical approaches are not so helpful in predicting and elucidation molecular structures of the reaction products. Thus, we will focus on the addition of atoms or groups that form single bonds to the fullerene core (i.e., atoms or radicals, “X” hereafter). Such reactions, on the one hand, often

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occur at rather high temperatures allowing close-to-equilibrium distribution of the products, and on the other hand, addition of such groups allows multiple possibilities from the point of the distribution of the groups on the surface of the fullerene. For instance, there are 23 possible isomers of C60 X2 ; 4,190 isomers of C60 X4 ; and 418,470 isomers of C60 X6 , and the number of possible isomers is growing rapidly with the further increase of the number of the added groups (Balasubramanian 1991). It is therefore desirable to find the principles which govern addition of the addends to the fullerenes, the task to be naturally addressed by theoreticians. It should be also noted that equilibrium or close-to-equilibrium distribution of the products in the addition reaction (i.e., thermodynamic control) is a necessary prerequisite for the conclusions based on the relative energies of isomers to be valid. Otherwise, kinetic factors can be equally or even more important than the relative stability of the isomers, and theoretical predictions of the products are severely complicated. All possible isomers which can be obtained by the addition of two hydrogen atoms to C60 (addition of the odd number of groups leads to the highly reactive radicals and therefore is not considered here) were studied at the PM3 level of theory by Dixon and coworkers (Matsuzawa et al. 1992). An AM1 study of the relative stabilities of C60 X2 isomers for X D H, F, Cl, Br, CH 3 , and t-C4 H9 was reported in 2003 (Clare and Kepert 2003a). Let us consider several possible isomers of C60 X2 (Fig. 11a). The simplest addition pathway is when two X groups are added to one double bond of C60 ; the rest of the  -system of the fullerene remains unchanged (in other words, this is 1,2-addition to a cyclohexatriene). If then two groups are attached to one hexagon in para position (1,4-addition to a cyclohexatriene), the  -system of the fullerene has to be adjusted by relocating one double bond to the pent/hex edge. All other variants of addition require more pronounced changes in the  -system of C60 and lead to the larger number of double bonds in pentagons

Fig. 11 (a–c) Fragments of the lowest energy isomers of C60 H2 : (a) 1,2-addition to hexatriene fragment; (b) 1,4-addition to hexatriene fragment; (c) 2,6-addition to naphtalene fragment; (d) relative energies of C60 H2 isomers computed at the AM1 level as a function of the number of double bonds in pentagons (DBIPs) (Based on the data from Clare and Kepert 2003a)

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(DBIPs). It was found that the number of DBIPs is an important parameter showing perfect correlation with the stability of the isomers of C60 X2 (Fig. 11b): the larger the number of DBIPs, the lower the stability of the given isomer (Matsuzawa et al. 1992). From the slope of the fitted line, the penalty for each DBIP in C60 H2 can be estimated as 36 and 37 kJ/mol at the PM3 and AM1 levels, respectively. The penalty is virtually independent on the nature of X, being 39 kJ/mol for C60 F2 and 37 kJ/mol for C60 (t-C4 H9 )2 at the AM1 level (Clare and Kepert 2003a). Obviously, “1,2”-addition should be the most preferable from the energetic point of view, and 1,2-C60 X2 is indeed the most stable isomer for X D H and F (Dixon et al. 1992; Clare and Kepert 2003a) in good agreement with the experimental structures of C60 H2 (Henderson and Cahill 1993) and C60 F2 (Boltalina et al. 2000b). However, in the 1,2-C60 X2 , the X groups are experiencing eclipsing interactions. With the increase of the size of the groups, the influence of the steric factor is increasing and can balance the destabilizing effect of DBIP. Indeed, either semiempirical or DFT calculations show that for the bulky groups (such as Br or CF3 ), 1,4-addition is more preferable (Dixon et al. 1992; Clare and Kepert 2003a; Goryunkov et al. 2003). Furthermore, for even larger groups such as t-C4 H9 , the isomer with two DBIPs and t-C4 H9 groups on different hexagons (Fig. 11d) is the most stable at the B3LYP/631G //AM1 level (at the same time, AM1 method predicts that 1,4-C60 (C4 H9 )2 is the lowest energy isomer) (Clare and Kepert 2003a). The studies of C60 X2 isomers allow one to conclude that the structures of the products of the multiple addition of X groups should be a result of the interplay of at least two factors: (1) destabilizing double bonds in pentagons and (2) destabilizing eclipsing interactions of bulky groups. It can be expected that for the relatively small atoms such as H or F, multiple 1,2-addition is preferable, while for the bulky groups, multiple 1,4-addition is to be expected. Detailed studies of the multiple additions of various groups (in particular, H and Br) to fullerenes performed by Clare and Kepert in 1990s using AM1 method confirmed these suggestions (Clare and Kepert 1993, 1994a, b, c, 1995a, b, 1999a, b, 2002a; Kepert and Clare 1996). In brief, the methodology of the authors included a search of the most stable isomers of C60 Xn , sorting out majority of the unstable structures, and then a search of the most stable isomers of C60 XnC2 based on the several most stable isomers of C60 Xn . Gradual increase of n allowed Clare and Kepert to cover broad range of compositions, reveal some general principles of the multiple addition to C60 and other fullerenes, and predict several addition patterns which were indeed found later in the experimental studies. In the remaining parts of this chapter, we will discuss several examples of the addition patterns to C60 and C70 .

Addition of H and F to C60 : Contiguous Addition, Benzene Rings, and Failures of AM1 Early on semiempirical calculations of C60 H2 and C60 F2 have shown that 1,2isomers are 18–19 kJ/mol lower in energy than 1,4-isomers, and then all other structures are substantially less stable (Dixon et al. 1992; Matsuzawa et al. 1992;

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Fig. 12 Addition patterns typical for small groups (H, F). (a) S and T addition motifs; (b) lowest energy Cs -isomer of C60 F8 ; (c) Cs -C60 F8 with S-motif; (d) Th -C60 X12 with isolated CX–CX edges; (e) crown C3v -C60 X18 with isolate benzene ring; (f) D5d -C60 X20 with S-type loop around equator; (g) T-C60 X36 ; (h) C3 -C60 X36 ; (i) C1 -C60 X36

Clare and Kepert 1993). Based on this fact, in their investigation of the multiple addition of hydrogen atoms to C60 , Clare and Kepert considered only 1,2-addition (addition to hex/hex edges). In the AM1 study (Clare and Kepert 1993), they have found that the most energetically favorable would be consequent addition of H2 molecules to hex/hex edges in a pathway, which eventually leads to Th -symmetric C60 H12 with all CH–CH edges separated from each other (Fig. 12d). While this addition pattern has never been experimentally verified for C60 H12 or C60 F12 , it was indeed proved for many hexakis-cycloadducts such as C60 (C(COOEt)2 )6 (Lamparth et al. 1995; Hirsch and Vostrowsky 2001). Further addition to hex/hex edges in Th -C60 H12 inevitably requires formation of longer CH–(CH)x –CH strings. It was, however, found that formation of C6 H6 is energetically unfavorable, and –(CH)x – strings should be built up from the edgesharing C6 H4 hexagons. Two motifs of the string growth, S and T (Fig. 12a), have been indicated (Clare and Kepert 1994b).

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Among many addition patterns studied by Clare and Kepert for C60 Hn (n  20), two are of particular interest. For C60 H18 , C3v -symmetric “crown” structure built from combination of S and T motifs (Fig. 12e) was shown to be one of the most stable isomers (Clare and Kepert 1994b). In this molecule, (CH)x – string forms a loop, which isolates one hexagon from the rest of the  -system. The isolated hexagon is highly aromatic, and the carbon cage is flattened in its neighborhood. Discovered first in 1994 in the theoretical study, in 1996 this addition pattern was proved by 1 H NMR spectroscopy for the experimentally available C60 H18 (Darwish et al. 1996), and later the same pattern was found in C60 F18 (Neretin et al. 2000). Consecutive extension of the S-motif leads to the closure of the –(CH)20 – string in a loop around the fullerene equator with formation of the D5d -symmetric C60 H20 (Fig. 12f) (Clare and Kepert 1994a). This addition pattern was found experimentally in C60 F20 (Boltalina et al. 2001), whose structure was recently confirmed by a singlecrystal X-ray diffraction study (Shustova et al. 2010). The crown motif with isolated benzene-like ring proposed for C60 H18 was then found common for highly hydrogenated C60 (Clare and Kepert 1994a, b, 1999a). In particular, C60 H36 was studied in detail because a compound with this composition was among the first C60 derivatives ever synthesized by different methods (Haufler et al. 1990; Attalla et al. 1993; Ruchardt et al. 1993), but its structure remained unclear for more than a decade. For C60 H36 , combination of several crown motifs can lead up to four isolated benzene ring as in the T-symmetric isomer (Fig. 12g), but in the AM1 studies, some isomers with two or three benzene rings were found to be equally stable (Clare and Kepert 1999a). Almost identical results were obtained at the AM1 level for isomers of C60 F36 (Clare and Kepert 1999a), whose structure also could not be unambiguously elucidated in the first synthetic studies. It was also found in the studies of C60 X36 and C60 X48 that double bonds in pentagons are not destabilizing anymore at high degrees of functionalization (Clare and Kepert 1997, 1999a, 2002a). For instance, the most stable isomers of C60 F48 have all their six double bonds in pentagons (Clare and Kepert 1997). In early 2000s, when routine DFT calculations have become feasible for the fullerene derivatives, Clare and Kepert reconsidered some of their earlier results by comparing AM1 to B3LYP/6-31G //AM1. One of the main conclusions of these studies was the understanding that one should treat results of AM1 calculations with caution. First, it was found that AM1 underestimated the energy difference between 1,2and 1,4-isomers of C60 X2 by ca 10 kJ/mol (at least, when compared to the results of DFT calculations) and the error accumulated at higher degrees of addition. As a result, while AM1 calculations predicted the C60 H6 , C60 H12 , and C60 H18 structures with, respectively, one, two, and three skew pentagonal pyramidal (SPP) motifs (Fig. 13d) to be the most stable isomers for their compositions (Clare and Kepert 1996), DFT calculations showed that these addition motifs are energetically unfavorable. For instance, C60 H18 with three SPP motifs is 73.8 kJ/mol more stable than the “crown” C3v -C60 H18 isomer at the AM1 level, but the latter is 256.8 kJ/mol more stable at the B3LYP/6-31G //AM1 level (Clare and Kepert 2003b). This agrees with the fact that hydro- or fluorofullerenes with SPP motifs have never been observed experimentally.

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Fig. 13 C60 Xn addition patterns typical for bulky groups (for C60 Brn and C60 (CF3 )n , PBE/TZ2Pcomputed relative energies (kJ/mol) are listed). (a–c) isomers of C60 X4 ; (d–e) isomers of C60 X6 ; (d–e) isomers of C60 X8 ; (h–j) isomers of C60 X12 ; (k–l) isomers of C60 X18 ; (m) Th -C60 X24 . Symmetry group of the isomer is indicated only when it is different from C1 . Ribbons of edgeshared C6 X2 -hexagons are shaded (except for C60 X18 , here shaded are one SPP and one isolated fulvene moiety, respectively)

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Second, DFT studies have shown that for the low degree of additions, the most stable are those isomers, which have –(CH)x – strings, rather than isolated CH– CH fragments (Clare et al. 2003a). At the AM1 level, the most stable isomer of C60 H4 is 13.4 kJ/mol level lower in energy than the Cs -symmetric isomer with C6 H4 fragment, while at the B3LYP/6-31G level, the latter is lower in energy by 15.3 kJ/mol. Likewise, the isomer of C60 H6 with the S-motif is 31.5 kJ/mol lower in energy at the B3LYP/6-31G //AM1 level than the isomer with three isolated CH–CH fragments, while at the AM1 level, the latter is more stable by 24 kJ/mol. For the isomers of C60 H8 , analogous energy differences increase to 59.7 and 31.1 kJ/mol, respectively. The preference of the contiguous addition at the early stages of fluorination was proved experimentally by 19 F NMR spectroscopy (Boltalina et al. 2002). It is instructive to show that although contiguous addition appears to be the rule supported by both experimental and DFT studies, the other possibilities should not be completely abandoned. In the experimental studies of low C60 fluorides, Cs symmetric isomer of C60 F8 was isolated (Boltalina et al. 2002). Cs symmetry is not compatible with the S-type string, and the authors proposed a contiguous addition pattern with T-motif and two additional fluorine atoms attached at both ends of the string. However, later calculations have shown that such a structure would by unstable. Sandall and Fowler performed a broad search of all possible structures of Cs symmetry compatible with NMR data (Sandall and Fowler 2003). After calculations of more than 500 isomers using MNDO, AM1, and PM3 methods and – for selected structures – Harree-Fock method with STO-3G and 6-31G basis sets, the authors have found that the most suitable isomer indeed has T-motif, but two additional fluorine atoms are added in a noncontiguous way with formation of 1,4C6 F2 hexagon (Fig. 12b). In a recent study, it was shown that two isomers of C60 F8 are actually formed in the synthesis, aforementioned isomer with Cs symmetry and another one, also with Cs symmetry (Goryunkov et al. 2006b). Exhaustive DFT PBE/TZ2P calculations performed by the authors confirmed assignment of the Cs isomer to the structure proposed by Sandall and Fowler and showed that another Cs -isomer is based on the S-motif. At the PBE/TZ2P level, S-Cs -isomer of C60 F8 is 5.5 kJ/mol lower in energy. The third failure of semiempirical methods revealed in the comparative AM1/DFT studies is that AM1 strongly underestimates stabilizing factor of the isolated benzene rings. Comparison of the relative energies of the isomers of C60 H36 with different number of benzene rings computed at the AM1 and B3LYP/631G //AM1 levels of theory shows that AM1-predicted relative energies should be adjusted by ca 60 kJ/mol per benzene ring to match the DFT-predicted values (Clare and Kepert 2002b). For instance, at the AM1 level, the relative energies of S6 (2 rings), C3 (3 rings), and T (4 rings) isomers of C60 H36 are 0.0, 19.4, and 52.0 kJ/mol, while at the B3LYP/6-31G //AM1 level, the values change to 0.0, 40.7, and 77.5 kJ/mol, respectively (Clare and Kepert 2002a). In a similar study of C60 H36 isomers at the PM3 and PBE/TZ2P levels of theory, it was found that PM3-computed relative energies should be corrected by approximately 67 kJ/mol per benzene ring to match the DFT results (Popov et al. 2004). For the isomers

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of C60 F36 , AM1-predicted relative energies should be corrected by approximately 30 kJ/mol per ring to match B3LYP/6-31G //AM1 energies (Clare and Kepert 2002b). At the same time, relative energies of the isomers with the same number of benzene ring are reliably predicted by semiempirical approaches (Clare and Kepert 2002b; Popov et al. 2006). Thus, semiempirical methods were unable to predict the most stable isomers of C60 H36 and C60 F36 . At the same time, DFT calculations correctly predict that T, C3 , and C1 isomers shown in Fig. 12g–i are the most stable isomers, in perfect agreement with the experimental results, which show that both C60 H36 and C60 F36 are obtained as mixtures of these three isomers (Boltalina et al. 1998; Gakh and Tuinman 2001; Gakh et al. 2003; Avent et al. 2002; Hitchcock and Taylor 2002; Popov et al. 2006). It should be noted that formation of isolated benzene rings is a very important stabilizing factor in the multiple addition to fullerenes, and it has been observed for many addends and many fullerenes. The failure of semiempirical methods in adequate prediction of the relative energies of isomers with such fragments is therefore a very serious problem and makes these methods hardly applicable when the broad search of the possible isomers is necessary. Good results demonstrated by DFTB in the studies of the fullerenes of different size show that this method might be useful for fullerene derivatives as well, but to our knowledge, benchmark calculations for fullerene derivatives with DFTB are not available yet.

Addition of Bulky Groups to C60 : Bromination and Perfluoroalkylation In contrast to H and F, bromine atom is sufficiently large to make eclipsing interactions strongly repulsive, so that 1,4-addition of two Br atoms is more energetically preferable than 1,2-addition either at the AM1 (E D 5.1 kJ/mol) or DFT levels of theory (2.4 kJ/mol at B3LYP/6-31G //AM1, 2.1 kJ/mol at LDA/DNP//AM1, and 14.0 kJ/mol at the PBE/TZ2P levels) (Dixon et al. 1992; Clare and Kepert 1995a, 2001). At the same time, the difference in the energy of 1,2- and 1,4-isomers is not very large, and hence the possibility of 1,2-addition should be also considered. Indeed, in the AM1 study of C60 Br4 isomers, it was found that two lowest energy isomers are results of the consecutive 1,4-additions, while the third most stable isomer was a result of a combined 1,2- and 1,4-additions (Fig. 13a–c; Clare and Kepert 1995a). Regression analysis of the relative energies of C60 Br4 isomers as a function of the number of different structural motifs (such as C6 Br, C6 Br2 , C6 Br3 , etc. hexagons) revealed that the most stable isomer of bromofullerenes might be obtained through formation of ribbons (also referred to as “strings”) of edge-sharing C6 Br2 hexagons, rather than by addition of pairs of bromine atoms to the distant parts of C60 . Thus, the most stable isomer has para-para-para string (p3 hereafter), while the second most stable isomer has para-meta-para string (pmp hereafter) (Fig. 13a, b). The relative energy of pmp isomer at the PBE/TZ2P (AM1) level of theory with respect to the p3 isomer is 0.1 (6.7) kJ/mol, while the next in stability order with one ortho- and one para-C6 Br2 hexagons is 9.9 (11.0) kJ/mol less stable

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(the DFT calculations reported hereafter for bromofullerenes at the PBE/TZ2P level with effective-core potential basis set for the Br atoms were performed specifically for this chapter). Further addition of Br atoms to the three most stable isomer of C60 Br4 confirmed the preference of the string formation (Clare and Kepert 1995a). However, the most stable isomer at the AM1 level has a special Cs -symmetric addition pattern titled by some authors as “skew pentagonal pyramid” (SPP, Fig. 13d). This addition pattern includes one pair of Br atoms bonded to neighboring carbon atoms and is 21.8 kJ/mol more stable than the second most stable isomer with a p3 mp ribbon (Fig. 13e). At the PBE/TZ2P level, the energy difference between SPP and p3 mp isomers is reduced to 9.6 kJ/mol. Experimentally isolated C60 Br6 indeed has the SPP pattern (Birkett et al. 1992), and this addition motif was also found for some other addends, including Cl (Birkett et al. 1993) and CF3 (Kareev et al. 2006). It should be noted that formation of SPP-C60 X6 derivatives for relatively large X is also favorable kinetically if the radical addition mechanism is adopted (Rogers and Fowler 1999). Next two added Br atoms continue the tendency of the ribbon formation. At the AM1 level, the most stable isomer of C60 Br8 has p3 mpmp ribbon (Fig. 13f), while DFT (PBE/TZ2P) shows that the isomer in which the ribbon is closed to a C2v -symmetric loop ((pm)4, Fig. 13g) is 9.3 kJ/mol more stable. The latter motif is indeed observed in the experimentally isolated C60 Br8 (Birkett et al. 1992). AM1 calculations have also shown that the isomers with the SPP motif and an additional para-C6 Br2 hexagon on the opposite part of C60 are also equally stable (Clare and Kepert 1995a). While there are no experimentally isolated C60 Brx compounds between C60 Br8 and C60 Br24 , theoretical studies of the intermediate compositions were still very useful for predicting the general trends in the addition of bulky groups (Clare and Kepert 1995a, b). In particular, a remarkable prediction was done for C60 Br12 . The authors have shown that the most stable isomers in SPP and ribbon series are the C2h -isomer with two SPP motifs on the opposite sides of C60 and the S6 symmetric isomer in which the ribbon is closed to a loop around the equator of the carbon cage (Fig. 13h, i). At the AM1 level, C2h isomer is 6.8 kJ/mol more stable, while PBE/TZ2P favors the S6 -isomer by 12.2 kJ/mol. Though not available experimentally for C60 Br12 , both addition patterns have been found for C60 (CF3 )12 derivatives (Popov et al. 2007b; Troyanov et al. 2006). In further addition of Br atoms, the competition between SPP motifs and ribbons is continued up to C60 Br18 , for which the isomer with three SPP moieties was predicted to be the lowest energy structure (Clare and Kepert 1995b). For even larger number of Br atoms, SPP motif does not provide stable structures anymore, and the most stable isomer of C60 Br24 is Th -symmetric structure in which all Br atoms are bonded to the non-neighboring carbon atoms (24 is the largest number of addends for which such a restriction is possible) (Clare and Kepert 1995a; Fowler et al. 1999). This isomer corresponds to the experimentally available structure of C60 Br24 (Tebbe et al. 1992). Significant progress achieved in the last years in the perfluoroalkylation of fullerenes (Boltalina et al. 2015) initiated exhaustive theoretical studies of C2n (CF3 )x

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isomers with the aim to find the most stable structure(s) for each composition. Since the studies were performed almost a decade after the AM1-based analysis of C60 Brx addition patterns by Clare and Kepert, they have strongly benefited from the serious progress in hardware. Instead of the maximum of about 100 structures reported in the early exploratory studies, up to thousands and tens of thousand isomers were routinely studied for CF3 derivatives, and results of the AM1 studies were then verified by DFT calculations. Besides, the isolation of C60 (CF3 )n derivatives with each even n from 2 to 18 (in many cases each composition is also presented by several isomers) enabled more detailed comparison between experimental and theoretical results. Both AM1 and DFT (PBE/TZ2P) unambiguously show that 1,4-isomer of C60 (CF3 )2 is much more stable than the 1,2-isomer, the DFT relative energy of the latter being 34.7 kJ/mol (Goryunkov et al. 2004a). The energy difference is more pronounced than for the C60 Br2 , which clearly shows that CF3 is bulkier than bromine. Therefore, in the compounds with larger number of CF3 groups, 1,2-addition is to be avoided. Indeed, the most stable isomers of C60 (CF3 )4 are p3 (E D 0.0 kJ/mol at the PBE/TZ2P level) and pmp (E D 8.2 kJ/mol) structures, just like for C60 Br4 ; however the o,p-C60 (CF3 )4 isomer, the third stable for C60 Br4 , is 36.3 kJ/mol less stable than p3 -C60 (CF3 )4 (for C60 Br4 its relative energy is 9.9 kJ/mol) (Goryunkov et al. 2004a). The pmp is the major isolable isomer of C60 (CF3 )4 , while the more stable p3 isomer is very reactive (Whitaker et al. 2013) and is usually isolated in the form of its epoxide C60 (CF3 )4 O (Goryunkov et al. 2003; Kareev et al. 2006). The trend revealed for C60 (CF3 )4 in comparison to C60 Br4 is further emphasized for six added groups. For CF3 , the ribbon addition motifs are noticeably more preferable than the SPP motif. The most stable isomer of C60 (CF3 )6 has p3 mp addition pattern, while the SPP isomer is 13.8 kJ/mol less stable (for C60 Br6 , SPP is 9.6 kJ/mol lower in energy) (Dorozhkin et al. 2007). The SPP-C60 (CF3 )6 still can be isolated, but its yield is ca 20 times lower than the yield of the major, p3 mp isomer (Kareev et al. 2006). Another difference between CF 3 and Br which becomes apparent for further additions is that CF3 groups are sufficiently bulky to experience repulsive interaction when two groups are in one pentagon, even though they are not bonded to the neighboring carbon atoms. Thus, the stable ribbon addition pattern of C60 Brx with C5 Br2 pentagons are destabilized for CF3 groups (although some stable isomer still can have them). For instance, the most stable C2v -symmetric addition pattern of C60 Br8 (9.3 kJ/mol more stable than p3 mpmp) is 46.2 kJ/mol less stable than the p3 mpmp isomer of C60 (CF3 )8 at the PBE/TZ2P level (Goryunkov et al. 2007). The p3 mpmp isomer itself is the second most stable (E D 2.6 kJ/mol, PBE/TZ2P), while the lowest energy was predicted for one of the isomers with p3 mp,p addition pattern (i.e., addition pattern of the lowest energy p3 mp-C60 (CF3 )6 isomer with an additional isolated para-C6 (CF3 )2 hexagon). Theoretical results agree well with experimental isolation of one p3 mpmp, three p3 mp,p, and one pmpmpmp isomers of C60 (CF3 )8 (Goryunkov et al. 2007; Popov et al. 2007b).

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Fortuitously, performance of the AM1 method for addition of bulky groups is much better than for the consecutive 1,2-additions of H and F, which encourages the use of AM1 for the prescreening of thousand isomers of C60 (CF3 )n. Although there is rather poor correlation between AM1 and PBE/TZ2P values (Fig. 14a), the trend in the energies of the most stable isomers is correctly predicted by AM1. For instance, when all isomers of C60 (CF3 ) 10 obtained by a combination 1,4-additions were studied at the AM1 level and then a cutoff of 40 kJ/mol was applied to the AM1 results for the consequent PBE/TZ2P calculations, the most stable isomers within the range of ca 15 kJ/mol could be identified (Popov et al. 2007b). Figure 14 shows addition patterns and relative energies of the eight most stable isomers of C60 (CF3 )10 . Except for one isomer, which has two symmetryrelated p3 m2 loops, all other structures have ribbon addition patterns similar to those of C60 (CF3 )8 . From the eight theoretically predicted structures in Figure 14, six isomers of C60 (CF3 )10 are characterized experimentally by single-crystal X-ray diffraction and/or 19 F NMR spectroscopy. The splitting of the 19 F NMR lines due to the through-space interaction of the fluorine atoms of CF3 groups in the same hexagon allows experimental identification of the lengths of the ribbon, and when combined with results of DFT calculations of the relative energies, these data allow unambiguous identification of the addition pattern. Molecular structures of several isomers of C60 (CF3 )10 were elucidated this way by a combination of NMR and DFT, and some of them were later confirmed by X-ray studies (Popov et al. 2007b). Two key addition motifs predicted by Clare and Kepert for C60 Br12 , double-SPP pattern and S6 -symmetic loop (Fig. 13h, i), have been identified experimentally for C60 (CF3 )12 (Popov et al. 2007b; Troyanov et al. 2006). Extensive AM1 and DFT calculations showed that the S6 -loop isomer is at least 19 kJ/mol more stable than all other isomer of C60 (CF3 )12 , while the C2h -(2  SPP) isomer is 32.6 kJ/mol less stable (Kareev et al. 2007). The isomer of C60 (CF3 )12 with ribbon motif and two C5 (CF3 )2 pentagons (Fig. 13j) has been found to be 39.8 kJ/mol less stable than the S6 -isomer, showing that kinetic factors can be also important in the isomeric distribution of C60 (CF3 )x derivatives even at 500 ı C (Kareev et al. 2007). When addition of more than 12 CF3 groups is considered, one has to consider that fullerenes have only 12 pentagons, and hence it is impossible to avoid formation of destabilizing C5 (CF3 )2 pentagons. Therefore, when CF3 groups are too crowded on the fullerene surface, results of 1,2-addition can be not as destabilizing as at the lower addition stages. Exhaustive AM1 and DFT calculations have shown that for C60 (CF3 )14 , the lowest energy isomer with one SPP fragment is only 0.4 kJ/mol less stable than the most stable ribbon isomer with two C5 (CF3 )2 pentagons; however, experimentally characterized isomers of C60 (CF3 )14 do not have SPP fragments (Omelyanyuk et al. 2007). Likewise, for C60 (CF3 )16 , SPP and ribbon isomers are equally stable; moreover, it was found that stable addition patterns with CF3 groups on adjacent carbon atoms can be realized without formation of the SPP moiety, and one of the experimentally characterized C60 (CF3 )16 isomers has o-C6 (CF3 )2 hexagon (Troyanov et al. 2007). For C60 (CF3 )18 , theoretical studies have shown that the most stable isomer has C3v -symmetry, two isolated benzene rings, three isolated fulvene fragments, nine C5 (CF3 )2 pentagons, and no CF3 groups on adjacent carbon

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Fig. 14 (a) Correlation between relative energies of C60 (CF3 )10 isomers computed at the AM1 and PBE/TZP levels of theory; (b–i) Schlegel diagrams of the eight lowest energy isomers of C60 (CF3 )10 (PBE/TZ2P-computed relative energies are also listed, based on the data from Popov et al. 2007a). Experimentally characterized compounds are framed in black (“X-ray” means single-crystal X-ray diffraction; “NMR” means 19 F NMR spectroscopy). Ribbons of edge-shared C6 (CF3 )2 -hexagons are shaded

atoms (Fig. 13l; Troyanov et al. 2007). DFT calculations show that this isomer is 89.5 kJ/mol more stable than the isomer with triple SPP motif, which also has one benzene ring (Fig. 13k). Moreover, even for C60 Br18 this isomer is 26.1 kJ/mol more stable than the structure with three SPP moieties. Recently, the isomer of C60 (CF3 )18 with this addition pattern has indeed been isolated and characterized by single-crystal X-ray diffraction (Samokhvalova et al. 2009). Up to now, 18 is the largest number of CF3 groups in structurally characterized C60 (CF3 )x derivatives;

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the Th -symmetric C60 X24 motif known for X D Br and Cl could not be reached yet for CF3 derivatives.

Addition to C70 and Higher Fullerenes Ih -C60 has only two types of C–C bonds, which can be straightforwardly classified according to their bond lengths as “single” and “double” bonds. For C70 and higher fullerenes, the symmetry in most cases is much lower, and the lengths of C–C bonds are more uniformly distributed. It is therefore almost impossible to make an unambiguous classification of the bond types, and the relevant chemical properties (such as addition pathways) are much harder to predict. The structure of C can be described as a combination of two C60 -like hemispheres on the poles separated by the belt of ten carbon atoms and five benzene-like rings around the equator (Fig. 15a). The bonds between the belt carbon atoms are significantly elongated compared to C60 values; besides, these atoms are located on triple hexagon junctions. Clare and Kepert reported AM1 study of the addition patterns of C70 Xn derivatives (n D 2–12, X D H, F, Br, C6 H5 ) (Clare and Kepert 1999b). For C70 H2 , the authors have found that 1,2-addition is energetically preferable in the pole region, while the lowest energy 1,4-addition occurs at the equator. The AM1-computed energy difference between the most stable 1,2- and 1,4-isomers is only 4.4 kJ/mol, which can be compared to 18 kJ/mol for C60 H2 . Such a considerable difference between the relative energies of C70 H2 and C60 H2 isomer can be explained by the fact that the 1,4-addition at the equator of C70 does not generate destabilizing double bond in any pentagon. For C70 H4 , the isomer with p3 -string at the equator (Fig. 15) is in fact 9.1 kJ/mol lower in energy than the isomer obtained by two 1,2-additions. The studies of C70 Hn with n D 6–10 further emphasized the preference of consecutive 1,4-additions with formation of the pn1 -ribbon along the equator of the carbon cage. The most stable C70 H6 and C70 H8 isomers in these series are, respectively, 28.0 and 54.3 kJ/mol lower in energy than the isomers obtained by 1,2-addition. Note that p5 and p7 -strings at the equator of C70 can be realized as either Cs or C2 -symmetric isomers. For C70 H6 , C2 -p5 isomer is 76.6 kJ/mol more stable than the Cs -p5 pattern (56.6 kJ/mol at the PBE/TZ2P level (Dorozhkin et al. 2006a), while for C70 H8 , the Cs -p7 isomer is 34.7 kJ/mol (21.6 kJ/mol at the PBE/TZ2P level) more stable than C2 -p7 . The most stable isomer at the next addition step is obtained by the closure of the p7 ribbon via addition of two atoms to pent/hex junction with formation of the Cs -p9 o-loop (Fig. 15), which cuts the  -system of the fullerene into two noncommunicating parts. This structure is 26.9 kJ/mol more stable than the C2 -p9 isomer, obtained by the growth of the equatorial ribbon via one more 1,4-addition step. Computations for the bulkier Br and C6 H5 groups have also shown that formation of the ribbon around the cage equator is the most referable addition motif in C70 (Clare and Kepert 1999b). Even for C6 H5 , the sterically strained Cs -p9 o-loop addition pattern is more stable for C70 X10 than C2 -p9 isomer. Results of these calculations agree very well with experimental data. Except for C70 (CF3 )10 discussed below, the majority

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Fig. 15 (a) Schlegel diagram of C; (b–i) addition patterns of selected C70 Xn (n D 2–10) derivatives and the relative energies of their isomer (kJ/mol, PBE/TZ2P level of theory) for X D H, Br, and CF3 ; ribbons of edge-shared C6 X2 -hexagons are shaded. Based on the data from (Dorozhkin et al. 2006a) and additional computations performed for this chapter

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of experimentally available C70 X10 compounds (X D H, Cl, Br, Ph, CH3 ) have Cs p9 o-loop (Birkett et al. 1995; Avent et al. 1997; Spielmann et al. 2000; Al-Matar et al. 2002; Troyanov et al. 2003), and only bulky OOt Bu groups form the decakis adduct with C2 -p9 addition pattern (Xiao et al. 2005). Some C70 Xn compounds with less than ten added groups have also been reported, and most of them also have equatorial string addition patterns. Thus, both computational and experimental data confirm that the most energetically stable addition motif for C70 is formation of the string from para-C6 X2 hexagons around the cage equator, and even for hydrogen this motif might be more preferable than the consecutive 1,2-additions. Trifluoromethylation of C70 deserves separate attention since, like for C60 , this reaction was studied recently in great details, and many compounds from C70 (CF3 )2 to C70 (CF3 )20 (each composition being presented by several isomers) have been isolated and structurally characterized (Goryunkov et al. 2005, 2006a; Kareev et al. 2005; Dorozhkin et al. 2006a, b; Ignat’eva et al. 2006; Popov et al. 2007a; Boltalina et al. 2015). The experimental studies were usually combined with and sometimes preceded by detailed theoretical analysis at the AM1 and PBE/TZ2P level. It was found that the two lowest energy isomers of C70 (CF3 )2 have a p-C6 (CF3 )2 hexagon on the pole of the molecule, and these structures are assigned to the experimentally available isomer (Dorozhkin et al. 2006a; Popov et al. 2007a). The most stable isomer with equatorial addition is 9.1 kJ/mol higher in energy at the PBE/TZ2P level of theory; meanwhile, the same method shows that the isomer of C70 H2 with equatorial addition is 7.9 kJ/mol more stable than the isomer with pC6 H2 hexagon on the pole. The reason for such a difference is most probably a much higher curvature of C70 cage at the poles as compared to the almost flat equator. Therefore, the distance between two CF3 groups is longer for p-C6 (CF3 )2 hexagon on the pole than for the hexagon on the equator; hence the repulsion between the groups is weaker on the pole. For C70 H2 , repulsion between the atoms is a less important factor, and the equatorial addition is more preferable. Addition on the pole is more preferable for C70 (CF3 )4 as well. Similar to C60 (CF3 )4 , the most stable isomers have Cs -p3 and C1 - pmp ribbons with the energy difference of only 1.0 kJ/mol (PBE/TZ2P) (Dorozhkin et al. 2006a). Both can be isolated experimentally; however the p3 isomer is kinetically unstable, and the isolable form is its epoxide C70 (CF3 )4 O (Popov et al. 2007a). DFT calculations have also shown that there are at least 15 other isomers of C70 (CF3 )4 within the range of 20 kJ/mol, including the isomer with Cs -p3 string at the equator (E D 6.9 kJ/mol, PBE/TZ2P), but no other structure has been characterized experimentally. Note that the model calculations of C70 H4 and C70 Br4 isomers at the PBE/TZ2P level have also shown that the equatorial addition is more energetically preferable for H and even Br (relative energy of the pole Cs -p3 isomer is 19.7 and 34.0 kJ/mol for Br and H, respectively, versus the equatorial Cs -p3 isomer). Starting from C70 (CF3 )6 , CF3 addition switches to the equatorial motif. Like for many other groups (H, Cl, Br, Ph), the lowest energy isomer has a C2 -p5 ribbon at the equator, and this is indeed the most abundant experimentally characterized isomer (Dorozhkin et al. 2006a). For C70 (CF3 )8 , the two most stable isomers have Cs -p7 and C2 -p7 (E D 6.0 kJ/mol, PBE/TZ2P) ribbons, and both isomers could

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be isolated and characterized by single-crystal X-ray diffraction (Goryunkov et al. 2005; Mutig et al. 2008). For C70 (CF3 )10 , the large size of CF3 groups again makes the lowest energy addition pattern different from that for other addends. Repulsion of CF3 groups on adjacent carbon atoms destabilizes the Cs -p9 o-loop isomer of C70 (CF3 )10 , and the most stable isomer is obtained if the Cs -p7 equatorial ribbon is continued by the mp fragment which is crawling to the pole again, as in the early stages of addition, and forming as a result C1 -p7 mp ribbon (Kareev et al. 2005). The C2 -p9 string is 12.3 kJ/mol less stable followed by Cs -p9 o-loop (12.4 kJ/mol). For comparison, the C1 -p7 mp isomer of C70 Br10 is 14.7 kJ/mol less stable than the Cs -p9 o-loop. In accordance with its DFT-predicted high thermodynamic stability, the C1 -p7 mp is the most abundant C70 (CF3 )10 isomer, and it can be obtained with unprecedented high yield (Popov et al. 2007a). C2 -p9 as well as three other less-stable minor isomers could be also structurally characterized, but their yield is much lower; the Cs-p9 oloop has never been found (Popov et al. 2007a). The stable addition pattern of the major C70 (CF3 )10 isomer is found in all except one lowest energy and experimentally isolated addition patterns of C70 (CF3 )12–18 . For instance, four isolated isomers of C70 (CF3 )12 have p7 mp,p patterns, while C70 (CF3 )16 and C70 (CF3 )18 have SPP fragment on the pole with p7 mp and p7 mpmp ribbons, respectively (Avdoshenko et al. 2006; Goryunkov et al. 2006a; Ignat’eva et al. 2006). One of the important findings revealed in the studies of the multiple addition to C70 is the fact that the isomers in which addends are bonded to the carbon atoms on the triple hexagon junctions (THJs) are very unstable. This rule is fulfilled for all C70 derivatives with low and medium addition rate, and from all known derivatives of C70 , the only violations of this rule are C70 F38 (Hitchcock et al. 2005) and C70 Cl28 (Troyanov et al. 2005). Moreover, it was found that this is also a strict rule for higher fullerenes (Troyanov and Kemnitz 2005; Shustova et al. 2006, 2007; Kareev et al. 2008b) when 12 or less groups are added. The energy penalty for addition of one CF3 group to THJ position was estimated as 60 kJ/mol (Boltalina et al. 2015). Thus, when considering the possible isomers of C2n Xm (m  12), one can exclude addition to triple hexagon junctions. This fact can dramatically reduce the number of possible isomers. For instance, in the experimental study of the trifluoromethylation of the “insoluble” small HOMO–LUMO gap fullerenes, the conditions were optimized to produce C2n (CF3 )12 as major products (Shustova et al. 2006). 19 F NMR spectroscopy then enabled identification of the symmetry of the derivatives and the relative location of the groups (i.e., number and length of the ribbons). Note that the cage isomers of the studied fullerenes were unknown. As a result, million isomers were possible for these derivatives. However, when the rule about the triple hexagon junctions was taken into account and combined with the condition that CF3 groups cannot share the same pentagon (12 is the largest number of the groups added to any fullerene which can be distributed with one group per pentagon), the number of possible isomers was reduced to only a few structures for all possible cage isomers, and the DFT studies of all compatible structures could easily identify the experimental structures and determine their

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carbon cage isomers (Shustova et al. 2006). Note that for higher degree of addition (more than 12 groups), the avoidance of THJ position by addends is not a strict rule anymore, and CF3 -bearing THJ carbons have been found in several derivatives of higher fullerenes (C76 (CF3 )14–18 , C94 (CF3 )20 ). Interestingly, in contrast to the empty fullerenes, multiple addition of CF3 groups to THJ carbon atoms was found for Sc3 N@C80 (Shustova et al. 2009, 2011) already at early stages of addition proving that the endohedral cluster and electron transfer to the carbon cage dramatically change chemical properties of fullerenes.

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Tsuchiya, T., Wakahara, T., Maeda, Y., Akasaka, T., Waelchli, M., Kato, T., Okubo, H., Mizorogi, N., Kobayashi, K., & Nagase, S. (2005). 2D NMR characterization of the La@C82 anion. Angewandte Chemie-International Edition, 44(21), 3282–3285. Valencia, R., Rodriguez-Fortea, A., & Poblet, J. M. (2007). Large fullerenes stabilized by encapsulation of metallic clusters. Chemical Communications, 40, 4161–4163. Valencia, R., Rodríguez-Fortea, A., & Poblet, J. M. (2008). Understanding the stabilization of metal carbide endohedral fullerenes M2 C2 @C82 and related systems. Journal of Physical Chemistry A, 112(20), 4550–4555. Wakabayashi, T., Kikuchi, K., Suzuki, S., Shiromaru, H., & Achiba, Y. (1994). Pressure-controlled selective isomer formation of fullerene-C78 . Journal of Physical Chemistry, 98(12), 3090–3091. Wakahara, T., Kobayashi, J., Yamada, M., Maeda, Y., Tsuchiya, T., Okamura, M., Akasaka, T., Waelchli, M., Kobayashi, K., Nagase, S., Kato, T., Kako, M., Yamamoto, K., & Kadish, K. M. (2004). Characterization of Ce@C82 and its anion. Journal of the American Chemical Society, 126(15), 4883–4887. Wakahara, T., Nikawa, H., Kikuchi, T., Nakahodo, T., Rahman, G. M. A., Tsuchiya, T., Maeda, Y., Akasaka, T., Yoza, K., Horn, E., Yamamoto, K., Mizorogi, N., Slanina, Z., & Nagase, S. (2006). La@C72 having a non-IPR carbon cage. Journal of the American Chemical Society, 128(44), 14228–14229. Wang, C. R., Kai, T., Tomiyama, T., Yoshida, T., Kobayashi, Y., Nishibori, E., Takata, M., Sakata, M., & Shinohara, H. (2000a). C66 fullerene encaging a scandium dimer. Nature, 408, 426–427. Wang, C. R., Sugai, T., Kai, T., Tomiyama, T., & Shinohara, H. (2000b). Production and isolation of an ellipsoidal C80 fullerene. Chemical Communications, 7, 557–558. Wang, C. R., Kai, T., Tomiyama, T., Yoshida, T., Kobayashi, Y., Nishibori, E., Takata, M., Sakata, M., & Shinohara, H. (2001). A scandium carbide endohedral metallofullerene: (Sc2 C2 )@C84 . Angewandte Chemie-International Edition, 40(2), 397–399. Wang, T.-S., Chen, N., Xiang, J.-F., Li, B., Wu, J.-Y., Xu, W., Jiang, L., Tan, K., Shu, C.-Y., Lu, X., & Wang, C.-R. (2009). Russian-doll-type metal carbide endofullerene: Synthesis, isolation, and characterization of Sc4 C2 @C80 . Journal of the American Chemical Society, 131(46), 16646– 16647. Wang, Z., Yang, H., Jiang, A., Liu, Z., Olmstead, M. M., & Balch, A. L. (2010). Structural similarities in Cs (16)-C86 and C2 (17)-C86 . Chemical Communications, 46(29), 5262–5264. Wang, S., Yang, S., Kemnitz, E., & Troyanov, S. I. (2016). Unusual chlorination patterns of three IPR isomers of C88 fullerene in C88 (7)Cl12/24 , C88 (17)Cl22 , and C88 (33)Cl12/14 . Chemistry – An Asian Journal. doi:10.1002/asia.201501152. Whitaker, J. B., Kuvychko, I. V., Shustova, N. B., Chen, Y.-S., Strauss, S. H., & Boltalina, O. V. (2013). An elusive fulvene 1,7,11,24-C60 (CF3 )4 and its unusual reactivity. Chemical Communications, 50, 1205–1208. Wu, J., & Hagelberg, F. (2008). Computational study on C80 enclosing mixed trimetallic nitride clusters of the form Gdx M3 -x N (M D Sc, Sm, Lu). Journal of Physical Chemistry C, 112(15), 5770–5777. Xiao, Z., Wang, F. D., Huang, S. H., Gan, L. B., Zhou, J., Yuan, G., Lu, M. J., & Pan, J. Q. (2005). Regiochemistry of [70]fullerene: Preparation of C70 ((OOBu)-t Bu)n , (n D 2, 4, 6, 8, 10) through both equatorial and cyclopentadienyl addition modes. Journal of Organic Chemistry, 70(6), 2060–2066. Xu, L., Cai, W. S., & Shao, X. G. (2006). Prediction of low-energy isomers of large fullerenes from C132 to C160 . Journal of Physical Chemistry A, 110(29), 9247–9253. Xu, L., Cai, W. S., & Shao, X. G. (2007). Performance of the semiempirical AM1, PM3, MNDO, and tight-binding methods in comparison with DFT method for the large fullerenes C116 –C120 . Journal of Molecular Structure-Theochem, 817(1–3), 35–41. Xu, L., Cai, W., & Shao, X. (2008). Systematic search for energetically favored isomers of large fullerenes C122 -C130 and C162 -C180 . Computational Materials Science, 41, 522–528. Yamada, M., Wakahara, T., Tsuchiya, T., Maeda, Y., Kako, M., Akasaka, T., Yoza, K., Horn, E., Mizorogi, N., & Nagase, S. (2008). Location of the metal atoms in Ce2 @C78 and its bis-silylated derivative. Chemical Communications, 5, 558–560.

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Yamada, M., Kurihara, H., Suzuki, M., Guo, J. D., Waelchli, M., Olmstead, M. M., Balch, A. L., Nagase, S., Maeda, Y., Hasegawa, T., Lu, X., & Akasaka, T. (2014). Sc2 @C66 revisited: An endohedral fullerene with scandium ions nestled within two unsaturated linear triquinanes. Journal of the American Chemical Society, 136(21), 7611–7614. Yang, S., Popov, A. A., & Dunsch, L. (2007a). The role of an asymmetric nitride cluster on a fullerene cage: The non-IPR endohedral DySc2 N@C76 . Journal of Physical Chemistry B, 111(49), 13659–13663. Yang, S. F., Popov, A. A., & Dunsch, L. (2007b). Violating the isolated pentagon rule (IPR): The endohedral non-IPR cage of Sc3 N@C70 . Angewandte Chemie-International Edition, 46(8), 1256–1259. Yang, S., Yoon, M., Hicke, C., Zhang, Z., & Wang, E. (2008). Electron transfer and localization in endohedral metallofullerenes: Ab initio density functional theory calculations. Physical Review B, 78, 115435. Yang, H., Beavers, C. M., Wang, Z., Jiang, A., Liu, Z., Jin, H., Mercado, B. Q., Olmstead, M. M., & Balch, A. L. (2010). Isolation of a small carbon nanotube: The surprising appearance of D5h (1)-C90 . Angewandte Chemie International Edition, 49(5), 886–890. Yang, H., Mercado, B. Q., Jin, H., Wang, Z., Jiang, A., Liu, Z., Beavers, C. M., Olmstead, M. M., & Balch, A. L. (2011). Fullerenes without symmetry: Crystallographic characterization of C1 (30)-C90 and C1 (32)-C90 . Chemical Communications, 47(7), 2068–2070. Yang, S., Wei, T., Kemnitz, E., & Troyanov, S. I. (2012a). The most stable IPR isomer of C88 fullerene, Cs -C88 (17), revealed by X-ray structures of C88 Cl16 and C88 Cl22 . Chemistry – An Asian Journal, 7(2), 290–293. Yang, S., Wei, T., Kemnitz, E., & Troyanov, S. I. (2012b). Four isomers of C96 fullerene structurally proven as C96 Cl22 and C96 Cl24 . Angewandte Chemie International Edition, 51(33), 8239–8242. Yang, S., Wei, T., & Troyanov, S. I. (2013). A new isomer of pristine higher fullerene Cs -C82 (4) captured by chlorination as C82 Cl20 . Chemistry – An Asian Journal, 8(2), 351–353. Yang, S., Wei, T., Kemnitz, E., & Troyanov, S. I. (2014). First isomers of pristine C104 fullerene structurally confirmed as chlorides, C104 (258)Cl16 and C104 (812)Cl24 . Chemistry – An Asian Journal, 9(1), 79–82. Yanov, I., Kholod, Y., Simeon, T., Kaczmarek, A., & Leszczynski, J. (2006). Local minima conformations of the Sc3 N@C80 endohedral complex: Ab initio quantum chemical study and suggestions for experimental verification. International Journal of Quantum Chemistry, 106(14), 2975–2980. Yumura, T., Sato, Y., Suenaga, K., & Iijima, S. (2005). Which do endohedral Ti2 C80 metallofullerenes prefer energetically: Ti2 @C80 or Ti2 C2 @C78 ? A theoretical study. Journal of Physical Chemistry B, 109(43), 20251–20255. Zalibera, M., Rapta, P., & Dunsch, L. (2007). In situ ESR–UV/VIS/NIR spectroelectrochemistry of an empty fullerene anion and cation: The C82 :3 isomer. Electrochemistry Communications, 9(12), 2843–2847. Zhang, J., Hao, C., Li, S. M., Mi, W. H., & Jin, P. (2007). Which configuration is more stable for La2 @C80 , D3d or D2h ? Recomputation with ZORA methods within ADF. Journal of Physical Chemistry C, 111(22), 7862–7867. Zhang, J., Bowles, F. L., Bearden, D. W., Ray, W. K., Fuhrer, T., Ye, Y., Dixon, C., Harich, K., Helm, R. F., Olmstead, M. M., Balch, A. L., & Dorn, H. C. (2013). A missing link in the transformation from asymmetric to symmetric metallofullerene cages implies a top-down fullerene formation mechanism. Nature Chemistry, 5(10), 880–885. Zhang, Y., Ghiassi, K. B., Deng, Q., Samoylova, N. A., Olmstead, M. M., Balch, A. L., & Popov, A. A. (2015). Synthesis and structure of LaSc2 N@Cs (hept)-C80 with one heptagon and thirteen pentagons. Angewandte Chemie International Edition, 52(2), 495–499. Zhao, X., Slanina, Z., Goto, H., & Osawa, E. (2003). Theoretical investigations on relative stabilities of fullerene C94 . Journal of Chemical Physics, 118(23), 10534–10540. Zheng, G. S., Irle, S., & Morokuma, K. (2005). Performance of the DFTB method in comparison to DFT and semiempirical methods for geometries and energies Of C20 –C86 fullerene isomers. Chemical Physics Letters, 412(1–3), 210–216.

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Zuo, T. M., Beavers, C. M., Duchamp, J. C., Campbell, A., Dorn, H. C., Olmstead, M. M., & Balch, A. L. (2007). Isolation and structural characterization of a family of endohedral fullerenes including the large, chiral cage fullerenes Tb3 N@C88 and Tb3 N@C86 as well as the Ih and D5h isomers of Tb3 N@C80 . Journal of the American Chemical Society, 129(7), 2035–2043. Zuo, T., Walker, K., Olmstead, M. M., Melin, F., Holloway, B. C., Echegoyen, L., Dorn, H. C., Chaur, M. N., Chancellor, C. J., Beavers, C. M., Balch, A. L., & Athans, A. J. (2008). New egg-shaped fullerenes: non-isolated pentagon structures of Tm3 N@Cs (51365)-C84 and Gd3 N@Cs (51365)-C84 . Chemical Communications, 2008(9), 1067–1069. Zywietz, T. K., Jiao, H., Schleyer, R., & de Meijere, A. (1998). Aromaticity and antiaromaticity in oligocyclic annelated five-membered ring systems. Journal of Organic Chemistry, 63(10), 3417–3422.

Structures and Electric Properties of Semiconductor clusters Panaghiotis Karamanis

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Properties of Semiconductor Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ground-State Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silicon Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III–V and II–VI Semiconductor Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric Polarizabilities and Hyperpolarizabilities of Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cluster (Hyper)Polarizabilities: Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semiconductor Cluster (Hyper)polarizabilities: General Trends and Selected Studies . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 3 5 7 12 18 18 21 27 32 33

Abstract

Materials that exhibit an electrical resistivity between that of conductor and insulator are called semiconductors. Devices based on semiconductor materials, such as transistors, solar cells, light-emitting diodes, digital integrated circuits, solar photovoltaics, and much more, are the base of modern electronics. Silicon is used in most of the semiconductor devices while other materials such as germanium, gallium arsenide, and silicon carbide are used for specialized applications. The obvious theoretical and technological importance of semiconductor materials has led to phenomenal success in making semiconductors with nearatomic precision such as quantum wells, wires, and dots. As a result, there is a lot of undergoing research in semiconductor clusters of small and medium

P. Karamanis () Groupe de Chimie Théorique et Réactivité, ECP, IPREM UMR 5254, Université de Pau et de Pays de l’Adour, PAU Cedex, France e-mail: [email protected]; [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_20-2

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sizes both experimentally and by means of computational chemistry since the miniaturization of devices still continues. In the next pages, we are going to learn which the most studied semiconductor clusters are, we will explore their basic structural features and visit some of the most representative ab initio studies that are considered as works of reference in this research realm. Also, we are going to be introduced to the theory of the electric properties applied in the case of clusters by visiting some of the most illustrative studies into this research area. It is one of the purposes of this presentation to underscore the strong connection between the electric properties of clusters and their structure.

Introduction About two decades ago Robert Pool stated: “Clusters are strange morsels of matter: when metals or semiconductors are shrunk down to clumps only 10 or 100 atoms in size, they become a “totally new class of materials” with potentially valuable applications” (Pool 1990). This distinctive behavior of clusters has triggered an explosion of scientific work in this area and it is in close relation with the dramatic development of a new field in science. This field is now widely known as “ nanoscience.” The term nanoscience comes from the terms “nano” and “science.” The word “nano” is used to define a specific length of one billionth of a meter. This size scale resides between bulk materials and typical molecular dimensions. Thus, nanoscience is associated with the study of structures, materials, and devices, which are larger than the typical molecules but extremely small, usually in the range from 0.1 to 100 nm. The increasing importance of this area and its rapid expansion is owed mainly to the requirements of modern technologies for a vast diversity of new materials with more than one function. The field of nanoscience is highly multidisciplinary and joins different disciplines such as chemistry, applied physics, materials science, colloidal science, device physics, supramolecular chemistry, and even more, mechanical and electrical engineering. As a matter of fact, nanoscience can be viewed as extension of most of existing sciences into the nanoscale or the upgrade of the existing molecular sciences to the nanoscale dimensions. Among nano-objects clusters occupy a large part of the big picture. The socalled nanoclusters apart from a bridge between molecules and solids are systems with properties of their own which are considerably different from those of the bulk materials and from those of their constituent parts (atoms and molecules). Under specific conditions those species can be considered as the building modules of nano-objects which are the foundations of nanoscience. Clusters are aggregates of atoms or molecules in the size of few nanometers and their constitution ranges between 10 and 106 atoms. They can be built from only one type of atoms such as the homogeneous silicon clusters, or from more than one element such as the heterogeneous gallium arsenide binary clusters. Atomic clusters can be met in neutral or charged forms while their atoms bind together by almost all of the main bond types such as covalent, ionic, van der Waals, and metallic.

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Amongst the different types of clusters that have been the subject of numerous studies in the past, clusters which originate from semiconductor materials have attracted considerable attention. In spite of their origin, these entities are not necessarily semiconductors. Their basic bonding features resemble those of the metals but they keep as well some of the covalent bonding characteristics which dominate in semiconductor nanocrystals. The purpose of this presentation is to bring together the most representative ab initio studies of the non-bulk-like structural and electric properties of semiconductor clusters. Both of these two different property classes are closely related to each other and have occupied the interest of the researcher in numerous studies.

Structural Properties of Semiconductor Clusters Excluding clusters built from carbon (fullerenes) then the most studied semiconductor clusters are those made from silicon (Si) and gallium arsenide (GaAs). GaAs clusters belong to the so-called III–V group of semiconductors which are formed between the elements of groups 13 and 15. Also, there is a significant amount of work focusing on the II–VI semiconductor clusters which are formed between the elements of the 12 and 15 groups. Especially for the latter clusters there is significant amount of reported experimental and theoretical work due to their unique optical properties. The most studied II–VI semiconductor clusters are those built from cadmium sulfide which have served as prototypes for studies of the effect of quantum confinement on electronic and optical properties of semiconductor nanoobjects. For instance, CdS quantum dots which are water soluble and biocompatible (Bruchez et al. 1998), are potentially useful as fluorescent biological labels since they are characterized by large optical gaps. Also, cadmium selenide and telluride clusters have attracted considerable attention due to their narrow bandgaps and large Bohr exciton radii. The last characteristic implies that nanostructures built from those systems are expected to exhibit strong quantum confinement and this feature makes both CdSe and CdTe nano-objects very interesting to explore their properties. Other kinds of semiconductor clusters that have been the subject of a notable number of computational studies are clusters built from GaN, GaP, InP, InAs, ZnO, ZnS, AlP, HgTe, Ge, SnTe, BeTe, BiTe, BN, BAs, and BP.

The Ground-State Structure In principle, the most favorable atom arrangement of a cluster should be the one that is characterized by the smallest total energy. This structure is generally called the ground-state structure and at zero or low enough temperatures should be the mathematical global minimum of the potential energy of a cluster as a function of the coordinates of the atomic centers (the potential energy surface, PES). The arrangement of the atoms in the ground-state structure is the one that is considered in the majority of cluster property studies, assuming that most of

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those properties are defined by the properties of the lowest energy configuration. As a result, a respectable number of various computational schemes have been developed and proposed in order to treat the ground-state structure problem in an accurate and efficient manner. However, the solution to this problem is neither obvious nor simple. In the contrary, it corresponds to a task of extremely high computational complexity. The problem becomes even harder given that the available experimental methods are not yet in the position to provide the necessary microscopic structural details for the complete understanding of clusters structures in order for the developed theoretical methods to be verified or improved. Also, the developed computational methods face certain difficulties that are mainly delivered from “practical” limitations of the algorithms used or restrictions of the quantum chemical methods in the precise calculation of the potential energy surface of a given cluster. Accordingly, in several cases the global minimum of the potential energy cluster surface exhibits a strong dependence on the quantum chemical method one uses in order to explore it (Karamanis et al. 2007b). A classical example of the above statement is the case of C20 This cluster is the smallest possible fullerene, and experiments suggest that it should be a planar ring (Castro et al. 2002; Dugourd et al. 1998; Prinzbach et al. 2000). However, detailed theoretical studies (Grossman et al. 1995) by means of quantum Monte Carlo, all-electron fixed-node quantum Monte Carlo (Sokolova et al. 2000), single- and multireference MP2 methods (Grimme and Mück-Lichtenfeld 2002) and high-level ab initio methods (An et al. 2005) pointed out the bowl and the cage structures as more stable. The answer to this puzzle is that both experiments and theory are correct since if one introduces finite-temperature corrections to the calculations then ring structure becomes more stable (see Fig. 1). Furthermore, some specific clusters show a peculiar behavior well known as “fluxional behavior.” This behavior is connected with the fact that at a given temperature the clusters are expected to be in some sort of thermal equilibriumm rapidly interconverting from one stable isomer to the other due to fast and sometimes drastic atomic rearrangements. During a structural rearrangement of this kind, the resulting “isomerization” leads to structures characterized by different bonding patterns. Thus, it would not be irrational for one to claim that there is no guarantee that the theoretically predicted ground states are favored in all situations and condition. In order to overcome such problems many studies extend their property investigations to stationary points which correspond to cluster conurations energetically Fig. 1 The three different structures of C20 competing for the ground state

Bowl

Cage

Ring

Structures and Electric Properties of Semiconductor clusters

Si7

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0.0

+0.006 eV

0.021 eV

0.030 eV

0.032 eV

+0.141 eV

+0.116 eV

+0.142 eV

+0.152 eV

+0.156 eV

+0.166 eV

+0.221 eV

+0.232 eV

Fig. 2 Ground state structures and low lying isomers of AISi7 in comparison with the lowest energy structures of Si7 and Si8

close to the ground state. These clusters are meta-stable species and their study is expected to complement the investigations which focus on the narrow area of the global minima. A demonstration of the numerous stable local minima that exist even for clusters of small size is given in Fig. 2 which depicts the ground-state structures of aluminum-doped clusters along with several low-lying isomers which are extremely close in energy (Karamanis et al. 2010).

Structural Determination There are two major classes of global computational methods: the unbiased and the seeded methods. The first class works independently of initial cluster configurations

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and is considered as the most significant and reliable but the more computationally demanding. The second class utilizes a set of initial structures as seeds and they are usually faster than the unbiased; however, there is always the possibility of disregarding configurations of low energy that are not expected. Another optional way of finding the most favorable structures of clusters is based on guiding principles and structural rules stability principles, such those that have been originally developed and tested over the years for boranes and carboranes and bisboranes. Although this kind of approach is more of an art than of an automatic algorithmic computational approach it has been utilized with remarkable success in various systems such as pure or doped silicon cluster, or clusters made from gallium arsenide, cadmium sulfide, selenide, or telluride (see for instance, Al-Laham and Raghavachari 1991, 1993; Lipscomb 1966; Zdetsis 2008, 2009; Williams 1992; Gutsev et al. 2008a, b; Raghavachari and Logovinsky 1985, Raghavachari and Rohlfing 1988; Matxain et al. 2000, 2001, 2003, 2004). In any case though, whatever the method chosen, one has to deal with the existence of a vast number of local minima even in the case of clusters composed by few atoms. Let us now see in brief some of the developed algorithms that have been used in cases that involve semiconductor clusters. The most commonly used computational schemes in finding the ground-state structures of semiconductor clusters are those based on genetic algorithms that simulate the evolution in nature through natural selection. This approach utilizes concepts such as chromosome mixing, mutations, and the selection of the fittest. Another family of algorithms is the so-called basin hopping which use canonical Monte Carlo simulations at a certain temperature. The basic idea of the basin hopping algorithms is to lower the energetic barriers that separate the various local minima of a cluster by leaving the number and the kinds of the local minima unchanged. This has been proven very efficient in scanning different areas of the potential energy surface and has been successfully applied in cases of clusters up to 100 atoms. A different approach which has been applied in a great variety of systems is the thermal simulated annealing. This method lets the system to evolve initially at a high temperature and then a cooling down procedure starts by reducing gradually the temperature of the system to zero temperature. At each step some neighbor structures are considered based on probability rules and then it is decided whether this new structures are accepted or not following certain energy criteria (e.g. by moving to states of lower energy). As a result, the search space which contains appropriate solutions to the problem considerably narrows down. However, if the free energy global minimum changes at low temperatures where dynamical relaxation is slow, the algorithms may become stuck in the structure corresponding to the high-temperature free energy global minimum. Thus, although this algorithm is easy to follow and understand because of its physical concepts, it is less efficient than methods based on genetic and basin hopping algorithms. Finally, another method that has been developed to treat the problem of rich potential energy surfaces close to the minimum is the so-called Global Search Algorithm of Minima (GSAM) (Marchal et al. 2009, 2010, 2011). In brief, the GSAM includes three major parts: the first part is devoted to the generation of an initial guess set of structures for a large number of cluster structures

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randomly generated by several techniques. The second part performs an automatic selection of the most appropriate structures through a special pattern recognition technique (Karamanis et al. 2009), which are expected to lead to different local minima. Finally, the third part comprises full geometry optimizations of the selected configurations. This last part yields the most stable cluster structures and the structure which corresponds to the ground state among them. Further information into the subject of the methods that have been used to treat cluster structural problems in a global manner can be found in the Refs. Bazterra et al. (2004), Biswas and Hamann (1986), Blaisten-Barojas and Levesque (1986), Hamad et al. (2005), Ho et al. (1998), Hossain et al. (2007), Jelski et al. (1991), Menon and Subbaswamy (1995), Nair et al. (2004), Pedroza and Da Silva (2007), Sokolova et al. (2000), Tekin and Hartke (2004), Yoo et al. (2004, 2008), Yoo and Zeng (2005, 2004).

Silicon Clusters Selected Structural Studies As we have already mentioned, among the various semiconductor clusters studied during the past two decades, those made from silicon have attracted most of the attention. A significant number of algorithms and computational strategies have been employed to identify the ground states and the low-lying isomers of silicon clusters of small and medium size with remarkable success. As a result, it is now a common sense that the silicon atoms of a cluster are “different” from those of the bulk material. It is evident that most of the studies on silicon clusters concern their neutral forms. However, it is impossible to neglect investigations on charged clusters since they have led to very significant results and conclusions (Lyon et al. 2009; Xiao et al. 2002; Nigam et al. 2004; Wei et al. 1997; Li et al. 2003, Li 2005; Zhou and Pan 2008). One of the most impressive investigations on this subject is the recent study reported by Lyon et al. who studied for the first time the experimental infrared spectra of positively charged silicon bare clusters up to 21 atoms in the gas phase. In this study the experimental spectra were compared to theoretical spectral predictions and the authors managed to make unambiguous structural assignments for these species (Fig. 3). This work is one of the most recent proofs confirming that the silicon clusters which have been predicted one way or another by theory also exist in the gas phase. In the case of the neutral clusters up to 13 atoms, the most stable structures are not planar except Si3 and Si4 Mainly they prefer polyhedral structures of rather high symmetry. Raghavachari and Logovinsky (1985), Fournier et al. (1992), Li (2005), and Yu et al. (2002) have established the basic structural patterns of those small species which nowadays are considered as systems of reference. For those species, Zhu and Zeng (2003) provided accurate geometries computed at high levels of theory. A recent study performed by Zdetsis (2007a, b) have showed that silicon clusters of 5 up to 13 atoms, and their dianions follow common structural

8 Fig. 3 Ground-state structures of neutral and charged silicon clusters. Stable structures of neutral silicon clusters

P. Karamanis Neutral

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Si4

Si6

Si8

Si7

Si9

Si10

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Structures and Electric Properties of Semiconductor clusters

9

motifs of the corresponding closo-boranes. Among the small clusters with 2 up to 13 atoms there are two species of remarkable interest. These species are the Si6 and Si10 which are considered as magic clusters because of their high stability. The six-atomic silicon cluster has attracted the attention of many researchers due the existence of three almost isoenergetic structures that compete for the ground state. Namely, the distorted (compressed) octahedron of D4h symmetry and two trigonal bipyramidal shapes of lower C2v symmetry, the edge-capped and facecapped trigonal bipyramids. On the other hand Si10 is also known as an extremely stable entity and it has been used as a building module of larger structures (Raghavachari and Rohlfing 1988; Mitas et al. 2000; Rohlfing and Raghavachari 1990; Grossman et al. 1995). From n D 14 the shapes of the most stable structures become prolate (Zhu et al. 2004; Yoo and Zeng 2005). An interesting structural feature of the stacked prolate structures is that they are built from blocks that in fact are smaller stable clusters such as the Si9 tricapped trigonal prism or the Si6 puckered-hexagonal-ring. For instance Si11 and Si15 are built upon the tricapped-trigonal-prism of Si9 while Si6 puckered-hexagonal-ring motifs can be identified for silicon clusters with 16–20 atoms. This is the most reliable explanation of their stability since it is related to the stability of their building blocks. This finding is of crucial importance since it may open new directions in the nanomaterial and single nanoparticle design. After n D 19, hollow near-spherical shapes start to show significant stability. From n D 24, the near-spherical structure with interior atoms becomes nearly degenerated with the prolate configurations (Jarrold and Bower 1992; Jarrold and Constant 1991; Kaxiras and Jackson 1993). However, up to n D 25, the stacked prolate structures are still the most stable. Finally, although clusters with more than 25 atoms are not very well studied and their ground-state structures are either unknown or under debate (Bai et al. 2006), there are recent studies that yielded very interesting results. For instance, Yoo and coworkers (Avramov et al. 2007; Yoo et al. 2008, 2004; Yoo and Zeng 2006, 2005) searched for generic structural features as well as patterns of structural evolution for the low-lying silicon clusters up to the size of 80 atoms by studying the structures and relative stability of different cluster shapes families. Their results indicate that for some cluster sizes such as Si39 , Si40 , and Si50 , the fullerene cage motifs consistently are more stable than other kind of cage structures.

The Si6 and Si36 Cases To demonstrate the difficulty in studying the structures and properties of clusters even of small size even when are built from only one element, and the complexity in interpreting computational and experimental results of diverse sources we shall revisit two distinctive cases: One concerns the small Si6 cluster and the other the larger Si36 As mentioned, Si6 is a particularly stable cluster and along with Si10 coincides with clusters found abundantly in the experiment. Figure 4 shows three optimized structures of Si6 at the CCSD(T) level and the natural atomic charges computed with the CCD density. It is obvious that although the two C2v isomers keep the characteristics of their symmetry group as they were defined in the staring point

10

P. Karamanis +0.001 kcal / mol

+0.014 kcal / mol

a

b

+0.13

c

+0.11 2.4115

90.0º + 0. 13

2.3973 2.3973 x

z

2.4107 94.8º

89.6º +0.13

–0.25

+0.13

+0.13 αxx = 200.55 αyy = 150.19 αzz = 200.55

+0.11 –0.26 2.4115 2.3872

+0.15

2.3872 +0.15 α = 200.58 +0.15 xx αyy = 150.33 αzz = 201.48

+0.09

+0.17

–0.26

2.4156

2.3804

2.4107 +0.13 αxx = 200.64 αyy = 150.35 αzz = 201.51

Fig. 4 CCSD(T)/6-311G(2d) optimized structures of the three isomers of Si6 atomic charges after a natural bond orbital analysis at the CCD level and MP2/6-311G(2d) polarizabilities (au). All drawn bondlengths are in Å and dihedral angles in degrees. Structure (a) is of D4h symmetry, structures (b) (face-capped) and (c) (edge-capped) of C2v symmetry.

geometries, they are almost identical with the distorted octahedron and their energy separations are extremely smaller than expected. The C2v face-capped isomer and the distorted octahedron are lying only 0.001 and 0.014 kcal/mol higher in energy than the edge-capped isomer, respectively. Additionally, MP2 polarizability tensor values (au) at the CCSD(T) optimized geometries (also included in Fig. 2) show that the two C2v structures are almost as polarizable as the D4h structure in all three directions. Consequently, the two trigonal bipyramid structures could be viewed as slightly distorted tetragonal bipyramidal shapes belonging to the C2v point group. A careful literature search shows that these three structures compete for the ground state. More specifically, all electron and core potential geometry optimizations at Hartree–Fock (HF) level by Raghavachari and Rohlfing prefer an edge-capped trigonal bipyramidal shape of C2v symmetry while MP2 geometry optimizations favor a distorted octahedron of D4h symmetry. On the other hand, higher order correlated method based on Möller–Plesset perturbation theory with the standard 6-31G(d) basis set by Zdetsis (2001) showed that the MP2 predicted structure of D4h symmetry is unstable. Additionally, density functional theory (DFT) (Karamanis et al. 2007b) shows an fluctuating preference between the C2v facecapped, C2v edge-capped trigonal bipyramids. It is evident that there is an obvious disagreement between different theoretical approaches. Let us now explore the most important experimental results concerning this problem. Raman experiments by Honea et al. (1993) of matrix isolated clusters, supported indirectly the MP2/6-31G(d) structure of D4h symmetry. Their conclusions were based on two arguments: The first was the number of visible bands in both spectra which characterize species of D4h symmetry, and second, the agreement between the experimental and scaled theoretical harmonic frequencies computed at the MP2/6-31G(d) level with the small 6-31G(d) standard basis set. The strongest arguments used to reject the other two candidates were two: first, geometry optimizations at MP2 level starting from either face-capped or

Structures and Electric Properties of Semiconductor clusters

11

edge-capped trigonal bipyramidal shapes collapsed to the distorted octahedron of D4h symmetry; second, the other two structures were predicted to have a larger number of active vibrational bands than observed in both experimental Raman and IR spectra. Obviously such a puzzling problem rarely has solely one answer. Indeed, the resolution to this puzzle is twofold. On one hand the source of the disagreement among the various methods used hides in the particular capabilities of each method in the treatment of the pseudo Jahn–Teller effect (PJTE) (Karamanis et al. 2007b). According to general Jahn–Teller theory the only source of instability in any molecular systems is the proper Jahn–Teller effect, for degenerated states (Renner Teller effect in linear systems), and pseudo-Jahn–Teller for non-degenerated ones. PJTE is a part of modern JT interaction theory (Bersuker 2001) and its importance in the formation of the equilibrium structures of clusters has been demonstrated by Garcia-Fernandez et al. (2006) and by Pushpa et al. (2004). In the Si6 case, geometry optimizations with methods based on Møller–Plesset perturbation theory, which have been demonstrated not to treat the PJTE correctly, predict that the distorted octahedron of D4h symmetry is the ground-state structure, while, methods which do provide PJTE treatment (DFT, HF) suggest that the distorted octahedron is unstable and undergoes PJTE distortions. On the other hand, the flat potential energy surface is connected with a fluxional behavior of Si6 (Zdetsis 2007b) caused by the couplings between the electronic motion and soft nuclear vibrations due to the second-order Jahn–Teller effect under the influence of “electron deficiency” and the resulting charge smoothing process. The soft nuclear motion that is implied by the calculation is shown in Fig. 5. Of course, this kind of discrepancies in cluster structures can hardly be identified by experimental means since in such a flexible (or fluxional) system with low barriers, the consideration of separate “conformers” is probably not appropriate at all, and experimental spectroscopic properties are just an average of all structures. This has been highlighted in the recent work of Fielicke et al. (2009) who used tunable far-infrared-vacuum-ultraviolet two-color ionization to obtain vibrational spectra of neutral silicon clusters in the gas phase. Their results show clearly that it is almost impossible to distinguish the three different structures by means of vibrational spectroscopy. Another example that demonstrates the complexity in determining the groundstate structures of larger clusters, and the importance of exploring the complete space when searching for atomic cluster is the case of Si36 (Bazterra et al. 2002; Sun et al. 2003). This cluster is one of the first large species that have been studied using first principles within the density functional framework (Sun et al. 2003). The structure that was initially proposed as the most stable was a stuffed fullerene-like configuration of near-spherical shape (Fig. 6 structure a). This structure seemed to confirm the up to that time experimental findings according to which a shape transition from prolate shapes to compact ones is occurring as the size of silicon clusters grows. Nonetheless, a subsequent study (Bazterra et al. 2002) based on a global approach using genetic algorithms, showed that a totally different structure lies lower in energy than the one previously adopted (Fig. 6 structure b). Once more, the structure of Si36 proposed by Bazterra et al. can be envisioned as an assembly of smaller, stable clusters.

12

P. Karamanis Distored octahedron (higher order saddle point)

Face-capped trigonal bipyramid (Saddle point)

Edge-capped trigonal bipyramid (minimum)

Edge-capped trigonal bipyramid (minimum) Face-capped trigonal bipyramid (Saddle point)

Fig. 5 Schematic representation of the three competing structures of Si6 based on the normal modes of the imaginary frequencies at CCSD(T) level of theory with the split valence Pople-like Basis set 6-311G(2d)

III–V and II–VI Semiconductor Clusters General Features The III–V and II–VI semiconductor clusters are isoelectronic with the elements of the group 13 and 15. For instance, gallium arsenide clusters of the type Gan Asn have the same number of electrons with Ge2n clusters while the analogue Aln Pn clusters are isoelectronic with the corresponding Si2n cluster species. For clusters of these two semiconductor families with even number of electrons, the preferred spin multiplicity is the singlet, where all electrons are in pairs. Triplets and larger spin multiplicities are always higher in energy (Gutsev et al. 2010). In the case of

Structures and Electric Properties of Semiconductor clusters

13

Fig. 6 Two competing structures of Si36

nonstoichiometric GaAs clusters which are species with even number of electrons, thus, open shell systems the preferred spin multiplicity is the doublet (Karamanis et al. 2009). The basic bonding pattern in the compound III–V and II–VI semiconductor clusters is the alternating arrangement of their atoms (Figs. 7 and 8). The alternating A-B type of bonding is a result of the polarity of these entities which is owed to the electronegativity difference between their atoms. As a result, the bonding in clusters formed by atoms with large electronegativity differences is expected to be more ionic than in clusters built from atoms characterized by smaller differences. For instance, the spectroscopic electronegativity differences 4spec converted to Pauling scaling between the As and the atoms Ga and In are 0.455 and 0.555, respectively. Accordingly, the bonding in III–V compound clusters built from As and Ga is expected to be of less ionic (or of more covalent character) than clusters built from In and As. However, although the alternating A-B type of bonding is generally observed for this class of semiconductor clusters due to the polarity of their bonding, there are some cases in which homo-atomic bonds are detected between the electronegative atoms. For example the ground-state structures of Ga4 As4 (Fig. 9) and Al5 P5 (Fig. 8) are characterized by As–As and P–P bonds. On the other hand, this bonding feature is not favorable for II–VI semiconductors which also do not favor endohedral structures.

Gallium Arsenide Clusters Combined experimental and computational studies have led to very interesting and important conclusions concerning microscopic feature of those systems such as their bonding and electronic structures. For instance, the revealed even/odd alternation in the photoionization cross section obtained with an ArF excimer laser (6.4 eV) indicated that clusters of any composition composed by even numbers of atoms (e.g., Ga1, As5 , Ga2 , As4 , Ga3 , As3 , Ga4 , As2 , Gas5 , As1 )have singlet ground states, whereas odd numbered clusters of any composition (e.g., Ga1, As5 , Ga1 , As4 , Ga2 , As3 , Ga3 , As2 , Ga4 , As1 ) have doublet ground states with an unpaired electron in a weakly bound HOMO. This sort of electronic structure clearly indicates that the structural and bonding features of the produced small GaAs clusters should be

14

P. Karamanis

As Ga

Ga2As2-D2h

Ga3As3-Cs

Ga9As9-C3h

Ga4As4-Ci

Ga5As5-Cs

Fig. 7 Established ground-state structures of Gan Asn clusters with n D 2–5 and 9. Such structure as for n D 9 is the most stable also in the cases of Al9 P9 and Cd9 (S, Se, Te)9

As

Ga (GaAs)27 S6 fullerene

(GaAs)36 C3h bulk-like (GaAs)24 S6 fullerene

Fig. 8 Fullerene and bulk-like structures of large GaAs clusters (Gutsev et al. 2010)

dramatically different from the tetrahedral sp3 hybridization of the bulk. Both of those indirect experimental conclusions–assumptions have been verified by all the later computational studies on those species which clearly showed that the singlet closed shell forms of the even-numbered GaAs clusters are the most stable than any other electronic configuration and their structures are completely different from the bulk (see: O’Brien et al. 1985; Lou et al. 1992; Karamanis et al. 2007a; Gutsev et al. 2008a, b). Let us now make a brief review of the most representative studies of those important species starting from GaAs clusters which are the most studied species

Structures and Electric Properties of Semiconductor clusters

15

P

Al Al

P

Al

Al

Al Al

Al

Al

P

P

P

P P

P

Al

Al2P2-D2h

Al

Al3P3-D3h Al

P

Al

P

Al P

P

Al3P3-Cs

P Al Al

Al

P Al

P

P

Al

Al4P4-Td

P

Al

Al

Al P

P

Al

P

P

Al

Al

Al5P5-Cl

Al P

P

Al6P6-D6d P

P

P

Al

Al

P Al

Al Al P

Al

Al

P Al

P

P

Al P

Al7P7-Cl

P

P

P Al

Al

P

Al Al

P

Al

P P Al

Al

Al P

Al Al P

Al Al P

Al8P8-S4

P

P

Al P

Al Al

Al P

P

Al9P9-D3h

Fig. 9 Structures of aluminum phosphide clusters

from those belonging to the III–V semiconductor family. Current and future applications of those species, in their nano or bulk, forms are very promising and interestingly diverse, extending from fuel cells Schaller and Klimov (2006) and optoelectronic sensors (Chen et al. 2004) to spintronics (spin-based electronics), medical diagnostics, and quantum computing (see Calarco et al. 2003; Michalet et al. 2005; Wolf et al. 2001). Pioneering studies performed by O’Brien et al. on supersonic cluster beams of GaAs clusters generated by laser vaporization of discs made from pure GaAs brought theory closer to the experimental reality by providing cluster species of sizes suitable for very precise theoretical computations. As a result, the structural and bonding features of small GaAs clusters have occupied the interest of the researchers in the early cluster computational studies. For instance, Graves and Scuseria (1991) studied the first members of these systems using the Hartree–Fock (HF) method. Liao and Balasubramanian (1992) by means of the multireference configuration interaction method with singles and doubles (MRCISD) made a reference estimation of the bond lengths and angles of (GaAs)2 ; Lou et al. (1992) studied the structures and the bonding of small GaAs clusters up to 10 atoms

16

P. Karamanis

while Andreoni within the Car–Parinello molecular dynamics approach studied the stoichiometric systems up to the ten-atomic Ga5 As5 . Later, Song et al. (1994) applied a fourth-order many body perturbation theory to study species from 2 to 4 atoms (Gan Asn with n D 2–4). The interest in studying those systems remains intense even the recent years and a respectable variety of different methods have been applied. For instance, Costales et al. (2002) applied a DFT-GGA method for the systems up to the trimer, Zhao et al. (2006), Zhao and Cao(2001), Zhao et al. (2000) used another level of DFT-GGA approach for (GaAs)n (n D 6–9) while Karamanis et al. (2007b) revisited the previous reported results using second-order many body perturbation theory (MP2) up to clusters with 18 atoms. More recently, Gutsev et al. (2008a, b) extended the earlier studies and reported structures up to 32 atoms with the DFT framework. In addition, there are some reported attempts in studying nonstoichiometric species and the influence of the composition on their properties. One of the most representative works in this field is the investigation of structural and electronic properties of Gam Asn clusters reported by Feng et al. (2007); Gutsev et al. (2010) studied the electronic and geometrical structures of neutral (GaAs)n clusters using density functional theory with generalized gradient approximation and relativistic effective core potentials for n D 17–40, 48, and 54. Also, they examined the energetic preference of tubular and non-tubular cages with increasing n and found that tubular cages are generally lower in total energy than the other cage isomers. Along with the cage structures, they probed structures cut off from a zinc blende lattice, tailoring the surface atoms in such a way as to have no dangling bonds. They found that the lowest energy state of (GaAs)36 has a bulk-like structure with 12 tetrahedrally coordinated inner atoms (Fig. 7). Finally, Gutsev and coworkers (Karamanis et al. 2011) studied the structure and properties of prolate (GaAs)n clusters up to 120 atoms corresponding to the (2, 2) and (3, 3) armchair and (6, 0) zigzag capped single-wall tubes within the generalized gradient approximation (DFT-GGA). It was found that the bandgap in all three series does not converge to the GaAs bulk value when the cluster length increases.

II–VI Semiconductor Clusters The synthesis of ZnO, ZnS, ZnSe, CdS, CdSe, and CdTe nanostructures which shape an important category of optically active materials with strong size dependence of their exciton energy (see: Alivisatos 1996; Swaminathan et al. 2006; Murray et al. 2000; Peng et al. 1998, 2000) have inspired most of the studies on II–VI semiconductor clusters. The respective nanomaterials can be synthesized in a great variety of sizes and shapes (spheres, rods, tetrapods or branched and core-shell structures) and they have been used for a wide assortment of applications, including quantum light emitting diodes pigments, biological tagging, and solar cells (Gur et al. 2005). Particularly, for CdSe clusters, correlations between their structure and properties have been highlighted by Jose et al. (2006). One of the most fascinating features of this class of semiconductor clusters is the size dependence of the HOMO–LUMO bandgap which is distinguishable in their luminescence properties. For instance, it has been shown that by adjusting the particle size of colloidal CdSe–CdS core–shell nanoparticles the fluorescence can be

Structures and Electric Properties of Semiconductor clusters

17

Fig. 10 Ring structure-types of MnXn (M D Zn, Cd and X D S, Se, Te) small clusters

tuned between blue and red. Motivated by those finding, in nanoparticles, Sanville et al. (2006) used direct laser ablation to produce ZnS, CdS, and CdSe clusters and found that clusters composed of 6 and 13 monomer units were ultrastable in all cases. Kasuya et al. (2004) by means of mass spectral analysis of CdSe nanoparticles found that (CdSe)33 and (CdSe)34 are so stable that they grow in preference to any other chemical compositions to produce macroscopic quantities with a simple solution method. First-principles calculations predict that these clusters are puckered (CdSe)28 -cages, with four- and six-membered rings based on the highly symmetric octahedral analogues of fullerenes, accommodating inside their framework either (CdSe)5 or (CdSe)6 to form a three-dimensional network with essentially heteropolar sp3 -bonding. Let us now see in brief the most cited computational studies for this very interesting class of cluster species. Behrman et al. (1994) have shown that ZnO clusters can form stable, fullerene-like structures while Hamad et al. (2005) provided evidences that ZnS clusters can form onion-like or alternatively “double bubble” structures. Matxain et al. (2000) studied systematically the structures of small ZnO clusters with up to nine atoms and demonstrated that three-dimensional structures may be envisioned as being built from Zn2 O2 and Zn3 O3 rings. Also, the same group (Matxain et al. 2001, 2003) have found that clusters built from ZnS, ZnSe, and CdO clusters can form stable one-dimensional ring structures (Fig. 10). For species made from Cd and the chalcogenides S, Se Gurin(1998) studied small Cdx Sy and Znx Sy (x, y  6) clusters, Troparevsky and Chelikowsky(2001), Troparevsky et al. (2002) reported structures of Cdn (S, Se)n with up to 8 atoms, Deglmann et al. (2002) used density functional theory potential surface investigations to study small CdS, and CdSe clusters, up to the heptamer while CdSe large clusters have been investigated further using large crystal structure sections with up to 198 atoms. Also, Matxain et al. (2004) and Jha et al. (2008) reported ring

18

Transition state structure with 2 degenerated imaginary freqs

P. Karamanis

Displacement patterns of the imaginary Frequencies

Distorted global minimum

Fig. 11 Displacement patterns and ground-state structure of Cd5 Te5

and spherical structures with up to 16 atoms while Sanville et al. (2006) obtained geometries and energies of the neutral and positively charged Mn Xn clusters up to n D 16 (M D Zn, Cd and X D S, Se). Their results showed that one-dimensional structures are favored for some cluster sizes while three-dimensional structures may be envisioned as being built from Cd2 (S, Se)2 and Cd3 (S, Se)3 rings. At this point it is important to stress that contrary to what is reported in several studies about CdX clusters, the five-atom rings are not totally planar. Instead, they maintain distorted envelope structures of cm1 symmetry due to the weakening of the steric repulsion between the metal atoms as the size of the ring increases. As, a result the planar structure is a transition point in each cluster’s potential energy surface characterized by soft imaginary frequencies frequencies of a few cm1 which indicate a very floppy system that is expected to exhibit fluxional behavior (Fig. 11). Lastly, small Cdn Ten and Hgn Ten clusters have attracted the attention of several groups (see for instance Wang et al. 2004, 2008, 2009). Especially HgTe clusters are very interesting since this material is considered as a semimetal with a small negative bandgap around D0.14 eV at 300 K and a semimetal-to-semiconductor can be produced by size quantization making HgTe nano-objects promising candidates as optical amplifiers (Fig. 12).

Electric Polarizabilities and Hyperpolarizabilities of Clusters Definitions and Theory The electric dipole (hyper)polarizabilities describe the response of a molecular system to an external electric field. Briefly, the polarizabilities (or linear polarizabilities) are used to express the ability of the electronic density of a molecular or cluster entity to distort under the influence of weak external fields such as the field generated by a charge in close distance. The hyperpolarizabilities (or nonlinear polarizabilities) describe the same response to laser beams. Large polarizabilities imply easily polarized molecules, whereas, small polarizabilities correspond to molecules in which the electronic density is considerably insensitive

Structures and Electric Properties of Semiconductor clusters

19

Cd X

6-6

10-10

7-7

8-8

9-9

12-12

Fig. 12 Typical cage structures of Cdn Xn (X D S, Se, Te) clusters

to electric field perturbations. Large hyperpolarizabilities involve molecules which can interact with intense electric fields in a “nonlinear” manner and the corresponding materials are known as nonlinear optical materials. The polarizabilities are associated with fundamental chemical concepts such as global softness (Vela and Gázquez 1990), basicity–acidity (Headley 1987), electronegativity (Nagle 1990), and stability (Hohm 2000), (Parr and Chattaraj 1991). On the other hand, the hyperpolarizabilities are closely related to the nonlinear optical behavior of materials in such processes as the second harmonic generation (SHG) (or frequency doubling), the third harmonic generation (THG), the sum frequency generation (SFG), the optical Kerr effect, and others which are expected to find application in a wide range of future technologies (see Bloembergen 1996 and references therein). For clusters, polarizability is one of the few microscopic quantities that are available from experiment (Backer 1997) and is linked both to macroscopic and microscopic features of the matter such as the bulk dielectric constants and chemical bonding, respectively. Hence, this property can provide valuable information about crucial microscopic cluster features and the behavior of low dimensional systems that may be used in nanotechnology applications where nano-objects are subjected in small electric perturbations. On the other hand, given that the macroscopic nonlinear optical properties of materials are governed by the microscopic molecular hyperpolarizabilities, the study of those properties on clusters (Leitsmann et al. 2005) may offer a new ground in the search of new nonlinear optical materials with potential applications in future nanostructure technologies. The nonlinear behavior could be employed for amplification, modulation, and changing the frequency of

20

P. Karamanis

optical signals, similar to how nonlinearities of valves and transistors are used to govern the processes in traditional electric chips and circuits. The microscopic (hyper)polarizabilities are studied by means of the so-called theory of the response functions which is of importance for all molecular and cluster entities (Roman et al. 2006). The most commonly used approach in studying the linear and nonlinear optical properties of clusters is the so-called semiclassical one. According to this approach a classical treatment is used to describe the response of the cluster to an external field (radiation) while the system itself is treated using the lows and techniques of quantum mechanics. This is done by using a Hamiltonian which combines both of the above treatments: 1b 1 b 1 b ˛ˇ F˛ˇ  ‚˛ˇ ı Faˇ ı C    H D H0  b a Fa  ‚ ˛ˇ Faˇ  3 15 105

(1)

H0 is the Hamiltonian which represents the free of the field system and the coefficients denoted with the Greek capital letters stand for the tensors of the dipole quadrupole, octupole, and hexcadecapole moments which are used to implement the field perturbation (F) into the new Hamiltonian. The expectation value after the implementation of this modified operator on the wavefunction is the perturbed energy of the system due to the influence of the field which in the case of a static field F can be written using a Taylor series expansion as 1 1 1 1 ˆ˛ˇ ı Faˇ ı C    aaˇ Fa Fˇ E p D E 0  a Fa  ‚˛ˇ F˛ˇ  ˛ˇ F˛ˇ  3 15 105 2 1 1 1 1  A˛;ˇ Fa;ˇ  C˛ˇ; ı Faˇ; ı  E˛;ˇ ı Fa;ˇ ı C     ˇ˛ˇ Fa Fˇ F 3 6 15 6 1 1  B˛ˇ; ı Fa Fˇ F ı  ˛ˇ ı Fa Fˇ F Fı C    6 24 (2) Ep is the energy of the atomic or molecular system in the presence of the static electric field (F), E0 is its energy in the absence of the field, ˛ corresponds to the permanent dipole moment of the system, ˛ ˛ˇ to the static dipole polarizability tensor and ˇ ˛ˇ ,  ˛ˇ ı to the first and second dipole hyperpolarizabilities, respectively. Greek subscripts denote tensor components and can be equal to x, y, and z and each repeated subscript implies summation over x, y, and z as follows: 1 1 aaˇ Fa Fˇ D 2 2

X

aa;ˇ Fa Fˇ

(3)

˛;ˇDx;y;z

In the case of a weak uniform (homogenous) field, the above expression becomes more simplified:

Structures and Electric Properties of Semiconductor clusters

21

1 1 1 E p D E 0  ˛ F˛  ˛˛ˇ Fa Fˇ  ˇ˛ˇ F˛ Fˇ F  ˛ˇ ı F˛ Fˇ F Fı C    2 6 24 (4) The properties that are routinely computed and discussed are the mean (or average) static dipole polarizability .˛/ ; the anisotropy (4˛) of the polarizability tensor, the vector component of the first hyperpolarizability tensor in the direction of the   ground state permanent dipole moment ˇjj ; the total first-order hyperpolarizability ˇ tot , and the scalar component of the second hyperpolarizability tensor  : Those quantities are related to the experiment and in terms of the Cartesian components are defined as ˛D ˛ D

 1 ˛xx C ˛yy C ˛zz 3

 1=2 h  2 2  1 ˛xx  ˛yy C .˛xx  ˛zz /2 C ˛zz  ˛yy 2  i1=2  2 2 2 C 6 ˛xy C ˛xz C ˛zy   3=5 ˇx x C ˇy y C ˇz z ˇjj D jj

(5)

(6)

(7)

where ˇx D ˇxxx C ˇxyy C ˇxzz ˇy D ˇyxx C ˇyyy C ˇyzz ˇz D ˇzxx C ˇzyy C ˇzzz 1=2  ˇt ot D ˇx2 C ˇy2 C ˇz2

(8)

   D 1=5 xxxx C yyyy C zzzz C 2xxyy C 2yyzz C 2xxzz

(9)

Cluster (Hyper)Polarizabilities: Computational Approach The majority of the undergoing theoretical (hyper)polarizability studies on semiconductor clusters rely on conventional ab-initio and density functional methods (DFT) As it is broadly accepted the ab initio approach leads to accurate (hyper)polarizability predictions and most of the reported attempts rely on a hierarchy of ab initio methods of increasing predictive capability such as the Hartree–Fock

22

P. Karamanis

approximation (HF), the Møller–Plesset (MP) many body perturbation theory, and also on the coupled cluster theory such as singles and doubles coupled-cluster (CCSD), and singles and doubles coupled-cluster with an estimate of connected triple excitations via a perturbational treatment (CCSD(T)) (further reading for those quantum chemical methods can be found in Szabo and Ostlun(1989), and Helgaker et al. (2000). Although this computational strategy is not always straightforward and in most of the cases computationally demanding, leads to reliable predictions of cluster (hyper)polarizabilities giving also crucial and detailed information about the electron correlation effects on the properties of interest. On the other hand the density functional approach (Koch and Holthausen 2000) is by far less computationally costly and allows the treatment of large systems; however, the reliability of DFT results rely heavily on the ability of the functional applied each time to describe the response of the system to an external electric field (see Karamanis et al. 2011 and references therein) Looking carefully at Eq. 4 it becomes obvious that the computation of polarizabilities and hyperpolarizabilities in the case of static fields is in fact a mathematical derivation issue of the cluster energy with respect to the applied field. Consequently, the best accuracy can be achieved by finding and using the analytic expressions of the following relations:  

@2 E @F 2



@3 E  @F 3 

@4 E  @F 4

 D astatic;F D0

(10)

D ˇstatic;F D0

(11)

D static;F D0

(12)





In the case of the linear polarizability, the majority of the available quantum chemical software is able to evaluate analytically the first field derivative of the energy yielding in a straightforward manner all components of the polarizability tensor. In most of the cases the type of the calculation has to be indicated in the command line of the input file and the polarizabilities of the system are returned after the calculation ends. Unfortunately no analytical evaluation of the high-order field derivatives of the energy is available in all commercial packages for all methods. What is more, for methods of high accuracy such as the CC methods, this option is not available even for the linear polarizabilities. To overcome this problem several numerical procedures have been developed. The most applied numerical solution to this mathematical problem is the one provided by the so called finite field approach. According to this approach, first a field perturbation of different magnitudes and along different directions is added to the Hamiltonian. Then the wavefunction is “solved” for each perturbation and the energy of the perturbed

Structures and Electric Properties of Semiconductor clusters

23 300

240 a

b

235 280

Mean dipole polarizability (a.u.)

230

n=5

225

260

Ga6 220

n=4

215 n=3 174

210 n=2 195 As6

172

n=1

190

170 CCSD(T)

CCSD

MP4

MP2

HF

CCSD(T)

CCSD

MP4

MP2

HF

185

Fig. 13 Method performance on the computation of the mean polarizabilities of Gan Asm clusters with m C n D 6 in comparison with the performance on the computation of the mean polarizabilities of Ga6 and As6 clusters

system is obtained. Lastly the differentiation of energy of the system with respect to the applied field is performed numerically and the coefficients of the Eq. 4 are obtained (Figs. 13 and 14). Let us now demonstrate a simple example of the above process which will show us how the Eq. 4 can be utilized in order to calculate all the components of the polarizability tensor for a cluster without symmetry (C1 point group). For this symmetry group there are three independent components of the dipole moment (x , y and z ) and the six for the dipole polarizability: three principal (axial) components ˛ xx , ˛ yy , and ˛ zz and three transversal ones ˛xy D ˛yx ; ˛zy D ˛yz ; and ˛ xz ˛ zx The three principal ones can be computed by applying a weak electric field (of 0.0005–0.005 au strength) along each Cartesian direction, as follows:

24

P. Karamanis Ga As

D3h

D4h

C5h

C2J

Cs

C1

Fig. 14 Gallium arsenic clusters of various symmetries

E .Fa /  E .Fa / 2Fa

(13)

2E.0/  E .Fa /  E .Fa / Fa2

(14)

a 

˛aa 

In the same manner, the transversal components can be obtained by applying fields in each ˛ˇ plane along each /4 diagonal between x, y, and z axis (this is done by applying fields of equal strengths (F˛ DFˇ ) at each xy, yz and xz planes). In this case one obtains the following equation: ˛aˇ

    E .Fa / C E Fˇ  E Fa ; Fˇ  E.0/  Fa Fˇ

(15)

In the case of centrosymmetric structures, the computation of the dipole polarizability becomes more straightforward since the permanent dipole moment and the transversal components of dipole polarizability of those systems are vanishing. In this case we have ˛aa 

E.0/  E .Fa / Fa2

(16)

Provided that a suitable field is chosen, the above computational approach leads to quite reliable polarizability values. The advantage of this approach is that it gives the flexibility to the user to compute the polarizabilities of a given system using a variety of post Hartree–Fock methods only by computing the energies of the free clusters with and without the field. The described approach can be applied to other response properties apart from the perturbed energies provided that the method used for their computations satisfy the Hellmann–Feynman theorem (Feynman 1939; Hellmann 1937) according to which the derivative of the total energy with respect to the field is equivalent with the expectation value of the derivative of the Hamiltonian with respect to the field. In this case, one can safely use the following relationships:

Structures and Electric Properties of Semiconductor clusters



@4 E  @F 4



 D

@3  @F 3





@3 E  @F 3

 



 D 

D

@2 E @F 2

@2 ˛ @F 2

@2  @F 2



 D



 D

  D

@ @F

@ˇ @F

@˛ @F

25

 D static;F D0

(17)

 D ˇstatic;F D0

(18)

 D astatic;F D0

(19)

These relations can be used in all cases where the approximation to the true wavefunction is variationally optimized with respect to the Hamiltonian. This holds in the case of the SCF approach and all the DFT methods in which the energy of the systems is obtained through an iterative process. Contrary, in the case of the finite-order Møller–Pleset perturbation theory, (MP2, MP3, MP4), which is not variational, the relations Eqs. 17–19 do not hold and the calculations of the properties of interest should be extracted using only the energy derivatives. Another key point in (hyper)polarizability calculations of clusters, as in all molecular systems, is the basis set choice. This has been clearly demonstrated in previous basis set studies on semiconductor clusters (see for instance Karamanis et al. 2006a, b, 2007a, 2008a; Maroulis et al. 2003, 2007; Papadopoulos et al. 2005, 2006). In general the polarizabilities are less sensitive to the basis set used than the first and second hyperpolarizabilities (see Fig. 15) Systematic basis set studies on cluster polarizabilities using basis sets of increasing size emphasize the importance of the diffuse s p Gaussian type functions (GTFs) and the effect of GTFs of polarization functions and functions of higher angular momentum such as ffunctions Also it has been illustrated that as the cluster size increases the overall basis set effect on the mean polarizability gradually decreases (Karamanis et al. 2011). Besides the methods and techniques described above there are also some theoretical methods that have been developed to treat the polarizabilities of clusters in a different manner from that described above. One of those schemes has been used by Jackson et al. (2007) and relies on partitioning the volume of a system into atomic volumes and using the charge distributions within the atomic volumes to define quantities such as atomic dipole moments and polarizabilities. In a second step each of the latter quantities are further decomposed into two components: the dipole (or local) and the charge-transfer component. The summation of each component over all of the atoms gives the local and charge-transfer parts of the total system dipole moment and polarizability. This treatment allows one to perform a site-specific analysis of dielectric properties of finite systems and, also, to obtain an insight about the contribution of each atom or of any desired building block of the cluster. The specific method has been applied to silicon clusters with up to 28 atoms which were subjected to a small external electric field. The obtained results showed clearly a strong dependence of the individual atomic polarizabilities, as well as their

P. Karamanis

(Mean polarizability per atom) / e2 a02 Eh–1

26

38

6-31G 6-31+G 6-31+G(d) 6-31+G(2d) 6-31+G(2df) 6-31+G(3df)

36 34 32 30 28 26 24 22 20 2

4

6

8

10 12 14 Number of Atoms

16

18

20

103 × 185 6-31G 6-31+G Mean dipole second hyperpolarizability \ au

180 +d

+2d

+3d

175

+3d2f +3df

170 +3d2f 80 +3df +3d

70

60 +d

+2d

103 × 50

Fig. 15 Basis set effect, at the Hartree–Fock level of theory, on the mean dipole polarizabilities per atom of the ground state structures of aluminum phosphide cluster of the type Aln Pn win n D 2–9, and basis set effect on the mean second hyperpolarizability of Al9 P9

dipole and charge-transfer components, on the site location of the atom within the cluster. For atoms at peripheral sites those properties are substantially larger than for atoms at the interior sites, and the more peripheral atoms are characterized by larger polarizabilities. The same conclusion was drawn later by Krishtal et al. (2010).

Structures and Electric Properties of Semiconductor clusters

27

An alternative methodology which uses electrostatic interaction schemes to predict the evolution of cluster polarizabilities has been developed and tested on silicon clusters by Guillaume et al. (2009). This approach accepts the electrostatic nature of the dominant intra- and intermolecular interactions of a bulk material which is under the influence of external electric fields. This bottom-up approach allows the treatment very large clusters, enabling to reach the convergence of the response of the bulk material up to a point where adding new cluster units in the system would not change significantly the response of the material. Further reading about the computational aspects of (hyper)polarizabilities on clusters and other molecules can be found in the following: Buckingham(1967), McLean and Yoshimine(1967), Marks and Ratner(1995), Kanis et al. (1994), Brédas et al. (1994), Bishop et al. (1997), Xenides(2006), Xenides and Maroulis(2000, 2006), Champagne et al. (2002), Maroulis(2008, 2004, 2003), Maroulis et al. (2007, 2003), Pouchan et al. (2004), Maroulis and Pouchan (2003), Papadopoulos and coworkers (Avramopoulos et al. 2004; Raptis et al. 1999; Reis et al. 2000); Karamanis and coworkers (Hohm et al. 2000; Kurtz et al. 1990; Luis et al. 2000).

Semiconductor Cluster (Hyper)polarizabilities: General Trends and Selected Studies General Trends Semiconductor cluster polarizabilities have been the subject of some very important experimental studies via beam-deflection techniques (Backer 1997; Schlecht et al. 1995; Schnell et al. 2003; Schäfer et al. 1996; Kim et al. 2005) while they have been extensively studied using quantum chemical and density functional theory. In this research realm, one of the areas intensively discussed is the evolution of the cluster’s polarizabilities per atom (PPA) with the cluster size. The PPA is obtained by dividing the mean polarizability of a given system by the number of its atoms. Such property offers a straightforward tool to compare the microscopic polarizability of a given cluster with the polarizability of the bulk (see Fig. 16) as the latter is obtained by the “hard sphere” model with the bulk dielectric constant via the Clausius–Mossotti relation :   3 "1 at ˛D (20) 4 " C 2 where at is the volume per atom in the unit cell of the bulk and " is the static dielectric constant of the bulk. Most of the obtained knowledge on the polarizabilities of semiconductor clusters has emerged as the result of extensive studies of such species as silicon and gallium arsenide which have served well as models for the better understanding of the (hyper)polarizability evolution of semiconductor clusters. Also other kinds of clusters built from different elements such as aluminum phosphide, cadmium sulphide, selenide, and telluride have attracted the attention of several researchers and the

28

P. Karamanis

42 As

(Mean polarizability per atom) / e2 a02 Eh–1

40

Ga

38 36 34 32 30 28

Bulk polarizability

26 24 22

4

6

8

10

Ga2As2

Ga3As3

Ga4As4

Ga5As5

Fig. 16 Comparison between computed polarizabilities of small GaAs clusters and the polarizability of the bulk. The polarizabilities have been computed at the MP2 level of theory

corresponding studies have led to interesting conclusions. Those studies suggest that cluster polarizabilities are very different from the known polarizabilities of bulk materials. In general the PPAs of small semiconductor cluster are larger than the polarizability of the bulk (Fig. 16). For instance, in all studied cases the theoretical predicted polarizabilities of small semiconductor clusters (e.g., Si, AlP, CdSe, CdS, GaAs) are found higher than the bulk polarizability and converge to the bulk limit from above. Also, the polarizability dependence of the cluster size is strong and as a rule cluster species exhibit strong and nonmonotonic size variations. In addition there is a strong dependence of the polarizability on the shape of the cluster. It has been shown that for species which maintain closed cage-like structures (i.e., close to the spherical shape), the polarizability per atom (PPA) decreases with clusters size (Fig. 15) (see for instance (Jackson et al. 2004, 2005; Karamanis et al. 2008b, c; Zhang et al. 2004). On the other hand the per atom polarizabilities of clusters which maintain prolate structures (e.g., tubular clusters) increase as a function of the

Structures and Electric Properties of Semiconductor clusters Lowest energy structures

230 Mean dipole Polarizability / au

29

225 1st low lying isomers

220 215

2nd low lying isomers

210 205 200

3rd low lying isomers

195 190 Ga2As4

Ga3As3

Ga4As2

Fig. 17 Composition-dependent polarizabilities of small Gan Asm with clusters n C m D 6 computed at the B2PLYP/aug-cc-pVDZ-PP level

cluster size (Jackson et al. 2005; Karamanis et al. 2008b, 2011). Another important issue is the dependence of the electric polarizabilities on the cluster bonding. It has been shown that semiconductor clusters which favor structures characterized by covalent bonds are expected to exhibit larger polarizabilities than clusters which tend to form ionic structures This dependence is significantly more intense in the case of the second hyperpolarizability. The general rule that holds in this case is that semiconductor clusters which are characterized by significant electron transfer from the electropositive atoms (ionic bonding) to the electronegative ones are less hyperpolarizable than species in which the electron transfer is smaller (covalent bonding) (Karamanis et al. 2008d; Krishtal et al. 2010). This variation explains the nonmonotonic size variations of the mean PPAs of small binary semiconductor clusters. Also, both polarizabilities and hyperpolarizabilities of clusters have been proven very sensitive to the cluster elemental constitution. For instance, InAs clusters are more (hyper)polarizable than GaAs, AlP, AlAs, InP, and GaP clusters while CdTe clusters, are far more (hyper)polarizable than CdS and CdSe clusters (Karamanis and Pouchan 2009). What is more, the strong correlation between the polarizabilities of semiconductor clusters and their composition is also significant. In the case of GaAs clusters it has been demonstrated both experimentally and theoretically that Ga-rich gallium arsenide clusters are more polarizable than Asrich clusters due to the larger polarizability of Ga (Fig. 17) (Karamanis et al. 2009). Lastly, it was shown that the cluster shape dominates the magnitude of the second hyperpolarizabilities of large clusters (Karamanis and Pouchan 2011).

Selected Studies Polarizabilities Let us now briefly explore some of the most representative studies in the subject of the polarizabilities of Si and GaAs clusters that have attracted considerable

30

P. Karamanis

attention. One of the first studies has been reported by Vasiliev et al. (1997). In that work the authors computed the polarizabilities of small Sin ; Gen .n  10/ and Gan Asm .n C m  8/ clusters and they found that the polarizabilities of all systems are higher than the values estimated from the “hard sphere” model using Eq. 20. Also they evidenced that the computed per atom polarizabilities tend to decrease with increasing the cluster size and they related this trend to the one observed in the case of metallic clusters, for which the polarizability bulk limit is approached from above. This observation was considered as an evidence for the “metallic-like” nature of small semiconductor clusters. Two similar studies about silicon clusters up to 20 atoms by Jackson et al. (1999) and up to 28 atoms by Deng et al. (2000) confirmed the reported trends regarding to the PPA evolution with size and the metallic nature of silicon semiconductor clusters. The same conclusion has been reported also in later studies by Bazterra et al. (2002) and Maroulis et al. (2003). In the case of GaAs clusters the early results reported by Vasiliev et al. (1997) have been confirmed by Torrens(2002) with an interacting-induced-dipoles polarization model and have been extended up to 18 atoms by Karamanis et al. (2007a, 2008). Furthermore, the polarizability ordering in the case of the III–V dimers has been established: In2 As2 > In2 P2 > Al2 As2 > Ga2 As2 > Al2 P2 > Ga2 P2 (Karamanis et al. 2008a). Hyperpolarizabilities In the domain of the nonlinear polarizabilities of semiconductor clusters (i.e., the second dipole hyperpolarizability and third polarizability) the reported studies are fewer than in the case of the linear polarizabilities. However, this field is very promising due to the properties that the compound semiconductors possess. With respect to their interaction with intense electric fields compound semiconductors are characterized by high sensitivity, high speed, and they may operate at a wide variety of wavelengths in the near-infrared range. In particular the compound III–V and II–VI semiconductors such as GaAs, InP, and CdS possess the two properties needed for the existence of the photorefractive effect, the photoconductivity, and the electro-optic effect. These properties are very crucial in electro-optical technology; and alloptical information technology, thus the study of clusters and nano-objects of those materials are essential, first for better understanding the properties of the bulk material and second for the research of new nano-materials with improved nonlinear optical properties (Bechstedt et al. 2007; Adolph and Bechstedt 1998; Hughes and Sipe 1996; Bergfeld and Daum 2003; Bloembergen 1996; Butcher and Cotter 1990; Powell et al. 2002; Vijayalakshmi et al. 2002). At the microscopic level, one of the first studies into this matter was reported by Korambath and Karna(2000) who studied the first dipole hyperpolarizabilities of GaN, GaP, and GaAs clusters within the ab initio time-dependent Hartree– Fock method using an even-tempered Gaussian basis set and demonstrated that the magnitudes of the calculated (hyper)polarizabilities depend on the size of the cluster. Lan et al. (2003) studied the hyperpolarizabilities of Ga3 As3 , Ga3 Sb3 In3 P3 , and In3 Sb3 clusters using a similar approach based on the time-dependent Hartree– Fock (TDHF) formalism combined with sum-over-states (SOS) method and found that the charge transferred from the - bonding to - antibonding orbitals between

Structures and Electric Properties of Semiconductor clusters

31

III and V group atoms under the influence of an electric field make a significant contribution to the second-order polarizability. The first ab initio attempt in treating the hyperpolarizabilities of semiconductor using methods of high sophistication such as CC and MP has been reported by Maroulis and Pouchan who explored the second hyperpolarizabilities of cadmium sulfide clusters. Their results showed that the mean second hyperpolarizability per atom of those species decreases rapidly with the cluster size. A second attempt on systems of increasing size has been reported by Lan et al. (2006). These authors studied several small GaAs clusters up to 10 atoms within the time-dependent density functional theory and reported that the dynamic behavior of ˇ and  show that the small Gan Asm clusters are expected as good candidates for nonlinear optical properties due to the avoidance of linear resonance photoabsorption. An application of the time-dependent DFT (TD-DFT) theory on the hyperpolarizabilities of clusters can be found in the work of Sen and Chakrabarti(2006) who studied the second hyperpolarizabilities of CdSe clusters. In their conclusions, it is highlighted that these clusters have a high nonlinear optical activity. Karamanis et al. (2007a) reported ab initio values at the HF, MP2 level of theory for stoichiometric GaAs clusters up to 18 atoms and showed that as the size of the cluster increases the per atom mean second hyperpolarizability decreases fast and converges to a specific value. Finally, the significance of the bonding on the nonlinear polarizabilities of small clusters has been demonstrated by Karamanis and Leszczynski(2008) in a hyperpolarizability study of stoichiometric aluminum phosphide clusters up to 18 atoms. The conclusions of investigations on aluminum phosphide clusters have been extended later to several types of III–V semiconductor in a comparative hyperpolarizability study by Karamanis et al. (2008a). In that study the relative mean static dipole hyperpolarizabilities of the X2 Y2 type (X D Al, Ga, In and Y D P, As) clusters have been established in several levels of theory,  W In2 As2 > In2 P2 Š Al2 As2 > Al2 P2 Š Ga2 As2 > Ga2 P2 : This classification follows the size ordering of the electropositive atoms. Ga is slightly smaller than Al, while In is far more larger than both Ga and Al. The atomic radius of Ga(3d10 as24 p1 ) is slightly smaller than the Al(3s2 3p1 ) atomic radius, despite its larger number of electrons. This is caused by the insufficient shielding that the 3d10 electrons provide to the 4s2 4p1 valence electrons of Ga. As a result, the valence electrons of Ga experience a larger attraction from the nuclear charge than the electrons of Al. Considering that the (hyper)polarizabilities are valence-related properties, the described peculiarity of the Ga atoms can explain qualitatively the smaller hyperpolarizabilities of both Ga2 P2 and Ga2 As2 compared to Al2 P2 and Al2 As2 clusters. The dynamic-second order hyperpolarizabilities of Si3 and Si4 have been explored by Lan et al. (2008) using the coupled cluster cubic response theory. These authors reported that Si3 and Si4 clusters exhibit wide non-resonant optical region. The same approach has been used by Lan and Feng(2009) in the study of the nonlinear response of another class of semiconductor clusters, namely, the SiCn and Sin C In this work it was demonstrated that the size dependence of the first-order hyperpolarizabilities of the SiCn clusters, which have approximate Siterminated linear chain geometry, is similar to that observed in -conjugated organic

32

P. Karamanis

Mean se cond hy perpola rizability

14 0 13 0 12 0 11 10 0 0 90 80 70 60 50 40 30 20 10 0

Cd9X9 X = Te

Cd3X3 X = Se X=S Cd2X2

Fig. 18 Comparisons between the second-order mean hyperpolarizabilities (divided by 103 ) of CdS, CdSe, and CdTe clusters computed at the HF/aug-cc-pVDZ-PP level of theory

molecules. Lastly concerning the (hyper)polarizabilities of other II–VI clusters apart from CdS there two reported studies on their hyperpolarizabilities. The first (Li et al. 2008) deals with the ground states and low-lying isomers of stoichiometric (ZnO)n clusters with n D 2–12 using finite field approach within density functional theory framework and the second (Karamanis et al. 2009) involves a comparative study of CdS, CdSe, CdTe with respect to their second hyperpolarizabilities (Fig. 18). Both of those studies demonstrate the structural and size dependence of those properties while the second one reveal that CdTe clusters are far more (hyper)polarizable than CdS and CdSe, verifying previous experimental work reported by Wu et al. (2003) which provided evidences that that CdTe nanoparticles exhibit stronger nonlinear optical properties than nanoparticles built from CdS and CdSe.

Concluding Remarks In this brief review, we made an attempt to present some of the most illustrative computational investigations of the structural properties of semiconductor clusters. We have chosen to present studies that had as subject cluster species which have attracted most of the attention of the researchers the last two decades and have served as models in the better understanding of cluster properties. Therefore it must

Structures and Electric Properties of Semiconductor clusters

33

be emphasized that although this chapter does not cover the vast number of studies that have been reported during the previous years, it can be utilized as starting material for those who are interested in contributing in this field, or simply, as a guide for those who wish to get some general information about the structural and bonding features of semiconductor clusters. Finally, in this chapter we have revisited most of the recent studies that focus on the electric properties of those species which also demonstrate in a comprehensive manner that clusters and electric properties such as the dipole polarizabilities and especially hyperpolarizabilities are strongly related.

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Xenides, D., & Maroulis, G. (2000). Basis set and electron correlation effects on the first and second static hyperpolarizability of SO2. Chemical Physics Letters, 319(5–6), 618–624. Xenides, D., & Maroulis, G. (2006). Electric polarizability and hyperpolarizability of BrCl(X1†C) Journal of Physics B: Atomic. Molecular and Optical Physics, 39(17), 3629– 3638. Xiao, C., Hagelberg, F., & Lester, W. A., Jr. (2002). Geometric, energetic, and bonding properties of neutral and charged copper-doped silicon clusters. Physical Review B - Condensed Matter and Materials Physics, 66(7), 754251–7542523. Yoo, S., & Zeng, X. C. (2005). Structures and stability of medium-sized silicon clusters. III Reexamination of motif transition in growth pattern from Si15 to Si20. Journal of Chemical Physics, 123(16), 1–6. Yoo, S., & Zeng, X. C. (2006). Structures and relative stability of medium-sized silicon clusters. IV. motif-based low-lying clusters Si21Si30. Journal of Chemical Physics, 124(5), 1–6. Yoo, S., Zhao, J., Wang, J., & Xiao, C. Z. (2004). Endohedral silicon fullerenes Sin (27  n  39). Journal of the American Chemical Society, 126(42), 13845–13849. Yoo, S., Shao, N., & Zeng, X. C. (2008). Structures and relative stability of medium- and largesized silicon clusters. VI. Fullerene cage motifs for low-lying clusters Si39 , Si40 , Si50 , Si60 , Si70 and Si80 . Journal of Chemical Physics, 128(10), 104316/ 1–104316/9. Yu, D. K., Zhang, R. Q., & Lee, S. T. (2002). Structural transition in nanosized silicon clusters. Physical Review B - Condensed Matter and Materials Physics, 65(24), 2454171–2454176. Zdetsis, A. D. (2001). The real structure of the Si6 cluster. Physical Review A. Atomic, Molecular, and Optical Physics, 64(2), 023202/1–023202/4. Zdetsis, A. D. (2007a). Analogy of silicon clusters with deltahedral boranes: how far can it go? reexamining the structure of sin and sin 2-, n D 5–13 clusters. Journal of Chemical Physics, 127(24), 244308/1–244308/6. Zdetsis, A. D. (2007b). Fluxional and aromatic behavior in small magic silicon clusters: a full ab initio study of Sin, Si1n, Si2n, and Si1Cn n D 6, 10 clusters. Journal of Chemical Physics, 127(1), 014314/1–014314/10. Zdetsis, A. D. (2008). High-stability hydrogenated silicon-carbon clusters: a full study of Si2C2H2 in comparison to Si2C 2, C2B2H4, and other similar species. Journal of Physical Chemistry A, 112(25), 5712–5719. Zdetsis, A. D. (2009). Silicon-bismuth and germanium-bismuth clusters of high stability. Journal of Physical Chemistry A, 113(44), 12079–12087. Zhang, D. Y., Bégué, D., & Pouchan, C. (2004). Density functional theory studies of correlations between structure, binding energy, and dipole polarizability in Si9 Si12. Chemical Physics Letters, 398(4–6), 283–286. Zhao, W., & Cao, P.-L. (2001). Study of the stable structures of Ga6 As6 cluster using FP-LMTO MD method. Physics Letters, Section A: General, Atomic and Solid State Physics, 288(1), 53–57. Zhao, W., Cao, P.-L., Li, B.-X., Song, B., & Nakamatsu, H. (2000). Study of the stable structures of Ga4 As4 cluster using FP-LMTO MD method. Physical Review B - Condensed Matter and Materials Physics, 62(24), 17138–17143. Zhao, J., Xie, R.-R., Zhou, X., Chen, X., & Lu, W. (2006). Formation of stable fullerenelike Gan Asn clusters (6  n  9): gradient-corrected density-functional theory and a genetic global optimization approach. Physical Review B - Condensed Matter and Materials Physics, 74(3), 035319/1–035319/2. Zhou, R. L., & Pan, B. C. (2008). Low-lying isomers of SiC n and Si n (n D 31–50) clusters. Journal of Chemical Physics, 128(23), 234302/1–234302/6. Zhu, X., & Zeng, X. C. (2003). Structures and stabilities of small silicon clusters: ab initio molecular-orbital calculations of Si7  Si11. Journal of Chemical Physics, 118(8), 3558–3570. Zhu, X. L., Zeng, X. C., Lei, Y. A., & Pan, B. (2004). Structures and stability of medium silicon clusters. II Ab initio molecular orbital calculations of Si12 Si20. Journal of Chemical Physics, 120(19), 8985–8995.

Structures, Energetics, and Spectroscopic Fingerprints of Water Clusters n D 2–24 Soohaeng Yoo and Sotiris S. Xantheas

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models of Intermolecular Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Models from Electronic Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Minimum Structures of then D 2–10 Water Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Global Minima of Medium-Sized Water Clusters in he Range 11  n  16 . . . . . . . . . . . . . . . The Transition from “All-Surface” to “Internally Solvated” Clusters at n D 17 . . . . . . . . . . . . . . . . Validation from Electronic Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectroscopic Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Family of Minima for (H2 O)20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Four Major Families of Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibrational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Few Water Cages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Pentagonal Dodecahedron (D-cage)(H2 O)20 Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Tetrakaidecahedron (T-Cage)(H2 O)24 Cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 8 14 17 18 19 19 21 22 22 22 24 27 28 30

Abstract

This chapter discusses the structures, energetics, and vibrational spectra of the first few (n  24) water clusters obtained from high-level electronic structure calculations. The results are discussed in the perspective of being used to parameterize/assess the accuracy of classical and quantum force fields for water. To this end, a general introduction with the classification of those force fields is presented. Several low-lying families of minima for the medium cluster sizes S. Yoo () • S.S. Xantheas Chemical & Material Sciences Division, Pacific Northwest National Laboratory, Richland, WA, USA e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_21-2

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S. Yoo and S.S. Xantheas

are considered. The transition from the “all surface” to the “fully coordinated” cluster structures occurring at n D 17 and its spectroscopic signature is presented. The various families of minima for n D 20 are discussed together with the lowenergy networks of the pentagonal dodecahedron (H2 O)20 water cage. Finally, the low-energy networks of the tetrakaidecahedron (T-cage) (H2 O)24 cluster are shown and their significance in the construction of periodic lattices of structure I (sI) of the hydrate lattices is discussed.

Introduction Water’s function as a universal solvent and its role in mediating several biological functions that are responsible for sustaining life has created tremendous interest in the understanding of its structure at the molecular level (Ball 2008). Due to the size of the simulation cells and the sampling time needed to compute many macroscopic properties, most of the initial simulations are performed using a classical force field, whereas several processes that involve chemistry are subsequently probed with electronic structure based methods. A significant effort has therefore been devoted toward the development of classical force fields for water (Robinson et al. 1996). Clusters of water molecules are useful in probing the intermolecular interactions at the microscopic level as well as providing information about the subtle energy differences that are associated with different bonding arrangements within a hydrogen-bonded network. They moreover render a quantitative picture of the nature and magnitude of the various components of the intermolecular interactions such as exchange, dispersion, induction, etc. They can finally serve as a vehicle for the study of the convergence of properties with increasing size. Over the last decade, there have been tremendous advances in the experimental (Cruzan et al. 1996a, b, 1997; Liu et al. 1994, 1996a, b, 1997a, b; Pugliano and Saykally 1992; Suzuki and Blake 1994; Viant et al. 1997) and theoretical (Xantheas 1994, 1996a; Xantheas and Dunning 1993a, b) studies of the structural, spectral, and energetic properties of water clusters, as well as refinements of the macroscopic structural experimental data for liquid water (Hura et al. 2000; Soper 2000; Sorenson et al. 2000). These developments have created a unique opportunity for incorporating molecular level information into classical force fields for water as well as using the available database of cluster energetics for the assessment of their accuracy. The process of developing interaction potentials is by no means straightforward, and the different approaches that are used to model the various components of the underlying physical interactions have their own advantages and shortcomings. Some of the outstanding issues that are associated with the development of models based on molecular level information are the shortage or scarcity of experimental data for clusters. To this end, an alternative approach based on the use of ab initio results for water clusters can be put into practice. Assuming that the route of parameterization of an interaction potential for water from the cluster results is adopted, the question naturally arises of the level of accuracy that needs to be attained so the models can produce meaningful and

Structures, Energetics, and Spectroscopic Fingerprints of Water Clusters n D 2–24

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accurate results for the macroscopic properties of extended systems such as the bulk liquid and ice. In the subsequent sections, we will present the results of ab-initio electronic structure calculations for cluster sizes up to 24 water molecules and we will address some important features of their structural and spectral properties. We will finally present results on the first few water cages that are the constituents of the three-dimensional hydrate networks.

Models of Intermolecular Interactions Intermolecular interactionsForce fields describing intermolecular interactions can be in general classified into two categories: (1) classical and (2) quantum models, the characteristics of which are outlined in Fig. 1. Classical potentials consist of force fields that employ an analytic expression, in conjunction with several adjustable parameters, for the potential energy surface (PES) as a function of the coordinates of the atoms in the system. In contrast, for quantum models, the PES is computed by considering both the position of the nuclei and the distribution of the electrons of the system, and it is obtained by solving Schrödinger’s equation. The choice of the adjustable parameters used in conjunction with classical potentials can result to either effective potentials that implicitly include the nuclear quantization and can therefore be used in conjunction with classical simulations (albeit only for the conditions they were parameterized for) or transferable ones that attempt to best approximate the Born–Oppenheimer PES and should be used in nuclear quantum statistical simulations. Representative examples of effective force fields for water consist of TIP4P (Jorgensen et al. 1983), SPC/E (Berendsen et al. 1987) (pair-wise additive), and Dang-Chang (DC) (Dang and Chang 1997) (polarizable, many-body). The “polarizable” potentials contain – in addition to the pairwise additive term – a classical induction (polarization) term that explicitly

Quantum

Classical

˜) (r˜ ; R

No electronic degrees of freedom

Wave function Energy Potential energy surface Zero-point energy Simulation methodology Transferability across different environments §

E =


YES

“Effective” Implicitly included Classical

Quantum

NO§

NO¶

Born-Oppenheimer Explicitly included Classical Quantum

NO

appropriate only for the environment that the ZPE has been effectively included

¶ double-counting

Fig. 1 Characteristics of classical and quantum potential models

YES

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(albeit approximately) accounts for many-body effects to infinite order. These effective potentials are fitted to reproduce bulk-phase experimental data (i.e., the enthalpy at T D 298 K and the radial distribution functions at ambient conditions) in classical molecular dynamics simulations of liquid water. Despite their simplicity, these models describe some experimental properties of liquid water at ambient conditions quite well but usually fail for conditions and environments that are not parameterized for, and so they have limited predictive power. On the other hand, force fields such as the Effective Fragment Potential (EFP) (Adamovic et al. 2003; Day et al. 1996; Gordon et al. 2001; Jensen and Gordon 1998; Netzloff and Gordon 2004), the SAPT-5 (Bukowski et al. 2007; Szalewicz et al. 2009), and the family of Thole-type models (TTM (Burnham et al. 1999), TTM-2F (Burnham and Xantheas 2002a), TTM2.1-F (Fanourgakis and Xantheas 2006), TTM3-F (Fanourgakis and Xantheas 2008), and TTM4-F (Burnham et al. 2008)) are examples of transferable potentials that are fitted to high-level first principles electronic structure results of the PESs of small water clusters, usually the water dimer and trimer, and should be used in conjunction with nuclear quantum statistical simulations. The availability of energetic information for water clusters (such as binding energies) is of paramount importance in both fitting transferable potentials as well as assessing their performance. Currently, binding energies of small water clusters, even the water dimer, are not available experimentally but can only be obtained from high-level electronic structure calculations.

Quantum Models from Electronic Structure Calculations Figure 2 shows the spectrum of methods that are used to describe intermolecular interactions ranging from the classical descriptions, such as Molecular Mechanics (MM), up to the highly correlated ones, including Coupled Cluster with Single,

Classical to Highly Correlated ...

MM

MD

QM/MM

DFT

HF

MP(2-4)

MCSCF

CCSD(T)

MRSDCI

...

Scaling O(N7)

O(N) Accuracy 100 kcal

0.1 kcal

Size 107 atoms

10 atoms

20 000 bf 1 000 000 atoms

5 000 atoms

6 000 bf 500 atoms

1 000 bf 100 atoms

Fig. 2 Models of intermolecular interactions: from classical to highly correlated

500 bf 10 atoms

Structures, Energetics, and Spectroscopic Fingerprints of Water Clusters n D 2–24

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Double, and perturbative estimates of Triple excitations [CCSD(T)] and Multi Reference Singles and Doubles Configuration Interaction (MRSDCI). Methods on the left part of the spectrum have the advantage of low formal scaling and as such allow for the treatment of large systems, but they are also associated with large inaccuracies. In contrast, methods to the right side of Fig. 2 have the disadvantage of high formal scaling (as much as  O(N9 ), where N is the system size) and are more appropriate for smaller systems for which high accuracy is achieved. Currently, the state of the art consists of the application of highly correlated methods such as CCSD(T) to a cluster of about 24 water molecules (1,100 basis functions and 270 electrons), a calculation that can sustain a performance of 1.4 Petaflop/s (Apra et al. 2009).

Electron Correlation and Orbital Basis Set The cluster energies are obtained from the solution of the nonrelativistic Schrödinger equation for each system. The expansion of the trial many-electron wave function delineates the level of theory (description of electron correlation), whereas the description of the constituent orbitals is associated with the choice of the orbital basis set. A recent review (Dunning 2000) outlines a path, which is based on hierarchical approaches in this double expansion in order to ensure convergence of both the correlation and basis set problems. It also describes the application of these hierarchical approaches to various chemical systems that are associated with very diverse bonding characteristics, such as covalent bonds, hydrogen bonds and weakly bound clusters. As regards the description of the electron correlation problem it has been recognized that the coupled cluster method (Bartlett and Purvis 1978; Cizek 1966, 1969; Coester 1958; Coester and Kümmel 1960; Kucharski and Bartlett 1992; Purvis and Bartlett 1982), which includes all possible single, double, triple, etc. excitations from a reference wave function, represents a viable route toward obtaining accurate energetics for hydrogen-bonded dimers (Halkier et al. 1999; Peterson and Dunning 1995). Among its variations, the CCSD(T) approximation (Raghavachari et al. 1989, 1990), which includes the effects of triple excitations perturbationally, represents an excellent compromise between accuracy and computational efficiency and has been recently been coined as the “gold standard of quantum chemistry.” The CCSD(T) approximation scales as N6 for the iterative solution of the CCSD part, plus an additional single N7 step for the perturbative estimate of the triple excitations. This represents a substantial additional expense over the widely used second-order perturbation level (Møller and Plesset 1934) of theory (MP2) which formally scales as N5 . For the basis set expansion, the correlation-consistent (cc-pV nZ) orbital basis sets of Dunning and coworkers (Dunning 1989; Dunning et al. 1998; Kendall et al. 1992), ranging from double to quintuple zeta quality (n D D, T, Q, 5), offer a systematic path in approaching the complete basis set (CBS) limit. These sets were constructed by grouping together all basis functions that contribute roughly equal amounts to the correlation energy of the atomic ground states. In this approach, functions are added to the basis sets in shells. The sets approach the complete basis

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set (CBS) limit, for each succeeding set in the series provides an evermore accurate description of both the atomic radial and angular spaces. The extension of those sets to include additional diffuse functions for each angular momentum function present in the standard basis yields the augmentedcorrelation consistent (aug-cc-pV nZ) sets. The exponents of those additional diffuse functions were optimized for the corresponding negative ions.

The Basis Set Superposition Error Correction The use of a finite basis set in electronic structure calculations as been known to overestimate the interaction energy (Clementi 1967) (E) of a weakly bound dimer complex, a quantity defined as ˛[ˇ

ˇ

E D EAB .AB/  EA˛ .A/  EB .B/;

(1)

where superscripts indicate the basis set; subscripts, the molecular system; and parentheses, the geometries for which the energies are obtained. In this notation, ˛[ˇ the term EAB .AB/ is the energy of the dimer (AB) at its equilibrium geometry AB with the basis set of the two fragments aUb. The overestimation comes from the fact that the basis functions that are centered on the one fragment help lower the energy of the other fragment and vice versa, a situation first termed “Basis Set Superposition Error” (BSSE) by Liu and McLean (1973). Boys and Bernardi (1970) originally proposed two approaches to circumvent this problem, introducing corrections via the “point counterpoise” (pCP) and the “function counterpoise” (fCP) methods, the latter being nowadays almost exclusively adopted in electronic structure calculations of intermolecular interaction energies (Chalasinski and Szczesniak 1994; van Duijneveldt et al. 1994). For a dimer of two interacting moieties (extension to larger clusters is straightforward), the fCP correction is: ˛[ˇ

˛[ˇ

˛[ˇ

E .fCP/ D EAB .AB/  EAB .A/  EAB .B/ :

(2)

Equations 1 and 2 will not converge to the same result as the basis set increases toward the Complete Basis Set (CBS) limit since the reference energies of fragments A and B are computed at different geometries, viz. the dimer vs. the isolated fragments. Alternatively, if the BSSE correction is estimated via the equation (Xantheas 1996b) ˛[ˇ

˛[ˇ

˛[ˇ

ˇ

E .BSSE/ D EAB .AB/  EAB .A/  EAB .B/ C E˛rel .A/ C Erel .B/

(3)

where ˛ ˛ Erel .A/ D EAB .A/  EA˛ .A/

ˇ

ˇ

ˇ

Erel .B/ D EAB .A/  EB .B/

(4)

(5)

Structures, Energetics, and Spectroscopic Fingerprints of Water Clusters n D 2–24

7

represent the energy penalty for distorting the fragments from their isolated geometries to the ones that adopt due to their interaction in the complex, then it is readily seen that after substitution of Eqs. 4 and 5 into Eq. 3 and collecting terms, the BSSE correction (Eq. 3) can be cast as: o n o n ˛[ˇ ˛[ˇ ˇ E .BSSE/ D E  EAB .A/  E˛AB .A/  EAB .B/  EAB .B/ :

(6)

Equations 1 and 6 do converge to the same result at the CBS limit since the terms in the brackets in Eq. 6 will numerically approach zero as the basis sets ’ and “ tend toward CBS, viz. lim

˛;ˇ!CBS

E .BSSE/ D

lim

˛;ˇ!CBS

E:

(7)

The importance of this correction, although previously recognized (Emsley et al. 1978; Smit et al. 1978; van Lenthe et al. 1987), has rarely been applied (Eggenberger et al. 1991; Kendall et al. 1989; Leclercq et al. 1983; Mayer and Surjan 1992; van Duijneveldt-van de Rijdt and van Duijneveldt 1992) until the problems arising from omitting it, especially for (1) strongly bound hydrogen-bonded complexes exhibiting large fragment relaxations and (2) calculations employing large basis sets, were noted. It should be emphasized that for some strongly hydrogen-bonded dimers such as F (H2 O) and OH (H2 O), the energy penalty term is quite large ( 5 kcal/mol) and its omission will lead to erroneous results as regards the magnitude and convergence of the uncorrected and BSSE-corrected values of the interaction energy as the basis set increases. This large energy penalty is caused by the strong interaction between the ion and water, a fact that results in a large elongation of the hydrogen-bonded OH bond causing the water molecule to adopt a configuration that is far from its isolated equilibrium geometry. A further consequence of this effect is the large red shift in the corresponding OH stretching vibration, a fact that has been experimentally verified.

Extrapolation to the Complete Basis Set Limit Application of the family of these sets to a variety of chemical systems (see Reference Dunning (2000) and references therein) – ranging from the very weakly bound (by around 0.01 kcal/mol) rare gas diatomics, to intermediate strength hydrogenbonded neutral (2–5 kcal/mol), singly charged ion-water (10–30 kcal/mol) clusters, and single-bond diatomics (50–100 kcal/mol), to very strong (>200 kcal/mol) multiply charged metal–water clusters and multiple bond diatomics – has permitted a heuristic extrapolation of the computed electronic energies and energy differences to the CBS limit. Among the various approaches that have been proposed (Bunge 1970; Fast et al. 1999; Feller 1992; Halkier et al. 1999; Klopper 1995; Martin 1996; Termath et al. 1991; Wilson and Dunning 1997; Xantheas and Dunning 1993c) in order to arrive at the CBS limit, we have relied on the following two: 1. A polynomial with inverse powers of 4 and 5 (4–5 polynomial):

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S. Yoo and S.S. Xantheas

E D ECBS C =.`max C 1/4 C .`max C 1/5

(8)

where `max is the value of the highest angular momentum function in the basis set and 2. An exponential dependence on the cardinal number of the basis set n (n D 2, 3, 4, 5 for the sets of double through quintuple zeta quality, respectively): E D ECBS C ˛  exp .ˇ  n/ :

(9)

It should be noted that in nearly every case, this “extrapolation” procedure only accounts for a very small change when compared to the “best” computed quantity with the largest basis set [usually the (aug)-cc-pV5Z or, computer resources permitting, the (aug)-cc-pV6Z]. This result suggests that effective convergence of the respective properties (such as structure and relative energies) has already been achieved with the largest basis sets of this family.

Global Minimum Structures of then D 2–10 Water Clusters Figure 3 shows the global minimum structures of the water clusters n D 2–5 and Fig. 4 shows the variation of their uncorrected and BSSE-corrected binding energies from MP2 calculations with the correlation-consistent basis sets (Xantheas et al. 2002). The CBS limit of 15.8 kcal/mol for the cyclic trimer was estimated from MP2 binding energies with basis sets up to aug-cc-pV5Z. It is worth noting that extrapolating the uncorrected and BSSE-corrected binding energies using Eqs. 8 and 9 yields CBS limits that are within 0.01 kcal/mol from each other, while the extrapolation process itself results in minimal ( 21) clusters. They can be used in conjunction with new approaches for the efficient sampling of the configuration space in order to obtain the low-lying isomers of clusters in this size regime. The structures of the small- and medium-sized water clusters are essential in order to assess the accuracy of those force fields in developing transferable models that can be used in different environments such as liquid water, ice, at aqueous interfaces, or around charged species or hydrophobic surfaces. Given the fact that accurate energetics for the interaction even between two water molecules is not currently available experimentally, the use of high-level first principles electronic structure calculations to obtain the cluster energetics represents an indispensable and currently irreplaceable path toward understanding aqueous environments.

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Acknowledgments This work was supported by the Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Sciences, U.S. Department of Energy. Battelle operates the Pacific Northwest National Laboratory for the U.S. Department of Energy. This research was performed in part using the Molecular Science Computing Facility (MSCF) in the Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of Energy’s Office of Biological and Environmental Research. Additional computer resources were provided by the Office of Basic Energy Sciences, US Department of Energy at the National Energy Research Scientific Computing Center, a U.S. Department of Energy’s Office of Science user facility at Lawrence Berkeley National Laboratory.

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Fundamental Structural, Electronic, and Chemical Properties of Carbon Nanostructures: Graphene, Fullerenes, Carbon Nanotubes, and Their Derivatives Tandabany C. Dinadayalane and Jerzy Leszczynski

Contents Introduction to Carbon Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natural Abundance of Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fullerene Nano-capsules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolated Pentagon Rule (IPR) in Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Defects in Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon Nanotubes (CNTs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Defects in Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Approaches Used to Study Carbon Nanostructures: An Overview . . . . . . . . . . . . . Structural, Electronic, and Chemical Properties of Graphene, Fullerenes, and SWCNTs . . . . . . Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogenation of Graphene With and Without Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giant Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Strain in Curved Polycyclic Systems: POAV and Pyramidalization Angle . . . . . . . . . . . . Stone–Wales Defect in C60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Studies on Vacancy Defects in Fullerene C60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Studies of Single-Walled Carbon Nanotubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Covalent Functionalization of SWCNTs: H and F Atom Chemisorptions . . . . . . . . . . . . . . . . . . . . Theoretical Studies on Common Defects in SWCNTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stone–Wales Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Ring Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single- and Di-vacancy Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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T.C. Dinadayalane () Department of Chemistry, Clark Atlanta University, Atlanta, GA, USA e-mail: [email protected] J. Leszczynski Interdisciplinary Center for Nanotoxicity, Department of Chemistry and Biochemistry, Jackson State University, Jackson, MS, USA e-mail: [email protected] © Springer Science+Business Media Dordrecht 2016 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_22-2

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Outlook of Potential Applications of Carbon Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter provides information on various carbon allotropes and in-depth details of structural, electronic, and chemical properties of graphene, fullerenes, and single-walled carbon nanotubes (SWCNTs). We have written an overview of different computational methods that were employed to understand various properties of carbon nanostructures. Importance of application of computational methods in exploring different sizes of fullerenes and their isomers is given. The concept of isolated pentagon rule (IPR) in fullerene chemistry has been revealed. The computational and experimental studies involving Stone–Wales (SW) and vacancy defects in fullerene structures are discussed in this chapter. The relationship between the local curvature and the reactivity of the defect-free and defective fullerene and single-walled carbon nanotubes has been revealed. We reviewed the influence of different defects in graphene on hydrogen addition. The viability of hydrogen and fluorine atom additions on the external surface of the SWCNTs is revealed using computational techniques. We have briefly pointed out the current utilization of carbon nanostructures and their potential applications.

Introduction to Carbon Nanostructures Carbon is one of the first few elements known in antiquity. The pure forms of this element include diamond and graphite, which have been known for few thousand years (http://www.nndc.bnl.gov/content/elements.html; Pierson 1993; Wikipedia – http://en.wikipedia.org/wiki/Carbon). Both of these materials are of immense importance in industry and in everyday life. Diamond and graphite are termed as giant structures since, by means of a powerful microscope, one could see millions and millions of atoms, all connected together in a regular array. Diamond would appear as a rigid and rather complex system like some enormous scaffolding construction. Carbon is also the major atomic building block for life. All lifeforms on Earth have carbon central to their composition. More than 10 million carbon-containing compounds are known. Compounds containing only carbon atoms, particularly nano-sized materials, are intriguing and attract attention of scientists working in various disciplines. Before 1985, scientists deemed that there were only three allotropes of carbon, namely, diamond, graphite, and amorphous carbon such as soot and charcoal. Soccer ball-shaped molecule comprising of 60 carbon atoms, C60 buckyball named fullerene, was discovered in 1985, and it is another interesting carbon allotrope (Kroto et al. 1985). Carbon nanotubes (CNTs), a spin-off product of fullerene, were reported in 1991 by Iijima (1991). Important well-known carbon materials are depicted in Fig. 1. The publication of transmission

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

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CARBON

Diamond Soot

Graphite

Fullerene

Fig. 1 Well-known carbon materials

electron microscope (TEM) images of CNTs by Iijima was a critical factor in convincing a broad community that “there is plenty of room at the bottom” and many new structures can exist at the nanoscale. Figure 2 shows eight allotropes of carbon. In addition to graphene, fullerenes, and carbon nanotubes, there are few other uncommon carbon nanostructures such as nanohorns (Iijima et al. 1999; Poonjarernsilp et al. 2009), nano-onions (Palkar et al. 2008; Zhou et al. 2009), nanobuds (He and Pan 2009; Nasibulin et al. 2007a, b; Wu and Zeng 2009), peapods (Launois et al. 2010; Li et al. 2009a; Smith et al. 1998), nanocups (Chun et al. 2009), and nanotori (Liu et al. 1997; Sano et al. 2001). We performed a quick search in SciFinder on “fullerene,” “carbon nanotubes,” and “graphene” to reveal their importance and growth in current science, engineering, and technology. We received a total of nearly 60,000 references for the word “fullerene,” 145,000 references for “carbon nanotubes,” and 110,000 for “graphene” when we searched these topics in November 2015. This is indicative that carbon nanomaterials have gained a momentum with the development of nanotechnology as the driving force of the modern science and engineering. Among various carbon nanostructures, CNTs play a special role in the nanotechnology era. The design and discovery of new materials is always exciting for the potential of new applications and properties (Cohen 1993; Serra et al. 1999). In this chapter, we aim to present an overview of carbon nanostructures, with a particular interest on structural, electronic, and chemical properties of graphene, fullerenes, and carbon nanotubes. Important topological defects in the graphene, fullerenes, and carbon nanotubes will be delineated. Thus, this chapter is intended to be an informative guide of carbon nanostructures and to provide description of current computational

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Fig. 2 Eight allotropes of carbon: (a) diamond, (b) graphite, (c) lonsdaleite, (d) C60 (buckminsterfullerene or buckyball), (e) C540 , (f) C70 , (g) amorphous carbon, and (h) single-walled carbon nanotube or buckytube (The picture adopted from Wikipedia – http://en.wikipedia.org/wiki/ Allotropes_of_carbon)

chemistry applications involving these species to facilitate the pursuit of both newcomers to this field and experienced researchers in this rapidly emerging area.

Graphene Carbon displays a unique feature of making a chemically stable two-dimensional (2D), one-atom-thick membrane called graphene in a three-dimensional (3D) world. Each carbon atom in graphene is covalently bonded to three other carbon atoms

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

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Graphene

Graphite

C60

Carbon nanotube

Fig. 3 Carbon-containing molecules (graphite, buckminsterfullerene (C60 ), and carbon nanotube) derived from graphene

with sp2 hybridization. Graphene is the thinnest known material and in the same time is the strongest material ever to be measured. It can sustain current densities six orders of magnitude higher than that of copper. It has extremely high strength and very high thermal conductivity and stiffness and is impermeable to gases (Geim 2009). There are many challenges and opportunities for graphene research because graphene is not a standard solid state material. It should be noted that electrons in graphene do not behave in the same way as in ordinary metals and semiconductors due to the unusual energy–momentum relation (Neto 2010). Well-known forms of carbon-containing molecules that derived from graphene are graphite, fullerene, and carbon nanotube, which are depicted in Fig. 3. Graphite consists of stacked layers of graphene sheets separated by 0.3 nm and is stabilized by weak van der Waals forces (He and Pan 2009). Buckminsterfullerene (C60 ) is formed from graphene balled into a sphere by including some pentagons and hexagons into the lattice (Kroto et al. 1985). The combined experimental and computational study showed the direct transformation of graphene to fullerene (Chuvilin et al. 2010). Carbon nanotubes can be viewed as rolled-up cylinders of graphene. Therefore, graphene can be called “the mother” of all these three sp2 carbon structures. It was presumed that planar graphene cannot exist in free state since they are unstable compared to curved structures such as soot, nanotubes, and fullerenes. This presumption has changed since Novoselov et al. prepared graphitic sheets including single graphene layer and studied their electronic properties (Novoselov et al. 2004, 2005a). The detailed information of growth and isolation of graphene has been provided in the recent review by Geim (2009). Graphene is a prospective material

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for nanoelectronics. The electron transport in graphene is described by Dirac-like equation (Geim and Novoselov 2007; Novoselov et al. 2005b; Ponomarenko et al. 2008). The experimental realization of graphene motivates several studies focusing on fundamental physics, materials science, and device applications (Abanin et al. 2006; Geim and Novoselov 2007; Novoselov et al. 2004; 2005a, b; Pereira et al. 2009; Ponomarenko et al. 2008). The studies pertinent to the chemistry of graphene sheets have also been reported (Abanin et al. 2006; Avouris et al. 2007; Geim and Novoselov 2007, Neto et al. 2009; Pereira et al. 2009). Graphene research is a hot topic in this decade, thanks to the recent advances in technology for growth, isolation, and characterization of graphene. Graphene sheets need not always be as perfect as one thinks. Various defects such as Stone–Wales (SW) (Stone and Wales 1986), vacancies (Carlson and Scheffler 2006), pore defects (Jiang et al. 2009), and substitution atoms (Miwa et al. 2008; Zhu et al. 2005) can occur in the thin graphene sheet. Like the creation of vacancies by knocking atoms out of the graphene sheet, surplus atoms can be found as adatoms on the graphene surface. Ad-dimer defect can be introduced to graphene and is characterized by two adjacent five-membered rings instead of two adjacent seven-membered rings in Stone–Wales defect. Therefore, ad-dimer defect is called inverse Stone–Wales (ISW) defect. Figure 4 depicts some of the common defects in graphene sheet. Experimental observations of defects in graphene have been reported recently (Meyer et al. 2008; Wang et al. 2008). Zettl and coworkers showed the direct image of Stone–Wales defects in graphene sheets using transmission electron microscopy (TEM) and explored their real-time dynamics. They found that the dynamics of defects in extended, two-dimensional graphene membranes are different than in closed-shell graphenes such as nanotubes or fullerenes (Meyer et al. 2008). High-resolution transmission electron microscopy (HRTEM) and atomic force microscopy (AFM) have been useful in identifying various defects in graphene. AFM and HRTEM images of graphene sheet with different defects are shown in Fig. 5. The effect of various defects on the physical and chemical properties of graphene was studied theoretically (Boukhvalov and Katsnelson 2008; Carpio et al. 2008; Duplock et al. 2004; Lherbier et al. 2008; Li et al. 2005). The characteristics of typical defects and their concentrations in graphene sheets are unclear. Computational and experimental studies concerning defects in graphene sheet are critically important for basic understanding of this novel system, and such understanding will be helpful for scientists who actively work on applications of graphene-based materials. Although the surface physics of graphene sheets is currently at the center of attention, its chemistry has remained largely unexplored. Like any other molecule, graphene can involve in chemical reactions. The chemical functionalization is probably one of the best approaches to detect imperfections in a graphene sheet (Boukhvalov and Katsnelson 2008). The functionalized graphene can be suitable for specific applications. Research on bended, folded, and scrolled graphene is rapidly growing now.

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Fig. 4 Defects in graphene sheet; the segment of graphene containing (a) the Stone–Wales (SW) defect; (b) a bivacancy; (c) a nitrogen substitution impurity; (d) an all-hydrogen saturated pore in graphene; (e) the pore electron density isosurface of all-hydrogen passivated porous graphene; (f) creation of a nitrogen-functionalized pore within a graphene sheet: the carbon atoms in the dotted circle are removed, and four dangling bonds are saturated by hydrogen, while the other four dangling bonds together with their carbon atoms are replaced by nitrogen atoms; (g) the hexagonally ordered porous graphene. The dotted lines indicate the unit cell of the porous graphene; (h) the pore electron density isosurface of nitrogen-functionalized porous graphene; (i) an inverse Stone–Wales (ISW) defect. Color code for (d), (f), and (g): C black, N green, H cyan. Isosurface is at 0.02 e/Å3 (The pictures were reprinted with permission from references Jiang et al. (2009) and Boukhvalov and Katsnelson (2008). Copyright 2008 and 2009 American Chemical Society)

Fullerenes Discovery of fullerene C60 and other fullerene molecules is discussed below. The fullerene era started in 1985; Kroto and his colleagues obtained cold carbon clusters when they carried out an experiment to simulate the condition of red giant star formation. With the use of the mass spectrometer, they found a large peak commensurate with 60 carbon atoms (Kroto et al. 1985). The molecule C60 was proposed to have a football structure, known to mathematicians as the truncated icosahedron. The shape is composed of 12 pentagons located around the vertices of an icosahedron and 20 hexagon rings placed at the centers of icosahedral faces. The C60 molecule was named “buckminsterfullerene” in honor of the renowned architect

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Fig. 5 (a) HRTEM image of a single graphene layer (atoms appear white). (b) Image of graphene with Stone–Wales defect (atomic configuration superimposed for easy recognition). (c) Image of vacancy defect with atomic configuration. (d) Defect image with atomic configuration consisting of four pentagons (green) and four heptagons (red). (e) Defect image with atomic configuration consisting of three pentagons (green) and three heptagons (red) (Pictures were reprinted with permission from reference Meyer et al. (2008). Copyright 2008 American Chemical Society)

Buckminster Fuller, who designed geodesic domes based on similar pentagonal and hexagonal structures. The carbon atoms in C60 fullerene are arranged in exactly the same way, albeit much smaller, as the patches of leather found on the common football (Fig. 6a). Since the remarkable discovery of fullerenes in 1985 (Kroto et al. 1985), these new carbon allotropes have received significant attention from the scientific community and still exhibit vast interest (Lu and Chen 2005; Thilgen and Diederich 2006). The 1996 Nobel Prize in Chemistry was awarded to Sir Harold W. Kroto, Robert F. Curl, and the late Richard E. Smalley for their discovery of fullerenes. Essentially, the most prominent representative of the fullerene family is C60 . In early 1990, a method was discovered for producing macroscopic amounts of this fascinating molecule (Krätschmer et al. 1990). This breakthrough allowed scientists to explore the properties of C60 and understand its chemistry. Krätschmer et al. characterized the fullerene C60 using mass spectroscopy, infrared spectroscopy, electron diffraction, and X-ray diffraction (Krätschmer et al. 1990). Both Kroto et al. (1985) and Krätschmer et al. (1990), by means of mass spectroscopy, also characterized the fullerene C70 . Pure C60 and C70 fullerenes were isolated and

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

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a

Icosahedron

C60

Soccer ball

b R2 R3

R1

Ellipsoidal shape

Rugby ball C70

Fig. 6 The structures of fullerenes C60 and C70 and their familiar shapes. C60 and C70 are in icosahedron (Ih ) and D5h point groups

separated by Kroto and coworkers (Taylor et al. 1990). The stable fullerenes of C60 and C70 were reported in the ratio of approximately 5:1. 13 C nuclear magnetic resonance (NMR) spectroscopy was used to characterize the fullerenes (Taylor et al. 1990). These two molecules are members of a homologous series of hollow closedcage molecules. The fullerene C70 belongs to a class of nonspherical fullerenes. It adopts an ellipsoidal shape (point group D5h ) and it looks like a “rugby ball” as shown in Fig. 6b. Existence of C60 was predicted by Eiji Osawa (1970). However, his prediction did not reach Europe or America since it was published in Japanese magazine. In the eighteenth century, the Swiss mathematician Leonhard Euler demonstrated that a geodesic structure must contain 12 pentagons to close into a spheroid, although the number of hexagons may vary. Later research by Smalley and his colleagues showed that there should exist an entire family of these geodesic-domeshaped carbon clusters (Kroto et al. 1985). Fullerenes form with an even number n  20 of three connected vertices, 3n/2 edges, 12 pentagonal faces, and (n20)/2 hexagonal faces (Fowler and Manolopoulos 1995; Kroto et al. 1985). Thus, C60 has 20 hexagons, whereas its “rugby ball”-shaped cousin C70 has 25 hexagons. As hexagons are added or removed, the molecule begins to lose its roundness. Giant fullerenes take on a pentagonal shape. Smaller fullerenes look like asteroids. One should note that all of the fullerenes have the same Gaussian curvature sign; therefore all of them have a convex surface. The buckminsterfullerene C60 , shown in

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Fig. 6, has a spherical-like shape and the full group of symmetry of the icosahedron (Ih ), which means that it could be rotated by the angle of 2 /5 around the center of each pentagon and reflected in the mirror located on each plane of its symmetry. Another class of spherical fullerenes like C140 and C260 lacks the mirror symmetry h; hence, their maximum symmetry group is icosahedral (I) (Terrones et al. 2002). The fullerenes C60 and C70 were identified in carbon flames, and their ratios depend on the temperature, pressure, carbon/oxygen ratio, and residence time in the flame (Howard et al. 1991). The molecular structure of C70 was deduced from electron diffraction using a simulated-annealing method (McKenzie et al. 1992). Scientists tried to understand the crystal structures of C60 and C70 using X-ray diffraction technique (David et al. 1991, 1992; Fischer et al. 1991; Valsakumar et al. 1993). At ambient temperature and pressure, C60 crystals have face-centered cubic (fcc) structure with a lattice constant of 14.17 Å (David et al. 1991), while the C70 crystals adopt to a hexagonal close-packed (hcp) structure with a D 10.1 Å and c D 17.0 Å (David et al. 1992). The average diameters of C60 and C70 fullerenes are about 7 and 9 Å, respectively. Since the discovery of C60 followed by C70 (Kroto et al. 1985; Taylor et al. 1990), different sizes of carbon cage fullerenes were revealed. In early 1990, the carbon cages of C76 , C84 , C90 , and C94 were characterized by mass spectrometry, 13 C NMR, electronic absorption (ultraviolet–visible), and vibrational (infrared) spectroscopy techniques (Diederich et al. 1991a). As compared with C60 and C70 , the isolation of higher fullerenes is really challenging, and their characterization is complicated by the presence of a varying number of isomers. Fullerenes are generally represented by a formula Cn , where n is an even number and denotes the number of carbon atoms present in the cage. Theoretical calculations predicted that fullerenes larger than C76 should have at least two isomeric forms (Manolopoulos and Fowler 1991). For fullerenes C84 and C96 , 24 and 187 distinct isomers were predicted, respectively (Manolopoulos and Fowler 1991, 1992). Three isomers for C78 and two isomers for C84 were isolated and characterized by 13 C NMR spectroscopy (Kikuchi et al. 1992a). Some of the isomers of C78 and C82 proposed by experimental 13 C NMR spectroscopy are depicted in Fig. 7. Many of the unique properties of fullerenes originate from their unusual cage structures. Therefore, determining the ground-state geometries of the fullerenes was considered to be an important step in understanding their unusual properties. Experimental works are very limited for higher fullerenes beyond C84 because such species are difficult to isolate in pure form in quantities suitable for comprehensive study. Synthesis of C60 , in isolable quantities, was achieved using flash vacuum pyrolysis (FVP) technique by Scott and coworkers by 12 steps in 2002, and no other fullerenes were formed as by-products (Scott et al. 2002). Larger fullerene C78 was synthesized using the same FVP technique used for C60 synthesis (Amsharov and Jensen 2008). This shows a promise for the synthesis of higher fullerenes. It is noteworthy to mention that there has been a lot of experimental and theoretical studies involving fragments of fullerenes, called “buckybowls” (Barth and Lawton 1966; Dinadayalane and Sastry 2001, 2002a, b; Dinadayalane et al. 2001, 2002, 2003, 2004; Mehta and Rao 1998; Mehta et al. 1997; Priyakumar and Sastry 2001a,

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

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a

C2v

C¢2v

D3

b

C2 (2)

C2 (1)

C2 (3)

c

C3v

C2v

Fig. 7 The structures of fullerene isomers suggested by the 13 C NMR measurements. (a) Three ' , C , and D point group. (b) Three structural candidates isomers of C78 fullerene with C2v 2v 3 for C82 fullerene with C2 symmetry. (c) Structures of C3v - and C2v -C82 isomers. (The picture was reprinted with permission from Macmillan Publishers Ltd.: Nature, reference Kikuchi et al. (1992a)), Copyright 1992)

b, c; Sakurai et al. 2003; Sastry and Priyakumar 2001; Sastry et al. 1993, 2000; Seiders et al. 1999; Sygula and Rabideau 1999; Wu and Siegel 2006). The smallest buckybowl “corannulene” was synthesized nearly 20 years prior to the discovery of fullerene C60 (Barth and Lawton 1966). Another small fragment of C60 called “sumanene” was successfully synthesized in 2003 after so many futile attempts by different groups (Sakurai et al. 2003). Fullerene can be classified into (a) classical fullerene and (b) nonclassical fullerene. The former one is a closed carbon cage containing 12 pentagons and any number of hexagons, while a nonclassical fullerene can have heptagons, octagons, and an additional number of pentagons or squares. Growing classical fullerenes from nonclassical fullerenes, for example, from C50 to C60 by the dimer addition, was proposed (Hernández et al. 2001). However, there is no clear experimental evidence for fullerene formation through this route. The experimental evidence was reported for the formation of fullerenes by collisional heating of carbon rings in the gas phase (Helden et al. 1993). Various mechanisms have been proposed so far

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C60

C180

C720

C960

C240

C1500

C540

C2160

Fig. 8 The structures of giant fullerenes (Reprinted with permission from Zope et al. (2008). Copyright 2008 by the American Physical Society)

for the formation of fullerenes. They can be divided into two major models: the pentagon road (PR) model (Klein and Schmalz 1990; Maruyama and Yamaguch 1998; Smalley 1992) and the fullerene road (FR) model (Heath 1991). Since the discovery of C60 , scientists showed vast interest in larger fullerenes. Therefore, the family of fullerenes increased, and now it also includes C70 , C76 , C78 , C82 , C84 , C86 , C88 , C90 , C92 , C94 , and C96 (Diederich et al. 1991a, b; Kikuchi et al. 1992b; Kimura et al. 1995; Miyake et al. 2000; Mizorogi and Aihara 2003; Taylor et al. 1992, 1993). The fundamental understanding of the size dependence of the closed carbon cage structures is important for tailoring these systems for possible nanotechnology applications. Larger fullerenes that have an icosahedral symmetry can also be constructed. This procedure generates 12 pentagons positioned around vertices of an icosahedron, while all other carbon rings are hexagonal. In general, there are two kinds of fullerenes with Ih symmetry, one being n D 60 k2 and the other n D 20 k2 , where n is the number of carbon atoms and k is any positive integer (Miyake et al. 2000). Figure 8 depicts some of the giant fullerene structures, where C180 and C720 belong to 180 k2 family of icosahedral fullerenes, but all other structures belong to 60 k2 family of icosahedral fullerenes. For more than a decade, these giant fullerenes have been fascinating molecules for theoreticians (Calaminici et al. 2009; Dulap and Zope 2006; Dunlap et al. 1991; Gueorguiev et al. 2004; Lopez-Urias et al. 2003; Tang and Huang 1995; Tang et al. 1993; Zope et al. 2008). The closed carbon cages smaller than C60 consist of adjacent pentagons. Such smaller fullerenes are predicted to have unusual electronic, magnetic, and mechanical properties that arise mainly from the high curvature of their molecular surface (Kadish and Ruoff 2002). A dodecahedron consisting of 20 carbon atoms with only pentagon rings is topologically the smallest possible fullerene. The

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

Cage

Bowl

13

Ring

Fig. 9 Isomers of C20 : cage-, bowl-, and ring-shaped structures (Reprinted with permission from Macmillan Publishers Ltd.: Nature, reference Prinzbach et al. (2000), Copyright 2000)

well-known isomers of C20 are cage, bowl, and ring as shown in Fig. 9. The bowl-shaped isomer is reminiscent of corannulene. The realization of the smallest carbon closed-cage C20 , which exclusively contains 12 pentagons, was doubtful until 2000. The C20 closed cage has extreme curvature and high reactivity, which led to doubts about its existence and stability (Wahl et al. 1993). Prinzbach et al. produced the smallest fullerene C20 from its perhydrogenated form in the gas phase and also obtained the bowl- and ring-shaped isomers for comparison purposes (Prinzbach et al. 2000). All these structures were characterized by photoelectron spectroscopy (PES) and their electron affinities vary significantly (Prinzbach et al. 2000). Theoretical calculations at different levels predicted dissimilar energetic ordering for these three isomers. However, all revealed very small relative energies of isomers. MP2 method predicted the fullerene to be the most stable followed by the bowl and then the ring, and this prediction is very similar to the calculations of density functional theory (DFT) using the local density approximation (LDA). Complete reversal of the stability ordering was obtained in the calculations with Becke–Lee–Yang–Parr (BLYP) functional. Some other DFT functionals predicted the bowl to be the most stable structure, closely followed by the fullerene isomer (Scuseria 1996). Hybrid density functional theory and time-dependent DFT formalism validated the synthesis of the smallest cage fullerene C20 by comparing the computed photoelectron spectra with the experimental results (Saito and Miyamoto 2001). Closed-cage structure of C36 was detected by mass spectroscopy in very early days of fullerene science (Kroto 1987; Rohlfing et al. 1994). Zettl’s group claimed the first preparation of C36 in the solid form (Piskoti et al. 1998). However, the existence of C36 has not been fully confirmed to date. C36 has 15 conventional fullerene isomers, out of which the D6h and D2d have a minimal number of pentagons (Fowler and Manolopoulos 1995). Therefore, these two are potential candidates for the most stable structure. In general, the number of isomers increases as the carbon cage size increases for these small fullerenes as shown in Fig. 10. Fullerenes from C20 to C58 have been extensively studied by theoreticians (Fowler and Manolopoulos 1995; Scuseria 1996; Shao et al. 2007). They have been predicted to have narrow HOMO–LUMO gaps and high reactivity. The structures,

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2 1

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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 # of carbon, n in Closed carbon cage (Cn)

Fig. 10 Number of isomers for the closed carbon cages from C20 to C58 (Data was taken from reference Fowler and Manolopoulos 1995))

aromaticity, reactivity, and other properties of fullerenes smaller than 60 carbon atoms were reviewed (Lu and Chen 2005). Readers may refer to the review by Lu and Chen and the references therein for detailed understanding and further knowledge if required (Lu and Chen 2005). Selected structures of smaller fullerenes and their isomers are depicted in Fig. 11. Schlegel diagram is commonly used by scientists to sketch the fullerenes in planar view, which is very helpful in identifying atoms and the CC bonding networks (Fowler and Heine 2001; Thilgen and Diederich 2006; Troyanov and Tamm 2009). Figure 12 depicts the Schlegel diagram for fullerenes C20 , C36 , C60 , and C70 .

Natural Abundance of Fullerenes Scientists discovered the presence of natural fullerenes on Earth. Interestingly, occurrence of fullerenes such as C60 and C70 was reported in shungite, a metaanthracite coal from a deposit near Shunga, Russia (Buseck et al. 1992). The presence of C60 at very low concentrations in Cretaceous–Tertiary boundary sites in New Zealand was published (Heymann et al. 1994). Fullerenes (C60 and C70 ) were found in a unit of shock-produced impact breccias (Onaping Formation) from the Sudbury impact structure in Ontario, Canada (Becker et al. 1994). The abundance of naturally occurring fullerenes was found in carbon materials, for example, coal, rocks, interstellar media, and even dinosaur eggs (Heymann et al. 2003).

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

C20

C28

C46(C2)

C52(C2)

C36(D6h)

C36(D2d)

C46(Ds)

C54(C2v)

C44(D3h)

C50(D3)

C56(D2)

C44(D2)

15

C44(D2)

C50(D5h)

C58(Cs /C 3v)

Fig. 11 Representative structures of smaller fullerenes and low-energy isomers are given for some of them. The pentagon–pentagon fusions are highlighted in blue color only for C52 , C54 , C56 , and C58 . The point groups are given in parentheses (Reprinted with permission from reference Lu and Chen (2005). Copyright 2005 American Chemical Society)

Fullerene Nano-capsules In the area of fullerene science, one should not forget to mention an interesting property of holding the atoms or ions or molecules inside the fullerene cage (Thilgen and Diederich 2006). Fullerenes are potential nano-capsules. Experimental detection of the nano-capsules of fullerenes such as La@C60 , La@C70 , La@C74 , La@C76 , La@C78 , La@C82 , and Ce2 @C80 was reported (Kessler et al. 1997; Kubozono et al. 1996; Moro et al. 1993; Saunders et al. 1993; Thilgen and Diederich 2006; Yamada et al. 2005, 2010). Fullerenes are known in the field of radioactive chemistry/physics. Radioactive nuclear materials can be stored by encapsulating inside the fullerenes. U@C28 , Gd@C60 , and Gd@C82 are few examples of encapsulated radioactive materials (Guo et al. 1992; Kubozono et al. 1996). The stability of these metallic fullerenes could bring the new effective solution of the radioactive waste elimination. For the substance enclosed in the fullerene nano-capsule, carbon atoms act like a defense shield, and the fullerene containers are good for protecting their contents from water and acid. The structures, stabilities, and reactivities of encapsulated fullerenes (nano-capsules) and doped fullerenes have been the subject of theoretical interest (Guo et al. 1992; Lu et al. 2000; Park et al. 2005; Simeon et al. 2005; Wang et al. 2003; Wu and Hagelberg 2008; Zhao and Pitzer 1996). The closed-cage “fullerenes” or “heterofullerenes” can be placed inside the

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Fig. 12 Schlegel diagrams of C20 , C36 , C60 , and C70 fullerenes. The point groups are given in parentheses, systematic numbering recommended by IUPAC (Pictures were reprinted with permission from reference Thilgen and Diederich (2006). Copyright 2006 American Chemical Society. Reference Fowler and Heine (2001). Copyright 2001 Royal Society of Chemistry)

single-walled carbon nanotubes, for example, C60 @SWCNT (Hirahara et al. 2000; Okada 2007; Smith et al. 1999). Leszczynski and coworkers have explored the mechanism of the catalytic activity of fullerene derivatives using reliable computational methods (Sulman et al. 1999; Yanov et al. 2004). Fullerenes are certainly worthy of scientific study because of their unique shape and intriguing properties.

Isolated Pentagon Rule (IPR) in Fullerenes A wide range of methods available for producing fullerenes concluded that C60 is the most abundant and is followed by C70 (Kadish and Ruoff 2002). The pristine C60 (Ih ) contains two different C–C bonds: the one at the junction of two six-membered rings and the other one at the junction of a five- and a six-membered rings. These two bonds are usually labeled as a [6,6] and [5,6] C–C bonds, respectively. The pristine

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

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C70 (D5h ) has eight distinguishable C–C bonds. It has been known to chemists that energetically it is not favorable to have two pentagons sharing the same CC bond. There are 1812 mathematical ways of forming a closed cage with 60 carbon atoms (isomers), but the buckminsterfullerene C60 (Ih ) is the most special and stable because all of its pentagons are isolated by hexagons. This condition is called the “isolated pentagon rule” (IPR), which tends to make fullerenes more stable (Fowler and Manolopoulos 1995). In fact, C60 is the smallest fullerene cage that obeys the isolated pentagon rule. Fullerenes C62 , C64 , C66 , and C68 do not satisfy the IPR. The next fullerene, which follows the IPR, is C70 (it has 8149 possible isomers). Also, C72 has an IPR structure. Most of the higher fullerenes have proven to follow IPR (Kroto 1987). Only one IPR-obeying isomer exists for C60 and for C70 (Fowler and Manolopoulos 1995), while the number of possible IPR isomers increases rapidly with increase in the size of the fullerenesas shown in Fig. 13. Fullerene C78 has five isomers that satisfy the IPR. Three isomers (two with C2v and one with D3 symmetry) out of these five were identified and characterized using 13 C NMR spectra (Kikuchi et al. 1992a). The fourth isomer (D3h -C78 ) has been recently separated and characterized in the form of C78 (CF3 )12 (Shustova et al. 2006, 2007), and the last one has been synthesized using FVP technique (Amsharov and Jensen 2008). Theoretical study predicted that

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C82 has nine IPR-satisfying isomers (Manolopoulos and Fowler 1991), out of which three isomers with C2 symmetry were experimentally characterized using 13 C NMR spectroscopy, which gave 41 NMR lines with nearly equal intensity (Kikuchi et al. 1992a). There are 24 geometric isomers satisfying IPR and 51,568 non-IPR isomers are possible for fullerene C84 (Fu et al. 2009). Earlier experimental 13 C NMR spectroscopy studies characterized two IPR isomers with D2 and D2d point groups (Kikuchi et al. 1992a; Taylor et al. 1993). Third isomer was also identified and reported (Achiba et al. 1995; Crassous et al. 1999). Pure D2 -C84 was synthesized (Dennis and Shinohara 1998). A theoretical study revealed that C84 cage is special in the family of fullerenes from C60 to C90 since the number of preferable isomers is larger than in the case of other fullerenes. The geometries of all of the 24 IPR isomers of fullerene C84 along with their point groups were reported (Okada and Saito 1996). Two IPR isomers of C86 out of possible 19 isomers were separated using multistage HPLC (high-performance liquid chromatography) (Miyake et al. 2000), and these two isomers were characterized to have C2 and Cs point groups by 13 C NMR spectroscopy (Taylor et al. 1993). Burda et al. showed the experimental evidence for the photoisomerization of higher fullerenes. They confirmed thetheoretical prediction that C86 has less number of IPR-satisfying isomers (19 isomers) than C84 (24 isomers) (Fig. 14; Burda et al. 2002). Experimental studies based on 13 C NMR spectroscopy revealed that fullerenes C88 and C92 possess 35 and 86 IPR isomers, respectively, and HPLC was used to separate the isomers (Achiba et al. 1996; Miyake et al. 2000; Tagmatarchis et al. 2002). Computational methods

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Fig. 14 Correlation between the number of isomers (dotted line as a visual guide) for each fullerene and the time constant for the formation of the lowest excited singlet state monitored at 550 nm (dots) (Reprinted with permission from reference Burda et al. (2002). Copyright 2002 American Chemical Society)

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were used to calculate relative energies of the IPR isomers and the 13 C NMR spectra of fullerenes (Shao et al. 2006; Slanina et al. 2000a, b; Sun 2003). The computed 13 C NMR spectra were interpreted with the available experimental data (Beavers et al. 2006; Chaur et al. 2009a, b; Melin et al. 2007; Rojas et al. 2007; Scheina and Friedrich 2008; Shao et al. 2006; Slanina et al. 2000a, b; Sun 2003; Xie et al. 2004). Theoretical calculations predicted that C88 (33) is one of the most stable IPR isomers (Shao et al. 2006). The number provided in the parenthesis is the isomer number. In consistent to the theoretical prediction, Troyanov and Tamm reported the isolation and X-ray crystal structures of trifluoromethyl derivatives of C88 (33) and C92 (82) fullerene isomers complying with the isolated pentagon rule (Troyanov and Tamm 2009). Rojas et al. showed the experimental evidence of the decreasing trend in the gasphase enthalpy of formation and strain energy per carbon atom as the size of the cluster increases. Thus, the fullerenes become more stable as they become larger in size (Rojas et al. 2007). Interestingly, molecules encapsulated inside the carbon cages stabilize the fullerene isomers that violate IPR (Beavers et al. 2006; Fu et al. 2009; Thilgen and Diederich 2006). Several IPR and non-IPR endohedral fullerenes (single metal, di-metal, or tri-metal nitride encapsulated fullerenes) were isolated and characterized experimentally (Beavers et al. 2006; Chaur et al. 2009a, b; Fu et al. 2009; Melin et al. 2007; Thilgen and Diederich 2006), and their isolation motivated significant theoretical interest (Fu et al. 2009; Park et al. 2005; We and Hagelberg 2008). The structures and relative energies of the IPR isomers of buckybowls were examined using computational methods (Dinadayalane and Sastry 2003). Head-totail exclusion rule was proposed in explaining the stability of carbon cage structures that obey the IPR (Scheina and Friedrich 2008). Fullerenes with less than 60 carbon atoms cannot have isolated pentagons and therefore they should be highly unstable and reactive. Xie et al. synthesized nonIPR D5h -C50 fullerene, which is a little sister of C60 , by introduction of chlorine atoms at the most reactive pentagon–pentagon vertex fusions. They confirmed the D5h -C50 structure by mass spectrometry, infrared, Raman, ultraviolet–visible, and fluorescence spectroscopic techniques (Xie et al. 2004). The report of novel small cage “Saturn-shaped” C50 Cl10 structure encourages the possibility of obtaining other small non-IPR fullerenes and their derivatives. The investigations of the properties and applications of small fullerenes and their derivatives are now open.

Common Defects in Fullerenes Stone and Wales examined rotation of CC bonds in various fullerene structures using approximate Huckel calculations. The 90ı rotation of CC bond in fullerene is called Stone–Wales (SW) or “pyracylene” rearrangement (Fig. 15; Stone and Wales 1986). Austin et al. reported that 94 % of all fullerene C60 isomers can rearrange to buckminsterfullerene by SW transformation (Austin et al. 1995). The C78 cage represents the smallest fullerene in which SW rearrangement can give

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a

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Fig. 15 (a) Stone–Wales or “pyracylene” transformation in fullerenes interchanges pentagons and hexagons; (b) Stone–Wales transformation of C2v isomer of C60 with two adjacent pentagons gives the most stable Ih buckminsterfullerene; the CC bond involved in 90ı rotation is highlighted, and the two adjacent pentagons are marked in the fullerene structure in left-hand side

stable IPR isomers, C78 :5 (D3h ) $ C78 :3 (C2v ) $ C78 :2 (C2v ) $ C78 :4 (D3h ), where the numbers 5, 3, 2, and 4 indicate the isomer numbers (Austin et al. 1995). In case of higher fullerenes, the number of IPR isomers that can be transformed one into another by SW rearrangement considerably increases. For example, the SW transformation gives 9 and 21 stable IPR isomers for C82 and C84 , respectively (Fowler and Manolopoulos 1995). The SW transformation is usually thought to be the possible mechanism for achieving fullerene isomers (Austin et al. 1995; Fowler and Manolopoulos 1995; Stone and Wales 1986). It was proposed that fullerenes can have seven-membered rings in addition to five- and six-membered rings (Taylor 1992). Troshin et al. isolated and characterized the C58 fullerene derivatives in which the cage structure contains the sevenmembered ring. The structures were characterized using mass spectrometry, IR, and NMR spectroscopy (Troshin et al. 2005). Smalley and coworkers found that laser irradiation can fragment C60 into C58 , C56 , C54 , and other smaller cages with even number of carbon atoms via losing C2 fragments (O’Brien et al. 1988). The formation of seven-membered rings was considered to play an important role in the fragmentation process of fullerenes (Murry et al. 1993). The laser desorption/ionization of products generated from the reactions of C60 with O3 gives the odd-numbered clusters such as C59 , C57 , C55 , and C53 (Christian et al. 1992; Deng et al. 1993). Vacancy defects destroy the original topology of five- and sixmembered rings in fullerenes (Christian et al. 1992; Deng et al. 1993; Hu and Ruckenstein 2003, 2004; Lee and Han 2004; Murry et al. 1993; O’Brien et al. 1988). They generate various sizes of rings such as four-, seven-, eight-, and ninemembered rings and also produce the new five- and six-membered rings depending on the number of carbon atom vacancies (Fig. 16; Christian et al. 1992; Deng et al.

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1993; Hu and Ruckenstein 2003, 2004; Lee and Han 2004; Murry et al. 1993; O’Brien et al. 1988).

Carbon Nanotubes (CNTs) Discovery and Classification of CNTs Modern “nanotechnology revolution” was flourished by the discovery of fullerenes and has been escalating since the isolation of multi- and single-walled carbon nanotubes. The detection of carbon nanotubes by Iijima in 1991 is one of the landmarks in nanotechnology (Iijima 1991). In the interview to Nature Nanotechnology, Iijima told that the discovery of carbon nanotubes was unexpected but not entirely accidental because he had accumulated a lot of experience in looking at short-range order in carbon species such as amorphous carbon and very thin graphite sheets (Iijima 2007). The discovery of buckminsterfullerene by Kroto, Curl, Smalley, and coworkers motivated Iijima’s interest in finding out new carbon allotropes (Kroto et al. 1985). There are two structural forms of carbon nanotubes: multi-walled carbon nanotubes (MWCNTs) and single-walled carbon nanotubes (SWCNTs). The former one was reported by Iijima (1991). The SWCNTs were reported independently by Iijima as well as Bethune groups (Bethune et al. 1993; Iijima and Ichihashi 1993). The existence of carbon nanotubes was reported as early as 1952 and also in 1976

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Fig. 17 Representative structures of (a) multi-walled and (b) single-walled carbon nanotubes

(Oberlin et al. 1976; Radushkevich and Lukyanovich 1952). However, those reports did not reach the wide range of scientific community because they were published in unpopular journals, and at that time, no fabrication process was known that would lead to the synthesis of macroscopic amounts of carbon nanotubes (Oberlin et al. 1976; Radushkevich and Lukyanovich 1952). Monthioux and Kuznetsov documented the history of carbon nanotube since 1952 (Monthioux and Kuznetsov 2006). Synthesis of carbon nanotube using coal pyrolysis was reported (Moothi et al. 2015). High-resolution electron microscopy (HREM) images of the CNTs showed the resemblance of a “Russian doll” structural model that is based on hollow concentric cylinders capped at both ends. The model structures of multi-walled and single-walled carbon nanotubes are shown in Fig. 17. A wide range of methods, such as arc evaporation of graphite, laser ablation, chemical vapor deposition (CVD), vapor phase decomposition or disproportionation of carbon-containing molecules, etc., have been reported for the synthesis of multi-walled and singlewalled carbon nanotubes (Dresselhaus et al. 2001). It remains unclear whether SWCNTs and MWCNTs are formed via the same mechanism. It is also unclear whether various methods used to produce carbon nanotubes are mechanistically consistent (Dresselhaus et al. 2001). For the transformation pathway, fullerenes are known to be a suitable carbon source for MWCNT growth under certain conditions (Suchanek et al. 2001). An ideal MWCNT consists of cylindrical tubes in which the neighboring tubes are weakly bonded through van der Waals forces. The MWCNT is incommensurate when each of its walls has its own chirality independent of other walls.

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SWCNT, which is a one-dimensional (1D) system, can be considered as the conceptual rolling of a section of two-dimensional (2D) graphene sheet into a seamless cylinder forming the nanotube. The structure of SWCNT is uniquely   described by two integers (n, m), which refer to the number of ! a 1 and ! a2 unit vectors of the 2D graphene lattice that are contained in the chiral vector,   Ch D n! a 1 C m! a 2 . The chiral vector determines whether the nanotube is a semiconductor, metal, or semimetal. From the (n, m) indices, one can calculate the nanotube diameter (dt ), the chirality or chiral angle ( ), the electronic energy bands, and the density of electronic states. The nanotube diameter (dt ) determines the number of carbon atoms in the circular cross section of the nanotube shell, one atom in thickness (Saito et al. 1998). The tube diameter and chiral angle can be written in terms of (n, m) as Tube diameter; dt D

p  p  3= acc m2 C mn C n2 :

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np o 3 m= .2n C m/ ;

where acc is the nearest-neighbor carbon atom distance of 1.421 Å. Among the large number of possible Ch vectors, there are two inequivalent highsymmetry directions. These are termed “zigzag” and “armchair” and are designated by (n,0) and (n,n), respectively. Either achiral (armchair and zigzag) or chiral SWCNTs can be constructed depending on the orientation of the six-membered rings with respect to the nanotube axis. Schematic representation of the structures of armchair, zigzag, and chiral SWCNTs is shown in Fig. 18. Theoretical studies in 1992 predicted that the electronic properties of “ideal” SWCNTs depend on the width and chirality of the tubes (Hamada et al. 1992; Mintmire et al. 1992; Saito et al. 1992). The electronic properties of an SWCNT vary in a periodic way between being metallic and semiconductor. SWCNTs are metals if (nm)/3 represent an integer; otherwise, they are called semiconductors (Dresselhaus et al. 2002). Several metallic (n, m) nanotubes have almost the same diameter dt (from 1.31 to 1.43 nm), but have different chiral angles:  D 0, 8.9, 14.7, 20.2, 24.8, and 30.0ı for nanotubes (18, 0), (15, 3), (14, 5), (13, 7), (11, 8), and (10, 10), respectively (Dresselhaus et al. 2002). Few people realize that CNTs constitute a large family with a wide variety of sizes and properties, which are determined by their structure and composition, including chirality, number of walls, ordering of the wall, defects, surface functionalization, and other features. Strano sorted out chiral SWCNTs into left-handed and right-handed tubes (Strano 2007). Significant progress has been made in the area of carbon nanotubes. Scientists are able to disperse, identify, sort, and now also isolate various types of carbon nanotubes (Arnold et al. 2006; Peng et al. 2007; Strano 2003, 2007). Specific methods were found to grow long SWCNTs and control the nanotube diameters (Lu et al. 2008; Zhang et al. 2008). Controlled synthesis of nanotubes

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Fig. 18 (a) The roll-up of graphene sheet into SWCNT; (b) picture shows how to roll up graphene sheet to generate three different types of SWCNTs; (c) (3,3) armchair SWCNT; (d) (5,0) zigzag SWCNT; (e) (4,2) chiral SWCNT; (f) three types of SWCNTs (armchair, zigzag, and chiral) with fullerene end caps. These can be viewed as the growth of SWCNTs by adding several layers of hexagonal rings at the middle of different fullerenes; (g) mirror image of the chiral SWCNT; the structures (c), (d), and (e) are given exactly same types of SWCNTs that are mentioned in (b). In (c)–(e), ¢ v and ¢ h indicate the vertical plane of symmetry and horizontal plane of symmetry, respectively. Further, in these three structures, the red line is the axis of rotation; the distance of one unit cell for these three types of SWCNTs is provided; the number of carbon atoms (x) in each layer and the number of layers (y) required for one unit cell is given as xy , for example, – 63 means six carbon atoms in each layer and three layers required for one unit cell of (3,3) armchair SWCNT (Pictures (b) and (g) were reprinted with permission from Macmillan Publishers Ltd.: Nature Nanotech., reference Strano (2007), Copyright 2007)

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opens up exciting opportunities in nanoscience and nanotechnology (Dai 2002). A range of methods was found for effective separation of metallic and semiconducting SWCNTs. Although some synthetic procedures have been known, they are not easy methods for synthesizing bulk quantities of metallic and semiconducting SWCNTs (Zhang et al. 2008). Scientists succeeded the preferential growth of SWCNTs with metallic conductivity (Rao et al. 2009a). Raman and electronic spectroscopy techniques are useful in characterizing metallic and semiconducting SWCNTs. The radial breathing mode (RBM) in Raman spectra of SWCNTs is helpful in determining the diameter and chiral indices (n, m) of the nanotubes (Dresselhaus et al. 2002, 2005, 2007; Harutyunyan et al. 2009; Rao et al. 2009a). Experimental resultspointed out decreasing band gap with increasing radius of the armchair SWCNTs (Fig. 19; Ouyang et al. 2002). Another breakthrough in carbon nanotube chemistry was accounted by Zhang and Zuo, who have determined a quantitative atomic structure of MWCNT containing five walls with diameter ranging from 17 to 46 Å and the CC bond lengths of individual SWCNTs using electron diffraction technique (Zhang and Zuo 2009). Their results indicate that there are three different bond lengths in chiral walls and two different bond lengths in achiral walls (Zhang and Zuo 2009). Electron diffraction technique was used in determination of atomic structure of SWCNTs and the chiral indices (n,m) of CNTs (Jiang et al. 2007; Qin 2007). Rao et al. have revealed the efficient growth of SWCNTs of diameter 1–3 nm from diamond nanoparticles and fullerenes (Rao et al. 2009b). Recently, experimental reports are available to create three-dimensional graphene-CNT hollow fibers with radially aligned CNTs seamlessly sheathed by a cylindrical graphene layer through a

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one-step chemical vapor deposition using an anodized aluminum wire template for efficient energy conversion and storage (Xue et al. 2015; Yu et al. 2014; Zhu et al. 2012). SWCNTs have stimulated vast interest due to their unique structural, mechanical, electronic, thermal, and chemical properties and their potential applications in diversified areas. There has been enormous growth in patents related to carbon nanotubes, fuelled by predictions that the market for nanotubes will be $9 billion by 2020. Between 1994 and 2006, it was estimated that 1865 nanotube-related patents were issued in the USA. Still, there is a cumulative backlog of more than 4500 patent applications relevant to CNT as reported in 2008 (MacKenzie et al. 2008). Since there are reports of the natural abundance of fullerenes (Becker et al. 1994; Buseck et al. 1992; Heymann et al. 1994, 2003), the issue of the natural occurrence of carbon nanotubes has also attracted the attention of researchers. In 2004, TEM images that appear to be MWCNTs isolated from a Greenland ice core were reported (Esquivel and Murr 2004). The images of hollow carbon fibers from oil-well samples were reported (Velasco-Santos et al. 2003). However, we do not have any evidence for naturally occurring SWCNTs.

Various Defects in Carbon Nanotubes Carbon nanotubes are not as perfect as they were thought to be earlier. Defects such as pentagons, heptagons, Stone–Wales defects, vacancies, adatoms, and dopants can occur in the nanotube during the growth or in processing and handling of the CNTs (Charlier 2002). Figure 20 depicts different types of defects in SWCNTs. Heptagon defects are found to play a crucial role in the topology of nanotube-based molecular junctions, for making X and Y type nanotube connections (Menon and Srivastava 1997). Long ago, theoretical studies proposed that pentagon–heptagon pair can be found in the intramolecular junctions of two SWCNT segments of different chiralities (Fig. 20d; Charlier et al. 1996; Chico et al. 1996). Experimental study revealed that ion irradiation-induced defects in the SWCNTs and the dangling bonds produced by irradiation are rapidly saturated (Chakraborty et al. 2007). Low-energy electron and photon also induce damage in SWCNTs. The defect formation in SWCNTs is strongly dependent on the nanotube diameter, suggesting that the curvature-induced strain energy plays a crucial role in the damage (Suzuki and Kobayashi 2007). The defect formation and healing are reversible processes (Berthe et al. 2007; Suzuki and Kobayashi 2007). The defects in the SWCNTs affect their electronic, optical, and chemical properties. A competition between the defect formation and healing at room temperature or even below was reported. Raman spectroscopy, electrical measurements, and photoluminescence (PL) spectroscopy were used to examine the defect formation. However, the type of defects was not confirmed. Chemically stable topological defect, Stone–Wales defect, was ruled out because the activation energy for the defect healing was quite small ( 1 eV). Lowenergy electron and photon can break CC bonds in SWCNTs, as it was concluded

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Fig. 20 (a) to (c) Nonchiral Haeckelite nanotubes of similar diameter; (b) nanotube segment containing only heptagons and pentagons paired symmetrically. (b) Nanotube segment exhibiting repetitive units of three agglomerated heptagons, surrounded by alternating pentagons and hexagons. (c) Nanotube segment containing pentalene and heptalene units bound together and surrounded by six-membered rings. (d) Atomic structure of an (8,0)-(7,1) intermolecular junction; the large red balls denote the atoms forming the pentagon–heptagon pair. (e) The SW transformation leading to the 5-7-7-5 defect, generated by rotation of a CC bond in a hexagonal network. (f) HRTEM image obtained for the atomic arrangement of the SW defect. (b) Simulated HRTEM image for the model shown in (f). (h) (5,5) armchair SWCNT with a Stone–Wales defect. (i) Ideal single-vacancy defect in (5,5) armchair SWCNT. (j) Ideal double-vacancy defect in (5,5) armchair SWCNT. (k) Defect (5,5) SWCNT with seven (n D 7)-, eight (n D 8)-, and nine (n D 9)-membered rings. (l) SWCNT doped with boron [B atoms are bonded to three C atoms; B in red spheres and C in blue spheres]. (m) SWCNT doped with nitrogen [N atoms are bonded to two C atoms; N in red spheres and C in blue spheres] (Pictures (a)–(d), (l), and (m) were reprinted with permission from reference Charlier (2002). Copyright 2002 American Chemical Society. Pictures (e)–(g) were reprinted with permission from Macmillan Publishers Ltd.: Nature, reference Suenaga et al. (2007), copyright 2007. Pictures (h–j) were reprinted with permission from reference Yang et al. (2006a). Copyright 2006 American Chemical Society. Picture (k) was reprinted with permission from Nishidate and Hasegawa (2005). Copyright 2005 by the American Physical Society)

based on energetic criterion. Thus, the experimental study proposed that the defects may be a vacancy and an adatom (Suzuki and Kobayashi 2007). The Stone–Wales defect is one of the important defects in carbon nanotubes. Stone and Wales showed that a dipole consisting of a pair of five- and sevenmembered rings could be created by 90ı rotation of a CC bond in a hexagonal

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network (Stone and Wales 1986). Such a dipole was later called Stone–Wales defect. SW transformation is thought to play an important role during the growth of carbon nanotubes. Miyamoto et al. reported an unambiguous identification of SW defect in carbon and boron nitride nanotubes using photoabsorption and vibrational spectroscopy (Miyamoto et al. 2004). Experimental vibrational frequency of 1962 cm1 was reported to be a signature in identifying SW defect in carbon nanotube (Miyamoto et al. 2004). Identifying and characterizing topological defects in SWCNTs are highly challenging tasks. A powerful microscope with high resolution and high sensitivity is required for characterizing the topological defects in CNTs. Using HRTEM, the first direct image of the pentagon–heptagon pair defect (Stone–Wales defect) in the SWCNT was reported (Suenaga et al. 2007). Computational studies examined the structures and defect formation energies of the SWCNTs with defects containing different sizes of rings (seven-, eight-, and ninemembered rings) (Nishidate and Hasegawa 2005) and different types of defects (Amorim et al. 2007; Dinadayalane and Leszczynski 2007a, b; Ding 2005; Wang et al. 2006; Yang et al. 2006a, b).

Computational Approaches Used to Study Carbon Nanostructures: An Overview Theory and computation play an important role in understanding structures and reactivity of carbon nanosystems such as graphene, fullerenes, and carbon nanotubes. Computational nanoscience often complements the experiments and is very useful for the design of novel carbon nanomaterials as well as predicting their properties. Theory is helpful in obtaining knowledge on the mechanism of reactions and fragmentations of carbon clusters. Thus, we provide an overview of the computational approaches employed to study various carbon nanostructures in this chapter. Carbon nanostructures are very large systems. Hence, performing very high-level quantum chemical calculations is not possible even when using modern supercomputers. Many-body empirical potentials, empirical tight-binding molecular dynamics, and local density functional (LDF) methods were used to perform electronic structure calculations of carbon nanosystems including fullerenes and model CNTs in early of the last decade (Robertson et al. 1992; Zhang et al. 1993). In the mid1990s, electronic structure calculations for large fullerenes with Ih point group were performed using Huckel approximation (Tang and Huang 1995). In the late 1990s, scientists performed geometry optimizations for large fullerenes using molecular mechanics (MM3), semiempirical methods (MNDO (Dewar and Thiel 1977), AM1 (Dewar et al. 1985) and PM3 (Stewart 1989)), and Semi-Ab Initio Model 1 (SAM1) (Dewar et al. 1993). The single-point energy calculations were affordable at that time using ab initio Hartree–Fock (HF) method in combination with small basis sets such as 3-21G and 4-31G (Slanina et al. 1997). The computing power has been tremendously increasing since 2000. Thus, currently theoreticians enjoy investigating medium-sized molecules using reliable quantum chemical methods

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and exploring carbon nanoclusters beyond molecular mechanics and semiempirical methods. The popular B3LYP functional, which is a combination of Becke’s threeparameter (B3) (Becke 1993) hybrid functional incorporating exact exchange with Lee, Yang, and Parr’s (LYP) (Lee et al. 1988) correlation functional, has been employed with small- and medium-sized basis sets like STO-3G, 3-21G, 4-31G, and 6-31G(d) for calculations on fullerenes and carbon nanotubes (Bettinger et al. 2003; Dinadayalane and Leszczynski 2007b; Feng et al. 2005; Matsuo et al. 2003; Yumura et al. 2005a, b; Zhou et al. 2004). Computational studies indicate that the B3LYP functional can yield reliable answers for the properties of carbon compounds and carbon nanostructures (Bettinger et al. 2003; Dinadayalane and Leszczynski 2007b; Feng et al. 2005; Matsuo et al. 2003; Yumura et al. 2005a, b; Zhou et al. 2004). PBE1PBE/6-311G(d) level has been used for calculating relative energies and 13 C NMR spectra of fullerene isomers (Shao et al. 2006, 2007). The PBE1PBE functional was concluded to be very reliable DFT functional since it yields the same relative energy ordering as the high-level coupled cluster calculations for the top three isomers of C20 (cage, bowl, and ring isomers) (An et al. 2005). In comparison with ab initio MP2 or CCSD methods, DFT is less timeconsuming and computationally feasible for large carbon nanosystems. For studying chemical reactivity in fullerenes and nanotubes, ONIOM approach is more costeffective than treating the whole molecule with DFT. ONIOM is a hybrid methodology in which the molecule is partitioned into two or more fragments. The most important part (one fragment) of the molecule is treated with high-level method, and the other parts are treated with low-level methods (Maseras and Morokuma 1995; Morokuma et al. 2006; Osuna et al. 2009). The performance of ONIOM approach by taking different density functional theory levels was examined against the experimental results for the Diels–Alder reaction between cyclopentadiene and C60 (Osuna et al. 2009). Two-layer ONIOM approach ONIOM(B3LYP/6-31G(d):AM1), where B3LYP/6-31G(d) and AM1 are used for high and low layers, was utilized to study chemisorption of alkoxide ions with the perfect and Stone–Wales defective armchair (5,5) SWCNTs of cap-ended and H-terminated structures (Wanbayor and Ruangpornvisuti 2008). Independent theoretical studies considered DFT methods in investigating the structures and properties of SWCNTs (Akdim et al. 2007; Amorim et al. 2007; Andzelm et al. 2006; Bettinger 2005; Dinadayalane and Leszczynski 2007a, b; Govind et al. 2008; Lu et al. 2005; Nishidate and Hasegawa 2005; Robertson et al. 1992; Wang et al. 2006; Yang et al. 2006a, b; Zhang et al. 1993). The B3LYP functional with double-— basis set was often employed to investigate the electronic structures of pristine and defective SWCNTs and also the influence of defects on functionalization of SWCNTs (Akdim et al. 2007; Andzelm et al. 2006; Dinadayalane and Leszczynski 2007b; Govind et al. 2008; Lu et al. 2005). Sometimes, more than one basis set was utilized for exploration of defective SWCNTs and the viability of metal adsorption in the defect tubes (Yang et al. 2006b). DFT with periodic boundary condition (PBC) as implemented in Gaussian 03 program package (Frisch et al. 2003) was used to examine the reactivity of

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T.C. Dinadayalane and J. Leszczynski

Stone–Wales defect in (5,5) and (10,0) SWCNTs (Bettinger 2005). The unit cell should be carefully chosen for the calculations involving PBC in order to simulate the tubes of infinite length. Popular DFT methods fail to provide reliable answers for  –  interactions involving fullerenes and other carbon clusters (Cuesta et al. 2006; Kar et al. 2008; Shukla and Leszczynski 2009). Although calculations at the MP2 and CCSD(T) levels are required to obtain very reliable results for  -stacking interactions (Dinadayalane et al. 2007a; Lee et al. 2007; Sinnokrot and Sherrill 2004), they are not possible for such large systems with current computational facilities. Recently developed meta-hybrid density functional (M06-2X) has been reported to be a promising functional to calculate the binding energies for  –  interactions involving large carbon nanostructures (Zhao and Truhlar 2007, 2008). Using powerful supercomputers, performing static and dynamic calculations at high-level ab initio and DFT methodologies is affordable for graphene and carbon nanotubes. Very recently, density functional theory (PBE functional, Perdew et al. 1996) calculations with plane-wave basis sets and periodic boundary conditions (PBCs) were employed to understand small-molecule interactions with the defective graphene sheets (Jiang et al. 2009). Vienna ab initio simulation package (VASP) has been used in several studies to perform static and dynamic calculations (Kresse and Furthmuller 1996a, b). Theoretical calculations are helpful to understand the electronic structure of graphene sheets and SWCNTs and their viability as ion separation systems and gas sensors (Jiang et al. 2009; Li et al. 2009b; Nishidate and Hasegawa 2005). The Stone–Wales defect formation energy for graphene and CNTs was calculated using DFT, invoking the local density approximation to the exchange-correlation potential as implemented in VASP (Ertekin et al. 2009). The mechanical properties of CNTs have been investigated by theoreticians for the last two decades (Avila and Lacerda 2008; Chandra et al. 2004; Dereli and Sungu 2007; Yakobson et al. 1996). An array of methods has been employed for computing the Young’s modulus of MWCNTs and different types of SWCNTs (armchair, zigzag, chiral). A wide range of Young’s modulus values has been reported in the literature (Avila and Lacerda 2008; Chandra et al. 2004; Dereli and Sungu 2007; Mielke et al. 2004; WenXing et al. 2004; Yakobson et al. 1996). Most of the molecular dynamics methods used so far are classical or tight binding (Avila and Lacerda 2008; Chandra et al. 2004; Dereli and Sungu 2007; WenXing et al. 2004; Yakobson et al. 1996). Quantum chemical calculations on mechanical properties of carbon nanotubes or graphene sheets are scarce since they are still highly timeconsuming (Mielke et al. 2004). It is not of our interest to discuss mechanical properties in this chapter since there are many papers and some of classic reviews on this subject available (Avila and Lacerda 2008; Chandra et al. 2004; Dereli and Sungu 2007; Mielke et al. 2004; WenXing et al. 2004; Yakobson et al. 1996). DFT and DFT-D approaches were used to study single and multiple Na adsorption and diffusion on graphene (Malyi et al. 2015). By using DFT and time-dependent DFT methods, one can obtain IR, Raman, NMR, and UV spectra. Recent advances in computer hardware and ab initio electronic structure methods have brought a substantial improvement in the capabilities of quantum chemists to predict and study the properties of carbon nanostructures. However, the application of state-of-the-art

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

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quantum chemical methods to study the structures and properties of large carbon nanoclusters (graphenes, fullerenes, and CNTs) is still a great challenge.

Structural, Electronic, and Chemical Properties of Graphene, Fullerenes, and SWCNTs Graphene An experimental investigation of mechanical properties of monolayer graphene reported a breaking strength of 40 N/m and the Young’s modulus of 1.0 TPa (Lee et al. 2008). Graphene displays a thermal conductivity of 5000 Wm1 K1 at room temperature (Balandin et al. 2008). Graphene chemistry is expected to play an important role in producing graphene-based materials. Computational study using the density functional theory with generalized gradient approximation revealed the cooperative effects of degenerate perturbation and uniaxial strain on band gap opening in graphene. Furthermore, the band gap width could be continuously tuned by controlling the strain (Jia et al. 2016). In this chapter, we outline the Stone– Wales defect in graphene, chemisorption process (covalent functionalization) on graphene, and the influence of defects on chemisorption; particular interest is given to hydrogen chemisorption. Chemical functionalization in graphene should produce new 2D systems with distinct electronic structures and different electrical, optical, and chemical properties. Chemical changes can probably be induced even locally. The first known example of hydrogenated graphene is graphane, which is a 2D hydrocarbon with one hydrogen atom attached to every site of the honeycomb lattice (Elias et al. 2009; Sofo et al. 2007). Stone–Wales defect is expected to enhance the tendency of graphitic layers to roll up into other carbon nanostructures such as fullerenes and nanotubes. Therefore, indepth understanding of Stone–Wales defect in graphene is required. It is known that pentagons and heptagons induce curvature in graphitic materials. In perfect graphene, the equilibrium CC bond length is reported as 1.42 Å using PBE functional with the plane-wave code CPMD (Hutter et al.). Further details of the calculations can be obtained from the paper of Ma et al. (2009). The CC bond shared by two heptagons of the SW defect in graphene is compressed to 1.32 Å using the same method. Density functional theory and quantum Monte Carlo simulations reveal that the structure of the SW defect in graphene is not simple. Ma et al. systematically studied the polycyclic hydrocarbon size dependence on the structural distortion caused by the Stone–Wales defect formation. They considered different systems ranging from the smallest analog of SW defect, azupyrene (C16 H10 ), to 1D tape-like structure of C50 H28 and, finally, to 2D planar cluster of C228 H38 . As known earlier, azupyrene is planar. The optimized bond length of the CC bond at the center of azupyrene is 1.38 Å, which is longer than the corresponding CC bond length of 1.32–1.33 Å observed for the SW defect in graphene (Ma et al. 2009). Large carbon clusters exhibit a tendency to buckle upon the creation of SW defects. Vibrational frequency calculations of the flat graphene sheet with the Stone–

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a 90 deg. rotation

Defect-free

Stone-Wales (SW) defect

b Top view

Side view sine-like buckled SW defect

cosine-like buckled SW defect

Fig. 21 (a) Stone–Wales transformation by 90ı rotation of C–C bond in graphene sheet; (b) top and side views of sine-like and cosine-like buckled SW defect graphene sheets (The pictures were reprinted with permission from Ma et al. (2009)). Copyright 2009 by the American Physical Society)

Wales defect reveal that the structure is not a local minimum, but instead has two imaginary frequencies. The true minimum is a sine-like structure in which the CC bond at the defect core is 0.01 Å longer than in the flat defect. Furthermore, many CC bonds are slightly elongated in the buckled structure compared to the flat defect structure. The cosine-like SW defect structure was obtained as a transition state connecting to sine-like SW defect structure. The optimized structures of sinelike and cosine-like SW defect graphenes are depicted in Fig. 21. Vibrational frequencies also revealed that the maximum phonon frequencies corresponding to the stretch of the rotated CC bond for the flat and buckled SW structure are 1880 and 1774 cm1 , respectively. The corresponding frequency computed for perfect graphene is 1612 cm1 (Ma et al. 2009). Theoretical study pointed out that for a graphene sheet of C228 H38 containing a SW defect, the sine-like buckled structure becomes more stable (by 10 meV) than the flat SW defect (Ma et al. 2009).

Hydrogenation of Graphene With and Without Defects Chemical modification of graphene has been less explored (Geim and Novoselov 2007). Attachment of atomic hydrogen to each site of the graphene lattice to create graphane is an elegant idea (Sofo et al. 2007). As a result, the hybridization of carbon atoms changes from sp2 to sp3 , thus removing the conducting p-bands and

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

a

33

b Wz 1

2

Nz

c Wa

y

z 1

2 Na

H C

Fig. 22 (a) Top (upper) and side (lower) view of a 2D graphane layer. Geometric structures of the (b) 7 zigzag and (c) 13 armchair graphane nanoribbons. The ribbons are periodic along the z direction. The ribbon widths are denoted by Wz and Wa , respectively (Reprinted with permission from reference Li et al. (2009b)). Copyright 2009 American Chemical Society

opening energy gap (Boukhvalov et al. 2008; Sofo et al. 2007). In experiment, the fully hydrogenated graphene called “graphane” was produced by exposing graphene to hydrogen plasma discharge. Raman spectroscopy and transmission electron microscopy confirmed the reversible hydrogenation of single-layer graphene (Elias et al. 2009). The structural and electronic properties of graphane were investigated using DFT PW91 functional with plane-wave basis set applying periodic boundary conditions as implemented in VASP (Li et al. 2009b). Computations revealed that hydrogenation of graphene nanoribbon is experimentally viable and the electronic properties of graphane are completely different from graphene nanoribbons. Figure 22 depicts the structures of graphane. Two types of graphane nanoribbons (zigzag and armchair edge) can be obtained by cutting the optimized graphane layer. The edge carbon atoms were all saturated with H atoms to avoid the effects of dangling bonds. The bond lengths of edge CC and CH bonds are almost as the inner CC (1.52 Å) and CH (1.11 Å) bonds. The calculated CC bond length is similar to the bond length of 1.53 Å in diamond (sp3 carbon atoms) and is longer than 1.42 Å characteristic of sp2 carbon in graphene. Both spin-unpolarized and spin-polarized computations yielded same energy for ground-state graphane nanoribbons (Li et al. 2009b). Figure 23a shows that computed band gap decreases monotonically with increasing ribbon width for both zigzag and armchair nanoribbons. Graphane nanoribbons are semiconductors. The formation energy increases with increasing ribbon width (Fig. 23b) irrespective of the type, indicating that narrow ribbons are more likely to form than the wider ribbons (Li et al. 2009b). Sofo et al. investigated the structures, formation energies, and vibrational frequencies of graphane using DFT with planewave basis set (Sofo et al. 2007). They found two favorable conformations of

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T.C. Dinadayalane and J. Leszczynski

a

Band gap (eV)

4.0

Zigzag Armchair

3.9 3.8 3.7 3.6 3.5 10

15

20 25 Width(Å)

30

35

b Formation energy (eV)

–0.056 –0.058

Zigzag Armchair

–0.060 –0.062 –0.064 –0.066 10

15

20 25 Width(Å)

30

35

Fig. 23 Variation of the band gap (a) and the formation energy (b) of zigzag (6  Nz  16) and armchair (10  Nz  27) graphane nanoribbons as a function of ribbon width. N is the number of zigzag chains for a zigzag ribbon and the number of dimer lines along the ribbon direction for an armchair ribbon (Reprinted with permission from reference Li et al. (2009b). Copyright 2009 American Chemical Society)

graphane: chair-like conformer with the hydrogen atoms alternating on both sides of the plane and the boat-like conformer with the hydrogen atoms alternating in pairs. Chair conformer has one type of CC bond (1.52 Å), while boat conformer possesses two different types of CC bonds (bond lengths of 1.52 Å and 1.56 Å). The boat conformer is less stable than the chair conformer due to the repulsion of the two hydrogen atoms bonded to first-neighbor carbon atoms on the same side of the sheet. This repulsion results in slightly longer CC bonds in boat conformer. Calculated CH bond stretching frequencies are 3026 cm1 and 2919 cm1 for the boat and chair conformers, respectively. These CH stretching modes are IR active, and they should be useful in characterizing these two types of conformers of graphane (Sofo et al. 2007). Molecular dynamics (MD) simulations using adaptive

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

a

0.5

b

e

f c g

Chemisorption energy (eV/atom)

d

35

h

0 –0.5

Stone-Wales

–1 0.5

i

0 –0.5 bivacancy

–1 0.5

j

0 –0.5 Nsubstitutional

–1 0

20 40 60 80 100 Coverage (%)

Fig. 24 Optimized geometric structures for graphene supercell containing (a) the Stone–Wales defect, (b) a bivacancy, and (c) a nitrogen substitution impurity. Optimized structures for the Stone– Wales (SW) defect functionalized by (d) 2, (e) 6, and (f) 14 hydrogen atoms and (g) completely covered by hydrogen. Green circles represent carbon atoms, violet circles represent hydrogen atoms, and blue circle represents nitrogen atom. Hydrogen atom chemisorption energy per atom as a function of coverage for a graphene sheet containing (h) a Stone–Wales (SW) defect, (i) a bivacancy, and (i) a nitrogen substitution impurity. The blue dashed line represents the results for the ideal infinite graphene sheet (Reprinted with permission from reference Boukhvalov and Katsnelson (2008). Copyright 2008 American Chemical Society)

intermolecular reactive empirical bond order (AIREBO) force field in LAMMPS package revealed the wrinkling characteristics in hydrogenated graphene annulus under circular shearing at the inner edge. Such hydrogenation-induced changes in topological and mechanical characteristics of graphene will be useful to develop novel graphene-based devices (Li et al. 2015). Using density functional calculations, Boukhvalov and Katsnelson have studied hydrogenation of graphene sheets with defects such as Stone–Wales (SW), bivacancies, nitrogen substitution impurities, and zigzag edges. They performed calculations for chemisorptions of hydrogen atoms on the defects in the graphene from low to high coverage. The optimized geometries of the graphene supercells with varioustypes of defects as well as their hydrogenated structures are depicted in Fig. 24, which also displays the computed chemisorption energy as the function of coverage for the graphene containing different defects. The chemisorption energy of a single hydrogen atom to the defect-free graphene was given as 1.5 eV, while those of 0.30 eV for SW defects, 0.93 eV for bivacancies, and 0.36 eV for substitution impurities of nitrogen in graphene were reported. This indicates the significant influence of defects on single hydrogen atom chemisorption energy in graphene. The calculated chemisorption energy for different nonequivalent carbon atoms of the graphene containing SW defect reveals that the chemisorption energy for the entire area surrounding the SW defect is lower compared to the perfect graphene.

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Further, the defects also decrease the chemisorption energy of two hydrogen atoms at adjacent positions compared to the defect-free graphene. It was reported that, for the complete coverage, the binding energy is smaller for the hydrogen chemisorption of graphene with defects than the perfect graphene. Thus, completely hydrogenated graphene is less stable with defects than without them (Boukhvalov and Katsnelson 2008). Graphenes with various kinds of defects may have different types of properties and applications. Therefore, obtaining knowledge on graphenes with defects is important. DFT calculations showed only physisorption of water molecule with perfect graphene, while the vacancy defect greatly assists the dissociative chemisorption of water molecules in the graphene (Cabrera-Sanfelix and Darling 2007; Kostov et al. 2005). There can be many possible reaction pathways for the dissociation of water molecule over defective sites in the graphene (Kostov et al. 2005). Computational studies provide evidence that defects such as Stone– Wales and vacancy strongly influence the chemisorption of functional groups in the graphene (Boukhvalov and Katsnelson 2008; Boukhvalov et al. 2008; CabreraSanfelix and Darling 2007; Kostov et al. 2005).

Fullerenes Computational Studies of Fullerene Isomers Computational methods were employed to systematically search and study the lowlying isomeric structures of fullerenes, and such thorough investigations have been useful to predict the best candidates for the lowest-energy structures of higher fullerenes because of the growing experimental interest (Shao et al. 2006, 2007; Slanina et al. 2000a, b; Sun 2003; Sun and Kertesz 2002; Zhao et al. 2004a, b). Fullerene C86 has 19 possible isomers obeying IPR, and all of these isomers were studied using B3LYP functional with different basis sets (Sun and Kertesz 2002). Among 19 isomers, the isomer 17 with C2 symmetry is the most stable followed by isomer 16 with Cs symmetry, and these two isomers are shown in Fig. 25. It should be noted that these two isomers were experimentally observed. At the B3LYP/6-31G level, isomer 16 was predicted to be about 6 kcal/mol less stable than isomer 17, albeit the former one has slightly larger HOMO–LUMO gap than the latter. The variation of relative stability at different theoretical levels for all 19 IPR-satisfying isomers of C86 is depicted in Fig. 26. The relative stabilities were calculated with respect to the lowest-energy isomer (17). The HF/3-21G level and the semiempirical AM1 Hamiltonian overestimate the relative energies compared to the density functional theory (DFT) levels (Sun and Kertesz 2002). Experimental study identified two isomers of fullerene C86 and characterized them using 13 C NMR spectroscopy (Miyake et al. 2000). Based on the experimental NMR spectra, C2 and Cs point groups were assigned for the two isomers, but there are more than single C2 and Cs isomers. Theoretical calculations play a crucial role in identifying the correct structure by comparing theoretical and experimental 13 C NMR spectra. The 13 C NMR chemical shifts were calculated for all of the 19

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

37

Fig. 25 Two experimentally observed IPR isomers of fullerene C86 . Their point groups are given in parentheses (Reprinted with permission from reference Sun and Kertesz (2002). Copyright 2002 Elsevier)

B3LYP / STO-3G 90

B3LYP / 3-21G B3LYP / 6-31G

80

AM1 HF / 3-21G

70 Relative energy (kcal / mol)

B3LYP / 6-31G* 60 50 40 30 20 10 0 –10 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Isomer Number

Fig. 26 The relative energy of IPR-satisfying isomers of C84 at various levels of theory. Isomers 1, 5, 7, 11, 12, and 13 have C1 symmetry. Isomers 2, 3, 4, 6, 14, and 17 possess C2 point group. Isomers 9 and 10 have C2v point group. Isomers 8, 15, and 16 have Cs point group. Isomers 18 and 19 possess C3 and D3 point groups, respectively (The data was taken from reference Okada and Saito (1996))

IPR isomers of C86 , except isomer 8. Theoretical 13 C NMR spectra complement the experimental spectra as evidenced from Fig. 27. Computational study revealed that isomer 17 has high thermodynamic and kinetic stability among the six IPR isomers of C84 possessing C2 point group. The computed NMR spectrum of isomer

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T.C. Dinadayalane and J. Leszczynski

a

C2 17 14 6 4 3 2

160 155 150 145 140 135 130 125 120 115 110

Chemical Shift (ppm)

b

Cs 16 15

170 165 160 155 150 145 140 135 130 125 120

Chemical Shift (ppm) Fig. 27 Experimental and theoretical 13 C NMR spectra of (a) C2 isomers of fullerene C86 . (b) Cs isomers of fullerene C86 . Theoretical spectra are labeled by isomer number and experimental spectrum labeled by symmetry (Reprinted with permission from reference Sun and Kertesz (2002). Copyright 2002 Elsevier)

17 supports the results of experimental spectrum. Among the Cs isomers, the second most stable isomer 16 has large HOMO–LUMO gap (Sun and Kertesz 2002). Isomers 6, 10, 11, 12, 13, and 18 were predicted to have relative energies less than 20 kcal/mol and moderate HOMO–LUMO gap, thus indicating the possibility of experimental realization (Sun and Kertesz 2002). Okada and Saito proposed the number of extractable fullerenes among the IPRsatisfying isomers of fullerenes from C60 to C90 . They found that C84 is unique since the number of preferable isomers is more than for other fullerenes and this was attributed to the abundant production of C84 after C60 and C70 . All 24 IPRsatisfying isomers of C84 were studied computationally (Okada and Saito 1996). A complete set of 187 isomers that obey IPR of fullerene C96 was systematically investigated using various theoretical methods including molecular mechanics (MM3), semiempirical (AM1, MNDO, and PM3), and quantum mechanical (HF/431G and B3LYP/6-31G) methods. All of the theoretical levels unequivocally predicted that isomer 183 with D2 point group is the lowest-energy one. The relative energies for some of the isomers were reported to be quite method sensitive and varied dramatically with different methods. The computational study highlighted the importance of the entropy effect in examining the relative stability of IPRobeying isomers of fullerene C96 (Zhao et al. 2004a). A large set of 450 IPR

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

39

isomers of C100 has been explored using the abovementioned semiempirical and molecular mechanics (MM3) methods. Systematic theoretical calculations predicted the isomer with D2 point group (isomer 449) as the lowest energy by all of the methods employed (Zhao et al. 2004a). Shao et al. searched the lowest-energy isomer of the fullerenes C38 to C80 and C112 to C120 . For the first set (C38 to C80 ), all IPR and all non-IPR isomers were considered, and only IPR-satisfying isomers were considered for the second set of fullerenes (C112 to C120 ). Thus, a large set of molecules was taken for optimizations at the semiempirical density functional-based tight-binding (DFTB) method and the single-point energy calculations at the DFT (Shao et al. 2007). It is known that the fullerene with large HOMO–LUMO gap and high symmetry is not necessarily the lowest-energy structure. The decreasing trend of HOMO–LUMO gap was reported with increasing the fullerene size (Shao et al. 2006, 2007). An unexpected manner of pentagonal adjacency was observed in the low-lying isomers in the series of fullerenes C38 to C80 (Shao et al. 2007). In a comprehensive computational study, Shao et al. identified 20 isomers as the best candidates for the lowest-energy structures. Among the 20 isomers, 10 isomers with relative energies less than 1 kcal/mol aredepicted in Fig. 28 (Shao et al. 2007). Thus, these 10 isomers can be observed experimentally. The 13 C NMR chemical shifts for these 10 isomers were calculated. Theoretical study predicted that C116 :6061 can be more easily isolated and characterized in the laboratory than other higher fullerenes from C112 to C120 (Shao et al. 2007). In a different study, Shao et al. proposed the seven best candidates of the lowest-energy isomers for the fullerenes C98 to C110 based on the systematic study using DFTB and DFT

C112:3299

C112:3336

C112:3189

C114:4462

C116:6061

D2

D2

C2

Cs

Th

C118:7924

C118:7308

C120:10253

C120:4811

C120:10243

C1

Cv

C2

Cs

C1

Fig. 28 Best candidates for the lowest-energy structure of higher fullerenes (C112 to C120 ). The isomer number and point group are given (Reprinted with permission from reference Shao et al. (2007). Copyright 2007 American Chemical Society)

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T.C. Dinadayalane and J. Leszczynski

methods. They pointed out that C102 (C1 : 603) and C108 (D2 : 1771) isomers can be easily synthesized (Shao et al. 2006). The concepts of cage connectivity and frontier  -orbitals play important roles to understand the relative stability of charged fullerene isomers without performing extensive quantum chemical calculations. This theoretical study correctly predicted the structures observed experimentally and explained why the isolated pentagon rule is often violated for fullerene anions, but the opposite is found for fullerene cations (Wang et al. 2015b). Fullerenes C50 and C50 Cl10 were computationally studied using B3LYP/631G(d) level due to the experimental report of the latter compound (Lu et al. 2004). The computational study thoroughly explored the structures, relative energies, HOMO–LUMO energies, and HOMO–LUMO gap for low-lying isomers of C50 and its anions. The computed IR, Raman, 13 C NMR, and UV–Vis spectra of the C50 Cl10 with D5h symmetry showed very good agreement with the reported experimental data. The pentagon–pentagon fusions were found to be the active sites of addition reactions in both D3 and D5h symmetric isomers of fullerene C50 . It was observed that HOMO and LUMO coefficients of C50 (D5h ) are distributed around the equatorial pentagon–pentagon fusion sites. This was given as a reason for the binding of Cl atoms around the equatorial pentagon–pentagon fusion sites of C50 yielded C50 Cl10 (Lu et al. 2004).

Giant Fullerenes Giant fullerenes have been the subject of theoretical interests (Calaminici et al. 2009; Dulap and Zope 2006; Dunlap et al. 1991; Gueorguiev et al. 2004; LopezUrias et al. 2003; Zope et al. 2008). The structures and stabilities of the giant fullerenes C180 , C240 , C320 , and C540 were investigated using high-level density functional theory calculations (Calaminici et al. 2009). The results of the uncorrected binding energy (in eV) per carbon atom for the giant fullerenes obtained using the VWN functional are depicted in Fig. 29. The inclusion of the basis set superposition error (BSSE) decreases the calculated binding energies but does not alter the trend. The increasing trend of binding energy indicates that the large fullerenes become more and more stable with increasing size. Fullerene C540 has a similar binding energy to diamond, giving the hope that such giant fullerenes could be prepared. However, the binding energy per carbon atom of the fullerene, C540 , is considerably lower than that of graphene (Calaminici et al. 2009). Gueorguiev et al. performed the calculations for giant fullerenes using semiclassical approximation LR-LCAO (linear response model in the framework of the linear combination of atomic orbitals). They reported the decreasing trend of HOMO–LUMO gaps (except C20 ) and the considerably large increase of the static polarizability as increasing the size of the fullerene cage (Fig. 30; Gueorguiev et al. 2004). The static dipole polarizability per atom in C2160 is three times larger than that in C60 (Dunlap et al. 1991).

Binding energy (eV)

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

41

8.80

8.75

8.70 C180

C240

C320

C540

Fullerene size

Fig. 29 Binding energy (in eV) for C180 , C240 , C320 , and C540 fullerenes. The calculations have been performed with the VWN functional in combination with DZVP basis sets (Reprinted with permission from reference Calaminici et al. (2009). Copyright 2009 American Chemical Society)

Local Strain in Curved Polycyclic Systems: POAV and Pyramidalization Angle Fullerenes experience large strain energy because of their spherical shape. The curvature-induced pyramidalization of the carbon atoms of fullerenes weakens the  -conjugation. The curved  -conjugation in carbon networks of fullerenes has not only  -character but also substantial s-character. The  -orbital axis vector (POAV) analysis developed by Haddon is useful in measuring the local curvature of the nonplanar conjugated organic molecules, fullerenes, and SWCNTs (Haddon 1993; Haddon and Scott 1986). In general, the sp2 -hybridized carbon atom prefers to be in the planar arrangement, but it is pyramidalized in fullerenes. The local strain of carbon framework in fullerenes and SWCNTs is reflected in the pyramidalization angle  P at the carbon atoms (Niyogi et al. 2002). The pyramidalization angle ( P ) equals to the difference between the  -orbital axis vector (POAV) and the normal right angle 90ı : thus,  P D ( ¢   90ı ), where the ™¢  is the angle between the  -orbital of the conjugated atom and the ¢-orbital of the surrounding atoms. As shown in Fig. 31, the pyramidalization angle is 0ı and 19.47ı for a planar sp2 hybridized carbon and a tetrahedral sp3 -hybridized carbon, respectively. All carbon atoms in the icosahedral C60 have the same  ¢  of 11.6ı . Pyramidalization angle of a carbon atom in fullerenes and SWCNTs is helpful in predicting the chemical reactivity (Akdim et al. 2007; Bettinger 2005; Dinadayalane and Leszczynski 2007a, b; Lu and Chen 2005; Lu et al. 2005). The larger pyramidalization angle of carbon atom indicates the higher reactivity toward addition reactions in the curved systems of fullerenes and SWCNTs. Curvature-induced pyramidalization and the  -orbital misalignment cause local strain in SWCNTs (Fig. 32). Hence, carbon atoms of SWCNTs are more reactive than that of a perfect

42

T.C. Dinadayalane and J. Leszczynski

a

b 2.5 HOMO–LUMO gap (eV)

Average Radius (R, A)

30 25 20 15 10 5

2.0 1.5 1.0 0.5 0.0

0 C20

C60

C240 C540 C960 C2160 C3840

C20

Fullerene cage (Cn)

C60

C240 C540 C960 C2160 C3840

Fullerene cage (Cn)

Static polarizability(α, A3)

c 32000 28000 24000 20000 16000 12000 8000 4000 0 C20

C60

C240 C540 C960 C2160 C3840

Fullerene cage (Cn)

Fig. 30 The variation of (a) radius (R, Å), (b) HOMO–LUMO gap (ev), and (c) static polarizability as increasing the size of the fullerene cage (C20 , C60 , C240 , C540 , C960 , C2160 , and C3840 ) (The data for the plots was taken from reference Gueorguiev et al. 2004))

graphene sheet (Niyogi et al. 2002; Park et al. 2003). Cyranski et al. studied the structures and energetics of the 12 lowest-energy isomers of neutral, closed-shell IPR fullerenes C60 -C96 using B3LYP/6-31G(d) level. They obtained the decreasing values of pyramidalization angles, while no regular trend was obtained for HOMO– LUMO gaps with increasing size of fullerenes (Fig. 32; Cyranski et al. 2004). Decachloro-derivative of C50 fullerene has been synthesized and experimental characterization confirmed the existence of C50 cage. Two C20 caps and five C2 units around the equator are present in the C50 core of C50 Cl10 (Xie et al. 2004). The calculated pyramidalization angle for the carbon atoms of the C20 caps of fullerene C50 ranges from 10.7 to 12.88ı , which are comparable to that of C60 (11.68ı ). However, a large pyramidalization angle (15.58ı ) is obtained for the equatorial C atoms (pentagon–pentagon fusion). Such large value was attributed to high reactivity of those carbon atoms in addition reactions to form exohedral adducts (Chen 2004; Lu et al. 2004). Such structural features were reasoned for the instability of a bare C50 cage and the stability of C50 Cl10 (Chen 2004).

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

a

p orbital

3

a = 13 ∑ a qp + 90° 1

s a1

5

1

b

2 0

a3 a2

3

1

1/4

3

2/5

2

5

3/6

2

4 1

6 3

CH4

c qsp = 90°

6

4

i =1

φ = 0° H2C = CH2

43

6 5

d

qsp = 109.47°

4

e

qsp = 101.6°

3 1 2

5 4 2

6

1 3 1/4

qP = 0º

qP = 19.47º

2

qP = 11.6º

5

6 3

φ = 21.3°

Fig. 31 (a) Pyramidalization angle ( P ) is defined by the angle between the  -orbital and ¢-bond minus 90ı so that  P D 0ı for a graphene sheet and  P D 19.47ı for sp3 -hybridized carbon. For practical reasons, we take the average of three  P values: (b)  P for a perfect planar sp2 -hybridized carbon atom (e.g., in C2 H4 ), (c)  P for a perfect tetrahedral sp3 -hybridized carbon atom (e.g., CH4 ), (d)  P for a nonplanar sp2 -hybridized carbon atom (e.g., C atom in C60 or SWCNT). (e) The  -orbital misalignment angle (¥) along the C1–C4 bond in the (5,5) SWCNT and the fullerene C60 (Pictures were reprinted with permission from references Lu and Chen (2005), Niyogi et al. (2002), and Park et al. (2003). Copyright 2002, 2003 and 2005 American Chemical Society)

b

12.0

HOMO–LUMO gap (in eV)

Pyramidalization angle (in degrees)

a

11.5 11.0 10.5 10.0 9.5

3.0 2.5 2.0 1.5 1.0 0.5

9.0 C60 C70 C72 C74 C76 C78 C80 C82 C84 C86 C90 C96

C60 C70 C72 C74 C76 C78 C80 C82 C84 C86 C90 C96

(Ih) (D5h) (D6d) (D3h) (D2) (C2v) (D2) (C2) (D2d) (C2) (C2) (D2)

(Ih) (D5h) (D6d) (D3h) (D2) (C2v) (D2) (C2) (D2d) (C2) (C2) (D2)

Fullerene Cage

Fullerene Cage

Fig. 32 (a) Variation of pyramidalization angle for the carbon atom of the most stable IPR fullerene as increasing the size of fullerene. (b) Variation of HOMO–LUMO gap for the most stable IPR fullerene as increasing the size of fullerene. The point groups are given in the parentheses. The values for these graphs were taken from reference Cyranski et al. (2004))

44

T.C. Dinadayalane and J. Leszczynski

a

b

1.467 1.474 {1.475} {1.469} 1.415 {1.421} 1.397 {1.408} 1.443 {1.447}

Stone-Wales-type C60 (C2v) 1.68 eV (38.7 kcal / mol)

C60 (Ih) 0.00 eV (0.0 kcal / mol)

1.475 {1.477}

1.353 {1.366}

c 1.425 {1.423}

1.458 {1.462}

d

1.478 {1.480}

1.431 {1.433}

1.474 {1.474} 1.604 {1.592}

1.615 {1.611}

1.467 {1.472}

1.379 {1.388}

2.236 {2.242} 2.181 {2.187}

4

1.484 {1.484}

2 3

1.509 {1.479} 1

1.401 {1.412}

1.245 {1.257}

1.648 {1.643}

d(C1-C4):2.276 {2.249} d(C1-C3):2.269 {2.265}

Fig. 33 (a) Buckminsterfullerene to C60 -C2v with Stone–Wales defect generated by the 90ı rotation of the CC bond in blue color of C60 (Ih ). (b) Optimized structure of the C2v symmetry isomer. (c) Structure of the C2 symmetry transition state for the concerted Stone– Wales transformation pathway. (d) Structure of the asymmetric transition state between carbene intermediate and C60 -C2v isomer. Bond lengths were obtained at the B3LYP/6-31G* and PBE/631G* (in curly brackets) levels of theory and are given in Å (Pictures were reprinted with permission from reference Bettinger et al. (2003) and Yumura et al. (2007). Copyright 2003 and 2007 American Chemical Society)

Stone–Wales Defect in C60 Fullereneisomers are likely to interconvert through Stone–Wales transformation (Stone and Wales 1986; Troyanov and Tamm 2009). Very recently, experimental study has reported that the chlorine-functionalized D2 -C76 IPR isomer transformed to non-IPR isomer, and this transformation was proposed to include seven single Stone–Wales rearrangements (Ioffe et al. 2009). Computational chemists strived to understand the energy barriers for the Stone–Wales transformation and the possible mechanisms involved in this rearrangement, particularly considering the C60 fullerene (Bettinger et al. 2003; Eggen et al. 1996; Yumura et al. 2007).

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

45

Stone–Wales transformation is a thermally forbidden rearrangement by following the orbital symmetry considerations of Woodward and Hoffmann (1969). The icosahedral C60 fullerene (buckminsterfullerene) gives an isomer of C60 with C2v point group that violates the isolated pentagon rule. Two different pathways, namely, concerted and stepwise pathways, and two different (symmetric and asymmetric) transition states were identified theoretically for the Stone–Wales transformation in C60 fullerene. The C60 -C2v isomer, which is a Stone–Wales-type defect structure with two adjacent pentagons, was reported to be less stable by 33.9–38.7 kcal/mol (1.47–1.68 eV) than the buckminsterfullerene using various density functional theory levels (Yumura et al. 2007). Bettinger et al. listed the CC bond lengths of C60 (Ih ) and the activation barrier for the Stone–Wales defect transformation through different transition states at various levels of theory. Computed geometries of buckminsterfullerene at different levels showed shorter [6,6] CC bond length than the [5,6] CC bond length, in consistent with experimental results (Bettinger et al. 2003). Figure 33 depicts the concerted C2 symmetric transition state and asymmetric transition state involved in SW transformation of C60 -Ih to C60 -C2v . The intrinsic reaction coordinate calculations by Bettinger et al. support the concerted pathway rather than stepwise pathway for the SW transformation in the C60 fullerene. Based on the computed activation energies, both concerted and stepwise pathways are highly competitive. The rigorous computational study of SW transformation in buckminsterfullerene revealed that the empirical schemes such as Tersoff–Brenner potentials and density functional-based tight binding (DFTB) underestimate the barrier heights, and semiempirical AM1 appears to be promising for such investigations (Bettinger et al. 2003).

Computational Studies on Vacancy Defects in Fullerene C60 Vacancy defects in fullerene C60 were studied using quantum chemical methods (Hu and Ruckenstein 2003, 2004; Lee and Han 2004). They were generated by removal of 1–4 carbon atoms in C60 as shown in Figs. 34 and 35. Different modes are possible to remove carbon atoms from C60 to generate vacancy defects; hence, different sizes of rings (four-, seven-, eight-, and nine-membered rings) were produced by removing carbon atoms in C60 . Removing one, two, three, and four adjacent carbon atoms from the C60 cluster generates two, three, three, and six different isomers for the C59 , C58 , C57 , and C56 clusters, respectively (Hu and Ruckenstein 2004; Lee and Han 2004). The odd-numbered clusters have unsaturated carbon, which favors being located in a six-membered ring rather than a fivemembered ring. Two-atom vacancies give structure with seven- and eight-membered rings, whereas one-atom vacancy gives the structure with nine-membered ring. Four-atom vacancies provide the most stable structure with only five- and sixmembered rings. Thus, increasing the number of vacancies need not increase the size of the hole (Hu and Ruckenstein 2003, 2004).

46

T.C. Dinadayalane and J. Leszczynski One atom vacancy defect C59

5-8 Two atoms vacancy defect

4-9

C58

5-5-7 Three atoms vacancy defect

4-4-8(6)

4-4-8(5)

5-6-7

5-5-9

C57

4-5-9

Fig. 34 B3LYP/6-31G(d) optimized structures of C59 , C58 , and C57 clusters. Description indicates highlighted rings. “A-B” denotes A- and B-membered ring. Circle denotes an unsaturated atom (Reprinted with permission from reference Lee and Han (2004). Copyright 2004, American Institute of Physics)

The singlet structures are more stable than the triplet ones for C58 cluster, while the reverse is true in the case of C57 clusters. The reported stabilization energy per atom at the B3LYP/6-311G(d)//B3LYP/6-31G(d) level is 2.18, 1.49, and 3.10 kcal/mol for the C59 , C58 , and C57 , respectively. Quantum chemical calculations provide relationship between structure and stability of the defect fullerene clusters (Hu and Ruckenstein 2003, 2004; Lee and Han 2004). In case of removal of four adjacent carbon atoms in C60 , additional five-membered rings are formed in geometry optimizations (e.g., isomer 1 in Fig. 35). The isomer 4 has only fiveand six-membered rings (12 five-membered rings and 18 six-membered rings) and was predicted to be the most stable among the isomers depicted in Fig. 35. The stability energy for the isomers generated by removing four carbon atoms has the following sequence: isomer 4 > isomer 3 > isomer 2 > isomer 5 > isomer 6 > isomer 1. All defect clusters have lower stability energy per atom than C60 . The removal of carbon atoms from C60 increases the HOMO and decreases the

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

47

mode 1: romoving atoms 1, 2, 3, 4

isomer 1

4(1)-5(13)-6(15)-9(1)

mode 2: romoving atoms 1, 2, 3, 6

isomer 2

4(1)-5(12)-6(16)-8(1)

mode 3: romoving atoms 1, 2, 3, 9

isomer 3

4(1)-5(11)-6(17)-7(1)

mode 4: romoving atoms 1, 2, 6, 7

isomer 4

5(12)-6(18)

mode 5: romoving atoms 2, 3, 6, 7

isomer 5

4(1)-5(12)-6(16)-8(1)

mode 6: romoving atoms 2, 3, 6, 9

isomer 6

3(1)-4(2)-5(9)-6(17)-10(1)

1 6 2

5

7 4

3 8 9

16

6

7

18 10

17

15

18 5

9

17

4

7

17

10 2

19

5

16 4

15

2

isomer 1

14

6

5

15 16 15

9

3

10 2

13

isomer 2

18

4

8

18

4

9

3

10

18 6

7

13 3

8

7 8

5

16 14

9

4 14

14

12

5

15

8

4

1 3

20

5

15 13

11 1

17 7

6

9

14 3

13

6 12

6

7

17

16 16

11

8

8

11

14 2

1 12

2 11

1

12

9 1

10 24

2

13 13

12

isomer 5

20 11

12 23

isomer 4

21

1

11

isomer 3

19

10

3

22

isomer 6

Fig. 35 Different modes to generate isomers of C60 with four vacancies by removing four adjacent atoms from the perfect C60 structure. Structures of isomers 1–6 of defect C60 with four vacancies. The ring size and the number of rings (in parentheses) for each isomer are given; for example, isomer 1-4(1)-5(13)-6(15)-9(1) means 1 four-membered, 13 five-membered, 15 six-membered, and 1 nine-membered rings (Reprinted with permission from reference Hu and Ruckenstein (2004). Copyright 2004, American Institute of Physics)

LUMO energy. Consequently, the defect structures exhibit lower HOMO–LUMO gap compared to C60 . No relationship was obtained between the stability energy per carbon atom and the HOMO–LUMO gap for the defective carbon clusters of C60 (Hu and Ruckenstein 2003).

Computational Studies of Single-Walled Carbon Nanotubes Computational chemists explored the structures, electronic properties, and reactivities of SWCNTs of varying lengths and diameters (Bettinger 2004; Dinadayalane and Leszczynski 2009; Dinadayalane et al. 2007b; Galano 2006; Kaczmarek et al. 2007; Matsuo et al. 2003; Niyogi et al. 2002; Yang et al. 2006c). They also tried to understand the influence of different defects on these properties at reliable theoretical methods within the limitations of hardware and software (Akdim et al. 2007; Andzelm et al. 2006; Bettinger 2005; Dinadayalane and Leszczynski 2007a, b; Govind et al. 2008; Lu et al. 2005; Nishidate and Hasegawa 2005; Wanbayor and Ruangpornvisuti 2008; Wang et al. 2006; Yang et al. 2006a, b). A series of finite-length hydrogen-terminated armchair SWCNTs have been computationally studied to obtain knowledge on the influence of diameter and length on the structural and electronic properties (Galano 2006). The optimized armchair (n,n) SWCNTs possess Dnh and Dnd point groups for ª/2 even and odd, respectively. The different

48

T.C. Dinadayalane and J. Leszczynski

Weighted average C-C distance

1.432 1.430 1.428 1.426 1.424 1.422 1.420 1.418 1.416

(3, 3)

(4, 4)

(5, 5)

(6, 6)

1.414 10

15

20

25

30 J

35

40

45

50

Fig. 36 Calculated weighted average values of the C–C distance as a function of the tube length for armchair (n,n) SWCNTs (Reprinted with permission from reference Galano (2006). Copyright 2006 Elsevier)

lengths of the armchair SWCNTs have the general formula C(2n)k H4n with kD ª/2. Galano considered (3,3), (4,4), (5,5), and (6,6) armchair SWCNTs with k of 6 to 26 (i.e., from 6 to 26 carbon atom layers). There are two types of bonds in the perfect (n,n) armchair SWCNTs; one is perpendicular to the tube axis (rI ) and another one is nearly parallel to the tube axis (rII ). The maximum difference between rI and rII was obtained in the case of the narrow diameter (3,3) tube. The influence of diameter on the weighted average values of the CC distances is larger than the influence of the tube length (Fig. 36; Galano 2006). The frontier molecular orbitals (HOMO and LUMO) play an important role in SWCNTs since they are helpful in predicting a number of ground-state properties of molecules. According to Huckel theory, the (n,n) armchair SWCNTs should be metallic (Saito et al. 1998), but the finite-length armchair SWCNTs are semiconducting with a finite size of the HOMO–LUMO gap (Cioslowski et al. 2002). The computed HOMO–LUMO gaps for (3,3) to (6,6) SWCNTs were reported to be lower than the corresponding value for fullerene C60 . The HOMO–LUMO gap oscillates as the tube length increases for all of these armchair tubes (Fig. 37). The behavior of narrow diameter (3,3) tube is different from other armchair SWCNTs (Galano 2006). Matsuo et al. classified the structures of finite-length armchair (5,5) and (6,6) SWCNTs as Kekule, incomplete Clar, and complete Clar networks depending on the exact length of the tubes. The (5,5) and (6,6) SWCNTs were elongated layer by layer of 10 and 12 carbon atoms, respectively (Matsuo et al. 2003). The local aromaticity of different lengths of the tubes was evaluated using the NICS (nucleusindependent chemical shift) calculations (GIAO-SCF/6-31G*//HF/6-31G* level).

HUMO-LUMO gap (eV)

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . . 1.8

1.8

1.6

1.6 (3, 3)

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0.0

0.0 10

20

30

40

50

1.8 HUMO-LUMO gap (eV)

(4, 4)

1.4

1.4

49

10

20

30

40

50

1.8

1.6

1.6

(5, 5)

1.4

(6, 6)

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0.0

0.0 10

20

30

40 J

50

10

20

30

40

50

J

Fig. 37 Variations of HOMO–LUMO gaps as increasing the tube length for the armchair SWCNTs. There is no periodicity in (3,3) tube and the shaded region indicated the broken periodicity in other tubes (Reprinted with permission from reference Galano (2006). Copyright 2006 Elsevier)

Matsuo et al. pointed out that the geometry of C50 H10 is similar to the equatorial belt of the fullerene C70 . Bond lengths of optimized structures exhibit oscillation with increase in tube length for both (5,5) and (6,6) armchair SWCNTs. The schematic structures of Kekule, incomplete Clar, and complete Clar networks for (5,5) and (6,6) SWCNTs are depicted in Fig. 38 along with the NICS values of dissimilar benzenoid rings. The energy of frontier molecular orbitals and HOMO–LUMO gap also oscillate as the length of the nanotube increases (Fig. 39). The Kekule structure shows larger HOMO–LUMO gap than the other two. It was reported that the band gap will eventually disappear at a certain tube length (Matsuo et al. 2003). The pyramidalization angle ( P ) and  -orbital misalignment angles are useful to gauge the reactivity of the carbon atom sites of SWCNTs. The end caps of SWCNTs resemble fullerene hemisphere; thus, the end caps are expected to be more reactive than sidewalls irrespective of the diameter of the SWCNTs. Carbon atoms in fullerene are more distorted than those in the corresponding SWCNTs. For example, the carbon atom of (10,10) armchair SWCNT has the pyramidalization angle ( P ) of about 3.0ı , while the carbon atom of the fullerene with corresponding

50

T.C. Dinadayalane and J. Leszczynski

(i) C40H20 [5, 5], C48H24 [6, 6] A

Kekule

c

b d

(ii) C50H20 [5, 5], C60H24 [6, 6] b d

B

e

(iii) C60H20 [5, 5], C72H24 [6, 6] c

B

b d e

f

(iv) C70H20 [5, 5], C84H24 [6, 6] B

c g

C

b d e f

(v) C80H20 [5, 5], C96H24 [6, 6] b

B

g

d f e

C

h

(vi) C90H20 [5, 5], C108H24 [6, 6]

B C D

B

–3.90 –12.30

–4.55 –13.09

A B

–9.09 –0.25

–10.59 –0.64

A B C

–8.64 –8.67 1.29

–9.84 –9.50 1.08

A –4.06 B –11.42 C –0.77

–5.36 –12.30 –1.26

a

A

Complete Clar

A

a c

A

Incomplete Clar

–9.09

a

A

Kekule

–8.62

a

A

Complete Clar

A

a c

A

Incomplete Clar

NICS [5, 5] [6, 6]

a

c g

b d f e h i

A B C D

–7.01 0.04 3.35 –8.54

–9.58 –1.55 1.43 –11.58

Fig. 38 Schematic structures and color-coded NICS maps of finite-length (5,5) and (6,6) SWCNTs. Hydrogen atoms are omitted for clarity. Chemical bonds are schematically represented by using single bond (solid single line; bond length > 1.43 Å), double bond (solid double line; bond length < 1.38 Å), single bond halfway to double bond (solid dashed line; 1.43 Å > bond length > 1.38 Å), and Clar structure (i.e., ideal benzene). NICS coding: red, aromatic < 4.5; blue, nonaromatic > 4.5 (Reprinted with permission from reference Matsuo et al. 2003. Copyright 2003 American Chemical Society)

radius (fullerene C240 ) has the  P of about 9.7ı (the hemisphere of fullerene C240 can be capped to (10,10) SWCNT) (Niyogi et al. 2002). Chen et al. mentioned that the pyramidalization angle of the C atoms of the sidewalls of SWCNTs is smaller compared to that of the fullerenes of same radius. As a consequence, the

6.5

–1.5

6.0

–2.0

5.5

–2.5

5.0

–3.0

4.5

–3.5

4.0

–6.0

3.5

–6.5

3.0

–7.0 –7.5

40

70

100 300 160 carbon number (= 10j)

190

LUMO (eV)

51

HOMD (eV)

HOMO / LUMO gap (eV)

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

–8.0

Fig. 39 Variation of HOMO–LUMO energies and HOMO–LUMO energy gap with increase in tube length of (5,5) armchair SWCNT (C10j H20 ) (Reprinted with permission from reference Matsuo et al. (2003). Copyright 2003 American Chemical Society)

covalent functionalization to SWCNTs is less favorable compared to fullerenes of same radius (Chen et al. 2003). The  -orbital misalignment is likely to be a main source of strain in the SWCNTs. For both armchair and zigzag SWCNTs, the pyramidalization angle and the  -orbital misalignment angle decrease with increase in diameter of the tube (Fig. 40; Niyogi et al. 2002).

Covalent Functionalization of SWCNTs: H and F Atom Chemisorptions The covalent functionalization of SWCNTs, which modifies the properties of the tubes, has become a challenging field of research for the past few years (Bettinger 2006; Hirsch 2002; Niyogi et al. 2002; Vostrowsky and Hirsch 2004). Functionalization of tubes is considered to be promising to produce carbon nanotube-based materials for selective applications (Bettinger 2006; Cho et al. 2008; Denis et al. 2009). The binding of hydrogen with SWCNTs has generated a lot of experimental and theoretical interests due to their potential application in hydrogen storage (Dillon et al. 1997; Dinadayalane and Leszczynski 2009; Dinadayalane et al. 2007b; Kaczmarek et al. 2007; Nikitin et al. 2005; Ormsby and King 2007; Yang et al. 2006c; Zhang et al. 2006). Scientists have tried to obtain knowledge on the mechanism of hydrogen adsorption in SWCNTs. They attempted to design the viable nanotube-based hydrogen storage material to meet the Department of Energy (DOE) target of 6.5 wt% at ambient temperature (Dillon et al. 1997; Dinadayalane

52

T.C. Dinadayalane and J. Leszczynski

b 6.0

Pyramidalization angle (degrees)

Pyramidalization angle (degrees)

a 5.5 5.0 4.5 4.0 3.5 3.0 2.5 (5, 5)

(7, 7)

(6, 6)

(8, 8)

5.5 5.0 4.5 4.0 3.5 3.0 2.5 (10, 0) (12, 0) (14, 0) (16, 0) (18, 0) (20, 0)

(9, 9) (10, 10)

armchair SWCNT

Zigzag SWCNT

d 22

π-orbital misalignment angle (degrees)

π-orbital misalignment angle (degrees)

c 20 18 16 14 12 10 8 (5, 5)

(6, 6)

(7, 7)

(8, 8)

(9, 9) (10, 10)

armchair SWCNT

22 20 18 16 14 12 10 8 (10, 0) (12, 0) (14, 0) (16, 0) (18, 0) (20, 0) Zigzag SWCNT

Fig. 40 The change of pyramidalization angle (a, b) at the carbon atom and the  -orbital misalignment angle (c, d) between two adjacent carbon atoms of armchair (a, c) and zigzag (b, d) SWCNTs. The  -orbital misalignment angle is zero for the carbon atoms of the circumferential bond in armchair tube and axial bond in zigzag tube (The data was taken from reference Niyogi et al. (2002))

and Leszczynski 2009; Dinadayalane et al. 2007b; Kaczmarek et al. 2007; Nikitin et al. 2005; Ormsby and King 2007; Yang et al. 2006c; Zhang et al. 2006). The experimental investigations reported the chemisorption of H atoms on the surface of SWCNTs as promising approach to meet DOE’s target of hydrogen storage (Nikitin et al. 2005; Zhang et al. 2006). Chemisorptions of hydrogen atoms on the surface of SWCNTs were investigated (Dinadayalane and Leszczynski 2009; Dinadayalane et al. 2007b; Kaczmarek et al. 2007; Ormsby and King 2007; Yang et al. 2006c). The covalent functionalization of SWCNTs by H atoms is a hot topic. Thus, we discuss the quantum chemical studies of the hydrogen chemisorption on different types of SWCNTs. Yang et al. studied, using DFT and ONIOM calculations, the chemisorption of atomic hydrogen(s) on the open-ended finite-sized (5,0), (7,0), and (9,0) zigzag and

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

53

Fig. 41 Finite-sized small carbon nanotube models of (5,0), (7,0), and (9,0) zigzag and (5,5) armchair SWCNTs considered for low occupancy of H chemisorptions (Reprinted with permission from reference Yang et al. (2006c)). Copyright 2006 American Chemical Society)

(5,5) armchair SWCNTs (Fig. 41; Yang et al. 2006c). They compared the binding energies obtained for nanotubes with results of the model graphene sheet in order to examine the effect of curvature. It was reported that the chemisorptions of H atoms to the exterior wall of the SWCNTs are more favorable than the interior walls. The H chemisorption has strong dependence of tube diameter and helicity or chirality in both interior (endohedral) and exterior (exohedral) addition. In case of single H atom addition, the binding energy (chemisorption energy), which is the reaction energy for H chemisorption with SWCNT, decreases with increase in tube diameter. In the chemisorption of two hydrogen atoms in the interior and exterior walls of (5,0) and (7,0) SWCNTs, two hydrogen atoms prefer to bind at alternate positions rather than adjacent positions. This was attributed to the crowding effect when two hydrogen atoms occupy in the adjacent positions. In the case of (5,0) SWCNT, chemisorption of ten hydrogen atoms (33 % coverage) decreases the magnitude of chemisorption energy, which is further decreased by an increase in the coverage to 50 %. Similar to the situation in zigzag SWCNTs, two hydrogen atoms prefer to attach at alternate carbon sites rather than adjacent sites in the graphene sheet. Significantly large deviation of chemisorption energy between the graphene sheet and zigzag SWCNTs (H atoms chemisorbed on the exterior wall) was reported. It was found that the chemisorptions of H atoms with small diameter SWCNTs are much more favorable than with the graphene sheet (Yang et al. 2006c). We investigated a single-hydrogen chemisorption, the preference of the positions (i.e., 1-2, 1-2’, 1-3, or 1-4 positions) for the chemisorption of two hydrogen atoms considering (3,3), (4,4), (5,5), and (6,6) armchair SWCNTs of 9 and 15 carbon atom layers. The SWCNTs of 15 carbon layers considered in our study are shown in Fig. 42. The addition of H atoms on the outer wall of SWCNT (exohedral addition) has only been considered in our study. We performed DFT calculations using B3LYP/6-31G(d) level for full geometry optimizations. The finite-length SWCNTs were capped with hydrogen atoms to avoid dangling bonds. The reaction energies for hydrogen chemisorption (Er ) on the external surface of

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T.C. Dinadayalane and J. Leszczynski

C2

C1C4 C3 C2

N33_15cl

C2 C1 C2

C4 C3

N44_15cl

C2 C1 C2

C4 C3

N55_15cl

C2

C1 C2

C4 C3

N66_15cl

Fig. 42 Structures of (3,3), (4,4), (5,5), and (6,6) armchair SWCNTs of 15 carbon layers (15 cl) considered for the chemisorption of one and two H atoms. The carbon atom sites for attachment of H atoms are shown (Reprinted with permission from reference Dinadayalane et al. (2007b)). Copyright 2007 American Chemical Society)

SWCNTs have been calculated using the formula Er D ESWCNTCnH – ESWCNT  nEH , where ESWCNTCnH denotes the total energy of hydrogen-chemisorbed nanotube, n represents the number of hydrogen atoms chemisorbed, and ESWCNT and EH correspond to the energies of pristine nanotube and the hydrogen atom, respectively. The reaction energy Er can also be considered as hydrogen chemisorption energy. The chemisorption of hydrogen is an exothermic process if the value of Er is negative (Dinadayalane et al. 2007b). We observed the rupture of circumferential C1C2 bond when two hydrogen atoms were chemisorbed in the case of (3,3) SWCNT of 15 carbon layers. As shown in Fig. 43, the reactions of single as well as two hydrogen chemisorptions on the surface of armchair SWCNTs are highly exothermic. The reaction energy for the addition of two H atoms is more than two times that of one H chemisorption except for H(1,3) addition. Our computational study revealed a competition between H(1,2) and H(1,4) addition in the case of (5,5) and (6,6) SWCNTs, but such competition was not seen in the case of narrow diameter (3,3) and (4,4) SWCNTs. Increasing the length of the tube has pronounced effect on the reaction energy of hydrogen chemisorption. In case of armchair SWCNTs, the chemisorption of two hydrogen atoms at alternate positions is thermodynamically less favored compared to H(1,2) and H(1,2’) additions regardless of the length and diameter of the tubes (Dinadayalane et al. 2007b). The least positional preference of H(1,3) for armchair SWCNTs is different compared to the results of zigzag-type nanotubes (Yang et al.

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . . Nanotubes with 9 carbon layers

Er (in kcal / mol)

–40

Nanotubes with 15 carbon layers

(3, 3) SWCNT (4, 4) SWCNT (5, 5) SWCNT (6, 6) SWCNT

–20 –40 Er (in kcal / mol)

–20

–60 –80 –100 –120 –140

55

(3, 3) SWCNT (4, 4) SWCNT (5, 5) SWCNT (6, 6) SWCNT

–60 –80 –100 –120 –140

–160

–160 H(1) H(1, 2′) H(1, 2) H(1, 3) H(1, 4)

H(1) H(1, 2′) H(1, 2) H(1, 3) H(1, 4)

Fig. 43 The variation of reaction energies at the B3LYP/6-31G(d) level for the chemisorption of one and two hydrogen atoms on the external surface of (3,3), (4,4), (5,5), and (6,6) armchair singlewalled carbon nanotubes (SWCNTs) (Reprinted with permission from reference Dinadayalane et al. (2007b)). Copyright 2007 American Chemical Society)

2006c; Dinadayalane et al. 2007b). We found that the H chemisorption on nanotubes of different diameters and the positions of two hydrogen atoms chemisorbed on the surface of armchair SWCNTs can be characterized by CH stretching frequencies of chemisorbed hydrogen atoms (Dinadayalane et al. 2007b). In the investigation of chemisorption of H atoms with (3,3) and (4,4) SWCNTs of different lengths, we found that changing the length of the nanotube has significant effect on the reaction energy of hydrogen chemisorption and HOMO–LUMO gap of pristine and hydrogen-chemisorbed SWCNTs (Kaczmarek et al. 2007). The reactivity pattern was predicted for the hydrogenation in chiral SWCNTs (Ormsby and King 2007). Investigations involving chiral SWCNTs are more challenging than for zigzag and armchair SWCNTs because single unit cell contains many atoms; consequently, more computational resources are required. Computational study demonstrated that hydrogenation of the fully benzenoid (12,9) SWCNT was significantly less energetic (by 8 kcal/mol per mol H2 ) than the hydrogenation of (12,7) and (12,8) SWCNTs (Fig. 44). Furthermore, the hydrogenation at an internal Clar double bond or bonds was reported to be more exothermic than at randomly selected internal bonds. Like other polycyclic aromatic hydrocarbons, hydrogenation of double bonds is energetically preferred over hydrogenation of aromatic sextets. The frontier molecular orbitals (HOMO and LUMO) of chiral SWCNTs have maximum amplitude at the double bonds suggesting that Clar’s model also predicts the kinetic reactivity. Thus, the frontier molecular orbitals are useful in predicting the favorable sites for hydrogenation in chiral SWCNTs (Ormsby and King 2007). In the early 2000s, experimentalists found that partial fluorination of the SWCNTs could be used as a technique for cutting the nanotubes of varying lengths. However, the mechanism of cutting the nanotube is not clear (Gu et al. 2002). In a computational study, Bettinger observed a strong oscillation of reaction energy for

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a (12, 8)

(12, 7)

–6.1 (kcal / mol) / mol H2

b

–6.6 (kcal / mol) / mol H2

c

(12, 7)

(12, 8) 90°

+1.6 (kcal / mol) / mol H2

(12, 7)

90°

Seams

90°

Seams

(12, 9)

90°

90°

90°

90°

90°

90°

(12, 8)

90°

(12, 9)

(12, 9) 90°

90°

d

Fig. 44 (a) Hydrogenation of equivalent vectors of (12,7), (12,8), and (12,9) chiral SWCNT segments (planar representation). (b) HOMO plotted on the isodensity surface for (12,7), (12,8), and (12,9) chiral SWCNTs. (c) LUMO plotted on the isodensity surface for (12,7), (12,8), and (12,9) chiral SWCNTs. (d) The hydrogenated model chiral SWCNT (side and top views). The locations of double bond are indicated in (b) and (c). HOMO and LUMO isodensity surface structures generated at AM1 method (Reprinted with permission from reference Ormsby and King (2007). Copyright 2007 American Chemical Society)

the addition of F atom on the external surface of (5,5) SWCNT of varying lengths (Fig. 45). The computed reaction energy oscillation ranges from 43 to 68 kcal/mol at the UB3LYP/6-31G(d) level using UPBE/3-21G optimized geometries. The shortest tube exhibited the highest exothermicity. The energy oscillation was reported to be

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

57

n

E/ k cal mol–1

–30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

–40

–50

–60 C30+10nH20 + F

F-C30+10nH20

(U)B3LYP / 6-31G* // (U)PBE / 3-21G –70

Fig. 45 Variation of reaction energy for the addition of F atom to (5,5) armchair SWCNT of different lengths (n – increasing number of carbon layers) (Reprinted with permission from reference Bettinger (2004)). Copyright 2004 American Chemical Society)

periodic with large exothermicity for the fully benzenoid frameworks, in agreement with their smaller band gaps compared to Kekule and incomplete Clar structures. Computational study demonstrated that the addition of F atom to the sidewall of SWCNT strongly depends on the length of the nanotube (Bettinger 2004). As observed in H atom addition (Dinadayalane et al. 2007b; Kaczmarek et al. 2007; Yang et al. 2006c), the F atom addition to the sidewalls of SWCNTs transforms the carbon atom hybridization from sp2 to sp3 (Bettinger 2004). The chemical reactivity of carbon nanotubes is governed by the local atomic structure. As mentioned earlier, the pyramidalization angle is an important parameter in predicting the chemical reactivity of SWCNTs. Park et al. predicted the hydrogenation and fluorination energies of each carbon from its pyramidalization angle for zigzag SWCNTs (Fig. 46). They formulated the Etotal for chemisorption of H and F atom on the external surface of zigzag tubes of different diameters as a function of pyramidalization angle of the binding site of tubes. They revealed that the metallic zigzag SWCNTs are slightly more reactive than the semiconducting SWCNTs. Furthermore, the fluorination is more viable than the hydrogenation (Park et al. 2003).

Theoretical Studies on Common Defects in SWCNTs Investigating the atomic defects is important in tailoring the electronic properties of SWCNTs. Recent experimental study reported a method to selectively modify the electronic properties of semiconductor SWCNTs by the creation and annihilation of point defects on their surface with the tip of a scanning tunneling microscope (STM) (Berthe et al. 2007). Such experimental study motivates theoreticians to explore the structures, energetics, reactivities, and electronic properties of SWCNTs containing different types of defects.

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a CNT + H

25

(13, 0)

(5, 0)

–4 –1

b

10 25

CNT + F

20

0

3

6 9 Initial Pyramidal Angle

15

(5, 0)

(6, 0)

(10, 0) (9, 0) (8, 0)

(15, 0)

–4

(12, 0)

(13, 0)

–2

–3

15

Pyramidal Angle of Final Structure

C60

20

(6, 0)

(15, 0)

(12, 0)

–3

Graphite

Total Reaction Energy (eV)

Graphite

–2

(10, 0) (9, 0) (8, 0)

–1

10 12

Fig. 46 DFT-computed Etotal () and pyramidalization angle ( P ) values (ı) for fully relaxed configurations and their estimated values (solid curves) for (a) hydrogenation and (b) fluorination (Reprinted with permission from reference Park et al. (2003). Copyright 2003 American Chemical Society)

Stone–Wales Defect The Stone–Wales defect can be created by 90ı rotation of one of the CC bonds in the hexagonal network of SWCNTs. Two types of CC bonds exist in each of armchair and zigzag SWCNTs. Therefore, one can generate Stone–Wales defect in two different orientations in both armchair and zigzag SWCNTs (Fig. 47). DFT calculations revealed that the formation energies of (5,5) SWD_II and (10,0) SWD_II are lower than those of (5,5) SWD_I and (10,0) SWD_I. The computed formation energies of (5,5) SWD_I, (5,5) SWD_II, (10,0) SWD_I, and (10,0) SWD_II are 66.4 (2.88 eV), 57.0 (2.47 eV), 68.9 (2.99 eV), and 63.0 (2.73 eV) kcal/mol, respectively. The formation energy was calculated as the relative energy of the Stone–Wales defective tube with respect to the corresponding defect-free SWCNT. It was reported that the formation of Stone–Wales defect causes no change in the HOMO–LUMO gaps (Yang et al. 2006a, b).

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . . C1-C2 circumferential C2-C3 nearly axial

90 deg. rotation of C1-C2 bond

(5, 5) SWD_I

(5, 5)

C1-C2 axial C1-C3 nearly axial

90 deg. rotation 90 deg. rotation (10, 0) of C2-C3 bond of C1-C3 bond

(5, 5) SWD_II

59

(10, 0) SWD_I

90 deg. rotation of C1-C2 bond

(10, 0) SWD_II

Fig. 47 Generation of Stone–Wales defect with different orientations in (5,5) armchair and (10,0) zigzag SWCNTs. The atoms in the Stone–Wales defect region are highlighted in yellow color (Reprinted with permission from reference Yang et al. (2006a). Copyright 2006 American Chemical Society. Reprinted with permission from Yang et al. (2006b). Copyright 2006, American Institute of Physics)

We investigated the structures, formation energies, and reactivities of Stone– Wales defect with two different orientations and different locations from the end of tube in armchair (5,5) SWCNTs of C80 H20 (I) and C100 H20 (II) (Fig. 48a; Dinadayalane and Leszczynski 2007b). We employed HF/4-31G, HF/6-31G(d), B3LYP/3-21G, and B3LYP/6-31G(d) levels of theory. HF/4-31G level overestimates the Stone–Wales defect formation energy compared to B3LYP/6-31G(d) level. Our study revealed that B3LYP/3-21G level, which provides reasonable energy estimation, may be employed for large nanotube systems to compute the defect formation energies when the calculations at the B3LYP functionals with large basis sets are prohibitive. Our study showed that the Stone–Wales defective (5,5) armchair SWCNTs generated by rotation of nearly axial bond (ABR) are more stable than those created by circumferential bond rotation (CBR) as shown in Fig. 48b. The SW defect structures generated by ABR show lower HOMO–LUMO gap than those created by CBR and the defect-free SWCNTs (Dinadayalane and Leszczynski 2007b). Bettinger demonstrated in a comprehensive computational study that some of the bonds of SW defect show higher reactivity than pristine tube; others are less reactive (Bettinger 2005). Computational studies explained the reactivity of carbon atom sites based on the pyramidalization angles (Akdim et al. 2007; Bettinger 2005; Lu et al. 2005). Lu et al. investigated addition of O, CH2 , and O3 across CC bonds of SW defective and defect-free armchair SWCNTs. They showed that the central CC bond of the SW defect in armchair SWCNT (SW defect generated by CBR) is chemically less reactive than that in the perfect tube, and it was attributed to small local curvature in the carbon atoms of central CC bond of the SW defect (Lu et al. 2005). We found that the values of pyramidalization angles do not completely

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a 5 1

9 7 6 8 2 3 4

6 54

I, C30H20 (D5d)

8 9 7 1 10 11 3 2 14 16 15 1312

I-SWD1-CBR, (C5)

9 1 10 11 2 14 5 4 316 15 12 13

I-SWD2-CBR, (C4)

8 9 1 10 11 3 2 14 12 16 15 13 7

6 54

12 13 14 11 15 6 1 3 3 16 98 7 6 54

8

7

6

I-SWD3-ABR, (C2)

8

11 12 9 10 8 7 1 2 13 14 6 5 4 3 15 16

I-SWD4-ABR, (C2)

9

7 1 10 3 2 14 11 5 4 16 15 1312 6

9 8 7 1 10 6 3 2 14 11 5 4 16 15 1312

11

9

10

5 1

6 7 2 3

8 4

II-SWD1-CBR, (C ) 5

13 14 11 12 15 10 1 2 3 16 9 8 7 6 5 4

II-SWD2-CBR, (C5)

10 11 12 9 1 13 2 14 8 7 3 15 6 5 4 16

II, C H (D ) 100 20 5d

II-SWD4-ABR, (C ) 2

II-SWD5-ABR, (C ) 1

II-SWD3-CBR, (C5)

12 13 14 11 15 10 1 2 3 16 7 6 9 8 5 4

II-SWD6-ABR, (C ) 1

SW defect formation energy (kcal / mol)

b 70 60 50 40 30 20 10

I-S

W I-S D1 W -CB D I-S 2- R W CB R D I-S 3-A W B R D 4A B R IISW D IISW 1-C B II- D1- R SW C B R II- D3SW C B II- D4 R SW -A B II- D5- R SW A B R D 6A B R

0

Fig. 48 (a) B3LYP/6-31G(d) level optimized structures of defect-free (5,5) armchair SWCNTs and the Stone–Wales defect tubes generated by the 90ı rotation of circumferential and nearly axial C–C bonds. (b) The Stone–Wales defect formation energy obtained at the B3LYP/6-31G(d) level (Reprinted with permission from reference Dinadayalane and Leszczynski (2007b). Copyright 2007 Elsevier)

explain the reactivity of different bonds of SW defect region for cycloaddition reactions and the reactivity may arise from various other reasons, in addition to topology. We concluded that the cycloaddition reactions across the CC bond

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

a

(5,5)

(5,5)

(5,5)

61

(5,5)

Defect free (8,0)

n=7 (8,0)

n=8 (8,0)

n=9 (8,0)

Defect formation energy (Efn, eV)

b LDA GGA

10 8 6 4 2 0

n=7 n=8 n=9 (5,5) SWCNT

n=7 n=8 n=9 (8,0) SWCNT

Defect ring size (n) Defect free

n=7

n=8

n=9

Fig. 49 (a) Fully relaxed structures of the defect-free and the defective (5,5) (upper panel) and (8,0) (lower panel) SWCNTs obtained by the GGA calculations. Gray balls and rods are the carbon atoms and bonds shorter than 1.5 Å, respectively. Carbon ring defects (given in red color) are indicated by the thick circles. (b) Defect formation energy (Enf ) of n-membered carbon rings for (5,5) and (8,0) calculated using LDA and GGA (The pictures in (a) and the data for (b) were taken with permission from Nishidate and Hasegawa (2005). Copyright 2005 by the American Physical Society)

shared by two heptagons (7-7 ring fusion) need not always be less reactive than the corresponding bond in the pristine structure and the reactivity of that bond depends on the orientation of the SWD in the SWCNTs (Dinadayalane and Leszczynski 2007b).

Topological Ring Defects Nishidate and Hasegawa calculated the formation energies of n-membered topological ring defects with n D 7 (heptagon), n D 8 (octagon), and n D 9 (enneagon) in (5,5) armchair and (8,0) zigzag SWCNTs (Fig. 49a). They used both local density approximation (LDA) and the generalized gradient approximation (GGA: PW91). The spin-polarized projector augmented-wave (PAW) implemented in VASP code was employed for calculations, and periodic boundary condition was used (Nishidate and Hasegawa 2005). The defect formation energy (Enf ) of n-membered rings was calculated as the energy difference between the total energy of defective SWCNT (Entot ) and that of n the pristine SWCNT (Etot ); i.e., Efn D Etot  Etot . The number of atomsof each defective SWCNT is the same as that of the corresponding SWCNT. Computational

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T.C. Dinadayalane and J. Leszczynski

study showed that the defect formation energy increases with increase in defect ring size (n) (Fig. 49b). In general, LDA method yielded higher defect formation energy than generalized gradient approximation. The SWCNTs were reported to be more fragile than the graphene sheet for defect formation. Distortion of the SWCNTs became larger as the defect ring size increases (Nishidate and Hasegawa 2005).

Single- and Di-vacancy Defects An ideal single vacancy (SV) with three dangling bonds (DBs) was generated by removing one carbon atom from the perfect (5,5) and (10,0) SWCNTs. Upon geometry optimization, an ideal SV with three dangling bonds rearranged into a pentagonal ring and one DB (Yang et al. 2006a). Hence, this defect is called a 5-1DB defect (Lu and Pan 2004; Yang et al. 2006a). In case of (5,5) SWCNT, the optimization of the ideal SV resulted in two different 5-1DB defects. The structures were named as (5,5) SV_I and (5,5) SV_II as shown in Fig. 50. The latter structure was reported to be energetically more favorable (by 1.20 eV) than the former one. A similar behavior was observed for (10,0) zigzag SWCNT. The bond length of

Ideal SV in (5,5)

(5,5) SV_I

(5,5) SV_II

Ideal SV in (10,0)

(10,0) SV_I

(10,0) SV_II 5 7 4 9 3 8 1 2

(5,5) DV_I

Ideal DV in (5,5)

Ideal DV in (5,5)

(5,5) DV_II

1

(10,0) DV_I

Ideal DV in (10,0)

Ideal DV in (10,0)

6

3 2

4 5 6

(10,0) DV_II

Fig. 50 Configurations of the single (SV) and double (DV) vacancies. Ideal SV and DV mean one and two carbon atoms removed from pristine SWCNT and the structures were not relaxed. Single and double vacancies in two different orientations (I and II) given for (5,5) and (10,0) SWCNTs. Carbon atoms in defect region are given in yellow color (Reprinted with permission form reference Yang et al. (2006a). Copyright 2006 American Chemical Society)

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

63

the new CC bond forming five-membered ring is 1.64 Å in (5,5) SV_I, while that of 1.55 Å was obtained for (5,5) SV_II (Yang et al. 2006a). Lu and Pan found using tight-binding calculations that the single-vacancy defect formation energy for (n,n) armchair SWCNTs increases monotonically with increasing tube radii. The formation energy curve of single-vacancy defects in the zigzag (n,0) SWCNTs is periodic, which is mainly characterized by metallic (n,0) tubes (such as (6,0), (9,0), (12,0), (15,0), etc.) (Lu and Pan 2004). An ideal di-vacancy can be generated by removing two carbon atoms. Two different orientations are possible because of the presence of two different types of bonds in both (5,5) armchair and (10,0) zigzag SWCNTs. Upon geometry optimizations, SWCNTs with ideal di-vacancies yielded 5-8-5 (five-eight-five-membered rings) defects in two different orientations in both armchair- and zigzag-type tubes. Computational study revealed that (5,5) DV_II and (10,0) DV_II configurations are energetically more favorable than (5,5) DV_I and (10,0) DV_I by 0.97 and 0.63 eV, respectively. (5,5) DV_II and (10,0) DV_II can be obtained by removing the carbon atoms with the dangling bond from (5,5) SV_II and (10,0) SV_II, which are the most stable configurations among the possible types of SVs in each type of tubes (Yang et al. 2006a). The di-vacancy in graphene as well as SWCNTs generates structures possessing two pentagons side by side with an octagon (585 structure) as a result of geometry optimization. The 585 configuration can reconstruct further into a complex structure composed of three pentagons and three heptagons, called 555777 defect structure. In fact, 555777 configuration is more stable than 585 configuration in graphene. Amorim et al. investigated the stability of these types of configurations, derived by di-vacancies, in armchair and zigzag SWCNTs considering different tube diameters (Amorim et al. 2007). The 585 defect in SWCNTs has two possible orientations with respect to the tube axis: perpendicular and tilted in armchair and parallel and tilted in zigzag SWCNTs. For the (5,5) SWCNT, the perpendicular orientation is less stable than the tilted one by 3.5 eV. In case of (8,0) SWCNT, the tilted orientation is less stable by 2.7 eV than the parallel one. Only the tilted and the parallel orientations of defects were considered for armchair and zigzag SWCNTs, respectively (Fig. 51; Amorim et al. 2007). In contrast to graphene, the 585 defect was predicted to be more stable than the 555777 defect in both armchair and zigzag SWCNTs. Both 585 and 555777 defects in nanotubes (both armchair and zigzag) are more stable than in graphene. The defect formation energy increases monotonically as the diameter of the armchair SWCNT increases (Fig. 52a). The energy difference between the 585 defect and the 555777 defect was computed to be 1.6 eV in case of zigzag tubes, while the difference was reported to be 0.7 eV for armchair SWCNTs. Zigzag SWCNTs exhibited oscillations in the formation energies and the oscillations were related to the alternation between semiconductor and metallic character of the (n,0) zigzag SWCNTs (Fig. 52b; Amorim et al. 2007).

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Fig. 51 Ball and stick models for the final geometries of the defects: (a) 585 and (b) 555777 in graphene; (c) 585 tilted and (d) 555777 in the (5,5) armchair SWCNT; (e) 585 parallel and (f) 555777 in the (8,0) zigzag SWCNT. The carbon atoms in pentagons are marked in red, the ones that complete either the octagons or the heptagons are colored in blue, and the ones at the center of the C3 symmetry operation in the 555777 defects are colored in green. All the others are colored gray (Reprinted with permission from reference Amorim et al. (2007). Copyright 2007 American Chemical Society)

Outlook of Potential Applications of Carbon Nanostructures Graphene is used as a base material for nanoelectromechanical systems (NEMS) due to its lightweight and stiffness properties (Bunch et al. 2007; Robinson et al. 2008). Functionalized graphene can be exploited for water splitting and hydrogen production. Scientists produce graphene-based materials with high structural and electronic quality for the preparation of transparent conducting electrodes for displays and touch screens. Solution processing and chemical vapor deposition are the ideal means to produce thin films that can be used as electrodes in energy devices such as solar panels, batteries, fuel cells, or in hydrogen storage (Bonaccorso et al. 2015). Graphene-based resonators have notable advantages in comparison with nanotubes. Reduced graphene oxide films are used to make drum resonators. The high Young’s modulus, extremely low mass, and large surface area make the graphene-based resonators ideally suited for use as mass, force, and charge sensors (Ekinci et al. 2004; Knobel and Cleland 2003; Lavrik and Datskos 2003). Graphene can be used for metallic transistor applications and ballistic transport. One of the potential applications of graphene sheet is its use as membrane for separation (Jiang et al. 2009). Graphene may be useful for electro- and magneto-optics (Geim 2009). Graphane (fully hydrogenated graphene) nanoribbons have quite promising applications in optics and opto-electronics due to the wide band gap. Graphene may also be used for transistor applications (Novoselov et al. 2004). Research into applications for carbon graphene nanosheets has focused on their uses as platforms for next-wave microchips, active materials in field emitter arrays for flat panel screen displays; in gas sensors (Wang et al. 2015a), biological sensors, and medical imaging devices; in solar energy cells; and in high-surface area electrodes for use

Fundamental Structural, Electronic, and Chemical Properties of Carbon. . .

a

7.6

≈ 0.8 eV

7.2 6.8

585 555777 585 - Graphene 555777 - Graphene

6.4 E1[eV]

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d[Å] Fig. 52 Formation energy vs. diameter for the 585 and 555777 defects in (a) armchair and (b) zigzag SWCNTs. The limits for graphene were given in both graphs (Reprinted with permission from reference Amorim et al. (2007). Copyright 2007 American Chemical Society)

in bioscience. Graphene-based materials are known for energy and environmental applications (Bo et al. 2015; Chen et al. 2015; Yuan and He 2015). Graphene is a possible replacement material where carbon nanotubes are presently used (Xia et al. 2009). Graphene-based liquid crystal devices (LCD) show excellent performance with high contrast ratio. Thus, LCDs might be graphene’s first realistic commercial application (Blake et al. 2008). Fullerenes hold possibilities of application in many areas including antiviral activity, enzyme inhibition, DNA cleavage, photodynamic therapy, electron transfer, ball bearings, lightweight batteries, new lubricants, nanoscale electrical switches, new plastics, antitumor therapy for cancer patients, and combustion science and astrophysics (Dresselhaus et al. 1996; Lebedeva et al. 2015). The fullerene derivatives obtained by attachment of electron donor moieties are used as photovoltaic

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devices. The supramolecular design of molecular assemblies involving fullerenes holds the possibility to reach new efficient photovoltaic devices (Hudhomme and Cousseau 2007). Fullerenes show promising biomedical applications (Bakry et al. 2007; Bosi et al. 2003; Mashino et al. 2003; Stoilova et al. 2007; Thrash et al. 1999). The fullerene derivatives showed antibacterial and antiproliferative activities; they inhibited bacteria and cancer cell growth effectively (Mashino et al. 2003). Cationic fullerenes were identified to work as antimicrobial photosensitizers. Bisfunctionalized C60 derivatives have shown the activity against HIV-1 and HIV-2 strains (Bosi et al. 2003). The antiviral activity of fullerene derivatives is based on several biological properties including their unique molecular architecture and antioxidant activity (Bakry et al. 2007). Fullerenes derivatized by hydrophilic moieties are capable of carrying drugs and genes for the cellular delivery (Thrash et al. 1999). The localization of the metallofullerol in bone might be a useful chemotherapeutic agent for treatment of leukemia and bone cancer (Thrash et al. 1999). Several potential applications have been proposed for carbon nanotubes, for example, conductive and high-strength composites, energy storage and energy conversion devices, sensors, field emission display and radiation sources, and nanotube-based semiconductor devices (Baughman et al. 2002; Sinha and Yeow 2005). Supercapacitors with carbon nanotube electrodes can be used for devices that require higher power capabilities than batteries. Nanotubes have potential application as hydrogen storage (Dinadayalane and Leszczynski 2009). CNTs can be added to aircraft to offer EMI (electromagnetic interference) shielding and lightning strike protection. They will also make the aircraft stronger and lighter, allowing for larger payloads and greater fuel efficiency. They may be used in commercial aircraft and in notebook computers to efficiently draw away generated heat without adding additional weight (Sinha and Yeow 2005). Nanotube films may be used by the automobile industry to make cars and trucks stronger yet lighter and, therefore, more fuel efficient. Three-dimensional graphene-CNT hollow fibers with radially aligned CNTs could be useful for efficient energy conversion and storage (Xue et al. 2015; Yu et al. 2014; Zhu et al. 2012). MWCNTs show great potential for use in nanofluidic devices because of their high mechanical strength and fluid transport ability at near-molecular length scales (Sinha and Yeow 2005). Due to the advantages of miniature size of the nanotube and the small amount of material required, the carbon nanotubes are being explored for chemical sensing applications. SWCNTs are promising materials for building highperformance nano-sensors and devices (Close et al. 2008). Defects in SWCNTs play an important role in chemical sensing applications (Robinson et al. 2006). CNTs can be used as implanted sensors to monitor pulse, temperature, blood glucose, and heart’s activity level and can also be used for repairing damaged cells or killing them by targeting tumors by chemical reactions (Sinha and Yeow 2005). Some of the selected applications of carbon nanotubes are shown in Fig. 53. Potential biological and biomedical applications of CNTs are under investigation (Dhar et al. 2008; Karousis et al. 2009; Liu et al. 2007; Sinha and Yeow 2005). Carbon nanotubes have potential to make miniature biological electronic devices,

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Fig. 53 (a) and (b) Electron micrographs of two different AFM cantilever tips, each with a nanotube attached; (a) an SEM (scanning electron microscope) micrograph of a nanotube. (b) A TEM (transmission electron microscope) micrograph of a nanotube (Reprinted with permission from reference Stevens et al. (2000). Copyright 2000 Institute of Physics). (c) Nanoelectromechanics of suspended nanotubes – experimental scheme for measuring the electromechanical property of the nanotube. (d) The SWCNT evolves into an n-type FET (field-effect transistor) after adsorption of PEI (polyethyleneimine). Pictures (c) and (d) were reprinted with permission from reference Dai (2002). Copyright 2002 American Chemical Society). (e) Folate receptor (FR)-mediated targeting and SWCNT-mediated delivery of Pt-containing complex (Reprinted with permission from reference Dhar et al. (2008). Copyright 2008 American Chemical Society)

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including probes and sensors (Sotiropoulou and Chaniotakis 2003; Stevens et al. 2000, 2004). Water-soluble peptidomimetic-functionalized carbon nanotubes have been reported to have antitrypsin activity (Dhar et al. 2008). Functionalized and water-soluble SWCNTs have been explored to find biological applications in the area of drug delivery (Dhar et al. 2008; Karousis et al. 2009; Liu et al. 2007). CNTs could be used as potential delivery tools for peptide-based synthetic vaccines. CNTs are currently being considered as suitable substrate for neuronal growth, as ion channel blockers, and as vectors for gene transfection (Sinha and Yeow 2005). Carbon nanotubes have provided possibilities for applications in nanotechnology. Continuous and optimistic research efforts in the area of carbon nanotubes are required to realize a lot of breakthrough commercial applications.

Summary and Outlook In this chapter, we provided vital information and up-to-date research on carbon nanostructures, particularly graphene, fullerenes, and carbon nanotubes, which are critical in the nanotechnology revolution. This chapter also covered the modeling aspects, especially the current trends of computational chemistry applications in understanding the structures, reactivity, and other properties of abovementioned carbon nanostructures, and their importance in supporting the experimental results. Many aspects of basic research and practical application requirements have been motivating both theoreticians and experimentalists to gain better understanding about the carbon nanostructures. Obtaining knowledge on a specific class of chemical reactions with graphene, fullerenes, and SWCNTs is required for making novel materials as well as producing carbon-based nanomaterials for specific applications. Computational investigations provide opportunity to understand the structures, binding of atoms/molecules with the carbon nanotubes. A systematic and careful computational chemistry approaches could have important implications for the rational design of novel CNT composite materials, novel nanotube-based sensors, as well as for the development of new chemical strategies for SWCNT functionalization. Strong interactions between experimentalists and theoreticians working in the area of carbon nanostructures will enhance the real-time applications rapidly. Future efforts should not only provide high-tech nano-devices but also address fundamental scientific questions. Further exciting developments in nanoscience and nanotechnology are expected. Acknowledgments This work was supported by the High Performance Computational Design of Novel Materials (HPCDNM) Project funded by the Department of Defense (DoD) through the US Army Engineer Research and Development Center (Vicksburg, MS) Contract # W912HZ-06-C0057 and by the Office of Naval Research (ONR) grant 08PRO2615-00/N00014-08-1-0324. JL acknowledges the support from the National Science Foundation (NSF) for the Interdisciplinary Center for Nanotoxicity (ICN) through CREST grant HRD-0833178. TCD acknowledges the start up support provided by the Clark Atlanta University.

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Yumura, T., Nozaki, D., Bandow, S., Yoshizawa, K., & Iijima, S. (2005a). End-cap effects on vibrational structures of finite-length carbon nanotubes. Journal of the American Chemical Society, 127, 11769–11776. Yumura, T., Sato, Y., Suenaga, K., & Iijima, S. (2005b). Which do endohedral Ti2 C80 metallofullerenes prefer energetically: Ti2 @C80 or Ti2 C2 @C78 ? A theoretical study. The Journal of Physical Chemistry B, 109, 20251–20255. Yumura, T., Kertesz, M., & Iijima, S. (2007). Local modifications of single-wall carbon nanotubes induced by bond formation with encapsulated fullerenes. The Journal of Physical Chemistry. B, 111, 1099–1109. Zhang, J., & Zuo, J. M. (2009). Structure and diameter-dependent bond lengths of a multi-walled carbon nanotube revealed by electron diffraction. Carbon, 47, 3515–3528. Zhang, B. L., Wang, C. Z., Ho, K. M., Xu, C. H., & Chan, C. T. (1993). The geometry of large fullerene cages: C72 to C102 . Journal of Chemical Physics, 98, 3095–3102. Zhang, G., Qi, P., Wang, X., Lu, Y., Mann, D., Li, X., & Dai, H. (2006). Hydrogenation and hydrocarbonation and etching of single-walled carbon nanotubes. Journal of the American Chemical Society, 128, 6026–6027. Zhang, H., Cao, G., Wang, Z., Yang, Y., Shi, Z., & Gu, Z. (2008). Influence of ethylene and hydrogen flow rates on the wall number, Crystallinity, and length of millimeter-long carbon nanotube array. The Journal of Physical Chemistry C, 112, 12706–12709. Zhao, K., & Pitzer, R. M. (1996). Electronic structure of C28 , Pa@C28 , and U@C28 . Journal of Physical Chemistry, 100, 4798–4802. Zhao, Y., & Truhlar, D. G. (2007). Size-selective supramolecular chemistry in a hydrocarbon nanoring. Journal of the American Chemical Society, 129, 8440–8442. Zhao, Y., & Truhlar, D. G. (2008). Computational characterization and modeling of buckyball tweezers: Density functional study of concave–convex interactions. Physical Chemistry Chemical Physics, 10, 2813–2818. Zhao, X., Slanina, Z., & Goto, H. (2004a). Theoretical studies on the relative stabilities of C96 IPR fullerenes. The Journal of Physical Chemistry. A, 108, 4479–4484. Zhao, X., Goto, H., & Slanina, Z. (2004b). C100 IPR fullerenes: Temperature-dependent relative stabilities based on the Gibbs function. Chemical Physics, 306, 93–104. Zhou, Z., Steigerwald, M., Hybertsen, M., Brus, L., & Friesner, R. A. (2004). Electronic structure of tubular aromatic molecules derived from the metallic (5,5) armchair single wall carbon nanotube. Journal of the American Chemical Society, 126, 3597–3607. Zhou, L., Gao, C., Zhu, D. D., Xu, W., Chen, F. F., Palkar, A., Echegoyen, L., & Kong, E. S.W. (2009). Facile functionalization of multilayer fullerenes (carbon nanoonions) by nitrene chemistry and grafting from strategy. Chemistry - European Journal, 15, 1389–1396. Zhu, Z. H., Hatori, H., Wang, S. B., & Lu, G. Q. (2005). Insights into hydrogen atom adsorption on and the electrochemical properties of nitrogen-substituted carbon materials. The Journal of Physical Chemistry. B, 109, 16744–16749. Zhu, Y., Li, L., Zhang, C., Casillas, G., Sun, Z., Yan, Z., Ruan, G., Peng, Z., Raji, A.-R., Kittrell, C., Hauge, R. H., & Tour, J. M. (2012). A seamless three-dimensional carbon nanotube graphene hybrid material. Nature Communications, 3, 1225. Zope, R. R., Baruah, T., Pederson, M. R., & Dunlap, B. I. (2008). Static dielectric response of icosahedral fullerenes from C60 to C2160 characterized by an all-electron density functional theory. Physical Review B, 77, 115452-1–115452-5.

Optical Properties of Quantum Dot Nano-composite Materials Studied by Solid-State Theory Calculations Ying Fu and Hans Ågren

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid-State kp Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitons in Nanostructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polarization and Optical Properties of Exciton-Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exciton-Polariton Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photonic Dispersion of QD Dimer Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lossless Dielectric Constant of QD Dimer Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple-Photon and Multiple-Carrier Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiphoton Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact Ionization and Auger Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 8 12 17 18 24 25 25 26 33 34

Abstract

This chapter reviews the fundamental concepts of excitons and excitonic polaritons and their extraordinary optical properties in quantum dot nano-composite materials. By starting with the optical excitation of an exciton in the nanostructure we show that the effective dielectric constant of the nanostructure becomes significantly modified due to the exciton generation and recombination, resulting

Y. Fu () Division of Theoretical Chemistry and Biology, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden e-mail: [email protected] H. Ågren Division of Theoretical Chemistry and Biology, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden Organic Chemistry, Bogdam Khmelnitskii National University, Cherkassy, Ukraine e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_23-2

1

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Y. Fu and H. Ågren

in high positive and negative dielectric constants. We also discuss single exciton generation by multiple photons and multiple exciton generation by single photon. All these nonlinear optical properties of quantum dot nano-composite materials offer novel possibilities and are expected to have deep impact in nanophotonics.

Introduction Materials with nanoscale features have the potential to revolutionize optoelectronics, permitting new and interesting device and system capabilities. The key design element is the geometry, that is, size and shape, which defines the properties of the nanostructured material and has led to the expectation of new and/or significantly improved physical, electrical, and optical properties. To precisely engineer light– matter interaction at the nanoscale, electron-hole pairs, that is, excitons, and their coupling with photons, exciton-polaritons, are becoming increasingly important. Excitons are of great interest in nanoscience because it has been discovered that their properties can be dictated by the size and the shape of a nanostructure in which they are confined, in addition to the constitution of the nanostructure. The new aspect of excitons that is prevalent in or even defining nanoscience is that the physical size and shape of the nanostructure are parameters that significantly influence their properties. Exciton polaritons were intensively studied in the 1960s and 1970s, and their manifestation in various optical phenomena, including light reflection and transmission, photoluminescence, and resonant light scattering are well established and documented by now, see, for example, the dedicated volume (Sturge and Rashba 1982) and references therein. We refer to Kavokin (2007) that presents detailed discussions of energy dispersion of exciton-polaritons in microcavities (largely about quantum wells, QWs), to Scholes and Rumbles (2006) for a very intuitive description of excitons in nanoscale systems and a detailed review about excitons in polymers and nanotubes, and to Weisbuch et al. (2000) for research and development of excitions and photons in confined structures from threedimensionally (3D) extended states (bulk) to atom-like quantum dots (QDs) and their applications in light emitting devices. In semiconductor technology, the geometric miniaturization of individual components and devices have been scaling according to Moore’s Law (International Technology Roadmap for Semiconductors; Fu et al. 2006a). Photonics has also been catching up (Thylén et al. 2006) but at a slow pace due to the limitations imposed by the diffraction limit. Resonant excitonic states in nanostructures, after coupling with light to form excitonic polaritons, provide a source for both high and negative dielectric constant (Fu et al. 2006b, 2008; Zia et al. 2006), thus offering novel possibilities to drastically increase the integration density and new functionalities in photonics at the nanoscale. Rapid growth of nanotechnology also greatly expands the clinical opportunities. Colloidal QDs are highly fluorescent with excellent photochemical stability, extreme brightness, and broad excitation spectral range, which have been introduced successfully in many fluorescence-based optical imaging applications (Medintz et al. 2005). Surface-modified and water-soluble

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

3

QDs open a new era in cell imaging and bio-targeting as transport vehicles for therapeutic drug delivery to different diseases such as cancer and atherosclerosis (Derfus et al. 2007; Molnár et al. 2010; Vashist et al. 2006). In this chapter we concentrate on nano-semiconductor systems, where it has become possible during the last 15 years to handle and control single excitons. More specifically we focus on excitons and exciton-polaritons in semiconductor QDs. This chapter presents first a theoretical background of the solid-state theory, the exciton and the coupling between the exciton and the photon (excitonic polariton). Optical properties of exciton polaritons in nanostructures and their photonic engineering are then discussed.

Solid-State kp Theory The theoretical description of electronic and optical properties of nanostructures is currently advancing using modern quantum methodologies such as pseudo potential, tight-binding, and atomistic semi-empirical pseudo potential methods (Allan and Delerue 2008; Franceschetti et al. 2006; Jiang et al. 2009; Rabani and Baer 2008). However, these studies have been limited to light atoms and small nanostructure sizes. Our target nanostructure systems are about 10 nm in diameter and even more where macroscopic micrometer scale solid-state theory is well developed. In isolated atoms the energy levels are sharply defined. When two atoms are brought close to each other their electron wave functions overlap. As a result of the interaction between the electrons, it turns out that each single state of the isolated atom splits into two states with different energies. The degree of splitting increases as the inter-atomic separation decreases. Similarly, if five atoms are placed in close proximity, then each original energy level splits into five new levels. The same process occurs in a solid, where there are roughly 1028 atoms/m3 : The energy levels associated with each state of the isolated atom spread into essentially continuous energy bands separated from each other by energy gaps. Before further examining the various properties of semiconductors it is extremely useful to examine the atomic structure of some of the elements which make up commonly used semiconductors as listed in Table 1. A very important conclusion can be drawn about the elements making up the semiconductors: The outmost valence electrons are made up of electrons in either the s- or p-type orbitals. While this conclusion is strictly true for elements in Table 1 Atomic structures of typical elements making up common semiconductors Element IV-Si IV-Ge III-Ga V-As II-Cd VI-Se

Core electrons 1s2 s2 2p6 1s2 2s2 2p6 3s2 3p6 3d10 1s2 2s2 2p6 3s2 3p6 3d10 1s2 2s2 2p6 3s2 3p6 3d10 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 1s2 2s2 2p6 3s2 3p6 3d10

Valence electrons 3s2 3p2 4s2 4p2 4s2 4p1 4s2 4p3 5s2 4s2 4p4

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Y. Fu and H. Ågren

the atomic form, it turns out that even in crystalline semiconductors the electrons in the valence band (VB) and conduction band (CB) retain this s- or p-type character. The core electrons are usually not of interest, except of some special characterization-type experiments. Note that II–VI compounds such as CdSe are normally complicated and their lattice structures are hexagonal. In general, it is found that when atoms exchange or share valence electrons so that the complement of quantum states is completed, they have a lower electrostatic energy for their combined electron patterns than when they are separate. For example, silicon has four valence electrons grouped in two closely spaced energy levels (3 s and 3 p, see Table 1); it can combine with other silicon atoms by sharing four valence electrons with four surrounding silicon atoms in an endless array. The atoms around any one atom are centered at the corners of a regular tetrahedron: the tetrahedral bond. This creates the diamond crystal structure. Essentially all semiconductors of interest for electronics and optoelectronics have the face-centered cubic (fcc) structure. However, they have two atoms per basis. The coordinates of the two basis atoms are (000) and (a/4) (111), indicated in d1 and d2 in Fig. 1b. Here a is normally denoted as the crystal lattice constant. If the two atoms of the basis are identical, the structure is called the diamond structure. Semiconductors such as silicon, germanium, and carbon fall into this category. If the two atoms are different, for example, GaAs, AlAs, and CdS, the structure is called zincblende. The intrinsic property of a crystal is that the environment around a given atom or group of atoms is exactly the same as the environment around another atom or similar group of atoms. Many of the properties of crystals and many of the theoretical techniques used to describe crystals derive from such a periodicity of crystalline structures. This suggests the use of Fourier analysis in the form of real space– reciprocal space or wave vector space duality for crystal problem discussions. Many concepts are best understood in terms of functions of the wave vector. We prefer to describe a wave with wavelength  as a plane wave with wave

a

b z

R1

z

a

R2

a

y

d2 R3

d1 x

y

x

Fig. 1 (a) Face-centered cubic (fcc) lattice with primitive translation vectors R1 , R2 , and R3 (b) Zincblende crystal structure where d1 and d2 denote two different atoms (can be identical such as in Si crystal in which case the lattice structure is called diamond). In both (a) and (b), a denotes the lattice constant

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

5

vector k of magnitude 2/ and propagation direction perpendicular to the wave front. The space of the wave vectors is called the reciprocal space. The kp method is much used in analyzing semiconductor nanostructures as it gives a good approximation of the states close to the  point (k D 0), which is the most relevant region for many device applications. The approximation is given with good accuracy, while in many nanostructure calculations, this is the only feasible numerical model that can be implemented (even though it is still much restricted due to the computational capability). The goal is essentially to solve the Schrödinger equation for a nanostructure to get the eigenenergies and their associated eigenvectors (wave functions). The wave functions usually have too high frequencies to be feasible to calculate explicitly using numerical methods on computers as an inordinately high number of grid points would be necessary to capture an acceptable numerical representation of the wave function. A solution is to separate the wave functions into an oscillatory part and a modulating part which is of the same scale as the heterostructure. This is the basic idea of the envelope function approximation – the modulating part is called the envelope function. We write the envelope function as a Bloch function: nk

.r/ D e ik:r unk .r/

(1)

where n is the state index and k is the wave vector. The Schrödinger equation for this wave function is simply 

p2 C V .r/ 2m0

 nk

.r/ D En .k/

nk

.r/

(2)

Substituting the factorization into the equation requires some care in evaluating the product of the momentum operator p and the two parts of the wave function. With p D i „r and knowing from vector calculus that re ikr D i ke ikr , the product of the momentum operator and the wave function becomes   i „r e ikr unk .r/ D i ¯e ikr .r C i k/ unk .r/ D e ikr .p C ¯k/ unk .r/

(3)

Applying the momentum operator a second time gives e ikr .p C ¯k/2 unk .r/, so the Schrödinger equation can be written in the following way that the oscillatory part will cancel out: # " 2 ikr .p C ¯k/ e C V .r/ unk .r/ D e ikr En .k/ unk .r/ (4) 2m0 Expanding the .p C ¯k/2 term gives 

 p2 ¯k  p ¯2 k 2 unk .r/ D En .k/ unk .r/ C V .r/ C C 2m0 m0 2mo

(5)

6

Y. Fu and H. Ågren 5 GaAs

InAs

3 2 "

Electron energy [eV]

4

Conduction bandedge

1

Eg

Valence bandedge

0 –1 –2

L

Γ

X

L

Γ

X

Fig. 2 Energy band structures of electrons in GaAs and InAs bulk materials

The first two terms are identical to the original Hamiltonian, so if the two other terms are treated as a small perturbation the Hamiltonian can be expressed as .H0 C H1 C H2 / unk .r/ D En .k/ unk .r/, and treat H1 D .¯=m0 / k  p and H2 D ¯2 k 2 =2m0 as first-order and second-order perturbations, respectively. If the equation is solved for k D 0 with only H0 remaining nonzero, the result is a set of eigenvectors un0 (r), typically at the optimal points such as the  point in Fig. 2 which shows the energy band structure of most common GaAs and InAs bulk materials. For the following discussion we express the eigenfunction using Dirac notation as jni, and eigenvalues En . These form a complete basis set with orthogonal eigenfunctions which can be used in perturbation calculations of states with k ¤ 0. As the atoms of the elements making up the semiconductors are brought together to form the crystal, the valence electronic states are perturbed by the presence of neighboring atoms. While the original atomic functions describing the valence electrons, of course, no longer are eigenstates of the problem, their characteristics can be used as a good approximate set of basis states to describe the “crystalline” electrons. For most semiconductor materials of interest, the atomic functions required to describe the outermost electrons are s, px , py , and Pz types, as shown by Table 1. A common approach of choosing the above jni, at the  point is to follow Kane (1957) and define a set of eight states, that is, unk (r) in Eq. 1 at k D 0; jS "i ; jX "i ; jY "i ; jZ "i ; jS #i ; jX #i ; jY #i ; jZ #i, where the arrows indicate spin up and down. jX, Y, Zi, denotes degenerate VBs while jSi denotes the CB. Other remote bands are not included unless specifically required. The eigen function for a given k is a linear combination of the basis functions: unk .r/ D

X m

cnm .k/ jmi

(6)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

7

so the objective is now to find the coefficients cnm that form our envelope parts together with the basis functions at . If we substitute this linear combination into the Schrödinger equation, we get H

X

cnm .k/ jmi D

m

X

cnm .k/ H jmi D En .k/

X

m

cnm .k/ jmi

(7)

m

Multiply this on the left with the conjugate of any, say j`i, of the basis functions, and we obtain D ˇ E X X ˇ (8) cnm .k/ h` jH j mi D En .k/ cnm .k/ `ˇm D En .k/ cn` .k/ m

m

The right-hand part is the result of wave function’s orthonormal property. Inserting the expanded kp Hamiltonian Eq. 5, multiplying with the conjugate, and integrating over an unit cell using the fact that the basis functions are orthonormal gives   X  ¯2 k 2 ¯ Em C ı`m C h` jk  pj mi cnm .k/ D En .k/ cn` .k/ 2m0 m0 m

(9)

The first-order approximation is thus En .k/ D En C

¯2 k 2 ; cnm .k/ D ınm 2m0

(10)

The second-order approximation is thus given by En .k/ D En C

¯2 k 2 ¯ X jh` jk  pj nij2 C 2m0 m0 En  E`

(11)

`¤n

The result can be expressed in terms of an effective mass m*: En .k/ D En C

X ¯2 ki kj 2mij i;j

(12)

where i, j D x, y, z, and ˝ ˇ ˇ ˛ m0 2 X hn jpi j `i ` ˇpj ˇ n D ıij C mij m0 En  E`

(13)

`¤n

Note here that the effective mass can be anisotropic which can be utilized for optical coupling in quantum well photodetection (Xu et al. 1994).

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Y. Fu and H. Ågren

It can be easily seen that a narrow bandgap, which leads to two states being close to each other, gives a small effective mass which agrees very well with the experimental data which indicates that InSb has both the smallest bandgap and the lowest effective mass. By a series of experiments the parameters for the kp Hamiltonian can be determined, and the final result is a matrix with many materialdependent parameters and no arbitrary parameters to adjust. Details of the kp Hamiltonian for III–V and II–VI semiconductor materials are well documented in reviews (Madelung 1991, 1992; Vurgaftman et al. 2001).

Excitons in Nanostructures After reviewing some solid state theory, we are now capable of modeling excitons and exciton-polaritons in semiconductor nanostructures. We first study and model the exciton in general, then excitons in QD-based nanostructures, and finally applications of QD excitons and exciton-polaritons. Details of exciton theory can be found in Dimmock (1967) and Fu and Willander (1999). As discussed in the previous section, the energy band structure of electrons in a semiconductor consists of energy bands separated by band gaps. At absolute zero temperature, a pure semiconductor is characterized by having only completely occupied and completely empty energy bands. The highest occupied bands are referred as VBs and the lowest unoccupied band is the CB. Let H0 be the Hamiltonian of the nanostructure under investigation, then H0

vkh

.r h / D Ev .kh /

vkh

.r h /

cke

.r e / D Ec .ke /

cke

.r e /

denotes the VB state and H0

the CB state. We consider an optical absorption process in which an allowed electric dipole (not ultra intense excitation) transition raises an electron from a filled VB state to an empty CB state. The properties of a system with an electron missing from a VB state vkh can be described by considering a single particle of positive charge in the state, that is, the VB hole. The exact expression of the interaction potential between the CB electron and VB hole is rather complicated. For commonly used semiconductor systems, and when the electron and hole are widely separated and moving relatively slowly with respect to one another, the Coulomb interaction between them is screened and given by Dimmock (1967) V .r e  r h / D 

e2 421 jr e  r h j

where 21 is the high-frequency dielectric constant.

(14)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

9

The exciton state is described by the two-particle wave function cke

.re /

vkh

.rh /

The total wave vector of the excited state is K D ke C kh . Because of the Coulomb interaction, cke .re / vkh .rh / does not represent an eigen state of the system. We approximate the eigen state of the exciton by a linear combination of electron-holestates X (15) AnK .cke ; vkh / cke .r e / vkh .r h / nK .r e ; r h / D cke ;ckh

where n is the quantum index of the exciton state. By introducing the Fourier transform of AnK (cke vkh ), nK

.r e; r h / D

1 X ike r e ikh r h e e AnK .cke ; vkh /  k ;k e

(16)

h

where  is the normalization volume, it is easy to show that differential equation ŒEc .i re /  Ev .i rh / C V .r e  r h /

nK

nK (re , rh )

.r e ; r h / D E

nK

satisfies the

.r e ; r h /

(17)

where Ec (ir e ) is the expression obtained by replacing ke in the power-series expansion of Ec (ke ) by ir e . The result is valid provided Ec (ke ) and Ev (kh ) are analytic and can be expanded in powers of ke and kh In the effective mass approximation of Eq. 12, the exciton Hamiltonian becomes Ho D 

¯2 re2 ¯2 rh2 e2    2me 2mh 421 jr e  r h j

(18)

where m*e and m*h are effective masses of electron and hole. It is and centre-of-mass coordinates re rh and  the relative  easier to introduce R D me r e C mh r h = me C mh to solve the above exciton Hamiltonian ¯2 rr2e r h ¯2 r 2 e2 Ho D    R     2mh 421 jr e  r h j 2 me C mh

(19)

where 1=mr D 1=me C 1=mh is the reduced effective mass. For three-dimensionally extended bulk material, the part of the wave function in R contains a factor of eiKR , the part in the relative coordinates contains a hydrogenic nK .r e ; r h / D nK .r e  r h / so that the exciton wave function and its energy are

10

Y. Fu and H. Ågren

e iK R

nK

.r e ; r h / ; EnK D

¯2 K 2 mr e 4 ;  2221 n2 2 me C mh 

(20)

and the wave function of the ground exciton state (n D 1) corresponding to the last two terms of the Hamiltonian Eq. 19 is 1K

1 .r e ; r h / D q e jr e r h j=aBr 3 aBr

(21)

where aBr D 421 ¯2 =mr e 2 is the Bohr radius of the exciton in the semiconductor material. In three-dimensionally confined nanostructures, the center of mass of the exciton is not mobile so that we simply let K D 0 in Eqs. 20 and 21. We now study the effect of excitons on the optical transition. The probability that the system makes an electronic transition from the ground state 0 to an excited (exciton) state nK (re , rh ) is proportional to ˇ˝ ˇ

nK

ˇ ˇ iqr ˇe a  p ˇ

0

˛ˇ2 ˇ

(22)

where p is the momentum operator, q and a are, respectively, the wave vector and polarization vector of the photon. By Eq. 15 we obtain h

nK

jpj

0i

X

D

AnK .cke ; vkh / h

cke

jpj 

vkh i

(23)

cke ;vkh

where  is the time-reversal operator and  vkh is the unoccupied VB state out of which the electron was excited. Denoting  vkh D vkh as the hole state and making use of the negligible q (the photon wave vector is negligibly small so that K D 0) so that ke D kh  k, h

n0

jpj

0i

D

X

An0 .ck; v  k/ h

ck

jpj

vk i

(24)

k

where we have neglected the contribution from all other bands except the VB and CB in the vicinity of the Fermi level. Under normal device operation conditions, h ck jpj vk i  pcv which is independent of k over the range of k involved, h

n0

jpj

0i

D pcv

X

Ano .ck  vk/

(25)

k

By Eq. 16 the transition probability is therefore proportional to jh

no

j pj

0 jij

2

D p2cv j

n0

.r e ; r h /j2r e Dr h

(26)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

11

Table 2 Excitons in common semiconductor (Fu et al. 2000; Madelung 1991, 1992) (low temperature for IV and III–V). denotes the band gap between VB and CB m*h (heavy hole) m*e (electron) Eg [eV] aBr [nm] E10 [meV]

Si 0.537 1.026 1.170 15.2 14.3

GaAs 0.51 0.067 1.519 11.6 4.2

AlAs 0.409 0.71 2.229 8.6 20.0

InAs 0.35 0.0239 0.418 38.1

CdSe 0.45/1.0 0.13 1.842 5.4 13.2

ZnO 0.59 0.28 3.435 1.8 60.0

Note that the optical transition is also constrained to K D 0 since the photon wave vector is negligibly small. For spherical state n0 (0) is nonzero only for s-type states of Eq. 21 (the reader can refer to the hydrogen atom in many quantum mechanics textbooks) and j n0 .0/j2 / 1=n3 so that jh

n0

jpj

0 ij

2

/

1 n3

(27)

which shows that only the low-energy exciton states (small n) can be probed optically for bulk materials. The exciton binding energy E10 in Eq. 20 for the ground exciton state (n D 1) in commonly used bulk semiconductors is less than 10 meV so that their room temperature luminescence is very weak (exciton thermally dissolved so that n is large). Moreover, exciton Bohr radii, that is, the effective spatial extensions of the excitons, in bulk materials are also quite extended, see Table 2. In high quality semiconductor nanostructures, quantum size effects prevail and subband formation strongly influences the exciton as the electron and hole are forced to stay in a very confined spatial area. Coulomb interaction between the electron and hole, that is, the exciton binding energy, becomes much increased. Even at room temperature the sharp exciton lines can be detected in the absorption spectra of AlGaAs/GaAs multiple QWs (Miller et al. 1982), which can hardly be observed in bulk GaAs samples. One of the reasons for the fast development of ZnO and related materials is their large excitonic binding of about 60 meV which binds the electron and hole strongly in bulk ZnO even at room temperature (Sun et al. 2002). A photonic switch operating by controlling exciton excitation in ZnO QWs via an optical near field has been reported, where ZnO QWs were in the form disks with a disk diameter of 80 nm and the heights of 2 nm (in a structure composed of one single-QW and three double-QW) and 3.5 nm in the one double-QW case (Yatsui et al. 2007, 2008). Another development based on the large exciton binding energy of ZnO was the room-temperature polariton laser (Zamfirescu et al. 2002). Equation 27 tells us that the major reason for studying and developing semiconductor nanostructures for optical applications refers to the forced quantum confinement. The exciton will not dissolve even when the exciton binding is comparable or smaller than the thermal energy. Light–matter interaction in the

12

Y. Fu and H. Ågren

forms of optical exciton generation and radiative recombination is much enhanced, resulting in many novel nanostructure-based optoelectronic applications.

Polarization and Optical Properties of Exciton-Polaritons In this section, we study the optical properties of excitons in nanostructures. We start with the detailed analysis of optical absorption in which an allowed electric dipole transition creates an exciton nK from a filled VB and in this scheme the wave function of the initial state (the filled VB) is simply unity denoted by 0 in the formalism of second quantization. Assume that our electron-hole system is initially in its ground state 0 for time t < 0. We switch on an external radiation of E(r, t) for t  0; the first-order perturbation Hamiltonian is Z V D

d .r/  E .r; t / dr

(28)

where d(r) is the dipole-moment operator given by, d .r/ D er e ı .r  r e / C er h ı .r  r h /

(29)

where e is the charge unit. We denote the wave function of the electron-hole system as jre , rh , ti, its time-dependent Schrödinger equation is then i¯

@ jr e ; r h ; t i D .H0 C V / jr e ; r h ; t i @t

(30)

where H0 is given by Eq. 18. By the first-order perturbation approximation where we consider only the exciton ground state 0 to one excited excitonic state nK (re , rh, t) the time-dependent wave function jr e ; r h ; t i D j

0 .t /i

C cnK .t / j

nK

.r e ; r h ; t /i

(31)

Here we simply assume that jcnK .t /j  1. We will discuss this more closely in the coming sections. We now formally denote both 0 (t) and nK (re , rh , t) as eigen functions of H0 such that H0 0 .t / D E0 0 .t / D ¯!0 0 .t / H0 nK .r e ; r h ; t / D EnK nK .r e ; r h ; t / D ¯!nK i!0 t 0 .t / D 0 e i!nK t nK .r e ; r h ; t / D nK .r e ; r h / e Substituting Eq. 31 into Eq. 30 leads to

nK

.r e ; r h ; t /

(32)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . . .t/ @ nK .r e ;r h ;t/ i ¯ @ @t0 .t/ C i ¯ @cnK nK .r e ; r h ; t / C i ¯cnK .t / @t @t D H0 0 .t / C cnK .t /H0 nK .r e ; r h ; t / C V 0 .t / C cnK .t /V

13

nK

.r e ; r h ; t /

By Eqs. 32, .t/ ¯!0 0 .t / C i ¯ @cnk nK .r e ; r h ; t / C ¯!nK cnK .t / nK .r e; r h ; t / @t D E0 0 .t / C cnK .t /EnK nK .r e ; r h ; t / C V 0 .t / C cnK .t /V nK .r e ; r h ; t /

which is i¯

@cnK .t / @t

nK

.r e ; r h ; t / D V

Multiplying the above equation by h 1 @cnK .t / D @t i¯



0 .t /

C cnK .t /V

nK (re , rh , t)j

nK

.r e; r h ; t /

(33)

we come to

ˇZ ˇ ˇ ˇ d .r/  E .r; t / dr ˇ nK .r e ; r h ; t / ˇ

 0 .t /

(34)

It can be shown (Fu and Willander 1999)

 ˇZ ˇ .r; / .r / .r / d  E t drj ; r ; r nK e h ˇ 0 e h Z  D !nKe m0 nK .r; r/ p cv  E .r; t / dr

where

nK (r, r)

is obtained from

1 e dcnK .t /  dt i ¯ !nK m0

Z

nK (re , rh )  nK

(35)

by setting re D rh D r. Thus,

.r; r/ pcv  E .r; t / dr e i.!nK !0 /t

(36)

Note that optical couplings among excitonic states are zero because of the quantum selection rules. As well, only a limited number of excitonic states have nonzero optical couplings with the ground state. In reality !0 D 0 since this is the reference state where the VB is completely filled and the CB is empty. Furthermore, we can very well describe the excitation radiation as E .r; t / D E .r; !/ e i!t , so that integrating the above equation we obtain immediately cnK .t / D

e e i.!nK !/t ¯ .!nK  ! C i  / !nK m0

Z

 nK

.r; r/ pcv  E .r; !/ dr

(37)

Before we move on, let us discuss the spatial distribution of the exciton wave function. Excitons are normally categorized into two types. The Frenkel exciton is an excited electronic state where the electron and the hole are situated in the same molecule or atom. Because of their small radius, the Frenkel excitons

14

Y. Fu and H. Ågren

usually are considered to be local. The large-radius Wannier–Mott exciton in bulk semiconductors is relatively weakly bound due to a typically large exciton Bohr radius. A hybrid state of Wannier–Mott exciton and Frenkel exciton in different heterostructure configurations involving QDs was found to exist at the interfaces of the QDs and the surrounding organic medium with complimentary properties of both kinds of excitons (Birman and Huong 2007). In heterostructures both the CB electrons and the VB holes will experience extra potential energies such as band offsets. For low-energy Frenkel excitons in a QD, the electrons and holes are confined within the same QD volume (type-I exciton) so that we will have a common potential energy V .jR  aj/ that confines the exciton, where R is the center of mass of the exciton (see Eq. 19), a denotes the center of the QD. For common semiconductor QDs, the radius of the QD, RQD , is in the order of the exciton Bohr radius aBr so that one may neglect the free motion of the center of mass of the electron-hole pair in the QD, that is, K D 0 in Eq. 20. For the ground state of the exciton we can approximate V .jR  aj/ D 0 when jR  aj < RQD and V .jR D aj/ D 1 elsewhere so that the wave function corresponding to the motion of the center of mass becomes (Gasiorowicz 1996)   1  jR  aj sin p RQD 2RQD jR  aj

(38)

for jr  aj  RQD . The wave function is zero elsewhere. We thus finally obtain the total wave function of the exciton ground state inside the QD p



1 2RQD jR  aj

sin

 jR  aj RQD

 q

1

e jr e r h j=aBr uc .r e / uv .r h /

(39)

3 aBr

Here Bloch functions uc (re ) and uv (rh ) at CB and VB bandedges are included, see Eqs. 1 and 6. The polarization of the excited exciton in a QD centered at a is P a .r; t / D

X

h

nK

.r e; r h ; t / jd .r/j

0

 .r e ; r h /i cnK .t /c0 .t / C c:c:

(40)

nK

where d is the dipole of the exciton given by Eq. 29. Since the wave function Eq. 39 of the exciton is confined and real, the polarization of the exciton in the QD excited by an external electromagnetic field E(r, t) is (Fu et al. 1997, 2000) P a .r; t / D

nK

X

e 2 pcv e i!t 2 ¯ .!nK  ! C i  / !nK m20 nK Z  0 0  0 0 .r; r/ nK r ; r p cv  E r ; ! dr

(41)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

15

By assuming small spatial variation of the excitation field within the QD, an effective permittivity is defined for the QD exciton polariton by writing D .r; !/ D –1 E .r; !/ C P .r; !/ D –QD .r; !/ E .r; !/

(42)

where 21 is the background dielectric constant of the QD material, and –QD .r; !/ D –1 1 C

X nK

!nK

sin ˛nK 2!LT  ! C i  ˛nK

! (43)

3 2 2 where ˛nK D  jr  aj =RQD : –1 !LT aBr D e 2 pcv =¯!nK m20 : !LT is normally referred to be the exciton longitudinal-transverse splitting. The above equation was obtained under the perturbation approximation and thus can also be accounted at a semiclassical level by adding a Lorentz-oscillator term to the dielectric function (Andreani et al. 2005). Notice that –QD (r, !) in the above equation is position-dependent. Making an average over the QD volume results in an effective dielectric constant

" 2QD .!/ D 21 1 C

X nK

6!LT  2 .!nK  ! C i  /

# (44)

For a normal system without any extra external modifications except the incident electromagnetic field of E(r, t) under investigation, the occupation of excited excion state, jcnK (t)j2 , is small (the first-order perturbation condition). Moreover, this occupation of the excited state is also much smaller than the ground state, see Eq. 31, so that an optical absorption dominates. This means that the optical field is absorbed during the exciton excitation following its propagation through the QD. The situation is normally referred to as optical loss, for example, in photodetectors. Optical gain is achieved in light-emitting and laser devices by population inversion. To achieve optical gain in the QD, we increase the occupation of the exciton state. Here we consider such a pumping process that the exciton population of the excited exciton state c(0) nK is finite at time t D 0 so the time-dependent wavefunction becomes h i i X h .0/ .0/ cnK C cnK .t / j nK .r e ; r h ; t /i jr e ; r h ; t i D c0 C c0 .t / j 0 .r e ; r h /i C nK

(45) due to the external electromagnetic field E(r, t). We can obtain (Fu et al. 2008) ˇ ˇ

3 ˇ .0/ ˇ X 6 2 ˇc0 ˇ 2  1 !LT 5 –QD .!/ D –1 41 C 2 .! /   ! C i  nK nK 2

(46)

16

Y. Fu and H. Ågren

Let us write " D "0 C i"00 ,  ˇ ˇ  3 ˇ .0/ ˇ2 .! 6 2  1  !/ ! ˇc ˇ nK LT 0 X 7 6 7 h i "0QD .!/ D "1 6 5 41 C 2 2 .! 2   !/ C  nK nK  ˇ ˇ  3 2 ˇ .0/ ˇ2 6 2  1 ! ˇc0 ˇ LT X 7 6 h i7 "00QD .!/ D "1 6 1 C 5 4 2 2 .! 2 nK  nK  !/ C  2

(47)

Since "00 represents the energy loss of the incident electromagnetic field, we observe ˇ ˇ2 ˇ ˇ2 ˇ .0/ ˇ ˇ .0/ ˇ that the QD is lossy when ˇc0 ˇ < 1=2, it becomes transparent when ˇc0 ˇ D 1=2, ˇ ˇ2 ˇ .0/ ˇ and optical gain will occur when ˇc0 ˇ > 1=2. At the same time, the polarization changes sign. Figure 3 shows the calculated dielectric constant for one QD before and after some fraction of the exciton states becomes populated. In this example, II–VI PbSe/ZnSe QD in a CdSe background is assumed and ¯!nKD 1:5eV. In Fig. 3 and throughout this chapter we write " D "0 C i"00 where "0 and "00 are expressed in units of "0 Here we observe the possibility of finding low-loss negative dielectric constant. As shown in Fig. 3, a huge effective permittivity is expected in the vicinity of ! nK , with which the energy dispersions of photonic crystals based on QD arrays were derived (Fu et al. 2000) to understand photonic band gaps observed in CdS

400

a

|C0(0)|2 = 1.0

b

|C0(0)|2 = 0.25

300

ε′ and ε″ [ε0]

200 ε′

ε′

100

ε″

0 ε″

–100 –200 1.49

1.50

1.51

1.49

1.50

1.51

Photon energy [eV]

Fig. 3 Dielectric coefficients of the QD before (a) and after (b) the excited exciton state in the QD becomes populated "1 D 12:8; ¯!nKD 1:5eV; ¯!LT D 5meV, and ¯ D 1 meV for typical II–VI semiconductor QDs, for example, colloidal PbSe/ZnSe QD

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

17

QDs embedded in fcc porous silica matrices (Vlasov et al. 1997). A close-packed 3D array of spherical CuCl QDs in air was shown by extended Maxwell-Garnett theory and rigorous layer-multiple-scattering method to have a negative refractive index within the region of the excitonic resonance (Yannopapas 2007). The structure was proposed for subwavelength imaging (the loss is however formidably high) (Yannopapas 2008).

Exciton-Polariton Photonic Crystals As shown by Eq. 43, the effective dielectric coefficient of the QD is strongly modified when the exciton state in the QD is resonantly excited by an incident electromagnetic field. The contrast between this exciton effective dielectric coefficient and the background material can thus be utilized to construct a photonic crystal when the QDs are positioned in space in a periodic manner. Equation 43 was coupled to the electromagnetic field to study the optical dispersion of exciton-polariton crystals (Fu et al. 2000; Ivchenko et al. 2000) stimulated by the experimental works of fcc structured silica opals (Vlasov et al. 1997). Maxwell equations describing the incident electromagnetic field in the QD photonic crystal, free of charges and free of drift-diffusion current, are written as r Œr E .r/ D 0 ! 2 D .r/ r:D .r/ D 0

(48)

in the MKS unit system, where E(r) is the electric field, D(r) is the displacement vector, and ! is the angular frequency of the incident electromagnetic field. Here we have assumed that the QD consists of uniform isotropic linear media, for which D D 21 E and B D 0 H. H is the magnetizing field and B is the magnetic field. 0 is the magnetic constant (permeability of free space). The nonlocal material equation relating D(r) and E(r) is (Fu et al. 1997) D .r/ D "1 E .r/ C

X

P a .r/

(49)

a

where Pa (r) is the polarization contribution from the excited exciton in a QD centered at a, that is, Eq. 41. We have neglected the overlap of exciton envelope functions centered at different QDs so that excitons excited in different QDs are assumed to be coupledX only via the electromagnetic field. By denoting P D P a .r/, it follows from the second Eq. 48 and Eq. 49 that a

1 r  E .r/ D  r  P .r/ " so that the first Eq. 48 can be rewritten as

(50)

18

Y. Fu and H. Ågren 0.50

a

0.25 0.00 –0.25 –0.50 Primitive cubic lattice –0.75 X 0.6

R

Γ

X

M

b

(ω–ω0) / ωLT

0.4

0.2

0.0 Face centered cubic lattice –0.2 X

U

Γ

L

X

W

K

Fig. 4 Photonic dispersion relation of QD photonic crystals. (a) Primitive cubic lattice. (b) Facecentered cubic lattice

  k02 1 P .r/ C 2 r Œr  P .r/ r E .r/ C k E .r/ D  "0 k 2

2

where k0 D !=c; k D k0 n D !n=c; and n D Bloch-like solutions of Eq. 51 satisfying

(51)

p "1 ="0 :

E q .r C a/ D e iqa E q .r/

(52)

is the photonic band structure. Figure 4 shows the dispersion relationships of primitive-cubic and fcc QD photonic crystals (Fu et al. 2000). An overall bandgap, most prominent along the ƒ(L) line (the [111] direction) is observed in the fcc lattice (Zeng et al. 2006a).

Photonic Dispersion of QD Dimer Systems 0

As demonstrated by Eqs. 47 and 3, "QD can be adjusted to be positively large as well as negative by tuning the initial occupation condition of the exciton state in the QD. However, optical loss is not avoidable when a single type of QDs is used.

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

19

2R2

τ

a

excited

2R1

lossy Fig. 5 Schematic QD dimer system in a face-centered cubic (fcc) lattice structure. R1 and R2 denote the radii of the QDs and a the lattice constant of the fcc lattice

To tackle the loss issue, we construct a QD dimer photonic crystal by positioning two types of spherical QDs, having radii R1 and R2 respectively, in a fcc lattice having a lattice constant a, see Fig. 5. The lossy QDs, that is, type-I QDs (exciton energies are ¯! 1 ) occupy the normal fcc lattice sites, while the already excited QDs, type-II QDs (¯! 2 ) are displaced from the type-I QDs by . Let cai be the occupation of the ground state of QD i positioned at lattice site a, that is, the valence band is completely filled and the conduction band is completely empty, the contribution of an excited exciton state in a QD ai to the dielectric polarization is given by

P ai .r/ D



e 2 pcv 2jcai j2  1 ¯ .!ai  ! C

Z ai .r; r/

2 i  / !ai m20

 ai

 0 0  r ; r pcv  E r 0 dr 0 (53)

by referring to Eqs. 41 and 46. We seek again for Bloch-like solutions of Eq. 51 satisfying Eq. 52. We expand the vector function Eq (r) in the Fourier series E q .r/ D

X g

e i.qCg/r E qCg

(54)

20

Y. Fu and H. Ågren

q Where g are the reciprocal lattice vectors. Denote a1 D 2jc1 j2  1; a2 D q 2jc2 j2  1; v0 D a3 =4 is the volume of the primitive unit cell of the fcc lattice, 3 i D ai i and t D " !LT aBr , the excitonic polarization becomes P .r/ D

X ai

t ai .r; r/ !i  ! C i 

Z

 0 0  0  r ; r E r dr 0 ai

(55)

Note that the excitonic wave function is real so that we have dropped off the complex conjugation of the wave function in the integrals in Eq. 53. The two integrals in the above equation can be transformed into Z Z

1;a .r/ E .r/ dr D e iqa

X

I1;qCg E qCg e iqa ƒ1

g

2;a .r/ E .r/ dr D e iqa

X

(56) I2;qCg E qCg e iqa ƒ2

g

Z

Z

i.qCg/r

dr; I2;qCg D 2;0 .r/ e i.qCg/.rC / dr. The where I1;qCg D 1;0 .r/ e X X sums 1;a .r/e iq:a and 2;aC .r/e iq:a satisfy the translational symmetry a aC

and can be presented as X a

X aC

1;a .r/ e iqa D

X g

2;aC .r/ e iqa D

e i.qCg/r

X g

e

 I1;qCg

v0 I i.qCg/r 2;qCg

(57)

v0

The linear equations for the space harmonics Eq Cg can thus be written  jq C gj 2  k 2 E qCg #  "   I1;qCg I2;qCg k02 t 1 2 ƒ1 C ƒ2 1  2 .q C g/ D v0 k !1  ! C i  !2  ! C i 

(58)

Let S .Q/˛ˇD ı˛ˇ  Q˛ Qˇ =k 2 ; ˛; ˇ D x; y; z; ı˛ˇ is the Kronecker symbol.

Dividing Eq. 58 by jq C gj2  k 2 , multiplying it by I1 ,q C g , and summing over g:

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

X

21

I1;qCg E qCg D ƒ1

g

" ˇ # ˇ  ˇI1;qCg ˇ 2 b I1;qCg I2;qCg S .q C g/ ƒ1 C ƒ2 D !2  ! C i  jq C gj 2  k 2 !1  ! C i  g c1 .!; q/ ƒ1 C M c2 .!; q/ ƒ2 DM k02 t v0

X

(59)

where X b ˇ2 k02 r S .q C g/ ˇˇ I1;qCg ˇ 2 2 .!1  ! C i  / v0 g jq C gj  k X b k02 r S .q C g/  c2 .!; q/ D M I1;qCg I2;qCg .!1  ! C i  / v0 g jq C gj2  k 2 c1 .!; q/ D M

(60)

 Similarly, dividing Eq. 58 by jq C gj 2  k 2 , multiplying it by I2 ,q C g , and summing over g, X

I2;qCg E qCg D ƒ2

g

" # ˇ ˇ  ˇI2;qCg ˇ 2 b I2;qCg I1;qCg S .q C g/ ƒ1 C ƒ2 D !2  ! C i  jq C gj 2  k 2 !1  ! C i  g c3 .!; q/ ƒ1 C M c4 .!; q/ ƒ2 DM k02 t v0

X

(61)

where X b k02 t S .q C g/  I2;qCg I1;qCg .!1  ! C i  / v0 g jq C gj2  k 2 X b ˇ2 k02 t S .q C g/ ˇˇ c4 .!; q/ D I2;qCg ˇ M 2 .!1  ! C i  / v0 g jq C gj  k 2 c3 .!; q/ D M

(62)

We arrive at the vector equations h i c1 .!; q/ ƒ1 D M c2 .!; q/ ƒ2 IM h i c4 .!; q/ ƒ2 D M c3 .!; q/ ƒ1 IM where I is a 3 3 unit matrix. We can rewrite the above equations as

(63)

22

Y. Fu and H. Ågren

0

1 1  M1;11 M1;12 M1;13 M2;11 M2;12 M2;13 B M M1;23 M2;21 M2;22 M2;23 C 1;21 1  M1;22 B C B C B  M1;31 M1;32 1  M1;33 M2;31 M2;32 M2;33 C DB C B M3;11 M3;12 M3;13 M4;11  1 M4;12 M4;13 C B C @ M3;21 M3;22 M3;23 M4;21 M4;22  1 M4;23 A M3;31 M3;32 M3;33 M4;31 M4;32 M4;33  1

M1;˛ˇ .; q/ D

8 !LT R13 ja1 j 2 11;˛ˇ .; q/ v0 .!1  ! C i  /

M2;˛ˇ .; q/ D

8 !LT .R1 R2 /3=2 a1 a2 12;˛ˇ .; q/ v0 .!2  ! C i  /

M3;˛ˇ .; q/ D

8 !LT .R1 R2 /3=2 a1 a2 21;˛ˇ .; q/ v0 .!1  ! C i  /

M4;˛ˇ .; q/ D st;˛ˇ .; q/ D

8 !LT R23 ja2 j 2 22;˛ˇ .; q/ v0 .!2  ! C i  / X f .jg C qj Rs / f .jg C qj Rt / S˛ˇ .g C q/ g

2 .g C q/  2

(64)

e i.st/.gCq/

 2 sin x f .x/ D x . 2  x 2 /

D

! !1

 .Q/ D

c jQj !1 n

(65)

where s, t D 1, 2. The exciton-polariton dispersion !(q) satisfies the equation Det jDj D 0

(66)

Note that by setting jc1 j2 D 0 and jc2 j2 D 0:5 (type-II QDs are totally transparent) we retrieve the case of a photonic crystal composed of only type-I QDs (Fu et al. 2000), see Fig. 4. By setting jc1 j2 D jc2 j2 D 0 and !1 D !2 we retrieve the case of a compound QD photonic crystal (Zeng et al. 2006b). The dispersion relationship !(q) for a QD dimer fcc lattice is presented in Fig. 6. The lattice is denoted by its lattice constant a, which is set to be (0.95aBragg ) (which is 116 nm for GaAs), where aBragg D c=!1 n. The lossy type-I QDs .¯!1 D 1:5 eV/ occupy the normal fcc lattice sites, while the excited type-II QDs .¯!2 D 1:503 eV/ are displaced by D .a=2; a=2; a=2/ : ¯!LT D 5 meV. Here we observe the modification of the photonic band structure of the QD dimer system

Optical Properties of Quantum Dot Nano-composite Materials Studied. . . 1.0

23

a

0.8 0.6 0.4

|c2|2 = 1.0

0.2 0.0 1.0

b

(ω–ω1)/ωLT

0.8

|c2|2 = 0.4

0.6 0.4 0.2 0.0 1.0

c

0.8 |c2|2 = 0.5

0.6 0.4 0.2 0.0 X

U

L

Γ

X

W

K

Fig. 6 Energy dispersion relationships of QD dimer systems in the fcc lattice. R1 D a=4; R2 D a=5; ¯!1 D 1:5 eV; ¯!2 D 1:503 eV; ¯!LT D 5 meV; a D 0:95aBragg ; D .a=2:a=2; a=2/ :jc1 j2 D 0. (ajc2 j2 D 1:0 (complete excitation), (bjc2 j2 D 0:4, (cjc2 j2 D 0:5 (type-II QDs are total transparent)

by pumping one type of the QDs (type-II QDs), which evolves from the one of only type-I QDs jc1 j2 D 0; i.e., at their ground exciton state) when type-II QDs are transparent [jc2 j2 D 0:5, Fig. 6c] to the compound system [jc2 j2 D 1:0, Fig. 6a]. We can observe modified but still characteristic features of the photonic dispersions of individual type-I and type-II QDs in their separate photonic crystal formats in the compound system, see Fig. 4. More specifically, the resonance features of type-I QDs around .!  !1 / =!LT D 0:3 in Fig. 6d become compressed by the radiative

24

Y. Fu and H. Ågren

interaction between type-I QDs and type-II QDs, and they are also shifted down to around 0.18 in Fig. 6a. By varying jc2 j2 from 0.0 to 1.0 we find that the solutions of Eq. 66 are symmetric with respect to jc2 j2 and 1  jc2 j2 when c1 D 0. Thus, Fig. 6a represents also the photonic dispersion of the QD dimer system when both type-I and type-II QDs are all initially at their ground exciton states, that is, jc1 j2 D jc2 j2 D 0. Thus the modification of the exciton state (from ground state to excited state) in one type of QDs in the dimer system does not affect the feature of the dispersion structure.

Lossless Dielectric Constant of QD Dimer Systems As shown by Fig. 4, the exciton polarization provides possibilities of generating positively high dielectric constant as well as negative dielectric constant at the cost of disturbing the external electromagnetic field (either loss or gain). At least in the photonic crystal composed of one type of QDs (either lossy or gain), the dielectric modulation is always accompanied by loss or gain. The idea of the QD dimer system discussed in the previous sub-section is to compensate the loss of type-I QDs by the gain of type-II QDs, while still maintaining dielectric modulation. More specifically, we require "00 D 0 at some frequencies in order to achieve lossless dielectric modulation for various optoelectronics applications. In the following, we consider a PbSe/ZnSe QD assembly such as Fig. 5 immersed in a medium of dielectric constant "1 . The macroscopic dielectric constant " for the ensemble of the QDs can be described by the dielectric theory of Maxwell-Garnett (1906) and Gittleman and Abeles (1977), which for two QD species can be written as "  "i "QD1  "i "QD2  "i D x1 C x2 " C 2"i "QD1 C 2"i "QD2 C 2"i

(67)

where x1 is the volume fraction of the i-th QD species. While Eq. 67 works best for x1 less than 0.4, there is evidence that useful information can be available with 0 higher concentrations (Cohen et al. 1973). Figure 7a shows the spectra of "QD and 00 "QD of the two types of QDs. For the combination of two types of QDs, that is, the QD dimer system, one type is lossy and the other gain, immersed in a conducting polymer with an effective dielectric constant "1 D 1.8 the optical spectrum is presented in Fig. 7b. Figure 7 shows that a mixture of two types of QDs such as Fig. 5 can produce an effective dielectric constant that is lossless and negative, thus permitting, in concept, arbitrarily small scaling of the optical mode volume in the field of nanophotonics (Fu et al. 2008). Another binary mixture of lossy QDs and plasmonic silver nanorods was also shown to have a negative " and compensated optical loss by using finitedifference time-domain calculations (Bratkovsky et al. 2008).

Optical Properties of Quantum Dot Nano-composite Materials Studied. . . 40

a 1.500 1.503

20 0

1.500

x10 1.503

–20 ε′ and ε″ [ε0]

25

ε′ and ε″ of two QDS –40 5 0

b

ε′ ε″

–5 Composite –10 1.495 1.490

1.500

1.505

1.510

Photon energy [eV]

ˇ ˇ2 Fig. 7 (a) Dielectric constants of type-I QDs at 1.500 eV and ˇc .0/ ˇ D 1:0, and type-II QDs at ˇ ˇ2 1.503 eV and ˇc .0/ ˇ D 0:0. Solid lines: "0 ; Dotted lines: "00 . Note that "00 is magnified by 10. (b) QD dimer system after immersing the two types of QDs in "1 D 1.8. The densities of two types of QDs are 7  1016 cm3 Adapted from Fu et al. ((2008)

Multiple-Photon and Multiple-Carrier Processes Multiphoton Process Another important aspect about the optical properties of QDs is the multiphoton process which has been widely applied in recent years in biological and medical imaging after the pioneer work of Goeppert-Mayer (1931), Lami et al. (1996), Helmchen et al. (1996), Yokoyama et al. (2006). The multiphoton process has largely been treated theoretically by steady-state perturbation approaches, for example, the scaling rules of multiphoton absorption by Wherrett (1984) and the analysis of two-photon excitation spectroscopy of CdSe QDs by Schmidt et al. (1996). Non-perturbation time-dependent Schrödinger equation was solved to analyze the ultrafast (fs) and ultra-intense (in many experiments the optical power of laser pulse peak can reach GW/cm2 ) dynamics of multiphoton processes in QDs that are highly fluorescent with excellent photochemical stability, extreme brightness, and broad excitation spectral range (Fu et al. 2006c, 2007). For spherical QDs the exciton wave function can be expressed by a radial function and spherical harmonics. The degeneracy of each exciton state is 2 .2` C 1/ ; where ` is the quantum number of the angular momentum. Figure 8 shows the exciton states (open circles) confined in the QD as well as their corresponding

26

Y. Fu and H. Ågren

[meV]

0.6

Light–matter interaction

0.7

0.5 0.4 0.3 0.2

2

4 6 Angula

8 10 12 r mome ntum

14

Ene

0.0 0

7 6 5 4 3 2 1 0

rgy sta of exc te [ eV] iton

0.1

16

Fig. 8 Exciton states (open circles) confined in the QD and their corresponding light-matter interactions (vertical lines). Adapted from Fu et al. (2009)

light–matter interactions (vertical lines). For spherical QDs, only exciton states having zero angular momenta .` D 0/ have nonzero relaxation energies. Moreover, only confined exciton states (i.e., both the electron and hole states that compose the exciton state under discussion are confined) have significant light–matter interaction. In the ultrashort regime, after a continuous-wave optical excitation is switched on at t D 0, Fig. 9 shows that the time to reach peak two-photon excitation is much shorter than the one-photon excitation. Moreover, the multiphoton excitation spectral range is very broad. From about 1.0 to 2.5 eV (i.e., two-photon excitation range), the calculated excited exciton rate is much larger than the occupation of the first-excited exciton so that the emission spectrum is expected to be dominated by the recombination of the ground-state exciton. It was concluded by the theoretical simulation that experimentally observed strong multiphoton excitations can be reproduced when optical transitions among all confined states in the QD and an additional few hundred extended states in the barrier are taken into account (Fu et al. 2006c). This agrees with the experimental results of multi-particle (Huxter and Scholes 2006) and multi-exciton dynamics (Suffczy´nski et al. 2006).

Impact Ionization and Auger Recombination Nozik proposed that impact ionization might be enhanced in semiconductor QDs to increase the efficiency of solar cells up to about 66 % (Nozik 2002). This was later verified experimentally by Schaller and Klimov (2004). Impact ionization was

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

One-photon

S = 1 GW / cm2

0.03

Exciton p

hotoexcit

ation

0.04

27

0.02

Two-photon

0.01

0.00 0.5

5

ot

Ph

1.0

4

on

1.5

er

en

3

gy

2.0

]

2

hω V]

[e

2.5

1 3.0

e

[fs

m

Ti

0

Fig. 9 Multiphoton excitation of QD exciton in an ultrashort period excited by a continuous wave light source switched on at t D 0. S denotes the optical power of the excitation light source. Adapted from Fu et al. (2006c)

reported to produce multiple exciton generation (MEG) per photon in a QD that results in a very high quantum yield (up to 300 % when the photon energy reaches four times the QD bandgap) in QD solar cells (Hanna et al. 2004). Multiple carrier extraction (>100 %) was observed at photon energies greater than 2.8 times the PbSe QD bandgap with about 210 % measured at 4.4 times the bandgap (Kim et al. 2008). In a recent communication in Nano Letters, Trinh et al. (2008) showed compelling support for carrier multiplication in PbSe QDs. We concentrate here on reviewing impact ionization and Auger recombination in QDs using the solid-state theory. As shown in Fig. 10, a highly photoexcited electron and hole pairs can evolve into a multiple-exciton state through impact ionization. The carrier–carrier interaction is expressed by the Coulomb potential in the form of V D

e2 4" jr 1  r 2 j

(68)

to account for the two-body interaction  of two electrons from an initial state  j .r 1 / i .r 2 / exp i Ej C Ei t =¯ , that is, one electron occupying a VB state

28

Y. Fu and H. Ågren Excited exciton

Biexciton

e1

2

CB Eg

e2

e1

1 h2

e2

VB

h1

h1

Fig. 10 Schematic depiction of impact ionization of a high-energy electron-hole pair. (1) Electron-hole pair (e1 and h1 ) is generated, for example, by a photon. (2) e2 gets excited from a VB state to a CB state via Coulomb interaction with e2 , leaving hole h2 behind. The reverse process is referred to be Auger recombination

 j .r 1 / exp iEj t =¯ and another electron occupying a CB state i

.r 2 / exp .iEi t =¯/

to a final state ˇ 1 ˇˇ p ˇ 2

.r 1 / n .r 2 / n

ˇ .r 1 / ˇˇ ˇ exp Œi .En C Em / t =¯ m .r 2 / m

(69)

in the CB. Notice that the two electrons in the final state are not distinguishable so that we use a Slater determinant. Other ionization pathways exist such as one electron occupying a low-energy VB state falls into h1 and the released energy excites another electron from the VB to the CB. Auger recombination can be expressed similarly. Before discussing the QD case, let us consider the computation of an Auger-type process in solid states that an incident high-energy CB electron with a wave vector k1 collides with a second electron that occupies a VB state k2 resulting in two CB 0 0 electrons k1 and k2 . The general expression for the scattering matrix element of this process is Z Z

 ck1

.r 1 / ˆvk2 .r 2 /

1 jr 1  r 2 j

ck10

.r 1 /

ck20

.r 2 / dr 1 dr 2

(70)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

29

where ck (r)[˚ vk (r)] denotes the total wave function of the CB electron (VB hole) state k. For semiconductor systems and within the effective mass approximation and  are ck .r/ D ck .r/ uck .r/ and ˆvk .r/ D vk .r/ uvk .r/, where envelope functions of the CB electron and VB hole, respectively, and u’s are periodic Bloch functions, the above expression becomes Z Z



 ck1

ck10

 .r 1 /uck1 .r 1 / vk .r 2 / uvk2 .r 2 / 2

.r 1 / uck10 .r 1 /

ck20

1 jr 1  r 2 j

.r 2 / uck20 .r 2 / dr 1 dr 2

(71)

(72)

See, for example, Eq. in Ridley (1988). Following overlap integrals are thus involved Z I1 D Z

cell

I2 D cell

uck1 .r 1 / uck10 .r 1 / dr 1 (73) uvk2 .r 2 / uck20 .r 2 / dr 2

in the evaluation of the Auger-type scattering processes. The first overlap integral can be approximated to unity. Because of the orthogonality of u functions for different bands but the same k, the second overlap integral I2 is, in crudest approximation, zero. This, however, is not correct since the periodic Bloch functions are functions of wave vectors. More specifically, we can write uck .r/ D uc .r/ C krk uc .r/ C    uck .r/ D uc .r/ C krk uc .r/ C   

(74)

for small k, where uc (r) Zand uv (r) are periodic Bloch functions at the CB and VB bandedges, respectively. cell

uv .r 2 / uc .r 2 / dr 2 D 0. Substituting these expressions

into the overlap integrals we obtain the squared overlap integral in terms of the heavy-hole mass mv (Landsberg and Adams 1973; Ridley 1988) jI2 j2 D

¯2 2Eg



1 1 C m0 mv



2

jk2  k02 j

(75)

where Eg is the energy band gap of the bulk material. By using the inverse Bohr radius of shallow impurities as a measure about the k values, Landsberg and Adams obtained jI2 j D 0:223 for shallow-impurity-assisted Auger-type processes in bulk CdS and jI2 j D 0:265 in GaAs (Landsberg and Adams 1973). Note that the inverse Bohr radius of shallow impurities in bulk semiconductors is small. For QDs under investigation, the effective Bohr radius of the electron and hole distribution is largely determined by the QD size, which is about 5 nm, that is, very small compared with

30

Y. Fu and H. Ågren

the Bohr radius of shallow impurities in bulk semiconductors (about 100 nm in CdS and GaAs Landsberg and Adams 1973). This results in a large value of jI2 j. Under this specific circumstance we approximate jI2 j D 1. In other words, the electrons and holes in QDs, described by effective masses in the presence of the QD confinement potentials, interact with each other via the Coulomb force of Eq. 68 (Abrahams 1954). The approach has been adopted for describing carrier interactions in many electronic devices such as tunnel junctions where the kinetic energies of relevant carriers are large, see for examples Takenaka et al. (1979) and Rodina et al. (2002). By the scattering Dtheoryˇ and the Golden rule (Landau and Lifshitz ˇ generalized E ˇb ˇ .t /ˇ j i  e wj i t=2 as the temporal development 1965) and denoting j i ˇT b.t / of state ji, it is easy to obtain T wj i D

2 X ˇˇ˝ j ¯

i

jV j

n

m

˛ˇ2  ˇ ı Ej C Ei  En  Em

(76)

nm¤j i

j i D ¯wj i =2 is the relaxation energy and 1/wji the decaying time. In numerical calculations the • functions in the above equation are replaced by h 2 i  and ji is to be calculated by the above j i = j2i C Ej C Ei  En  Em equation in a self-consistent way that the left and right sides of Eq. 76 equal after knowing the interaction values of h j i jVj n m ij2 . Numerical calculations of the Coulomb interaction Eq. 68 are not trivial. They were simplified in Allan and Delerue (2008) by using a screened Coulomb potential involving one-electron wave functions which were derived for bulk semiconductor materials (Landsberg 1991). To calculate the impact ionization interaction from an initial state of R`1 .r 1 / Y`1 m1 .1 ; 1 / R`2 .r 2 / Y`2 m2 .2 ; 2 / to a final state R`3 .r 1 / Y`3 m3 .1 ; 1 / R`4 .r 2 / Y`4 m4 .2 ; 2 / (the first term one in Eq. 69), we notice 1 4 jr 1  r 2 j

D

1 X ` X `D0 mD`

r`C1 `m

(77)

where r < D min fjr 1 j ; jr 2 jg and r > max fjr 1 j ; jr 2 jg so that the impact ionization energy consists of ` r< `C1 r>

R`1 .r 1 / R`2 .r 2 / R`3 .r 1 / R`4 .r 2 / Y`1 m1 .1; 1 / Y`2 m2 .2; 2 /

 .2 ; 2 / Y`m .1 ; 1 / Y`3 m3 .1 ; 1 / Y`4 m4 .2 ; 2 / Y`m

(78)

for which we utilize the addition theorem for spherical harmonics (Cohen-Tannoudji et al. 1991)

Optical Properties of Quantum Dot Nano-composite Materials Studied. . .

Y`1 m1 Y`2 m2 D

`X 1 C`2 `Dj`1 `2 j

s

.2`1 C 1/ .2`2 C/ 4 .2` C 1/

31

(79)

C . `0j `1 0I `2 0/ C . `m1 C m2 j `1 m1 I `2 m2 / Y `m1 Cm2 where C . `m1 C m2 j `1 m1 I `2 m2 / is the CG-coefficient. A more detailed analysis shows a few selection rules for ` 0 s and m’s such as m4 m3 D m1 m3 . However, these selection rules can be easily fulfilled in nanoscale QDs because of the large number of available states confined in the QDs (see further discussion below). The most important qualitative selection rules refer to the radial functions that Z Z

r 0 Electric field > 0

Electric field = 0

α

β

β

L

M

c

R

β

L

M

R

0.5

0.4

1 0.2

d

Δα,Δβ (ev)

0.3

0 0.1

0

2 wEext (V)

4

0 0

0.1

0.2

0.3

Eext (V Å–1)

Fig. 4 (a) Schematic density of states diagram of the electronic states of a zigzag GNR in the absence of an applied electric field. (b) Schematic density of states diagram in the presence of a transverse electric field. The electrostatic potential on the left edge is lowered, whereas the one on the right edge is raised. (c) Dependence of half-metallicity on system size. Red denotes the bandgap of ’-spin, and blue the gap of “-spin as function of Eext for the 8-GNR (filled circles), 11-GNR (open circles), 16-GNR (squares), and 32-GNR (triangles). The rescaled gaps for the various widths collapse to a single function as shown in the inset (Reprinted with permission from Macmillan Publishers Ltd: Nature (Son et al. 2006b), © (2006))

12

V. Barone et al.

Modeling the Optical Spectrum of Single-Walled Carbon Nanotubes and Graphene Nanoribbons Single-Walled Carbon Nanotubes Characterization methods for single-walled carbon nanotubes based on optical absorption have attracted much attention during the past few years. Experiments based on photoluminescence (Bachilo et al. 2002; Weisman and Bachilo 2003), resonant Raman spectroscopy (Fantini et al. 2004; Telg et al. 2004), and Rayleigh scattering (Sfeir et al. 2006) have been reported in which optical transitions are obtained as fingerprints of a certain (n, m) SWNT. Optical transitions were first studied theoretically within the tight-binding approach considering the excitations as inter-band transitions. Within this framework, the optical spectra is generally obtained by utilizing the random-phase approximation (RPA) for the imaginary part of the dielectric function – (for a review, see Onida et al. 2002) Im .–/ D

1 X X ˇˇ ˝ ˇ !2 o;u

k o

jpj

˛ˇ

k ˇ2 u ˇ

  ı –k0  –ku  ! ;

(1)

k

where p is the linear momentum operator and the indices o and u stand for occupied and unoccupied Bloch orbitals, respectively. An example of the band structure and optical spectrum for a semiconducting tube obtained from DFT calculations is shown in Fig. 5. Allowed optical transitions, marked with arrows in the band structure of panel (a), produce a peak in the optical spectrum (panel (b)). Within the RPA the first-order optical transition corresponds to the fundamental gap of the semiconducting SWNT (generally dipole-allowed). The relation between the optical transitions and the diameter of the nanotubes (Kataura plot) (Kataura et al. 1999) was used as a useful guide for experimentalists in characterizing nanotubes samples. However, Kataura plots based on conventional TB calculations present serious limitations (Bachilo et al. 2002). DFT calculations utilizing local and semi-local functionals are well known to underestimate the bandgap of semiconductors significantly. This holds also true for higher order optical transitions. Hybrid functionals have been shown to improve significantly the description of the bandgap in semiconducting materials (Heyd and Scuseria 2004; Heyd et al. 2005). In the specific case of semiconducting nanotubes, commonly employed hybrid functionals, like B3LYP or PBE0 significantly overestimate the bandgap (Barone et al. 2005b). Despite the excitonic character of optical transitions in SWNTs (Spataru et al. 2004), the hybrid meta-GGA TPSS and the screened-exchange hybrid HSE provide excellent agreement with experiments for the optical gap (Barone et al. 2005b) as depicted in Fig. 6 where the mean error for the first-order optical transition with respect to experimental values in a set of ten semiconducting nanotubes is shown. It is worth pointing out, however, that DFT-based approaches cannot predict the strength of exciton binding energies due to the mean-field nature of the approximation.

Modeling of Quasi-One-Dimensional Carbon Nanostructures with Density. . .

a

13

b E11 400

–3

Im(ε)

Energy (eV)

300 –4

E22

200

–5 100

E33

E44

–6 0 Γ

K

1

0

k

3 4 2 Photon energy (eV)

5

6

Fig. 5 Band structure (a) and optical spectrum (b) of the semiconducting (10,8) SWNT (Reprinted (Adapted) with permission from Barone et al. (2005b). © (2005) American Chemical Society)

0.35

B3LYP

0.05 HSE

TPSSh

TPSS

–0.15

PBE

–0.05 LSDA

Mean Error (eV)

0.15

PBEh

0.25

–0.25

–0.35

Fig. 6 E11 mean errors in a set of ten semiconducting SWNTs calculated with different functionals

14

V. Barone et al.

First-principlescalculations including electron–electron interactions beyond the mean field theory have shown the excitonic character of optical transitions in SWNTs with large exciton binding energies (of up to 1 eV for the (8,0) SWNT) (Spataru et al. 2004). These predictions have been corroborated by experiments (Shaver et al. 2007; Wang et al. 2005). Unfortunately, these calculations are too demanding for routine calculations in large diameter tubes, and unlikely to be practical to study the effect of defects and functionalization. The success of the hybrid functionals HSE and TPSSh in predicting the peak position of optical transitions has been attributed to unknown error cancellations (Kümmel and Kronik 2008; Spataru et al. 2008). However, excitonic effects in metallic nanotubes are up to two orders of magnitude smaller than in semiconducting tubes (Deslippe et al. 2007) and remarkably, the same hybrid functionals that are able to describe the optical peaks in semiconducting tubes also produce excellent results in metallic tubes (Barone et al. 2005a). As a first example of the predictive capabilities of these functionals, we show in Fig. 7 calculated first-order optical transitions, E11 as a function of the corresponding experimental values in a set of five semiconducting and five metallic chiral nanotubes. All non-hybrid functionals employed here (LDA, PBE, and TPSS) underestimate E11 in metallic tubes by approximately 0.3 eV. This error is comparable to the error for E11 in semiconducting tubes. The best overall

2.6 2.4

LDA TPSS HSE

2.2 Calculated E 11 (eV)

PBE TPSSh

2.0 1.8 Semiconductor 1.6 1.4 Metal

1.2 1.0 0.8 0.6 1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Experimental E11 (eV)

Fig. 7 Calculated versus experimental first optical excitation energies for semiconducting and metallic SWNTs (Reprinted (Adapted) with permission from Barone et al. (2005a). © (2005) American Chemical Society)

Modeling of Quasi-One-Dimensional Carbon Nanostructures with Density. . .

15

Table 2 First-order optical transitions (eV) in metallic and semiconducting tubes calculated using the hybrid TPSSh and HSE functionals (Barone et al. 2005a, b), GW plus electron–hole interactions (GW C e–h) (Spataru et al. 2008), and experimental values Tube TPSSh Semiconductor (10,0) 1.04 (11,0) 1.19 Metallic (12,0) 2.25 (10,10) 1.89

HSE

GW C e–h

Exp.

Reference (Exp.)

0.97 1.12

1.00 1.21

1.07 1.20

Bachilo et al. (2002) Bachilo et al. (2002)

2.24 1.86

2.25 1.84

2.16 1.89

Fantini et al. (2004) Fantini et al. (2004)

performance is achieved by the hybrids TPSSh and HSE, which yield comparable first-order transitions in the case of metallic SWNTs. As a second example, we compare in Table 2 first-order transitions calculated using the hybrid TPSSh and HSE functionals (Barone et al. 2005a), and calculations considering GW plus electron–hole interactions (GW C e–h) (Spataru et al. 2008), with experimental values. Here, it is worth to point out the results obtained with hybrid functionals, that predict peak positions in agreement with more complex quasiparticle and excitonic effects approaches. An explanation for this behavior has been recently presented by Brothers et al. (2008).

Graphene Nanoribbons Theoretical and experimental studies reveal that the optical peaks of AGNRs might be utilized as tools to determine the nature of their edges (Barone et al. 2006; Pimenta et al. 2007). The first calculations of the optical spectrum of GNRs was presented by Barone et al. (2006) by means of DFT using the screened-exchange hybrid HSE functional. As expected from an inter-band transitions framework, first optical excitations present the corresponding oscillations as a function of the width. Second-order transitions also exhibit these oscillations, as shown in Fig. 8. For armchair ribbons with hydrogen terminations there is a shift between the oscillation of the first and second optical transition energies such that the local maxima of E11 coincide with the local minima of E22 . This is expected to give rise to a doublet in the optical spectrum. A similar doublet is expected to appear for bare GNRs with widths smaller than 1.2 nm. Nevertheless, for larger widths, the bandgap oscillations of the first and second optical transition energies of bare ribbons are in phase and the doublet is expected to disappear. This effect should provide a practical way of revealing information on the size and the nature of the edges of GNRs. Manybody approaches later pointed out that due to the one-dimensional nature of GNRs, and like in the case of SWNTs, their optical spectrum is dominated by excitonic effects with binding energies as high as 1.4 eV for a 1.2 nm wide ribbon (Prezzi et al. 2008; Yang et al. 2007).

16

V. Barone et al. 4.5

4.5 Bare E11 E22

4.0

E11 (eV)

3.5

E11 E22

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

H-terminated

4.0

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 Width (nm)

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 Width (nm)

Fig. 8 Dependence of the first (E11 ) and second (E22 ) optical transition energies on the width of bare (left panel) and hydrogen-terminated (right panel) GNRs, at the HSE/6-31G* level of theory (Reprinted (Adapted) with permission from Barone et al. (2006). © (2006) American Chemical Society)

Chemistry at the Edges of Graphene The question of what is at the edges of graphene has fascinated researchers in the area (Radovic and Bockrath 2005), especially, since the possibility of finding magnetic edge states was openly discussed in terms of the type of chemical bond that is expected to appear for bare and hydrogen-terminated edges (Radovic and Bockrath 2005). The question of how magnetic and electronic properties of nanoribbons depend upon chemical functionalization and doping has been partially addressed by several studies. One of the first studies along this line of research was presented by Hod et al. (2007a). In this study, the authors assumed that in most synthetic scenarios GNRs edges will most likely be oxidized with an unknown effect on their electronic properties. The oxidation schemes considered in these calculations included hydroxyl, lactone, ketone, and ether groups. The authors have shown that these oxidized ribbons are, in general, more stable than hydrogenterminated GNRs (see Fig. 9). These configurations maintain a spin-polarized ground state with antiferromagnetic ordering localized at the edges, similar to the fully hydrogenated counterparts. Edge oxidation has been found to lower the onset electric field required to induce half-metallic behavior (see section “Graphene Nanoribbons”) and extend the overall field range at which the systems remain halfmetallic. When the edges of the ribbon are fully or partially hydrogenated, the field intensity needed to switch the system to the half-metallic regime is about 0.4 V/Å and the range at which the half-metallic behavior is maintained is of 0.3 V/Å. Nevertheless, when the edges are fully oxidized, the system turns half-metallic at a

Modeling of Quasi-One-Dimensional Carbon Nanostructures with Density. . .

17

a

Hydroxyl I

b

Hydroxyl II

Lactone

Ketone

PBE

0.25

Ether I

Ether II

Ether (I)

HSE

0.20

dG (eV / atom)

0.15 0.10 0.05

Hydrogen

Ether (II)

0.00 –0.05 –0.10

Hydroxyl (I)

Ketone Lactone

–0.15 –0.20

Hydroxyl (II)

Fig. 9 (a) Scheme of the oxygen- containing groups considered. (b) Relative stability of the different chemical groups at the edges as calculated using the enthalpy of formation with respect to molecular oxygen and hydrogen, and graphene as shown in Hod et al. (2007a) (Reprinted (Adapted) with permission from Hod et al. (2007a). © (2007) American Chemical Society)

lower field intensity (0.2 V/Å) and the range of half-metallic behavior is doubled to 0.6 V/Å. Unfortunately, it is found that oxygen-containing groups at the edges have a minor effect on the energy difference between the antiferromagnetic ground state and the above-lying ferromagnetic state. This indicates a weak site-to-site exchange coupling and therefore oxygen terminations do not increase the spin coherence length in these systems.

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Gunlycke et al. (2007) also studied the effect of oxygen and imine groups and found that transport and magnetic properties are greatly affected by the nature of the chemical groups at the edges. The electronic properties of ribbons with edge carbons presenting two hydrogen or two fluorine atoms were also studied and it was found that in narrow GNRs the sp3 type of edge carbon results in a non-spin-polarized state (Kudin 2008). Kan et al. (2007) studied the effect of combining donor and acceptor groups at different edges of zigzag nanoribbons using DFT in the GGA. Within this approach, the authors found a half-metallic behavior without an external field for some particular chemical decorations. Cervantes-Sodi et al. (2008a, b) investigated the electronic properties of chemically modified ribbons and observed that chemical modifications of zigzag ribbons can lift the spin degeneracy which promotes a semiconducting-metal transition, or a half-semiconducting state, with the two spin channels having a different bandgap, or a spin-polarized half-semiconducting state, where the spins in the valence and conduction bands are oppositely polarized. The authors find that edge functionalization studied in their work gives electronic states a few eV away from the Fermi level in armchair ribbons and does not significantly affect their bandgap. Lee and Cho (2009) presented calculations based on the local spin density approximation and found that edge-oxidated ZGNRs present a metallic behavior. This was rationalized in terms of the electronegativity of O with respect to C. However, it needs to be stressed that while LSDA or PBE calculations might yield a metallic solution, hybrid functionals still predict all oxidation schemes to be semiconducting (Hod et al. 2007a). Chemical functionalization of rectangular graphene nanodots has been studied in detail using the DFT-B3LYP approach (Zheng and Duley 2008). This study shows that edge chemical modifications in finite ribbons significantly alter their electronic structure. Finite size effects in this type of dots will be further discussed in the next section. It is interesting to note that besides chemical functionalization at the edges of graphene it is possible to introduce adatoms on the graphene surface. Usually, this type of interaction is governed by a charge transfer mechanism. Rigo et al. (2009) presented DFT calculations on Ni adsorption on graphene nanoribbons. The adsorption takes place preferentially along the edges of zigzag nanoribbons and the interaction of the adatom with the carbon backbone quenches the magnetization of the latter in the neighborhood of the adsorption site. Other transition metal atoms adsorbed on GNRs were studied using DFT (Sevincli et al. 2008). Interestingly, Fe or Ti adsorption makes certain armchair GNRs half-metallic with a 100 % spin polarization at the Fermi level. These results indicate that the properties of graphene nanoribbons can be strongly modified through the adsorption of 3 d transition metal atoms. Alkaline and alkali metal adsorption on graphene nanoribbons were found to exhibit a strong site-dependent interaction (Choi and Jhi 2008). Similar results have been obtained by Uthaisar et al. (2009) when considering the interaction of Li adatoms and graphene nanoribbons. The strength of the interaction is much larger in zigzag GNRs than in armchairs and occurs preferentially along the edges. This enhancement is rationalized in terms of the larger number of electron acceptor states in ZGNRs compared to AGNRs. Energy barriers for Li migration also present

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important characteristics along the edges which can result in faster kinetics than in regular graphene (Uthaisar and Barone 2010). In addition, recent computational studies suggest lithium doping as a possible route for bandgap engineering of graphitic systems (Krepel and Hod 2011).

Finite Size Effects in Low-Dimensional Graphitic Materials Finite size effects play a central role in dictating the electronic properties of materials at the nanoscale. Due to their unique electronic structure, quasi-zerodimensional (quantum dots) graphitic structures may exhibit fascinating physical phenomena, which are absent in their quasi-one-dimensional (nanowires, nanotubes, and nanoribbons) counterparts. Many factors govern the effect of reduced dimensions on the electronic properties of nanoscale materials. Here we focus on two such important factors, which are strongly manifested in the electronic characteristics of graphitic materials, namely, quantum confinement and edge effects: 1. Quantum confinement is related to the boundary conditions enforced on the electronic wave function by the finite size of the system. When the typical deBroglie wavelength associated with the Fermi electrons becomes comparable to the dimensions of the system, its electronic and optical properties deviate substantially from those of the bulk system. As the confining dimension decreases and reaches this limit (which is typically within the nanometer regime) the energy spectrum turns discrete and the energy gap becomes size dependent. 2. Edge effects in graphitic materials are dominated by localized states, which are physically located at the boundaries of the system and energetically positioned in the vicinity of the Fermi energy. These states influence not only the electronic properties of these systems but also their chemical reactivity.

Quantum Confinement in Graphitic Systems As shown in previous sections, combining the unique electronic structure of the two-dimensional graphene sheet with the quantum confinement in quasi-onedimensional graphene derivatives, results in unique electronic properties, which are governed by the specific geometry and dimensions of the relevant system. In section “Structure–Property Relations in Single-Walled Carbon Nanotubes and Graphene Nanoribbons,” it was shown that one of the most notable examples of such effects is found in the strong dependence of the electronic character of carbon nanotubes on their specific diameter and chirality. This diversity in the electronic structure obtained from a single material just by changing its spatial symmetry, is one of the most promising characteristics of carbon nanotubes for applications as basic components in future nanoscale electronic devices.

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Recent experimental procedures have allowed the production of ultra-short carbon nanotubes (Chen et al. 2006a, b; Gu et al. 2002; Javey et al. 2004; Khabashesku et al. 2002; Mickelson et al. 1998; Nakamura et al. 2003). The electronic structure of these quasi-zero-dimensional systems is expected to be considerably different from their elongated counterparts, since the reduction of dimensionality implies additional confinement restrictions which may result in the emergence of new and interesting physical phenomena. Several theoretical investigations have addressed the importance of quantum confinement on the electronic properties of finite carbon nanotubes (Baldoni et al. 2007; Li et al. 2002; Liu et al. 2001; Rochefort et al. 1999b). Using a variety of methods including the HF approximation, semiempirical calculations, and the GGA, Rochefort et al. (1999b) have investigated the change in bandgap, density of states, and binding energies as a function of the length of armchair carbon nanotubes. In contrast with the metallic character of the infinite tubes, finite segments shorter than 10 nm are predicted to present a considerable bandgap, which vanishes as the length of the tube increases. Interestingly, the convergence to the infinite metallic system is non-monotonic, and pronounced threefold bandgap oscillations occur as the length of the tube is extended (see Fig. 10). These bandgap oscillations were associated with the periodic changes in the bonding characteristics of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital), which are a direct consequence of the quantum confinement of the   electrons along the tube axis. As one may expect, the calculated bandgaps strongly depend on the specific computational method, where the HF and semiempirical approximations predict bandgap values which are four to six times larger than those obtained by the generalized gradient DFT and extended Huckel calculations. Nevertheless, the general characteristics of the bandgap behavior and the existence of the bandgap oscillations are predicted by all methods. Similar results have been obtained using the hybrid B3LYP functional for open-ended and capped finite nanotube sections (Li et al. 2002). Expanding the study to the case of zigzag and chiral nanotubes, Liu et al. (2001) have used extended Huckel calculations to show the dependence of the bandgap oscillations on the chirality of the finite nanotube segment. As shown in Fig. 11, as the chirality of the tube changes gradually from armchair to zigzag, the amplitude of the oscillations reduces, and almost vanishes for the case of (12,0) zigzag nanotube segments. This leads to the interesting conclusion that the metallic character of the infinite armchair nanotubes is replaced by HOMO–LUMO gap oscillations for finite armchair nanotube segments, while the oscillatory bandgap nature of infinite zigzag nanotubes is replaced by a vanishing HOMO–LUMO gap for the finite zigzag nanotube segments. An interesting interpretation of the behavior of the HOMO and LUMO levels of finite carbon nanotube segments as a function of their length was recently given using Clar sextet theory (Baldoni et al. 2007). As discussed above, when an infinite graphene sheet is cut to form a quasione-dimensional graphene nanoribbon with a finite width and infinite length, the  -electrons wave function is confined along the direction perpendicular to the axis of the ribbon and is forced to vanish at large distances along this direction. These “particle in a box” like boundary conditions induce discretization of the

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Fig. 10 Variation of the band gap of a (6,6) nanotube as a function of its length determined using different computational techniques (Reprinted (Adapted) with permission from Rochefort et al. (1999b). © (1999) American Chemical Society)

two-dimensional dispersion relation into a set of one-dimensional bands. This discretization induces bandgap dependence on the width of the obtained ribbons. The dimensionality of these systems may be further reduced to formgraphene nanodots, which can be viewed as molecular derivatives of graphene. This extra confinement has been recently shown to strongly impact the electronic structure of these systems (Hod et al. 2008; Shemella et al. 2007). Figure 12 presents the HOMO-LUMO gaps as a function of the length and width of a large number of graphene quantum dots calculated using the local density approximation (upper left panel), the PBE flavor of the generalized gradient correction (upper right panel), and the screened-exchange hybrid HSE functional (lower left panel). The studied graphene derivatives are rectangular in shape and denoted by N  M where N and M are the number of hydrogen atoms passivating the armchair and zigzag edge, respectively. As in the case of infinite armchair graphene nanoribbons, an oscillatory

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(8, 4)

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(6, 6) (7, 5)

(6, 6) 1

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Fig. 11 The bandgap as a function of the (even) number of SWNTs sections (Reprinted with permission from Liu et al. (2001). © (2001) by the American Physical Society)

behavior of the energy gap as a function of the length of the ribbon is observed. The periodicity of these oscillations appears to be somewhat different than the threefold period obtained for the infinitely long counterparts. This, however, is a result of the fact that in order to prevent dangling carbon bonds, the width step taken for the finite systems is twice as large than the one taken in the infinitely long armchair ribbon calculations. The amplitude of the oscillations is found to be considerably damped due to the finite size of the ribbons. As expected, when the length of the armchair edge (N) is increased, the oscillation amplitude increases as well. It is interesting to note that, in general, the HOMO–LUMO gap is inversely proportional to the width (N) and the length (M) of the finite GNR in accordance with the semimetallic graphene sheet limit. Therefore, in order to obtain energy gap tailoring capability, one will have to consider GNRs with long armchair edges (large N values) and short zigzag edges (small M values). This will increase the amplitude of the energy gap oscillations while maintaining overall higher gap values.

Edge Effects in Graphitic Systems The effects of localized edge states on the electronic properties of quasi-onedimensional systems have been discussed in previous sections. The question arises of whether similar effects can be observed in quasi-zero-dimensional systems. It is well established that small molecular derivatives of graphene, such as different types of polyaromatic hydrocarbons, have a closed shell nonmagnetic ground state. On the other hand, infinite ZGNRs present a spin-polarized ground state. This suggests

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Fig. 12 HOMO–LUMO gap values for three sets of graphene nanodots, as calculated by the local spin density approximation (upper left panel), PBE functional (upper right panel), and the HSE functional (lower left panel) (Reprinted with permission from Hod et al. (2008). © (2008) by the American Physical Society)

that there exists a critical size at which molecular graphene derivatives become spin polarized. One of the earliest studies addressing this issue dates back to more than two decades ago (Stein and Brown 1987). Using Huckel theory it was found that finite graphene flakes exhibit electronic edge states along the zigzag edges. More recently, several studies, based on DFT calculations have investigated this question in detail. As a first step for obtaining quasi-zero-dimensional graphene derivatives, it is convenient to consider truncated graphene nanoribbons in the form of rectangular nanodots (Hod et al. 2008; Jiang et al. 2007; Kan et al. 2007; Rudberg et al. 2007; Shemella et al. 2007). Surprisingly, it was found that even molecular scale graphene derivatives, such as the bisanthrene (phenanthro[1,10,9,8-opqra]perylene) isomer of the C28 H14 molecule and C36 H16 (tetrabenzo[bc,ef,kl,no]coronene), are predicted to present a spin polarized ground state as shown in Fig. 13 (Hod et al. 2008; Jiang et al. 2007; Kan et al. 2007). For the smaller molecules, this was further verified using a complete active space self-consistent field many-body-wave function approach (Jiang et al. 2007). Furthermore, the application of an in-plane electric field perpendicular to the zigzag edge was found to turn the finite systems

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a

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am Se

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Fig. 13 (a)–(e) Isosurface spin densities of the antiferromagnetic ground state of several molecular graphene derivatives as obtained using the HSE functional and the 6-31G** basis set. (f) and (g) represent diradical and hexaradical Clar structures of bisanthrene (Reprinted with permission from Hod et al. (2008). © (2008) by the American Physical Society)

into molecular scale half-metals (Hod et al. 2008; Kan et al. 2007), similar to the case of infinite ZGNRs. When considering more complicated graphene derivatives such as triangular graphene flakes (Ezawa 2007; Fernandez-Rossier and Palacios 2007), the combination of the local antiferromagnetic spin ordering on adjacent carbon sites and the edge geometry of the triangular structure results in spin frustration. This leads to a metallic ground state with an overall ferromagnetic character and a finite magnetic moment. The results of tight-binding Ising model (Ezawa 2007) and Hubbard model (Fernandez-Rossier and Palacios 2007) Hamiltonians are consistent with the predictions of DFT (Fernandez-Rossier and Palacios 2007). These can be further rationalized by Lieb’s theorem (Lieb 1989) regarding the total spin S of the exact ground state of the Hubbard model in bipartite lattices. The honeycomb lattice of graphene is formed by two triangular interpenetrating sublattices, A and B. Since triangular nanostructures have more atoms in one sublattice, NA > NB the total spin S of the ground state is 2 S D NA  NB > 0. Here, the main contribution to the magnetic moment comes from edge states around the zigzag edges of

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the triangular structure. Such molecular graphene derivatives were suggested to function as permanent magnets in future nanoscale memory devices. Naturally, the effect of edge states reduces as the surface to edge ratio becomes larger. An interesting question to address is at what length would the effect of the edges disappear and the electronic structure of the system becomes essentially identical to that of the infinite system? In a recent study, this question was addressed using a divide-and-conquer (D&C) DFT approach, which enables the efficient and accurate calculation of the electronic properties and charge transport through finite elongated systems (Hod et al. 2006). Within this approach, the Hamiltonian H is given in a localized basis set representation by a block-tridiagonal matrix, where the first and last diagonal blocks correspond to the two terminating units of the ribbon (see upper panel of Fig. 14). The remaining diagonal blocks correspond to the central part of the GNR which is composed of a replicated unit cell. The terminating units and the replicated central part unit cell are chosen to be long enough such that the block-tridiagonal representation of H (and the overlap matrix S) is valid. The terminating unit diagonal Hamiltonian blocks and their coupling to the central part are evaluated via a molecular calculation involving the two terminating units and one unit cell cut out of the central part. The replicated unit cell blocks of the central part and the coupling between two such adjacent blocks are approximated to be constant along the GNR and obtained from a periodic boundary conditions calculation. The resulting block-tridiagonal matrix E S  H is then partially inverted for each value of the energy E using an efficient algorithm, to obtain the relevant Green’s function blocks needed for the DOS and transport calculations using the following formula 1 .E/ D  I m fT r ŒG r .E/Sg ; 

(2)

where G r .E/ D ŒES  H1 . A detailed account of the D&C method can be found in Hod et al. (2006). To study these edge effects in graphene, three graphene nanoribbons with consecutive widths were considered and their DOS as a function of the ribbons’ length were calculated using this D&C approach (Hod et al. 2007b). As an example, Fig. 14 presents the DOS of a finite graphene nanoribbon of different lengths. It can be seen that for graphene nanodots up to 12 nm in length, the DOS resembles that of a finite molecular system characterized by a discrete set of energy levels. When the ribbon is further elongated to 38 nm, a constant DOS around the Fermi energy of the infinite system arises and typical Van Hove singularities start to build up. For this specific system, at a length of 72 nm most of the features that are related to the edge states disappear, and the DOS almost perfectly matches that of an infinite armchair graphene nanoribbon. The rate of convergence of the electronic properties of finite systems to those of the infinite counterparts will depend on the type of the system and the specific interactions that govern its electronic character. Therefore, the divide-and-conquer method allows the quantitative estimation of this rate based on state-of-the-art DFT approximations.

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0.04

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Fig. 14 Upper panel: A finite elongated graphene nanoribbon. Lower panels: DOS of the finite GNR at different lengths. The red curve in the lowermost panel is the DOS of the infinite system (Reprinted with permission from Hod et al. (2007b). © (2006) by the American Physical Society)

We now turn back to the case of carbon nanotubes. When a zigzag nanotube is cut into finite segments, the zigzag edges are exposed. We have shown in the previous section that quantum confinement effects may play an important role in determining the electronic character of such structures. Similar to the case of graphene nanoribbons, these effects are now accompanied by the formation of spinpolarized edges states which put their own fingerprints on the electronic structures of the system (see Fig. 15). There has been a debate in the literature regarding the energetic stability of these edge states, and whether they would appear for all finite zigzag nanotubes regardless of their diameter (Higuchi et al. 2004; Kim et al. 2003; Okada and Oshiyama 2003). It is now well accepted that all finite zigzag SWNT segments are expected to present a spin-polarized ground state (Hod and Scuseria 2008; Kim et al. 2003; Mananes et al. 2008). Furthermore, as for ZGNRs, the application of an axial electric field will drive the system into a half-metallic state, thus forming a perfect spin filter. If one now eliminates only one zigzag edge of the tube by capping it with a half-fullerene, spin frustration results in a

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Fig. 15 Antiferromagnetic-type ground state spin density maps of the (7,0) (upper left panel), (8,0) (upper right panel), (9,0) (lower left panel), (10,0) (lower right panel) finite zigzag SWNT segments as obtained using the HSE functional with the 6-31G** basis set. Red and blue isosurfaces indicate the two spin flavors with an isovalue of 0.0015 a0 3 (Reprinted (Adapted) with permission from Hod and Scuseria (2008). © (2008) American Chemical Society)

spin-polarized ferromagnetic ground state (Kim et al. 2003), which resembles the case of triangular graphene derivatives discussed above and forms a molecular magnet bearing a permanent magnetic moment. Two-dimensional graphene and its lower dimensional derivatives present a diversity of electronic behaviors, making them particularly attractive as building blocks for future nano-devices. Simplified model Hamiltonian approaches may give important insights on the general physical trends in these systems. Nevertheless, density functional theory in general, and the screen exchange hybrid functional approximation, in particular, seem to be excellent tools to quantitatively study the structure–function relations in these systems and the effects of external perturbations such as chemical substitutions and electric and magnetic fields.

Electromechanical Properties of One-Dimensional Graphitic Structures Electromechanical devices are based on systems for which the mechanical properties can be controlled via the application of external electric potentials and/or the electronic properties may be altered via induced mechanical deformations. Such devices can be scaled down to form microelectromechanical systems (MEMS), which are small integrated devices or systems that combine electrical and mechanical components. Their dimensions may vary from the submicron level up to a few millimeters. By fabricating miniature mechanical elements such as beams, gears, diaphragms, and springs MEMS enabled the realization of diverse applications

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including ink-jet-printer cartridges, accelerometers, miniature robots, microengines, locks, inertial sensors, microtransmissions, micromirrors, micro actuators, optical scanners, fluid pumps, transducers, and chemical, pressure, and flow sensors. Nanoelectromechanical (NEMS) systems present the ultimate miniaturization of such devices, scaling them further down to the molecular level. Apart from the reduced dimensions, these systems are characterized by lower energy consumption and increased sensitivity toward external perturbations. Furthermore, due to their quantum-mechanical nature, molecular-sized NEMS may present unique physical properties that cannot be obtained from their microscopic and macroscopic counterparts. The unique electronic and mechanical properties of carbon nanotubes have marked them as promising candidates for key components in NEMS. In recent years, several theoretical investigations as well as experimental realization of NEMS have been presented (Cao et al. 2003; Cohen-Karni et al. 2006; Fennimore et al. 2003; Gomez-Navarro et al. 2004; Hall et al. 2007; Maiti 2003; Minot et al. 2003; Nagapriya et al. 2008; Paulson et al. 1999; Rueckes et al. 2000; Sazonova et al. 2004; Semet et al. 2005; Stampfer et al. 2006; Tombler et al. 2000).

Carbon Nanotubes in NEMS Applications The key components in most of the suggested setups are suspended carbon nanotubes bridging the gap between two conducting electrodes without making contact with the underlying surface (see Fig. 16). One may then induce mechanical deformations via manipulation of an external nanoscale tip while simultaneously measuring the changes in the conductance of the system. Strong reversible electromechanical response of carbon nanotubes under combined bending and stretching deformations have been recorded by several experimental groups (Cao et al. 2003; Maiti 2003; Minot et al. 2003; Semet et al. 2005; Stampfer et al. 2006; Tombler et al. 2000). In Fig. 17, the change in the SWNT’s resistance as a function of the depression applied to the top cantilever (Fig. 16) is plotted. The resistance of the system increases by more than an order of magnitude over a depression range of less than 4 nm, suggesting that such setups may be used for highly sensitive and reliable displacement sensors. A similar setup, where a floating pedal is attached to a suspended Fig. 16 SEM image of a cantilever-SWNT-based NEMS sensing device. The white arrows indicate the location of the suspended SWNT (Reprinted (Adapted) with permission from Stampfer et al. (2006). © (2006) American Chemical Society)

a

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multi-walled nanotube, has enabled the investigation of the electromechanical response of single (Hall et al. 2007) and multi-walled (Cohen-Karni et al. 2006) nanotubes under the application of torsional deformations. In this setup, an atomic force microscope (AFM) tip presses down on the floating pedal. Since the SWNT is fixed at both edges to the conducting electrodes, the central section experiences torsion. This change in helicity of the system induces considerable variations in the electronic conductance. Interestingly, pronounced bandgap oscillations are observed as a function of the deflection angle of the pedal. The origin of these oscillations has been associated with the distortion of the first Brillouin zone (fBZ), which results in the shifting of the Dirac points (Cohen-Karni et al. 2006; Nagapriya et al. 2008). As the Dirac points shift within the fBZ they may approach an allowed electronic band thus increasing the conductance of the system. Once the Dirac point crosses the allowed band the conductance reaches a maximum value. Further deformation results in shifting the Dirac points away from the allowed band and thus a decrease in the conductance. This process is repeated anytime the Dirac point crosses one of the many parallel allowed bands. Other prototype devices such as nanotube actuators (Fennimore et al. 2003; Gomez-Navarro et al. 2004), and nonvolatile memory components based on SWNTs junctions (Rueckes et al. 2000) have been

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successfully fabricated as well, demonstrating the potential of SWNTs as nanoscale electromechanical devices. Aiming to obtain a microscopic understanding of these experimental observations and to guide further experiments with carbon nanotubes under various mechanical deformation modes, many theoretical investigations have been performed (Maiti 2009). Due to the complexity of the deformed systems, most of the computational treatments were based on semiempirical or DFT-based TB calculations. The effects of axial stretching (Heyd et al. 1997; Jiang et al. 2004; Kane and Mele 1997; Kleiner and Eggert 2001; Yang et al. 1999), bending (Kane and Mele 1997; Liu et al. 2000; Nardelli 1999; Nardelli and Bernholc 1999; Rochefort et al. 1998, 1999a; Yang and Han 2000), torsion (Jiang et al. 2004; Kleiner and Eggert 2001; Rochefort et al. 1999a; Yang and Han 2000; Yang et al. 1999; Zhang et al. 2009), and radial compression (Lammert et al. 2000; Lu et al. 2003; Peng and Cho 2002; Svizhenko et al. 2005) were studied extensively showing high sensitivity of the electronic properties and transport characteristics of the deformed nanotubes to the applied deformations. Several DFT calculations were conducted to study the effect of axial strain (Guo et al. 2005), bending (Maiti 2001; Maiti et al. 2002), radial compression (Mehrez et al. 2005; Wu et al. 2004), and torsional deformations in carbon nanotubes (Nagapriya et al. 2008): 1. Axialstrain: While isolating the effect of axial stretching and compression is a challenging experimental task (Cao et al. 2003), pure axial deformations are easy to study using modern computational tools (Guo et al. 2005). Bandgap variations

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Fig. 18 Calculated bandgap variation of zigzag (9,0), (12,0), (15,0), and (27,0) SWNTs as a function of axial strain (Reprinted with permission from Guo et al. (2005). © (2005) American Institute of Physics)

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due to axial compression and stretching were studied using the local density approximation. As it can be seen in Fig. 18 for strains up to 3 %, linear variations in the bandgap are predicted. The slope of the bandgap response depends on the tube type and on the sign of the axial deformation. For all systems studied, the relative changes in bandgap are of the order of 0.5 eV and are definitely measurable in experimental conditions. 2. Bending: The effect of pure bending and AFM tip-induced bending was studied using the Harris functional approximation (Maiti 2001; Maiti et al. 2002). It was shown that both pure bending and tip-induced compression maintain the hexagonal sp2 lattice structure up to relatively high bending angles. Armchair tubes remain significantly conducting even at large deformations. However, metallic zigzag tubes display a dramatic drop in conductance, in particular under tip-induced deformations. 3. Compression: Radial compression was studied via a combined molecular dynamics and density functional theory-based nonequilibrium Green’s function approach (Wu et al. 2004). Reversible pressure-induced metal-to-semiconductor transitions of armchair SWNTs were predicted suggesting that SWNTs may be used as miniature sensitive pressure detectors (Mehrez et al. 2005; Wu et al. 2004). 4. Torsion: While the torsional electromechanical response was successfully explained via the deformation of the Brilluoin zone, another possible explanation for the bandgap oscillations may be the periodic formation of Moire patterns due to registry mismatch between the different walls of the multi-walled tubes, which have a different curvature. In order to rule out this possibility, DFT calculations within the LSDA were used to show that the interlayer coupling is weak and the electronic structure of the individual walls resembles that of single-walled nanotubes for different intertube orientations (Nagapriya et al. 2008). As can be seen from all of the examples given above, the study of the physical properties of nanotubes in general and their electromechanical behavior in particular requires an intimate relation between theory and experiment, where the theoretical tools enable an atomic scale understanding of the experimental observations and provide guidelines for the design of new experiments. As we have seen above, when considering finite size effects and chemical functionalization, DFT presents new insights that are not captured within the simple TB approximations. When considering the electromechanical response of SWNTs to different mechanical deformations, TB calculations seem to give a reliable description. Nevertheless, calculations based on DFT are required for comparison and validation purposes.

Graphene Nanoribbons in NEMS Applications Soon after the successful isolation of a single graphene sheet (Novoselov et al. 2004) the first experimental realization of an electromechanical device based on graphene nanoribbons was presented (Bunch et al. 2007). In this setup, atomic

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thick layers of graphene were suspended above predesigned trenches carved in the underlying silicon oxide surface while bridging the gap between gold electrodes. Both optical and electrical actuation procedures were demonstrated resulting in vibrational frequencies in the MHz regime. High charge sensitivities suggested the possibility of using similar devices as ultrasensitive mass and force detection (Bunch et al. 2007). Following this novel experimental fabrication and manipulation of graphene nanoribbons as nanoelectromechanical components, several other studies have explored the mechanical (Frank et al. 2007; Poot and van der Zant 2008) and electromechanical response of these systems (Milaninia et al. 2009). The effects of uniaxial strains in isolated graphene nanoribbons were recently studied in details using the generalized gradient approximation of the exchange correlation functional within the framework of DFT (Faccio et al. 2009; Sun et al. 2008). It was found that the electronic properties of zigzag GNRs are not sensitive to uniaxial strain, while the energy gap of armchair nanoribbons displays an oscillatory pattern as a function of the applied strain. By comparison to TB calculations, it was deduced that the nearest-neighbor hopping terms between atomic sites within the carbon hexagonal lattice are responsible for the observed electromechanical response. To simulate the effects of bending and torsional deformations in suspended graphene nanoribbons, DFT calculations utilizing the screened-exchange HSE density functional were performed (Hod and Scuseria 2009). A large set of short armchair graphene nanoribbons was considered, where narrow strips of atoms close to the zigzag edges of the ribbon were fixed to simulate a doubly clamped suspended nanoribbon (see Fig. 19). The effect of a driven deformation due to an external tip were simulated by constraining a strip of hexagons at the central part of the ribbon to be either depressed or rotated with respect to the fixed edge atoms. Extreme mechanical deformations, way beyond the linear response regime, were applied to the full set of nanoribbons studied. Most of the nanoribbons considered stayed stable under depressions of 1 nm as well as torsional angles of 90ı maintaining their spin-polarized ground state character. As it can be seen in Fig. 20, pronounced electromechanical responses for both bending and torsion (torsion not presented in Fig. 20) have been obtained including evidence of dimension-dependent bandgap oscillations similar to the case of SWNTs (Cohen-Karni et al. 2006; Nagapriya et al. 2008). Fig. 19 An artist view of an electromechanical device based on graphene nanoribbons. An external tip may induce bending and torsion of the nanoribbon (Reprinted (Adapted) with permission from Hod and Scuseria (2009). © (2009) American Chemical Society)

Egap (eV)

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1.2

1.4

1.0

1.2

0.8

03 × 14

04 × 14

03 × 20

04 × 20

03 × 28

04 × 28

0

2.5

1 5

7.5

10

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Egap (eV)

1.2

1.0

0.8

05 × 14 05 × 20

0.5

05 × 28

0

2.5

5

7.5

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Depression (Å)

Fig. 20 HOMO–LUMO gap variations as a function of the depression applied to the central region of a doubly clamped GNR. Different panels and line colors represent results for GNRs of different widths and lengths, respectively (Reprinted (Adapted) with permission from Hod and Scuseria (2009). © (2009) American Chemical Society)

Apart from indicating the promise that graphene nanoribbons hold from an electromechanical perspective, such calculations based on state-of-the-art density functional theory approaches, enhance the molecular scale understanding of the physical processes governing the behavior of these intriguing materials. Furthermore, theory and computations may guide future experiments to pursue promising scientific and technological routes toward which efforts and resources should be directed.

Concluding Remarks In this chapter, we have given several examples of how theoretical investigations can be applied to elucidate the behavior of carbon nanostructures with emphasis on the understanding of the basic physical mechanisms that take place at the molecular level. In particular, we have shown that density functional theory is a powerful tool to this end, and we have provided several examples where density functional theory has been utilized to investigate electronic and structural properties of graphene

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nanoribbons, carbon nanotubes, and other low-dimensional graphitic materials. We hope that this chapter will furnish the reader with an ample background on this growing field and that it will serve as a starting point for readers interested in pursuing research in this exciting field. Acknowledgments VB acknowledges the donors of The American Chemical Society Petroleum Research Fund for support through the award ACS PRF#49427-UNI6. OH acknowledges the support of the Israel Science Foundation (Grant 1313/08), the Center for Nanoscience and Nanotechnology at Tel-Aviv University, the Lise Meitner-Minerva Center for Computational Quantum Chemistry, and the European Community’s Seventh Framework Programme FP7/20072013 under grant agreement no. 249225. JEP acknowledges support from NSF DMR Award #DMR-0906617.

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Variation of the Surface to Bulk Contribution to Cluster Properties Antonis N. Andriotis, Zacharias G. Fthenakis, and Madhu Menon

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tight-Binding Molecular Dynamics Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collinear Magnetic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 1: Inclusion of Randomness in the Direction of the Atomic MagneticMoments . . . . . . . Step 2: Inclusion of Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 3: Inclusion of Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 4 6 6 7 9 10 11 16 17

Abstract

Recent computer simulations have indicated that there is a linear relationship between the melting and the Curie temperatures for Nin (n  201) clusters. In this chapter, it is argued that this result is a consequence of the fact that the surface and the core (bulk) contributions to the cluster properties vary with the cluster size in an analogous way. The universal aspect of this result is also discussed. Among the many interesting consequences resulting from this relationship is the intriguing possibility of the coexistence of melting and magnetization. As demonstrated, these conclusions have as their origin the major contribution coming from the melting/magnetization ratio arising from surface effects and appear to overshadow all other contributions. As a result, this can be quantified A.N. Andriotis () • Z.G. Fthenakis • M. Menon Institute of Electronic Structure and Laser, FORTH, Heraklion, Crete, Greece Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, KY, USA e-mail: [email protected]; [email protected]; [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_25-2

1

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with approximate methods which are suitable for describing any major surface contribution to a cluster property.

Introduction As the cluster size increases, the cluster properties evolve toward their bulk counterparts. The understanding of this evolution is of fundamental importance not only from the perspective of basic science but also from the technological viewpoint. At a very approximate level, one can claim that the cluster properties can be described in terms of their surface and core (bulk) contributions and due to the fact that the surface to bulk ratio gets smaller as the cluster size increases. Consequently, it is natural to expect that the cluster properties will evolve to their corresponding bulk-phase ones for large cluster sizes. As one particular example, we mention the melting temperature of large clusters. According to the proposed model, the following functional relationship for the variation of the melting temperature, T cl melt,N , of a cluster as the number N of its atoms increases has been suggested: cl bulk Tmelt;N D Tmelt  ımelt N 1=3 ;

(1)

where T bulk melt is the melting temperature of the corresponding bulk phase and •melt is a constant that depends on N (Garcia-Rodeja et al. 1994; Gunes and Erkoc 2000; Lee et al. 2001; Nayak et al. 1998; Qi et al. 2001; Rey et al. 1993; Sun and Gong 1998). Correction terms in N2/3 and N1 powers to the above expression have also been suggested (Doye and Calvo 2001). The term proportional to N1/3 in Eq. 1 reflects the surface to bulk contribution to the melting temperature. Equation 1 was found to describe reasonably well the experimental findings in the large-size regime (N > 500). However, for clusters of smaller size, (especially for clusters with number of atoms N  50), Eq. 1 does not ensure a quantitative description of the variation of T cl melt,N with the cluster size (see, e.g., Baletto and Ferrando 2005; Lee et al. 2001; Nayak et al. 1998; Qi et al. 2001). This is because for small clusters, (1) the surface-to-volume contribution to T cl melt,N is very large and (2) is very sensitive to the variations of the surface structure and the cluster geometry (symmetry), as both of these characteristics get altered as the cluster size changes. Another property that has attracted much interest recently is the one pertaining to the evolution of the magnetic features of the magnetic clusters as the cluster size and temperature increase. This is mainly because of the potential applications of the magnetic grains in fabricating new materials for advanced magnetic storage devices and other applications (Bansman et al. 2005). Recently, the results of our computer simulations led us to the conclusion that cl the Curie, Tcl C,N , and the melting, Tmelt,N , temperature of NiN clusters consisting of N atoms are linearly related over a large range of cluster sizes (N  201), and this relationship is quantified by the following equation (Andriotis et al. 2007) :

Variation of the Surface to Bulk Contribution to Cluster Properties cl cl Tmelt;N D ˛TC;N C ˇ2;

3

(2)

where ˛ and ˇ are constants. Least square fitting of our results leads to ˛ D 0.5414 and ˇ D 443.82 ı K. This relationship was suggested to be related to the ratio of the surface to bulk contributions to the cluster melting as well as to the average magnetic moment per cluster atom, i;N .T /i D 1; : : : ; N . It was then claimed that this relationship could be justified within the mean field theory applied separately to the surface and core regions of the cluster (Andriotis et al. 2007). A cursory thought seems to suggest that any type of direct relationship between cl T cl melt,N and T C,N to be fortuitous since melting and magnetic order seem to reflect completely different aspects of the crystal potential. Therefore, Eq. 2 cannot be considered as one which reestablishes a valid physical relationship between these two cluster properties. cl Furthermore, the variation of both T cl melt,N and T C,N with the cluster size constitute separate distinct and complicated projects for both theory and experiment. This is because significant contributions to both of these physical quantities have their origin in surface as well as cluster-core (bulk) effects which, at first look, affect cl T cl melt,N and T C,N differently. These contributions include: the effects of the cohesive energy, the shape (symmetry), the size, the surface to bulk ratio, the surface tension, the temperature of the cluster, etc. In early reports, it was found that T cl melt,N is usually smaller than the corresponding bulk value. Of interest is the result applied for large enough spherical clusters of radius R: cl bulk Tmelt;N =Tmelt D 1  km =R

(3)

where km is a material-dependent constant (see, e.g., Buffat and Borel 1976 and references therein). A similar expression describing the variation of the Curie temperature with the cluster size was also found within the phenomenological Landau–Ginsburg–Devonshire theory (Huang et al. 2000), i.e., cl TC;N =TCbulk D 1  kc =R

(4)

Use of Eqs. 3 and 4 gives the following values for ˛ and ˇ in Eq. 2: ˛D

kc TCbulk bulk km Tmelt

(5)

and   kc bulk TCbulk : ˇ D TCbulk  ˛Tmelt D 1 km

(6)

Diep and collaborators (1989) using Monte Carlo simulations studied the effect of the magnetic interactions on the melting temperature of a cluster. Although their

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investigation was limited to very small clusters MN ; .N 2 Œ7; 17 / of transition metals M, however, the conclusions they arrived at are very important. In particular, among others, they have found that the incorporation of the magnetic interactions leaves the cluster structure unchanged. However, magnetic interactions lead to sharper peaks in the specific heat (and, therefore, to more precise determination of melting and Curie temperatures), a slight increase in the melting temperature, and a slight cluster-volume contraction (magnetostriction). After examining more carefully the results of Diep et al. (1989) (included in their Fig. 12), we find that cl the relation of T cl melt,N and T C,N is more or less linear with the exception of the data of the very small clusters, i.e., for N 2 Œ7; 8; 9. The linear relationship between cl T cl melt,N and T C,N was also suggested recently on the basis of semiempirical and approximate phenomenological model descriptions (see Yang and Jiang 2005 and references therein). All these descriptions support our findings (i.e., Eq. 2), which are based on a firm quantum mechanical model procedure as outlined in the following. In this work, we investigate the implications of such a relationship which seems to specify a universal aspect of the surface contribution to the cluster properties.

The Model The investigation of clusters of medium and intermediate sizes consisting of transition metal atoms poses a severe challenge in terms of computer capacities and computer time. For this reason, approximate schemes have been employed with most pronounced being those based on empirical classical potentials. However, these models have limited applicability when there is a need to understand more about the electronic structure of these systems and follow its changes as the cluster size increases and approaches the bulk phase. It is thus necessary to use methods with firm ab initio footing while at the same time not sacrificing computational efficiency. One such method is based on the Tight-Binding approximation which we have adopted in our work. In this section, we discuss briefly the implementation of this approach in order to model the temperature and magnetic features of transition metal clusters. We will firstly give a brief overview of our TB computational scheme that we used for systems at zero temperature. In the following sections, we will describe the generalization of our method enabling the inclusion of magnetic and temperature effects.

Tight-Binding Molecular Dynamics Methodology The details of our Generalized Tight-Binding Molecular Dynamics (GTBMD) scheme can be found in Andriotis and Menon (1998) and Menon and Subbaswamy (1997). The GTBMD method makes explicit use of the nonorthogonality of the orbitals resulting in a transferable scheme that works well in the range all the way from a few atoms to the condensed solid. The scheme also includes d-electron

Variation of the Surface to Bulk Contribution to Cluster Properties

5

interactions enabling dynamic treatment of magnetic effects in transition metal systems. Here, we give a brief overview. The total energy U is written in its general form as a sum of several terms (Andriotis and Menon 1998), U D Uel C Urep C U0 ;

(7)

where Uel is the sum of the one-electron energies En for the occupied states: Uel D

Xocc n

En :

(8)

In the tight-binding scheme, En is obtained by solving the characteristic equation: .H  En S/ Cn D 0;

(9)

where H is the Hamiltonian of the system and S the overlap matrix. The Hellmann–Feynman theorem for obtaining the electronic part of the force is given by Menon and Subbaswamy (1997), Cn @En D @x

 @H

 n @S C  E n @x : Cn SCn

@x

(10)

The total energy expression also derives contributions from ion–ion repulsion interactions. This is approximated by a sum of pairwise repulsive terms and included in Urep . This sum also contains the corrections arising from the double counting of electron–electron interactions in Uel (Andriotis and Menon 1998). U0 is a constant that merely shifts the zero of energy. The contribution to the total force from Urep is rather straightforward. One can then easily do molecular dynamics simulations by numerically solving Newton’s equation, m

d 2x @U D Fx D  2 dt @x

(11)

to obtain x as a function of time. Our TBMD scheme for a binary system consisting of elements A and B is based on a minimal set of five adjustable parameters for each pair (A,A), (B,B), and (A,B). These parameters are determined by fitting to experimental data for quantities such as the bond length, the vibrational frequency, and the binding energy of the dimers A2 , B2 , AB; the cohesive energy of the corresponding bulk states of the A, B, AB materials; and the energy level spacing of the lowest magnetic states of the dimer and trimer binary clusters consisting of atoms of A and B type. In the absence of experimental data, we fit to data of small clusters obtained using ab initio methods as described in the following subsection. It is apparent that only five parameters are required in the case of a single species system. The generalization to a system containing more than two kind of atoms is also plausible within this approach.

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The fixed set of TB parameters are obtained from the universal scheme proposed by Harrison (1980) suitably scaled with respect to the interatomic distance (Andriotis and Menon 1998).

Collinear Magnetic Effects In order to calculate the Curie temperature of a magnetic cluster, it is necessary to include non-collinear magnetic effects in our model description. These are introduced by extending our zero temperature (ZT) Tight-Binding Molecular Dynamics (TBMD) approach at the Hubbard-U level of approximation (Andriotis and Menon 1998) which we used to study magnetic clusters in the collinear magnetic approximation. According to this collinear model, an exchange-splitting parameter s(i) 0 is introduced which is proportional to the intra-site Coulomb interaction U. This specifies the energy splitting between spin-up and spin-down electrons in the ithatom in accordance with results obtained by ab initio methods. Thus, within this model, a site-diagonal spin-dependent Hamiltonian term V(i) spin is introduced which has the form: ! .i/ 0 s0 .i/ Vspin D (12) .i/ 0 s0 In this model, it is assumed that all atomic magnetic moments (MMs) (of the cluster atoms) are collinear to the z-axis of a local xyz-system assigned to the ith clusteratom. The generalization of this model to include non-collinear effects is achieved in three steps. In the first step, we include the randomness in the directions of the atomic MMs. In the second, we include the spin-orbit interaction, and in the third, we include the temperature effects. For the sake of completeness, we briefly discuss this generalization in the following.

Step 1: Inclusion of Randomness in the Direction of the Atomic MagneticMoments In the first step, it is assumed that the deviation of the direction of the MM, i , of the ith cluster-atom from the Z-axis of the global coordinate system XYZ is specified by the polar angles ( i ,  i ) defined with respect to this XYZ system. As a result, the potential V(i) spin , originally defined in the local coordinate system xyz of the ith (i),global

atom, is transformed to its expression Vspin in the global system XYZ as follows (Anderson and Hasegava 1955; Uhl et al. 1994): .i/;global

Vspin

.i/

D „ .i ; i / Vspin „ .i ; i / ;

(13)

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7

where „( i ,  i ) is the standard spin-1/2-rotation matrix :  „ .i ; i / D

e ii =2 cos i =2  e ii =2 sin i =2

e ii =2 sin i =2 e ii =2 cos i =2

 (14)

It is assumed that the Z-axis of the global system can be arbitrarily chosen, and a usual choice is to take Z in alignment with the easy axis of the system.

Step 2: Inclusion of Spin-Orbit Interaction In the second step, we introduce the Spin-Orbit (SO) interaction, V(i) SO , in the ith.i/ .i/ .i/ .i/ atom within the L– S coupling scheme, i.e., VSO D  L  S where, œ(i) is the spin-orbit coupling constant for the ith-atom, L(i) its orbital angular momentum along the Z-axis, and S(i) its total spin in the direction of i . Details of the implementation of the Spin-Orbit interaction within our TBMD method have been reported elsewhere (Andriotis and Menon 2004). In the presence of a magnetic field, B (assumed to be along the direction specified by the polar angles ( 0 ,  0 ) with respect to XYZ-system), the atomic MMs of the cluster-atoms tend to become parallel to the direction of B. The average projection of the MMs of the cluster-atoms, cl , along the direction of B (which is the experimentally measured quantity) is, ˇ ˇ ˇ 1 ˇˇXNcl cl D i cos i ˇˇ ; ˇ i Ncl

(15)

where Ncl is the number of cluster-atoms and cos i D cos 0 cos i C sin 0 sin i cos .0  i /. In a different formulation within the Hubbard-U model approximation to the e-e correlations, the spin-mixing interaction may be derived from a Coulombtype Hamiltonian term of the form (Kato and Kokubo 1994; Ojeda et al. 1999): P   Vsmix D U l l ;l cl cl , where cl (cl ) is the creation (annihilation) operator for an electron with spin ¢ at site l and l ;l denote the electron density matrix  elements, i.e., l ;l D< cl cl >. It can be easily verified that the Hamiltonian (i),global term Vsmix is equivalent to that given by Eq. 13, i.e., Vspin .

Evaluation of the TB representation of the SO-interaction In order to proceed with the evaluation of the TB matrix elements of the SOinteraction, we write the SO-term as S L D Lx Sx C Ly Sy C Lz Sz

(16)

and compute the matrix elements with respect to the basis set. We are using a basis set of atomic orbitals yilm (r), where the index i specifies the atom on which the

8

A.N Andriotis et al.

atomic orbital (AO) is centered, l specifies the angular momentum, m is used to count the various d-orbitals (i.e., dxy ; dxz ; dyz ; dx 2 y 2 ; dz2 ), and denotes the spin. In terms of a linear superposition of these basis functions, the single electron wave functions take the form X (17) ‰i .r/ D Cilm yilm .r/ ; l;m

where Cilm denote the coefficients which are to be determined from the diagonalization of the Hamiltonian. The spin operators of Eq. 16 refer to the global system and care has to be exercised as they act on the local spin states, the latter related with the former according to Eqs. 13 and 14. For example, < "i 0 jSx j #i 0 > D

1 cos i sin i ; 2

(18)

where the prime indicates the local functions and ", # indicate spin-up and spindown states, respectively. The matrix elements of the orbital angular momentum operators are obtained by operating on the angular part of the wave functions which, in our TBMD formalism are described by the Cubic harmonics. As an example, we write down the expression of the average value of the z-component of the orbital magnetic moment, Lz , in terms of its matrix elements: X Z ? .r/ Lz ‰i .r/ (19) < Lz >D d r‰i

i

or < Lz >D

X Xocc X

i

X l0m

l;m

C? C 0 0 0 ilm il m

Z

? .r/ Lz yil 0 m0 .r/ d ryilm

(20) Assuming orthogonality of AOs centered at different atoms, and orthonormal basis functions centered at one particular atom, we finally obtain the following expression for the d-orbital contribution to the orbital magnetic moment hLdz i: < Ldz >D

X Xocc X

i

X lD2

mm0

? il mm0 Cilm Cilm0

(21)

where il mm0 are constants easily calculated by applying the relation: Lz Ylm D mYlm

(22)

where Ylm are the spherical harmonics. For example, a straightforward calculation of the constants il mm0 shows that the only nonzero matrix elements of Lz are the

Variation of the Surface to Bulk Contribution to Cluster Properties

9

following: < dx 2 y 2 jLz j dxy >D 2i

(23)

< yz jLz j zx >D i

(24)

and

Combining the above, we construct the spin-dependent TB-representation of the SO-interaction term and add this to the other Hamiltonian terms.

Step 3: Inclusion of Temperature Effects In the third step, our ZT-TBMD method has been extended by incorporating the Nose-bath (Nose 1984) and the Multiple Histogram approximations (Fanourgakis et al. 1997), so as to be applicable to cluster studies at finite temperatures in an efficient way (Andriotis et al. 2006, 2007; Fthenakis et al. 2003). This generalization allows one to calculate the caloric curve for the cluster and use this to study the effect of temperature on the structural, electronic, and magnetic properties of transition metal clusters and binary systems containing transition metal and semiconductor atoms. The method has been used to study the variation of structural and magnetic properties with temperature as well as to obtain the caloric curves of the Ni-clusters (Andriotis et al. 2006, 2007; Fthenakis et al. 2003). Upon thermalization at temperature T, a cluster can be described by the canonical probability distribution function of total energy, PT (E), which specifies the probability that the system will be found in the energy interval ŒE; E C E at the specified temperature T. The distribution function corresponding to this temperature, within the canonical ensemble description, is (see Fanourgakis et al. 1997; Schmidt et al. 1997 and references therein): PT .E/ D

Π.E/ e E=kB T nT .E/ D NT ZT

;

(25)

where nT (E) is the number of states in the energy interval [E, E C E], NT is the total number of accessible states, kB is Boltzmann’s constant,  (E) the number of all the P energy in the interval ŒE; E C E (i.e., given by P different states with ZT D Ei exp .ˇEi / D i .Ei / exp .ˇEi /) and ZT the partition function at temperature T. A molecular dynamics (MD) simulation at a given temperature T provides numerical values for nT (E) at every accessible energy E. Having obtained these, we make use of the proposed Multiple Histogram Method (MHM) (Weerasinghe and Amar 1993), and obtain the partition functions ZTj for a finite set of temperatures Tj ; j D 1; : : : ; M .M  200/ and the entropy terms S .Ei / D kB ln Œ .Ei /

10

A.N Andriotis et al.

(within an additive constant) for a much larger set of energy values Ei ; i D 1; : : : ; N .N  6; 000/ (Weerasinghe and Amar 1993). Having obtained the quantities nT (E), ZT (E) and ST (E), we can then describe all the thermodynamic properties of the clusters and, in particular, the variation with temperature of their structural, electronic, and magnetic properties.

Computational Approach The computation of the magnetic features of the clusters is performed within the above described non-collinear TBMD scheme as this allows for a full quantum MD relaxation of systems containing several hundred transition metal atoms while incorporating magnetic effects dynamically. More specifically, it includes: (1) e- e correlation effects at the Hubbard-U approximation (Andriotis and Menon 1998), (2) the spin-orbit interaction (in the L  S approximation), and (3) non-collinear magnetic effects (Andriotis and Menon 1998, 2004). Furthermore, the effect of temperature (Andriotis et al. 2006; Fthenakis et al. 2003) is included in the formalism while making use of the full s, p, d basis set and contains many unique features that make it ideally suited for the treatment of transition metal (TM) and semiconducting materials. For large-scale simulations we have developed a parallel algorithm that enables molecular dynamics simulations of systems containing atoms in excess of a thousand. This method is more accurate than the order-N methods that are being used at present to treat systems of these sizes. The successful application of our collinear TB scheme (see, e.g., Andriotis et al. 1998, 1999, 2000; Lathiotakis et al. 1996) guarantees similar success for the present generalization as well. Finally, we also mention that this approach has been suitably adapted in order for exclusive use in studying the transport properties (based on our computational codes as described in Andriotis and Menon 2001; Andriotis et al. 2002). While the TBMD computational approach generalized in such a way is suitable for calculating the magnetic properties of the clusters (of a specific geometry), its use for relaxing the structure of particularly large clusters at nonzero temperatures has been found inefficient due to the extreme computational complexity. This is because the thermodynamic equilibration of a crystal requires sufficiently long MD relaxation time (of the order of 2–6 ns which is translated into 10–40 million MD steps with each step being of the order of a femtosecond). In order to make our computations feasible, we firstly reach the thermodynamic equilibrium at each temperature using the classical Sutton–Chen interatomic potential (Sutton and Chen 1990) appropriately fitted to the TBMD results (Fthenakis et al. (unpublished)). It should be noted that for Ni, the classical Sutton–Chen potential (Sutton and Chen 1990) was found to give results closer in agreement with our TBMD method than any other classical potentials in use for Ni (Erkos 2001; Fthenakis et al. (unpublished)). While reaching the thermodynamic equilibrium, we apply our generalized TB formalism every 100 time-steps in order to calculate the MM of the cluster. In the MM calculation, the structure of the cluster is assumed frozen (as obtained within the classical potential approach at that particular

Variation of the Surface to Bulk Contribution to Cluster Properties

11

time step), and the calculation of the MM of the cluster is repeated for a large number, N (i) ran , of atomic spin configurations taken randomly over the ith-structural .i/ configuration (Nran  120  300 for every i-time step in the present calculations). In view of these results, Eq. 15 is generalized as follows: ˇ ˇ   .i / j 1 ˇˇXNcl XNran j j  Ei E0 =kB T ˇˇ cl D i cos i e ˇ; j KNcl ˇ i

(26)

under the assumption that each spin configuration contributes to the magnetic state    j j 1  Ei E0 =kB T of the ith geometric configuration with probability PM Ei D K e where, KD

.i / XNran

j

e

  j  Ei E0 =kB T

:

(27)

The index j in Eqs. 26 and 27 denotes quantities evaluated at the particular jth random atomic spin configuration of the ith structural configuration of the cluster. E0 is taken to be the energy of the ferromagnetically aligned atomic-spin configuration. In this way, we take the average over the low-lying spin configurations of the cluster of a particular (frozen) geometric structure (i.e., as calculated at the specific time step). Finally, the thermodynamic average of cl given by Eq. 26 over the various cluster geometric structures (i.e., time-steps) is obtained with the help of the probability PT (E) as given by Eq. 25. Having obtained the temperature dependence of the average magnetic moment cl (T) per cluster atom, we proceed with the calculation of the temperature dependence of the specific heat of each cluster by taking the derivative dfcl (T)g2 /dT of the corresponding cl (T) curves. Following Gerion et al. (2000), we obtain the Curie temperature Tcl C,N for each cluster by locating the maximum of the magnetic contribution to CV . This is repeated for a number of clusters of various sizes.

Results and Discussion In the present work, our focus is on the properties of the magnetic transition metal clusters and, in particular, on Ni clusters for which experimental data are available for comparison. Using the procedures discussed above, we have calculated the cl melting, Tcl melt,N , and the Curie, TC,N , temperatures of the NiN clusters for N  201. These results were discussed recently in Andriotis et al. (2007). The melting temperature Tcl melt,N has been derived on the basis of the Lindemann index using the caloric curve as obtained from the classical potential MD simulations. The Curie temperature Tcl C,N has been derived according to the quantum mechanical procedure discussed above. Both were found to increase with the cluster size tending to their corresponding bulk values as the size of the clusters increases, in good agreement with the existing experimental data (Andriotis et al. 2007).

12

A.N Andriotis et al. 2000 Eqn. 1 Eqn. 1 (for α = 1 and β = 0) From Ref. [12]

Tmelt(°K)

1500

1000

500

0

0

200

400

600

800

1000

1200

Tc(°K) cl Fig. 1 Plot of T cl melt,N as a function of T C,N for the NiN clusters studied in the present work, N D 43, 80, 147, 177, 201 (solid line). The dotted black line denoted Eq. 2 while and dashed black line describes Eq. 2 setting ˛ D 1 and ˇ D 0. Points above (below) the dashed line have Tmelt > TC (Tmelt < TC ) respectively

cl The correlation between Tcl melt,N and TC,N obtained from our results is shown in Fig. 1 by the black solid line. This demonstrates and supports the validity of Eq. 2. We discuss the universal aspect of this correlation and the conclusions that can be derived from in the following. Firstly, it should be noted that the conclusions one can arrive at from the obtained relationship between melting and Curie temperatures of magnetic clusters depend crucially on the accuracy with which cluster melting temperatures are determined. This is a major issue as it has been extensively discussed in the literature (see, e.g., Baletto and Ferrando 2005; Qi et al. 2001; Sun and Gong 1998 and references therein). This is because surface melting of small particles occurs in a continuous manner over a broad temperature range in contradistinction to the melting of the solid-core (bulk-like) which occurs at a specific critical temperature (Garrigos et al. 1989). For this reason, it has been proposed that in order for Tcl melt,N to be described quantitatively, the surface effects are usually treated separately from the core effects by introducing the surface thickness, t0 , as a free parameter and calculate Tcl melt,N by expressing firstly the heat of the cluster fusion in terms of t0 and the cluster radius (see, e.g., Lai et al. 1996). A more commonly used approach for determining Tcl melt,N is by employing the Lindemann’s criterion, an approach followed in the present work (Fthenakis et al. 2003). Such a calculation is subject

Variation of the Surface to Bulk Contribution to Cluster Properties

13

to the limitations and the accuracy of this method. In particular, the so derived melting temperatures depend strongly on the choice of the classical potential used (Andriotis et al. 2007; Fthenakis et al. 2003). This explains why the reported values for the melting temperature of Ni clusters cover a wide range. Nevertheless, the Sutton–Chen classical potential, employed in the present work for Ni, leads to accurate values for surface energies, vacancy energy, stacking fault energies, and bulk melting temperature in very good agreement with experiment (Qi et al. 2001). The accuracy of Lindemann’s criterion depends also on the steepness of its variation with temperature at phase transition and on the specification of its percentage increase which should be adequate to discriminate surface (partial) from all-cluster melting. In the present case, we assign Tcl melt,N to the temperature at which Lindemann’s index starts increasing. This allows us to obtain the onset of cluster melting and to derive the melting temperature corresponding to a cluster phase in which unmolten parts with a possible magnetic order are still present. Similarly, the determination of T cl C,N depends crucially on the accurate location of the maximum of the heat capacity variation with temperature. This was demonstrated in our previous report (Andriotis et al. 2007) when discussing the Tcl melt,N and T cl results of Ni and their deviation from the prediction of Eq. 2. Additionally, 43 C,N the surface energy contribution to the free energy of the cluster may be another reason that small cluster temperatures cannot be extrapolated to bulk phase values (Qi et al. 2001). Following these clarifications, we next discuss some obvious and hidden consequences of Eq. 2. One of the first conclusion that can be deduced from these results is that the Curie temperature, T cl C,N , of a magnetic cluster has to follow a sizecl dependence relationship analogous to that of Tcl melt,N . That is, if Tmelt,N is given by cl Eq. 1, then T C,N has to follow the following equation: cl TC;N D TCbulk  ıC N 1=3 ;

(28)

where •C is a constant that may have an N-dependence as •melt . In fact, assuming the validity of Eqs. 1 and 28 and taking the ratio between Eqs. 1 and 28 by parts, we obtain: cl TC;N  TCbulk cl Tmelt;N



bulk Tmelt

D

ıC D ı0 ; ımelt

(29)

where •0 is a constant which is expected to depend on N. It is apparent that Eq. 29 has exactly the form of our Eq. 2, suggesting that the N-dependence of the constant •0 is very weak. It is worth noting that the numerical justification of Eq. 29 by our results as expressed by Eq. 2 cannot ensure that the trends described by Eqs. 1 and 28 and which are possibly valid for large clusters can be extrapolated to small clusters as well. The conclusion that comes from Eq. 2 is that whatever the functional relationship between the melting temperature of a cluster and its size

14

A.N Andriotis et al.

(not necessarily limited to that of Eq. 1) is, the functional relationship followed by Tcl melt,N should dictate the relationship between the Curie temperature with its size as well. This hypothesis is supported by the results of Diep and collaborators (Diep et al. 1989) who found that the incorporation of the magnetic interactions leaves the cluster structure unchanged, thereby justifying our computational procedure. One may argue that the use of two noncomparable methods, i.e., that of a classical potential MD simulation for calculating the melting temperature and a quantum mechanical approximation for calculating the magnetic moments and the Curie temperature of a cluster, cannot lead to results that can be correlated. We addressed this issue by fitting the classical Sutton–Chen potential to the data for small clusters in such a way that resulted in TBMD and fitted Sutton–Chen potential simulations giving the same structural properties for small Ni clusters. Obtaining similar structural results by both methods appears to confirm the validity of the classical potential MD simulations for our present purpose. Furthermore, in order to resolve any reservations and ambiguities about the consistency of our conclusions derived from the use of mutually inconsistent methods in calculating the melting and Curie temperatures, it is demonstrated in the following that a calculation of the melting temperatures within our TBMD approximation is in excellent agreement with the results of the classical potential approximation used in the derivation of Eq. 2. Following exactly the same procedure as the one we used to calculate the average magnetic moment per cluster atom (and from this the Curie temperature) (Andriotis et al. 2007), we calculate the average total energy, , of each cluster at its thermodynamic equilibrium at a series of temperatures T. An average over Nran random spin configurations over the cluster atoms is taken at each kth time step for Ncl time steps (Nran is taken approximately between 100 and 200). Finally, these spin-averaged values are averaged over time. That is, PNran k .E k E0 /=kB T i 1 XNcl 1 iD1 Ei e < ET >D PNrun .E k E0 /=kB T : kD1 Ncl Nran e i

(30)

iD1

where E0 is a reference energy (Andriotis et al. 2007). For completeness, it is recalled that X i;k i;k Eik D "j C Erep (31) j

i,k where "i,k j denote the eigenvalues of the TB cluster Hamiltonian and Erep , the sum of the repulsive interactions (Andriotis and Menon 1998) of the cluster at the ith random spin configuration and the kth time step. As in the case of the calculation of the Curie temperatures, the averaging process over time in Eq. 30 is performed while reaching the thermodynamic equilibrium every 100 time-steps. In Figs. 2 and 3, we present our TB results for the variation with temperature of for the Ni147 and Ni201 clusters. For Ni147 , it is observed that the onset of a phase change appears at  450 ı K while the melting starts at 700 ı K. For

Variation of the Surface to Bulk Contribution to Cluster Properties

15

–420

Total Energy (eV)

Ni147: TB-Present work

–430

–440

–450 0

200

400 600 Temperature (°K)

800

Fig. 2 Numerical results obtained within the TBMD method for the variation with temperature of the average total energy of the Ni147 cluster. Straight lines are least square fits to portions of the numerical data –570 TB-Present work

Total Energy (eV)

–580

–590

–600

–610

0

200

400

600

800

1000

1200

Temperature (°K)

Fig. 3 The same as in Fig. 2 but for the Ni201 cluster

Ni201 , the melting takes place at 800 ı K (taken to be the midpoint of the “parallel” shift of the two linear parts of the thermodynamic curve. The so obtained melting temperatures appear to be in excellent agreement with the results found using the classical potential method and the Lindemann criterion according to which ı ı cl cl Tmelt;N D147 D 700 K and Tmelt;N D201 D 780 K.

16

A.N Andriotis et al.

Further headway can be made if Eq. 2 is taken to be the zeroth-order approximation of a piece-wise function of N in analogy to similar findings (Gunes and Erkoc 2000; Qi et al. 2001) for the expression for T cl melt,N given by Eq. 1. In this view, the results of Diep et al. (1989) (see Fig. 12 of their work) in the extreme case of very small clusters (N 2 [7,17]) can lend support to our results and conclusions. The appearance of the nonzero constant term at the right-hand side of Eq. 2 cl indicates that there is a possibility for T cl C,N to be greater than T melt,N . In particular, cl cl Eq. 2 predicts that T C,N could be greater than T melt,N if cl TCbulk > TC;N >

ˇ : 1˛

(32)

However, according to our results, the above inequality does not hold for the Ni ı ı clusters since ˇ= .1  ˛/  968 K, a value much greater than TCbulk D 631 K. This is demonstrated in Fig. 1 with the indicated crossing of the dotted and dashed black lines with the former describing Eq. 2 and the latter describing the same equation cl cl taking ˛ D 1 and ˇ D 0 (i.e., corresponding to the TC;N D Tmelt;N case). This incompatibility can be taken as an indication that the functional forms dictated by Eqs. 1 and 28 are not valid for the entire range of the cluster sizes. The nonzero value of the constant “ has another consequence; it predicts that the ratio

cl Tmelt;N cl TC;N

depends on the cluster size and, in fact, increases as the cluster size

decreases. If the variation of T cl melt,N is assumed as given by Eq. 1, the predictions of cl Eq. 2 lead to the conclusion that T cl C,N decreases at a slower rate than T melt,N as the cluster size decreases and, therefore, the “melting temperatures” of partially molten clusters can be found to be lower than the Curie temperatures as Eq. 2 implies. However, such a conclusion has to be taken with care as the validity of Eq. 1 over the entire range of cluster sizes is not valid.

Conclusion We have presented results for the variation with the cluster size of the melting and Curie temperatures of Nin , n  201, clusters. Two complimentary methods were used, i.e., the classical MD employing the Sutton–Chen potential and the TBMD for obtaining the melting and the Curie temperatures of the clusters. We have demonstrated that by fitting the classical potential to the results of the TB description in the case of small clusters, we can achieve excellent agreement between the results of the two methods referring to the structural properties and the estimation of the melting temperatures of the clusters. Our results demonstrate without any ambiguity that the variation of the cluster properties with the cluster size exhibits strong dependence on the ratio of the surface to bulk (core) contributions, the latter appearing to have the same functional cl dependence on the cluster size for both T cl C,N and T melt,N .

Variation of the Surface to Bulk Contribution to Cluster Properties

17

In view of the established dependence of the melting temperature of a cluster on its surface to volume contribution (as, for example, Eq. 1), our conclusion can be interpreted as an indication of a universal aspect of the surface to volume contribution to the cluster properties. This justifies previous findings based on approximate and semiempirical approximations. Acknowledgments The present work is supported by grants from US-DOE (DE-FG0200ER45817 and DE-FG02-07ER46375).

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Fthenakis, Z., Andriotis, A. N., & Menon, M. (2003a). Temperature evolution of structural and magnetic properties of transition metal clusters. Journal of Chemical Physics, 119, 10911. Fthenakis, Z., Andriotis, A. N., & Menon, M. (2003b). Understanding the structure of metal encapsulated Si cages and nanotubes. Journal of Chemical Physics, 119, 10911. Garcia-Rodeja, J., Rey, C., Gallego, L. J., & Alonso, J. A. (1994). Molecular-dynamics study of the structures, binding energies, and melting of clusters of fcc transition and noble metals using the voter and chen version of the embedded-atom model. Physical Review B, 49, 8495. Garrigos, R., Cheyssac, P., & Kofman, R. (1989). Melting for lead particles of very small sizes: Influence of surface phenomena. Zeitschrift für Physik D, 12, 497. Gerion, D., Hirt, A., Billas, I. M. L., Chatelain, A., & de Heer, W. A. (2000). Experimental specific heat of iron, cobalt, and nickel clusters studied in a molecular beam. Physical Review B, 62, 7491. Gunes, B., & Erkoc, S. (2000). Melting and fragmentation of nickel nanparticles: Moleculardynamics simulations. International Journal of Modern Physics, 11, 1567. Harrison, W. (1980). Electronic structure and properties of solids. San Francisco: W. H. Freeman. Huang, H., Sun, C. Q., & Hing, P. (2000). Surface bond contraction and its effect on the nanometric sized lead zirconate titanate. Journal of Physics: Condensed Matter, 12, L127. Kato, M., & Kokubo, F. (1994). Partially antiferromagnetic state in the triangular hubbard model. Physical Review B, 49, 8864. Lai, S. L., Guo, J. Y., Petrova, V., Ramanath, G., & Allen, L. H. (1996). Size-dependent melting properties of small tin particles: Nanocalorimetric measurements. Physical Review Letters, 77, 99. Lathiotakis, N. N., Andriotis, A. N., Menon, M., & Connolly, J. (1996). Tight binding molecular dynamics study of Ni clusters. Journal of Chemical Physics, 104, 992. Lee, Y. J., Lee, E. K., Kim, S., & Nieminen, R. M. (2001). Effect of potential energy distribution on the melting of clusters. Physical Review Letters, 86, 999. Menon, M., & Subbaswamy, K. R. (1997). Nonorthogonal tight-binding molecular-dynamics scheme for silicon with improved transferability. Physical Review B, 55, 9231. Nayak, S. K., Khanna, S. N., Rao, B. K., & Jena, P. (1998). Thermodynamics of small nickel clusters. Journal of Physics: Condensed Matter, 10, 10853. Nose, S. (1984). A unified formulation of the constant temperature molecular dynamics methods. Journal of Chemical Physics, 81, 511. Ojeda, M. A., Dorantes-Davila, J., & Pastor, G. (1999). Noncollinear cluster magnetism in the framework of the hubbard model. Physical Review B, 60, 6121. Qi, Y., Cagin, T., Johnson, W. L., & Goddard, W. A. (2001). Melting and crystallization in Ni nanoclusters: The mesoscale regime. Journal of Chemical Physics, 115, 385. Rey, C., Gallego, L. J., Garcia-Rogeja, J., Alonso, J. A., & Iniguez, M. P. (1993). Moleculardynamics study of the binding energy and melting of transition-metal clusters. Physical Review B, 48, 8253. Schmidt, M., Kusche, R., Kronmuller, W., von Issendorff, B., & Haberland, H. (1997). Experimental determination of the melting point and heat capacity for a free cluster of 139 sodium atoms. Physical Review Letters, 79, 99. Sun, D. Y., & Gong, X. G. (1998). Structural properties and glass transition in Aln clusters. Physical Review B, 57, 4730. Sutton, A. P., & Chen, J. (1990). Long-range Finnis-Sinclair potentials. Philosophical Magazine Letters, 61, 139. Uhl, M., Sanrdatskii, L. M., & Kubler, J. (1994). Spin fluctuations in ”-fe and in fe3 pt invar from local-density-functional calculations. Physical Review B, 50, 291. Weerasinghe, S., & Amar, F. G. (1993). Absolute classical densities of states for very anharmonic systems and applications to the evaporation of rare gas clusters. Journal of Chemical Physics, 98, 4967. Yang, C. C., & Jiang, Q. (2005). Size and interface effects on critical temperatures of ferromagnetic, ferroelectric and superconductive nanocrystals. Acta Materialia, 53, 3305.

Theoretical Studies of Structural and Electronic Properties of Clusters Michael Springborg

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Only Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ni Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bimetallic Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clusters on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Only Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Both Nuclei and Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Na and Au Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HAlO Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AB Semiconductor Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metcars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 6 7 7 11 17 20 25 27 28 28 29 35 40 49 51 54

Abstract

Clusters contain more than just some few atoms but not so many that they can be considered as being infinite. By varying their size, their properties can often be varied in a more or less controllable way. Often, however, the precise relation between size and property is largely unknown: the sizes of the systems are below the thermodynamic limit so that simple scaling laws do not apply. Theoretical

M. Springborg () Physical and Theoretical Chemistry, University of Saarland, Saarbrücken, Germany School of Materials Science and Engineering, Tianjin University, Tianjin, China e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_26-2

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M. Springborg

studies of such systems can provide relevant information, although in many cases idealized systems have to be treated. The challenge of such calculations is the combination of the relatively large size of the systems together with an often unknown structure. In this presentation, different theoretical methods for circumventing these problems shall be discussed. They shall be illustrated through applications on various types of clusters. These include isolated metal clusters with one or two types of atoms, metal clusters deposited on a surface, nanostructured HAlO, semiconductor nanoparticles, and metallocarbohedrenes. Special emphasis is put on the construction of descriptors that can be used in identifying general trends.

Introduction Nanostructures are materials whose spatial extension in 1, 2, or 3 dimensions is roughly of the order of at most some 100 nm. Thereby material properties different from those of the macroscopic materials we know from our daily life emerge. According to the simplest description of the properties of some given material, we may apply scaling laws so that, e.g., the electrical resistance scales with the inverse cross section and linearly with the length of the system. With such scaling laws, it is indirectly assumed that the materials form some homogeneous continua and, accordingly, that the fact can be ignored that on the atomic length scale the materials are not at all homogeneous. However, since typical interatomic bond lengths are of the order of some tenths of a nm, a proper description of the material properties of nanostructures often cannot be based on the above assumption that the materials are homogeneous. Thus, for nanomaterials, one can no longer apply the abovementioned scaling laws, i.e., the systems of interest have sizes far from the thermodynamic limit and one has left the scaling regime so that “every single atom counts.” Indeed, the fact that the material properties depend, in some cases even strongly, on the size of the system for nanostructures is the reason for the large interest in such systems. Here, both the prospects related to possible practical applications of these materials as well as the wish to understand in detail the size– property relations are motivations behind this interest. The present chapter is devoted to the results of our theoretical studies of clusters, i.e., nanostructures that are finite in all three spatial dimensions. Therefore, their number of surface atoms relative to their total number of atoms is large, and of the same reason, finite-size effects show up. In addition, they most often contain just one or, at most, some few types of atoms, with the possible existence of surfactants as an exception. The surfactants are radicals that are used in saturating dangling bonds at the surfaces and, thus, in stabilizing the materials. For theoretical studies, it is important to notice that these systems are fairly large, but not so large that they can be considered as being infinite and periodic and that, consequently, they may have a quite low symmetry so that in a theoretical calculation chemically identical atoms may have to be treated as being different since they have different surroundings.

Theoretical Studies of Structural and Electronic Properties of Clusters

3

In experimental studies, such clusters/colloids are often produced and studied in a solution or on the surface of some substrate. Moreover, as mentioned above, they are often stabilized through ligands. In total, this means that the systems often are interacting with other systems, this being the solvent, the substrate, or the ligands. This is contrasted by theoretical studies that often consider isolated nanoparticles in gas phase that, in addition, often are naked (i.e., without ligands). Another difference between experiment and theory is that the precise size of the experimentally studied systems often is only approximately known, whereas, per construction, theoretical studies consider well-defined systems, although for those one often assumes a certain structure that may or may not be realistic. Thus, experimental and theoretical studies are rarely competing but instead complementing each other. When attempting to use electronic-structure methods in studying the properties of clusters, in particular the size dependence of those where one, accordingly, will consider a larger range of sizes, one very fast encounters some fundamental properties related to theoretical calculations. We shall here briefly outline those. In a typical electronic-structure calculation, the number and types of the atoms for the system that shall be studied are known. Then (cf. Fig. 1), some initial structure is chosen and various properties for this structure are calculated. These properties are first of all the total energy but may also be others like the forces acting on the nuclei (i.e., the derivatives of the total energy with respect to the nuclear Fig. 1 A flowchart for a typical electronic-structure calculation

start

choose structure

calculate properties

new structure ? no stop

yes

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M. Springborg

coordinates). Also available experimental information may be sought calculated. Subsequently, one may ask whether a new structure shall be studied. The reasons for doing so could be that the calculated forces are not approximately vanishing, that the calculated properties differ significantly from those obtained in experiment, or that one simply wants to explore further parts of the structure space. For larger systems, this approach is related with two serious problems. First, if the system of interest contains N nuclei and M electrons, the computational costs for calculating the properties for one single structure scale as the size of the system to some power, i.e., like M k or N k , where k is some power from 2 and upward. For some of the most popular approaches, like density-functional and Hartree– Fock methods (see, e.g., Springborg 2000), k is typically 3, but for more advanced methods, k may be larger than 7. Thus, in this case, it should be obvious that for just intermediately large systems, the calculation of the properties for just one single geometry may become computationally very costly. Independently of the scaling of the computational needs as a function of the size of the system, another complementary problem causes additional complications. This problem is related to the fact that the number of nonequivalent minima on the total-energy surface as a function of structure grows very fast with the size of the system. In fact, it has been shown (Wille and Vennik 1985) that the number of local total-energy minima grows faster than any polynomial in N or, alternatively expressed, that the determination of the global total-energy minimum is a so-called NP-hard problem. A model system that is so simple that detailed studies can be performed is that of a cluster of identical atoms for which it is assumed that the total energy can be written as a sum of pair potentials, each one being a simple Lennard–Jones potential. This system was studied by Tsai and Jordan (1993), who found a rapidly increasing number of nonequivalent metastable structures as a function of size. In Fig. 2 their results have been fitted with an exponential, a  exp.bN /, and it is seen that the fit follows the calculated results fairly close. The fit gave a D 0:00341 and b D 0:983,

2000

Number

Fig. 2 A schematic representation of the number of nonequivalent local total-energy minima for Lennard–Jones clusters (the black circles) together with a fit with an exponential (the full curve)

1000

0 10

15 N

Theoretical Studies of Structural and Electronic Properties of Clusters

5

which in turn means that for N D 55 the fit predicts that of the order of 1021 nonequivalent minima exist. This result points directly to a central issue of this presentation, i.e., how can we determine the structure for a system with at least some 10s of atoms and for which no, or only very limited, information on the structure can be used. The ground-state structure is obtained by minimizing the total energy. Employing the Born–Oppenheimer approximation, the nuclei are treated as classical and not quantum particles, and there is no variational principle that can be used in systematically getting arbitrarily close to the true ground-state structure, even when assuming that the total energy can be calculated accurately for any structure. Instead, most often one has to change the structure “by hand” until one feels confident that the structure of the lowest total energy has been identified. But it shall be stressed that there is absolutely no approach that with absolute certainty can guarantee that precisely that structure has been found. Trying to prove so would require complete knowledge about the total-energy surface as a function of all internal degrees of freedom. This information is beyond whatever will be available, even if the computers keep on becoming more and more powerful and the programs more and more efficient. That this information will never become available can be seen by, e.g., considering the largest possible computer, i.e., the complete universe. Lloyd (2002) estimated that the complete universe, viewed as an enormous quantum computer that has been operating since the Big Bang, could have performed of the order of 10120 binary operations. Although this number is very large, it is also finite. Therefore, it means that considering, e.g., a system of 42 atoms, assuming that a total-energy calculation for this system and a given structure is just one single binary operation, and considering just 10 different values for each internal structural degree of freedom, exactly 10120 calculations are needed for obtaining this, relatively crude, total-energy hypersurface. Thus, the calculations would have required the use of the complete universe since the Big Bang! In total, it is obvious that for any but the absolutely smallest systems it is not possible to explore anything but very limited parts of the total-energy hypersurface and, moreover, that it is never possible to be absolutely sure that the true global totalenergy minimum has been found. Any total-energy minimum may provide a total energy that is close to that of the global minimum, although the structures of the two may be markedly different, simply due to the very large number of local totalenergy minima. It is also obvious that any attempt to identify the global total-energy minimum has to be based on some kind of qualified search in the multidimensional structure space. It is one of the purposes of this presentation to discuss some of the approaches that we are using in searching for the global total-energy minimum in the structure space. Another purpose is to present various descriptors/tools that we have derived in order to extract chemical or physical information from the calculated properties.

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M. Springborg

We emphasize that we shall concentrate on the results of our own studies. For more general reviews the reader is referred to, e.g., Baletto and Ferrando (2005), Ferrando et al. (2008), and Springborg (2009).

Basics In the most general case, our goal is to calculate the properties of a system containing N nuclei and M electrons. We shall denote the nuclear positions R1 ; R2 ;    ; RN and those of the electrons r1 ; r2 ;    ; rM . Moreover, we use Hartree atomic units and set accordingly me D jej D 40 D „  1. The mass and charge of the kth nucleus is then Mk and Zk , respectively. The combined coordinate xi denotes the position and spin coordinate of the i th electron. In the absence of external interactions and relativistic effects, the Hamilton operator for this system can then be written as a sum of five terms, HO D HO k n C HO ke C HO nn C HO ee C HO en ;

(1)

with HO k n D 

N X 1 r2 2Mk Rk kD1

HO ke D 

M X 1 i D1

2

rr2i

N Zk Zl 1 X HO nn D 2 jRk  Rl j k¤lD1

M 1 1 X HO ee D 2 jri  rj j i ¤j D1

HO en D 

N X M X kD1 i D1

Zk ; jRk  ri j

(2)

i.e., the kinetic energy operator for the nuclei, that for the electrons, and the three potential energy operators for the nucleus–nucleus, the electron–electron, and the electron–nucleus interactions, respectively. The time-independent Schrödinger equation becomes then HO  .x1 ; x2 ;    ; xM ; R1 ; R2 ;    ; RN / D E   .x1 ; x2 ;    ; xM ; R1 ; R2 ;    ; RN /: (3) As indicated above, we shall assume that the Born–Oppenheimer approximation is accurate. Then, HO k n is ignored, and furthermore, the wave function  that depends

Theoretical Studies of Structural and Electronic Properties of Clusters

7

functionally on both electronic and nuclear coordinates is written as a product of two functions,  .x1 ; x2 ;    ; xM ; R1 ; R2 ;    ; RN / D n .R1 ; R2 ;    ; RN /  e .R1 ; R2 ;    ; RN I x1 ; x2 ;    ; xM /: (4) Here, n is the wave function for the nuclei (which for the present study is irrelevant) and e is the electronic wave function. e depends functionally on the electronic coordinates but also parametrically on the nuclear coordinates (implying that different structures have different electronic wave functions). e is calculated from the electronic Schrödinger equation HO e e D Ee e

(5)

HO e D HO ke C HO ee C HO en :

(6)

with

Moreover, the total energy is E D Ee C HO nn D E.R1 ; R2 ;    ; RN /;

(7)

i.e., a function of the structure of the system. Although the Born–Oppenheimer approximation does lead to some computational simplifications, solving the resulting equations is still computational demanding, so that when aiming at studying many structures for larger systems (as is the goal of the present work), it can easily become necessary to invoke additional approximations. Then, the type of approximation will depend on the system and scientific issue at hand, as we shall illustrate below.

Only Nuclei Methods Ultimately, the goal is to study the total energy, E, as a function of structure, i.e., E.R1 ; R2 ;    ; RN /. The determination of the electronic orbitals, their energies, and the total electronic energy, Ee , provides thereby additional information that may be highly relevant for other purposes, but may also be considered as a complication that might be circumvented. The use of approximate expressions (i.e., force fields) of the form X .1/ X .2/ E.R1 ; R2 ;    ; RN / ' Et .i / C Et .i /;t .j /.Ri ; Rj / i

C

X ij k

ij .3/

Et .i /;t .j /;t .k/.Ri ; Rj ; Rk / C   

(8)

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M. Springborg

represents one possibility. Here, t.i / is the atom type of the i th atom, and the total energy has been split into atomic parts as well as 2-, 3-,    body interactions. The various functions E .i / are then represented in some approximate, analytical, or numerical form. Such approximations are very useful for the study of the structural properties of a larger number of larger systems, but may, of course, suffer from inaccuracies due to the approximate nature of the total-energy expression. The Lennard–Jones clusters discussed above represents one such expression, where only 1- and 2-body terms are included. Some further examples are discussed in Springborg (2006). When applying such approaches to clusters, it is important to choose the method according to the chemical bonding of the system of interest. In metallic systems, the electrons are considered as being delocalized and packing effects are often the main driving force for the structure. On the other hand, for many semiconductors and insulators, the bonds between the atoms are partly covalent and partly ionic, so that directional bonding and electron transfers are important, i.e., a precise description of the electronic degrees of freedom is important. Yet other types of systems include, e.g., molecular crystals with weak interactions between the molecules, but here we are concerned with the first two types of systems. One approach that we have been using for clusters of metal atoms is based on the embedded-atom method (EAM) of Daw, Baskes, and Foiles (Daw and Baskes 1983, 1984; Daw et al. 1993; Foiles et al. 1986). According to this method, the total energy of the system is written as a sum over atomic energies, Etot D

N X

Ei

(9)

i D1

with Ei D Fi .ih / C

N 1 X ij .jRi  Rj j/ 2

(10)

i ¤j D1

being the energy of the i th atom. In Eq. (10) the first term is the so-called embedding energy, which is obtained by considering each atom as an impurity embedded into a host provided by the rest of the atoms. The second term describes electron–electron interactions and is represented in terms of short-range pair potentials. The local density at site i is assumed being a superposition of atomic electron densities, ih D

X

ja .jRi  Rj j/

(11)

j .¤i /

where ja .r/ is the spherically averaged atomic electron density provided by atom j at the distance r.

Theoretical Studies of Structural and Electronic Properties of Clusters

9

With the EAM, the total-energy expression of Eq. (8) contains also contributions from P -body interactions with P being the number of nearest neighbors of the atoms of the system of interest. Other potentials that also include many-body interactions are the Gupta, Murrell–Mottram, Sutton–Chen, and Cleri–Rosato potentials among others. Since we have used the Gupta, Sutton– Chen, and Cleri–Rosato potentials in some studies, we shall here discuss these briefly. With the Gupta potential (Gupta 1981), the total energy is written in terms of repulsive and attractive many-body terms, Etot D

N X ŒV r .i /  V m .i /

(12)

i D1

where V r .i / D

N X j D1 .¤i /

   jRi  Rj j 1 A.a; b/ exp  p.a; b/ r0 .a; b/

(13)

and  V .i / D m

N X j D1 .¤i /





jRi  Rj j 1  .a; b/ exp  2q.a; b/ r0 .a; b/ 2

12 :

(14)

The summations run over all (N ) atoms of the system of interest. Furthermore, A, r0 , , p, and q are fitted to experimental values of the cohesive energy, lattice parameters, and elastic constants for the crystal at 0 K. The Sutton–Chen potentials are based on the empirical N -body potentials that have been developed by Finnis and Sinclair (1984) for the description of cohesion in metals. With these potentials, the total internal energy is represented by a cohesive functional of pair interactions and a predominantly repulsive pair potential. The main difference of such a potential to a pair potential is that using only pair potentials, the calculated force exerted by one atom on another depends on the interatomic distance only, whereas in the present case it depends on all neighbors of both atoms. Finnis–Sinclair potentials are of relatively short range and extend only to third neighbors in fcc crystals. The long-range modification constitutes the Sutton–Chen potential. This potential has an extra r16 van der Waals tail and, accordingly, the following form 2 E D 4

1 XX 2

i

j ¤i

3 Xp V .jRi  Rj j/  c i 5 i

(15)

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M. Springborg

with n a V .jRi  Rj j/ D jRi  Rj j m X a i D : jRi  Rj j 

(16)

j ¤i

With the empirical many-body potential given by Cleri and Rosato (1993), the cohesive energy of the system is written as Ec D

X

.ERi C EBi /;

(17)

i

where ERi D

X



Ae

p



jRi Rj j 1 r0

j

EBi D 

8 0; and for a spherical structure, p D 0. Sieck and coworkers (2003) studied various different isomers corresponding to three fixed sizes, namely, N D 25, 29, and 35. In Fig. 3, the cohesive energy versus the prolateness parameter for the low-lying isomers is displayed. An investigation into the most stable structures clearly indicates the tendency to go from elongated to spherical structures as the clusters grow in size.

Cohesive energy (eV/atom)

−3.96 −3.97 −3.98 −3.99 −4 −4.01 −4.02

Si25

−4.03

Si39

Si29

−4.04 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Prolateness

Fig. 3 Cohesive energy versus prolateness parameter defined in Eq. 3 for 30 lowest lying isomers of Si25 , Si29 , and Si35a (Reprinted with permission from Sieck et al. (2003). © (2003) by WileyVCH)

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An important point is noted by Sieck and coworkers (2003), that different methods, even different flavors of the same method (LDA/GGA in DFT), may disagree about the exact energetic ordering of clusters; however, usually the lowest energy structures are local minima in all of them. As an example to the multiscale methods mentioned above, Bulusu and coworkers (2005) used a basin-hopping algorithm combined with DFT to explore low-lying minima for Ge12 –Ge20 . Global minima for the Ge clusters in this size range are compared to minima of Si clusters. Various checks were performed to make sure different starting points yield the same global minima, such as different seeding patterns. Unlike most studies in the literature, zero-point motion is taken into account while calculating the binding energies. In the same work (Bulusu et al. 2005) of Si and Ge, clusters in the size range 12–20 were compared and clusters of both Si and Ge were shown to have prolate geometries. However, their growth patterns were found to diverge at N D 13. The global minima for Ge clusters of size 12–16 are obtained by adding atoms to the Ge9 tetracapped trigonal prism (TTP) structure familiar from Si clusters (Bulusu et al. 2005). For low energy-Si clusters, TTP-to-sixfold puckered ring (six/six) transition occurs at N D 16 and clusters with higher number of atoms all contain the six/six pattern. In contrast, TTP-to-six/six pattern may occur at N D 19 for Ge clusters, and at N D 20, the magic number cluster Ge10 appears to be the preferred structural motif. According to Shvartsburg and Jarrold (1999), clusters of Sn up to n  35 follow the trend of germanium and prolate silicon clusters (Shvartsburg and Jarrold 1999). For N > 20, the mobility of Sn clusters shows larger fluctuations than either Si or Ge. This might indicate the presence of multiple isomers of Sn clusters in this size range (Shvartsburg and Jarrold 1999). Even though the ˛ ) ˇ transition occurs between 286 and 310 K in the bulk, Sn clusters do not undergo such a transition even at higher temperatures. In fact, mobility measurements show that Sn clusters do not show significant changes in structure for a very broad temperature range. Transition from prolate to spherical growth in Sn clusters is not abrupt like the transition in Si and Ge clusters but occurs in steps. Clusters of N  35 adopt a stacked prolate morphology much like Si and Ge clusters. This is unexpected because these highly noncompact structures are suitable for covalent materials, whereas bulk Sn under ambient conditions is a metal. In a sense, the covalent-tometal transition that occurs between the fourth and fifth row of the periodic table for the carbon series in the bulk fails to occur in their clusters. Shvartsburg and Jarrold C (2000) also find that Pb n and Pbn clusters display different magic numbers. Going down the group IV in the periodic table, a comprehensive set of data is given by Shvartsburg and Jarrold (2000). Their findings reveal that Pb clusters are structurally different from Si, Ge, and Sn cluster for N < 25. While the latter clusters have low mobilities in this size range indicating a prolate structure, Pb clusters exhibit much higher mobilities meaning they are quasispherical in shape. For N > 25, Si clusters also display mobilities in accord with a spherical shape. However, their mobilities are considerably smaller than the Pb clusters in the same

Modeling of Nanostructures

11

size range. This is attributed to the densely packed nature of the Pb clusters in contrast with the open, cage-like arrangement of the Si clusters. Transition metals form the largest group of elements considered in this section. Even as bulk materials, they exhibit very interesting and diverse properties, in particular magnetism. Interesting questions therefore arise concerning whether such properties as magnetism are maintained or altered in clusters (Briere et al. 2002; Kabir et al. 2006). Appearance of magnetism in the clusters of 4d elements (such as Ru, Rh, and Pd) is a very interesting phenomenon as these elements are nonmagnetic in the bulk. Clusters of 3d elements (such as Fe, Co, and Ni), which are already magnetic in the bulk, exhibit enhanced magnetic moments in the cluster form due to narrower band widths and the increased localization of the electrons (Kumar and Kawazoe 2002; Pawluk et al. 2005). Indeed, a recent Stern–Gerlach experiment revealed that Mn clusters in the range N D 11–99 display ferromagnetic ordering even though no such ordering is observed in the bulk phase (Knickelbein 2001). These results reveal PES minima at N D 13 and N D 19 and PES maxima at N D 15 and N D 23–25. We should mention here that most of the magnetism studies mentioned in this section take into account only the electronic spin contribution to the magnetic moment. This can be done only in the cases where the spin-orbit coupling can be neglected (Bansmanna et al. 2005). In the work by Rodríguez-López and coworkers (2003), Co clusters in the range N D 4–60 were studied by means of an evolutive algorithm based on the Gupta potential and tight binding. Experiments reveal that much like the Cr clusters (Payne et al. 2006), different isomers of Co clusters coexist with distinct magnetic moments. In this work two sets of isomers are identified for each size – the lowest and the second-lowest lying. For the lowest lying isomers, an icosahedral growth is observed with structures derived by adding atoms to the main icosahedral sizes at N D 7, 13, 19, 23, 26, 34, 43, and 55. For the second isomers, no particular growth pattern was identified. The stability of these sizes was also confirmed by the second energy difference E2 in addition to other intermediate icosahedral structures. Relative thermodynamic populations of the lowest energy isomers were used to simulate possible experimental conditions. Isomers coexist particularly evenly between sizes of enhanced stability. This is due to the influence of the entropic contribution of the low-frequency normal modes of the isomers to the free energy. For both the global minimum and the second isomer, a nonmonotonic decrease is observed with increasing size. The greatest difference between the two sets of clusters arises in the range N D 20–40, which corresponds to the range where the average interatomic distance and average coordination of the two sets show significant difference. In this size range, two effects seem to compete for determining the magnetization of the two sets of configurations. On the one hand, the average coordination is higher for the global minima, which should result in lower magnetic moments for the global minima. On the other hand, the average nearest-neighbor distance is higher for the global minima, which should yield higher magnetic moments. The results indicate that the average coordination number effect dominates.

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For noble and transition metals, the interactions between atoms are not pairwise and simple empirical potentials are inappropriate (Barreteau et al. 2000). Therefore incorporating many body effects into the potentials is essential. Moreover, for magnetism studies, ab initio methods need to be employed, which render global optimization efforts extremely computation intensive. Therefore, most results we shall quote here will be based on restricted searches of the potential energy surface. In the work by Barreteau and coworkers (2000), for instance, the relative stability of cuboctahedra and Mackay icosahedra is determined for Rh and Pd clusters for N D 13, 55, 147, 309, and 561 using a tight-binding method. Since both structures have an identical sequence of magic numbers, it is interesting to determine the transition size. A continuous transition is possible from the Mackay icosahedron to the cuboctahedron, and for the Rh and Pd clusters, this pathway is explored. An analysis of the Mackay transition from the cuboctahedron to the icosahedron reveals that for N D 13, the cuboctahedron is unstable for both Rh and Pd, becoming metastable for larger sizes with an increasing activation barrier with size. The magnetic moment of rhodium was found to disappear for sizes more than 100 atoms, and palladium clusters were found to be hardly magnetic. In Mn clusters, on the other hand, magnetism plays an important role in determining the ground state structures. As mentioned in the work by Briere and coworkers (2002), many spin isomers can lie close in energy. In this work, a few local geometric configurations of N D 13, 15, 19, and 23 were studied using spinpolarized calculations. At all sizes except N D 15, the structure with the lowest energy was found to be icosahedral. For N D 15, a bcc configuration was found to be favorable. In terms of spin, all the structures were found to be ferrimagnetic with alternating domains of different spin configurations (see Fig. 4). Except for N D 15, the mean value of the integrated spin density was found to decrease with increasing size. In the DFT work by Kabir et al. (2006), the ground state structure for Mn13 was found to be the icosahedron with the two pentagonal rings that are coupled antiferromagnetically. Therefore, the resulting magnetization is small, namely, 0.23B /atom. This magnetization is considerably smaller than the neighboring sizes 12 and 14. The N D 14 structure differs from the N D 13 structure by a single capping atom. However, the presence of this atom changes the magnetization arrangement considerably. In this case the pentagonal rings are ferromagnetically coupled and the magnetization is 1.29B /atom. The case of Mn15 is worth mentioning because of the discrepancy between two DFT studies by Briere and coworkers (2002) and Kabir et al. (2006). In the latter, the ground state structure was found to be icosahedral with a magnetic moment of 0.87B /atom, whereas Briere et al. (2002) found a bcc structure with 020B /atom. For N D 19, a double icosahedron was observed, which again has a smaller magnetic moment, 1.10 B /atom, than its neighboring clusters. The central pentagonal ring is AFM coupled to the neighboring pentagonal rings. This behavior is persistent in the N D 20 cluster, which has a magnetic moment of 1.50B /atom. In the range N D 11  20, spin segregation is observed, where like spins tend to cluster. The binding energy is observed to increase monotonically with

Modeling of Nanostructures

Mn13

Mn19

13 Mn15 bcc

Mn15 icosahedral

Mn23

Fig. 4 Lowest energy structures of some Mn clusters. Relative spin alignments are marked with dark for spin up and light for spin down (Reprinted with permission from Briere et al. (2002). © (2002) by the American Physical Society)

increasing size. This is due to the increased sp. bonding. However, when compared with other transition metals, Mn clusters remain weakly bound. An interesting property of Ta clusters was demonstrated in a recent study (Fa et al. 2006), where ferroelectricity and ferromagnetism were proven to coexist. Initial structures were obtained by simulated annealing using an empirical potential. These structures were later reoptimized with DFT calculations. The magic numbers for Ta clusters were found to be 4, 7, 10, 15, and 22. It was therefore deduced that Ta clusters do not prefer icosahedral growth. For N D 13, for instance, the lowest energy structure among those studied was found to be a distorted five-capped hexagonal bipyramid. For N D 19, the most stable structure is decahedral in contrast with the double icosahedron, which was found to be stable for many other clusters. No perfectly symmetric structures were found indicating that Jahn–Teller distortions play an important role in determining the ground state structures of Ta clusters. In the size range studied in this work (Fa et al. 2006), the atomic packing shows differences such that each size behaves like an individual system rather than steps of a continuous growth sequence. In addition, electronic dipole moment and magnetic moment were also calculated. The electronic dipole moment was found to have the same trend as the inverse coordination number, which is a parameter that reflects the asymmetry of the cluster. This agreement is attributed to the strong correlation between the structure and the electronic dipole moment of the clusters. Odd-N Ta clusters also display a magnetic moment of about 1B , which suggests the possibility of the coexistence of ferroelectricity and ferromagnetism. The growth pattern of Ta was found to be very similar to that of Nb. However, when compared with vanadium clusters, this similarity is absent (Fa et al. 2006).

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In a similar work by Fa and coworkers (2005), Nb clusters were also found to display ferroelectricity supported by a recent experimental study. This is an important discovery because ferroelectricity was never observed in single-element bulk materials. For N  38, the electric dipole moment exhibits even-odd oscillations. This suggests that there is a strong correlation between the structure and the ferroelectricity. In the work by Pawluk and coworkers (2005), the structure and stability of several Ir clusters were studied using DFT. Rather than using a global optimization algorithm, possible configurations both truncated from bulk and built independently were relaxed locally. The results indicate that Ir clusters mostly prefer cube-like structures up to N D 13, except for N D 11, which assumes an elongated structure. At N D 13, the lowest structure among the ones studied is the icosahedron. This is in contrast with Ru and Pt, which prefer simple cubic structures. When compared to clusters cut from the fcc bulk, simple cubic structures turn out to be more stable up to a size of N D 48. This transition occurs near N D 40 in Ru and N D 13 for Pt. Another interesting property studied in this work is the fluidity of clusters. The results indicate that while the Pt clusters exhibit a more fluidlike character and will thus easily coalesce with other clusters, Ir clusters are more rigid and have less tendency toward coalescence. An interesting experimental result concerning Pt clusters was reported by Liu and coworkers (2006), who found that Pt13 clusters exhibit substantial magnetism (about 0.65B /atom) even though bulk Pt is not magnetic. Although for such clusters as Ni and Au, the icosahedral structure is calculated to be metastable with stability decreasing with size, experiment reveals that icosahedral structures are found in clusters containing several thousands of atoms. This means that kinetic effects are also very important in determining the structure of a cluster (Gafner et al. 2004). In order to simulate these kinetic effects, a 555 Ni cluster was studied using tight binding by Gafner and coworkers in (2004). The cluster was heated to 1,800 K (Tmelt D 1; 145 K) and subsequently cooled to 300 K. The melting and crystallization curves are determined from a sudden change in the potential energy as a function of temperature. They found that slow cooling results in an fcc structure, whereas fast cooling results in the formation of a metastable icosahedral structure. In the study by Köhler and coworkers (2006), the potential surface of a few sizes (N D 13; 53–57) of Fe clusters was mapped out with respect to magnetization and volume change using a DFT-based tight-binding scheme. Icosahedra were found to be the most stable structures for the magic numbers N D 13 and N D 55. Two local minima were observed for the N D 13 icosahedron, one ferromagnetic and one antiferromagnetic. The PESs for clusters with N D 53–57 were mapped out using a genetic algorithm-based procedure. Derivatives of the N D 55 structure were considered for N D 53, 54, 56, and 57. Generally icosahedra and icosahedronderived structures have relatively small magnetic moments. The structures without apparent symmetry show higher magnetic moment than icosahedra. No ferromagnetic ordering was found for the Fe55 cluster.

Modeling of Nanostructures

15

Tiago and coworkers provide a very comprehensive explanation of the origin of magnetism in small Fe clusters in their article (Tiago et al. 2006). In the Fe atom, the magnetic moment is a result of exchange splitting. The 3d" states are occupied by five electrons, while the 3d# states are occupied by a single electron, which results in a rather high magnetic moment of 4B . When the atoms come together to form a crystal, hybridization of the large 4 s bands and the 3d bands reduces the magnetism down to 2.2B . In clusters, hybridization is not so strong because of the reduced coordination numbers of the surface atoms. Because this hybridization depends on orientation, clusters with faceted surfaces are expected to have different magnetic properties than those with irregular faces. According to Tiago and coworkers (2006), this effect is the likely cause of the nonmonotonic suppression of magnetic moment as a function of size. Two classes of Fe clusters were considered in this work: faceted and nonfaceted. Nonfaceted structures are nearly spherical in shape and faceted structures are built using the conventional layer-by-layer growth model. The magnetic moment was calculated as the expectation value of the total angular momentum, M D

B Œgs hSz i C hLz i ; ¯

(4)

where gs D 2 is the gyromagnetic ratio of the electron. The magnetic moments of the clusters as a function of size are displayed in Fig. 5 for all classes of clusters considered. Suppression of the magnetic moment with increasing size is observed, in good agreement with experiment. Clusters with faceted surfaces indeed have a lower magnetic moment due to more efficient hybridization although the correlation of shape and magnetic moment is not always well defined. Icosahedral structures are predicted to have magnetic moments lower than bcc clusters. Another good example of disagreement between different methods is given in Krüger and coworkers where Pd clusters having several high-symmetry structures including icosahedra, octahedra, and cuboctahedra were optimized using both the LDA and GGA in the DFT framework. GGA yields larger bond lengths in accord with the general expectation. But the difference is almost independent of size. Icosahedral structures tend to yield larger bond lengths than octahedral and cuboctahedral structures, also displaying a flatter variation with increasing coordination number. The accuracy of this breathing mode relaxation was confirmed by comparing to a full relaxation of the Pd55 cluster. The Pd13 cluster was found in the work of Kumar and Kawazoe (2002) to have an icosahedral structure with a 0.62B /atom magnetic moment. This result agrees with some of the earlier studies and disagrees with some others. The central atom in the cluster is found to have a smaller magnetic moment than the surface atom only by a very small amount unlike the large difference in the Mn13 study of Kabir and coworkers (2006). For N D 14 and 15, icosahedron-derived structures are favored over other high-symmetry structures. The magnetic moments for both sizes remain around 0.5B /atom. Going from N D 16 to the double icosahedron at N D 19, the magnetic moment decreases from 0.38B /atom to 0.32B /atom. As

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a

Atom-Centered Clusters

3.2

2.8

2.4

Magnetic Moment (μB/atom)

0

100

b

200

300

400

Bridge-Centered Clusters

3.2

2.8

2.4

0

c 3.2

100

200

300

400

Icosahedral Clusters

2.8

2.4

0

100 200 300 Cluster Size (Number of Atoms)

400

Fig. 5 Magnetic moments of faceted and nonfaceted Fe clusters in the icosahedral and bcc configurations: (a) atom-centered bcc, (b) bridge-centered bcc, and (c) icosahedral. Experimental data is displayed as black diamonds for comparison (Reprinted with permission from Tiago et al. (2006). © (2002) by the American Physical Society)

the cluster sizes increase, there is a decrease in local (atomic) magnetic moments. For N D 55 and N D 147, the lowest lying state was found to be the Mackay icosahedra. Cubic and decahedral structures were found to have the next highest energies, with 0.29B /atom and 0.41B /atom, respectively. For the Pd55 cluster, structures with smaller magnetization lie very close to the ground state and would therefore be accessible at room temperature. So magnetic order can be easily lost under experimental conditions. In contrast to the results for Mn (Kabir et al. 2006), the icosahedral structures have a higher magnetic moment than other high-symmetry structures.

Modeling of Nanostructures

17

In work of Nava and coworkers (2003), Pd clusters were studied using the spinpolarized DFT method in the range N D 2–309. The N D 13 cluster was found to have an icosahedral structure with a high spin state. It is seen to undergo a very slight Jahn–Teller distortion, which increases the cohesive energy only by 0.01 eV. The truncated decahedron and the cuboctahedron are found to be less stable. Yao and coworkers (2007) utilized results from simulated annealing of Ni clusters using an empirical potential as starting configurations to further optimize them using a DFT code. This is another technique often used in cluster literature. In this work, Ni clusters with N D 10–27 atoms were found to attain icosahedronlike structures with the N D 13 cluster being a perfect icosahedron and N D 19 the double icosahedron. Clusters in the range N D 28–40 have very complicated structures because they are in the transition region between the N D 13 icosahedron and the N D 55 icosahedron. Around N D 19 and N D 55, clusters are mostly formed by adding or removing a few atoms from the corresponding perfect icosahedra. Dips are observed in the magnetic moment for N D 13 and 55 as expected from the compact structure of these clusters. While all of the studies mentioned above deal with collinear spin, which singles out one direction along which the magnetic moment may be oriented, a very important class of clusters with high magnetic moments displays noncollinear spin. In these clusters, the magnetic moment is allowed point in an arbitrary direction and thus presents a new degree of freedom. A good example for the investigation of this effect is presented in the work by Du et al. (2010) where six-atom clusters of Co and Mn atoms with several different compositions but always in the octahedral geometry were investigated within the DFT theory including the noncollinear spin formalism. Low-lying isomers with up to more than 40ı of average degree of noncollinearity were identified. For certain compositions and geometries, different tendencies for magnetic coupling (AFM vs. FM) are also found to cause a certain degree of spin frustration. In the work by Zhang and coworkers (2004), Mo clusters were studied within the DFT framework. An interesting result was that the initial icosahedral structure for the N D 13 cluster was found to undergo a very large distortion. This distortion was explained by the tendency of Mo clusters to form Mo dimers. The strength of the bonds in Mo13 is covalent; therefore, at this size, the cluster shows nonmetallicity. At N D 55, the icosahedral structure is found to again undergo a large distortion. However in contrast to N D 13, the alternate Oh structure was not found to be much more stable than the distorted icosahedron. This was explained by the decrease of nonmetallicity with size.

1D Structures: Nanotubes, Nanowires, and Nanorods In nanoscience literature, the name one-dimensionalwas coined to describe systems where one of the dimensions is several orders of magnitude larger than the other two dimensions. Much like the zero-dimensional case, the border between one- and two-dimensional systems is ill defined and system dependent. For this reason, two-

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dimensional systems exhibit the same richness in structural and electronic properties as in the zero-dimensional case unlike their bulk counterparts. One-dimensional structures may, in the broadest classification, be divided into nanotubes, nanowires, and nanoribbonsGraphene nanoribbons (GNRs) (Law et al. 2004; Hyun et al. 2013). Nanotubes are simply described as two-dimensional materials (such as graphene, BN sheets, TiO2 sheets, and many others) seamlessly rolled into a hollow cylinder although the actual fabrication usually follows a much more involved procedure. Nanotubes may be regarded as a unique subset of one-dimensional structures that do not have a surface and are thus devoid of surface effects. In this respect they are analogous to the fullerenes in the zerodimensional case. Nanowireson the other hand are extremely thin wires that are grown or extracted along well-defined crystal directions and may have widely different surfaces. The surfaces, however, rarely remain in their bulk configuration and often reconstruct to reduce strain or saturate broken bonds. This procedure is highly size and material specific, and nanowires therefore display a large variety of strongly surface-dependent properties (Dasgupta et al. 2014; Kempa et al. 2013). Finally, nanoribbons are thin strips of two-dimensional materials such as graphene. The edges may reconstruct or be saturated with different species to modify their properties (Ozdamar and Erkoc 2012; Ince and Erkoc 2011; Kilic and Erkoc 2013b). In addition to structural and electronic properties that are explored in zerodimensional materials, one-dimensional materials also exhibit rather interesting elastic properties. We shall begin this section with a brief review of elastic considerations regarding one-dimensional nanomaterials and afterward move onto structural and electronic properties.

Elastic and Structural Properties In nanowire applications such as AFM tips, NEMS, and MEMS, which make use of mechanical properties, it is crucial to have a good understanding of the evolution of elastic properties all the way down to the nanoscale. Elastic properties that are ordinarily under investigation include elastic moduli, plasticity, crack propagation, buckling, and breaking points. There are numerous examples of experimental determination of elastic properties of nanowires in the literature. A recent study by Barth and coworkers (2009) determines the Young’s modulus of SnO2 nanowires by anchoring and bending them with the help of an AFM tip. By mapping the bending amplitude to the Young’s modulus through classical elasticity formulae, the Young’s modulus is estimated at around 110 GPa for the samples studied. Contrary to the macroscopic scale, the elastic properties of nanoscale onedimensional systems are often seen to depend on their physical dimensions. This intriguing fact brings forth the necessity of studying, among others, the elastic properties of such systems as a function of their size. In fact, the theoretical literature is very rich in examples of such studies. Empirical potentials are particularly suitable for studies of size dependence since systems with large numbers of atoms may be

Modeling of Nanostructures

19

handled with relative ease, allowing the determination of convergence of elastic properties of nanomaterials to those of their bulk counterparts. A related question to this fact is to what extent the laws of continuum elasticity can be applied to nanoscale systems, which has received much attention both theoretically and experimentally. A recent study by Rudd and Lee (2008) opens with this very same question, where the Young’s modulus of Ta h001i and Si h001i nanowires has been determined using a Finnis and Sinclair (1984) and Stillinger and Weber (1985) potential, respectively, in a molecular dynamics simulation. Nanowire radii were explored up to approximately 10 nm and both materials were found to display a strong dependence on the wire width eventually converging to the bulk value. Interestingly, the convergence trends are opposite where the Young’s modulus of the Si nanowires increases while that of Ta nanowires decreases with radius. In the same study, DFT calculations were also performed on Si nanowires, partially confirming the outcome of the Stillinger–Weber calculations. However, the match with continuum theory was found to be much better for the DFT results. Size dependence is also demonstrated in a study by Hu and coworkers (2008) on ZnO nanowires and nanotubes where they use an exp-6-type empirical potential including Coulomb interaction for Young’s modulus calculations. The Young’s moduli of nanowires and nanotubes show a very strong dependence on the radius and wall thickness, respectively. Due to the extremely high surface-to-volume ratio in nanowires, the particular surfaces that are exposed at the outer edges of the nanowires play a crucial effect in the determination of their elastic properties (Ozdamar and Erkoc 2013; Kilic and Erkoc 2013a; Yagli and Erkoc 2014). Recent evidence for this fact was demonstrated by Wang and Li (2008) in their DFT-parameterized model study of Ag, Au, and ZnO nanowires with different surface terminations. Their results show discernible albeit small differences in the size dependence of the Young’s modulus for different surface terminations of the nanowires in question. Nanotubes have also received a great deal of attention from researchers due to their extraordinary elastic properties. They have been shown to possess an unusually high axial stiffness in addition to very high reversibility under large distortions. In addition to the large body of literature on experiments probing the elastic properties of nanotubes, many theoretical studies have also been conducted. Due to their varying radii and chirality, nanotubes present endless possibilities for the investigation of their elastic properties. Liang and Upmanyu (2006), for instance, have studied the radius (or equivalently curvature) and chirality dependence of the torsion induced by applied axial strength of the nanotubes. Their studies, which utilize the widely used second-generation reactive empirical bond-order potential (Brenner 1989), revealed a torsional response of up to 0.75 nm, which varied remarkably for different radii and chiral angles. Conversely, as reported in several early studies on nanotubes employing DFT calculations, Young’s modulus is known to be largely independent of the chirality. Elastic properties of nanotubes of many materials other than C have also been theoretically explored. In an exhaustive work, Baumeier and coworkers (2007) used DFT calculations to survey such properties as strain energy and Young’s

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modulus of SiC, BN, and BeO nanotubes. Baumeier and coworkers observe that being composed of two atomic species instead of one as in carbon nanotubes, the nanotubes made out of these materials exhibit a different relaxation pattern for the relevant anions and cations, suggesting possible different behavior than carbon nanotubes. Nevertheless, the results show that at least for (n, n) and (n, 0) nanotubes of similar radii, the behavior is similar. All three materials display a decreasing Young’s modulus as a function of radius converging to the sheet value for large radii. Among the three materials, BN nanotubes display a significantly high Young’s modulus, followed by SiC and finally by BeO nanotubes. The interwall attraction in multi-walled nanotubes opens up another possible avenue for the study of elastic properties. Zhang and coworkers (2008b) have studied the Young’s modulus, Poisson ratio, and the buckling point of multi-walled nanotubes using a combination of second-generation reactive bond-order potential to model intralayer bonding and a Lennard-Jones potential for the interlayer interaction. The multi-walled nanotubes studied were divided into two sets, the first formed by embedding increasingly smaller (n, n) tubes into a large (20, 20) nanotube (up to four walls) and the second by placing a (5, 5) tube into increasingly larger (n, n) tubes (again up to four walls). A molecular dynamics algorithm was used during their calculation and the dependence of the moduli in addition to the buckled morphologies was presented. Young’s modulus and Poisson’s ratio turn out to follow a different trend for the two sets considered. While the set that grows inward displays increasing (decreasing) Young’s modulus (Poisson ratio) for increasing number of walls, the set that grows outward follows the opposite trend. As mentioned in the introduction to this section, one way to saturate the surface bonds of nanowires is through passivation by different species. The elastic behavior of passivated versus unpassivated nanowires raises an interesting question. Lee and Rudd (2007) studied several elastic properties of H-passivated nanowires of a large range of radii varying between 0.61 and 3.92 nm (see Fig. 6) using DFT. As expected, the Young moduli of H-passivated Si nanowires mostly follow predictions from continuum models regardless of the varying proportions of h100i and h001i surfaces exposed. This is attributed to the fact that the bulk-like covalent bonding character at the surfaces is preserved when H is used to passivate the dangling bonds at the surface. Although microscopic modeling is of utmost importance in the understanding of nanoscale materials, there are a number of experimental situations of interest that cannot be handled by these time-intensive methods due to their large size. For large enough systems (such as large portions of nanotubes suspended over a trench), continuum methods may be employed (Ustunel et al. 2005). However, complete coarse graining is also not always a viable choice since one then loses detailed information on locally nonhomogeneous regions of the system such as defects and local deformations. In such cases, multiscale methods which apply different methods at different scales of the system are the methods of choice. An illustrative example was studied by Maiti (2008) where a micromechanical sensor made out of a nanotube was deformed by a Li needle. The bent but undeformed portions of the nanotube were modeled by a coarse-grained molecular mechanics simulation, while

Modeling of Nanostructures

9Si 12H

0.61nm

21

21Si 20H

0.92 nm

49Si 28H

57Si 36H

1.39 nm

1.49 nm

25Si 20H

1.00 nm

405Si 100H 205Si 68H 109Si 52H 0.3 0.2 0.1 2.05nm

0.0 2.80nm 3.92nm

Fig. 6 H-passivated Si nanowires of different sizes studied by Lee et al. (Reprinted with permission from Lee and Rudd (2007). © (2007) by the American Physical Society)

the highly deformed midsection (enclosed in a box) was modeled using a quantum mechanical method. The two methods were then matched at the interface of the two regions.

Structural Properties The novel electronic properties of nanotubes, nanowires, and nanobelts are inextricably linked to their structural properties. This is largely due to their high surface-to-volume ratio, where the bonding on the surface structure determines the electronic states which in turn determines such properties of the system as conductivity and magnetism. As mentioned in the introductory section, nanowires nowadays may be manufactured from a great variety of materials. As fabricated, it is experimentally difficult, if not impossible, to intuitively infer their surface structure. Simulations on the other hand provide an inexpensive yet accurate way of studying this relation between structure and electronic properties. The methods of preference in nanowire modeling are generally tight binding and to a larger extent DFT, since electronic properties depend sensitively on the structure requiring accurate calculations. Once a material (or combination of materials) is

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chosen, the important parameters of nanowire modeling are the particular surfaces surrounding the nanowire and passivation, that is, the saturation of the bonds at the surfaces. Two of the most commonly studied surface terminations are surface reconstruction and H passivation. Due to the axial periodicity of one-dimensional nanomaterials, plane wavebased methods which are traditionally used for crystals are very often employed. In the work of Arantes and Fazzio (2007), where they study free and passivated Ge nanowires, the band gap of passivated and unpassivated nanowires was determined using a plane wave-based GGA-DFT method. The nanowires were grown in the h110i and h111i directions, and their band gaps were calculated as a function of nanowire diameter. In spite of the well-known underestimation of the band gap by LDA and GGA methods, a trend can be obtained rather reliably. The band gaps are seen to vary with respect to direction and size. Though not often encountered, global search algorithms are also used for determining the structure of nanowires. Especially in the case of nanowires with smaller radii, the structure may be so different from bulk as to prevent any a priori prediction. Chan and coworkers (2006) conducted a genetic algorithm search based on the formation energy of H-terminated Si NWs where the formation energy is defined as f D .E  H nH / =n  ;

(5)

where E is the total energy of the NW in question, H and  are the chemical potentials of H and Si, respectively, and n is the number of Si atoms. The genetic algorithm was conducted in two stages. In the first stage, a long evolution through several generations was conducted using an empirical potential (of the Hansel– Vogel type). In the second stage, the outcome structures of the evolution for each size were relaxed using a DFT algorithm. The result, reminiscent of cluster structures, is that certain sizes of Si NWs are preferred over other sizes. These structures are once again termed magic sizes. In general, structures with even number of Si atoms are preferred over those with an odd number. The most stable structures are observed either in a platelike form where chains of Si hexagons join together to form flat structures, or a hexagon-shaped cross section. One of the most important parameters in determining the geometric and therefore the electronic structure of nanowires is surface termination. The high-energy dangling bonds at the surface may either be saturated by a rearrangement and rebonding of the surface atoms or by attaching electropositive species (such as H). In general, while the reconstruction may significantly alter the structural and electronic properties of the material, passivation by other species leaves these properties relatively unaltered. In the recent work of Migas and Borisenko (2008), h001ioriented Si nanowires were passivated by O, F, and H, and the structural and a large number of different sizes and geometries were studied using DFT. Under different combinations and coverages by these elements, the band gap of the Si nanowires considered (only nanowires with rhombic cross sections were considered) was

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found to vary between 1.20 and 1.96 eV, which shows that termination can be used as a means to control the electronic properties. Countless other studies in understanding the structure of experimentally relevant one-dimensional structures have been done. Bi nanowires (Qi et al. 2008), CdSe nanorods with hexagonal and triangular cross sections (Sadowski and Ramprasad 2007), and Te nanowires (Ghosh et al. 2007) are some studies that may be mentioned. In addition to materials that have already been manufactured, researchers have also been interested in the possible existence and properties of nanoscale systems that have not yet been experimentally realized. Rathi and Ray (2008), for instance, have examined the possibility of Si–Ge nanotubes, while Qi and coworkers (2008) investigated Bi nanotubes and hollow Bi rods. Recently it was demonstrated by model calculations that the stable carbon nanotube structure might be possible with non-graphene-like form (Erkoc 2004a). Furthermore, nanorod structures constructed from benzene rings only (called as benzorods) may also be possible. Their structural and electronic properties were investigated by performing model calculations (Erkoc 2003; Malcioglu and Erkoc 2004).

Electronic, Magnetic, and Optical Properties As a result of the immense variation in structural properties of one-dimensional structures, one observes an equally diverse spectra of electronic properties. Perhaps the most intriguing property of one- and two-dimensional nanoscale systems is that after the required geometrical deformations in order to reach their equilibrium, one might observe a stark difference between the newly formed system and its bulk counterpart. A material which is an insulator in bulk may become a conductor when taken to the nanoscale. Similarly one- or two-dimensional nanostructures of a nonmagnetic material may have a nonzero magnetic moment. These differences usually stem from the new states introduced into the electronic structure of the material by terminating structures such as surfaces, steps, edges, and corners. The interest in these emergent properties is due to the possibilities of integrating these small-scale materials into technological applications and controlling their properties by controlling the structure. Due to the ease of interfacing with current technology, Si nanowires have perhaps been the most intensely studied system. Rurali and Lorente (2005), for instance, explored a large range of surface reconstructions for h100i Si nanowires with a small radius of about 1.5 nm. They discovered that for certain reconstructions, the nanowires develop conducting states in their band gap, while for others the semiconducting behavior is retained. This is a prime example of effects of confinement on the electronic structure of a system. Another parameter that has a strong effect on the electronic properties of nanowires is surface termination. As demonstrated by Rurali (2005) that both Si and C-terminated H-passivated SiC nanowires have a larger band gap than that of bulk SiC. As discussed by Rurali, this is purely a confinement effect since passivation prevents reconstruction and the related introduction of gap states. However, when

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the nanowires are allowed to reconstruct, both species of SiC nanowires are seen to become conducting due to the surface states introduced into the band gap. In addition to the conventional carbon nanotubes, several other materials were investigated as viable candidates for nanotube structures. BN (Chopra et al. 1995) and more recently (Sun et al. 2002) SiC nanotubes are two such materials which have been successfully synthesized in the nanotube form. Following their synthesis Gao and Kang (2008) conducted a DFT-PAW study of undoped and N-doped SiC nanotubes of varying sizes. For each size the nanotubes were doped initially with 2 and 4 N atoms and their structural and electronic properties were studied. For each doping level, several possibilities were investigated and the most stable structure was identified. As an extreme case, the case in which all the C atoms were replaced with N atoms, in other words SiN nanotubes, was considered. These nanotubes, instead of being circular, were found to have a staggered or starlike cross section. In all the cases considered, the nanotubes were found to be semiconducting with an indirect band gap. Recent model calculations on binary compounds BN (Erkoc 2001b), GaN (Erkoc et al. 2004), InP (Erkoc 2004b), and ZnO (Erkoc and Kokten 2005) nanotubes give reasonable results comparable with experimental findings. Although by nature rather free of defects, the few existing defects in graphene and nanotubes change the electronic structure of their host material drastically (Pekoz and Erkoc 2008). In spite of the several experimental methods that have been developed to locate and study the properties of such defects, theoretical methods are an indispensable tool for creating isolated defects of the desired nature and studying their effects on electronic properties. A particularly interesting question is the stability of well-characterized graphene defects in nanotubes of varying radius and chirality. Since a graphene sheet can be viewed as a nanotube with an infinitely large radius, the formation energy of any nanotube defect should tend to the equivalent defect on a graphene sheet. Amorim et al. (2007) demonstrated an example of this behavior by studying the so-called 555777 defect, which is a combination of three pentagons and three heptagons. This defect is formed in two steps. In the first step, two divacancies coalesce to form a new defect (585) composed of two pentagons and an octagon. This step is followed by a further structural change, which yields the 555777 defect, see Fig. 7. In graphene the 555777 defect is found to be more stable than the 585 defect by 0.8 eV. The same defects were then created in zigzag and armchair nanotubes of radius in the range of 7–15 Å. As in the graphene case, the 555777 defect is found to be more stable in all the nanotubes studied. The formation energy is, however, lower than in the graphene case. The expected tendency toward the corresponding graphene values is seen in both cases and convergence is estimated to occur around a radius of 40 Å. The effect of defects on the electronic properties of their host substance is illustrated in this work by the conductance graph calculated using the Green’s function density functional theory. The results indicate that the presence of both the 585 and 555777 reduces the conductance considerably while at the same time displaying different voltage dependence.

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Fig. 7 585 (a) and 555777 (b) defects in graphene and 585 (c, e) and 555777 (d, f) defects in a nanotube in different orientations (Reprinted with permission from Amorim et al. (2007). © (2007) by the American Chemical Society)

Another investigation of defects in nanotubes for the purpose of application as a gas sensor was conducted by Andzelm and coworkers (2006). The particular question at hand is the binding of NH3 molecule to nanotubes and whether or not binding is enhanced by defects. Three defects are considered: a Stone–Wales defect (a defect formed by rotating a given bond by 90ı resulting in the formation of two pentagons and two heptagons), a monovacancy, and an interstitial C atom placed on top of a bridge. In addition, the case of an O2 molecule dissociated on a SW defect was also considered to mimic the environmental effects. Two different orientations, straight and chiral, for the SW defect were considered. All-electron DFT calculations reveal that the monovacancy is the most stable. Calculation of reaction barriers also reveals that defects with preabsorbed O dissociate NH3 into NH2 and H. The heterostructure problem, which has been widely studied in the bulk form, is becoming an increasingly popular topic also in the one-dimensional systems. The band alignment problem has been addressed recently in a DFT study conducted by Kagimura and coworkers (2007) where a Si–Ge interface was studied. The model for such a system is shown in Fig. 8. The band states contributed by the surface dangling bonds were investigated as possible candidates for the induction of a potential well. The search for nanoscale materials that exhibit spontaneous magnetization has become an increasingly rich field in the past decade. Several materials such as doped nanotubes and nanowires, defective graphene, and nanoribbons of two-dimensional materials can be itemized as candidates considered in these studies. In order to identify the suitability of a material for and electron  spontaneous   magnetization  r r are compared. If there and n# ! density in the two spin channels, n" ! is a significant difference in this distribution, the material is nominated for use in magnetic applications. If in addition the density of states in the two spin channels shows different characteristics at the Fermi level such that in one channel there is

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Fig. 8 A Si–Ge interface shaped into a nanotube. (a) Side view, (b) cross view (Adapted from Kagimura et al. 2007)

a significantly larger number of states than the other, then the material provides promise also in spintronic applications. One such candidate material for magnetic applications is BN sheets and nanotubes doped with several different elements. In a recent example, Li and coworkers (2008a), in their DFT study on BN nanotubes with one, two, and three H atoms adsorbed at different locations, observed that some of the configurations considered may give rise to a magnetic moment of up to 2.0. The origin of this magnetic moment is evident from the band structures where the contribution of band gap states is due to only one of the spin channels. The states that are seen in the gap are also shown visually and their origin is unambiguously identified as due to the adsorbed hydrogen. A rather novel and intriguing application of nanomaterials and its investigation using theoretical modeling is the subject of a recent work by de Oliveira and Miwa (2009). Their study discusses a phenomenon called self-purification of nanomaterials, namely, the expulsion of foreign species from the surface. Self-purification is attributed to the fact that nanoscale materials have a lower incorporation rate of impurities compared to their bulk counterpart. In this work SiC nanowires are studied as another potential example of self-purifying materials. B and N impurities were planted at different positions inside and on the surface of three SiC nanowire configurations h100i Si-coated, h100i C coated, and h111i Si and C coated at locations ranging from the center to the surface. The formation energies of the configurations thus formed were then calculated and compared. As a result, the self-purification process was found to be favorable for B-doped SiC nanowires irrespective of their orientation. The B atom was found to segregate favorably to the surface and thus expelled. The N impurities, however, were found to prefer the sites in the core of the nanowires, and therefore, N-doped SiC nanowires were found to fail as self-purifying materials.

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One-dimensional systems are actively sought after also for their optical properties. Using the GW method, which is an accurate method for studying excited states, Bruno and coworkers (2007) demonstrated that optical properties of Si and Ge nanowires depend not only on the nanowire diameter but also on the orientation.

2D Structures: Graphene and Derivatives Due to the remarkable ability of carbon to exist in different hybridization states, carbon-based materials display an unusually rich variety. Diamond, graphene, cagelike molecules and perhaps most notably carbon nanotubes are examples of the large selection of possibilities. Graphene, which is a single layer of graphite, may be thought of as the building block of most of the allotropes listed above (Fig. 9). Nanotubes are geometrically rolled up versions of graphene, and fullerenes can be formed by introducing

Fig. 9 Graphene is a 2D building material for carbon materials of all other dimensionalities. It can be wrapped up into 0D buckyballs, rolled into 1D nanotubes, or stacked into 3D graphite (Reprinted with permission from Geim and Novoselov (2007). © (2007) by Nature Publishing Group)

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topological defects (heptagons and pentagons) into the perfect honeycomb lattice structure of graphene in order to introduce positive and negative curvature (Freitag 2008; Peres et al. 2006b; Kim and Rogers 2008; Tang and Zhou 2013). Regardless of the immense attention that graphene received in the theoretical literature, it wasn’t until 2004 (Novoselov et al. 2004) that a single layer of graphene was isolated experimentally. Since two-dimensional crystals were proven to be unstable theoretically (Landau and Lifshitz 1980; Mermin 1968), the discovery of free-standing graphene came as a surprise (Meyer et al. 2007). This apparent discrepancy, however, was lifted when upon closer inspection, the isolated graphene sheets were not perfectly flat but had corrugations reaching up to 1 nm in size (Rusanov 2014). Since the discovery and isolation of graphene was achieved, many other materials were found to form two-dimensional structures. In this chapter, we give an overview of carbon-based two-dimensional materials including graphene, graphene nanoribbons, nanobelts, and strips in addition to two-dimensional structures of several other materials (Singh et al. 2011; Bonaccorso et al. 2015; Niu and Li 2015).

Graphene, Nanosheets, Nanoribbons, Nanobelts, and Nanostrips With the advent of several sophisticated experimental techniques, the twodimensional confinement of graphene layers was further extended to one dimensional in the form of nanoribbons. Nanoribbons are narrow strips of graphene that may exhibit quasi-metallic or semiconducting behavior depending on the geometry of their edges. Much like nanotubes, graphene nanoribbons (GNRs) are also termed zigzag or armchair based on the directionality of the bonds with respect to the long axis (see Fig. 10). Due to the dependence of their electronic properties on their geometry, it is important to control the morphology and crystallinity of these edges for practical purposes. It has been experimentally shown that (Jia et al. 2009) controlled formation of sharp zigzag and armchair edges in graphitic nanoribbons is possible by Joule heating. During Joule heating and electron beam irradiation, carbon atoms are vaporized, and subsequently sharp edges and step-edge arrays are stabilized, mostly with either zigzag- or armchair-edged configurations. In addition to the edge geometry, the electronic properties, in particular the band gap, of nanoribbons also depend on their width (Han et al. 2007; Wu and Zeng 2008). The magnetic properties may also be severely altered upon reduction of size to a graphene fragment (Wang et al. 2008) which results in the emergence of giant spin moments. Another path for controlling the electronic properties of nanoribbons is by an application of an external electric field or by chemical doping of the pristine samples. Half-metallicity, for instance, which has several applications in spintronics (Wu and Zeng 2008) may be introduced through functionalization with such species as H, COOH, OH, NO2 , NH3 , and CH3 (Son et al. 2006). Straight GNRs with zigzag, armchair, or mixed edges are proven to be semiconducting by the experiment. In addition, GNRs can be sculpted by attaching two

Modeling of Nanostructures

29

Fig. 10 (a) Armchair graphene ribbon model. (b) Zigzag graphene ribbon model

Fig. 11 The model structure of a sawtooth-like GNR (Adapted from Wu and Zeng 2008)

segments together that are manufactured to make a 120ı angle with each other thereby forming a sawtooth-like nanoribbon (Fig. 11). The structure of a sawtoothlike GNR can be characterized by two integers (w, l). The first integer denotes the width of the nanoribbon, while the second integer describes its periodic length (Wu and Zeng 2008). Stability is an important issue for a material which is intended to be used as a building block of device applications. Even though perfect two-dimensional crystals are proven to be unstable, graphene is found to be stabilized by corrugations in the third dimension. Understanding the effect of GNR width on the stability is therefore also a central issue for possible applications. Molecular dynamics computer simulations using empirical interatomic potentials predict that structural stability of graphene nanoribbons show dependence on size (width) and edge orientation (Dugan and Erkoc 2008; Ozdamar and Erkoc 2012). Figures 12 and 13 show, respectively, the relaxed structures of armchair- and zigzag-edged GNRs with various widths. In the recent nanoscale literature, a wealth of materials other than C has been identified, both theoretically and experimentally, as viable candidates for future use as two-dimensional devices. Boron nitride, for instance, having electronic properties that resemble carbon can exist in a hexagonal structure h-BN similar to the graphite-layered geometry.

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Fig. 12 Armchair-edged GNRs. Left column shows top view and right column shows tilted view. GNRs of width five hexagonal rings are shown. 1 K, 300 K, and final temperature images are given (Adapted from Dugan and Erkoc 2008)

Fig. 13 Zigzag-edged GNRs. Left column shows top view and right column shows tilted view. Ribbons of width five hexagonal rings are shown. 1 K, 300 K, and final temperature images are given (Adapted from Dugan and Erkoc 2008)

Much like graphene sheets, BN sheets can be grown on more or less lattice-matched transition metal surfaces (Corso et al. 2004; Huda and Kleinman 2006). A model BN sheet is shown in Fig. 14. BN, being a member of the III-V semiconductor family, indicates the possibility of nanoribbons or nanobelts made of other semiconductor families. Indeed, several oxide and other II-VI family nanobelts have been discovered and reported (Pan et al. 2001) (ZnO, SnO2 , CdO, Ga2 O3 , PbO2 , ZnS, CdSe, and ZnSe). Some of the oxides in this family such as ZnO and SnO2 , owing to their polarity and crystal structure, deform in novel morphologies such as rings, springs, and spirals in order to bring together the positive and negative charges, counteracting the charge imbalance (Yang and Wang 2006). These structures comprise a versatile set of nanomaterials that are promising candidates for various applications such as sensors, resonators, and transducers.

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Fig. 14 The geometric structure of BN sheet (Adapted from Venkataramanan et al. 2009)

Nanobelts, a term coined by researchers working on these structures (Wang 2004), are described as nanowires with not only a well-defined growth direction but also well-defined top, bottom, and side edges and cross section. The nanobelt structures are usually obtained from functional oxides, which are semiconductor materials, such as ZnO; Ga2 O3 ; tSnO2 ; oSnO2 ; In2 O3 ; CdO; and PbO2 . Pure metal nanobelt structures are also possible; Zn is one of the metals that form fine nanobelt structures (Wang 2004). ZnO, being one of the most versatile materials in nanoscale research alongside with carbon and BN, has once again been investigated at great depth in the context of nanobelts (Kulkarni et al. 2005; Wang 2004). Molecular dynamics computer simulations using empirical interatomic potentials (Kulkarni et al. 2005) reveal that ZnO nanobelts display properties that depend on their size and orientation. Depending on the growth direction, ZnO nanobelts may exhibit an interesting shell structure or a simple surface reconstruction (Wang 2009; Ozgur et al. 2005). Nanostrips are similar to nanobelts; they are used with the same meaning with nanobelts. All these nanostructures (ribbons, belts, strips) are ideal materials for building nano-sized devices and sensors (Lin et al. 2009). In addition to the materials mentioned above, recent theoretical studies have proposed a wealth of two-dimensional structures that are composed of less common materials. One such example is the recently proposed B2 C graphene (Wu et

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Fig. 15 (a) A C2V  CB4 motif structure. Each carbon atom is bonded with four boron atoms. Optimized structures of two types B2 C nanoribbons with two different edge configurations, referred as type I (b) and type II (c) (Adapted from Wu et al. 2009)

al. 2009). The optimized B2 C graphene structure is displayed in Fig. 15. Two neighboring CB4 motifs share two common boron atoms, giving rise to a hexagon and a rhombus. The mean B–C and B–B bond length is 1.557 and 1.685 Å, V respectively. The sheet is slightly corrugated with a distance of only  0:085 A. The corrugation is a result of an excess of 2p electrons normal to the sheet relative to the gas-phase CB4 molecule. The quasi-one-dimensional B2 C nanoribbons are finite-size graphene with parallel edges. The width of the B2 C nanoribbon is defined by the number of C atoms normal to the long axis of the ribbon. As shown in Fig. 15, type I and II B2 C nanoribbons are displayed. The dangling bonds at the edges of B2 C nanoribbons can be passivated by either H atoms or CH groups. TiO2 with its uses in solar cell applications and surface catalysis has received much attention both theoretically and experimentally. Its nanostructures are of equal interest to technology due to their chemical inertness, endurance, strong oxidizing power, large surface area, high photocatalytic activity, non-toxicity, and low production cost. The titania nanostructures are constructed by “cutting” of TiO2 monolayers into nanostrips and by rolling them into cylindrical nanotubes or nanorolls (Enyashin and Seifert 2005). There are two different topological nanostrip models constructed from titania. One model is obtained from (101) surface of titania, called as anatase layer, and the second model is obtained from (010) surface of titania, called as lepidocrocite layer. A view of the structures of (101) and (010)

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Fig. 16 Monolayers of TiO2 in anatase (top (a) and side (b)) and lepidocrocite polymorphs (top (c) and side (d))

Fig. 17 Idealized 1212 (a) and reconstructed 1221. (b) Boron sheets

TiO2 layers is shown in Fig. 16. By rolling of these strips, various nanostructures can be generated, such as nanotubes and nanospirals (nanorolls) (Enyashin and Seifert 2005). Non-carbon elemental sheet structures have also been investigated. Lau et al. (2006) proposed four possible configuration models for the boron sheet. According to this study, the flat form, denoted f1212g, seen in Fig. 17 is a triangular network, while the buckled f1212gb and pair-buckled f1212gpb configurations include chainwise and pairwise out-of-plane displacements. Finally, a reconstructed f1212g configuration is investigated with inversion symmetry in the unit cell. It can be considered as a triangular–square–triangular unit network. The DFT calculations reveal (Lau et al. 2006) that the reconstructed f1221g configuration is the most stable configuration by 0.23 eV/atom relative to the idealized f1212g configuration. Both the f1212gb and f1212gpb configurations tend to converge to the idealized f1212g configuration when relaxed during the geometry optimization.

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Energy

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Fig. 18 The band structure of a representative three-dimensional solid (left) is parabolic, with a band gap between the lower-energy valence band and the higher-energy conduction band. The energy bands of 2D graphene (right) are smooth-sided cones, which meet at the Dirac point

Electronic and Mechanical Properties Graphene has a unique and curious band structure which can be approximated by a double cone close to the six Fermi points at the corners of the Brillouin zone (see Fig. 18). Commonly referred to as Dirac electrons, the conduction electrons follow a linear energy-momentum dispersion and have a rather large velocity. The conduction in graphene is enabled by the delocalized  -electrons above and below the plane. Due to their relative detachment from the tightly knit planar network, these electrons are free to move along the graphene sheet with rather high mobility. This is of course a rather desirable property for devices used in electronics (Li et al. 2008b). A good yet simple method for understanding the band structure of graphene is the tight-binding formalism (Neto et al. 2009). Graphene is made out of carbon atoms arranged in hexagonal structure, as shown in Fig. 19. The structure can be seen as a triangular lattice with a basis of two atoms per unit cell. The lattice vectors can be written as p  a p  a 3; 3 3;  3 ; a1 D a2 D 2 2 V is the carbon–carbon distance. The reciprocal-lattice vectors are where a  1:42 A given by b1 D

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The two points K and K 0 at the corners of the graphene Brillouin zone (BZ) are named Dirac points. Their positions in momentum space are given by  KD

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where ai, (ai, ) annihilates (creates) an electron with spin  . D"; #/ on-site Ri on sublattice A (an equivalent definition is used for sublattice B), t . 2:8 eV/ is the nearest-neighbor hopping energy (hopping between different sublattices), and t 0 . 0:1 eV/ is the next nearest-neighbor hopping energy (hopping in the same sublattice). The Hamiltonian in Eq. 7 is solved at various momenta and the energy bands are obtained as follows: p E˙ .k/ D ˙t 3 C f .k/  t 0 f .k/ (7) with ! p   p  3 3 ky a cos kx a ; f .k/ D 2 cos 3ky a C 4 cos 2 2

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where the plus sign applies to the upper .  / and the minus sign the lower () band. As illustrated by this simple model, graphene is a semimetal or a zero-gap semiconductor. At low temperatures it does not possess superconducting properties; however, as demonstrated by Pathak et al. (2008) using variational Monte Carlo, there is a possibility that doped graphene may superconduct. The graphene nanoribbons (GNRs) discussed in the previous sections may also be modeled rather easily using the tight-binding formalism taking as the basis the usual Schrödinger’s equation (Ezawa 2006) or the massless particle Dirac equation (Sasaki et al. 2006). These models predict that armchair GNRs can be either metallic or semiconducting depending on their widths and that zigzag GNRs with zigzagshaped edges are metallic regardless of the width. The edges of GNRs are also suitable sites for chemical functionalization (Wang et al. 2007). Due to the existence of dangling bonds at the edges, the electronic properties of GNRs may be controlled by modifying these bonds by addition of other species. The results of first-principles calculations using linear combination of atomic orbital density functional theory (DFT) method predict that the electronic band structures and band gaps of the GNRs show a dependence on the edge structure of the nanoribbons (Wu and Zeng 2008). Clearly, the electronic band structures are sensitive to the edge structure of the nanoribbons. All straight GNRs are semiconducting. Two distinct features can be seen concerning the band gap. The band gap of GNR with zigzag edges slightly decreases with increasing the width w, while that of GNR with armchair edges varies periodically as a function of w. Like the straight GNRs, the calculated electronic band structures of the sawtoothlike GNRs (see Fig. 20) also show semiconducting characteristics with direct band gap. More interestingly, even though the sawtooth-like GNRs have zigzag edges, their band gaps show similar oscillatory behavior as those of straight GNRs with armchair edges, which depend on the width w. However, for most nanoribbons, their band gap reduces monotonically with increasing periodic length l. The band gaps of nanoribbons with w D 1, 2, or 4 reduce much rapidly, whereas those of nanoribbons with w D 3 or 5 reduce gradually. For (2, l) sawtooth-like nanoribbons, the band gap approaches zero rapidly as the l increases (Wu and Zeng 2008). These results show that the band gaps of the sawtooth-like GNRs are much more sensitive to their geometric structures, which is an opportunity for tuning the band gap. Quantization of electric conductance under the action of an external magnetic field displays a rather interesting trend for GNRs. In the exhaustive tight-binding study conducted by Peres et al. (2006a), the number of plateaus in the quantized conductance was found to be even in armchair GNTs, while the same number is odd for the zigzag edge. Rosales et al. (2008) investigated theoretically the effects of side-attached one-dimensional chains of hexagons pinned at the edges of the GNRs. These onedimensional chains could be useful to simulate, qualitatively, the effects on the electronic transport of GNRs when benzene-based organic molecules are attached to the edges of the ribbons. They propose a simple scheme to reveal the main electronic

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Fig. 20 (a) The electronic band structures of several sawtooth-like GNRs with various w and l. The band gaps versus (b) the width w and (c) the periodic length l (Reprinted with permission from Wu and Zeng (2008). © (2008) by Springer)

properties and the changes in the conductance of such decorated planar structures. For simplicity, they consider armchair and zigzag nanoribbons and linear polyaromatic hydrocarbons (LPHC) and poly(paraphenylene), as the organic molecules. The attached molecules are simulated by simple one-dimensional carbon hexagonal structures connected to the GNRs. These nanostructures are described using a single-band tight-binding Hamiltonian, and their electronic conductance and density of states are calculated within the Green’s function formalism based on real-space renormalization techniques (Rosales et al. 2008). As revealed by the theoretical analysis conducted by Nakabayashi et al., GNRs are a solution to the difficulty of producing an effective graphene switch for turning off the current. Zigzag nanotubes due to their peculiar band structures may be utilized for better current control rather than graphene. A single layer of graphene has no band gap, which makes it difficult to control current. However, Nakabayashi et al. (2009) found that when a sheet of graphene was in the form of nanoribbons

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just a few nanometers wide, its electronic structure changed so that currents could be controlled in radically different ways in comparison to conventional semiconductor devices. Nakabayashi et al. (2009) showed that when a nanoribbon was cut so that its edges formed a zigzag structure with an even number of zigzag chains across its width, and an electrostatic barrier potential applied along part of its length, then it behaved as a so-called band-selective filter, preferentially scattering charges into either even- or odd-numbered bands of its electronic structure depending on the potential. And when two such filters with different potentials were connected in series, they showed it should be possible to completely shut off the flow of current through the nanoribbons. Tight-binding studies have revealed countless interesting electronic properties regarding GNRs. GNRs, much like single-walled carbon nanotubes, can display metallic or semiconducting properties depending on their orientation and width. Similar to armchair nanotubes, zigzag ribbons are all metallic and may have magnetic properties. Bare and H-terminated ribbons, as studied by Barone et al., may show such effects as gap oscillations which make them a viable choice for possible band structure engineering applications. The results of this study, which is a careful investigation of GNRs of several orientations, widths, and terminations, are displayed in Fig. 21 along with the details of the models used and the band gap profiles of the corresponding models. GNRs have localized edge states located near the Fermi level. By terminating the edges with different species, one can change the character of these states and consequently the properties of GNRs. In fact, a first-principles calculation within spin-unrestricted local-density functional formalism on zigzag-edged graphene nanostrips terminated with hydrogen and oxygen atoms as well as hydroxyl and imine groups shows that these different species have a significant impact on the electronic structure of these strips near the Fermi level (Gunlycke et al. 2007). Zigzag-edged nanostrips terminated with hydrogen atoms or hydroxyl groups exhibit spin polarization, while the nanostrips terminated with oxygen or imine groups are unpolarized. These differences of course result in very different conductance characteristics for these systems. In further support of the aforementioned evidence, Zhang and Yang (2009) confirmed in their linear combination of atomic orbitals (LCAO) tight-binding study that H-terminated armchair GNTs exhibit size dependence in their conductance properties. On the other hand, ZnO nanoribbons show different characteristics from that of carbon nanoribbons. ZnO nanoribbons grown along the [0001] direction can form two different planar monolayer, zigzag and armchair ribbons. First-principles calculations of zinc oxide nanoribbons show that the stability of armchair-edged structures is greater than the zigzag-edged configurations. Furthermore, singlelayered armchair ribbons are semiconductors, whereas the zigzag counterparts are metallic (Botello-Mendez et al. 2007). An effect that is not present in purely carbonbased systems is the presence of two differently charged species offering different edge configurations. For the zigzag ZnO ribbons, for instance, the exposed atoms are oxygen atoms while the hidden are zinc. The opposite edge has the inverse

Modeling of Nanostructures

39

a 1.5

f = 23.4°

0.0

b Energy Gap (eV)

1.5

c

d

f = 13.9°

0.0 1.5

f = 8.9°

0.0 1.5

f = 4.7°

0.0

e

f = 0°

1.5 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 L (nm)

Fig. 21 Left: A representative set of semiconducting hydrogen-terminated GNRs, created by “unfolding” and “cutting” different types of CNTs. (a) A GNR with a chiral angle of D 23:4ı created by unfolding and cutting a C(6,4). (b) A GNR with a chiral angle of D 13:9ı created by unfolding and cutting a C(6,2). (c) A GNR with a chiral angle of D 8:9ı created by unfolding and cutting a C(20,4). (d) A GNR with a chiral angle of D 4:7ı created by unfolding and cutting a C(10,1). (e) An armchair GNR . D 0ı / created by unfolding and cutting a zigzag CNT. Right: Dependence of the band gap on the width of hydrogen passivated chiral GNRs. The different panels correspond to the different CNRs presented in the left (Reprinted with permission from Barone et al. (2006). © (2006) by the American Chemical Society)

structure. The armchair ribbons are characterized by a ZnO pair at the outer edge and another pair at the inner edge. After relaxation, the far edge oxygen ions tend to shift outward on both sides of the ribbon. As in the case of graphene materials, understanding the effects of edge-dimensional variation on band gap energies will provide the new insights into the fundamental principle of architecture design of nanodevices fabricated with ZnO-nanostructured materials. A simple tight-binding model can be used to investigate the electronic properties of nonpolar ZnO nanobelts following the procedure described below. Consider a simplified unit cell model (see Fig. 22) along the nanobelt growth z direction with Zn and O in the xy plane. This is the actual case for the ultrathin ZnO film oriented along the [001] direction. In this case, the tight-binding Hamiltonian is given by (Yang and Wang 2006) H D

X r2A

"r ar ar C

X r2B

"r br br

X X   C br arCSi r;i arC brCSi C i;r r2A;B i D1;2;

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H. Toffoli et al. Zn (A) O(B) my

mx

Fig. 22 Schematic illustration showing a tight-binding unit cell for a ZnO nanobelt (Adapted from Yang and Wang 2006)





where operators ar (ar ) and br (br ) create (annihilate) a state at sites A (for Zn) and B (for O) at their coordinates r, respectively. " denotes the on-site self-interaction energy, whereas  is the interatomic coupling term. Similarly, the wave functions for Zn and O at their respective lattice sites can be written as a linear combination of Bloch waves along the z direction and standing waves along x and y as the following (Yang and Wang 2006), My Mx X ˇ X X  C ˇˇ ˇ cp e i kz zm sin .kx mx a/ sin ky my b am ˇZn; kx ; ky >D ˇ0 > x my mx D1 my D1 zm

Ny Nx X ˇ ˇ X X  ˇ ˇ cp e i kz zn sin .kx nx a/ sin ky ny b anCx ny ˇ0 > ˇO; kx ; ky >D nx D1 ny D1 zn

in which lattice vectors a and b are as defined in Fig. 19 and cp and cq are coefficients. To simplify the model, one may consider only the coupling term between the nearest neighboring sites. In this case, on the same xy plane, the O-sites take the following coordinates around a Zn center: nx D mx  1; mx and mx C 1I with ny D my  1; my and my C 1 in correspondence. Along the yz plane, there exist two O atoms that directly interact with the center Zn with zn D zm ˙ c, respectively (Yang and Wang 2006). In addition to the conventional strip-like geometry, other configurations based on GNRs have also been proposed as a part of a search for favorable transport properties. Various graphene nanojunctions based on the GNRs have been studied, such as L-shaped, Z-shaped, and T-shaped GRN junctions (Areshkin et al. 2007; Chen et al. 2008; Jayasekera and Mintmire 2007). Of particular interest is the hybrid junctions which are mixtures of armchair and zigzag GNRs. Some studies have shown that the transport properties of these hybrid GNR junctions are very sensitive to the details of the junction region (Xie et al. 2009). A model calculation based on tight binding and Green’s function methods for L-shaped GNRs shows that

Modeling of Nanostructures

a

41

NB

b

NB

N1

N1

NA

NA

N2

Fig. 23 (a) A deformed LGNR (called LGNR1) where a triangle graphene flake (in the dashedline triangle) is connected to the inside corner of a standard right-angle LGNR. (b) A deformed LGNR (called LGNR2) in which a triangle graphene flake is cut from the outside corner of LGNR1 in (a). NA and NZ represent the widths of the semi-infinite AGNR and ZGNR, respectively; N1 represents the side length of added triangle flake, while N2 represents the side length of cut triangle flake (Adapted from Xie et al. 2009)

the corner geometry of the L-shaped junction has great influence on the electron transport around the Fermi energy (Xie et al. 2009). The L-shaped GNR models considered are shown in Fig. 23. The calculated conductance plots as a function of electron energy are shown in Fig. 24. Interestingly the results reveal that the corner is the decisive factor for the electronic properties of these junctions. As more carbon atoms are added to the inner side of the corner, the junction moves from being metal to a semiconductor, developing a band gap. Carbon nanoribbons have also interesting mechanical properties. For instance, Raman spectra show a remarkable dependence on ribbon width (W) (Zhou and Dong 2008). For the GNRs whose widths are larger than 25 Å, the radial breathing-like mode (RBLM) frequencies follow the 1/W rule: ! D 3086:97 

1 C 1:08 in 1=cm: W

But for the narrow p GNRs whose widths are less than 25 Å, their RBLM frequencies follow the 1= W rule: 1 0 ! D 1407:81  p 164:38 in 1=cm: W A unified fitting function has been proposed (Zhou and Dong 2008), which can be suitable for all the GNRs: ! D 1584:24 

1 1 C 351:98  p  10:0 in 1=cm: W W

42

H. Toffoli et al. a 1

G (2e2 / n)

1

0

b N1 = 2

c

1

0 1

0 –0.5

a

N2 = 0

0 1

b

N2 = 2

0 1

c

N2 = 4

0 1

d

N2 = 6

1

G (2e2 / n)

0

N1 = 0

N1 = 4

d N1 = 6

0 –0.5 0.0 E (t)

0.5

0.0 E (t)

0.5

Fig. 24 Left: The conductance as a function of electron energy for the LGNR1 in Fig. 23a with different N1 . Other parameters are NA D 11 and NZ D 7. Right: The conductance as a function of electron energy for the LGNR2 in Fig. 23b with N1 D 6 and different N2 . Other parameters are NA D 11 and NZ D 7 (Reprinted with permission from Xie et al. (2009). © (2009) by Elsevier)

Another interesting application of graphene is the antidot lattices. Antidot lattices are triangular arrangements of holes in an otherwise perfect graphene sheet. In each hexagonal unit cell of the lattice, a circular hole is introduced whose radius R can be adjusted along with the length of the hexagonal unit cell, L. Previously known applications of antidot lattices involve semiconductor lattices, while in a recent publication, the possibility of creating antidot structures on graphene using e-beam lithography has been reported (Pedersen et al. 2008b). The antidot lattice causes graphene to become a semiconductor, introducing a tuning parameter which is the dot size and shape. In addition spin qubit states can be formed in the antidot lattice by manipulating the antidot arrangement. Pedersen et al. (2008a) demonstrated that the hole shape plays a crucial role in determining the electronic structure. A triangular hole instead of a circular hole results in the appearance of a metallic state at the Fermi level. The different band structures are illustrated in Fig. 25.

Magnetic and Optical Properties Lower-dimensional systems are expected to show electronic, magnetic, and optical properties that are not observed in their bulk counterparts. In two-dimensional structures, the inherent or emergent magnetic properties depend on many factors including shape, size, and the interaction between subregions (Jensen and Pastor 2003; Zhao et al. 2014).

Modeling of Nanostructures

43

1.2

0.8

Energy [eV]

0.4

0.0

–0.4

–0.8

–1.2

Γ

M

K

Γ

M

K

Γ

Fig. 25 Band structures of a f10, 3g antidot lattice and similar structure having a triangular hole with zigzag edges. Note the dispersionless band at 0 eV in the triangular case (Reprinted with permission from Pedersen et al. (2008a). © (2008) by the American Physical Society)

GNRs are a typical example of emergent and controllable magnetic systems. Spin-polarized first-principles calculations by Gorjizadeh et al. (2008) have shown that doping GNRs with 3d transition metals result in the appearance of FM or AFM states. In particular Cr and Co provide a large magnetic moment and doping with Fe or Mn at low densities yields half-metallic ribbons. As discussed in the work by Enoki and Takai, as the dimension of the system decreases, the contribution coming from the edge states increases in proportion (Enoki and Takai 2009). The localized edge states in GNR, which are dispersionless states that appear at the Fermi level, give rise to interesting magnetic properties. From the application’s point of view, the magnetic properties of graphene nanoribbons are more interesting and studied both experimentally and theoretically. Particularly quasi-1D GNRs have magnetic properties depending on their size and symmetry. These are edge states of nanoribbons with opposite spin polarization and band gaps varying with the width of the ribbon (Topsakal et al. 2008). Using first-principles plane wave calculations within the DFT method, Topsakal et al. (2008) predict that in addition to edge states, electronic and magnetic properties of graphene nanoribbons can also be affected by defect-induced states. In particular, when H-saturated holes are introduced into the GNT, the band structure is modified dramatically altering in return the electronic and magnetic properties. Similarly, vacancies and divacancies induce metallization and magnetization in nonmagnetic semiconducting nanoribbons due to the spin polarization of local

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H. Toffoli et al.

a

a

b

1

0 EF −1

μ = 0 μB

BAND ENERGY (eV)

Relaxed

(MAGNETIZATION)

BAND ENERGY (eV)

1

(METALIZATION)

BANDS

2

AGNR (22;5) +1V

Unrelaxed

AGNR (22;5) +2V

b

BANDS

2

EF

0

−1

μ = 1 μB

2

Γ

kx

X

2

Γ

kx

X

Fig. 26 (a) Metallization of the semiconducting AGNR(22) by the formation of divacancies with repeat period of l D 5. (b) Magnetization of the nonmagnetic AGNR(22) by a defect due to the single carbon atom vacancy with the same repeat periodicity. Isosurfaces around the vacancy correspond to the difference of the total charge density of different spin directions. Solid (blue) and dashed (red) lines are for spin-up and spin-down bands; solid (black) lines are nonmagnetic bands (Reprinted with permission from Topsakal et al. (2008). © (2008) by the American Physical Society)

defect states. Antiferromagnetic ground state of semiconducting zigzag ribbons can change to ferrimagnetic state upon creation of vacancy defects. In this study, the changes in electronic properties are studied as a function of the location and the geometry of the vacancies in different types of armchair GNRs. Due to the spin polarization of localized states and their interaction with edge states, magnetization may be introduced into the GNRs. Some of the representative results of this work are displayed in Fig. 26. Not only armchair but also zigzag nanoribbons are strongly affected by defects due to single and multiple vacancies (Topsakal et al. 2008). When coupled with the magnetic edge states of the zigzag nanoribbons, the vacancy defect brings about additional changes. The magnetic state and energy band structure of these ribbons depend on the type and geometry of the defects. In a ZGNR(14;8), a single vacancy formation energy is lowered by 0.53 eV when the defect is situated at the edge rather than at the center of the ribbon. Furthermore, two defects associated with two separated vacancy and a defect associated with a relaxed divacancy exhibit similar behavior (Topsakal et al. 2008). On the other hand, Pisani et al. (2007) investigated theoretically the electronic and magnetic properties of zigzag graphene nanoribbons by performing firstprinciples calculations within DFT formalism. They predict that the electronic

Modeling of Nanostructures

45

structure of graphene ribbons with zigzag edges is unstable with respect to magnetic polarization of the edge states. As can be seen from the model calculations of Topsakal et al. (2008, 2007), the energy band gaps and magnetic states of graphene nanoribbons can be modified by defects due to single or multiple vacancies. Electronic and magnetic properties of finite length graphene nanoribbons also show similar behavior as infinite length nanoribbons. Finite length ribbons are usually referred to as graphene quantum dots (GQD). Tang et al. (2008) investigated theoretically the electronic and magnetic properties of a square graphene quantum dot. Electronic eigen-states of a GQD terminated by both zigzag and armchair edges are derived in the theoretical framework of the Dirac equation. They find that the Dirac equation can determine the eigen-energy spectrum of a GQD with good accuracy. By using the Hartree– Fock mean-field approach, they studied the size dependence of the magnetic properties of GQDs. They find that there exists a critical width between the two zigzag edges for the onset of the stable magnetic ordering. On the other hand, when such a width increases further, the magnetic ground state energy of a charge neutral GQD tends to a saturated value (Tang et al. 2008). Magnetic properties of graphene show a sensitive dependence on single-atom defects; defect concentration and packing play an important role in magnetism. Singh and Kroll (2009) investigated the magnetism in graphene due to single-atom defects by using spin-polarized density functional theory calculations. Interestingly, they find that while the magnetic moment per defect due to substitutional atoms and vacancies depends on the defect density, it is independent of defect density for adatoms. The graphene sheet with B adatoms is found to be nonmagnetic, but with C and N adatoms, it is magnetic. The adatom defects cause a distortion of the graphene sheet in their vicinity. The distortion in graphene due to C and N adatoms is significant, while the distortion due to B adatoms is very small. The vacancy and substitutional atom (B, N) defects in graphene are planar in the sense that there is in-plane displacement of C atoms near the vacancy and substitutional defects. Upon relaxation the displacement of C atoms and the formation of pentagons near the vacancy site due to Jahn–Teller distortion depend upon the density and packing geometry of vacancies (Singh and Kroll 2009). The defect models considered by Singh and Kroll are shown in Fig. 27.

Fig. 27 The model samples of (a) ideal graphene, (b) graphene with one atom vacancy, (c) graphene with one substitutional atom, and (d) graphene with one adatom defect (Adapted from Singh and Kroll 2009)

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The optical absorption coefficient is one of the most important quantities in solids and is closely related to the electronic band structure. The Kubo formula-based calculations of Zhang et al. indicate that at high frequencies, the optical spectrum of graphene is highly anisotropic (Zhang et al. 2008a). It has also been shown that the weak optical response in graphene nanoribbons can be significantly enhanced in an applied magnetic field (Liu et al. 2008). Another effect important for determining the optical properties of graphenerelated materials is spin-orbit coupling. Although spin-orbit coupling is negligible at high frequencies, it is found to have a significant enhancing effect on the optical absorption at lower frequencies as shown in a direct solution of the spin-orbit Hamiltonian in the effective mass approximation by Wright et al. (2009).

Adsorption Phenomena Adsorption of adatoms on graphene and/or graphene nanoribbons plays an important role in functionalizing graphene materials for various device applications, such as gas sensors and spin valves. Transition metal (TM) atom-decorated graphene shows different magnetic properties depending on the concentration and the species of TM atoms. There are several coverage models, such as one TM atom adsorbed on either .2  2/ or .4  4/ unit cells on only one side as well as on both sides, namely, above and below the graphene. Sevincli et al. (2008) investigated the possible adsorption sites of TM atoms on graphene and GNR. The geometrical configurations of the structures they considered include bridge (over a C–C bond), atop (on top of a C atom), and on center (over the center of a hexagon) adsorption sites for both perfect graphene and for AGNR of various width Na . In relation to another very important application, doping of transition elements was found to increase the hydrogen storage properties of materials (Shevlin and Guo 2006). Especially nickel and rhodium are widely used in the hydrogenation reaction and also in the synthesis of BN sheets. Considering the potential application of Ni and Rh nanoparticles in hydrogen storage and in catalysis, Venkataramanan et al. (2009) investigated the interaction of Ni and Rh atom on the BN sheets through first-principles calculations. They also analyzed the interaction between hydrogen molecules and the metal atoms adsorbed on the BN sheets, which might be useful to maximize the hydrogen storage capacity. Hydrogen adsorption studies over the metal-doped BN sheets shows that both Ni and Rh atoms can hold three hydrogen molecules. In the case of Ni-doped BN sheet, all three hydrogen molecules are chemically bound and are intact. In the case of Rh-doped BN sheets, the first hydrogen molecule dissociates and the remaining hydrogen molecules are bound to the metal atom. The absorption energy for the first hydrogen was found to have a larger value for the Rh-doped BN sheet, whereas for the second and third hydrogen molecules, the absorption energies were higher for the Ni-doped BN sheets. Upon addition of a fourth hydrogen molecule to the Nidoped BN sheets, the fourth hydrogen molecule moved to a distance of 3.855 Å. In the case of Rh-doped BN sheet, the Rh atom detached and acted in a similar way to

Modeling of Nanostructures

47

a cluster. Thus Ni atoms are more stable on BN sheets and have higher absorption energy compared to the Rh-doped BN sheets (Venkataramanan et al. 2009). Metallized graphene can be a potential high-capacity hydrogen storage medium. Graphene is metallized through charge donation by adsorbed Li atoms to its * bands. Each positively charged Li ion can bind up to four H2 by polarizing these molecules. The storage capacity up to the gravimetric density of 12.8 wt% is possible with a favorable average H2 binding energy of 0.21 eV (Ataca et al. 2008). Water and gas molecules adsorbed on nanoscale graphene play the role of defects which facilitate the tunability of the band gap and allow one to control the magnetic ordering of localized states at the edges (Berashevich and Chakraborty 2009). The adsorbed molecules push the ’-spin (up) and “-spin (down) states of graphene to the opposite (zigzag) edges such that the ’- and “-spin states are localized at different zigzag edges. This breaks the symmetry that results in the opening of a large gap. The efficiency of the wavefunction displacement depends strongly on the type of molecules adsorbed on graphene (Berashevich and Chakraborty 2009). The influence of adsorption of water on the electronic and magnetic properties of graphene is based on calculation of the spin-polarized density functional theory, and the results of the calculations are depicted in Fig. 28.

a

spin density

α(HOMO–4)state

β(HOMO–4)state

spin density

α(HOMO)state

β(HOMO)state

b

Fig. 28 Plots of the spin density and electron density for the ’-spin (up) and “-spin (down) states in nanoscale graphene. For the spin density, different colors indicate the ’- and “-states, while for the electron density distribution (’- and “-states), the different colors correspond to different signs of the molecular orbital lobes. (a) The electron density distribution in pure graphene. (b) The electron density distribution for graphene with adsorbed water (Reprinted with permission from Berashevich and Chakraborty (2009). © (2009) by the American Physical Society)

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Quantum Cluster Theory for the Polarizable Continuum Model (PCM) Roberto Cammi and Jacopo Tomasi

Contents Introduction: Quantum Cluster Theory and Solvation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coupled-Cluster Theory for the Polarizable Continuum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . The PCM-CC Reference State and the PCM-CC Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . The PCM Hartree-Fock Reference State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coupled-Cluster PCM Free-Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The PTDE and the PTE Coupled-Cluster Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coupled-Cluster PTDE Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coupled-Cluster PTE Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Explicit PCM-CCSD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCM-CCSD Analytical Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCM-CCSD Analytical Gradients with Relaxed MO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Equation-of-Motion Coupled-Cluster Theory for the Polarizable Continuum Model . . . . . The Excited-State PCM-CC Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCM-EOM-CCSD Analytical Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The PCM-EOM-CCSD Equations for the Z Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EOM ..... The Analytical Gradients of the PCM-EOM-CCSD Free-Energy Functional GK The Response Functions Theory for Molecular Solutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Variational Time-Dependent Theory for the Polarizable Continuum Model . . . . . . . . . . . . The Response Functions of Molecular Solutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coupled-Cluster Response Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: The Solute-Solvent PCM Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 7 8 8 10 10 11 12 13 15 17 18 19 22 22 23 25 25 28 29 34 35 36

R. Cammi () Dipartimento di Chimica, Università di Parma, Parma, Italy e-mail: [email protected] J. Tomasi Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Pisa, Italy e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_28-2

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Abstract

Recent extensions of the coupled-cluster (CC) theory for molecular solute described within the polarizable continuum model (PCM) were summarized. The advances covered in this review regard (i) the analytical gradients for the PCMCC theory at the single and double excitation level, (ii) the analytical gradients for the PCM-EOM-CC theory at the single and double excitation level for the descriptions of the excited-state properties of molecular solute, and (iii) the coupled-cluster theory for the linear and quadratic molecular response functions of molecular solutes. These computational advances can be profitably used to study molecular processes in condensed phase, where both the accuracy of the QM descriptions and the influence of the environment play a critical role.

Introduction: Quantum Cluster Theory and Solvation In this paper, we present some aspects of our recent work on the extension of computing methods connected to the polarizable continuum model (PCM). Our persevering attention to PCM can be in some sense immediately justified, having us proposed this method years ago and having continued to develop it in the years. In another sense, a justification is required and may be expressed under the form of an answer to a question: why to pass to implement PCM with sophisticated procedures as those given by the coupled-cluster (CC) theory Coupled-cluster is the top level at which quantum mechanical (QM) calculations on molecules can be nowadays performed (Bartlett and Musiał 2007); PCM is characterized by a drastic simplification of the material system by replacing the degrees of freedom of the solvent molecules with a two body integral operator. Apparently the two methods are operating at very different scales of accuracy, but actually they can be profitably coupled. To show it, we pass to examine some features of PCM. This rapid examination of some technical aspects of the methods used in PCM will be also useful as an introductory section to our exposition. The methodology on which PCM is based is that of an effective Hamiltonian expressed in terms of interaction integral operators depending on the solution of a nonlinear Schrödinger equation. The method is powerful and versatile, allowing applications of very different nature. A good number of such applications have been elaborated in the years by us and by other groups, and an undetermined, but large, number of new applications looks promising. Many among them are going beyond the field of dilute solutions; others address the description of solvent effects under special conditions, a few regard higher levels of the QM theory. We are here considering an application of this last type. The features of PCM we have to examine are so limited to the developments of the QM ab initio version of the procedures (other versions regard semiempirical and semiclassical descriptions). PCM was born as a method to describe electrostatic solvation effects on the electronic distribution of a molecule (Miertuš et al. 1981). The method was conceived for ab initio QM calculations, using a technique to solve the electrostatic problem never used before in molecular sciences, a technique based on the use of

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

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the concept of apparent charges. The integral interaction operator is thus reduced to the Coulomb interaction with a charge distribution spread on the surface of the solute and represented by a set of point charges. Solute-solvent interactions are not limited to the electrostatic one. The nonelectrostatic terms have been introduced in PCM with the aid of a phenomenological (and effective) Hamiltonian (Bonaccorsi et al. 1982). Each term of the interaction has a different physical meaning but a similar formal expression, given by an integral operator with kernel QX .r; r 0 /, where X stays for one of the contribution to the interaction energy: electrostatic, repulsion, dispersion, cavity formation. The effective Hamiltonian mixes quantum and classical components. In particular, the medium which appears in the kernel of the QX .r; r 0 / operators has to be precisely defined, in agreement with the physical conditions established for the experiments the computational model has to describe. The formal elaboration of the physical components of the model has to be followed by the elaboration of the corresponding mathematical procedures. In this step, several “concepts” of quantum and classical origin have to be introduced: examples are the molecular surface and the notion of charging process. We shall use here the concept of charging process as a guide in our exposition. Charging processes are of general use in physics, but rarely mentioned in molecular quantum mechanics; more explicit is its use in statistical thermodynamics to which our model is connected. The basic task in molecular QM, that of computing energy and charge distribution of an isolated molecule at a clamped nuclear geometry, corresponds to a charging process. The analogous process in solution with an effective continuum Hamiltonian presents however important differences. The medium defining the liquid portion of the material systems is changed (polarized) by charging processes. A portion of the energy gained in a reversible charging process is so lost. The remaining portion of the energy is the free energy of the system, measured from a reference state composed by the unperturbed medium and by the appropriate assembly of noninteracting nuclei and electrons giving origin to the solute molecule. This process has a classical counterpart and can be derived on QM grounds without invoking physical arguments (a derivation is based on the variational theorem with an effective Hamiltonian linearly dependent on the solution) (Tomasi and Persico 1994). The picture of the charging process can be used for the phenomenological partition we have introduced a few sentences above. This is a practiced used in statistical thermodynamics. The contributions can be described in an ordered sequence of charging processes; at the end of each process, the medium is modified (polarized), and the so polarized medium is the starting point for the following charging process. The selected order of charging influences the result, because couplings in this procedure are neglected and they change according the ordering, but with an appropriate selection of the sequence, the results of the solvation energy are well reproduced with a computational cost decidedly lower than that of the full calculation (Tomasi et al. 2003).

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The option of decoupling the non-electrostatic part from the main calculation has also another practical advantage that we shall exploit in the following of this paper, namely, of provisionally discard the non-electrostatic terms during the elaboration of a new electrostatic procedure. The non-electrostatic contributions to the energy are numerically large, but feebly coupled with the electrostatic one. We stress, however, that decoupling and simplified descriptions of single components are not strictly necessary: ab initio procedures computing everything in a unique calculation exists (Curutchet et al. 2006). We have so far examined some points of the original PCM procedure. Other elements of interest for the present paper, drawn from the numerous procedures implemented and tested later (Tomasi et al. 2005), will be here considered. These points regard the improvements in the description of the solute’s electronic wave function and the description of excited states, including nonequilibrium solvent effects. Improvements in the quantum description of a solute mostly regard the electron correlation effects. There are, in principle, no restrictions to introduce in the formal framework of our effective Hamiltonian any version of QM methods describing electron correlation. Calculations including a description of correlation at a low level are nowadays routinely performed, but for calculations at high levels, the convenience of keeping low computational cot becomes a necessity. There are features in the effective Hamiltonians for continuum solvation promising substantial reductions in some specific versions of post Hartree-Fock calculations. In the present paper, we shall make use of a decoupling of the QM charging process elaborated years ago to better study the behavior of the many-body perturbation scheme in continuum solution and later adopted to simplify calculations in the series of MPn calculations (Olivares del Valle and Tomasi 1991). The perturbation series may be expanded on the electronic configurations of the molecular solute’s electron distribution computed in vacuo and then used for the calculation of solvent reaction potential and solvation-free energy; this approximation is called PTD, being a version of the perturbation theory based on the electronic density. Another scheme is based on the expansion of the molecular solute’s electron distribution computed with PCM at the Hartree-Fock level and used to compute solvent reaction potential and energy; this approximation is called PTE. The comparison of the results obtained with the two schemes gave insights on the relative importance of the decoupled electron correlation and solvation potential effects on the solute. A more complete procedure couples the two effects into a single iterative computational scheme, with the correlated electronic density used to make the solvent field self-consistent; the procedure is called PTED. We shall examine in a following section the PTE and PTDE version for the coupled-cluster method. In this paper, we shall also consider the first basic aspects of the dynamical, i.e., nonequilibrium, behavior of liquid media. The subject embraces a very large variety of phenomena, requiring different formulations of the continuum models. We cite here an aspect, relatively simple, related to a phenomenon occurring in a span of time relatively short with respect to the characteristic relaxation times (CRT) of the solvent: the vertical electronic transitions in solutes (Kim and Hynes 1990; Basilevsky and Chudinov 1990; Marcus 1992).

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

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Electronic transitions are very fast, and the solvent molecules have no time to rearrange themselves during this phenomenon, saving for the electronic component of the solvent polarization which has a CRT of the same order of magnitude of the vertical electronic transition. This effect means that the energy of the vertical transition (a free energy, we remark) has a component due to the solvent interaction limited to changes in the fast electronic polarization. The effects of the electronic transition continue in the time, but these do not regard the vertical process and have to be described with a different formulation of the continuum method. There is to remark, for the sake of completeness, that there are two theoretical models, respectively, named Marcus and Pekar partitions, which describe this simple process, differing in the intermediate stages of the elaboration of the problem, but arriving at the same result when the correct expression for the nonequilibrium free-energy functional is used. The PCM versions of the two models have been presented in papers Aguilar et al. (1993) and Cammi and Tomasi (1995b). For a comment, see also Aguilar (2001). A detailed comparison of the formulas of two models has been reported in section 5.1 of Tomasi et al. (2005). The Pekar partition will be used in the following sections of this paper; the Marcus partition is in this case less convenient. Quantum models in condensed systems often present alternative computational routes merging into a unique form when the effect of the medium is neglected. In coding the formulas for problems, a great attention must be paid to the correct inclusion of the corrections to satisfy the free-energy requirements. This last remark also holds for the topic we shall consider now: the calculation of the electronic excitation energies. Two approaches are generally used to get the excitation energies: a version based on the CI approach or a version based on the response theory. Both approaches have in condensed media problems not present in the corresponding calculations in vacuo. The roots of a CI calculation give in vacuo the energies of the various states; in solution these values have to be corrected for the effect of the solvent electronic polarization. Also the corresponding description of the so defined electronic states has to be modified (with an iterative procedure), because mutual polarization effects occur at the same time. This operation of polarization has to be separately performed for each electronic state, being the effective Hamiltonian of each excited state different from the others. The method is said “state specific” (SS), and it constitutes the main way for the study of excited-state potential surfaces. The second approach is based on a Hamiltonian with explicit time dependence, provided with an opportune form of the of the time-dependent variation principle (the Frenkel principle is generally used Cammi et al. 1996). From this starting point, linear and nonlinear response functions are derived. The linear response functions (LR) nowadays are amply used both in gas phase and in solution for the characterization of excited states and of molecular properties. The complete equivalence between SS and LR results for the excitation energies was universally accepted, but recently it was shown (Kongsted et al. 2002; Cammi et al. 2005; Corni et al. 2005) that this equivalence is valid only in vacuo and that LR results can be seriously in error in solution. A computational strategy to reduce this

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error was proposed (Caricato et al. 2006; Improta et al. 2006), efficient enough to allow the exploitation of the less expensive LR scheme. It is convenient to mention here another aspect of the continuum solvation theory that will be used in the following pages: the calculation of analytical derivatives. It is essential for computational methods addressing chemical problems the availability of analytic expressions for derivatives. The variety of derivatives appearing in molecular calculations is quite large, but we limit ourselves to consider derivatives of the energy with respect to nuclear coordinates. The derivatives in solution have important additional features with respect to the derivatives in vacuo. In solution it is compelling to compute derivatives taking into account partial derivation with respect to cavity parameters. An accurate description of the cavity shape is compulsory: examples of bad results due to inaccurate descriptions of the cavity are present in every field of application – from the determination of the molecular structure to the equilibrium among conformers to spectroscopic properties. Spherical cavities have to be used only as a first-order approximation; more subtle are the errors with cavities badly defined or treated in a simplified way during the calculations. The problems of accurate handing of the derivatives of the cavity parameters become more complex and computing expensive with the increase of the order of the derivative and of the computational level. The problem of the numerical stability of the second derivatives obliged us in the past to change the definition of the cavity elements. The analytical expressions of the first and second derivatives with respect to nuclear coordinates have been elaborated for the present version, but we shall use in the second part of this paper the first derivatives only. The methodological and computational bases of PCM have been elaborated more extensively than shown in this exposition, but what said would be sufficient to show the care necessary to develop a reliable version of CC for PCM. Nothing has been said about the numerical accuracy of PCM results. The definition of the accuracy for a method allowing numerous different applications is a problem; it is almost compulsory to separately consider different classes of phenomena. The largest number of applications of solvent methods regards the solvation energy. Actually there is no need for this property of an accurate description of the electronic distribution, and in fact many methods are based on semiclassical approximations. There are however difficult cases in which there is the need of an accurate description of geometry and electronic charge distribution: in these cases, ab initio methods of sufficient accuracy are necessary. There are several methods satisfying this criterion, and all perform well. A recent comparison (Cramer and Truhlar 2008; Klamt et al. 2009) gives some indications, even if the really difficult cases were not much abundant in these comparisons. Another point to remark is that the best fitting with experimental date was obtained using “calibrated parameters.” We have not talked in the preceding pages of empirical calibration of parameters over experimental data, because we have not used this procedure in the elaboration of PCM methods (and we are not using it in this paper). Calibration of parameters is necessary for explicitly declared semiempirical methods, but it is risky for methods aspiring to generality because ad hoc parameterizations are limited to one (or

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

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few) property and performed over a single specific definition of the computational parameters (QM method, basis set, etc.). Anyway, versions of PCM parameterized for solvation energies exist, and they behave very well. More interesting is the consideration of accuracy for more complex problems of spectroscopic type, which actually constitute the direct motivation of this work. PCM is among the best methods in this field, with features not present on other methods, but the need of having description of the electronic structure of the chromophore more accurate than those currently used in these studies is in our opinion quite manifest. There has been a preceding attempt, several years ago, of using a CC wave function modified by continuum solvent effects, just for the calculation of molecular properties. The proposal was made by Mikkelsen and coworkers (Christiansen and Mikkelsen 1999) using a spherical cavity with a multipolar expansion of the solvent reaction potential. This attempt was bold and interesting, but the continuum model was forcibly simple, with the perspective of giving results, when applied to more complex chromophores, in which a gain in the description of the solute is accompanied by a poorer description of the solvent effect. Time was not yet ripe for efficient applications of the approach to systems of real spectroscopic interest. This is the answer to the question put at the beginning of this introduction. There is the need for some specific scopes of molecular calculations more accurate than those currently available, and the more convenient way to arrive at this level of accuracy apparently is given by an accurate (in the formulation and in the execution) ab initio continuum code coupled to an efficient procedure of the coupled-cluster family. The first principles of molecular dynamics at a high level of quantum description are not feasible. The use of computer simulations at a lower level accompanied by a final calculation at the CC level is a risky procedure, requiring extensive validation. In this introduction, we have not examined the details of the coupled-cluster procedures because we have used in the implementation standard aspects of it, on which particularly comments were not necessary (Bartlett and Musiał 2007). Details and comments will be necessary when an extension of the present study shall be published.

The Coupled-Cluster Theory for the Polarizable Continuum Model The coupled-cluster (CC) theory for the PCM model exploits the formalism of the coupled-cluster energy functional. As for the case of isolated molecules, this formalism allows to recast the coupled-cluster method for PCM into a pseudovariational problem. However, the physical status of this energy functional has an important difference with respect to isolated molecules as it corresponds to the free energy of the whole solute-solvent system (Christiansen and Mikkelsen 1999; Cammi 2009), as we have already said in the Introduction.

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In this section, we review both the formal derivation of the PCM-CC equations and corresponding explicit equations at the single and double excitation level (PCMCCSD).

The PCM-CC Reference State and the PCM-CC Energy Functional The coupled-cluster wave function of the molecular solute is written as the exponential ansatz: jC C >D e T jHF >

(1)

where jHF > is the reference state given by the Hartree-Fock ground state of the solvated molecule and the cluster operator T is given as a sum of all possible excitation operators over the N electrons: T D T1 C T2 C : : : C TN 1 X ab:::   Tn D tij ::: aa ai ab aj : : : .nŠ/2

(2)

ai bj :::

weighted by the amplitude tia ; tijab , etc. The excitation operators are here represented 



as products of second-quantization electron creation (ai ; ab ) and annihilation operators (ai ; ab ). As usual, indexes .i; j; k; : : :/ and .a; b; c; : : :/ denote, respectively, occupied and vacant spin orbitals MO, while .p; q; r; : : :/ denote general spin orbitals.

The PCM Hartree-Fock Reference State The Hartree-Fock reference state jHF > is obtained by the stationarity condition of PCM Hartree-Fock free-energy functional GHF (Cammi and Tomasi 1995a): GHF D< HF jH o C 1=2 < HF jQjHF > VjHF >

(3)

b > b V is where H o is the Hamiltonian of the isolated solute and < HF jQjHF b the solute-solvent interaction term, being < HF jQjHF > the solvent polarization charges computed with the Hartree-Fock state reference state jHF > and b V the molecular electrostatic potential operator of the solute (see Appendix A); the onehalf factor in front of the solute-solvent potential term is due the energy loss during the process of polarization of the dielectric representing the solvent, as already said in the introduction. In an N -electron system with spin orbitals expanded over a set of atomic orbitals (AO), f ;  ; : : :g, G HF may be written as

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

G HF D

X

HF P .h C j / C



9

 1 X HF HF  P P hjji C B; C VQNN 2 

(4) where h are the matrix elements, in the AO basis, of the one-electron core operator, hjji are the antisymmetrized combination of regular two-electron HF repulsion integrals (ERIs), and P indicates the elements of the Hartree-Fock density matrix. The one-electron matrix elements j and the pseudo two-electron integrals B; describe the solute-solvent interactions within the PCM-Fock operator. The one-particle AO integrals j and the pseudo two-electrons integrals B; represent, respectively, the interactions with the nuclear and with the electronic components of the ASC charges. The solvent integrals j , B; may be expressed in the following form j D v  qN uc

(5)

B; D v  q

(6)

where v is a vector collecting the AO integrals of the electrostatic potential operator evaluated at the positions sk of the ASC charges, qN uc is a vector collecting the ASC charges produced by the nuclear charge distribution, and q is a vector collecting the apparent charges produced by the elementary charge distribution  .r/ .r/. The apparent surface charges may be obtained in terms of the electrostatic potential of the solute charge distributions by using the integral equation formalism of the PCM (Cancès et al. 1997). The last term of Eq. (4), VQNN represents the interaction between the nuclei and the nuclear component of the apparent surface charges, VQNN D 1=2vN uc  qNuc . Requiring that GHF be stationary with respect to the variation of MO expansion coefficients, we obtain the PCM-HF equations: X

P CM .f  p S /cp D 0

(7)

 P CM where S and f are, respectively, the matrix elements of the overlap matrix and of the PCM-Fock matrix, in the AO basis, and p and cp are, respectively, the orbital energy and the expansion coefficients of the p MO. The PCM-Fock matrix elements are given by: P CM f D .h C j / C G .PHF / C X .PHF /

(8)

where G .PHF / are the matrix elements of the effective Coulomb-exchange twoelectron operator, while X .PHF / are the matrix elements of the solvent operator representing an effective Coulomb two-electron mediated by the polarization of the solvent.

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The Coupled-Cluster PCM Free-Energy Functional To define the PCM-CC free-energy functional, we introduce the bra coupled-cluster wave function: < CQC j D< HF j.1 C ƒ/e T

(9)

where ƒ is a de-excitation operator, ƒ D ƒ1 C ƒ2 C : : : ƒn D

X ij k:::  1   abc::: ai aa aj ab ak ac : : : 2 .nŠ/

(10)

ij kabc:::

ij

ia ; ab ; : : : are the de-excitation amplitudes. In terms of the bra and ket wave functions, the coupled-cluster free-energy functional (Christiansen and Mikkelsen 1999; Cammi 2009) may be written as GC C .ƒ; T / D< HF j.1 C ƒ/e T H0 e T jHF > 1N N C Q.T; ƒ/  V.T; ƒ/ 2

(11)

where < HF j.1 C ƒ/e T H0 e T jHF > is the energy functional for the isolated N and Q N are, respectively, the coupled-cluster expectation value of molecules and V b the PCM operators Q and b V (Bartlett and Musiał 2007): N Q.T; ƒ/ D< HF j.1 C ƒ/e T Qe T jHF >

(12)

N V.T; ƒ/ D< HF j.1 C ƒ/e T Ve T jHF >

(13)

We note that arguments of the expectation values (12) and (13) denote an explicit functional dependence on the T and ƒ amplitudes. These amplitudes, which define completely the coupled-cluster wave function of the molecular solute, can be determined imposing the stationary conditions on the energy functional GC C .ƒ; T /.

The PTDE and the PTE Coupled-Cluster Schemes To develop a coupled-cluster theory within the PTE and PTDE schemes described in the Introduction, the free-energy functional GC C .ƒ; T / of Eq. (11) is partitioned into a Hartree-Fock component GHF and into a coupled-cluster component GC C .ƒ; T /: GC C .ƒ; T / D GC C .ƒ; T / C GHF

(14)

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

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where GHF is the HF contribution, defined in Eq. (3), and GC C .ƒT / denotes the correlative CC contribution:   1N T H .0/N C QN .ƒ; T /  VN e T jHF > GC C .ƒ; T / D< HF j.1 C ƒ/e 2 (15) The partition of Eq. (14) is based on the so-called normal ordered operators N N .ƒ; T /  VN is the component H .0/N ; VN and ; QN (Cammi 2009). Specifically, Q of the solvent reaction potential due to the correlation CC electronic density, and H .0/N is the normal ordered form of Hamiltonian of the solute in presence of the N HF . frozen Hartree-Fock reaction field Q

The Coupled-Cluster PTDE Scheme Within the PTDE scheme, the coupled-cluster electronic density is used to make the solvent reaction field self-consistent. This corresponds to make the free-energy functional GC C .ƒ; T / stationary with respect to the ƒ and the T amplitudes. The stationarity with respect to the ƒ amplitudes gives the equation for the selfconsistent T amplitudes: @GC C ij ::: @ab:::

˚  D< HF j i  aj  b : : : e T H .ƒ; T /N e T jHF >D 0

(16)

In turn, the stationarity with respect to the T amplitudes gives the equation for the self-consistent ƒ parameters: ˚    T @GC C T D< HF j.1 C ƒ/e ŒH .ƒ; T / ; a i a j : : :

e jHF >D 0 (17) N @tijab::: ::: In Eqs. (16) and (17), H .ƒ; T /N is the Hamiltonian of the molecular solute in the PTDE scheme: N N .ƒ; T /  VN H .ƒ; T /N D H .0/N C Q

(18)

where H .0/N is the normal ordered form of Hamiltonian of the solute in presence N N .ƒ; T / b of the frozen HF reaction field and Q VN is the coupled-cluster component of the solvent reaction potential, as described above. The equation for the T amplitudes has a clear physical meaning: it corresponds to projection in the manifold spanned by all the orthonormal excitations to jHF > of the coupled-cluster Schrodinger equation for the molecular solute: N N .ƒ; T /  VN e T jHF >D EC C e T jHF > ŒH .0/N C Q where EC C is given by

(19)

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EC C D< HF j.1 C ƒ/e T H .ƒ; T /N e T jHF >

(20)

The coupled-cluster eigenvalue EC C differs from the free-energy functional GC C .ƒ; T / by the work spent during the charging process of the coupled-cluster reaction field, i.e., N N .ƒ; T /  V N N .ƒ; T / GC C D EC C  12 Q

(21)

N N .ƒ; T / is the coupled-cluster expectation value of the electrostatic where V potential. Being the Hamiltonian H .ƒ; T /N dependent on both T and ƒ parameters, the corresponding PTDE equations (16) and (17) are coupled, and they must be solved in a iterative and self-consistent way.

The Coupled-Cluster PTE Scheme In the PTE coupled-cluster approximation, the electronic distribution is computed with PCM at the Hartree-Fock level. This approximation is easily obtained from the above PTDE equations by neglecting all the contributions related to the coupledN N .ƒ; T /  VN . Specifically, the PTE Hamiltonian is the cluster solvent operator Q Hamiltonian H .0/N , which contains the fixed reaction potential of the solute at the HF level, and the free-energy functional is given by GCP CTE D ECPCTE D< HF j.1 C ƒ/e T H .0/N e T jHF >

(22)

The stationary condition of GCP CTE leads to the following T equations: @GCP CTE ij ::: @ab:::

˚  D< HF j i  aj  b : : : e T H .0/N e T jHF >D 0

(23)

and of the ƒ equations ˚  @GC C D< HF j.1 C ƒ/e T H .0/N ; a i a j : : : e T jHF >D 0 ab::: @tij :::

(24)

Within the PTE approximation, the PCM coupled-cluster equations (16) and (17) have the same structure of the coupled-cluster equations of the isolated molecules and the T , and ƒ amplitudes equations are not more coupled. As a consequence, we don’t need to solve the ƒ equations to compute the energy functional GCP CTE . In the next section, we describe how the basic PTDE and PTE equations lead to explicit expressions suitable for the implementation in quantum chemistry computational codes.

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

13

The Explicit PCM-CCSD Equations The explicit form of the PCM-CC equations (16), (17), (18), (19), (20), (21), (22), (23) and (24) has been recently presented (Cammi 2009) for the coupled-cluster single and double (CCSD) level, which restricts the T and ƒ operators to excitation operators T1 ,T2 and to the de-excitation operators ƒ1 , ƒ2 , respectively. The key entities are the similarity-transformed PTE Hamiltonian H.0/ D e T H .0/N e T and the similarity-transformed molecular electrostatic potential operators V D e T b VN e T . Both operators can be expressed as a terminated BackerCampbell-Hausdorff expansions. Specifically, H.0/ terminates at the fourfold commutator, because it has at most two-particle interactions:

1

1 1

1 1

1 H.0/ D H .0/N C ŒH .0/N ; T C ŒŒH .0/N ; T T

2 1 1 C ŒŒŒH .0/N ; T T T C ŒŒŒŒH .0/N ; T ; T T T D .H .0/N exp.T //C 3Š 4Š (25) while V terminates at the twofold commutator as it has at most one-particle interactions: 1 VN ; T T D .VN exp.T //C V Db VN C Œb VN ; T C ŒŒb 2

(26)

In Eqs. (25) and (26), the subscript c indicates that only the so-called “connected” terms are considered, when the operators are expressed in terms of normal ordered product of second-quantization creation and annihilation operators (Bartlett and Musiał 2007). The similarly transformed coupled-cluster PTDE-PCM Hamiltonian can then be written in the connected form H.ƒ; T / D .H .0/N exp.T //C C .VN exp.T //C

(27)

and the explicit PCM-CC equations may be obtained by substitution into Eqs. (16), (17), (18), (19), and (20). The corresponding integrals may be obtained by an algebraic approach, which exploits the Wick theorem (Bartlett and Musiał 2007) for normal product operators with H .T; ƒ/N written in the second quantized form  X X P CM N N .ƒ; T /vpq fp  qgC 1 CQ < pqjjrs > fp  q  srg fp;q 4 p;q p;q;r;s (28) The explicit equation for the correlation energy EC C is obtained as

H .T; ƒ/N D

EC C



1 O D < HF jH .T; ƒ/N T2 C T1 C T12 jHF > 2

(29)

14

R. Cammi and J. Tomasi

with the corresponding algebraic from given by N N .ƒ; T /  vi a tia C 1 < ij jjab > ijab EC C D Q 4

(30)

where ijab D tijab C tia tjb  tib tja . The explicit expression for the T1 ,T2 equations are given by

1 2 1 3 O jH .T; ƒ/N 1 C T2 C T1 C T1 T2 C T1 C T1 jHF >c 0 2 3Š

1 D 2Š 3Š 1 2 1 2 O 0 Dc 2 3Š 4Š Dc

(33)

and T a O 0 D< HF jHO .T; ƒ/N e T jab ij >c C < HF j.ƒ1 C ƒ2 /.H .T; ƒ/N e /c ji >c X C < HF jHO .T; ƒ/N e T jck > cDa;bIkDi;j

The algebraic form for the T1 ,T2 and ƒ1 ; ƒ2 equations is reported elsewhere (Cammi 2009). The above PTDE PCM-CC explicit equations can be easily implemented in any computational quantum code. However, the PTDE coupled-cluster procedure

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

15

is different with respect to a calculation on isolate molecules, as anticipate above. N N .ƒ; T /vme enters, as an The electron correlation component of the reaction field Q effective one-body contribution, both in the T and in the ƒ equations, which must be solved simultaneously by using a recursive procedure. The coupling between the T; ƒ equations has obvious computational consequences, with an increase of computational cost with respect to a coupled-cluster calculation for isolated molecules. In fact, in this case, it is not necessary to solve the ƒ equation to obtain the CC energy and wave function. The explicit equations for the PCM-CC-PTE approximation can be easily obtained from the PTDE equations by neglecting all terms involving the coupledcluster components of the solvent reaction potential. This makes the computational cost of the PTE approximation similar to a coupled-cluster calculation for isolated molecules.

PCM-CCSD Analytical Gradients The variational properties of the PCM-CC energy functional can be exploited to obtain analytical expressions of the energy derivative. For reasons of length, in this section, the exposition will be limited to the analytical first derivatives only. The interested reader may found the analytical second derivative expressions in Cammi (2009). The differentiation with respect to a perturbational parameter ˛, of the stationary PCM-CC functional GC C , leads to an expression which avoids the first derivative of the T; ƒ amplitudes. The first derivative @GC C =@˛ D GC˛ C can be expressed in the following form: GC˛ C D < HF j.1 C ƒ/je T ŒH .0/˛N e T jHF >

N N .ƒ; T / C 12 < HF j.1 C ƒ/je T Q˛N e T jHF > V 1 N T ˛ T C QN .ƒ; T / < HF j.1 C ƒ/je V e jHF > 2

(35)

N

where the upperscript ˛ of the various normal ordered operators denotes the total derivative of their second-quantization form. The differentiated Hamiltonian H .0/˛N in the second quantized form is given by H .0/˛N D

X pq

˚   1X ˚  P CM;˛ p q C fpq < pqjjrs >˛ p  q  sr 4 pqrs

(36)

P CM;˛ where fp;q and < pqjjrs >˛ denote, respectively, the total first derivatives of the PCM-Fock matrix elements and of the usual two-electron integrals, both in the MO basis. P CM;˛ The derivate matrix elements fp;q may be written in terms of the derivatives of the constituting MO integrals

16

R. Cammi and J. Tomasi

P CM;˛ ˛ fpq D h˛pq C jpq C

occ X ˛ .< pj jjqj >˛ CBjj;pq /

(37)

j ˛ ˛ where the derivatives of the solvent integrals jpq and Bjj;pq are given by ˛ jpq D v˛pq  qN uc C vpq  q˛N uc

(38)

˛ Bpq;rs D v˛  qrs C v  q˛rs

(39)

where q˛p;q v˛p;q are, respectively, the differentiated apparent charge integrals and the differentiated electrostatic potential integrals (Cammi and Tomasi 1994). The differentiated operators Q˛N and V˛N of Eq. (35) are given by: Q˛N D

X

˚  q˛pq p  q

(40)

˚  v˛pq p  q :

(41)

pq

V˛N D

X pq

Introducing Eqs. (36), (37), (38), (39), (40), and (41), the analytical gradients GC˛ C can be written in the following explicit form: GC˛ C D

X

C C resp

fabP CM;˛ ab

C

ij

ab

C

X ai

X

C C resp

faiP CM;˛ ai

C

C C resp

fijP CM;˛ ij

X

(42)

C C resp

fi Pa CM;˛ i a

ia

1X ˛ C C resp C B C C resp rs 2 pqrs pq;rs pq C

1X < pqjjrs >˛ rspq 4 pqrs

where rspq are elements of the effective two-particle density matrix (Bartlett and Musiał 2007). Equation (42) is the most general form of GC˛ C from which we can obtain explicit computational expressions. To proceed further, as in the case of isolated molecules, we have to consider two alternative forms of the PCM-CCSD analytical derivatives: the so-called nonrelaxed MO form, which neglects the orbital relaxation effects, and the so-called relaxed MO form, which includes these effects (Gauss 1999). Which of the two forms must be used is a controversial issue in the CC property calculations of isolated molecules. The use of unrelaxed derivatives has been advocated by Koch and Jörgensen (1990) for the calculation of electrical properties. On the other, the use of the orbital relaxation effect is mandatory in all

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

17

cases where perturbation-dependent basis functions are employed. For geometrical derivatives, only the MO-relaxed derivatives provide a correct description of the potential energy surface. In the following subsection, we present the orbital relaxed form, the gradients GC˛ C . The interested reader may found the corresponding formulation for the unrelaxed MO gradients in Cammi (2009).

PCM-CCSD Analytical Gradients with Relaxed MO When the MO are allowed to relax under the perturbation, the occupied-virtual P CM;˛ block of the derivative of the PCM-Fock matrix, fa;i , vanishes by virtue of the Brillouin’s theorem. Therefore, the contribution to the gradients GC˛ C of Eq. (42) due to the derivative PCM-Fock matrix consists only of an occupied-occupied and virtual-virtual blocks: X P CM;˛ C C resp X P CM;˛ C C resp GC˛ C D fab ab C fij ij ij

ab

1X ˛ C C resp C C resp C .q  vrs C qpq  v˛rs / pq rs 2 pqrs pq C

1X < pqjjrs >˛ rspq 4 pqrs

(43)

For the evaluation of the PCM-CC gradient, the first derivatives of the PCMFock matrix and of the other one- and two-electron MO integrals are needed. These derivatives lead to terms involving the skeleton derivatives of the MO integrals, i.e., integrals which involve the derivative of the corresponding AO integrals and also lead to terms involving the derivative of the MO coefficients. Then, the gradients GC˛ C may be written as: GC˛ C D

X

C C resp

ab

fabP CM;Œ˛ C

X

C C resp

ij

fijP CM;Œ˛

ij

ab

h

i 1 X C C resp Œ˛ N Œ˛

N N  vrs qrs  VN C Q rs 2 rs X X 0 ˛ C

pq;rs < pqjjrs >Œ˛ C Ipq Upq

C

pqrs

(44)

pq

a are where the upperscript Œ˛ denotes skeleton derivatives of the MO integrals; Umi 0 the derivatives of the MO coefficients cp , in the MO basis; and Ipq are auxiliary matrix elements whose expressions are given in Cammi (2009). a The explicit evaluation of the Umi derivatives may be avoided by using the orthonormality constraints of the MO and by using the interchange (Z-vector)

18

R. Cammi and J. Tomasi

method of Handy and Schaefer properly extended to the PCM framework (Cammi et al. 1999). The resulting PCM-CCSD gradients may be written as GC˛ C D

X

C C resp

ab

fabP CM;Œ˛ C

X

C C resp

ij

fijP CM;Œ˛ C

X

ij

ab

MOresp

ai

faiP CM;Œ˛

ij

h

i

X 1 X C C resp Œ˛ N Œ˛

N N  vrs qrs  VN C Q C rs

pq;rs < pqjjrs >Œ˛

2 rs pqrs X X X Œ˛

Œ˛

Œ˛

C Iij0 Sij C Iab Sab  2 Ii0a Sai (45) C

ij

ab

ai

MOresp

where the matrix elements ai are obtained as solution of the PCM-Z vector MOresp P CM 0 equation and Ii a D Ii a  ai fi i and Iij0 are given by Iij0 D Iij 

X

 MOresp < ei jjmj > C < i mjjje > C2Bem;ij em

em

Finally, the expression of the PCM-CCSD gradients can be reverted from the MO to the AO representation, for an efficient computational implementation. The corresponding AO form is GC˛ C D

X

  X C C MO ˛ 0 ˛ h˛ C j C  I S



C C

(46)



X

1 C C resp C C resp ˛ C C MO HF .  P C   /B 2  X

0

; < jj >˛

 0 C C MO HF where ; D ; C  P . Equation (46) refers to the PCM-CCSD analytical gradients within the PTDE scheme, as it contains terms involving coupled-cluster contribution to the solvent reaction field. By neglecting these contributions, we obtain the corresponding analytical gradients within the PTE-approximated scheme.

The Equation-of-Motion Coupled-Cluster Theory for the Polarizable Continuum Model In the previous sections, we have shown the coupled-cluster theory to the description of the ground state properties of molecular solutes. In this section, we present the coupled-cluster theory for the description of the excited states of solvated molecules.

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

19

We have already anticipated that the polarizable continuum model supports two approaches for the description of the excited states of molecular solutes, a version based on the CI approach (state-specific SS) and a version based on the response theory, and that these approaches have in condensed media problems not present in the corresponding calculations in vacuo. Each approach has advantages and disadvantages. The LR approach is computationally more convenient, as it gives the whole spectrum of the exited states of interest in a single calculation, but is physically biased. In fact, in the LR approach, the solute-solvent interaction contains a term related to the one-particle transition densities of the solute connecting the reference state adopted in the LR calculation, which usually corresponds to the electronic ground state, to the excited electronic state. The SS approach is computationally more expensive, as it requires a separate calculation for each of excited states of interest, but is physically unbiased. In fact, in the SS approach, the solute-solvent interaction is determined by the effective oneparticle electron density of the excited states. An excited-state CC method for continuum solvation models has been developed by Mikkelsen and co-worker (1999), based on the LR approach. In this section, we present an excited- state CC state-specific (SS) method for the PCM. As already discussed in the Introduction, we will limit ourselves to a formulation of the excitedstate PCM-CC to describe vertical absorption processes (the emission processes require a slightly different formulation) and in particular of absorption processes from the electronic ground state to a generic excited state. For this processes, we can give two versions: a first version in which the solvent is in equilibrium polarization with both states (state 0 and state K) and a second version in which the polarization of the K-th state is in nonequilibrium for the reasons discussed in the introduction. The formulation that we here report refers to the equilibrium formulation. For the nonequilibrium formulation, there are two versions: the formulation corresponding to the version II (Pekar) can be directly obtained from the equilibrium case by substituting the polarization charges of the solvent that appear in Eq. (52) for the equilibrium case, with those corresponding to the nonequilibrium case (Tomasi et al. 2005) and the version I of the nonequilibrium case requiring a slightly more elaborated formulation.

The Excited-State PCM-CC Wave Functions In the excited-state PCM-CC approximation, the excited electronic states are represented by a linear (CI-like) expansion buildup on the coupled-cluster wave function for the ground state. The corresponding expansion coefficients are then determined by solving a nonlinear eigenvalue problem. The eigenvalue equations are determined by a variational procedure involving the PCM-CC free-energy functional. The EOM-CC ket wave function (Nakatsuji and Hirao 1978; Stanton and Bartlett 1993) for the K-th state is defined as

20

R. Cammi and J. Tomasi

j‰K >DRK e T jHF >

(47)

where e T jHF > is the coupled-cluster state obtained by solving the PCMPTE equation (see Eq. (23) in section “The PTDE and the PTE Coupled-Cluster Schemes”) and RK is a quasi-particle excitation operator RK D RK;0 C RK;1 C RK;2 C : : : X 1    rijabc::: RK;n D 2 k::: .K/aa ai ab aj ac ak    nŠ

(48)

ij kl:::abc:::

The corresponding EOM-CC bra wave function is given by Q K j D < HF jLK e T D< LK jRL > D ıKL = < LK jRK >

(52)

1 N EOM N EOM  VK C Q 2 K C < HF jZe T HN .0/exp.T /jHF > Here the first term on the right side represents the EOM-CC energy of the state on N EOM  interest in presence of the fixed HF reaction potential, while the second term Q K EOM N N EOM V is the EOM-CC solute-solvent interaction contribution. Specifically, Q K K N EOM are the EOM-CC expectation value, respectively, of the polarization and V K charges and of the electrostatic potential for the K-th state: N EOM D< HF jLK e T b Q VN e T RK jHF > = < LK jRK > K N EOM D< HF jLK e T Q bN e T RK jHF > = < LK jRK > V K

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

21

EOM The last term of GK in Eq. (52) introduces the constraint for the ground state coupled-cluster wave function and contains the de-excitation operator Z given by

Z D Z1 C Z2 C : : : X 1    Zn D 2 ijabc::: k::: ai aa aj ab ak ac    nŠ

(53)

ij kl:::abc:::

The presence of the coupled-cluster constraint term is only relevant for the evaluation of analytical gradient of the PCM-EOM-CC energy functional.

The PCM-EOM-CC Eigenvalue Equations Imposing that GPEOM CM be stationary with respect to the RK and LK amplitudes, we obtain a right-hand and a left-hand eigenvalue equation:

EOM where HK Hamiltonian:

EOM HK RK jHF >D EKEOM RK jHF >

(54)

EOM < HF jLK EKEOM D< HF jLK HK

(55)

is the similarity-transformed state-specific PCM-EOM-CC EOM HK D e T HNEOM .K/e T D HNEOM .K/

(56)

N EOM  VN HNEOM .K/ D HN .0/ C Q K

(57)

with

where the first term of state-specific Hamiltonian describes the solute in presence N EOM  VN , represents of the fixed HF polarization charges, while the second term, Q K the interaction of the solute with the polarization charges produced by the solute in the excited-state K-th. The left and right eigenvalues and the EOM-CC energy can be obtained from the EOM matrix representing the non-Hermitian Hamiltonian HK in a suitable functional space. We limit ourselves to consider the case of the functional space associated to the coupled-cluster reference function at the CCSD level. The EOM-CCSD and the electronic states are associated with the diagonal EOM representation of HK in the subspace jp >, spanned by the jHF > determinant a and by its single ji > and double jab ij > excited determinants (i.e., jp >D jHF > ˚jai > ˚jab > ). Equations (54) and (55) can be converted to a non-Hermitian ij CI-like eigenvalue problem N R D E EOM R HEOM K K K K

(58)

22

R. Cammi and J. Tomasi

N D L E EOM LL HEOM K K K

(59)

where Rk and Lk represent the vectors of coefficients for the chosen excited state N is the PCM-EOM matrix Hamiltonian. The elements of matrix HEOM N and HEOM K K can be evaluated by standard methods. Details are given elsewhere (Cammi et al. 2010; Cammi 2010). N can be determined using a nonThe manifold of the eigenvectors of matrix HEOM K Hermitian modification of Davidson’s method (Hirao and Nakatsuji 1982). From this manifold, we can extract the RK and LK eigenvectors and the corresponding eigenvalue EKEOM . The remaining eigenvectors of the manifold are not of interest for the topics treated in this chapter. Finally, the free-energy functional value for the state of interest may be obtained in terms of the EOM-CC eigenvalue EKEOM as 1 EOM D EKEOM  VEOM  QEOM GK K 2 K

(60)

We have so completed the first step of the PCM-EOM-CC procedure, leading to the complete determination of the EOM-CC eigenfunctions and free energy for the target state K. We note that in this first step we have only exploited the stationarity EOM condition of GK with respect to the left and right eigenvectors. As already said, the stationary conditions on the Z and T amplitude are not necessary at this EOM stage, but they are instead necessary to obtain analytical derivatives of the GK functional.

PCM-EOM-CCSD Analytical Gradients Analytical first derivative of the free-energy PCM-EOM functional G EOM may be easily obtained by introducing the stationarity with respect to the Z and the T amplitudes. This avoids the evaluation of the first derivative of these parameters with respect to the external perturbations.

The PCM-EOM-CCSD Equations for the Z Amplitudes The stationary condition for the Z amplitudes leads to the PTE coupled-cluster equations for the reference ground state, which are assumed to be satisfied (see Eq. (23)); the stationary condition for the T amplitudes gives in turn the equations for the Z amplitudes. The derivative of G EOM with respect to the T amplitudes may be written as EOM  EOM  @GK D< HF jL HK ; O RjHF > @t

C < HF jZ ŒH.0/; 

O jHF >D 0

(61)

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

23

EOM where O is the excitation operator associated to the amplitude coefficient t , HK is the similarity-transformed EOM Hamiltonian, and H.0/ is the PTE similaritytransformed Hamiltonian of Eq. (25); the square brackets denote a commutator operator. Equation (61) can be rewritten as an explicit linear system of equation for the Z amplitudes:

@GPEOM CM N D< HF jZjg >< gjH.0/jg >  < HF j„jg >D 0 @t

(62)

N .0/jg >1 < HF jZjg >D  < HF j„jg >< gjH

(63)

or

where jg > denotes the subspace of the single and double excited Slater determiN N N nants, jg >D jai > ˚jab ij >, H .0/ is defined as H.0/ D H.0/ < 0jH .0/j0 >, and „ denotes the de-excitation operator „ D „11 C „2 C : : : X 1    „n D 2 ijabc::: k::: ai aa aj ab ak ac    nŠ

(64) (65)

ij kl:::abc:::

EOM whose amplitudes „n are given as matrix elements of the operator RHK jq >< a ab qjR between < HF j and jg >D ji > ˚jij > EOM jq >< qjRjg > < HF j„jg >D< HF jRHK

(66)

The explicit equation of the amplitudes n contains explicit PCM contributions which are given elsewhere (Cammi et al. 2010; Cammi 2010).

The Analytical Gradients of the PCM-EOM-CCSD Free-Energy Functional GEOM K EOM The derivatives of free-energy functional GK may then written as EOM @GK D< HF j.LK C Z/e T HN .0/e T RK jHF > @˛ 1 C < HF jLK e T V˛N e T RK jHF > QEOM K 2 1 C < HF jLK e T Q˛N e T RK jHF > VEOM K 2

(67)

24

R. Cammi and J. Tomasi

This expression can be recasted in a form that involves contractions between effective one- and two-particle EOM-CCSD density matrices and differentiated oneand two-electron MO integrals as EOM X @GK P CM;˛ EOM D fpq pq .K/ @˛ pq

C

1X ˛ EOM NR B EOM NR rs .K/ 2 pqrs pq;rs pq

C

1X < pqjjrs >˛ EOM .K/ 4 pqrs

(68)

where the Einstein summation convention is followed. The density matrices, EOM NR , EOM , and EOM , are defined by Stanton (1993): EOM NR .K/ D< HF jLŒp  qexp.T / c RjHF > pq EOM pq .K/ D EOM NR C < HF jZŒp  qexp.T / c jHF > EOM .K/ D< HF jLŒp  q  sre T c RjHF > C < HF jZŒp  q  sre T c jHF >

pqrs

where the subscript c denotes a limitation to connected diagrams. The effective density EOM NR is the usual reduced one-particle density (expectation value of p  q), while the effective density matrices EOM and EOM contain also correction terms that involve the  amplitude to account for the response of T to the perturbation. The PCM-EOM-CCSD analytical gradients (52) have the same form of PCMCCSD analytical gradients (42). Then, following the same procedure described for the PCM-CCSD gradients, it is easy to show that the differentiation of the MO integrals in Eq. (52) leads to terms involving the derivatives of the MO coefficients. In turn, the perturbative MO coefficients can be avoided using exploiting the PCM Z-vector technique (Cammi et al. 1999) to obtain a perturbation-independent oneMOresp particle  . The corresponding expression of the PCM-EOM analytical gradients in the AO basis is given by  X  EOM X @GK EOM MO ˛ 0 ˛ C D  .K/ h˛ C j I S @˛   X

1 EOM NR EOM NR ˛ EOM MO HF .  P C   /B 2  X 0 C

; < jj >˛ (69) C



Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

25

EOM MO where the additional one-particle density matrix  is defined as EOM MO EOM  D  C  . The equation of the PCM-EOM-CC analytical gradients contains additional oneelectron MO derivative integrals with respect to the corresponding gradients for isolated molecules (Stanton 1993). As these solvation terms can be evaluated with a small effort, the PCM-EOM-CC analytical gradients can be performed for all the molecular systems for which are feasible the EOM-CCSD gradient calculations in gas phase. MOresp

The Response Functions Theory for Molecular Solutes The response functions theory for molecules in the PCM solvation method (Cammi 2013) is an extension of the response theory for isolated molecules (Olsen and Jørgensen 1995; Helgaker et al. 2012), including this latter as a limit case. The extension from isolated to solvated molecules involves new aspects not present in the first case. These aspects are mainly related to the coupling between the interactions of the molecular solute both with the medium (i.e., the solvent) and with an external field which is in general time-dependent. A first aspect concerns the nonlinearity in the time-dependent QM problem. More specifically, the timedependent QM problem requires a proper extension of the time-dependent QM variation principle. A second aspect regards the time dependence in the solutesolvent responsive interaction, which must be described in a nonequilibrium solvation scheme (Marcus 1956). Finally, a third aspect regards the problem of the connection of the response functions of the molecular solutes with the corresponding macroscopic counterparts (i.e., the macroscopic susceptibilities) measured in the experiments.1 For reasons of available space, in the following, we will focus on the first two of these aspects (nonlinearity and nonequilibrium), while for the third aspect (macroscopic susceptibilities), we refer the interested reader to Cammi et al. (2000a) and Cammi and Mennucci (2007).

The Variational Time-Dependent Theory for the Polarizable Continuum Model Let us consider the time-dependent Schrödinger equation for a molecular solute i

@ N N j‰.t/ >D H j‰.t/ > @t

(70)

where H is the effective time-dependent Hamiltonian

1

The problem, historically known as local field problem, is due to the fact that the molecular solutes are locally subjected to perturbing fields which are different to that measured by the experimenter.

26

R. Cammi and J. Tomasi

N H D H 0 C Q.‰I t/  V C V 0 .t/

(71)

N t/  V is the potential where H 0 is the Hamiltonian of the isolated molecule, Q.‰I energy term representing the solute-solvent electrostatic interaction, and V 0 .t/ is a general time-dependent external perturbation (We assume that V 0 .t/ is applied adiabatically so that it vanishes at t D 1). In Eq. (71), Q.‰I t/ represents the polarization charges induced by the solute on the boundary of the cavity hosting the solute itself within the dielectric medium representing the solvent, the dot represents a vectorial inner product, and VO is a vectorial operator representing the electrostatic potential of the solute at the boundary cavity. The solvent polarization charges collected in Q.‰I t/ depend parametrically on time, as they are determined by the first-order density matrix N ‰  .t/‰.t/ of the molecular solute; the most general form of Q.‰I t/ must take into account the nonequilibrium solvation effects related to the intrinsic dynamics of solvent polarization. In the presence of a general time-dependent perturbation V 0 .t/, the timedependent polarization charges Q.‰I t/ may be formally defined as: N Q.‰I t/ D

Z

t

< ‰.t 0 /jQ.t  t 0 /j‰.t 0 / > dt 0

(72)

1

where Q.t  t 0 / is an apparent charge operator nonlocal in time, describing a general nonequilibrium solvation regime. In a nonequilibrium solvation regime, the description of polarization of the medium cannot be given in terms of a single value of the dielectric permittivity of the medium (as in the case of the equilibrium solvation), but it requires in principle knowledge of the whole spectrum of the frequency-dependent dielectric permittivity .!/ (Marcus 1956).2 A convenient approach to face both the nonlinear QM problem (70) and the nonequilibrium solvation problem (72) is given by the time-dependent quasi-freeenergy formalism.

The Time-Dependent Quasi-Free-Energy and Its Variational Principle It has been shown (Cammi and Tomasi 1996) that the time-dependent nonlinear Schrödinger (70) can be obtained from the Hamilton principle when a suitable QM Lagrangian density is defined and that this Lagrangian density implies an extension of the time-dependent Frenkel’s variational principle (Frenkel 1934) to time-dependent nonlinear Hamiltonians (70). N Let us consider the time-dependent wave function ‰.t/ > expressed in the phaseisolated form When the external perturbation V .t /0 is limited to static components (! D 0) and to components in the field of the optical frequencies, the description of the nonequilibrium polarization of the medium can be given in terms of a two-valued dielectric permittivity, 0 and 1 corresponding, respectively, to the static dielectric permittivity and to the dielectric permittivity at optical frequencies.

2

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

N j‰.t/ >D e iF .t / j‰.t/ >

27

(73)

where F .t/ is a function of time and ‰.t/ is the so-called regular wave function, which depends only parametrically on time and which reduces in the unperturbed limit (i.e., t D 1) to the time-independent wave function describing a stationary state of the molecular solute. The phase F .t/ of Eq. (73) is determined by substitution into Eq. (70): @ FP .t/ D< ‰.t/jH  i j‰.t/ > @t while the regular wave function ‰.t/ satisfies a generalized time-dependent Frenkel variational principles (Frenkel 1934): ıG.t/ C i

@ < ‰jı‰ >D 0 @t

(74)

where G.t/ is the time-dependent quasi-free-energy functional 1N @ G.t/ D< ‰.t/jH o C Q.‰I t/  V  i j‰.t/ > 2 @t

(75)

Let us now consider the case of a periodic time-dependent perturbation: V .t C T I / D V .tI /

(76)

with period T , frequency ! D 2 , and perturbation strength . In this case, the T phase-isolated form of the wave function acquires the same periodicity T : j‰.t C T / >D j‰.t/ > :

(77)

and by a time integration over the characteristic period T 3 , the variational condition (74) for the regular wave function (77) reduces to: 1N @ ıfG.t/gT D ıf< ‰.t/jH o C Q.‰I t/  V  i j‰.t/gT D 0 2 @t

(78)

being fG.t/gT time-averaged quasi-free-energy functional: fG.t/gT D

3

1 T

Z

T =2

T =2

1N @ < ‰.t/jH o C Q.‰I t/  V  i j‰.t/ > dt 2 @t

(79)

The time-average over a period T for a general time-dependent function g.t / is defined as R T =2 fg.t /gT D T1 T =2 g.t /dt )

28

R. Cammi and J. Tomasi

The time-averaged quasi-free-energy fG.t/gT (79) satisfies a generalization of the time-dependent Hellmann-Feynman theorem. Specifically, if the Hamiltonian (71) contains an external perturbation V .!/ with amplitude and periodicity T D 2=!, we have: d fG.t /gT d /

˚  D h‰.t/j @H @ j‰.t/i T D fh‰.t/jV .!/j‰.t/igT

(80)

where h‰.t/jV .!/j‰.t/i corresponds to the time-dependent expectation value of observable associated with the perturbing operator V .!/ and the time-average f: : :gT is related to its Fourier components having frequency !X . The time-dependent stationary conditions (78) and the corresponding HellmannFeynman theorem (80) are the basic equations for the determination of the response functions of the molecular solutes in the presence of periodic external perturbations.

The Response Functions of Molecular Solutes The molecular response functions of molecular solutes are defined in relation to a general time-dependent perturbation V 0 .t/ with several components (L) each having multiple harmonic frequencies (Ki ): V 0 .t/ D

Ki L X X

e .i !j t / Xi .!j /Xi

(81)

i D1 j DKi

where Xi is the i -th perturbation operator, having amplitude Xi .!i / in correspondence to the frequency !i . The response functions are defined as the coefficients of a Fourier-perturbation expansion of the time-dependent expectation values of a generic observable Y of the molecular system with respect to the perturbation V .t/ (Olsen and Jørgensen 1995): < ‰.t/jY j‰.t/ >D< Y >0 C

L;K Xi

e .i !j t / >!j Xi .!j /C

i;j

1 C 2

X

L;M;Ki ;Kj

e .i .!k C!l /t / >!k C!l Xi .!k / Xj .!l / C   

i;j;k;l

(82) where >!Xi represents the linear response function describing the contribution of Y .t/ of first order in the perturbation Xi having frequency !Xi and >!Xi C!Xj denotes the quadratic response function describing

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

29

Table 1 Selected response functions >!Y ;!Z ;::: of molecular solutes. The operators are ˛ , Cartesian components of the electric dipole operator, ‚˛;ˇ , element of the electric quadrupolar tensor, m˛ Cartesian component of the magnetic dipole operator Response function >! (Cammi et al. 1996) i >! (Mennucci et al. 2002; Caricato 2013b) >! (Rizzo et al. 2007) >! (Cammi et al. 1996) >!˛ ;!ˇ ;!ı (Cammi et al. 1996)  >!;0;0 C >!;0;0 (Cappelli et al. 2003)

Property Dipole polarizability ˛˛;ˇ .!I !/ 0 Optical rotation tensor G˛;ˇ .!/ Quadrupolar polarizability tensor ˛˛;ˇ .!I !/ First polarizability ˇ˛;ˇ; .!˛ I !ˇ ; ! / Second polarizability ˇ˛;ˇ; .!˛ I !ˇ ; ! /

Frequency dependent mixed electric magnetic hyper-magnetizability ˛;ˇ; ;ı .!I !; 0; 0/

the contribution of Y .t/ quadratic in the perturbations Xi ; Xj with frequencies !Xi ; !Xj and in a similar way are defined the higher-order response functions >!Xi C!Xj C!Xk ::: (see Table 1). The molecular response functions (82) may be determined by means of the Hellmann-Feynman theorem (80) when the time-averaged free-energy functional (79) is expanded in orders of the perturbation V 0 .t/ and the variational condition (78) is satisfied at the various order. The response function theory for molecular solutes within PCM has been developed for linear and nonlinear response functions and at several QM levels, including the variational wave function methods (SCF,MCSCF) and the density functional theory (Cammi et al. 1996; Cammi and Mennucci 1999; Tomasi et al. 1999; Cammi et al. 2000b; Cossi and Barone 2001; Mennucci et al. 2002; Cammi et al. 2003; Cappelli et al. 2003, 2005; Frediani et al. 2005; Jansík et al. 2006; Rizzo et al. 2007; Zhao et al. 2007; Ferrighi et al. 2007; Lin et al. 2008) and more recently at the coupled-cluster level (Cammi 2012a, b; Caricato 2013b). In the following, we will present the general aspects of the theory and its specific formulation at the coupled-cluster level.

The Coupled-Cluster Response Functions The time-dependent coupled-cluster wave function may be expressed as (Christiansen et al. 1998) jC C .t/ >D e iF .t / e T .t / jHF >

(83)

30

R. Cammi and J. Tomasi

where the reference state jHF > is the fixed, time-independent Hartree-Fock ground state of the solvated molecules4 , the phase factor F .t/ is a function of time, and T .t/ is the time-dependent cluster excitation operator. The coupled-cluster phase factor F .t/ is determined as:

N N .t/  VN C V .t/  i @ e T jHF > FP .t/ D< HF j.1 C ƒ/e T H .0/N C Q @t N N .t/ Here ƒ.t/ is the time-dependent coupled-cluster de-excitation operator, and Q is the vector collecting the time-dependent apparent charges: Z t N < HF j.1 C ƒ.t//e T .t / QN .t; t 0 /e T .t / jHF > dt 0 (84) QN .t/ D 1

0

where QN .t; t / is an apparent charge operator nonlocal with respect to the time (see Eq. (72) and ƒ is a time-dependent de-excitation operator. The time-dependent coupled-cluster T .t/ and ƒ.t/ amplitudes are determined using the quasi-free-energy functional Z 1 T =2 fGC C .t/gT D < HF j.1 C ƒ.t//e T .t / ŒH .0/N T T =2  1N @ e T .t/jHF > dt C QN .t/  VN C V .t/  i (85) 2 @t (where the time integration is over a common multiple of periods, T, such that V .t C T / D V .t/ Christiansen et al. 1998), with the variational condition ıfGC C .t/gT D 0

(86)

The variational Eq. (86) is solved using an additional perturbative approximation in which the coupled-cluster T and ƒ amplitudes are expanded with respect to the components of the perturbation V 0 .t/:

with: T .1/ D

X i

T .1/

T .t/ D T .0/ C T .1/ C T .2/ : : :

(87a)

ƒ.t/ D ƒ.0/ C ƒ.1/ C ƒ.2/ : : :

(87b)

exp.i !i t/

X

X .!i /T X .!i /

x

X 1X D exp.i .!i C !j /t/ X .!i / Y .!j /T X Y .!i ; !j / : : : 2 i;j X;Y

(88a)

4 (here, we consider the molecular orbital (MO) unrelaxed approach in which the reference state does not depend on the perturbation V 0 .t / (Christiansen et al. 1998))

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

ƒ.1/ D ƒ.0/ C

X

exp.i !i t/

X

X .!i /ƒX .!i /

x

i

ƒ.2/

31

X 1X D exp.i .!i C !j /t/ X .!i / Y .!j /ƒX Y .!i ; !j / : : : 2 i;j X;Y

(88b)

The expansions (88a), (88b) lead to an expansion of the time-averaged of the quasi-free-energy functional (85): o o o n n n .0/ .1/ .2/ fG.t/C C gT D GC C C GC C C GC C C : : : T

T

T

(89)

o n .n/ where the n-th order term GC C is defined as T

n

.n/

GC C

o T

D

1 XX X .!i / Y .!j /     G X Y ::: .!i ; !j ;    / nŠ i;j;::: x;y;z

(90)

where the Fourier components G X Y ::: .!i ; !j ;    / are defined as G

X Y :::

.!i ; !j ;    / D

˚  d n G .n/ T d i d j : : :

(91)

with !i C !j C    D 0. The perturbation expansion of Eq. (89) is then introduced into the time-average variational criterion (86), and separating by orders, we arrive to a set of variational criteria on the Fourier components G X Y ::: .!i ; !j ;    / ıG X Y ::: .!i ; !j ;    / D 0

(92)

that determine the Fourier coefficients T X Y  .!i ; !j ;   / and ƒX Y  .!i ; !j ;   / for the T and ƒ amplitudes (88a) and (88b). When the variational conditions (92) are satisfied, to the Fourier components GQ X Y ::: .!i ; !j ;    / correspond to the response functions of the molecular solutes: >!i >!i :: :

D GQ X Y .!i ; !j / D GQ X Y Z .!i ; !j ; !k / : D ::

>!i D GQ X Y Z::: .!i ; !j ; : : : /

(93)

32

R. Cammi and J. Tomasi

where the tilde denotes stationarity. The response functions (93) satisfy a generalized .2n C 1/ rule with respect to the perturbed T and ƒ amplitudes. In the following subsection, we give a detailed account of the explicit form of the coupled-cluster response equations from which the linear and quadratic response functions can be obtained.

Coupled-Cluster Response Equations The first-order equations for the coupled-cluster Fourier amplitudes ƒX .˙!/ T X .˙!/ are determined from the stationarity of the second-order time-averaged quasi-free-energy G X Y .!; !/: G X Y .!; !/ D P .X Y / < HF j.1 C ƒ/e T jŒX; T Y .!/ e T jHF > C < HF j.1 C ƒ/e T ŒŒHN ; T X .!/ ; T Y .!/ e T jHF > !P .X Y / < HF jƒX .!/T Y .!/jHF > (94) X T Y T CP .X Y / < HF jƒ .!/e ŒY C ŒHN ; T .!/ e jHF > 1 NX NY 2 P .X Y /QN .!/  VN .!/ N .0/  V is the unperturbed Hamiltonian of the molecular Here, HN D HN .0/ C Q N N X Y solute; QN .!/ and VN .!/ are given by T X T QNX ŒQj!j N .!/ D < HF j.1 C ƒ/e N ; T .!/ e jHF > j!j C < HF jƒX .!/e T QN e T jHF >

(95a)

VNYN .!/ D < HF j.1 C ƒ/e T ŒVN ; T Y .!/ e T jHF > C < HF jƒY .!/e T VN e T jHF >

(95b)

In Eq. (95a) the polarization charges operator Qj!j N is defined as Qj!j N D T.!/  VN

(96)

where T.!/ is the frequency-dependent PCM response matrix evaluated with the dielectric permittivity .!/ of the solvent at the frequency !. • The equations for the first-order T X .˙!/ amplitudes are obtained from the stationary condition ıG X Y .!; !/ D 0 with respect to the first-order ƒX .˙!/ amplitudes:  0 D < HF jp e T YQ .!/e T jHF >  C < HF jp Œe T HN e T ; T Y .!/  !T Y .!/jHF >

where YQ .!/ is an effective perturbing operator:

(97)

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

YQ .!/ D Y C QNYN .!/  VN

33

(98)

Note that this operator indicates that to the direct perturbation with the external field, represented by the operator Y , it is added an indirect source of perturbation due to the effect of the direct perturbation itself on the solute-solvent interaction QNYN .!/  VN . • The equations for the first-order ƒX .˙!/ amplitudes are obtained from the stationary condition ıG X Y .!; !/ D 0 with respect to the first-order T X .˙!/ amplitudes: 0 D < HF j.1 C ƒ/e T ŒYQ .!/; p e T jHF > (99) C < HF j.1 C ƒ/e T ŒŒHN ; p ; T Y .!/ e T jHF > Y Y T T ! < HF jƒ .!/p jHF > C < HF jƒ .!/e ŒHN ; p e jHF >

Coupled-Cluster Linear and Quadratic Response Functions • The linear response functions >! can then be obtained from Eq. (93) by introducing the stationary condition (97) in to the second-order quasi-freeenergy (94) >! D GQ X Y D

1 ˙! C fP .X Y / < HF j.1 C ƒ/e T jŒX; T Y .!/ e T jHF > 2

C < HF j.1 C ƒ/e T jŒŒHN ; T X .!/ ; T Y .!/ e T jHF > 1 X T P .X Y / < HF j.1 C ƒ/e T ŒQj!j N ; T .!/ e jHF > 2  < HF j.1 C ƒ/e T ŒVN ; T Y .!/ e T jHF > 1 T Y T P .X Y / < HF jƒX .!/Qj!j N e jHF >  < HF jƒ .!/VN e jHF >g 2 (100) where C ˙! is a symmetrization operator with respect to sign change of the frequencies to ensure that only the real part of the response function is retained and P .x; y/ is a permutation operator between x and y. Alternatively, introducing the stationary condition (99) into the second-order quasi-free-energy (94), we can write the linear response functions as >! D

1 ˙! C f< HF j.1 C ƒ/e T ŒX; T Y .!/ e T jHF > 2

C < HF jƒY .!/e T Xe T jHF >g

(101)

34

R. Cammi and J. Tomasi

• The quadratic response functions >!Y ;!Y can be obtained in agreement with the .2n C 1/ rule as >!Y ;!Y D GQ X Y Z D

1 ˙! 3 C fP .X Y Z/ 2 C P 6 .X Y Z/ < HF jƒX .!X /e T ŒY; T Z .!Z / e T jHF >

C < HF j.1 C ƒ/e T ŒŒŒHN ; T X .!X / ; T Y .!Y / ; T Z .!Z / e T jHF > C P 3 .X Y Z/ < HF jƒX .!X /e T ŒŒHN ; T Y .!Y / ; T Z .!Z / e T jHF >   1 3 Q Y Z .!Y C!Z / g Q X Y .!X C!Y /  VNZ .!Z /CQNX .!X /  V P .X Y Z/ Q N N N N 2 (102) where P 3 .xyz/ and P 6 .xyz/ denote, respectively, the cyclic permutation Q X Y .!X C operator and the full permutation operator of their arguments, while Q N 1 YZ Q !Y / and VN .!Y C !Z / are defined as T X C!Y j QY QX ŒŒQj! ; T X .!X /; T Y .!Y / e T jHF > N .!X C !Y / D< HF j.1 C ƒ/e N Y T Zj C P .X Y / < HF jƒX .!X /e T ŒQj! N ; T .!Y / e jHF >

Q Z .! C ! / D< HF j.1 C ƒ/e T ŒŒV ; T X .! /; T Y .! / e T jHF > VYN Y Z N X Y P .X Y / < HF jƒX .!X /e T ŒVN ; T Y .!Y / e T jHF > (103)

Conclusions We have presented a short review on recent progresses toward the extension of the QM coupled-cluster level for the study of the electronic structure and properties of molecular solutes with the Polarizable Continuum Model framework (Cammi 2009). Specifically, we have presented (i) the detailed expression for the evaluation of the analytical gradients for the PCM-CC theory at the single and double excitation level, for the ground states; (ii) the expressions of the analytical gradients for the PCM-EOM-CC theory at the single and double excitation level for the descriptions

j!X C!Y j The polarization charges operator QN is evaluated from Eq. (96) using the dielectric permittivity .j!X C !Y j/ of the solvent at the frequency j!X C !Y j. Therefore, Eq. (103) is able to describe the nonequilibrium solvation effects in the quadratic response function describing general second-order molecular processes (Cammi et al. 1996).

1

Quantum Cluster Theory for the Polarizable Continuum Model (PCM)

35

of the excited-state properties of molecular solute; and (iii) the expressions of the linear and quadratic PCM-CC response functions. Several implementations of these PCM-CC theories have been presented in the last years within the Gaussian suite of programs (Frisch et al. 2009b; Frisch et al. 2009a). Caricato et al. have presented implementations of the PCM-CC analytical gradients, of the PCM-EOC-CC gradients, and of the linear response functions PCM-CC (Caricato et al. 2010, 2011; Caricato 2011; Caricato and Scalmani 2011; Caricato 2012a, b, 2013b, a, 2014). Fukuda et al. have presented various implementations of the PCM-CC and PCM-EOM-CC methods and of the corresponding analytical derivatives within the framework of SAC/SACCI methods (Cammi et al. 2010; Fukuda et al. 2011, 2012; Fukuda and Ehara 2013; Fukuda et al. 2014; Fukuda and Ehara 2014; Fukuda et al. 2015). We hope that these computational progresses can be profitably used to study molecular processes in condensed phase, where both the accuracy of the QM descriptions and the influence of the environment play a critical role, as in photoionization processes, electronic transitions, and charge transfer reactions.

Appendix A: The Solute-Solvent PCM Operator In the Polarizable Continuum Model for salvation, the molecular solute is hosted into a cavity of a polarizable dielectric medium representing the solvent. The cavity is accurately modeled on the shape of the molecular solute (Miertuš et al. 1981), and the dielectric medium is characterized by the dielectric permittivity of the bulk solvent. The physics of the model is very simple. The solute charge distribution polarizes the dielectric medium which in turn acts back on the solute, in a process mutual polarization that continues until self-consistence is reached. The polarization of the solvent is represented by an apparent charge distribution (ASC) spread on the cavity surface. In computational practice, the ASC is discretized to a set of point charges, and the solute-solvent interaction is expressed as in terms of the interaction between these and the charge distribution of the molecular solute. In this framework, the PCM solute-solvent interaction operator (see Eq. (3)) can be defined in terms of a molecular electrostatic operators V and of a charge operator Q describing the PCM solute-solvent interaction operator (Cammi et al. 2002, 2005). V is a vector collecting the molecular electrostatic potential operator, evaluated at the positions of the ASC charges: .V/j D vel .sj / C vnuc .sj /

(104)

where vel .sj / and vnuc .sj / are the electrostatic potential of the electrons and of the nuclei, respectively, evaluated at the position sj of the j -th ASC point charge. In particular,

36

R. Cammi and J. Tomasi

vel .sj / D

N X i

1 jri  sj j

(105)

where N is the number of electrons of the molecular solute and ri is the vector position of the i -th electron. A similar expression holds for the nuclei contribution vO nuc .sj /. The apparent charges vector operator Q can be formally defined in a different way, depending on the different version of the BEM. In the integral equation formalism (IEF) version of the PCM (Cancès et al. 1997), which is here adopted, Q may be expressed as a linear transformation of the vector operator V: .Q/j D

N TS X

Tj l .b V/l

(106)

l

where Tj l are elements of the BEM square matrix. The matrix is dimensioned as the ASC charges, and it depends on the geometry of surface cavity and on the dielectric b operator .< ‰jQj‰ > permittivity of the medium. The expectation value of the Q / is a vector collecting the actual ASC charges induced by the molecular solute described by the wave function ‰. b >/  b The solute-solvent interaction operator .< ‰jQj‰ V of Eq. (2) of the text is b >/, then the inner product between the expectation values of the ASC, .< ‰jQj‰ and the molecular electrostatic operator V: < ‰jQj‰ > V D

N TS X

b j j‰ > .b < ‰j.Q/ V/j

(107)

j

where < ‰j.Q/j j‰ > denotes the expectation value of the j -th apparent surface charge.

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Bonaccorsi, R., Ghio, C., & Tomasi, J. (1982). The effect of the solvent on electronic transition and other properties of molecular solutes. In R. Carbó, (Ed.), Current aspects of quantum chemistry (p. 407). Amsterdam: Elsevier. Cammi, R. (2009). Quantum cluster theory for the polarizable continuum model. I. The CCSD level with analytical first and second derivatives. Journal of Chemical Physics, 131, 164104. Cammi, R. (2010). Coupled-Cluster Theories for the Polarizable Continuum Model. II. Analytical Gradients for Excited States of Molecular Solutes by the Equation of Motion Coupled-Cluster Method, International Journal of Quantum Chemistry, 110, 3040. Cammi, R. (2012a). Coupled-cluster theory for the polarizable continuum model. III. A response theory for molecules in solution. International Journal of Quantum Chemistry, 112, 2547. Cammi, R. (2012b). Recent Advances in the Coupled-Cluster Analytical Derivatives Theory for Molecules in Solution Described With the Polarizable Continuum Model (PCM). Advances in Quantum Chemistry, 64, 1. Cammi, R. (2013). Molecular response functions for the polarizable continuum model. Cham: Springer. Cammi, R., & Mennucci, B. (1999). Linear response theory for the polarizable continuum model. Journal of Chemical Physics, 110, 9877. Cammi, R., & Mennucci, B. (2007). Macroscopic Nonlinear Optical Properties from Cavity Models. In B. Mennucci & R. Cammi (Ed.) Continuum solvation models in chemical physics (p. 238). Chichester: Wiley. Cammi, R., & Tomasi, J. (1994). Analytical derivatives for molecular solutes. I. Hartree–Fock energy first derivatives with respect to external parameters in the polarizable continuum model. Journal of Chemical Physics, 100, 7495. Cammi, R., & Tomasi, J. (1995a). Remarks on the use of the apparent surface charges (ASC) methods in solvation problems: Iterative versus matrix-inversion procedures and the renormalization of the apparent charges. Journal of Computational Chemistry, 16, 1449. Cammi, R., & Tomasi, J. (1995b). Nonequilibrium solvation theory for the polarizable continuum model: A new formulation at the SCF level with application to the case of the frequencydependent linear electric response function. International Journal of Quantum Chemistry: Quantum Chemistry Symposium, 29, 465. Cammi, R., & Tomasi, J. (1996). Time-dependent variational principle for nonlinear Hamiltonians and its application to molecules in the liquid phase. International Journal of Quantum Chemistry, 60, 297. Cammi, R., Cossi, M., Mennucci, B., & Tomasi, J. (1996). Analytical Hartree–Fock calculation of the dynamical polarizabilities ˛, ˇ, and of molecules in solution. Journal of Chemical Physics, 105, 10556. Cammi, R., Mennucci, B., & Tomasi, J. (1999). Second-Order Møller–Plesset Analytical Derivatives for the Polarizable Continuum Model Using the Relaxed Density Approach. Journal of Physical Chemistry A, 103, 9100. Cammi, R., Mennucci, B., & Tomasi, J. (2000a). An Attempt To Bridge the Gap between Computation and Experiment for Nonlinear Optical Properties: Macroscopic Susceptibilities in Solution. Journal of Physical Chemistry A, 104, 4690. Cammi, R., Mennucci, B., & Tomasi, J. (2000b). Fast Evaluation of Geometries and Properties of Excited Molecules in Solution: A Tamm-Dancoff Model with Application to 4Dimethylaminobenzonitrile. Journal of Physical Chemistry A, 104, 5631. Cammi, R., Corni, S., Mennucci, B., Tomasi, J., Ruud, K., & Mikkelsen, K. V. (2002). A secondorder, quadratically convergent multiconfigurational self-consistent field polarizable continuum model for equilibrium and nonequilibrium solvation. Journal of Chemical Physics, 117, 13. Cammi, R., Frediani, L., Mennucci, B., & Rudd, K. (2003). Multiconfigurational self-consistent field linear response for the polarizable continuum model: Theory and application to ground and excited-state polarizabilities of para-nitroaniline in solution. Journal of Chemical Physics, 119 5818. Cammi, R., Corni, S., Mennucci, B., & Tomasi, J. (2005). Electronic excitation energies of molecules in solution: State specific and linear response methods for nonequilibrium continuum solvation models. Journal of Chemical Physics, 122, 101513.

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Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry B.F. Minaev, Hans Ågren, and V.O. Minaeva

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O2 Interaction with Heme, FAD, and Oxidases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Spin and the Angular Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Oxygen Structure and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Prohibition of Dioxygen Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cytochrome Oxidases and Related Heme-Containing Enzymes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dioxygen Reaction with Glucose Oxidase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dioxygen Binding to Heme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Orbit Coupling in O2 -HemeInteraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Spin in Biochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Magnetic Field Effects in Biochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B. F. Minaev () Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden Organic Chemistry, Bogdam Khmelnitskii National University, Cherkassy, Ukraine e-mail: [email protected] V. O. Minaeva Organic Chemistry, Bogdam Khmelnitskii National University, Cherkassy, Ukraine e-mail: [email protected] H. Ågren Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden e-mail: [email protected]

© Springer Science+Business Media Dordrecht 2016 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_29-2

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Abstract

We review the general concept of nonadiabatic quantum spin transitions in biochemistry. A few important examples are highlighted to illustrate the concept: the role of spin effects in oxidases, cytochromes, in dioxygen binding to heme, in photosynthesis, and in tentative models of consciousness. The most thoroughly studied of these effects are connected with dioxygen activation by enzymes. Discussion on the mechanisms of overcoming spin prohibitions in dioxygen reactions with flavin-dependent oxygenases and with hemoglobin and myoglobin is presented in some detail. We consider spin-orbit coupling (SOC) between the starting triplet state from the entrance channel of the O2 binding to glucose oxidase, to ferrous heme, and the final singlet open-shell state in these intermediates. Both triplet (T) and singlet (S) states in these examples are dominated by the radical-pair structures DC  O 2 induced by charge transfer; the peculiarities of their orbital configurations are essential for the SOC analysis. An account of specific SOC in the open  g -shell of dioxygen helps to explain the probability of T-S transitions in the active site near the transition state. Simulated potential energy surface cross-sections along the reaction coordinates for these multiplets, calculated by density functional theory, agree with the notion of a relatively strong SOC induced inside the oxygen moiety by an orbital angular momentum change in the  g -shell during the T-S transition. The SOC model explains well the efficient spin inversion during the O2 binding with heme and glucose oxidase, which constitutes a key mechanism for understanding metabolism. Other examples of nontrivial roles of spin effects in biochemistry are briefly discussed.

Introduction Many enzymes, for example, those that contain copper or iron-heme active sites (cytochromes, peroxidases), are paramagnetic and undergo spin transitions depending on the change of the metal site oxidation (Franzen 2002; Jensen and Ryde 2004; Prabhakar et al. 2003; Shaik et al. 2002; Shikama 2006; Sigfridsson and Ryde 2002; Stryer 1995). Oxygen molecules contain two unpaired valence electrons (Sawyer 1991) and are therefore also paramagnetic because of the nonzero total electron spin. The complete oxidation of organic materials by dioxygen in order to transfer them into CO2 and water is highly exothermic (Sawyer 1991), but the kinetic barriers preclude, fortunately, this spontaneous combustion of living matter at normal conditions in the open air because of the spin selection rules (Prabhakar et al. 2003). Like mass and charge, spin is an integral property of the electron, however, spin reveals itself through the relativistic quantum equation as a pure quantum process associated with the structure of space-time, thus it is more fundamental. The role of this fundamental quantum feature of the electron in biochemistry could be extremely important, especially with respect to the mystery of the brain, the nature

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of consciousness (Penrose 1994), and oxygen activation by enzymes (Franzen 2002; Jensen and Ryde 2004; Prabhakar et al. 2003; Sawyer 1991; Shaik et al. 2002; Shikama 2006; Sigfridsson and Ryde 2002; Stryer 1995). When O2 reaches a red blood cell in the alveoli of the lung, where the gas exchange of carbon dioxide and oxygen takes place, the hemoglobin binds O2 in such a strange way that the total electron spin of the whole heme-O2 system is not conserved (Sigfridsson and Ryde 2002). In ordinary chemistry the spin conservation rule is usually fulfilled (Sawyer 1991; Shikama 2006; Stryer 1995), which is in agreement with the general conservation law of physics (Sawyer 1991). But the change of the electron spin is a rather common case in biochemistry of enzymes, in the respiration cycle and in other processes connected with dioxygen consumption (Blomberg et al. 2005; Friedman and Campbell 1987; Lane 2002; Minaev 2007; Minaev et al. 2007; Silva and Ramos 2008; Strickland and Harwey 2007). The spin change has to be compensated by an orbital angular momentum change in order to follow the general conservation law of the total angular momentum. The present review will show that this requirement is fulfilled in a number of enzymatic processes that play key roles in biochemistry; this is accomplished by an analysis of spin-orbit coupling perturbations. With the appearance of photosynthesis on the primitive earth, a dramatic change in life occurred when living cells adapted themselves from anaerobic life to the aerobic form (Shikama 2006). Anaerobic glycolysis represented a successful attempt to extract some of the chemical energy from glucose. In contrast, the complete oxidation of glucose to CO2 and H2 O by utilizing oxygen molecules was the most significant and crucial advancement in cellular metabolism. It thus allowed organisms to enhance their ability to utilize the chemical energy of sugar for a most effective consumption. In light of the high efficiency of energy conservation, this dramatic change from anaerobic to aerobic systems was designated as an “oxygen revolution” (Lane 2002; Penrose 1994; Shikama 2006; Stryer 1995). The most essential feature of this revolution was the involvement of spin-catalysis and quantum spin effects in ordinary chemistry (Minaev 2007). In fact, both processes, photosynthesis and respiration, are spin-forbidden; from one side we have diamagnetic reactants and the paramagnetic species from the other side 6CO2 C chlorophyll 6H2 O C E      !  C6 H12 O6 C 6O2 . In order to synthesize O2 with chlorophyll from green plants beneath the sun and to use oxygen for glucose oxidation, nature has designed very smart chemical manipulations that allow spin prohibition in dioxygen activation to be overcome (Minaev 2007). With this revolution a large number of important enzymes that are based on spin-catalysis principles (Blomberg et al. 2000a; Jensen et al. 2005; Lane 2002; Minaev 2002, 2004, 2007, 2010; Minaev et al. 2007; Petrich et al. 1988; Prabhakar et al. 2002, 2004; Silva and Ramos 2008) have been developed and spread (Blomberg et al. 2000b; Klinman 2001; Minaev and Lunell 1993; Sheldon 1993).

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O2 Interaction with Heme, FAD, and Oxidases Some oxidases include flavin adenine dinucleotide (FAD), which can activate O2 to produce diamagnetic products without radical chain reaction steps (Minaev 2002; Stryer 1995). For example, glucose oxidase (GO) is a well-known enzyme that can bind dioxygen directly from air and produce hydrogen peroxide in the metabolic cycle (Prabhakar et al. 2003). An X-ray structure of GO from Aspergillus niger is available (Klinman 2001) and used for simulation of the active site in theoretical modeling (Prabhakar et al. 2002). The histidine residue has been assigned an important role in the substrate oxidation by the GO active site. In the heme biosynthesis pathway a deficient activity of the enzymes often leads to excessive excretion of porphyrins (Silva and Ramos 2008). These toxic products can induce porphyria diseases due to decreased activity of coproporphyrinogen III oxidase (CPO), which catalyzes the oxidative decarboxylation of propionic acid chains. This oxidase is similar to GO in that they both activate dioxygen without using metals and reducing agents. However, mechanisms by which these two oxidases produce diamagnetic peroxides from paramagnetic dioxygen are not properly understood on experimental grounds (Klinman 2001; Prabhakar et al. 2002; Silva and Ramos 2008), while some theoretical predictions seem to shed light on the specific quantum nature of such chemical transformations (Minaev 2007; Minaev et al. 2007; Prabhakar et al. 2003; Silva and Ramos 2008). Density functional theory (DFT) calculations (Prabhakar et al. 2002, 2003; Silva and Ramos 2008) indicate the common feature of spin transitions in these oxidases’ mechanisms. Hemoglobin and myoglobin are important globular proteins that reversibly bind the O2 molecule. Both proteins contain ferrous iron of a heme group, which is usually simulated by Fe(II) porphyrin, where the iron ion is tetra-coordinated to the nitrogen atoms of the tetra-pyrrole rings (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002). The proximal histidine residue from the protein side chain is also bound to the Fe(II) ion leaving one empty position in the octahedral coordination sphere around the ferrous iron. Hemoglobin and myoglobin bind several small gaseous molecules (CO and NO) besides dioxygen (Sigfridsson and Ryde 2002). The binding of these diatomic ligands to heme has been studied in biochemistry for over a 100 years (Shikama 2006). Dissociation of CO, NO, and O2 from the iron of heme occurs by both photolysis and thermolysis. In the former case, the fate of the diatomic ligands depends on the competition between geminate recombination and protein relaxation. Binding to heme occurs by the reverse of the thermal process (Franzen 2002). It is well established that the structure of surrounding protein affects the ligand binding ability of the heme group; carbon monoxide binds to free heme in solution 105 times more effectively than dioxygen, but in myoglobin this ratio is strongly reduced (Franzen 2002). Thus, myoglobin seems to favor O2 binding compared to CO (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002). Such discrimination is of vital importance: otherwise we would suffocate from CO produced in our body during metabolism. The reason for the

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discrimination has been connected with different geometric parameters of O2 and CO binding to the heme iron: the Fe–C–O bond is linear, whereas the Fe–O– O bond is bent (Franzen 2002). Both hemoglobin and myoglobin have another, distal, histidine ligand (His-64) positioned above the Fe(II) ion, but too far away to coordinate directly to the iron. His-64 is in the right position to affect the Fe–CO group and to produce a tension. This idea was supported by early crystal structures, showing an Fe–C–O angle of 120–130ı (Shikama 2006), an idea that has penetrated into the textbooks (Shikama 2006; Stryer 1995). Some newer X-ray measurements, IR spectra, and DFT calculations indicate that the Fe–C–O angle is nearly linear in the heme models; thus one finds the idea of a strongly bent Fe–C–O unit as the reason for the discrimination but which nowadays is agreed to be incorrect (Sigfridsson and Ryde 2002). Recent DFT calculations show that the FeO2 group is much more polar than the FeCO group in heme models (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002). The electronic structure of the FeP complex with dioxygen is close to that of a superoxide anion bound to a ferric ion, and this charge-transfer complex is in the singlet spin state (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002). Thus, electrostatic interaction between the distal histidine and the Fe(III)–O–O group is stronger than that for the Fe–CO group; therefore, the protein discriminates between O2 and CO by hydrogen bonding and electrostatic interaction in myoglobin (Franzen 2002; Jensen and Ryde 2004; Sigfridsson and Ryde 2002). Another important factor in such discrimination is determined by electron spin (Franzen 2002; Jensen and Ryde 2004; Minaev 2010; Shikama 2006; Sigfridsson and Ryde 2002; Stryer 1995). The O2 binding to heme is highly non-exponential at ambient temperature with a rapid phase of 3 ps, a longer phase of similar geminate recombination (20–200 ps), and a slow bimolecular process of 1 ms; all phases are spin-dependent processes (Franzen 2002; Jensen and Ryde 2004; Petrich et al. 1988; Shikama 2006). At this point we need to consider it in more detail.

Definition of Spin and the Angular Momentum Conservation Spin is derived from the Dirac equation and is a general intrinsic quantum property of electrons (Esposito 1999; Hagan et al. 2002; Hameroff and Penrose 1996; Hu and Wu 2004; Penrose Spin is an angular with a length of the p 1960). p momentum  p 2 E momentum vector S D S .S C 1/„ D 3=2 „, where the spin quantum number for one electron is equal to S D 1/2 and two projection MS D ˙ .1=2/ „ described by ’ and “ wave functions exist (the doublet state) (Esposito 1999). The role of electron spin in chemistry is rationalized in terms of spin-valence concept: a covalent chemical bond is formed when atomic orbitals of two electrons with opposite spins overlap each over (Stryer 1995). This spin wave function (’“  “’) is antisymmetric with respect to permutation of two electrons. The total spin of such pair is zero (S D 0) and the state is singlet (only one state; no spin, no intrinsic magnetic moment, the molecule is diamagnetic). When spins are parallel the total spin has a quantum number S D 1 in the equation for the length of the total spin

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p p p angular momentum SE 2 D S .S C 1/„ D 2„ and there are three spin states (’’, ““, and ’“ C “’) with different projections on z-axis MS D ˙„; 0 (triplet state) (Penrose 1960). For a simple covalent ¢-bond the triplet state is unstable and such simple molecule can exist only in the singlet state with two opposite spins. That is why almost all stable organic molecules contain an even number of electrons, which can be divided into ’ and “ pairs. They are diamagnetic, have paired spins and are in the singlet ground state. Such diamagnetic substances usually do not change zero spin during chemical reactions (Blomberg et al. 2000a, 2005; Friedman and Campbell 1987; Jensen et al. 2005; Klinman 2001; Lane 2002; Minaev 2002, 2007, 2010; Minaev et al. 2007; Petrich et al. 1988; Prabhakar et al. 2002, 2004; Sawyer 1991; Silva and Ramos 2008; Strickland and Harwey 2007). The absence of the total electronic spin in stable molecules leads to the illusion that spin is not important in organic chemistry and biochemistry. In fact, the total electronic spin is the main regulating factor in many metabolic processes catalyzed by metal-organic enzymes, such as cytochromes (Blomberg et al. 2005; Minaev 2007; Minaev et al. 2007; Shaik et al. 2002; Stryer 1995), copper-aminoxidase (Prabhakar et al. 2003; 2004), and even in metal-free glucose oxidase (Minaev 2002; Prabhakar et al. 2002, 2003). Spin inversion is especially important in many dioxygen reactions, binding to heme (Jensen and Ryde 2004), combustion, and respiration (Minaev 2007). One can propose that the importance of spin inversion is also reflected in the Perutz model of the hemoglobin cooperativity (Franzen 2002; Minaev et al. 2007). Dioxygen molecule is a well-known exception to the general rule: unlike many chemically stable organic compounds, the O2 molecule has the triplet ground state (Sawyer 1991). According to Hund’s rule, two unpaired electrons in two degenerate  g,x - and  g,y -orbitals have a lower repulsive energy in the triplet state compared to the singlet one. Because of this, oxygen is paramagnetic, it has intrinsic magnetic moment due to spins of two unpaired electrons and its addition to organic compounds is spin forbidden: starting reactants have the total spin S D 1 (from the O2 ), whereas the oxidation products are diamagnetic (S D 0). This is the reason why organic matter may exist in the oxygen-rich atmosphere (Sawyer 1991). Because of the spin prohibition, combustion of organic fuels requires activation in the form of high-temperature ignition stage (Prabhakar et al. 2002; Sawyer 1991), i.e., generation of primary radicals. Reaction of radical R (the triplet, S D 1/2, doublet state) with O2 molecule is spin-allowed, since the starting reactants (O2 C R• ) and product (RO2 ) both have the doublet states (reactants also possess quartet state, S D 3/2, being non-reactive), O2 R RO2  ŒŒ" Œ" C Œ# ) ŒŒ"# Œ"

(1)

which provide the radical chain character of the combustion reactions (Minaev 2007).

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

7

The quantum cell ["] denotes (Eq. 1) a molecular orbital (MO) with ’ spin; double brackets embrace one molecule. The radical RO2 can decompose into radical RO• and biradical • O• thus providing a branching chain reaction. In radical chain combustion the energy is released in the form of heat and light without any specific control (until fuel exhaustion). Clearly, such mechanism of oxidation by molecular oxygen cannot be realized in living cells. Cells meet their energy needs in the course of metabolic processes using strictly controlled energy of oxidation of organic compounds in their reactions with dioxygen, overcoming spin prohibition without high-temperature ignition step of a radical chain (Minaev 2007). An aerobic life evolved due to quantum kinetic prohibitions to reactions of paramagnetic oxygen with diamagnetic organic substances. The main reason for sluggish O2 reactivity at the ambient conditions is the spin prohibition, namely, the starting reagents have two unpaired spins (from O2 molecule), while in diamagnetic oxidation products (CO2 , H2 O, N2 ) all spins are always paired. Overcoming this prohibition by generating radicals to interact with dioxygen (like in combustion) is not possible in living matter. Since the cells cannot resist large temperature gradients, they have to transform the energy released through oxidation to some kind of chemical energy prior to dissipation in the form of heat. This occurs by combining oxidation with ATP synthesis. All versatile energy-supplying metabolic processes and reactions occur under subtle enzymatic regulation, which is often spin-dependent (Blomberg et al. 2000a; Jensen et al. 2005; Minaev 2002, 2007, 2010; Petrich et al. 1988; Prabhakar et al. 2002, 2004).

Molecular Oxygen Structure and Spectra The O2 molecule in the triplet ground state has the following electronic configuration (Sawyer 1991) (1 g )2 (1 u )2 (2 g )2 (2 u )2 (3 g )2 (1 u )4 (1 g )2 (Fig. 1). The two outer electrons in two degenerate 1 g –MO’s provide the lowest triplet state of the type ["]["], where the quantum cells [][] denote the degenerate  g -orbitals. These two unpaired electrons in antibonding  g -MOs (Fig. 1) are responsible for the specific character of the dioxygen interaction with radicals (combustion) and chemically stable diamagnetic compounds (slow oxidation), as outlined above. Two antibonding  g -vacancies make it possible to transform dioxygen into O2  and O2 anions, the formation of the latter being strongly dependent on the 2 presence of electron donors (enzymes) and magnetic perturbations that affect the spin prohibition. There are four possible quantum states (Eq. 2) for the electronic configuration mentioned above. For the imaginary form of two degenerate 1 g –MO’s: 1 ˙ g D § .r; ™/ e ˙i¥ , where ™ is an angle between the radius vector (r) and the molecular axis (z), ¥ is the rotation angle of the radius vector about the z axis, the quantum states are represented by the scheme (Minaev 2007):

8

B.F. Minaev et al.

Energy 3σu

2πg,x 2πg,y

2πu,x 2πu,y 3σg

2σu 2s

2s 2σg

1σu 1s 1σg AO

MO

AO

Fig. 1 Molecular orbital configuration of dioxygen

ˇŒ" Œ" ˇŒ"# Œ ˇ3  ˇ1 ˇ † g > g >

Œ Œ"# ˇŒ" Œ# ˇ1 ˇ g > ˇˇ1 †C > :

(2)

g

ˇ ˇ Exchange interaction stabilizes the triplet state ˇ3 † g >, while the degenerate singlet ˇ ˇ ˇ1 ˇ state ˇ g > is higher in energy by 22 kcal/mol, and the ˇ1 †C g > state is the ˇ ˇ3  uppermost one (37 kcal/mol above the ground triplet ˇ †g > state). Interelectronic repulsion between the two electrons in the two closely space-distributed 1 g –MO’s, expressed through exchange  integrals, is relatively high. This is an additional reason 3  for a barrier in the O2 †g reactions with the closed shell molecules. ˇ ˇ ˇ ˇ1 C The ˇ3 † g > and ˇ †g > states are mixed by weak internal magnetic perturbation, the so-called spin-orbit coupling (SOC) (Minaev 2007; Minaev and Lunell 1993). Some small admixture of one state (about 1.3 %) is present in the other state, when SOC is additionally accounted for in the nonrelativistic Schrödinger equation. Such a small correction is very important because it makes the forbidden

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

9

S-T transitions become allowed and can explain large difference in intensities of 1 3  1 C the atmospheric oxygen bands 3 † g  g .1270 nm/ and †g  †g (754 nm) (Minaev 1980, 1989, 2007; Minaev and Ågren 1995, 1996,; Minaev and Ågren 1997; Minaev and Minaeva 2001; Minaev et al. 1993; 1996; 1997; 2008; Ogilby 1999; Paterson et al. 2006; Schweitzer and Schmidt 2003). In the following analysis we want to understand the spin-dependent mechanism of the O2 binding step once dioxygen has moved from the solvent through the protein chains and reached either the myoglobin distal cavity near the ferrous-heme cofactor or the flavin moiety in the active site of the GO enzyme. In both cases a similar SOC effect of the S-T states mixing is responsible for the biochemically important oxygen activation (Minaev 2002, 2007; Minaev and Ågren 1995; Prabhakar et al. 2002). The heme-O2 binding step has been extensively studied after flash-photolysis (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002), which shows interesting kinetic features, like non-homogeneous decay, recombination barriers, etc., and indicates complicated spin-dependence (especially in comparison with NO and CO binding to myoglobin). Kinetics of dioxygen reaction with GO has been investigated by studies of deuterium isotope effect during oxidation half reaction (Klinman 2001; Prabhakar et al. 2002). Explanation of such spin-dependence is the main purpose of the present review. In the beginning we want to consider some general features of spin-effect manifestation in chemistry and in dioxygen activation.

Spin-Prohibition of Dioxygen Reactions Spins can undergo “depairing” when exposed to light; in (Eq. 3) an electron goes from a doubly occupied orbital Œ"# of the ground singlet state to a vacant MO [] with the simultaneous spin flip to produce the triplet excited state (Minaev 2007): ŒŒ"# Œ C h ) ŒŒ" Œ"

(3)

According to the Pauli principle, both spins can be parallel (total spin S D 1) in such an excited state when two electrons occupy two different MOs. This triplet state has three possible orientations of the total spin vector, thus the singlet ! triplet excitation includes three possible transitions to three spin sublevels. All of them are spin-forbidden. This is a very strict prohibition, since it can be removed only by influence of magnetic interactions that are much weaker in general than the electric interactions. The latter determine the energetics of chemical bonding, electronic “depairing” excitation and the pathways of chemical reactions. The spin influences the energy through exchange interaction (Minaev 2007). A weak SOC slightly mixes the singlet and triplet states of molecules, which provides a non-zero rate for the T ! S transitions. The late are observed in the form of phosphorescence and are well known as important quenching processes in photochemistry (Minaev 2004, 2007; Minaev and Minaev 2005). The nonradiative T-S transitions also play an important role in dark reactions, in particular, in catalysis (Metz and Solomon 2001;

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B.F. Minaev et al.

Minaev 2002). Weak SOC acts as a “key” needed to open a “heavy door”: that is, the system chooses a pathway of chemical reaction with low activation barrier in the triplet state instead of overcoming a high activation barrier in the singlet state. We remind that the exchange integral appears with different signs in the energy expression for the S and T states, namely, two radicals form a chemical bond in the S state and repel each other in the T-state (Friedman and Campbell 1987). The different behavior of the S and T states is important not only for radical reactions, but also for many chemical transformations that include spin “depairing” during bond scission or proceed through biradical intermediates. This often occurs in catalysis by transition metal compounds (Minaev et al. 2007; Prabhakar et al. 2004; Sheldon 1993), especially in hemoproteins (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002; Stryer 1995). The complete occupation of the degenerate-shell, Eq. 2, requires adding two additional electrons in order to become diamagnetic species, like usual chemical substance. Therefore, the oxygen molecule has a very strong tendency to take electrons from other substances and to make the complete electron-pairing in its unoccupied orbitals. This leads to the sequential formation of highly reactive or 2 toxic oxygen species such as the superoxide anion O 2 , peroxide anion O2 , as by-products of many normal cellular metabolisms. Consequently, the development of enzymes to protect cells against such “oxidant stress” was of great urgency to aerobic organisms. This resulted in the ubiquitous occurrence of superoxide dismutase, catalase, peroxidase, and so on (Shikama 2006).

Cytochrome Oxidases and Related Heme-Containing Enzymes Many heme-containing enzymes share the same active-species of the iron-ion types, but at the same time they exhibit significant differences in biochemical properties and reactivity with complicated spin-dependent kinetics (Burnold and Solomon 1999; de Winter and Boxer 2003; Kumar et al. 2005; Metz and Solomon 2001; Orlova et al. 2003; Shaik et al. 1998, 2002, 2005). Some of these heme-containing enzymes, cytochrome P450, cytochrome c oxidase, and horseradish peroxidase (HRP), are utilized as the active site of a high-valent oxo-iron heme species, called compound I (Shaik et al. 2002). Shaik et al. (Kumar et al. 2005; Shaik et al. 2002, 2005) proposed that the diverse reactions catalyzed by cytochrome P450 could be explained in terms of the difference in reactivity between the high-spin and low-spin states of the compound I. Their proposed two-state reactivity concept (Kumar et al. 2005; Shaik et al. 2002, 2005) has been widely used for the cytochrome P450 family and based on spin conversion transition between these states during reactions. HRP, being similar to cytochrome P450 by crystal structure of the active site, differs in axial (proximal) ligand bound to the iron; P450 has a thiolate ligand of a cysteinate side chain, whereas HRP binds an imidazole group of a histidine residue side chain (Kumar et al. 2005). Although HRP functions predominantly as an electron sink (Kumar et al. 2005), it is also known to perform oxygentransfer reactions (Afanasyeva et al. 2006; Chalkias et al. 2008; Shaik et al. 2002).

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

11

Generally, however, the reactivity of HRP enzymes is sluggish compared with the P450 enzymes family; sulfoxidation is the most efficient while C-H hydroxylation is rather sluggish. This originates mainly in the much smaller substrate-binding pocket in HRP (Afanasyeva et al. 2006; Shaik et al. 2002). The recent DFT study of sulfoxidation by HRP and by cytochrome P450 reveals a spin-state selection dependency on the proximal ligand: for thiolate ligand the reaction prefers the high-spin quartet state path while for the imidazole ligand the reaction mechanism involves a two-state reactivity (Kumar et al. 2005). This spin pattern also obeys orbital-selection rules, which are derived from the MO involvement of the ligands into the key oxo-iron porphyrin orbitals (Kumar et al. 2005; Shaik et al. 2005). In the case of the two-state reactivity the SOC-induced spin-transitions between the high-spin and low-spin reaction pathways could be magnetic-field dependent, if the viscosity of the media is sufficiently high (Afanasyeva et al. 2006; Buchachenko and Kouznetsov 2008; Buchachenko et al. 2005; Chalkias et al. 2008; Grissom 1995; Minaev 1983; Minaev and Lunell 1993; Serebrennikov and Minaev 1987). This is because SOC in the active sites is highly anisotropic and provides different rate constants for different spin-sublevels in zero field (Minaev 2002; Minaev and Lunell 1993), which become modified in the external field making the reaction to be magnetic-field dependent (Buchachenko and Kouznetsov 2008; Serebrennikov and Minaev 1987). Such a dependence is found for the HRP-catalyzed oxidation of NADH, which can be presented by the schemes (Afanasyeva et al. 2006):

Per

2C

HRP C NADH ! Per2C C NADHC ! C O2 ! Comp III CNAD ! Comp I C NADH ! Comp II;

(4)

where compound II represents the Fe(IV) D O group connected with the porphyrin ring, compound I represents Fe(IV) D O connected with the radical cation of porphyrin, and compound III represents the porphyrin-containing Fe(IV)–OOH group. Here porphyrin denotes the protoporphyrin IX linked with protein chain by the histidine residue. The scheme (Eq. 4) is compatible with the magnetic-field dependence of the rate constants measured by stop-flow spectroscopy (Afanasyeva et al. 2006). The catalytic cycle starts with electron transfer from NADH to native HRP (Per3C ) to produce NADHC• radical and the ferroperoxidase intermediate (Per2C ); (Per3C ) actually means here the ferric porphyrin-Fe(III). Interconversion of NADH to form NADC occurs by hydride transfer, in which the HC ion and two electrons are transferred between the C(4) carbon atom of nicotinamide ring and substrate (Minaev et al. 1999). Catalytic cycle of NADH oxidation in the presence of hydrogen peroxide begins with a two-electron oxidation of the HRP enzyme to form FeIV and the porphyrin cation radical; this compound I is a highly reactive species that can accept one electron from NADH to form a NADHC• radical cation, which can undergo deprotonation to yield NAD• and compound II (Afanasyeva et al. 2006; Grissom 1995). The process is competitive with reverse electron transfer to regenerate NADH and Comp I, as shown by chemically induced dynamic nuclear polarization (CIDNP) of one of the C(4) protons

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B.F. Minaev et al.

in the NMR spectrum of NADH in presence of HRP (Afanasyeva et al. 2006). An alternative explanation of CIDNP is also possible (Minaev 2002; Serebrennikov and Minaev 1987), based on account of anisotropic SOC in compound I (Minaev 2010; Minaev and Lunell 1993; Minaev et al. 2007). The magnetic-field dependence of the enzymatic reaction helps to reveal electron spin correlation among the catalytic states. Paramagnetic intermediates in the HRP catalytic cycle are difficult to detect, since direct observation by the EPR method depends on trapping unstable kinetically relevant radicals that may be transient or present at insignificant concentrations. Compound I is illusive in the sense that it does not accumulate in the cycle. This transient radical was inferred by cryogenic EPR/ENDOR technique (Davydov et al. 2008). The cryoreduction of the EPR-silent compound II produces Fe(III) species retaining the structure of precursor [Fe(IV) D O]2C or [Fe(IV) D OH]3C ; thus the EPR spectra of cryoreduced HRP II provide evidence of the low-spin hydroxyFe(III) heme species (Davydov et al. 2008; Kumar et al. 2005). The first stage (Eq. 4) is thought to be magnitosensitive (Afanasyeva et al. 2006). Electron transfer from NADH to native HRP produces the quartet (Q) radical pair of Per2C and NADHC• , which is unreactive toward recombination. The quartet – doublet (D) transition produces a radical pair that can recombine and quench the enzymatic process. It is claimed (Afanasyeva et al. 2006) that the Q-D transition is governed by isotropic hyperfine interaction and radical g-factors in terms of radical pair theory (Kumar et al. 2005). This Q-D transition is presented in the left part of the following scheme: Œ" Œ" Œ" HFC ! Œ" Œ" Œ# Œ" Œ" Œ" SOC ! Œ"# Œ  Œ" Q D Q D radical pair theory spin -orbit coupling

(5)

Here the set of three orbitals represents the triplet state of the Per2C active site and one unpaired spin in the NADHC• radical. The ferro-porphyrin intermediate (Per2C ) is known now to have a quintet ground state (Davydov et al. 2008) as follows from DFT calculations (Shaik et al. 2005). Thus, the main idea of the magnetic field mechanism (Afanasyeva et al. 2006) gives rise to some doubt. The most important objection is connected with the possible account of SOC, illustrated on the right side of the scheme (Eq. 5). It is possible to construct a number of configurations that could be definitely admixed to the open shell ground states of the Per2C active site, which provides a strong SOC between Q-D states of the radical pair (Eq. 4). Such an SOC-induced mechanism could be applied to a number of magnetic field effects in metal-containing enzymes (Davydov et al. 2008; Engstrom et al. 1998, 2000; Gegear et al. 2008; Hoff 1986; Johnsen and Lohmann 2005; Maeda et al. 2008). In the context of dioxygen activation we have to mention enzymes with binuclear metal centers, like cytochrome oxidase (Blomberg et al. 2000a, 2005), hemerythrin, and hemocyanin (Burnold and Solomon 1999; Metz and Solomon 2001). Generally speaking, the enzymes containing iron and copper active sites (Metz and Solomon 2001; Prabhakar et al. 2004) play key roles in dioxygen activation by generation of a

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

13

peroxo intermediate; either O2 is reduced by two electrons, provided by a binuclear metal site, or one electron is provided by metal and a second electron by a cofactor (Prabhakar et al. 2004). Cytochrome oxidase catalyzes the four-electron reduction of oxygen molecule to water (Stryer 1995). No intermediates were detected in the reaction O2 C 4e C 4HC D 2H2 O. However, many experimental measurements (Stryer 1995) have proved the formation of O2 2 . The reaction center of cytochrome oxidase includes one heme ferrous ion and one copper ion (Blomberg et al. 2000a). The oxygen molecule binds to the heme Fe2C cation and to the CuC ion that donates one electron each to form an O2 2 anion. This provides a way to overcome the major obstacle to oxygen activation, that is, spin inversion (T-S transition). The O 2 and O2 species have the following ground state configurations (in addition to Fig. 1): 2 Œ"# Œ" Œ" Œ"# Œ"# Œ#" ˇ ˇ2 ˇ ˇ …g > O ˇ 2 …g > O2 ˇˇ1 †C > : O g 2 2 2

(6)

Since the O2 2 dianion has a filled electron shell, the ground state of this species is totally symmetric and characterized by the term 1 †C g . Transfer of two electrons causes the ground-state term, 3 † of the O molecule (Fig. 1), to transform 2 g 1 C smoothly to the term †g of the dianion. The spin-orbit coupling between the states 3  †g and 1 †C g is symmetry-allowed (Minaev 1980, 2007); therefore, the reduction O2 ! O2 2 is also symmetry allowed with inclusion of SOC. Transition of the active site of cytochrome oxidase to the singlet state removes spin prohibition for subsequent fast chemical reactions up to formation ˇ of stable diamagnetic products ˇ (Minaev 2010). The orbital doubly degenerate ˇ2 …g > ground state of the O 2 ion is split by strong SOC (Minaev 2007), since it has a nonzero orbital angular momentum. This is an important key aspect of many enzymatic O2 -activation reactions, considered in the following sections.

Dioxygen Reaction with Glucose Oxidase It is often assumed (Sawyer 1991; Shikama 2006; Stryer 1995) that one can overcome the spin prohibition to oxidation of organic substrates with atmospheric oxygen by successive addition of single electrons and protons in the successive reduction of O2 . Further reactions of diamagnetic hydrogen peroxide, produced in such reduction, are spin allowed. It is assumed (Sawyer 1991; Sheldon 1993; Stryer 1995) that removal of spin prohibition in such reactions proceeds as in the case of radical-chain oxidation, where the spin prohibition can be removed upon formation of primary radicals. It is important to stress a fundamental difference between the enzymatic reactions involving radicals and the radical reactions in chain oxidation processes. In the latter case radicals go to the bulk of the gaseous plasma flame (or in the solution bulk) and no longer retain the “spin memory” about precursors. All participants of biochemical oxidation reactions, i.e., dioxygen and

14

B.F. Minaev et al.

a

b

c

HN

+

R

R

R

HN

NH

+

N NH

NH H

O2

N

H N 1

N H

O NH

5

H3C

H R

R

R H3C

O O

O2–

O

H3C H3C

N

. +

H N 1

O

H 3C

N

N 1

+ NH

5

N H

O

NH

5

H 3C

N H

O

O

Fig. 2 An active center of the glucosoxidase model. The first phase of the catalytic cycle (not shown) includes the FAD to FADH2 reduction. (a) starting model of the dioxygen entrance to the active site; (b) electron transfer stage and spin transition in the radical pair; (c) hydrogen peroxide production. The final stage (not shown) includes the proton transfer from the N5 atom to the histidine residue along the hydrogen-bond network in peptide chain of the enzyme

electron transfer agents, are confined within the same active site of enzyme. If an electron is transferred to the oxygen molecule from a diamagnetic enzyme M, i.e., C O2 C M ! O 2 C M , it produces a triplet radical ion-pair (triplet precursor), all spins remain correlated, the “spin memory” is retained, and the spin prohibition to subsequent reactions of the radical ion-pair thus generated is not removed and cannot lead to a singlet product. For example, reaction of O2 with glucose oxidase (Klinman 2001; Minaev 2002; Prabhakar et al. 2002, 2003) involves flavin adenine dinucleotide (FAD) and includes two stages; namely, glucose oxidation to glucosolactone with reduction of FAD to FADH2 and the reverse cycle FADH2 ! FAD, with reduction of O2 to H2 O2 . From the standpoint of dioxygen activation it is interesting to consider only the second stage (Fig. 2). In this model the protonated histidine residue (His516) is included, which is in close proximity to FAD as follows from X-ray analysis (Klinman 2001). O2 can occupy the cavity between FADH2 and His516 (Fig. 2a). DFT calculations indicate that an electron is transferred immediately from the reduced cofactor to dioxygen (Fig. 2b); the electron transfer process is determined by low ionization potential of FADH2 , relatively high electron affinity of O2 (0.45 eV), and attraction of the O 2 to the protonated histidine. After formation  of a triplet radical pair, FADH2 C O2 ! FADHC 2 C O2 , the T ! S transition has to occur in order to provide the final diamagnetic products FAD C H2 O2 , to which the singlet-spin stage (Fig. 2c) precedes.

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

15

After the T ! S transition in the ion-radical pair (Fig. 2b), an ordinary chemical transformation on the singlet state PES occurs (Fig. 2c). It involves abstraction of hydrogen atom from the N1 atom of FADHC and a proton abstraction from the 2 nearest histidine residue in order to create H2 O2 by reduction of the superoxide anion. This process, accompanied by the formation of hydrogen peroxide, can occur only in the singlet state. The final phase of the catalytic cycle (not shown in Fig. 2) consists of a subsequent proton transfer from FADHC ion back to histidine across the system of H-bonds in water-protein chain (Prabhakar et al. 2003). This proton transfer does not change the number of active electrons and the closed-shell spin state. The T ! S transition has been explained (Prabhakar et al. 2003, 2004) by a relatively large SOC between the S and T states of the radical pairs (Fig. 2b), which have different orbital structures inside the superoxide ion. As one can see in the orbital-configuration scheme (Fig. 2), the T ! S transition includes an electron jump from one  g,x molecular orbital of the dioxygen to another  g,y orbital (Spin on the FADHC 2 moiety is non-active during the transition). Such transformation is equivalent to orbital rotation, or to a torque, which creates a transient magnetic field; finally this magnetic field induces a spin flip (Prabhakar et al. 2003, 2004). Thus, the reductive activation of dioxygen by GO is favored by an increased electron affinity of O2 due to the proximity of protonated His516-residue and a fast electron transfer followed by the T ! S transition induced by strong SOC in superoxide ion. This simple consideration is supported by direct quantum-mechanical calculations of the SOC integrals (Minaev 2010; Silva and Ramos 2008). Exactly the same mechanism of the T ! S transition is applied for explanation of activity of coproporphyrinogen III oxidase (CPO) in catalysis of the oxidative decarboxylation during the heme biosynthesis (Silva and Ramos 2008). Silva and Ramos have supported the so-called Lash’s model of CPO oxidation by dioxygen (Silva and Ramos 2008) and obtained the T-S crossing point that lies only 8 kcal/mol above the reactant state. They have calculated the SOC integral between T and S potential energy surfaces crossing to be equal to 77.45 cm1 in an excellent agreement with the earlier prediction for GO reaction (Minaev 2002; Prabhakar et al. 2002). A similar mechanism of the SOC during the T ! S transition has been used to explain bioluminescence induced by the luciferase enzyme (Orlova et al. 2003). Attachment of triplet dioxygen to the closed shell luciferin to yield a singlet state intermediate (which finally provides bioluminescence) is a spin-forbidden process (Orlova et al. 2003). It is important to the use of previous GO model (Minaev 2002; Prabhakar et al. 2002) that a critically significant protonated histidine residue resides in the proximity of the acidic group of luciferin (Orlova et al. 2003), quite similar to the GO active site (Fig. 2). Photodestruction of this histidine inactivates luciferase completely, thus it is found plausible that the SOC-induced spin-flip in superoxide ion (Prabhakar et al. 2002) operates in the case of oxygenation by luciferase (Orlova et al. 2003). In the following we want to show that a similar mechanism of SOC enhancement by charge transfer can be applied for spin-dependent reaction of dioxygen binding to heme.

16

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Fig. 3 A simple model of Fe(II)-porphine-NH3 -O2 model of oxyhemoglobin (heme-cofactor coordinated with O2 . Simulates the heme (with NH3 as a model of a proximal histidine residue)

Dioxygen Binding to Heme The O2 binding with myoglobin model was studied recently by DFT methods (Blomberg et al. 2005; Franzen 2002; Jensen and Ryde 2004; Jensen et al. 2005; Minaev 2007; Minaev et al. 1993, 2007; Sigfridsson and Ryde 2002; Strickland and Harwey 2007). Fully relaxed potential energy curves (PEC) were calculated for the seven lowest electronic states in Ref. Sigfridsson and Ryde (2002), Strickland and Harwey (2007), and Blomberg et al. (2005), while the PEC for spin states S D 0, 1, 2, 3 at fixed geometry as a functions of the Fe–O2 distances were presented in Ref. Franzen (2002). In this work we have recalculated some points of the fixed Fe–O distance (1.8, 2.0 and 2.5 Å) with full geometry optimization of all other parameters 0 00 for all possible multiplets and accounting for different symmetries (A and A ) for the singlet and triplet states. The model of oxyheme is shown in Fig. 3, which includes Fe(II)-porphyrin coordinated with NH3 as a model of proximal histidine residue. Its calculation gives the similar singlet ground state as for the simplified Fe(II)-Porphin-Imidazole-O2 -model, which poses Cs symmetry (Sigfridsson and Ryde 2002). The simpler Fe(II)–Porphine–NH3 –O2 model (Fig. 3) has been used in this work for DFT calculations in the vicinity of the equilibrium. The B3LYP/631G method (Frisch et al. 2003) has been employed and the results quite close to those presented in Ref. Sigfridsson and Ryde (2002) have been obtained. For the Fe(II)–Porphyrin–NH3 –O2 model, shown in Fig. 3, all vibrational frequencies and their intensity in the infrared and Raman spectra have been calculated. Many normal modes are similar to those calculated in Refs. Minaev et al. (2007) and Minaev (2002). An additional Fe–O2 stretching vibrational frequency is calculated at 539 cm1 , which agrees qualitatively well with the resonance Raman band, observed at 567 cm1 for oxy-hemoglonin by Soret excitation (Potter et al. 1987). This indicates reliability of the chosen model and of the DFT method used in this work.

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17

In the entrance channel of the O2 binding reaction to heme we have a number of different multiplets. At the infinite separation the deoxyheme has a quintet ground state with the triplet state being very close in energy. This is in agreement with the experimental data (Shikama 2006), showing that the isolated deoxyheme is a high-spin quintet (Friedman and Campbell 1987). The optimized structure of this Fe(II)P complex with NH3 at the fifth coordination position agrees with the X-ray analysis of the crystal structure of deoxymyoglobin. In this case the porphyrin Fe–N distances (2.08 Å) are larger than in the low-spin states of deoxymyoglobin (2.0 Å) and the iron ion is above the porphyrin ring plane by 0.3 Å in agreement with the Xray data (0.36 Å) (Friedman and Campbell 1987; Sigfridsson and Ryde 2002). This illustrates the well-known fact that the high-spin iron ion Fe(II)(5 D) is too large to fit into the porphyrin ring cavity (Jensen and Ryde 2004). When this   deoxyheme interacts with the triplet ground state dioxygen O2 X3 †g  , there are six unpaired electrons (two from O2 plus 4 from heme). Their interaction can provide a variety of possible total spin states. The maximum 00 spin corresponds to the septet (7 A , S D 5) state, when both subsystems have 00 parallel spins. If they are anti-parallel, the triplet state 3 A occurs. At long Fe–O distances (R > 2. 5 Å) these states are degenerate because of the absence of exchange 00 interactions between O2 and heme. The intermediate quintet 5 A state is also 00 degenerate together with the triplet and septet. The A symmetry is determined by 0 00 the oxygen degenerate  g orbitals, which have the a and a symmetry, respectively, with respect to the plane, which contains the O2 and Fe–N bond in ammonia molecule. In general, all spin states that occur at each random collision of heme and O2 should lead to oxygen binding. But the rate constants would be spin-dependent and different for the triplet, quintet, and septet states (even for different spin sublevels of one multiplet). More detailed information about O2 binding has been obtained in flash-photolysis studies of O2 dissociation from heme, when the fate of dioxygen depends on the competition between intrinsic recombination rate constant and protein relaxation, as well as the O2 escape from the protein (Shikama 2006). The greatly different recombination dynamics of O2 , CO, and NO molecules with heme have been attributed to spin states of each ligand and to their possible combinations with the iron spin (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002). The observed recombination kinetics can be influenced by protein dynamics, if the intrinsic recombination rate constant is slower than this dynamics. By studies of viscosity and temperature dependence, the general averaged time scale for the recombination rate constant can be estimated (Friedman and Campbell 1987; Jensen and Ryde 2004; Shikama 2006). It is interesting to compare O2 , CO, and NO molecules in this respect (Blomberg et al. 2005; Friedman and Campbell 1987; Strickland and Harwey 2007). The CO recombination with heme is a single exponential process characterized by a slow rate constant (k  106 s 1 ) at the ambient temperature and a low solvent viscosity (below viscosity of globin). This intrinsic (geminate) recombination rate is slower than both protein relaxation and CO escape, thus the recombination yield is very small (0.04) (Shikama 2006). The rebinding of the NO stable radical

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(doublet ground state, S D 1/2) is characterized by two-exponential kinetics with the rapid (k1  108 s 1 ) and slow (k2  5106 s 1 ) rate constants under ambient conditions (Friedman and Campbell 1987; Shikama 2006). Since the ground state heme has the quintet spin state (S D 2), there are two starting states: (S D 2½ and S D 1½) depending on mutual spin orientation of two species, while the recombination NO-heme product has the doublet state (S D 1/2) (Friedman and Campbell 1987; Strickland and Harwey 2007). Both types of geminate recombination require spin change. The slow process includes two-step spin-flip transformations (S D 2½) ! (S D 1½) ! (S D ½), since spin-orbit coupling can mix states and induces spin transition with the selection rule S D 1 (Minaev 2010). The rapid recombination occurs in one step (S D 1½) ! (S D ½). The CO molecule is diamagnetic; all spins are paired, the total spin is zero. The heme-CO adduct is also diamagnetic (S D 0). Since the ground state heme has the quintet spin state (S D 2), the geminate recombination reaction is doubly spin forbidden. First it should be a quintet-triplet transition, which needs to overcome an additional activation barrier (besides the spin flip, induced by SOC) and then a final triplet-singlet transition. This is the reason why the CO recombination with heme is so slow, in spite of the high binding energy (Franzen 2002; Jensen and Ryde 2004; Shikama 2006; Sigfridsson and Ryde 2002; Strickland and Harwey 2007). It is interesting to make the comparison: inorganic ferric ion (Fe3C ) has some catalytic activity for the decomposition of H2 O2 into water and oxygen. When the ion is incorporated into the porphyrin molecule to form heme, the molecule is about a thousand times more effective than Fe3C alone. If the protein component of the enzyme catalase then adds to the heme, the catalytic efficiency increases by a further factor of 109 times (Shikama 2006). This means that paramagnetic spin catalysis in combination with the SOC-induced spin-catalysis provide some kind of synergetic effect.

Spin-Orbit Coupling in O2 -HemeInteraction In this review we shall consider dioxygen binding in more detail, since the O2 molecule is the main subject of spin-dependent biochemical phenomena in a general context. The ground state of the oxyheme product is an open-shell singlet in agreement with EPR experiment and Mossbauer spectra (Friedman and Campbell 1987; Sigfridsson and Ryde 2002) (the closed-shell singlet has been obtained in a number of calculations (Jensen et al. 2005; Shikama 2006), however, this result has since been revised (Blomberg et al. 2005; Strickland and Harwey 2007)). Thus, the reaction of O2 binding to heme is spin forbidden. At least the T-S transition has to occur (Franzen 2002). Such spin flip can be induced by spin-orbit coupling (SOC) between the T and S states. One has to calculate the matrix element of the SOC operator, which can be presented in the effective single-electron approximation as

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry H

SO D

X A

A

X i

lEi;A  sEi D

X i

BEi  sEi D

19

X  Bi;x si;x C Bi;y si;y C Bi;z si;z : i

(7) In (Eq. 7) —A is a SOC constant for atom A (—0 D 153 cm1 ), lEi;A ; sEi – are the orbital and spin angular momentum operators for the i-th electron, respectively. This is the effective single-electron SOC approximation, which proved to be useful in many spectroscopic and chemical applications including oxygen spin-forbidden atmospheric bands (Minaev 1989; Minaev and Ågren 1997; Minaev et al. 1996, 1997, 2008; Ogilby 1999; Paterson et al. 2006; Schweitzer and Schmidt 2003) and spin-forbidden enzyme reactions (Minaev 2002, 2007, 2010; Minaev et al. 2007; Prabhakar et al. 2003, 2004). Deoxyheme has a quintet ground state (four spins are unpaired, S D 2) (Minaev et al. 2007; Sigfridsson and Ryde 2002) and the adduct with the triplet dioxygen (two unpaired spins) would be expected to have either six (4 C 2 D 6) unpaired spins or two (4–2 D 2) unpaired spins depending on ferromagnetic or antiferromagnetic relative orientation of the two magnetic moments of the deoxiheme and O2 . An intermediate quintet spin state is also possible for the ground state species coupling. The triplet state of deoxyheme, being very close in energy, produces adducts with the triplet O2 , which could be either quintet (S D 2), triplet (S D 1), or singlet (S D 0) depending on if it is ferromagnetic or antiferromagnetic coupling of two species. Thus, only triplet deoxyheme could provide the ground singlet state product in the process of the antiferromagnetic coupling with the triplet O2 in a spin-allowed oxyheme formation without spin flip. Spin transition from the ground quintet to the close-lying triplet deoxyheme can be induced by SOC in the third shell of the iron ion. In this case the primary electronic reorganization takes place in the ferrous ion at the equilibrium between the quintet and triplet states already before dioxygen approaches the deoxyheme (Franzen 2002; Jensen and Ryde 2004). All other recombination processes include spin flip induced during heme–O2 interaction; they seem to be more important for dioxygen binding (Blomberg et al. 2005; Franzen 2002; Jensen and Ryde 2004; Minaev 2007; Minaev et al. 2007; Shikama 2006; Strickland and Harwey 2007). Such natural heme – O2 reactions could start with 00 the 3 A (2) state, which is repulsive at shorter distance (R < 2.5 Å) or with the 00 septet 7 A (1) state (Jensen and Ryde 2004) (both are the ground state of the entrance channel heme C O2 and go in parallel with some other multiplets until the short distances limit (2.5–3 Å)). The energy gap is about 0.1 eV at these limits in agreement with Ref. Jensen and Ryde (2004). The optimized singlet ground state 1 0 A (1) in the reaction product is lower in energy than other multiplets at least by 0.4 eV; this oxyheme is an open-shell singlet of a complicated orbital and spin structure (Franzen 2002; Jensen and Ryde 2004; Sigfridsson and Ryde 2002). It has a short Fe–O distance (1.81 Å) (Jensen and Ryde 2004) (reproduced in our DFT calculations, 1.84 Å) in contrast to the high-spin states (2–2.7 Å). Our result is close to the Fe3C  O 2 radical-pair structure in agreement with other DFT calculations and with Weiss’ model (Franzen 2002; Jensen and Ryde 2004; Shikama 2006;

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Sigfridsson and Ryde 2002). The spin densities in oxyheme are equal to 0.94, 0.31, and 0.72 for the Fe–O–O chain, respectively, at the optimized bond angle of 118ı . The O–O bond distance (1.36 Å) and vibration frequency (1110 cm1 ) correspond better to superoxide ion (Jensen and Ryde 2004; Minaev 2007). Potential energy surfaces for O2 binding with the heme model (deoxy-Feporphyrin bound with imidazole) have been calculated by Jensen et al. (Jensen and Ryde 2004), who optimized by DFT method the Cs symmetry restricted reaction with fixed Fe–O bond length, which was systematically increased point by point. Jensen et al. (Jensen and Ryde 2004) found that the spin-change occurs easily due to a broad crossing region of five electronic states. Similar PES crossing are obtained in other studies (Franzen 2002; Minaev et al. 1993; Strickland and Harwey 2007). Accounts of our data and the results of Refs. Sigfridsson and Ryde (2002), Franzen (2002), Jensen and Ryde (2004), and Strickland and Harwey (2007) allow us to consider the following scenario of the O2 binding to myoglobin. At the intermediate 00 distances 2.5–3 Å the starting 3 A (2) state from the entrance channel transfers to the triplet Fe3C  O there are few crossing points between 2 radical-pair. In0 0this region 0 S and T states, including the 3 A (2) -1 A (1) states crossing, where spin change could occur (Franzen 2002; Jensen and Ryde 2004; Sigfridsson and Ryde 2002). A 00 0 simplified electronic structure of the 3 A (2) and 1 A (1) states near the crossing of the potential energy surfaces is presented in scheme (7). Outer electrons of the ground triplet state dioxygen in two degenerate  g –MO’s provide a scheme ["]["]; electron transfer from Fe2C to O2 in order to produce the radical pair Fe3C  O 2 (Eq. 8) can be accomplished by the occupation of either  g,x - or  g,y -orbitals. Both radical pairs could be in T and S states; all four states are almost degenerate at the intermediate distances. The most interesting spin states are those presented in scheme (Eq. 8), since they correspond to the desired T ! S transition and to the final product of the O2 binding by heme. The scheme (Eq. 8) is equivalent to the scheme (Eq. 5) and the same explanation for the high SOC matrix element (Minaev 2002; Prabhakar et al. 2002) is relevant. SOC

Œ" :::::::::: Œ#" Œ" ! Œ" :::::::::: Œ# Œ"# Fe3C :::::::::O Fe3C :::::::::O 2 2 Singlet;1 A0 .1/ Triplet;3 A00 .2/

(8)

In this model SOC perturbation occurs entirely in the oxygen moiety and the Fe(III) ion is magnetically silent. The starting triplet radical pair corresponds 00 to ˇa charge-transfer described by 3 A (2) wave function 3 ‰C Tx D    (CT)   state ˇ < ˇ.3d ˛/ g;x ˇ g;x ˛ g;y ˛ ˇ, that is, transfer of an electron to the  g,x -orbital of O2 molecule, whereas the singlet radical ˇpair corresponds    to a CT   state described ˇ 0 by 1 A (1) wave function 1 ‰C Ty D < ˇ.3d ˛/ g;x ˇ g;y ˛ g;y ˇ ˇ and a transfer of an electron to another degenerate orbital of oxygen,  g,y . Here < means a proper anti-symmetrization of the wave function. Spin-orbit coupling between these CT states is the maximum possible for a system comprised of such light atoms like oxygen (Minaev 2002, 2004, 2007). The considered matrix element of the SOC operator (Eq. 7) is easily estimated to be equal:

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

ˇ Z ˇ1 0 ˇ ˇ ˇ A .1/ > D D

1

(9) The value is identical to those calculated for similar charge-transfer states including the GO (Minaev 2002; Prabhakar et al. 2002) and coproporphyrinogen III oxidase (Silva and Ramos 2008). This SOC matrix element (Eq. 9) is very close to a value of about 80 cm1 , postulated in estimation of the Landau-Zener rate constant for T-S transitions in spin-dependent O2 and NO binding to heme (Franzen 2002; Friedman and Campbell 1987; Petrich et al. 1988). With such a proposal for the generally unknown SOC integral (Shaik et al. 2002) a quite reasonable estimation for the spin-dependent rate constants of the CO, O2 , and NO recombination in heme proteins are obtained (Franzen 2002; Petrich et al. 1988). For CO binding to heme a quintet-triplet-singlet step-wise transition is necessary, which explains the million times slower recombination rate in this case in comparison with the O2 and NO recombination in heme proteins (Franzen 2002; Shaik et al. 2002; Sigfridsson and Ryde 2002). The gradient difference at the location of crossing points that enters the denominator of the Landau-Zener expression of the rate constant for spin transition is quite small (0.1–0.2 eV/Å) for O2 and NO binding to heme (Franzen 2002; Jensen and Ryde 2004), thus the topology of the binding curves supports a rapid recombination of both ligands to hemoglobin and myoglobin. The rapid NO rebinding to heme (k1  108 s1 ) includes one-step quartet-doublet transition; since the NO radical has one outer electron at the degenerate  x and  y orbitals, a quite similar theory of SOC in quasi-degenerate charge transfer states, like that, presented in Eqs. 8 and 9, can be applied. The only difference is that the SOC integral (9) now includes  x and  y orbitals of the NO molecule and thus is slightly smaller (about 60 icm1 ). The rate constant of spin transition in the Landau-Zener model is determined by the square of the SOC integral. This explains that the rapid NO rebinding rate constant is about three times slower than the rapid rate constant of the O2 recombination in heme proteins (Franzen 2002; Jensen and Ryde 2004; Shikama 2006). All previous analyses of SOC effects in hemoproteins were based on the assumption that the SOC integral in dioxygen binding to heme is determined by the iron ion and no attempt at direct calculation has been done (Blomberg et al. 2005; Franzen 2002; Jensen and Ryde 2004; Sigfridsson and Ryde 2002; Strickland and Harwey 2007). As follows from our simple analysis, the SOC integral (Eq. 9) is determined entirely by SOC in oxygen molecule and is connected with the degeneracy of the two  g,x - and  g,y -orbitals in the open-shell of dioxygen. This enhancement of SOC effect by inclusion of charge-transfer to O2 and involvement of superoxide-ion structure seems to be quite general in biochemistry (Minaev 2007, 2010); it is applied also to those enzymes that have no transition atoms, like glucose oxidase, or coproporphyrinogen III oxidase (Minaev 2007; Prabhakar et al. 2002; Silva and Ramos 2008). It is important to stress that the rapid T-S transition in O2 binding to heme is not only forbidden by spin, but also by orbital symmetry 00 0 (it includes the A – A symmetry change). Such double prohibition is necessary

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in order to make the spin change in chemical reaction to be effectively allowed (Minaev and Lunell 1993). The iron ion can also contribute to the SOC integral in the O2 binding to heme. In general, contributions from different atoms can cancel each other in SOC calculations; it happens when orbital rotation for different atoms occurs in opposite directions along the same axis during the T-S transition (Minaev 2007; Minaev and Lunell 1993). This type of SOC suppression is known in the form of the destructive external heavy atom effect (Minaev 2010). However, such an effect cannot be applied for SOC analysis in heme-O2 interaction. The main axis of the third-orbitals rotation in ferrous ion during the T-S transition is perpendicular to the porphyrin tetrapyrrole plane, while the SOC contribution from dioxygen is determined by the  g,x - and  g,y -orbitals’ rotation around the O–O axis and the two axes do not coincide. The angle of 118ı between these axes excludes possibility of cancellation: only the sum of the squares of the x,y,z projections, of the type given by Eq. 9, contributes to the final SOC value, which determines the rate constant of the T-S transition in terms of the Landau-Zener approximation (Harvey 2004; Kondo and Yoshizawa 2003; Minaev 2002; Minaev and Ågren 1995; Poli and Harvey 2003).

The Role of Spin in Biochemistry We can see that spin-dependent quantum effects and transformations are important in a number of biochemical processes (Afanasyeva et al. 2006; Blomberg et al. 2000a, 2005; Buchachenko 1977; Buchachenko and Kouznetsov 2008; Buchachenko et al. 2005; Burnold and Solomon 1999; Chalkias et al. 2008; Davydov et al. 2008; de Winter and Boxer 2003; Franzen 2002; Grissom 1995; Jensen and Ryde 2004; Jensen et al. 2005; Kumar et al. 2005; Lane 2002; Metz and Solomon 2001; Minaev 1983, 2002, 2007, 2010; Minaev et al. 1999, 2007; Orlova et al. 2003; Petrich et al. 1988; Prabhakar et al. 2002, 2003, 2004; Serebrennikov and Minaev 1987; Shaik et al. 1998, 2002, 2005; Sigfridsson and Ryde 2002; Silva and Ramos 2008; Strickland and Harwey 2007). Spin is a very fundamental quantum phenomenon associated with the structure of space-time (Esposito 1999; Hameroff and Penrose 1996; Hu and Wu 2004; Penrose 1960). Various models of elementary particles in modern physics and even space-time itself are built with spinors (Esposito 1999; Penrose 1960). Spin of the Dirac electron is qualitatively shown to be responsible for all known quantum effects and the quantum potential is a pure consequence of “internal motion” evidencing that the quantum behavior is a direct consequence of the fundamental existence of spin (Penrose 1960). These results have been expanded recently by deriving a spin-dependent gauge transformation between the Hamilton-Jacoby equation of classical mechanics and the time-dependent Schrödinger equation of quantum mechanics that is a function of the quantum potential in Bohm mechanics (Esposito 1999). It is quite natural that the most mysterious peculiarities of living matter, the neuron network machinery of brain, hemoglobin, cytochromes, and oxydases, cannot avoid utilizing such a fundamental property of electrons like spin and quantum behavior. The occurrence

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

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of non-zero electron spin in any bio-system indicates that we are at a level of the life science where classical concepts are insufficient for its proper description and understanding. Electron spin is much more important than nuclear spin since it determines the exchange interaction, the most significant part of chemical forces, which are finally responsible for the functions of cells and for metabolism processes.

External Magnetic Field Effects in Biochemistry Regardless of the comments above, we need to speak about nuclear spins, since they are everywhere (at least in the form of protons) and because they also can be connected with quantum effects in bio-systems. Recent experiments (Buchachenko and Kouznetsov 2008; Buchachenko et al. 2005) demonstrate that intramitochondrial nucleotide phosphorylation is a nuclear spin controlled process because the magnetic magnesium isotope 25 Mg(II) increases the rate of mitochondrial ATP synthesis in comparison with the spinless nonmagnetic 24 Mg(II), 26 Mg(II) ions. Such nuclear spin isotope effect is usually interpreted in terms of radical pair theory for separated spins in solvent cage (Buchachenko 1977), but an alternative explanation based on electronic spin-transition in the active center (Minaev 1983; Minaev and Ågren 1995; Serebrennikov and Minaev 1987) is also possible. New developments in magnetic resonance imaging of the brain demonstrate that induced quantum coherences of proton spins separated by long distances ranging from thousands to millions of nanometers (from 1 m to 1 mm) are sustained for milliseconds and longer (Hu and Wu 2004). While these quantum couplings are not the type of quantum processes that are likely to prove useful in brain function (Hagan et al. 2002), they nonetheless show that biology can take advantage of mesoscopic quantum coherence in clever ways. These quantum modes are not the stable entangled superpositions required for quantum computations in brain, but the proton spins’ coherence can indeed survive in DNA, peptides, and in the brain’s milieu (Hameroff and Penrose 1996; Hu and Wu 2004). Some new theory of consciousness is based on an idea that spin is the “mind-pixel” (Hagan et al. 2002; Hameroff and Penrose 1996; Hu and Wu 2004); it postulates that consciousness is intrinsically connected to quantum spin, since the latter is the origin of quantum effects in Bohm formalism and a fundamental quantum process associated with the structure of space-time (Esposito 1999; Hagan et al. 2002; Hameroff and Penrose 1996; Hu and Wu 2004; Penrose 1960). Spin involvement in the theory of consciousness can be connected also with memory storage and magnetic perturbations. External magnetic field effects (MFE) in biochemistry are well known (Afanasyeva et al. 2006; Buchachenko and Kouznetsov 2008; Buchachenko et al. 2005; Chalkias et al. 2008; Davydov et al. 2008; Engstrom et al. 1998, 2000; Grissom 1995; Hoff 1986; Minaev et al. 1999; Serebrennikov and Minaev 1987) and could be analyzed in terms of competition with the internal magnetic interactions including SOC as the most important one (Minaev 1983; Minaev and Ågren 1995;

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Minaev and Lunell 1993) and hyperfine coupling (Burnold and Solomon 1999; Metz and Solomon 2001; Minaev 2004); the spin-dependent exchange interactions are also involved in the MFE theories as well as the whole spin-catalysis concept (Minaev and Ågren 1996; Orlova et al. 2003). MFE on enzymatic reactions have been first interpreted (Grissom 1995) entirely in the context of RPT (Buchachenko 1977). It was clear since the 1980a that RPT cannot explain all variety of MFE in chemistry and biology and SOC effects could be taken into account (Minaev 1980, 1983). But only recently, a number of publications (Afanasyeva et al. 2006; Chalkias et al. 2008; Minaev 2002; Minaev and Ågren 1995) examined the external MFE in terms of SOC effects on the kinetics of enzymatic reactions. Activity increase of horseradish peroxidase (HRP) in the presence of magnetic particles (Fe3 O4 ) (Chalkias et al. 2008) and external MFE in enzymatic oxidation of NADH by HRP (Afanasyeva et al. 2006) are the most interesting examples. Interconversion of NADH to form NADHC• and NADC occurs by hydride transfer, in which HC ion and two electrons are transferred between the C(4) carbon atom of nicotineamide ring and substrate (Afanasyeva et al. 2006). Catalytic cycle of NADH oxidation in the presence of hydrogen peroxide begins with a two-electron oxidation of the HRP enzyme to form FeIV and the porphyrin cation radical; this compound I is a highly reactive species that can accept one electron from NADH to form NADHC• radical cation, which can undergo deprotonation to yield NAD• and compound II (Afanasyeva et al. 2006; Grissom 1995). MFE on the photosynthetic light-harvesting reaction center in bacterium Rhodopseudomonas sphaeroides was the first successful observation and interpretation of spin-dependent intermediates in biology (Grissom 1995; Hoff 1986). The quantum yield of the triplet chromophore decreases by 50 % at 0.05 T. As the applied magnetic field increases, the Zeeman interaction between MS D ˙1 spin sublevels grows and causes a decrease of hyperfine coupling between triplet T(˙1) and singlet S states, thus only T(0)-S states can interconvert. At high magnetic fields greater than 0.5 T the quantum yield of the triplet chromophore begins to increase slowly and attains parity with the zero-field (B D 0) value at B D 3 Tesla. At higher fields a net increase is detected, which is explained by T-S transitions via g mechanism (Afanasyeva et al. 2006; Buchachenko 1977; Engstrom et al. 2000). Such biphasic MFE dependence on B is typical for many photochemical (Hoff 1986; Serebrennikov and Minaev 1987) and enzymatic (Afanasyeva et al. 2006; Grissom 1995) reactions that occur through the radical pair mechanism. No total MFE on the photosynthetic reaction center would be expected since the total quantum yield is composed of a series of coupled vectorial processes that create an irreversible free energy cascade and drive proton pumping (Grissom 1995; Hoff 1986). For similar reasons it is impossible to expect other MFE with biological relevance. For example, MFE is possible, in principle, for the spin inversion step in GO, if the rotation of reactants in the step shown on Fig. 2b would be frozen. The T-S transitions for different MS D˙1,0 spin sublevels are characterized by different rate constants even in zero field in the fixed molecular frame. But rotation at room temperature quenches all anisotropy of zero-field splitting and SOC-induced

Spin-Orbit Coupling in Enzymatic Reactions and the Role of Spin in Biochemistry

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rate constants (Serebrennikov and Minaev 1987). That is why no MFE has been observed for glucose oxidase in vivo (Chalkias et al. 2008). Many mammals, reptiles, birds, insects, and fish are known to use the Earth’s magnetic field for orientation and navigation (Gegear et al. 2008; Johnsen and Lohmann 2005; Maeda et al. 2008). Birds have been studied very intensively, but biophysics and neurobiology of their avian magnetoreception are still poorly understood (Grissom 1995; Johnsen and Lohmann 2005). Maeda et al. (2008) have recently proposed a caratenoid-porphyrin model system to demonstrate that the lifetime of a photochemically induced radical pair is changed by MFE with a low magnetic field of about 50 T, which is comparable with the Earth’s magnetic field. Radiofrequency MFE can disrupt the ability of birds to orient and also has profound influence on radical pair reactions in vitro, which supports the RPT concept (Maeda et al. 2008). The recent DFT study of the electronic mechanisms of oxidative cleavage of the O–O bond by apocarotenoid oxygenase (ACO), which occurs in dark condition and includes a number of biradical and radical stages, is important additional support to the caratenoid-porphyrin model of Maeda et al. (2008). In this context it is interesting to pay attention to molecular oxygen. Each O2 molecule contains two unpaired electron spins and is a paramagnetic species capable of producing a fluctuating magnetic field along its diffusing pathway. Thus, O2 serves as a natural contrast agent in magnetic resonance imaging (Hu and Wu 2004). The existence of unpaired electrons in stable molecules is very rare indeed. Dioxygen is the only paramagnetic species found in large quantities in the brain (besides the nitric oxide) and in other aerobic tissues. The singlet excited 1 g oxygen (Minaev 2007) is a metastable species with a relatively short lifetime (few microseconds in water) (Schweitzer and Schmidt 2003) and long diffusion path (Ogilby 1999); it could serve as a light signal transmitter or transfer other types of information along definite distance in tissue. Generation and quenching of the singlet 1 g oxygen are governed by SOC and other quantum perturbations (Minaev 1980, 1989, 2007; Minaev and Ågren 1995, 1996; Minaev and Ågren 1997; Minaev and Minaev 2005; Minaev and Minaeva 2001; Minaev et al. 1993, 1996, 1997, 2008; Ogilby 1999; Paterson et al. 2006; Schweitzer and Schmidt 2003). But the ground state triplet3 †g oxygen bound and transported by hemoglobin is a source of many other quantum effects besides those described above.

Spin Neuroscience Conventional neuroscience has been unable to provide a complete understanding of cognitive processes (Esposito 1999; Hagan et al. 2002; Hameroff and Penrose 1996; Hu and Wu 2004; Penrose 1960). It has been accepted that the brain can be simulated as a neural network organized by the principles of classical physics (Hameroff and Penrose 1996; Shaik et al. 2002). Such an approach has delivered successful implementation of learning and memory and promoted optimism that a sufficiently complex artificial neural network would (in principle, at least) reproduce the full extent of brain processes involved in cognition and consciousness (Hagan

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et al. 2002). But the functioning of the nervous system lie outside the realm of classical mechanics; one finds ample support for this in analysis of sensory organs of vision, the operation of which is quantized at levels varying from the reception of individual photons by the retina to thousands of quanta in the auditory system (Hagan et al. 2002; Hameroff and Penrose 1996). It is also proposed that synaptic signal transmission has a quantum character (Hagan et al. 2002). The PenroseHameroff orchestrated objective reduction (OOR) model (Hameroff and Penrose 1996) of quantum computation in microtubules within neurons of the brain is compatible with modern physiology and can generate testable predictions. Coherent superpositions of tubulin proteins are unstable in the OOR model and subject to selfcollapse under quantum criterion. The phase of quantum superposition/computation is a preconscious process and each self-collapse event corresponds to “moment of conscious experience.” An apparent shortcoming of the Penrose-Hameroff OOR model and other models of quantum processes relevant to consciousness is the question of environmental decoherence (Hagan et al. 2002). The dioxygen utilization in living matter provides many interesting manifestations of spin-dependent quantum effects that are under a control by very weak relativistic perturbations (Minaev 2007). This illustrates a long-lived quantum coherence in O2 binding to hemoglobin (Franzen 2002; Friedman and Campbell 1987; Shikama 2006). Spin coherence can occur in photosynthetic systems, in oxidases, and in cytochromes. It depends on hyperfine interactions and spin–spin coupling in the ensembles of paramagnetic complexes. The decoherence can be induced by spin-flip and this review illustrates how to calculate and interpret such phenomena. It seems promising that the SOC-induced spin effects connected with dioxygen activation in bio-systems could be useful in other areas of life science.

Conclusions In this review we consider spin effects in a number of biochemical processes with special attention to O2 activation by heme, oxidases, and cytochromes. The universal role of the spin-orbit coupling (SOC) effect in such activations is stressed. Oxidation of organic materials by dioxygen, to give CO2 and water, is very favorable (exothermic); fortunately, the unfavorable kinetics preclude this spontaneous combustion of living matter into “a puff of smoke” (Sheldon 1993). The reason for the sluggish O2 reactivity is a spin barrier: the direct reaction of triplet dioxygen with singlet organic molecules to give stable diamagnetic products is a spin-forbidden process with a very low rate. The common way of circumventing this kinetic barrier via a free radical pathway (combustion) is impossible for living cells. The reaction of singlet (diamagnetic) molecules with 3 O2 forming two radicals is a spin-allowed process. Usually it is highly endothermic; Sheldon provides as an example of such a reaction observed at moderate temperature the activation of 3 O2 by flavin-dependent oxygenases (Sheldon 1993). He argues that it is possible only for very reactive substrates (flavins) that form resonance-stabilized free radicals,

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e.g., reduced flavins, like FADHC• , and HO•2 as a counterpart. We have stressed that, even in this case, the complete oxidation in the enzyme (e.g., glucose oxidase) is spin-forbidden if the HO•2 radical is not the final product that leaves the active center of enzyme and goes into the bulk of the cell, but still participates in further oxidative transformations (Minaev 2002). We have shown that flavin-dependent oxygenases provide a very efficient way to overcome the spin-prohibition by spin-orbit coupling perturbation at the stage of electron transfer (Eq. 4). A second way to overcome the obstacle of spin conservation is for 3 O2 to combine with a paramagnetic transition metal ion (Burnold and Solomon 1999; Metz and Solomon 2001; Prabhakar et al. 2003, 2004). Such reactions are spinallowed and governed by exchange interactions (exchange-induced spin-catalysis) (Minaev 2007; Prabhakar et al. 2004). Peroxo-, dioxo-, and superoxo-complexes with metals in different oxidation degree are considered during the search of “dream reactions” for selective industrial catalysts (Sheldon 1993). The expectation that the resulting metal-dioxygen complexes may react selectively with organic substrates at moderate temperature forms a background for the extensive studies of metalcatalyzed oxidation during the last four decades (Prabhakar et al. 2002, 2004). In this respect the reactions of 3 O2 with hemoglobin and myoglobin at moderate temperature are very peculiar, since they are still spin-forbidden and quite efficient (Franzen 2002; Jensen and Ryde 2004; Sigfridsson and Ryde 2002). We have recalculated some potential energy surface cross-sections for different multiplets along the heme-O2 binding reaction coordinate and obtained results in agreement with Refs. Sigfridsson and Ryde (2002), Franzen (2002), and Jensen and Ryde (2004). The Fe(II)porphine molecule coordinated with ammonia molecules (Fig. 3, NH3 as a model of the proximal histidine) are used as in other similar studies (Blomberg et al. 2005; Franzen 2002; Jensen et al. 2005; Sigfridsson and Ryde 2002; Strickland and Harwey 2007). The more realistic protoporphyrin IX model provides similar result for the ground state of the heme active site. Results of previous works (Blomberg et al. 2005; Franzen 2002; Jensen and Ryde 2004; Minaev 2007; Minaev et al. 2007; Sigfridsson and Ryde 2002; Strickland and Harwey 2007) indicate that the main reason for the facilitated binding of O2 to heme is a broad crossing region of the relevant spin states, which provides significant spin transition probabilities. They have shown that porphyrin is an ideal Fe(II) ligand for the spin-flip problem, because it tunes the spin states to be close in energy, giving parallel binding potential energy surfaces, small activation energies and large transition probabilities in terms of the Landau-Zener approach (Prabhakar et al. 2003). But none of these studies (Blomberg et al. 2005; Franzen 2002; Jensen and Ryde 2004; Jensen et al. 2005; Sigfridsson and Ryde 2002; Strickland and Harwey 2007) have considered the reason for relatively large spin-orbit coupling, which induces the necessary spin flip in the heme-O2 binding reaction; a general assumption that the SOC integral at Fe(II) ion of about 80 cm1 , postulated in Ref. Petrich et al. (1988), has been put forward instead. We have shown that such SOC integral (Eq. 9) is determined entirely by SOC in the oxygen moiety and is connected with the two  g,x - and  g,y orbitals in the open-shell of dioxygen. This is also connected with charge transfer

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1 (CT) and with the Fe3C  O 2 radical-pair structure of the ground state A (1) 3 00 and the close-lying A (2) state near the crossing of the potential energy surfaces, 00 see scheme (Eq. 8). This scheme indicates that the triplet and singlet states, 3 A 1 0 (2) and A (1), differ by a single electron jump inside O2 from the  g,x MO to the  g,y orbital. Such transformation is equivalent to the electronic orbital rotation, i.e., a torque, which creates transient magnetic field during the T-S transition and this magnetic field is responsible for the spin flip. In this model the magnetic perturbation occurs entirely in the oxygen moiety and the iron ion is silent. Account of SOC in the third-shell of the metal has to increase the total SOC integral. From a broader perspective, the non-zero electron spin and the concepts of quantum mechanics play a fundamental role in our understanding of the mystery of life. Spin of dioxygen is a property of the ground state of the molecule and protects O2 from its involvement in the realm of ordinary chemistry, where the brute force of activation energy just governs all spin-allowed biochemical transformations of diamagnetic species. Dioxygen is protected from brute force via the X3 †g  a1 g energy gap (22 kcal/mol) (Hagan et al. 2002) and via spin-prohibition; at the same time O2 is extremely fragile and easily activated by the presence of small amount of radicals or by magnetic perturbations (Minaev 2007). Such quantum spin protection is very important for biological systems, which operate at room temperature and are extraordinary complex, diverse, noisy, and “wet.” The quantum spin protection of dioxygen reactivity and its dependence on weak internal and external magnetic perturbations is important in connection with a fundamental fact, namely that biological systems are open driven systems. Self-organization in such biosystems should involve additional and very common mechanisms of self control, like ST transitions in the O2 activating enzymes, cytochromes, heme, photosynthetic centers, and probably in neural networks. A fundamental conclusion of this review is an assertion of the fact that the functioning of various enzymes is controlled by spin conversion determined by internal and external magnetic forces. The most important magnetic interaction is spin-orbit coupling and we indicate a simple physical origin of this perturbation as responsible for the functionality of enzymes like glucose oxidase, coproporphyrinogen oxidase, luciferase, and myoglobin, hemoglobin, and other hemoproteins.

Acknowledgments This work is supported by the State Foundation of Fundamental Investigations (DFFD) of Ukraine, the project F25.5/008, and by Visby project No D 01403/2007.

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Blomberg, M. R. A., Siegbahn, P. E. M., Babcock, G. T., & Wikstrom, M. (2000b). O–O bond splitting mechanism in cytochrome oxidase. Journal of Inorganic Biochemistry, 80, 261. Blomberg, L. M., Blomberg, M. R. A., & Siegbahn, P. E. M. (2005). A theoretical study of the binding of O2 , NO and CO to heme proteins. Journal of Inorganic Biochemistry, 99, 949. Buchachenko, A. L. (1977). Enrichment of magnetic isotopes–new method of investigation of chemical reaction mechanisms. Russian Journal of Physical Chemistry, 51, 2461. Buchachenko, A. L., & Kouznetsov, D. A. (2008). Magnetic field affects enzymatic ATP synthesis. Journal of the American Chemical Society, 130, 12868. Buchachenko, A. L., Kouznetsov, D. A., Orlova, M. A., & Markarian, A. A. (2005). Spin biochemistry: Intramitochondrial nucleotide phosphorylation is a magnesium nuclear spin controlled process. Mitochondrion, 5, 67. Burnold, T. C., & Solomon, E. I. (1999). Reversible dioxygen binding to hemerythrin. Journal of the American Chemical Society, 121, 8288. Chalkias, N. G., Kahawong, P., & Giannelis, E. P. (2008). Activity increase of horseradish peroxidase in the presence of magnetic particles. Journal of the American Chemical Society, 130, 2910. Davydov, R., Osborne, R. L., Kim, S. H., Dawson, J. H., & Hoffman, B. M. (2008). EPR and ENDOR studies of cryoreduced compound I. Biochemistry, 47, 5147. de Winter, A., & Boxer, S. G. (2003). Energetics of primary charge separation in bacterial photosynthetic reaction center mutants: Triplet decay in large magnetic field. Journal of Physical Chemistry A, 107, 3341. Engstrom, M., Minaev, B. F., Vahtras, O., & Agren, H. (1998). MCSCF linear response calculations of electronic g-factor and spin-rotational coupling constants for diatomics. Chemical Physics Letters, 237, 149. Engstrom, M., Himo, F., Graslund, A., Minaev, B. F., Vahtras, O., & Ågren, H. (2000). H-bonding to the tyrosyl radical analyzed by ab initio g-tensor calculations. Journal of Physical Chemistry A, 104, 5149. Esposito, S. (1999). On the role of spin in quantum mechanics. Foundations of Physics Letters, 12, 165. Franzen, S. (2002). Spin-dependent mechanism for diatomic ligand binding to heme. Proceedings of the National Academy of Sciences of the United States of America, 99, 16754. Friedman, J., & Campbell, B. (1987). Structural dynamics and reactivity in hemoglobin. New York: Springer. Frisch, M. J., Trucks, G. W., Schlegel, H. B., et al. (2003). Gaussian 03, Revision B. 03.. Pittsburg: Gaussian Inc. Gegear, R. J., Conelman, A., & Waddell, S. (2008). Cryptochrome mediates light-dependent magnetosensitivity in Drosophila. Nature, 454, 1014. Grissom, C. B. (1995). Magnetic field effects in biology: A survey of possible mechanisms with emphasis on radical-pair recombination. Chemical Reviews, 95, 3. Hagan, S., Hameroff, S. R., & Tuszynski, J. A. (2002). Quantum computation in brain microtubules. Physical Review E, 65, 061901. Hameroff, S., & Penrose, R. (1996). Conscious events as orchestrated space-time selections. Journal of Consciousness Studies, 3, 36. Harvey, J. N. (2004). Spin-forbidden CO ligand recombination in myoglobin. Faraday Discussions, 127, 165. Hoff, A. J. (1986). Magnetic interactions between photosynthetic reactants. Photochemistry and Photobiology, 43, 727. Hu, H. P., & Wu, M. X. (2004). Spin-mediated consciousness theory. Medical Hypotheses, 63, 633. Jensen, K. J., & Ryde, U. (2004). How O2 binds to heme. Journal of Biological Chemistry, 279, 14561. Jensen, K. J., Ross, B. O., & Ryde, U. (2005). The CAS PT2 study of oxymioglobin model. Journal of Inorganic Biochemistry, 99, 45. Johnsen, S., & Lohmann, K. J. (2005). The physics and neurobiology of magnetoreception. Nature Reviews Neuroscience, 6, 703.

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Minaev, B. F., Minaeva, V. A., & Vasenko, O. M. (2007). Spin states of the Fe(II)-pporphin molecule: Quantum-chemical study by DFT method. Ukrainica Bioorganica Acta, 1, 24. Minaev, B. F., Minaeva, V. A., & Evtuhov, Y. V. (2008). Quantum-chemical study of the singlet oxygen emission. International Journal of Quantum Chemistry, 108, 500. Ogilby, P. R. (1999). Singlet oxygen. Accounts of Chemical Research, 32, 512. Orlova, G., Goddard, J. D., & Brovko, L. Y. (2003). Theoretical study of the amazing firefly bioluminescence. Journal of the American Chemical Society, 125, 6962. Paterson, M. J., Christiansen, O., Jensen, F., & Ogilby, P. R. (2006). Singlet oxygen. Photochemistry and Photobiology, 82, 1136. Penrose, R. (1960). A spinor approach to general relativity. Annals of Physics, 10, 171. Penrose, R. (1994). Shadows of the mind. Oxford: Oxford University Press. Petrich, J. W., Poyart, C., & Martin, J. L. (1988). Spin-forbidden binding of O2 to hemoglobin. Biochemistry, 27, 4049. Poli, R., & Harvey, J. N. (2003). Spin-forbidden chemical reactions. Chemical Society Reviews, 32, 1. Potter, W. T., Ticker, M. P., & Caughey, W. S. (1987). Resonance Raman spectra of myoglobin. Biochemistry, 26, 4699. Prabhakar, R., Siegbahn, P. E. M., Minaev, B. F., & Ågren, H. (2002). Activation of triplet dioxygen by glucose oxifase: Spin-orbit coupling in the superoxide ion. Journal of Physical Chemistry B, 106, 3742. Prabhakar, R., Siegbahn, P. E. M., & Minaev, B. F. (2003). A theoretical study of the dioxygen activation by glucose oxidase and by copper amine oxidase. Biochimica et Biophysica Acta, 1647, 173. Prabhakar, R., Siegbahn, P. E. M., Minaev, B. F., & Agren, H. (2004). Spin transition during H2 O2 formation in the oxidative half-reaction of copper amine oxidase. Journal of Physical Chemistry B, 108, 13882. Sawyer, D. T. (1991). Oxygen chemistry. New York: Oxford University Press. Schweitzer, C., & Schmidt, R. (2003). Physical mechanisms of generation and deactivation of singlet oxygen. Chemical Reviews, 103, 1685. Serebrennikov, Y. A., & Minaev, B. F. (1987). Magnetic field effects due to spin-orbit coupling in transient intermediates. Chemical Physics, 114, 359. Shaik, S., Filatov, M., Schroder, D., & Schvarz, H. (1998). Electronic structure makes a difference: Cytochrome P-450 mediated hydroxylation of hydrocarbons as a two-state reactivity paradigm. Chemistry–A European Journal, 4, 193. Shaik, S., Ogliaro, F., de Visser, S. P., Schwarz, H., & Schroeder, D. (2002). Two state reactivity mechanism of hydroxylation and epoxidation by cytochrome P450 revealed by theory. Current Opinion in Chemical Biology, 6, 556. Shaik, S., Kumar, D., de Viser, S. P., Altun, A., & Tiel, W. (2005). Theoretical perspective on the structure of cytochrome P450 enzymes. Chemical Reviews, 105, 2279. Sheldon, R. A. (1993). A history of oxygen activation. In D. Barton et al. (Eds.), The Activation of dioxygen and homogeneous catalytic oxidation. New York: Plenum. Shikama, K. (2006). Nature of the FeO2 bonding in myoglobin and hemoglobin. A new molecular paradigm. Progress in Biophysics and Molecular Biology, 91, 83. Sigfridsson, E., & Ryde, U. (2002). Theoretical study of discrimination between O2 and CO by myoglobin. Journal of Inorganic Chemistry, 91, 101. Silva, P. J., & Ramos, M. J. (2008). A comparative DFT study of the reaction mechanism of the O2 -depemdemt coproporphyrinogen III oxidase. Bioorganic Medical Chemistry, 16, 2726. Strickland, N., & Harwey, J. N. (2007). Spin-forbidden ligand binding to the ferrous-heme group. Journal of Physical Chemistry B, 111, 841. Stryer, L. (1995). Biochemistry (4th ed., p. 152). New York: Freeman.

Protein Modeling G. Náray-Szabó, A. Perczel, A. Láng, and D.K. Menyhárd

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvent-Accessible Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Protein Hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ligand Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Protein-Protein Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enzyme Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3 8 12 12 14 16 17 17 20 20 21 21 22 25 29 33 33

Abstract

Proteins play a crucial role in biological processes; therefore, understanding their structure and function is very important. In this chapter, we give an overview on computer models of proteins. First, we treat both major experimental

G. Náray-Szabó () • A. Perczel • A. Láng • D.K. Menyhárd Laboratory of Structural Chemistry and Biology, Institute of Chemistry, Eötvös Loránd University and MTA-ELTE Protein Modelling Research Group, Budapest, Hungary e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Science+Business Media Dordrecht 2015 J. Leszczynski et al. (eds.), Handbook of Computational Chemistry, DOI 10.1007/978-94-007-6169-8_30-2

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structure determination methods, X-ray diffraction and NMR spectroscopy. In subsequent sections, computer modeling techniques as well as their application to the construction of explicit models are discussed. An overview on molecular mechanics and structure prediction is followed by an overview of molecular graphics methods of structure representation. Protein electrostatics and the concept of the solvent-accessible surface are treated in detail. We devote a special section to dynamics, where time scales of molecular motions, structures, and interactions are discussed. Protein in relation to its surroundings is especially important, so protein hydration, ligand binding, and protein-protein interactions receive special attention. The case study of podocin provides an example for the successful application of molecular dynamics to a complex issue. At last, computer modeling of enzyme mechanisms is discussed. It is demonstrated that protein representation by computers arrived to a very high degree of sophistication and reliability; therefore, even lots of experimental studies make use of such models. A list with a large number of up-to-date bibliographic references helps the reader to get informed on further details.

Introduction In order to obtain versatile models, we attribute a definite spatial structure and shape to molecules, which may, however, change in time. In case of proteins, the structure is too complicated and complex; thus, the use of computer graphics is unavoidable. In this chapter, we give an overview on the experimental and computational methods, which provide information on protein structure and allow deriving its relation to function. Neither of the techniques currently available is able to yield full information. X-ray diffraction works only for crystalline samples, NMR techniques refer to restraints, which determine a manifold of related but nonidentical structures, while modeling still lacks full reliability. Accordingly, in order to obtain thorough information on protein structure, various techniques should be combined, although in several cases even a single technique provides a model, which is adequate for some considerations. The present chapter may contribute to an overview on up-to-date techniques with reference to more detailed reviews on specific subjects, like the book published recently (Náray-Szabó 2014).

Structure Determination Both experimental and modeling techniques are available for the determination of the three-dimensional structure of proteins. They vary in performance and neither of them offers a unique tool to be applied for any type of proteins and their complexes. In most cases, publications refer to a single method only, structures are determined either by X-ray diffraction, nuclear magnetic resonance (NMR) spectroscopy or, eventually, by computer modeling. In order to get a more realistic, broad and rather complete information on structural and dynamic properties of proteins, a comparison of data obtained by a variety of methods is required.

Protein Modeling

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Experiment The first protein structures were determined by X-ray diffraction techniques in the late 1950s, and due to intensive work 95,375 structures are available now in the Protein Data Bank (as of 2015). Structure determination by NMR spectroscopy is also rapidly expanding; the number of deposited structures is 10,789 (Protein Data Bank 2015).

Protein Crystallography Protein structures can be visualized at the atomic level by X-ray crystallography. An appropriate crystal has to be irradiated and the obtained diffraction pattern is recorded and analysed by mathematical tools. The result is an electron density map with relative maxima at the position of atomic nuclei. Using classical chemical information, the density map can be evaluated and the three-dimensional molecular model of the protein can be constructed. In order to determine the structure of a protein, the following steps have to be performed (Brandén and Tooze 1999; Drenth 2007). Sample preparation and crystallization. Its goal is to produce a well-ordered crystal that is large enough to provide an appropriate diffraction pattern when irradiated with X-rays. It is inherently difficult because of the fragile nature of protein crystals. Proteins have irregularly shaped surfaces, which form large channels within the crystal. The noncovalent bonds that hold together the lattice must often be formed by flexible amino-acid residues of the protein surface. The success of protein crystallization depends on several factors, like purity, pH, concentration, temperature, precipitants, or additives. In order to ensure sufficient homogeneity, the protein should be pure to at least 97 %. It is necessary to screen a reasonable number of conditions, for that at least 200 l of material in a concentration of about 10 mg/ml is needed. The most commonly used techniques for crystallization are the hanging- and sitting-drop methods (Horsefield and Neutze 2006), although alternative approaches also have emerged, e.g., the microbatch method (Luft et al. 2003; Merlino et al. 2011). Efficient high-throughput methods have been developed to speed up the process and to determine optimal crystallization conditions (Chayen 2007). Data collection. Having an appropriate crystal, its symmetry, the unit cell parameters and the resolution limit have to be determined. Then, it will be irradiated by an X-ray beam and continuously rotated by a small angle. Thus, the X-ray diffraction pattern containing relevant information on the position of scattering centers can be recorded. For crystals of lower quality, application of synchrotron radiation is recommended, where the beam intensity is greater; therefore, data collection times are shorter, sometimes less than 10 min. Synchrotron radiation is also the basis of time-resolved crystallography (Bourgeois and Royant 2005; Petsko and Ringe 2015). The relatively slow reaction rates allow following changes in atomic positions during the enzymatic process.

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Structure determination. The relationship between the scattering angle and the distance between planes passing through the atoms in the crystal is given by the Bragg’s law. The electron density is related to the diffraction pattern by a mathematical function termed inverse Fourier transform. In order to compute this, both the amplitude and the phase of the diffracted waves should be known. Since experimental measurement of the relative phase angles is impossible, we face the so-called phase problem, often the most serious bottleneck in determining a new structure. A possible technique to solve this issue is the isomorphous replacement, i.e., to provide a crystal, which is nearly identical to the one studied, except that a few atoms have been replaced or added (e.g., using selenomethionine instead of methionine). If these atoms have a large atomic number, they will strongly perturb the diffraction pattern and allow deducing possible values for the phase angles. A similar technique applies the multiple-wavelength anomalous dispersion (MAD). By changing the wavelength of the X-rays, the degree to which the anomalously scattering atoms perturb the diffraction pattern also changes. This gives the same kind of information as isomorphous replacement. Model building. If the studied crystals were perfectly ordered, all the atoms would scatter in phase and the electron density map would have peaks at each of the atomic positions. However, in electron density maps of most proteins, atoms are not resolved from each another; we need a model to be fitted to the electron density. Fitting models to density requires the use of computer graphics programs such as COOT (Emsley and Cowtan 2004) and as a result the electron density map is interpreted in terms of a set of atoms. In most cases, the protein backbone is fitted first then the side-chain atoms are positioned if the resolution permits. The result is dependent on the resolution and the quality of the phases (Fig. 1). Often regions of high flexibility (e.g., surface lysine side chains) are not visible since the corresponding electron density is smeared. Fig. 1 Atomic (1.2 Å) resolution electron density map of a section of the complex between crayfish trypsin and an inhibitor from the desert locust Schistocerca gregaria (SGTI) (V. Harmat, unpublished)

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Refinement and validation. An atomic model can be improved by refinement, in which it is adjusted to the measured diffraction data. Refinement improves the phases leading to clearer maps and better models. The quality of an atomic model can be judged by the so-called R-factor, which is the average fractional error in the calculated amplitude compared to the observed one. Since, in general, there are insufficient diffraction data, these may be completed by restraints on geometry, which keep the bond lengths, angles, and close contacts within a reasonable range. Main-chain torsion angles are difficult to restrain but the distribution of these angles is strongly restricted as shown by the Ramachandran plot (Hovmöller et al. 2002), which is thus an additional indicator of the quality of a structure. Neutron diffraction provides special information on proteins, a special method requiring high thermal-neutron fluxes, which can be obtained only from nuclear reactors (Fitter et al. 2006; Dambrot 2012). Hydrogen atoms can be precisely located, which is almost impossible by X-ray diffraction. A diffraction experiment can be performed on a crystal; the results can be evaluated similarly as done for the X-ray technique.

Nuclear Magnetic Resonance Spectroscopy Nuclear magnetic resonance (NMR) spectroscopy is a versatile tool of modern structural biology, which allows determining protein structures even in some cases, where X-ray diffraction fails. It has become an independent method of routine structure determination for proteins with up to about 300 residues. Structure determination by NMR consists of multiple consecutive steps. Sample preparation and stability testing. For successful NMR experiments, a protein solution of high purity (>95 %), stability (over a week), and appropriate concentration (0.1–1 mM) is needed; the total sample volume and mass should be 350–550 l and 3–30 mg, respectively. 2,2-dimethyl-2-silapentane-5-sulfonic acid (DSS) rather than tetramethylsilane (TMS) is used as a reference compound. Although both natural and synthetic polypeptides and proteins can be used, most samples are expressed via a suitable prokaryotic or eukaryotic cellular or cell-free recombinant technique. The latter method eliminates most toxic effects attributed to the overproduction of recombinant proteins. Biotechnological approaches have the common advantage of large-scale protein production, specific or nonspecific stable isotope labeling, and easy sequence variation. Data acquisition (Schorn and Taylor 2004). Some magnetically active nuclei (e.g., 1 H, 13 C, or 15 N) experiencing a large magnetic field (10–20 T) possess two energy states. The energy difference between these states is large enough to be detectable if the resonance criterion is met during a radiofrequency irradiation (at or very near to the Larmour-frequency of the nucleus in question) as long as relaxation mechanisms are long enough. For example, the resonance frequencies, mostly between 500 and 950 MHz for 1 H nuclei, referring to the energy of the absorption are proportional to the strength of the static magnetic field. Depending on the local chemical environment, nuclei of the same type in a molecule resonate at slightly different frequencies. The relative shift from a reference (DSS for proteins in aqueous solution) resonance frequency is called the chemical shift. Peptides and

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proteins from natural sources are typically restricted to utilizing resonances based solely on protons; however, to get more information and/or to decrease spectral crowding/overlap and hence to decrease assignment ambiguity, further NMR active nuclei have to be introduced or enriched in the molecule (mostly 13 C and 15 N). In ideal signal dispersion, each magnetically active atom has its unique chemical shift by which it can be recognized and assigned. Assignment of each measured chemical shift to a single atom (or a group of atoms, e.g., a methyl group) is a mandatory step toward structure determination. Resonance frequency assignment is a very complicated and laborious procedure and often needs to introduce further “dimensions” into measurements but provides useful information on protein structure, dynamics, interaction, and ligand binding. Beside the most common direct dimension, typically associated with 1 H, experiments exploiting other nuclei (vide supra) are run to generate additional dimensions to minimize spectral overlap. For structure determination, two major types of NMR experiments are in everyday use: (i) the one based on coherence transfer between chemically connected atoms via indirect (through bond or J) coupling (e.g., COSY, TOCSY, HSQC, HNCO, HNCA, : : : ) and (ii) those relaying on magnetization transfer between spatially close atoms (through space or direct coupling); see below. The first set is for the identification of connected chemical shifts, while the second one is to localize spatial proximity. In both cases, it is the electron that mediates these interactions. Resonance frequency assignment. The basis of the analysis of NMR-spectra is the distinction of real and artificial (background) signals; in other words, a large signal-to-noise ratio is needed, which is met if using concentrated samples. Then, one has to find out which chemical shift corresponds to which nuclei within the macromolecule. Several different protocols were invented to achieve this tedious goal, primarily depending on the type of isotope labeling introduced for proteins. Homonuclear techniques are used for unlabeled typically smaller molecules by taking advantage of a series of specific experiments. Resonance assignment for proteins of medium size (5 < MW