Advances in Quantum Chemistry, Volume 75 presents work and reviews of current progress in computational quantum mechanic

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*Table of contents : Content: Series PagePage iiCopyrightPage ivContributorsPages ix-xPrefacePages xi-xiiJohn R. Sabin, Erkki J. BrändasChapter One - Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom SystemsPages 1-26Barak Hirshberg, R. Benny GerberChapter Two - Electron–Ion Impact Energy Transfer in Nanoplasmas of Coulomb Exploding ClustersPages 27-52Isidore Last, Joshua JortnerChapter Three - Molecular Properties of Sandwiched Molecules Between Electrodes and NanoparticlesPages 53-102Stine T. Olsen, Asbjørn Bols, Thorsten Hansen, Kurt V. MikkelsenChapter Four - Criterion for the Validity of D'Alembert's Equations of MotionPages 103-116John W. PerramChapter Five - A Time-Dependent Density Functional Theory Study of the Impact of Ligand Passivation on the Plasmonic Behavior of Ag NanoclustersPages 117-145Adam P. Ashwell, Mark A. Ratner, George C. SchatzChapter Six - Switching Activity of Allosteric Modulators Controlled by a Cluster of Residues Forming a Pressure Point in the mGluR5 GPCR: A Computational InvestigationPages 147-174Michael Sabio, Sid TopiolChapter Seven - Singlet Fission: Optimization of Chromophore Dimer GeometryPages 175-227Eric A. Buchanan, Zdeněk Havlas, Josef MichlChapter Eight - Continuum Contributions to Dipole Oscillator-Strength Sum Rules for Hydrogen in Finite Basis SetsPages 229-241Jens Oddershede, John F. Ogilvie, Stephan P.A. Sauer, John R. SabinChapter Nine - Features of Nearly Spherical Electronic SystemsPages 243-266Jan LinderbergIndexPages 267-272*

EDITORIAL BOARD Remigio Cabrera-Trujillo (UNAM, Mexico) Hazel Cox (UK) Frank Jensen (Aarhus, Denmark) Mel Levy (Durham, NC, USA) Jan Linderberg (Aarhus, Denmark) Svetlana A. Malinovskaya (Hoboken, NJ, USA) William H. Miller (Berkeley, CA, USA) John W. Mintmire (Stillwater, OK, USA) Manoj K. Mishra (Mumbai, India) Jens Oddershede (Odense, Denmark) Josef Paldus (Waterloo, Canada) Pekka Pyykko (Helsinki, Finland) Mark Ratner (Evanston, IL, USA) Dennis R. Salahub (Calgary, Canada) Henry F. Schaefer III (Athens, GA, USA) John Stanton (Austin, TX, USA) Alia Tadjer (Sofia, Bulgaria) Harel Weinstein (New York, NY, USA)

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CONTRIBUTORS Adam P. Ashwell Northwestern University, Evanston, IL, United States Asbjørn Bols Department of Chemistry, University of Copenhagen, Copenhagen, Denmark Eric A. Buchanan University of Colorado, Boulder, CO, United States R. Benny Gerber The Hebrew University of Jerusalem, Jerusalem, Israel; University of California, Irvine, CA, United States Thorsten Hansen Department of Chemistry, University of Copenhagen, Copenhagen, Denmark Zdeneˇk Havlas University of Colorado, Boulder, CO, United States; Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic Barak Hirshberg The Hebrew University of Jerusalem, Jerusalem, Israel Joshua Jortner School of Chemistry, Tel Aviv University, Tel Aviv, Israel Isidore Last School of Chemistry, Tel Aviv University, Tel Aviv, Israel Jan Linderberg Aarhus University, Aarhus C, Denmark; Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, UT; Quantum Theory Project, University of Florida, Gainesville, FL, United States Josef Michl University of Colorado, Boulder, CO, United States; Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic Kurt V. Mikkelsen Department of Chemistry, University of Copenhagen, Copenhagen, Denmark Jens Oddershede University of Southern Denmark, Odense, Denmark; University of Florida, Gainesville, FL, United States John F. Ogilvie University of Southern Denmark, Odense, Denmark; Simon Fraser University, Burnaby, BC, Canada; Universidad de Costa Rica, San Jose, Costa Rica

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Contributors

Stine T. Olsen Department of Chemistry, University of Copenhagen, Copenhagen, Denmark John W. Perram School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, Australia Mark A. Ratner Northwestern University, Evanston, IL, United States John R. Sabin University of Southern Denmark, Odense, Denmark; University of Florida, Gainesville, FL, United States Michael Sabio Stevens Institute of Technology, Center for Healthcare Innovation, Hoboken; 3D-2Drug, Fair Lawn, NJ, United States Stephan P.A. Sauer University of Copenhagen, Copenhagen, Denmark George C. Schatz Northwestern University, Evanston, IL, United States Sid Topiol Stevens Institute of Technology, Center for Healthcare Innovation, Hoboken; 3D-2Drug, Fair Lawn, NJ, United States

PREFACE

It is with the greatest pleasure that we present this volume in tribute to Mark Ratner on the occasion of his 75th birthday for his seminal contributions to theoretical chemistry during the last half century. Mark Ratner began his career as an undergraduate in chemistry at Harvard University, where he earned the AB degree in 1964. He then did a PhD in chemistry at Northwestern University in 1969, working with Prof. G. Ludwig Hofacker, followed by postdoctoral work with Prof. Jan ˚ rhus, Denmark. Mark took his first job at New York Linderberg in A University, and in 1975 moved to Northwestern University where he remains. Presently, Mark Ratner is Lawrence B. Dumas Distinguished University Professor Emeritus Departments of Chemistry, Materials Science, and Engineering at Northwestern University. While at Northwestern, he has held several positions such as Chairman of the Chemistry Department and Associate Dean of the College of Arts and Sciences. Mark Ratner is a very personable fellow, which makes him an ideal colleague to work with. He is friendly and helpful, and, perhaps most important, an original and deep thinker in many aspects of theoretical chemistry; he leads a scientific life dominated by a quest for understanding of chemical structure and function at the nanoscale and the theory that will describe and predict such in many nanoscale subfields. These characteristics have led to a world-class research program under Mark Ratner’s direction. Mark Ratner is not only a producer of world-class research, but he is prolific. Presently he has produced over 800 technical papers, 5 edited volumes, and 4 books. xi

xii

Preface

It should be noted that of the books, two of them are coauthored by his son, Daniel Ratner, and the other two are written together with George Schatz. In addition, he has given over 100 named lectures around the world from Denmark, to Israel, to Japan, and on. Perhaps Mark Ratner’s best known work is in the field of molecular electronics. In 1974, Mark Ratner and his at that time student, Ari Aviram, produced a work on molecular rectifiers [Bull. Am. Phys. Soc. 19, 341 (1974); Chem. Phys. Lett. 29, 277 (1974)] which is probably his best known work. This was the first time it had been shown, as inconceivable as it was by most experts in the field, that a single molecule could act as a rectifier. From their work, a whole field of molecular electronics has blossomed. However, Mark Ratner’s work has not been confined to this particular subfield, as he has significantly contributed in many other related areas. For instance, he has contributed substantially to work on molecular electron transfer and electron transport in organic polymers as well as to various aspects of molecular dynamics and relaxation. Mark Ratner’s achievements have won him many honors. Most prestigious, he is a member of the National Academy of Science (2002) and a foreign member of the Royal Danish Academy of Science (2004). In addition he has received honorary Doctor of Science degrees from the Hebrew University of Jerusalem (2004) and the University of Copenhagen (2010). In 2010, Mark Ratner was also celebrated and honored with a festschrift published in the Journal of Physical Chemistry C (volume 114, #48). It has certainly been a great honor for those of us whom have had interaction, scientific and social, with Mark Ratner, to collect this tribute to him. He has been a vibrant member of the Quantum Chemistry Club of those individuals who have practiced the science for nearly a half century, and we wish him the very best for the future. The papers presented here are written by Mark Ratner’s friends and coworkers to demonstrate their respect for him and his work. Best wishes, Mark! By: JOHN R. SABIN € ERKKI J. BRANDAS

CHAPTER ONE

Mean-Field Methods for TimeDependent Quantum Dynamics of Many-Atom Systems Barak Hirshberg*, R. Benny Gerber*,†,1 *The Hebrew University of Jerusalem, Jerusalem, Israel † University of California, Irvine, CA, United States 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Theoretical Framework, Advantages, and Limitations 2.1 Vibrational Self-Consistent Field 2.2 Time-Dependent Self-Consistent Field 2.3 Mixed Quantum–Classical Methods 2.4 Classical Separable Potentials 3. Recent Developments and Applications 3.1 Ab Initio Classical Separable Potentials 3.2 Mixed Quantum–Classical Methods 4. Summary and Future Directions Acknowledgments References

2 5 5 6 8 10 12 12 18 20 22 22

Abstract Methods that can accurately describe the quantum dynamics of large molecular systems have many potential applications. Since numerical solution of the time-dependent €dinger equation is only possible for systems with very few atoms, approximate Schro methods are essential. This paper describes the development of such methods for this challenging time-dependent many-body quantum mechanical problem. Specifically, we focus on the development of mean-field theories, to which Mark Ratner has contributed greatly over the years, such as the time-dependent self-consistent field method, mixed quantum–classical methods, and the classical separable potentials method. The advantages and limitations of the different variants of mean-field theories are highlighted. Recent developments, aimed at applying mean-field methods for large systems, and their applications are presented. Issues where further methodological advancement is desirable are discussed. Examining the tools available so far, and the

Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.01.002

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Barak Hirshberg and R. Benny Gerber

recent progress, we conclude there are promising perspectives for future development of mean-field theories for quantum dynamics with applications to realistic systems in important chemical and physical processes.

1. INTRODUCTION Classical molecular dynamics, in which the motion of the atoms is described using Newton’s equations of motion, is one of the most widely applied methods in chemistry. This includes methods in which the interactions between the atoms are described using empirical force-fields1 as well as ab initio molecular dynamics (AIMD),2–5 which directly employs potential energy surfaces (PES) obtained from quantum chemical methods to the classical nuclear equations of motion. The scope of applications is overwhelming and ranges from the study of biological molecules,6,7 properties of water and ice,8–10 materials science,11 chemical reaction dynamics,12 and many more. Describing the motion of the nuclei using classical mechanics is often an excellent approximation at typical chemical conditions, e.g., room temperature. However, in many cases, nuclear quantum effects (NQE) such as zero-point energy (ZPE), tunneling, and quantum interference may play a significant role, especially for light atoms. Important example is liquid water and ice, and a detailed review on the importance of NQE for water and ice was published recently by Ceriotti et al.13 Other examples include molecules embedded in superfluid helium droplets,14,15 the kinetic isotope effect in enzyme catalysis,16–18 and more. As a result, for these processes and others, a quantum description of the nuclei is essential. Ideally, one would like to numerically solve the time-dependent Schr€ odinger equation (TDSE) for the nuclei using a suitable Born– Oppenheimer (BO) PES. However, due to the exponential scaling of the computational effort with the number of degrees of freedom,19 this is an extremely difficult task. State-of-the-art calculations in full dimensionality are currently limited to systems with approximately 4–6 atoms,20 such as the recent study of electron photodetachment from F(H2O).21 Since most chemical and physical processes of interest occur in larger systems, approximate methods for quantum nuclear dynamics are necessary. Several different approximate methods are available which address, conceptually, NQE. This includes semiclassical methods such as the semiclassical initial value representation method by Miller22 and the Gaussian wavepacket method by Heller23,24 as well as methods based on the

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

3

Path-integral formulation of quantum mechanics, such as centroid molecular dynamics25,26 and ring-polymer molecular dynamics (RPMD).27 This overview will focus on a particular class of approaches, mean-field methods for quantum nuclear dynamics, such as the time-dependent self-consistent field method (TDSCF),28–30 the classical separable potentials (CSP) method,31 and mixed quantum–classical methods.28,32 The reason for this focus is that mean-field methods have been very successful in describing certain time-independent many body problems, such as vibrational spectroscopy. Unlike time-dependent methods, mean-field theories such as the vibrational self-consistent field (VSCF) method33,34 are now routinely applied using ab initio PES35 to many-atom systems36 including, recently, even peptides.37,38 This suggests that the same may be possible for meanfield time-dependent methods, in spite of the added computational effort due to the time-dependence. Several review articles were previously published on approximate theories for quantum nuclear dynamics: A masterful review on many of the methods discussed in this paper was written by Makri39 over 15 years ago. Excellent reviews focusing on specific methods not discussed in this paper were also written recently, such as the reviews by Habershon et al. on RPMD27 and by Meyer on the multiconfigurational time-dependent Hartree method.40 It is the purpose of this paper to give an updated account on the state of the field, with particular emphasis on mean-field methods, on their implementation to systems of substantial numbers of degrees of freedom, on the possibility of using directly ab initio potentials in these methods and on the connection between time-dependent and time-independent mean-field methods. In addition, we limit the scope of this paper to situations in which the BO approximation is adequate and a single adiabatic PES is considered. Perhaps the first mean-field theory used to describe time-dependent quantum dynamics is the TDSCF method which originates from the work of Dirac.41 One of its first applications to chemical physics was to the vibrational predissociation of a van der Waals molecule, I2Ne.28 In this method, a separable ansatz for the nuclear wave function is assumed, typically using normal coordinates. Applying the Dirac–Frenkel time-dependent variational principle42,43 to this ansatz results in a set of coupled equations for the vibrational modes. Each mode evolves in time under the influence of a mean potential which is the quantum mechanical average of the full PES over all other degrees of freedom. Classical and semiclassical variants of TDSCF were also introduced.28,44 TDSCF was shown to give good

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Barak Hirshberg and R. Benny Gerber

agreement with numerical solution of the nuclear TDSE for several processes such as vibrational predissociation of I2He45 as well as photodissociation of Xe-HI46 and Ar-HCl.47 While the computational complexity of the TDSCF method is significantly reduced in comparison to numerical solution of the TDSE, the evaluation of the mean potentials involves multidimensional integrals over all modes of vibration except one, which are the computational bottleneck of TDSCF. As a result, TDSCF was only applied to relatively small systems. An approximation which, when justified, can significantly decrease the computational complexity of TDSCF is the treatment of a small subset of the system using TDSCF, while treating the other degrees of freedom in the classical limit of TDSCF. This family of methods is often referred to as mixed quantum–classical methods.29 More precisely, the quantum degrees of freedom evolve in time according to a TDSCF-like equation; however, the mean potential for the quantum subsystem is obtained from classical trajectories. Accordingly, the classical degrees of freedom are not described by a wave function, and instead evolve in time classically under the influence of a force which is the quantum mechanical average of the full potential over the quantum degrees of freedom. Using quantum–classical TDSCF and employing an additional approximation, describing each mode using Gaussian wavepackets, low temperature clusters such as Ar13 and (H2O)548 and inelastic scattering of argon off (H2O)1149 were studied. Another approach which was successful in incorporating quantum effects in significantly larger systems is the CSP method.31 In this method, which is based on the TDSCF method, the quantum mechanical mean potential governing the time-dependent dynamics of each mode is replaced by an average potential obtained from classical trajectories simulations, using properly sampled initial conditions. A key point here, as well as in mixed quantum–classical dynamics, is that the initial conditions sampled for the classical trajectory simulations represent the classical analog for the quantum mechanical process studied. The main difference between CSP and mixed quantum–classical methods is that in CSP while all degrees of freedom are described quantum mechanically, there is no quantum force acting on the nuclei in the classical trajectories used to build the effective potentials for each mode. The average potentials obtained from classical molecular dynamics simulations implicitly couple the different modes of vibration to each other through the time-dependence of the potential. As a result, the single-mode equations are not solved self-consistently in the CSP method. While the lack of self-consistency may give rise to errors, it was shown that,

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

5

for several test cases, CSP gives results in good agreement with TDSCF results.31 Using CSP, much larger systems were studied such as the electron photodetachment in C60 .50 In this paper, we will also present a recent extension of CSP, denoted ab initio CSP (AICSP),51 which uses potentials directly obtained from quantum chemical methods. Recent application to test the validity of AICSP to vibrational spectroscopy of realistic systems will be discussed. The remainder of the paper is organized as follows: Section 2 presents the theory of VSCF, TDSCF, mixed quantum–classical methods, and the CSP method. Section 3 describes new methodological developments, which are aimed at describing large systems using ab initio PES, and their recent applications. Section 4 of the paper summarizes and presents current challenges of mean-field methods and possible future directions.

2. THEORETICAL FRAMEWORK, ADVANTAGES, AND LIMITATIONS 2.1 Vibrational Self-Consistent Field We start by briefly discussing the VSCF method and several issues involved in its practical application, which are also relevant for the development of timedependent mean-field methods.The time-independent nuclear Schr€ odinger equation is given by ^ ðnÞ ¼ E ðnÞ XðnÞ HX

(1)

where Χ (n) and E(n) are the total nuclear wave functions and energies, respec^ is the Hamiltonian of the system. In tively, corresponding to state n and H the framework of VSCF, neglecting rotational–vibrational coupling, the Hamiltonian is written in mass-weighted normal coordinates N 2X

^ ¼ ℏ H 2

@2 + V ðQ1 , …, QN Þ @Qj2 j¼1

(2)

where Q1, …, QN are the mass-weighted normal coordinates, V(Q1, …, QN) is the full PES, and N represents the number of vibrational degrees of freedom. While normal coordinates are often a good choice for describing vibrational spectroscopy, other coordinate systems can be beneficial, such as local coordinates52 or internal coordinates.53 For a detailed discussion of the optimal coordinates choice for VSCF, see the review by Roy and Gerber36 and references therein.

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Next, it is assumed that the total nuclear wave function can be represented by a simple product of single-mode functions Χ ðnÞ ðQ1 , …, QN Þ ¼

N Y χ j ðnÞ Qj

(3)

j¼1

Applying the time-independent variational principle results in the VSCF single-mode equations

ℏ2 @ 2 χ k ðnÞ ðQk Þ ðnÞ + V k ðQk Þχ k ðnÞ ðQk Þ ¼ εk ðnÞ χ k ðnÞ ðQk Þ 2 @Qk2

(4)

where the effective single-mode potential is given by the quantum mechanical average of the full PES over all other modes of vibration * + Y N N Y ð n Þ ð n Þ ð n Þ V k ðQk Þ ¼ χ j Qj V ðQ1 , …, QN Þ χ j Qj (5) j6¼k j6¼k Since the effective potential for each mode depends on all other modes, the VSCF equations are solved self-consistently. The main computational difficulty in applying VSCF lies in the evaluation of the multidimensional integrals involved in calculating the effective potentials. This computational complexity is significantly reduced if the potential can be approximated, to a reasonable extent, by a pairwise approximation: V ðQ1 , …, QN Þ ¼

N X

N X N X coup Qj + Vij Qi , Qj

diag

Vj

i

j¼1

(6)

j>i

The effective potentials then only involve integrals over 2 degrees of freedom. For many cases, the level of agreement obtained using this approximation with experimental results is very good.54 If needed,55 higher order coupling between specific modes can also be introduced.

2.2 Time-Dependent Self-Consistent Field In essence, TDSCF is the time-dependent analog of VSCF. Accordingly, the starting point for deriving the TDSCF equations is the time-dependent nuclear Schr€ odinger equation iℏ

@Χ ^ ¼ HΧ @t

(7)

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

7

^ as in the previous section, is the Hamiltonian of the system and Χ where H, is the total nuclear wave function. The question arises, which coordinates should be used to describe the time-dependent dynamics of the system? For VSCF, which is mainly designed for vibrational spectroscopy, normal coordinates are often a very suitable choice, despite the fact other choices may improve the accuracy for specific modes. However, the choice of coordinates for TDSCF is not as straightforward and the selection of optimal coordinates for TDSCF has been far less studied than for VSCF. To start with, it is not generally obvious that the dynamics can be restricted to the vicinity of a single minimum. Jungwirth et al. suggested56 a simple procedure, based on a set of classical trajectories, to evaluate the suitability of different coordinates for TDSCF and applied it to the relaxation dynamics following electron photodetachment in clusters such as IAr12. The coordinates used should be those that maximize the separability of the different modes during the time-dependent process. For processes which are dominated by a single minimum on the PES, one expects normal coordinates to be a reasonable choice, similarly to VSCF. In addition, one can also use normal modes at the transition state57–59 which can provide a good description of the transition state dynamics. There is no general method to determine optimal coordinates for all cases. We confine our attention to dynamics around a single minimum or saddle point. For convenience, we will use normal coordinates Q1, …, QN to present the TDSCF equations. The Hamiltonian is then given by Eq. (2). Assuming that the total nuclear wave function Χ(Q1, …, QN, t) can be represented by a single product of time-dependent single-mode functions χ j(Qj, t) ΧðQ1 , …, QN , t Þ ¼

N Y χ j Qj , t

(8)

j¼1

And applying the Dirac–Frenkel variational principle42,43 ^ iℏ hδΧjH

@ jΧi ¼ 0 @t

(9)

where δΧ is the variation in Χ with respect to the single-mode function χ k, we get the following TDSCF single-mode equations iℏ

@ϕk ðQk , t Þ ℏ2 @ 2 ϕk ðQk , tÞ + V TDSCF ðQk , t Þϕk ðQk , t Þ ¼ k 2 @t @Qk2

(10)

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Barak Hirshberg and R. Benny Gerber

where ϕk(Qk, t) differs from χ k(Qk, t) by a physically insignificant purely time-dependent phase factor and the single-mode effective potential V TDSCF ðQk , t Þ is given by k * + N N Y Y TDSCF V k ðQk , tÞ χ j Qj , t V ðQ1 , …, QN Þ χ j Qj , t (11) j6¼k j6¼k Similarly to VSCF, the main computational bottleneck for applying TDSCF is the evaluation of the multidimensional integrals that appear in the expression for the effective single-mode potentials. The complexity, however, is greater than for VSCF, since the integrals have to be reevaluated at every time step. Similarly to VSCF, the computational demand can be significantly reduced by using the pairwise approximation, expanding the potential according to Eq. (6). This approach for reducing the computational complexity of the TDSCF method has not been explored extensively so far. It is likely this will be somewhat less successful than in VSCF, because the configurational space relevant for dynamics is in practice larger. While the integrals that appear in the TDSCF equations still need to be evaluated at every time step, it will undoubtedly significantly increase the size of systems that can be studied using TDSCF. We note that while the validity of the pairwise approximation for time-dependent processes needs to be reevaluated, the success of this approximation within the framework of VSCF suggests it may be useful in time-dependent problems which are dominated by a single minimum on the PES.

2.3 Mixed Quantum–Classical Methods In cases when the density of states is high for certain modes, it is possible to describe these modes, to a good approximation, by classical mechanics while retaining the quantum description for all other degrees of freedom. This is the central idea behind mixed quantum–classical methods. It is clear that using this approximation, the computational complexity can be reduced, since the quantum mechanical problem will be of lower dimension. Here, we briefly highlight some central aspects of the theory. For an authoritative review on mixed quantum–classical methods, the reader is referred to the review by Tully.32 One possible starting point for mixed quantum–classical methods is the TDSCF equations given by Eqs. (10) and (11). Then, for a mode which can be approximately described by classical mechanics, we follow the usual

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

9

procedure60 of writing the single-mode wave function using the polar representation (given in Eq. (12)), deriving the equations for the amplitude Aj(Qj, t) and phase Sj(Qj, t) and taking the classical limit in which ℏ ! 0. i ϕj Qj , t ¼ Aj Qj , t eℏSj ðQj , tÞ

(12)

Plugging Eq. (12) into Eq. (10) and separating the real and imaginary parts, we get the equations for the amplitude and phase 2 @Sj Qj , t 1 @Sj Qj , t + V TDSCF Qj , t + j 2 @t @Qj @ 2 A j Qj , t ℏ2 (13) ¼ @Qj2 2Aj Qj , t @Aj Qj , t @Aj Qj , t @Sj Qj , t ¼ @t @Qj @Qj @ 2 Sj Qj , t 1 Aj Qj , t 2 @Qj2

(14)

So far, Eqs. (13) and (14) are completely analogous to the TDSCF equations. Now, however, we can take the classical limit to get an equation for the classical phase SC j (Qj, t) !2 C @SjC Qj , t 1 @Sj Qj , t + V TDSCF Qj , t ¼ 0 (15) + j 2 @t @Qj The equation for the classical amplitude is identical to Eq. (14), except that the classical phase appears instead of the quantum mechanical phase in the equation. Eq. (15) is simply the Hamilton–Jacobi equation for a particle evolving in time classically under the influence of the potential V TDSCF Qj , t . We note that V TDSCF Qj , t is, as before, the quantum j j mechanical average of the full potential over all other degrees of freedom. Thus, the classical modes are evolved under the influence of a force which originates from the other quantum degrees of freedom. Accordingly, the TDSCF equations for the quantum mechanical modes are not changed, but the averaging of the potential over the classical modes is now done using classical trajectories. We note that in the derivation given here, starting from the TDSCF equations and taking the classical limit, complete separability is

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assumed between all modes, including between one classical mode to another, for example. This is not a general requirement of mixed quantum–classical methods. An alternative derivation which is often used to present mixed quantum–classical methods only assumes general separability between the classical and quantum degrees of freedom, which results in the following equations of motion for the classical nuclei, written in Cartesian coordinates, and can be viewed as a specialization of the Ehrenfest theorem Z ! ! MI R€I ðtÞ ¼ rI drψ*ðr, tÞV ðr, RÞψ ðr, tÞ (16) Here, ψ(r, t) is the nonseparable wavefunction for the quantum nuclei !

and R I ðtÞ are the instantaneous position vectors of the classical nuclei. The equations of motion for the quantum nuclei are given by iℏ

@ψ ðr, tÞ ^ ¼ T r + V^ ðr, RðtÞÞ ψ ðr, tÞ @t

(17)

where T^ r is the kinetic energy operator for the quantum nuclei and V^ is the potential energy operator. In both derivations, the quantum mechanical equations are solved selfconsistently with the equations of motion for the classical subsystem. We note that while mixed quantum–classical methods allow simulations of large systems, the quantum mechanical subsystem must still be small. In cases where the energy or action for the classical subsystem is high, describing some modes classically and others quantum mechanically works very well, as shown for the photodissociation of HI in Xe clusters.61,62 However, due to the fact that some modes are not described by a wave function, significant errors can arise, such as negative absorption cross sections.63,64 Other possible discrepancies can results from the fact that the energy for the classical modes can drop below the ZPE of the mode.

2.4 Classical Separable Potentials The CSP method was suggested as an alternative procedure to mixed quantum–classical dynamics in order to describe large systems. The main assumption is that the effective single-mode potentials that appear in the TDSCF equations can be replaced by an average potential obtained from classical trajectories simulations. The potential is then given by

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

V CSP k ðQk , t Þ ¼

Ntraj X α Wα V Q1α ðtÞ,…, Qk , …, QN ðt Þ

11

(18)

α¼1

where Ntraj is the number of trajectories performed, Wα is the averaging weight given to each trajectory, and Qαj (t) is the value of normal mode Qj at time t from trajectory α. The CSP equations are then given by @ϕk ðQk , tÞ ℏ2 @ 2 ϕk ðQk , tÞ CSP + V k ðQk , tÞϕk ðQk , tÞ iℏ ¼ 2 @t @Qk2

(19)

In essence, this approximation means that each degree of freedom is described quantum mechanically, evolving in time under the influence of a mean-field potential which is generated as if all other modes of vibration are classical. The main difference between CSP and mixed quantum–classical dynamics is the absence of a self-consistent quantum force in the classical trajectories used to build the effective potentials for the quantum degrees of freedom. Because the effective single-mode potentials no longer depend explicitly on all other normal modes, the CSP equations are not solved self-consistently. While this may give rise to errors it was shown, as was mentioned previously, that for several test cases CSP gives results in good agreement with TDSCF,31 at least for short timescales. Another important difference is that all degrees of freedom in CSP are treated quantum mechanically. A key point in the CSP method, similarly to mixed quantum–classical dynamics, is that the initial conditions used to generate the classical trajectories must represent the classical analog to the initial quantum state of the system. For example, it is often possible to approximate the initial quantum state using a product of harmonic single-mode functions obtained from the PES at t ¼ 0. Then, the initial coordinates are sampled proportionally to the square of the wave function of each normal mode. The initial momenta can be determined from the harmonic ZPE of each mode. In other cases, if one is interested in representing a specific quantum state, which is not approximated well by a product of harmonic functions, the initial coordinates and momenta can be sampled from the Wigner distribution65 obtained from the state. If the initial quantum state represents a single highly excited mode, while all others are in their vibrational ground state, the initial conditions for the excited mode can be sampled using the classical probability distribution for the harmonic oscillator.

12

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The main limitation of the CSP method, compared to TDSCF and mixed quantum–classical methods, is the lack of self-consistency. However, the agreement with the TDSCF results previously mentioned suggests that is does not introduce significant errors. Its main advantage is that it is probably the only method presented in this paper which allows a quantum mechanical description of all nuclear modes, while being applicable to systems of realistic size. The fact that on the course to a quantum mechanical treatment one can obtain insights from the classical trajectories may also be useful.

3. RECENT DEVELOPMENTS AND APPLICATIONS Despite the various methods described in the previous section, there is no method available today which routinely allows quantum mechanical simulations of nuclear dynamics for large molecules, certainly when ab initio PES are used. Here, we describe recent theoretical developments and applications with particular aim to treat large systems using directly potentials from quantum chemical methods. We present detailed results only for work done recently in our group, but also summarize promising recent advancements done by other groups.

3.1 Ab Initio Classical Separable Potentials 3.1.1 Treatment of Stationary States Classical trajectories simulations, using either empirical force-fields36 or ab initio potentials,66,67 are often used to calculate the vibrational spectra of molecules and extended systems. The main advantages of classical MD simulations are that they are applicable to large systems, including condensed phases, as well as for long simulation times. However, empirical force fields often do not provide the necessary accuracy, for example, to be able to determine different amino acids conformers using vibrational spectroscopy.36 While AIMD simulations often perform better, their range of applicability for calculating vibrational spectra is limited to relatively high temperatures and it is not clear that their relative success does not result from a fortuitous cancelation of errors. As a result, it is desirable to have a quantum mechanical method that can describe stationary vibrational states and is applicable to large systems using ab initio PES. In addition, we stress that stationary vibrational spectroscopy offers a very strict test for quantum mechanical methods, due to the highly accurate experimental results, and the availability of data for comparison.

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

13

In order to accurately treat large systems, we recently developed51 the AICSP method which combines CSP with potentials obtained directly from quantum chemical methods. While the method is generally developed for time-dependent nonequilibrium processes, it is also useful in calculating stationary states. In order to describe stationary states using the AICSP method, one must determine a procedure to calculate the appropriate effective potentials for each mode. The key point here is to select the initial conditions for the classical trajectories simulations in a way which, at least approximately, will be representative of a stationary state of the system at T ¼ 0 K. In the vibrational ground state, the system has energy equal to the quantum mechanical ZPE and no energy flow is possible between the modes. One way to choose the initial conditions for the classical trajectories, used to build the effective potentials, is to simply give each mode a classical energy which is equal to its quantum mechanical ZPE. However, while in quantum mechanical simulations this ZPE cannot flow between the modes, classically, due to the energy differences between the modes, the energy will be transferred between the modes to a significant extent. Since these dynamics will be significantly different than found using a quantum mechanical description, this procedure seems inadequate for describing a stationary state. Alternatively, if one is interested in a specific vibrational mode, it is possible to choose the initial conditions such as to give only this mode its ZPE and keep all other modes initially at the classical equilibrium positions. This procedure, however, greatly underestimates the dynamical coupling between the different vibrational modes. As a result, in describing stationary states within the framework of AICSP, a better approach seems to be to choose an initial effective “classical ZPE” for each mode. This initial classical ZPE is chosen to be identical for all vibrational modes, in the spirit of the equipartition of the energy, to avoid, at least initially, significant energy flow between the modes. This, however, is not sufficient since this initial total energy must be small enough as to avoid unphysical processes at zero temperature, such as isomerization or chemical transformations. As a result, the effective classical ZPE, given initially to each mode, was chosen to be the harmonic ZPE for the softest vibrational mode of the system. The use of the harmonic approximation at this point is not essential or unique and we stress that the classical trajectories still include all anharmonic effects. After sampling initial coordinates and momenta using the procedure described above, which approximately describe a stationary state, classical MD simulations are performed and the effective single-mode potentials

14

Barak Hirshberg and R. Benny Gerber

are determined from Eq. (18). In order to obtain stationary frequencies and wavefunctions for each mode, a reasonable approach seems to be to average the time-dependent effective potentials over a time period τ, to obtain effective time-independent potentials, given by Eq. (20). τ should obviously be long compared with the time-scale of the relevant mode 1 e CSP V k ð Qk Þ ¼ τ

Zτ

V CSP k ðQk , t Þ:

(20)

0

Then, the one-dimensional time-independent Schr€ odinger for each mode can be solved on a grid to obtain the anharmonic frequencies and wavefunctions

e k ðQk Þ ℏ2 @ 2 ϕ e e e CSP +V k ðQk Þϕk ðQk Þ ¼ Ek ϕk ðQk Þ: 2 2 @Qk

(21)

3.1.2 Dynamical States TDSCF, CSP, and mixed quantum–classical methods have mostly been used to describe, naturally, the dynamics of nonequilibrium processes. While AICSP has not yet been applied to nonstationary states, it is an interesting challenge to apply it to time-dependent problems, e.g., calculating vibrational linewidths and lineshapes. We note here that if the initial state is not a stationary state, a very different procedure must be used for the sampling of initial conditions. The main idea is again to translate the initial quantum state to a set of corresponding initial coordinates and momenta for the classical trajectories. As was mentioned previously, there are several possible procedures to do so, depending on the process described. For situations in which a well-defined initial quantum state can be specified, such as photodissociation or electron photodetachment processes in which the initial state is determined by the Franck–Condon principle, the Wigner representation65 of the initial state can be used. Alternatively, in cases where the process is dominated by a single minimum on the PES, a separable approximation (typically the harmonic approximation) can be used for the initial state. In this procedure, initial coordinates can be sampled in proportion to the square of the harmonic single-mode functions, while the initial momenta can be determined from the single-mode energies.

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

15

3.1.3 Combining CSP With Ab Initio Potentials In all previous applications of CSP, empirical potentials were used. While for some specific processes, accurate potentials of this kind may exist, for the majority of problems, no such reliable potentials exist and the construction of accurate empirical potentials for complex systems is a difficult task. One exception is the application of CSP using an approximate density functional theory (DFT) method based on the local density approximation by Knospe and Jungwirth50 to the electron photodetachment from C 60. The recent development of AICSP51 allows for combining the CSP method with much better potentials, for example, potentials based on second-order M€ oller– Plesset perturbation theory68 (MP2) or state-of-the-art long range corrected hybrid DFT functionals with empirical dispersion corrections, such as ωB97X-D.69 The ability to use ab initio PES increases the accuracy of the CSP method, as well as the applicability of CSP to systems for which empirical potentials are not available. As explained previously, the application of AICSP involves several steps, mapping the initial quantum state to a set of initial coordinates and momenta for the classical simulations, performing the classical trajectories, building the effective potentials, and solving the single-mode time-dependent equation for each mode. When combining CSP with ab initio potentials, the classical trajectories and potential building steps become significantly more computationally intensive than when empirical potentials are used. However, the potential building step is entirely independent for each mode. As a result, parallelizing the potential building step on many CPUs is a trivial task, which suggests that this may allow application of AICSP to very large systems. Application of AICSP to increasingly large systems is limited to cases for which AIMD simulations can be performed. However, classical trajectories are required only for time scales involved in the quantum process under study, which are usually relatively short. In addition, the power of the AIMD method and codes for its efficient implementation are becoming increasingly available. 3.1.4 Applications The recent application of AICSP51 focused on CH, NH and OH stretching modes in amino acids such as glycine, alanine, and proline as well as the guanine–cytosine (G–C) pair of nucleobases. We note that while these systems are already nontrivial in size (for example, the G–C complex has 81 vibrational modes), admittedly, the AICSP method was developed with the intent of applying it to much larger systems, such as peptides. However, if the method performs well for amino acids, the building blocks of peptides

16

Barak Hirshberg and R. Benny Gerber

and proteins, this will encourage applications also for other biological molecules. The anharmonic frequencies obtained using AICSP were in very good agreement with experimentally measured values.51 Using MP2 potentials, the deviation from experimental values for nonhydrogen-bonded OH and NH stretching frequencies was smaller than 1%. For the hydrogenbonded OH stretches, the agreement was good, with deviations from experimental values being approximately between 2% and 4%. In this overview, we focus on features seen in the time-dependent effective potentials as well as on the properties of the effective potential averaged over time, from which the stationary vibrational frequencies are obtained. The differences between the effective single-mode potentials, the harmonic approximation, and the diagonal anharmonic potential, given by Eq. (22), are also discussed. We present specific results for conformer I of alanine, shown in Fig. 1, as a representative example

Fig. 1 Time-dependent single-mode effective potentials for different vibrational modes of conformer I of alanine. Panel (A) presents results for the symmetric NH2 stretch, panel (B) presents results for the asymmetric NH2 stretch, and panel (C) presents results for the OH stretch. Panel (D) shows the optimized geometry for conformer I of alanine, carbon atoms are shown in gray, oxygen atoms in red, nitrogen atoms in blue, and hydrogen atoms in white. Calculations are performed using potentials at the MP2/aug-cc-pVDZ level of theory.

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

diag

Vk ðQk Þ ¼ V ð0, …, Qk , …, 0Þ

17

(22)

While the first application of AICSP was to stationary states, as was mentioned earlier, analyzing the time-dependent potentials still provides dynamical information on classical vibrational energy flow in the system. Fig. 1 shows the different time-dependent single-mode effective potentials for several modes of conformer I of alanine. Several interesting features can be seen: First, fluctuations in the time-dependent effective potentials are seen initially for all modes. However, for approximately 600 fs these fluctuations are rather mild. Then, for the symmetric NH2 stretching mode and the OH stretching mode, the fluctuations seem to increase and the potential becomes softer than at t ¼ 0. This suggests that for short timescales, due to the initial conditions chosen, energy flow between the modes is indeed minor. However, after a certain period of time, energy starts to flow significantly between some of the modes, which results in larger fluctuations of the effective potentials. These increased fluctuations are not seen for the asymmetric NH2 stretching mode, which fluctuates at a similar amplitude up to a simulation time of 1 ps. This may suggest that the OH stretching mode and the symmetric NH2 stretch are more significantly coupled to other modes in the system. The significant energy flow between the modes may also provide information on vibrational linewidths, and we intend to pursue such analysis in the near future. Fig. 2 shows different time-averaged effective potentials obtained using Eq. (20) from the time-dependent single-mode potentials presented in Fig. 1. The shape of time-averaged effective potentials provides information on the nature of the different modes. For example, it shows that the OH stretch is significantly more anharmonic than the NH2 stretches. This agrees with experimental measurements70 since a significant redshift of approximately 170 cm1 between the harmonic frequencies calculated at the MP2 level of theory and the experimental value is seen for the OH stretching mode, while a much lower shift of approximately 30–55 cm1 is seen for the NH2 stretching modes. As was mentioned previously, the stationary frequencies obtained using the time-averaged potentials are in very good agreement with experimental measurements. A comparison of the time-averaged effective potentials with the harmonic potential and the diagonal anharmonic potential also provides additional information on the different modes. The comparison for the different modes of conformer I of alanine is given in Fig. 3. The main feature seen is that the time-averaged potentials are often very similar to the diagonal

18

Barak Hirshberg and R. Benny Gerber

Fig. 2 Time-averaged effective single-mode potentials obtained using Eq. (20) for different vibrational modes of conformer I of alanine. Blue diamonds represent the symmetric NH2 stretch, orange circles the asymmetric NH2 stretch, and yellow squares present the OH stretching mode. Results were obtained using MP2/aug-cc-pVDZ level of theory.

potential. This is not surprising since, especially for fundamentals, the diagonal accounts for a significant portion of the anharmonic correction, which is often called the intrinsic anharmonicity. The time averaging over the classical trajectories accounts for the different interactions between the modes. This effect is subtle, but very important in order to obtain spectroscopically accurate results for modes with significant anharmonicity, such as OH stretches. A comparison to the harmonic potentials shows that indeed, the asymmetric NH2 stretch can be well approximated by the harmonic approximation even for energies well above the fundamental transition. This agrees with the small deviation of the harmonic approximation from the experiment for this mode which is about 1% at the MP2 level of theory. On the other hand, this analysis also emphasizes the significant anharmonic character of the OH stretching mode for which anharmonic treatment is essential.

3.2 Mixed Quantum–Classical Methods In recent years, several important advancements were made in applying mixed quantum–classical methods to large molecules, some using ab initio PES. Here, we briefly discuss these developments and their applications.

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

19

Fig. 3 Time-averaged single-mode effective potentials (blue circles), the diagonal anharmonic potential given by Eq. (22) (orange squares), and the harmonic potential (yellow diamonds) for different normal modes of alanine I. Panel (A) presents results for the symmetric NH2 stretch, panel (B) presents results for the asymmetric NH2 stretch, and panel (C) presents results for the OH Stretch. Results were obtained using MP2/ aug-cc-pVDZ level of theory.

3.2.1 Quantum Wavepacket AIMD The quantum wavepacket AIMD (QWAIMD) method was developed by Iyengar and coworkers.71–73 In essence, it combines the mixed quantum– classical method described in Section 2 with ab initio PES. Using this method, the system is divided into three subsystems: the electrons, which are treated quantum mechanically, most of the nuclei, which are treated

20

Barak Hirshberg and R. Benny Gerber

classically; and selected nuclei, which are treated quantum mechanically. Similarly to the AICSP method, the BO approximation is used and the electrons are treated through the time-independent Schr€ odinger equation. The equations of motion for the classical nuclei are solved using the atomcentered density matrix propagation scheme,74–76 which is computationally efficient, with the aim of describing large systems. Initial applications treated the classical nuclei using pure quantum chemical potentials such as B3LYP for relatively small systems. However, later the method was also combined with a QM/MM description77 of the classical nuclei in order to perform efficient simulations for large systems. Applications of the method were performed for calculating vibrational spectra in small molecules, such as ClHCl,72 for hydrogen tunneling in a model for an active site of a protein78 and for shared proton dynamics in a complex between phenol and trimethylamine.71 More recently, QWAIMD was introduced for periodic systems,79 with application to a two-dimensional model of concentrated solid hydrochloric acid. While the method seems very promising for describing large systems, the general difficulties which are common to all mixed quantum–classical methods may also be relevant to QWAIMD. Mainly, since only few atoms in the system are treated quantum mechanically, errors may arise when calculating observables, such absorption cross sections.63,64 Another development direction in the field of mixed quantum–classical dynamics is the treatment of open quantum systems.80–82 For example, several methods were developed based on the quantum–classical Louiville equation. For a detailed description of the different methods, see the reviews by Karpal.32,80 Very recently, Kananenka et al. suggested an efficient method to describe the long-time dynamics of an open quantum system using the transfer tensor method. QWAIMD was also recently extended to treat coupled electron-nuclear dynamics of open systems through a partitioning of the total Schr€ odinger equation into local domains that interact with each other through absorbing or emitting potentials.83 This method was then applied to a model of a molecular wire.

4. SUMMARY AND FUTURE DIRECTIONS Mean-field methods for quantum molecular dynamics have gained renewed interest in recent years. One important reason for this is that they allow a computationally efficient framework to describe the interactions

Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems

21

between the nuclei using ab initio potentials directly. Such potentials are required in order to accurately describe complex systems, for which empirical potentials are not generally available. Several developments in this spirit, presented in this paper are (1) The AICSP method, which combines the CSP method for quantum nuclear dynamics with ab initio PES, and (2) the QWAIMD method, which is a mixed quantum–classical method where the trajectories for the classical subsystem are performed using potentials obtained directly from quantum chemical methods. AICSP was applied recently to the calculation of stationary anharmonic vibrational frequencies of amino acids and the G–C pair of nucleobases,51 for which ab initio potentials are essential in order to obtain quantitative agreement with experimental measurements. QWAIMD was applied, for example, to study hydrogen tunneling in a model for an enzyme active site.78 These recent developments open a way to several exciting future directions and to a wide range of applications, such as ultrafast timedomain spectroscopy measurements, the calculation of vibrational lineshapes, electron photodetachment in large systems, and transition state dynamics. For example, while several applications, mainly by McCoy and coworkers,57–59 were performed using mixed quantum–semiclassical TDSCF to study transition state dynamics of ClHCl in Ar clusters, the application of mean-field methods for transition state dynamics has so far been limited mainly to systems for which an accurate empirical potential in the vicinity of the transition state was available. The direct use of ab initio potentials can significantly increase the range of applicability of mean-field methods to study transition state dynamics in different systems. This may also allow quantitative comparison of mean-field methods with experimental measurements, which has not been done extensively in earlier studies. At the same time, a major challenge that remains is finding extensions of the method. Currently, methods such as AICSP are very much limited to cases for which normal coordinates can be used and the dynamics are dominated by a single minimum on the potential. Application of these methods for cases in which more than one local minimum is involved is problematic, since the separable ansatz will no longer be adequate. It is an interesting challenge to develop methods that use other coordinates which maintain the separability of the nuclear wavefunction for longer timescales and more complicated dynamics. In these cases, techniques used for stationary vibrational spectroscopy, such as the use of internal coordinates53 or local coordinates52 may prove very useful.

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ACKNOWLEDGMENTS We dedicate this paper to Mark Ratner. R.B.G. gratefully acknowledges very stimulating collaborations with Mark, including in the field of this paper, and many illuminating discussions and exchanges of ideas. B.H. is supported through an Adams Fellowship of the Israel Academy of Sciences and Humanities. We thank Dr. Laura McCaslin for her comments on the manuscript.

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[37] Roy, T. K.; Sharma, R.; Gerber, R. B. First-Principles Anharmonic Quantum Calculations for Peptides Spectroscopy: VSCF Calculations and Comparison with Experiments. Phys. Chem. Chem. Phys. 2015, 18(3), 1607–1614. [38] Roy, T. K.; Kopysov, V.; Nagornova, N. S.; Rizzo, T. R.; Boyarkin, O. V.; Gerber, R. B. Conformational Structures of a Decapeptide Validated by First Principles Calculations and Cold Ion Spectroscopy. ChemPhysChem 2015, 16(7), 1374–1378. [39] Makri, N. Time-Dependent Quantum Methods for Large Systems. Annu. Rev. Phys. Chem. 1999, 50, 167–191. [40] Meyer, H. D. Studying Molecular Quantum Dynamics with the Multiconfiguration Time-Dependent Hartree Method. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2012, 2(2), 351–374. [41] Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Math. Proc. Cambridge Philos. Soc. 1930, 26(3), 376. [42] L€ owdin, P. O.; Mukherjee, P. K. Some Comments on the Time-Dependent Variation Principle. Chem. Phys. Lett. 1972, 14(1), 1–7. [43] McLachlan, A. D. A Variational Solution of the Time-Dependent Schrodinger Equation. Mol. Phys. 1964, 8(1), 39–44. [44] Jin, S.; Sparber, C.; Zhou, Z. On the Classical Limit of a Time-Dependent SelfConsistent Field System: Analysis and Computation. Kinet. Relat. Model. 2017, 10(1), 263–298. [45] Bisseling, R. H.; Kosloff, R.; Gerber, R. B.; Ratner, M. A.; Gibson, L.; Cerjan, C. Exact Time-Dependent Quantum Mechanical Dissociation Dynamics of I2He: Comparison of Exact Time-Dependent Quantum Calculation with the Quantum TimeDependent Self-Consistent Field (TDSCF) Approximation. J. Chem. Phys. 1987, 87(5), 2760. [46] Alimi, R.; Gerber, R. B.; Hammerich, A. D.; Kosloff, R.; Ratner, M. A. Validity of Time-Dependent Self-Consistent-Field (TDSCF) Approximations for Unimolecular Dynamics: A Test for Photodissociation of the Xe–HI Cluster. J. Chem. Phys. 1990, 93(1990), 6484. [47] Garcı´a-Vela, A.; Gerber, R. B. Three-Dimensional Quantum Wave Packet Study of the Ar–HCl Photodissociation: A Comparison between Time-Dependent SelfConsistent-Field and Exact Treatments. J. Chem. Phys. 1995, 103(9), 3463. [48] Fredj, E.; Gerber, R. B.; Ratner, M. A. Semiclassical Molecular Dynamics Simulations of Low-Temperature Clusters: Applications to (Ar)13; (Ne)13; (H2O)n, n ¼ 2,3,5. J. Chem. Phys. 1996, 105(3), 1121. [49] Fredj, E.; Gerber, R. B.; Ratner, M. A. Quantum Mechanical Simulations of Inelastic Scattering in Collisions of Large Clusters: Ar + (H2O)11. J. Chem. Phys. 1998, 109(12), 4833–4842. [50] Knospe, O.; Jungwirth, P. Electron Photodetachment in C-60(-): Quantum Molecular Dynamics with a Non-Empirical, “on-the-Fly” Calculated Potential. Chem. Phys. Lett. 2000, 317(6), 529–534. [51] Hirshberg, B.; Sagiv, L.; Gerber, R. B. Approximate Quantum Dynamics Using Ab Initio Classical Separable Potentials: Spectroscopic Applications. J. Chem. Theory Comput. 2017, http://dx.doi.org/10.1021/acs.jctc.6b01129 (just accepted manuscript). [52] Cheng, X.; Steele, R. P. Efficient Anharmonic Vibrational Spectroscopy for Large Molecules Using Local-Mode Coordinates. J. Chem. Phys. 2014, 141(10), 104105. [53] Suwan, I.; Gerber, R. B. VSCF in Internal Coordinates and the Calculation of Anharmonic Torsional Mode Transitions. Chem. Phys. 2010, 373(3), 267–273. [54] Roy, T. K.; Carrington, T.; Gerber, R. B. Approximate First-Principles Anharmonic Calculations of Polyatomic Spectra Using MP2 and B3LYP Potentials: Comparisons with Experiment. J. Phys. Chem. A 2014, 118(33), 6730–6739.

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[55] Horn, T. R.; Gerber, R. B.; Valentini, J. J.; Ratner, M. A. Vibrational States and Structures of Ar_3: The Role of 3-Body Forces. J. Chem. Phys. 1991, 94, 6728. [56] Jungwirth, P.; Roeselova, M.; Gerber, R. B. Optimal Coordinates for Separable Approximations in Quantum Dynamics of Polyatomic Systems: Coordinate Choice Criteria and Error Estimates. J. Chem. Phys. 1999, 110(20), 9833. [57] Lavender, H. B.; McCoy, A. B. Transition State Dynamics of Arn(ClHCl) (N ¼ 0 5): Effects of Complex Formation on the Dynamics and Spectroscopy. J. Phys. Chem. A 2000, 104(3), 644–651. [58] McCoy, A. B.; Gerber, R. B.; Ratner, M. A. A Quantitative Approximation for the Quantum Dynamics of Hydrogen Transfer: Transition State Dynamics and Decay in ClHCl-. J. Chem. Phys. 1994, 101(3), 1975. [59] McCoy, A. B. Transition State Dynamics of Chemical Reactions in Clusters: A SixDimensional Study of Ar(ClHCl). J. Chem. Phys. 1995, 103(3), 986. [60] Messiah, A. Quantum Mechanics. Dover Books on Physics, 1961, Dover publications, ISBN: 9780486409245, https://books.google.co.il/books?id¼fCdjuJ2XPNQC. [61] Alimi, R.; Gerber, R. Solvation Effects on Chemical Reaction Dynamics in Clusters: Photodissociation of HI in Xe_{N}HI. Phys. Rev. Lett. 1990, 64(12), 1453–1456. [62] Gerber, R. B.; Alimi, R. Quantum Effects in Molecular Reaction Dynamics in Solids: Photodissociation of HI in Solid Xe. Chem. Phys. Lett. 1990, 173(4), 393–396. [63] Haug, K.; Metiu, H. Absorption Spectrum Calculations Using Mixed QuantumGaussian Wave Packet Dynamics. J. Chem. Phys. 1993, 99(9), 6253. [64] Haug, K.; Metiu, H. A Test of the Possibility of Calculating Absorption Spectra by Mixed Quantum-Classical Methods. J. Chem. Phys. 1992, 97(7), 4781–4791. [65] Hillery, M.; O’Connell, R. F.; Scully, M. O.; Wigner, E. P. Distribution Functions in Physics: Fundamentals. Phys. Rep. 1984, 106(3), 121–167. [66] Thomas, M.; Brehm, M.; Fligg, R.; V€ ohringer, P.; Kirchner, B. Computing Vibrational Spectra from Ab Initio Molecular Dynamics. Phys. Chem. Chem. Phys. 2013, 15(18), 6608–6622. [67] Gaigeot, M.-P.; Martinez, M.; Vuilleumier, R. Infrared Spectroscopy in the Gas and Liquid Phase from First Principle Molecular Dynamics Simulations: Application to Small Peptides. Mol. Phys. 2007, 105(19–22), 2857–2878. [68] Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46(7), 618–622. [69] Lin, Y. S.; Li, G. De; Mao, S. P.; Chai, J. Da. Long-Range Corrected Hybrid Density Functionals with Improved Dispersion Corrections. J. Chem. Theory Comput. 2013, 9(1), 263–272. [70] Stepanian, S.; Reva, I. Conformational Behavior of Alfa-Alanine. Matrix-Isolation Infrared and Theoretical DFT and Ab Initio Study. J. Phys. Chem. Rev. 1998, 102(97), 4623–4629. [71] Iyengar, S. S.; Jakowski, J. Quantum Wave Packet Ab Initio Molecular Dynamics: An Approach to Study Quantum Dynamics in Large Systems. J. Chem. Phys. 2005, 122(11), 114105. [72] Sumner, I.; Iyengar, S. S. Quantum Wavepacket Ab Initio Molecular Dynamics: An Approach for Computing Dynamically Averaged Vibrational Spectra Including Critical Nuclear Quantum Effects. J. Phys. Chem. A 2007, 111(41), 10313–10324. [73] Iyengar, S. S. Ab Initio Dynamics with Wave-Packets and Density Matrices. Theor. Chem. Acc. 2006, 116(1–3), 326–337. [74] Schlegel, H. B.; Millam, J. M.; Iyengar, S. S.; Voth, G. A.; Daniels, A. D.; Scuseria, G. E.; Frisch, M. J. Ab Initio Molecular Dynamics: Propagating the Density Matrix with Gaussian Orbitals. J. Chem. Phys. 2001, 114(22), 9758–9763. [75] Iyengar, S. S.; Schlegel, H. B.; Millam, J. M.; Voth, G. A.; Scuseria, G. E.; Frisch, M. J. Ab Initio Molecular Dynamics: Propagating the Density Matrix with Gaussian Orbitals.

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II. Generalizations Based on Mass-Weighting, Idempotency, Energy Conservation and Choice of Initial Conditions. J. Chem. Phys. 2001, 115(22), 10291–10302. Schlegel, H. B.; Iyengar, S. S.; Li, X.; Millam, J. M.; Voth, G. A.; Scuseria, G. E.; Frisch, M. J. Ab Initio Molecular Dynamics: Propagating the Density Matrix with Gaussian Orbitals. III. Comparison with Born-Oppenheimer Dynamics. J. Chem. Phys. 2002, 117(19), 8694–8704. Sumner, I.; Iyengar, S. S. Combining Quantum Wavepacket Ab Initio Molecular Dynamics with QM/MM and QM/QM Techniques: Implementation Blending ONIOM and Empirical Valence Bond Theory. J. Chem. Phys. 2008, 129(5), 1–16. Iyengar, S. S.; Sumner, I.; Jakowski, J. Hydrogen Tunneling in an Enzyme Active Site: A Quantum Wavepacket Dynamical Perspective. J. Phys. Chem. B 2008, 112(25), 7601–7613. Li, X.; Iyengar, S. S. Quantum Wavepacket Ab Initio Molecular Dynamics for Extended Systems. J. Phys. Chem. A 2011, 115(23), 6269–6284. Kapral, R. Progress in the Theory of Mixed Quantum-Classical Dynamics. Annu. Rev. Phys. Chem. 2006, 57(1), 129–157. Kapral, R. Quantum Dynamics in Open Quantum-Classical Systems. J. Phys. Condens. Matter 2015, 27, 73201. Kananenka, A. A.; Hsieh, C.-Y.; Cao, J.; Geva, E. Accurate Long-Time Mixed Quantum-Classical Liouville Dynamics via the Transfer Tensor Method. J. Phys. Chem. Lett. 2016, 7(23), 4809–4814. Pacheco, A. B.; Iyengar, S. S. Multistage Ab Initio Quantum Wavepacket Dynamics for Electronic Structure and Dynamics in Open Systems: Momentum Representation, Coupled Electron-Nuclear Dynamics, and External Fields. J. Chem. Phys. 2011, 134(7), 1–15.

CHAPTER TWO

Electron–Ion Impact Energy Transfer in Nanoplasmas of Coulomb Exploding Clusters Isidore Last, Joshua Jortner1 School of Chemistry, Tel Aviv University, Tel Aviv, Israel 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. A Microscopic Model for the Repulsive Impact Energy Transfer 3. The Simulation Methodology 4. Simulations of EIET 5. Dynamics of the EIET 6. Energetics of EIET 7. The Consequences of Coulomb Interactions 8. Concluding Remarks Acknowledgment References

28 30 34 36 38 40 44 47 50 50

Abstract Novel features of analysis and control of nanoplasma dynamics are manifested in elemental and molecular clusters irradiated by a near-infrared intense ultraintense laser pulse, where the laser energy pumped to the nanoplasma electrons is transferred to the cluster ions by Coulomb explosion (CE) and by electron–ion impact mechanisms. The contribution of the electron–ion impact was studied by a microscopic model, together with molecular dynamics simulations of the electron–ion kinetic energy transfer in the course of the electron–ion collision events. The simulations were performed for ionic (He+)N, (Ne+)N, and (Ne4+)N clusters containing weakly charged ions, as well as for (H+)N and (He2+)N clusters consisting of bare nuclei and electrons. The clusters were subjected to femtosecond (τ ¼ 30 fs) laser pulses with peak intensities of IM ¼ 1015– 1017 W cm2. The force Fimp, generated by the electron impact kinetic energy transfer was found to decrease strongly with the exploding cluster radius R, i.e., Fimp ∝ Rη, with η 4–6. The electron impact energy transferred to the periphery ions of clusters (in the size domain of N ¼ 104–106) made up less than 2.5% of the maximal ion energy. The laser energy transfer to the nanoplasma involves the dominating contribution of the Coulomb energy and a minor contribution of the electron impact, with the cluster expansion and decay being governed by the CE mechanism.

Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.01.003

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2017 Elsevier Inc. All rights reserved.

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Isidore Last and Joshua Jortner

1. INTRODUCTION Plasmas formed in nanoscale matter, e.g., large molecules and clusters, driven by intense laser pulses, are of considerable interest for the analysis and control of nanoplasma dynamics involving laser-matter energy transfer.1–7 In elemental and molecular clusters driven by an intense near-infrared laser pulse (with peak intensity IM > 1015 W cm2), the energy transfer to the clusters is realized through the interaction of the laser radiation with the cluster electrons.1–6 The first stage of this interaction involves an inner ionization process providing the partial or complete removal of electrons from the cluster atoms.1,3 The electrons released from the atoms initially form a nanoplasma within the cluster.3–6 The interaction of the nanoplasma electrons with laser radiation provides an increase of the energy of the electrons, with a portion of this energy being transferred to the cluster ions.2–6 There are two feasible, but distinct, processes of energy transfer from electrons to ions in a laser-driven cluster: (I) Coulomb explosion (CE) is generated in the course of outer ionization when nanoplasma electrons within the cluster leave due to their excess energy, or are directly removed by the force of the laser field. The withdrawal of the nanoplasma electrons results in a positively charged cluster, generating interionic repulsive Coulomb forces that provide cluster expansion and decay.1–4,6 The forces determining the CE motion of ions involve long-range Coulomb interactions. (II) Direct kinetic energy transfer from fast, high-energy (keV) electrons to the slowly moving ions. This energy transfer process involves electron–ion binary collisions. These electron impact energy transfer (EIET) processes correspond to short-time (attosecond) and to ˚ ) events. As the motion of the ions in exploding short-distance (1–2 A clusters is close to being spherically symmetric, only the events affecting the radial motion of the ions are of importance. EIET occurs when the electron–ion interaction is sufficiently strong to perturb the ion motion. The generation of EIET events is determined by the nature of the electron–ion interactions, which fall into two categories. First, short-range repulsive interactions prevail when the electron impact parameter b for the electron–ion collision is smaller than the ion radius ra. The interaction responsible for the EIET is then generated by “repulsive collisions.” Such repulsive collisions are expected to

Electron–Ion Impact Energy Transfer in Nanoplasmas

29

dominate for nanoplasmas produced in exploding clusters containing weakly charged ions, e.g., (He+)N, (Ne+)N, and (Ne4+)N, with ionic radii being relatively close to those of the corresponding neutral atoms. Second, short-distance Coulomb interactions prevail when b > ra. This interaction is responsible for the EIET generated by “Coulomb collisions,” which are realized in nanoplasmas produced in exploding clusters containing bare nuclei (ra ¼ 0), such as (H+)N and (He2+)N. In those clusters EIET is driven only by Coulomb collisions. On the other hand, in a cluster with large ionic radii both repulsive and Coulomb EIET may prevail. The direct kinetic energy transfer from electrons to ions was often described in terms of thermal hydrodynamic pressure.4–23 This description of the hydrodynamic pressure in clusters is based on a macroscopic model without a specification of the electron–ion interactions, which is distinct from the microscopic treatment explored herein.24 We present a theoretical–computational study of energy transfer from electrons to ions in clusters driven by an intense infrared laser field, and its dynamic and energetic implications. In Section 3 we focused on the exploration of EIET in (He+)N, (Ne+)N, and (Ne4+)N clusters consisting of ions with relatively large radii, while in Section 2 we advance a theoretical model describing the repulsive EIET mechanism for (H+)N and (He2+)N clusters. The numerical determination of EIET, including both repulsion and Coulomb mechanisms, was performed by molecular dynamics (MD) simulations (Sections 3 and 4). These simulations are not connected to any theoretical model, such as the macroscopic model for hydrodynamic pressure5,9 or for CE.1,3–6 The electron–ion potential for (H+)N and (He2+)N clusters used herein was presented by a Coulomb potential with a smoothening term, which was used in previous simulations.25,26 For the (He+)N, (Ne+)N, and (Ne4+)N clusters, the electron– ion potentials are presented as the sum of a short-range repulsive potential and the usual Coulomb potential. The good agreement attained between the microscopic theoretical model (Section II) and the results of numerical MD simulations (Sections 5 and 6) for the EIET energies and their time dependence provides strong support for the validity of the microscopic model for EIET. From the comparison of forces driving EIET with the Coulomb force, we show that a proper description of the laser energy transfer to the nanoplasma ions involves the dominating contribution of the Coulomb force (Section 7).

30

Isidore Last and Joshua Jortner

2. A MICROSCOPIC MODEL FOR THE REPULSIVE IMPACT ENERGY TRANSFER It is important to emphasize that the EIET considered herein does not correspond to a thermal-like system but rather to an expanding, spherically symmetric, system with radial motion of the heavy particles (ions). In such a system, an impact electron with a large angle between the electron and ion velocities performs a lateral-like collision and is not supposed to affect much the radial motion of the ion. Only an electron–ion collision, which is close to radial, may contribute significantly to the electron–ion energy transfer. As this collision somewhat reminds the one-dimensional front collision, we shall consider here the properties of the electron–ion front (one-dimensional) collisions. The one-dimensional collisional energy transfer between an electron (energy Ee, mass me) and a slower moving ion (energy Ea, mass ma), with a very small electron–atom mass ratio of η ¼ me/ma (η 5.45 104 for real atoms), is given by Δ0 ¼ 4ηEe 4ðηEe Ea Þ1=2 4ηEa

(1)

The plus sign for the second term on the RHS of Eq. (1) corresponds to the case of the electron moving in the same direction as the ion, always providing an ion energy increase. The minus sign for the second term on the RHS of Eq. (1) corresponds to the particles moving in opposite directions, providing either an increase or a decrease of the ion energy. In the domain of ðηEe Þ1=2 ≪ðEa Þ1=2 ≪ðEe =ηÞ1=2

(2)

ηve ≪va ≪ve ,

ð20 Þ

or

where ve and va are the velocities of the electron and of the ion, respectively, and both the first and the third terms on the RHS of Eq. (1) are negligibly small. Ignoring these two terms in Eq. (1), we are left with the equation Δ0 ¼ 4ðηEe Ea Þ1=2 ,

(3)

which demonstrates the proportionality of Δ0 to the square root of both the electron and the ion energies, without alluding to the specific mechanism for the acquisition of these energies. In the case of a Ne atom, η ¼ 2.72 105,

31

Electron–Ion Impact Energy Transfer in Nanoplasmas

and for a realistic electron energy of Ee ¼ 1 keV the validity conditions for Eq. (2) are fulfilled when (Ea)1/2 falls in the wide range of 0.16 ≪ (Ea/eV) ≪ 6000. For a rough estimate, we replace these two inequalities in Eq. (2) by the two equalities (Ea/eV)1/2 ¼ 0.16z and (Ea/eV)1/2 ¼ 6000/z, with the numerical factor z ¼ 5 marking the limits of the inequality. The ions’ energy then spans the very wide domain of 0.64 eV ≪ Ea ≪ 1.4 MeV, validating the use of Eq. (3). According to Eq. (3), the ion gains energy by the front collision when the electron outgoing from the cluster is running after it (plus sign) and loses energy when the electron and ion are running against each other (minus sign). We suggest that in the general case of three-dimensional collisions, the energy transfer Δ averaged over the collision configurations remains proportional to Δ0, Eq. (3), so that Δ ¼ 4ΩðηEe Ea Þ1=2

(4)

with the proportionality parameter Ω < 1. Using Eq. (4) we can express the rate of the ion energy change provided by the EIET by (5) dEimp =dt ¼ Δ νð +Þ νðÞ where ν(+) and ν() are the electron–ion collision frequencies for the outgoing and for the ingoing electrons, respectively. From Eqs. (4) and (5) we infer that dEimp =dt ¼ 4ΩðηEe Ea Þ1=2 νð +Þ νðÞ (6) which can be expressed in the form h i ð +Þ ðÞ dEimp =dt ¼ d Eimp Eimp =dt

(6a)

where ðÞ

dEimp =dt ¼ 4ΩðηEe Ea Þ1=2 vðÞ

(6b)

The frequencies ν(+) and ν() provide the electron–ion impact energies

ð +Þ Eimp

ðÞ

and Eimp , respectively, with the net impact energy ð +Þ

ðÞ

Eimp ¼ Eimp Eimp

being obtained from the integration of Eqs. (6a) and (6b).

(7)

32

Isidore Last and Joshua Jortner

In what follows we shall consider clusters with identical and fixed ionic charges qa. Initially, the clusters are assumed to be neutral with the number of unbound electrons qaN, where N is the number of the cluster atoms. The outer ionization level, expressed in terms of the efficiency noi ¼ Noi/qaN of the outer ionization process in the expanding cluster, is given by noi ¼ 1 Ne =qa N

(8)

where Ne is the number of electrons inside the cluster and Noi ¼ qaN–Ne is the number of electrons removed from the cluster.27 The Ne electrons remaining in the cluster, together with the N ions, form a charged nanoplasma. The effect of the EIET is expected to be of importance in clusters with weak outer ionization, i.e., noi ≪ 1, and, consequently, with Ne being close to qaN. Taking Ne ¼ qaN and assuming that the distribution of electrons inside the cluster is homogeneous, we obtain the electron density ρe ¼ qa ρa ðR0 =RÞ3 ,

(9)

where ρa is the atomic density of a neutral cluster prior to irradiation, R0 is the initial neutral cluster radius, and R is the radius of the expanding cluster. In what follows we shall consider the EIET to ions at the surface of the expanding cluster. The frequencies ν(+) and ν() of the electron–ion collisions generated by the outgoing and by the ingoing electrons, respectively, Eq. (6), are assumed to be proportional to the electron density, Eq. (9), to the average velocity of electrons moving inside the cluster ve, and to the electron–ion collision cross section σ. It is convenient to present these frequencies in the form νð +Þ ¼ Θð +Þ sin 2 ðα=2Þρa qa σve ðR0 =RÞ3

(10a)

νðÞ ¼ ΘðÞ sin 2 ðα=2Þρa qa σve ðR0 =RÞ

(10b)

3

where Θ(+) and Θ() are coefficients which can be estimated by simulations, and sin2(α/2) is the solid angle formed by a collision angle α between the radius vector of an ion and the velocity vector of an electron. In such a presentation of the frequencies, the case of Θ(+) ¼ Θ() ¼ 1 implies that all colliding electrons with a collision angle smaller than α provide the energy transfer and that this energy transfer per collision is equal to Δ, Eq. (4). Expressing the velocity of electrons ve in Eqs. (10a) and (10b) by the average energy Ee of the cluster electrons inside the expanding cluster, with Ea being the energy of the ions at the surface of the cluster, we obtain the time derivative, Eq. (6), of the net impact energy Eimp in the form

33

Electron–Ion Impact Energy Transfer in Nanoplasmas

dEimp =dt ¼ 4√2 sin 2 ðα=2Þψ m1=2 ρa qa σ ðR0 =RÞ3 Ee Ea1=2 a

(11)

with Ψ ¼ Ω(Θ(+)Θ()) being estimated from our simulations. Using the ˚ for distances, A ˚ 3 for densities, fs for time, and eV for energies units of A one obtains e1=2 ρa qa σ ðR0 =RÞ3 Ee Ea1=2 dEimp =dt ¼ 0:554 sin 2 ðα=2Þψ m a

(11a)

with m ea being expressed in atomic masses. For the angle α in Eqs. (11) and (11a), we arbitrarily chose α ¼ 0.2π (α ¼ 36°) with sin2(α/2) ¼ 0.0955. According to Eq. (11), the impact energy time derivative dEimp/dt is proportional to the electrons’ energy Ee and to the square root of the ion energy Ea and is inversely proportional to R3. The time dependence of dEimp/dt is complicated, as both Ea and R increase monotonously with increasing t. Regarding the effect of ion dynamics on dEimp/dt, we can introduce in Eq. (11) the ion velocity va ¼ (2Ea/ma)1/2 to get dEimp =dt ¼ 4 sin 2 ðα=2Þψ ρa qa σva ðR0 =RÞ3 Ee

(11b)

Replacing va in Eq. (11b) by dR/dt results in the energy transfer Eimp(R) (when the cluster radius attains the value R at time t) being obtained from the integral Eimp ðRÞ ¼ 4 sin

2

ðα=2Þψρa qa R03

ðR

3 σEe =ðR0 Þ dR0

(12)

R0

The final energy transfer E˜imp is determined by the integration of Eq. (12), which results in Eeimp ¼ Eimp ð∞Þ

(12a)

with 4sin2(α/2) ¼ 0.318 for α ¼ 0.2π. In what follows we shall denote the dynamic (time t and radius R) dependent energy E(R;t) by E, while the final e energy will be denoted by E. The electron impact force Fimp ¼ dEimp/dR is obtained from Eq. (11b) in the form Fimp ¼ 4 sin 2 ðα=2Þ ψρa qa σEe ðR0 =RÞ3

(13)

It is important to note that the time dependence of the force Fimp, Eq. 13, is determined only by the three variables, R(t), Ee(t), and σ(Ee). To

34

Isidore Last and Joshua Jortner

determine the dependence of the force Fimp on the cluster radius R, Eq. (13), one needs to know two variables only, i.e., Ee(R) and σ. It is of interest to compare the impact force Fimp with the electrostatic Coulomb forces, FCo, acting on the periphery ions. The Coulomb force for the surface ions of a homogeneous sphere with a uniform distribution of N ions and (1–noi)N electrons is given by FCo ¼ Bnoi q2a N =R2 ,

(14) ˚ 1

˚ , and FCo is expressed in eVA units. Eq. (14) ignores where B ¼ 14.4 eVA the microstructure of a cluster describing it as a continuous charged sphere. Eq. (14) differs from the expression for the electrostatic force provided by Ditmire et al.,5 due to the difference in the cluster nanoplasma charge distributions, which was taken by Ditmire et al.5 as a surface distribution and as a homogeneous (uniform) space distribution in the present work. The R dependence of the electrostatic force, Eq. (14), is distinct from the impact force, Eq. (13). For the simplest case of noi and Ee being independent of R, we infer that FCo decreases as R2, while Fimp decreases more strongly as R3. Our model provides the theoretical framework for the description of the dynamics and energetics of the nanoplasma driven by the collisional impact force, Eq. (13), and by the electrostatic force, Eq. (14). These theoretical results will now be confronted with the results of MD simulations.

3. THE SIMULATION METHODOLOGY In order to treat large clusters consisting of N ¼ 104–106 atoms, we applied a scaled electron and ions dynamics (SEID) simulation scheme.28–31 The SEID simulations consider a scaled cluster with a reduced number of composite particles, i.e., pseudoparticles (also called macroparticles30). The scaling parameter s (>1) is equal to the ratio between the number of e of the pseudoparticles, with real particles N, and of the number N e s ¼ N =N : The s values used for our calculations are sufficiently low to ensure the accuracy of the simulation results, including the EIET energies. We consider a linear polarized IR laser radiation of λ ¼ 840 nm. The laser pulses are taken to be of Gaussian shape with a peak intensity of IM and a pulse width τ.28,32 We used short pulses with τ ¼ 30 fs. Our approach excludes the inner ionization mechanisms of cluster atoms, which involve laser field ionization, as manifested by the barrier suppression ionization

Electron–Ion Impact Energy Transfer in Nanoplasmas

35

(BSI),4,33 and ionization by electron impact.4,23,34 At the onset of the simulations, the cluster is described as a neutral nanoplasma, consisting of a sphere with an initial cluster radius R0, which contains uniformly distributed atomic ions and quasi-free electrons.32,35 The ionic charges qa are assumed to be identical for all ions and are fixed in time. The SEID simulation results provided the time-resolved data with information on the cluster structure, on outer ionization of the nanoplasma (specified by noi, Eq. (8)), and on electrons and ions energetics of dynamics. The potential between electrons and ions with an ionic core, e.g., He+, + Ne , and Ne4+ ions, is presented as the sum of a usual Coulomb potential UCo and a short-range repulsive potential Urep36 U ðr Þ ¼ UCo ðr Þ + Urep ðr Þ

(15)

The repulsive potential is presented as a power function Urep ðr Þ ¼ Ua ðra =r Þ8

(15a)

with ra being the ion radius estimated by using Slater’s rules.37 The parameter Ua is determined by the condition that at the ion boundary, i.e., at r ¼ ra, the repulsive force 8Ua/ra is equal to the attractive Coulomb force, so that Ua ¼ Bqa/8ra. Generally speaking, the ion radius ra is distinct from the 2 . rcol electron–ion collision radius rcol of the collision cross section σ ¼ πrcol was determined as the distance which makes the potential Urep(rcol) equal to the energy Ee of the colliding electrons. Using this condition and Eq. (15a) one obtains the collision cross section σ ¼ πra2 ðUa =Ee Þ1=4

(16)

In our simulations, σ is an Ee dependent variable, which is expected to vary in the course of cluster expansion. The radii ra of the ions and the Ua potentials of Eq. (15a) were estimated ˚ and Ua ¼ 1.18 eV for as ra ¼ 1.19 A˚ and Ua ¼ 1.51 eV for He+, ra ¼ 1.53 A + 4+ ˚ Ne , and ra ¼ 1.34 A and Ua ¼ 5.37 eV for Ne . The initial atomic (ionic) densities used here correspond to those of the neutral clusters: ρa ¼ 0.0218 A˚3 with a neutral atom radius of 2.0 A˚ for HeN, and ˚ 3 with a neutral atom radius of 1.7 A ˚ for NeN. ρa ¼ 0.036 A The production of ions with a fixed charge, i.e., He+, Ne+, and Ne4+, was described in the framework of the BSI model, with the charge qa of an ion being determined univocally by the local laser radiation intensity I.33 In the case of helium, for example, BSI provides neutral atoms (qa ¼ 0) at

36

Isidore Last and Joshua Jortner

I < 1.5 1015 W cm2, singly ionized ions (qa ¼ 1) at 1.5 1015 < I < 8.8 1015 W cm2, and doubly ionized (qa ¼ 2) ions at I > 8.8 1015 W cm2. The BSI ionization thresholds of a neon atom are as follow: I ¼ 8.6 1014, 2.8 1015, 7.2 1015, 2.2 1016, 2.0 1017 W cm2 for qa ¼ 1, 2, 3, 4, 8, respectively. For Ne+, Ne2+, and Ne3+ ions the dependence of the BSI ionization thresholds on qa is partly supported by gas-phase experimental data.38 In addition to the BSI, the inner Coulomb fields within the charged cluster may contribute to the local fields, resulting in the enhancement of ionization due to the ignition effects.33,39 These effects of electron impact and ionization ignition may violate the simple BSI behavior by increasing the ionization level of the ions and affecting the distribution of the ion charges.2 In rare gas clusters a broad charge distribution is usually exhibited, in accord with both simulations and experimental results.2,5,31,40–45 The use of fixed ion charges for (He+)N, (Ne+)N, and (Ne4+)N constitutes a simplified model. In addition, we also performed a set of calculations for the forces and energetics in the real systems (H+)N and (He2+)N, using a smoothened Coulomb potential U(r) ∝ [r4 + α4]1/4, with α ¼ 0.5 A˚.25

4. SIMULATIONS OF EIET New procedures were advanced and implemented for the simulations of EIET. In our simulations, the EIET is calculated for periphery ions located at a distance R (R0 at the onset of expansion) from the cluster center. In the spherically expanding clusters the motion of the ions is essentially radial. In order to single out the EIET contribution against a background of other interactions a spatial procedure was applied, assuming that in the absence of the EIET events the kinetic energy of an ion changes smoothly. An EIET event takes place when an electron comes so close to the ion that the strong force of the repulsive potential, Eq. (15a), or of the short-distance Coulomb interaction, begins to affect the ion energy on an ultrashort collisional time scale of several attoseconds. It follows that the EIET event is identified as a sudden and short violation of the smooth character of the time dependence of the ion energy. The search for the violation of the smooth behavior of the ion energy is performed by tracing the trajectory of an ion, and by searching for discontinuities in the kinetic energy E(t) of an ion, as manifested in the sudden increase (or decrease) in dE/dt. The end of the

37

Electron–Ion Impact Energy Transfer in Nanoplasmas

EIET event is identified as the instant t when E(t) returns to manifest a smooth behavior. The energy change during the EIET event is taken to be the impact energy Δimp. When the Δimp energy is positive (being considered to be caused by the outgoing electrons), it is added to the impact energy ð +Þ ð +Þ ð +Þ Eimp Eimp ¼ Eimp + Δimp , Δimp > 0 . The negative Δimp energy (being connected with the effect of the ingoing electrons) is added to the impact ðÞ ðÞ ðÞ energy Eimp Eimp ¼ Eimp + Δimp , Δimp < 0 . The energies thus calculated ð +Þ

ðÞ

are Eimp and Eimp per ion and are averaged over the periphery ions. The procedure described earlier provides the time dependence of the ð +Þ

ðÞ

ð +Þ

ðÞ

average Eimp and Eimp energies. The Eimp and Eimp energies at the end of ð +Þ the simulations are taken to represent the final impact energies Eeimp and ðÞ Eeimp of the cluster explosion (see Section 5). Using Eq. (7) we determine the net impact energy Eimp both for the time-dependent function Eimp(t) and for the final (t ¼ ∞) energy Eeimp. ð +Þ

ðÞ

In addition to the impact energies Eimp and Eimp , the simulations also provide the average energy Ea of the periphery ions and the average energy Ee of the electrons located inside the cluster, as well as the number Ne of these electrons. Using Ne one can calculate the outer ionization efficiency noi, Eq. (8). The values Ea, Ee, and noi are of importance in analyzing the theoretical expressions of Section 2. The total averaged force F at the periphery is given by F ¼ Fimp + FCo, where Fimp is the impact force and FCo is the Coulomb force, both obtained from simulations. Presenting Eimp(R) and Ea(R) as functions of the radius R of the expanding cluster, it is possible to identify the total force F and the impact force Fimp as the derivatives of Ea(R) and of Eimp(R), so that F ¼ dEa ðRÞ=dR Fimp ¼ dEimp ðRÞ=dR

(17a) (17b)

FCo ¼ F Fimp

(17c)

These forces, Eqs. (17a)–(17c), are time dependent, each being calculated at the corresponding value of the expanding cluster radius. The physical significance of the Coulomb force FCo of Eq. (17c) is distinct from the macroscopic description of Eq. (14). The procedure of the EIET simulations described earlier may also be used in the absence of the repulsion potential, Eq. (15a), in particular in

38

Isidore Last and Joshua Jortner

the case of (H+)N and (He2+)N clusters, when Coulomb interaction prevails. Using the smoothened Coulomb potential (Section 3) in the EIET simulations for the (H+)N and (He2+)N clusters, we found that these simulations are considerably more time-consuming than in the case of ions with relatively large ion radii (Section 3).

5. DYNAMICS OF THE EIET For the exploration of the dynamics of the EIET, we shall consider here the simulation results for the largest (Ne+)N cluster (R0 ¼ 320 A˚) subjected to the laser intensity of IM ¼ 5 1015 W cm2. The time dependence ð +Þ

ðÞ

of the two components of the EIET, Eimp and Eimp , and of the net energy Eimp, Eq. (7), for the ions at the periphery of the expanding cluster (i.e., close to R), were obtained from the procedure of Section 4 and are presented in Fig. 1 in the time interval from t ¼ 10 fs (11 fs after the simulation onset) up to t ¼ 260 fs. Fig. 1 also presents the time dependence of the cluster radius in terms of the ratio R/R0, of the average electrons energy Ee, and of the periphery ions energy Ea (see Section 4). According to the simulation results (Fig. 1) the efficient generation of the ð +Þ

ðÞ

electron impact energies Eimp , Eimp , and Eimp starts in the vicinity of the laser intensity peak, t 0, when the average electron energy Ee and the periphery ion energy Ea start to increase noticeably (Fig. 1). At the very beginning of the ion expansion, at t ¼ 10 fs, the energy Eimp accounts for about 20% of the ion energy Ea (Fig. 1), contributing markedly to the ion motion. This Eimp contribution may generate some asymmetry in the ion motion.12,14 The longer time interval of 10 fs < t < 60 fs is characterized by a strong ð +Þ

ðÞ

increase of the energies Eimp , Eimp , and Eimp. This is in qualitative agreement with Eq. (11), as in this interval both the electron energy Ee and the ion energy Ea are large, while the radius R is still close to R0 (Fig. 1). At t > 60 fs the increase of the ion energy Ea slows down, the electron energy Ee decreases, and the ratio R/R0 begins to increase significantly (Fig. 1). As a result, and in accordance with Eq. (11), the increase of the impact energies slows down (Fig. 1) and the Eimp value at the end of the simulation tend 250 fs (Fig. 1) can be considered as the final (asymptotic) energy E˜imp at t ¼ ∞. On the other hand, the time dependence of the ion energy Ea(t) manifests a noticeable increase also in the terminal domain (Fig. 1). Due to this behavior, the final ion energy E˜a significantly exceeds the simulation

39

Electron–Ion Impact Energy Transfer in Nanoplasmas

R/R0 1.02

1200

1.26

1.70

2.21

(Ne+)N 1000

2.76

3.33

(+)

R0 = 320 Å IM = 5 × 1015 W cm2

14

E imp

12

Ea

10

800

600

Eimp

400

(–) E imp

6 4

200

2

Ee 0

–50

0

50

100

t(fs)

150

E(KeV)

E(eV)

8

200

250

0 300

Fig. 1 The dynamics of EIET in (Ne+)N clusters (R0 ¼ 320 Å, N ¼ 4.96 106) portraying the ð +Þ

ðÞ

time dependence of the impact energies Eimp and Eimp for the outgoing and the ingoing electrons, respectively, and for the net impact energy Eimp, Eq. (7). The cluster is driven by a Gaussian laser pulse with peak intensity IM ¼ 5 1015 W cm2 at t ¼ 0 and temporal width τ ¼ 30 fs. Data are presented for the time interval t ¼ 10–260 fs. The averaged electron energies Ee(t) (lower dashed curve marked Ee) and the periphery ions average energy Ea(t) (upper dashed curve marked Ea) were obtained from MD simulations. The dynamics of the cluster expansion is expressed in terms of the time dependence ð +Þ

(lower coordinate) of R(t)/R0 (upper coordinate). The EIET simulation results for Eimp (), ðÞ

for Eimp (), and for Eimp (●), Section 4, are marked by points. The results of the microscopic model for the net EIET impact energy, Eq. (12) (Section 2 and data fitting of Section 5), are marked by a solid curve.

end Ea(tend) ion energy. According to our rough estimate E˜a is larger than Ea(tend) by about 30%. Using the average electron energies Ee(R), the periphery ion energies Ea, the collision cross sections σ, Eq. (16), and the cluster radii ratio R/R0 provided by the simulations, it is possible to calculate the theoretical dEimp/dt derivative, Eq. (11), by numerical integration of the Eimp(R) (or Eimp(t)) function, Eq. (12). This procedure requires data for the coefficient Ψ of Eqs. (11) and (12). This coefficient, Ψ ¼ 0.125, was found by the fitting of the Eimp(t) function obtained by the simulations (Fig. 1) and was

40

Isidore Last and Joshua Jortner

performed for t > 11 fs, excluding the onset of the cluster expansion. The fitted theoretical Eimp values, marked by the solid line in Fig. 1, demonstrate an excellent agreement with the simulation results, which are marked as points in Fig. 1. We conclude that the theoretical treatment (Section 2) provides an adequate description of the dynamics of the EIET. This impressive agreement between the theoretical results based on our microscopic model for EIET (Section 2) and the simulation data inspires confidence in the validity of the microscopic model.

6. ENERGETICS OF EIET We shall now consider the final impact energies provided by our simulations (Section 4) for different clusters. The final impact energy compoð +Þ ðÞ ð +Þ ðÞ nents Eeimp and Eeimp are given by Eeimp (tend) and Eeimp (tend) at the end of ð +Þ ðÞ the simulations (Section 5). In Fig. 2 we portray Eeimp and Eeimp , together

400 He

0.016

0.008 100

0

40

~

0.012

Eimp /EM

200

Ne

~ Eimp ~ (+) Eimp ~ (–) Eimp ~ Eimp/EM

~

Eimp(eV)

300

0.020

0.004

80

120

160

200

240

0 280

R0(Å) ð +Þ Fig. 2 The cluster size dependence of the final electron impact energy components e E imp ðÞ and e E imp and of the final net energy e E imp, obtained from simulations (Section 4). Data are presented for (He+)N clusters (□) in the initial size domain R0 ¼ 90–270 Å, N ¼ 6.7 104– 1.81 106, driven by a laser pulse with IM ¼ 1015 W cm2, and for (Ne+)N clusters (■) in the initial size domain R0 ¼ 40–160 Å, N ¼ 1.04 104–6.28 105, driven by a laser pulse E imp/EM, where EM is the final maximal with IM ¼ 1016 W cm2. The scaled EIET parameter e ion kinetic energy obtained from simulations (Section 4), is presented (on the right vertical scale) for the (He+)N ionic clusters.

Electron–Ion Impact Energy Transfer in Nanoplasmas

41

ð +Þ ðÞ ˚, with the net energy Eeimp ¼ Eeimp Eeimp for relatively small, R0 160 A + 16 2 + (Ne )N clusters (at IM ¼ 10 W cm ), and for all (He )N clusters (at ð +Þ ðÞ IM ¼ 1015 W cm2). As expected, the Eeimp , Eeimp , and E˜imp energies increase

monotonously with increasing the cluster initial radius R0. Unexpectedly, ðÞ however, the Eeimp component was found to be only slightly smaller than ð +Þ

the dominant Eeimp component, providing a significant damping of the final net energy Eeimp. The analyses of all our results show that the Eimp energy ð +Þ usually makes up for approximately 30%–50% of the Eeimp energy. This finding indicates that the EIET cannot be exclusively considered as generated by the outgoing electrons. The dependence of the final net energy Eeimp on the laser peak intensity IM is presented in Fig. 3A for (Ne+)N clusters with radii R0 ¼ 80, 160, 240, ˚ , and for (Ne4+)N clusters with R0 ¼ 80 A˚. The energy Eeimp of the 320 A ˚ , at IM ¼ 1016 W cm2 demonstrates a largest (Ne+)N cluster, R0 ¼ 320 A maximum of about 630 eV. A significantly lower maximum is exhibited for the smaller, R0 ¼ 240 A˚, cluster, whereas the even smaller clusters, ˚ , demonstrate a monotonous decrease with increasing R0 ¼ 160 and 80 A IM. In contrast to the case of (Ne+)N, the energy Eeimp of the multicharged ˚ , increases monotonously with increasing ions’ cluster (Ne4+)N, R0 ¼ 80 A 16 2 IM. At IM ¼ 10 W cm the energy of the (Ne4+)N cluster is about 270 eV (Fig. 3A), more than one order of magnitude higher than for the case of the (Ne+)N cluster with the same size. Using the Eeimp values obtained by the simulations (Fig. 3A), it is possible to estimate the coefficients of Ψ in Eqs. (11)–(13) by a numerical integration of Eq. (12) with Ee(R) and σ, Eq. (16), provided by our simulations. For (Ne+)N clusters at IM ¼ 5 1015 W cm2 we obtained the coefficients ˚ , respectively, while at Ψ ¼ 0.093 and Ψ ¼ 0.137 for R0 ¼ 240 and 320 A 16 2 ˚ we obtained the coefficient Ψ ¼ 0.088. IM ¼ 10 W cm for R0 ¼ 320 A These Ψ coefficients demonstrate a relatively moderate spread, in support of the theoretical treatment of Section 2. In order to monitor the contribution of the effect of EIET on the cluster explosion process, we will introduce the scaled EIET parameter ε ¼ Eeimp =EM

(18)

with EM being the experimental observable for the final (t ¼ ∞) maximal kinetic energy of the ions produced by cluster explosion. The values of ε

42

Isidore Last and Joshua Jortner

A

700 320 Å

(Ne+)N (Ne4+)N

600

~

Eimp(eV)

500 400

80 Å

300 200

240 Å

100

160 Å 80 Å

0 1015

16

1017

10

1018

–2

IM(W cm ) B

0.030 320 Å

0.025

(Ne+)N

160 Å

(Ne4+)N 80 Å

240 Å

0.015

~

Eimp/EM

0.020

0.010 80 Å 0.005 0 1015

1016

1017

1018

IM(W cm–2)

Fig. 3 The dependence of the final net EIET energy on the laser peak intensity IM, obtained from simulations (Section 4). Data are presented for (Ne+)N clusters (solid lines) and for (Ne4+)N clusters (dashed line). The initial cluster radii are marked on the curves. E imp/EM, Eq. (18), where EM is the (A) Data for e E imp. (B) Data for the scaled EIET parameter e final maximal ion kinetic energy.

obtained by simulations are presented for the (Ne+)N and (Ne4+)N clusters in Fig. 3B and for the (He+)N clusters in Fig. 2. The highest parameter ε, which was obtained for the largest (Ne+)N cluster subjected to the lowest, IM ¼ 5 1015 W cm2, intensity, is as small as 0.024. The (He+)N clusters provide even a smaller maximal value of ε ¼ 0.018 (Fig. 2). From these numerical data, we infer that in all cases considered by us the effect of EIET can be considered as being of minor importance.

43

Electron–Ion Impact Energy Transfer in Nanoplasmas

The moderately large parameter ε, indicating a noticeable contribution of EIET, can be attained not only by a cluster size increase, as mentioned in some studies,8,40,46–50 but also by a laser intensity decrease,51 in accordance with our results (Figs. 2 and 3B). Our results also indicate a correlation between ε and the outer ionization efficiency noi, Eq. (8). This correlation ˚ ) and is demonstrated in Fig. 4 for two (Ne+)N clusters (R0 ¼ 320 and 160 A 4+ + for all (Ne )N and (He )N clusters (with the ε values of Figs. 2 and 3B). The data presented in Fig. 4 reveal an increase of ε with decreasing noi(tp), the outer ionization level at the end of the pulse. The increase of ε becomes steep at small noi(tp), i.e., at noi(tp) < 0.05 for (He+)N clusters and at noi(tp) < 0.1 for (Ne+)N clusters (Fig. 4). Fig. 4 also presents the ε parameters for a (H+)N, ˚ cluster, both at the laser R0 ¼ 77 A˚ cluster, and for a (He2+)N, R0 ¼ 90 A 15 2 peak intensity of IM ¼ 10 W cm . According to our rather limited data (Fig. 4), the ε parameters provided by the “pure” Coulomb collisions are lower than the ε parameters provided in the presence of repulsive collisions at the same value of noi. In the domain of low outer ionization parameters noi, the ε parameters for clusters with repulsive collisions (Section 3) increase 0.030 320 Å

(Ne+)N (Ne4+)N

160 Å

(He+)N

0.020

(He2+)N 90 Å

~ Eimp/EM

(H+)N 77 Å

0.010

80 Å 0

0.2

0.4

0.6

0.8

1.0

noi(tp)

Fig. 4 The dependence of the scaled EIET parameter e E imp/EM, Eq. (18), on the outer ionization level noi(tp), Eq. (8), obtained at the end of the laser pulse (time tp). Data presented for (Ne+)N (—■—) and (Ne4+)N (--■--) clusters containing weakly charged ions, with the fixed cluster radii R0 being marked on the lines and the laser peak intensity range being IM ¼ 1015–1017 W cm2, and for (He+)N ( □ ) clusters with cluster radii in the range R0 ¼ 45–270 Å at a fixed laser peak intensity IM ¼ 1015 W cm2. Data are also presented for clusters containing bare nuclei: for (H+)N clusters with R0 ¼ 77 Å at IM ¼ 1015 W cm2 and for (He2+)N clusters with R0 ¼ 90 Å at IM ¼ 1015 W cm2.

44

Isidore Last and Joshua Jortner

in the sequence of the ratio of the ionic/neutral atom radii, i.e., He+(0.60), Ne4+(0.79), and Ne+(0.90). Such a pattern of the ε parameters seems to indicate the increase of the EIET contribution when the repulsive electron–ion interactions become more significant. The small values of ε, with a low contribution of Eeimp, can be traced both to the short-range character of the impact force Fimp and to its low value. According to Eq. (13), the decrease of the force Fimp for a cluster with an initial radius R0 with increasing R ( R0) is even stronger than R3, due to the decrease (at least after the laser pulse end) of the electron energy Ee (Fig. 1). Such a behavior of Fimp is in contrast with the behavior of the Coulomb force, FCo, given for the electrostatic theory by Eq. (14), which for fixed noi decreases as R2. In Fig. 5 we present the simulation results for Fimp and FCo, based on Eqs. (17a)–(17c). In addition, Fig. 5 also presents the R dependence of the electron average energy Ee. The simulation results of Fig. 5 are given for the following three clusters: the (He+)N cluster with the radius ˚ (at IM ¼ 1015 W cm2) and ε ¼ 0.018 (Fig. 5A); the (Ne+)N R0 ¼ 270 A cluster with R0 ¼ 320 A˚ (at IM ¼ 3 1015 W cm2) and ε ¼ 0.027 ˚ (at IM ¼ 1016 W cm2) (Fig. 5B); and the (Ne4+)N cluster with R0 ¼ 80 A and ε ¼ 0.074 (Fig. 5C). According to Fig. 5, the Ee energy and all forces decrease during the cluster expansion with increasing R, except for the very beginning of the expansion. In accord with the theoretical results discussed earlier, the EIET force Fimp decreases much more steeply with increasing R than the Coulomb force FCo. At the beginning of the Coulomb expansion, the Fimp force is relatively large (Fig. 5), and may affect the ion motion, resulting in some new effects, e.g., expansion anisotropy in CE.8

7. THE CONSEQUENCES OF COULOMB INTERACTIONS An important contribution to the dynamics and energetics of an exploding cluster containing a nanoplasma is provided by Coulomb interactions. To assess the feasibility of our estimates of the electrostatic forces, we compare in Fig. 5 the Coulomb forces FCo obtained from the simulations, Eqs. (17a)–(17c), with the results of the electrostatic model, Eq. (14), and with the noi data being obtained from simulations. The simulation results of FCo are rather close, but systematically lower (by about 20%–33%) for large R/R0 than the results of the electrostatic model. The lowering of the simulation data may be attributed to the descriptions of

45

Electron–Ion Impact Energy Transfer in Nanoplasmas

A

103

103 Ee

102

+

(He )N R0 = 270 Å IM = 1015 W cm–2

FCo

10

Ee(eV)

F(eV/Å)

102

1 Fimp

tp 10–1 1 B

2 R/R0

3

4 104

103 Ee

103

FCo

102 10

Ee(eV)

F(eV/Å)

102

(Ne+)N R0 = 320 Å

1 tp 10–1

IM = 5 × 1015 W cm–2

Fimp

1

2 R/R0

3

4

C 103

103 Ee FCo

F(eV/Å)

10

102

10

Ee(eV)

2

(Ne4+)N

1

R0 = 80 Å IM = 1016 W cm–2

Fimp

tp

10–1 1

Fig. 5 See legend on next page.

2 R/R0

3

4

46

Isidore Last and Joshua Jortner

the average forces sampled for the peripheral ions in this case, while the electrostatic model considers the forces acting on the ions with maximal energies. To provide a complete description of the electrostatic interaction we consider the final Coulomb energy EeCo ¼

ð∞

dR0 FCo ðR0 Þ

(19)

R0

with FCo(R0 ) being taken from the electrostatic model, Eq. (14). Data for EeCo and for the maximal ion energy EM for a large number of expanding clusters are given in Table 1. As is evident from Table 1, EeCo is nearly identical with the maximal ion energy EM. For (He+)N and (Ne+)N clusters the ratio EeCo =EM falls in the range 0.93–1.22, while for the (Ne4+)N cluster this ratio is somewhat higher, falling in the range 1.08–1.31. Additional data were obtained for (H+)N clusters where EeCo =EM ¼ 0:85 0:95, and for (He2+)N clusters where EeCo =EeM ¼ 0:93 0:98 (Table 1), pointing toward the generality of this result. The data for (H+)N and (He2+)N clusters are of particular interest, as they can be produced and interrogated under real-life conditions, in distinction to the (He+)N, (Ne+)N, and (Ne4+)N model systems. From this analysis, we conclude that the major dominating contribution to the maximal ion energy, as obtained from the simulations, involves the electrostatic average energy calculated from the analytical model, Eq. (14). This satisfactory agreement between the simulation results for EM and the analytical model for EeCo also inspires confidence in the validity of the charge distribution used for the derivation of Eq. (14), which considerably differs from that used by Ditmire et al.5,9 Fig. 5 The dynamics of the forces in expanding clusters representing the dependence of the impact force Fimp and of the Coulomb force FCo on R/R0. Simulation results for Fimp () and for FCo (□) are based on Eqs. (17a)–(17c). The Coulomb force FCo was also calculated from the electrostatic model, Eq. (14), with the time dependence data for the outer ionization level noi being obtained from simulations (solid curve). Also presented are data for the dynamics of the averaged electron energy Ee (▲) obtained from our simulations. The laser driving the cluster (with pulse width τ ¼ 30 fs) is characterized by peak intensity values IM, marked on the figures. The time tp, corresponding to the end of the pulse, is marked by vertical arrows on the figures. (A) (He+)N cluster, R0 ¼ 270 Å and N ¼ 1.81 106. (B) (Ne+)N cluster, R0 ¼ 320 Å and N ¼ 4.96 106. (C) (Ne4+)N cluster, R0 ¼ 80 Å and N ¼ 7.76 104.

47

Electron–Ion Impact Energy Transfer in Nanoplasmas

Table 1 Initial Structure, Outer Ionization Level, and Energetics of Expanding Clusters Containing Nanoplasmas, Which Consist of Electrons Together With Ions or Bare Nuclei e R0 Å noi(tp) a EM(keV) b E Co/EM c Cluster IM(W cm22)

(He+)N

1015 1015 3 1015 1015 1015 1015

360 270 360 180 90 45

0.053 0.068 0.112 0.127 0.356 0.794

5.63 4.46 15.16 4.42 3.44 2.10

1.22 1.21 1.11 1.09 1.03 0.93

(Ne+)N

1015 5 1015 1016 3 1015

160 320 160 41

0.128 0.143 0.441 0.821

4.75 19.94 22.47 2.89

1.20 1.17 1.04 1.03

(Ne4+)N

1016 1016 1016

160 80 41

0.095 0.221 0.467

64.15 36.92 20.37

1.31 1.19 1.08

(H+)N

1015 1015 1015 3 1016

120 85 50 85

0.056 0.104 0.314 0.939

2.51 2.20 2.09 14.50

0.95 0.94 0.85 0.93

(He2+)N

1015 1015 5 1015

90 45 45

0.082 0.315 05 0.770

3.72 3. 6.65

0.93 0.96 0.98

a

Outer ionization levels, Eq. (8), at the end of the pulse (time tp). Maximal energy of the ions. c The final Coulomb energy EeCo, from Eqs. (14) and (19), with noi obtained from simulations. b

8. CONCLUDING REMARKS We explored energy transfer within a nanoplasma produced in an elemental cluster driven by an ultraintense laser field. In this broad context, the following issues should be addressed: 1. What are the mechanisms of kinetic energy transfer to ions in laser-irradiated nanoplasmas? 2. What are the forces acting on the ions, which govern the energetics and dynamics of the nanoplasma? 3. What is the proper description of the energetics of energy transfer and cluster expansion?

48

Isidore Last and Joshua Jortner

4. What are the experimental criteria for the distinction between the mechanisms of cluster expansion? We considered and analyzed two distinct microscopic mechanisms, which govern energy transfer from electrons to ions and Coulomb pressure. These microscopic mechanisms involve electron–ion impact EIET and electrostatic Coulomb interactions (Co). Two procedures were advanced and applied for the exploration of these distinct forces and energies: (A) Analytical models with input parameters obtained from simulation data and (B) MD simulation results. The microscopic treatment of the dynamics and energetics of exploding clusters, which is based on simulations, is expected to be valid for a broad range of cluster, laser, and dynamic parameters. The forces calculated by procedures (A) and (B) were subsequently used for the calculations of the corresponding energies. Our treatment provides quantitative data for the hierarchy of the energetics and dynamics. The microscopic impact force Fimp and the impact energy Eimp were calculated by procedure (A), Eqs. (11) and (13), with the parameter Ee being calculated from the simulation data as well as by procedure (B), Sections 3 and 4, and Eq. (17b). There is good agreement between the results of procedures (A) and (B). EIET was treated by a simple theoretical model based on energy transfer in front electron–ion collisions. This model describes well, at least semiquantitatively, the simulation results for EIET energetics and dynamics. From this agreement between the approximate quantum mechanical theory and simulations for EIET, we infer that EIET events are generated mainly by electrons with a motion which is close to radial. The electrostatic Coulomb forces FCo and energies EeCo were calculated from procedure (A), Eq. (14), with noi being obtained from simulation data and from procedure (B), Sections 5 and 6, and Eq. (17c). Again, there is good agreement (within 10%–30%) between the data obtained by the two procedures (Figs. 5A–C). On the basis of theoretical models and MD simulations, we established a hierarchy of the force and of the energies. For the relevant forces, we obtained a general relation during the cluster expansion Fimp ðRÞ < FCo ðRÞ

(20)

for all values of R0 < R < ∞. More informative are the data for the energetics expressed in terms of the hierarchy of the two final energies Eeimp ≪EeCo

(21)

Electron–Ion Impact Energy Transfer in Nanoplasmas

49

for all the expanding clusters studied herein. A quantitative description of the hierarchy of the energetics for the five clusters studied herein is presented in terms of the maximal values of the final energies Eeimp =EM ≲0:025; noi tp ’ 0:2 +0:2 EeCo =EM ’ 1:0 ; all noi tp 0:1

(22a) (22b)

The most important conclusion emerging from our data is that the electrostatic Coulomb forces provide the major contribution to the energetics and dynamics of cluster expansion and decay, providing the dominant contribution to the CE mechanism. The MD simulation (procedure (B)) directly incorporates all microscopic interactions. It is feasible to represent the total force acting on an ion as F ¼ FCo + Fimp, where both components are obtained from procedure (B) and FCo ≫ Fimp. A good, though approximate, description of the forces is to use the dominant electrostatic term for the electrostatic Coulomb interactions to account for the forces and energetics in energy transfer and cluster explosion. From these analytical and simulation data, we infer that in view of the dominant Coulomb contribution, together with the weak EIET contribution, the cluster expansion has to be dominantly described as a CE. This finding is supported by our macroscopic estimates of the contribution of the Coulomb forces to the ions energy. The Coulomb energy EeCo obtained from a macroscopic electrostatic model is close to the total microscopic maximal ion energy EM (Table 1), not leaving much space for the contribution from other effects. Our results infer the exclusive contribution of the (major) Coulomb and (small) electron impact effects to the energetics and dynamics of cluster expansion. Our results contradict the conclusions inferred from previous studies7,10,12–18,21,22,49 asserting the presence of so-called hydrodynamic explosion of hydrodynamic pressure. An alternative description of direct energy transfer from electrons to ions was previously advanced in terms of the macroscopic thermal hydrodynamic pressure model of Ditmire et al.,5,9 where the nanoplasma dynamics was described by the hydrodynamic pressure together with Coulomb pressure. In our yet unpublished work,24 we demonstrated that the commonly used macroscopic hydrodynamic pressure model is distinct from the microscopic model, exploring the “transition” from the microscopic EIET to the macroscopic description of nanoplasma electron dynamics.

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ACKNOWLEDGMENT This chapter is dedicated to Mark A. Ratner, who changed our perception of the chemical sciences, on the occasion of his 75th birthday.

REFERENCES 1. Last, I.; Jortner, J. Quasi-Resonance Ionization of Large Multicharged Clusters in a Strong Laser Field. Phys. Rev. A 1999, 60, 2215. 2. Heidenreich, A.; Last, I.; Jortner, J. Cluster Dynamics in Ultraintense Laser Fields. With Andreas Heidenreich and Isidore Last. Analysis and Control of Ultrafast Photoinduced Reactions. In Analysis and Control of Ultrafast Photo-induced Reactions. Springer Series in Chemical Physics; K€ uhn, O., W€ oste, L., Eds.; Springer-Verlag: Berlin, Heidelberg, 2007; pp 575–617. 3. Fennel, T.; Meiwes-Broer, K. H.; Reinhard, P. G.; Dinh, P. M.; Saraud, E. Laser-Driven Nonlinear Cluster Dynamics. Rev. Mod. Phys. 2010, 82, 1793. 4. Krainov, V. P.; Smirnov, M. B. Cluster Beams in the Super-Intense Femtosecond Laser Pulse. Phys. Rep. 2002, 370, 237. 5. Ditmire, T.; Donelly, T.; Rubenchik, A. M.; Falcone, R. W.; Perry, M. D. Interaction of Intense Laser Pulses With Atomic Clusters. Phys. Rev. A 1996, 53, 3379. 6. Ditmire, T. Simulation of Exploding Clusters Ionized by High-Intensity Femtosecond laser Pulses. Phys. Rev. A 1998, 57, R4094. 7. Peano, E.; Martins, J. L.; Fonseca, R. A.; Silva, L. O.; Coppa, G.; Peinetti, F.; Mulas, R. Dynamics and Control of the Expansion of Finite-Size Plasmas Produced in Ultraintense Laser-Matter Interactions. Phys. Plasmas 2007, 14, 056704. 8. Krishnamurthy, M.; Mathur, D.; Kumarappan, V. Anisotropic “Charge-Flipping” Acceleration of Highly Charged Ions From Clusters in Strong Optical Fields. Phys. Rev. A 2004, 69, 033202. 9. Ditmire, T.; Springate, E.; Tisch, J. W. G.; Shao, Y. L.; Mason, M. B.; Hay, N.; Marangos, J. P.; Hutchinson, M. H. R. Explosion of Atomic Clusters Heated by High-Intensity Femtosecond Laser Pulses. Phys. Rev. A 1998, 57, 369. 10. Zweiback, J.; Ditmire, T.; Perry, M. D. Femtosecond Time-Resolved Studies of the Dynamics of Noble-Gas Cluster Explosions. Phys. Rev. A 1999, 59, R3166. 11. Liu, J.; Li, R.; Zhu, P.; Xu, Z.; Liu, J. Modified Hydrodynamic Model and Its Application in the Investigation of Laser-Cluster Interactions. Phys. Rev. A 2001, 64, 033426. 12. Kumarappan, V.; Krishnamurthy, M.; Mathur, D. Two-Dimensional Effects in the Hydrodynamic Expansion of Xenon Clusters Under Intense Laser Irradiation. Phys. Rev. A 2002, 66, 033203. 13. Kim, K. Y.; Alexeev, I.; Parra, E.; Milchberg, H. M. Time-Resolved Explosion of Intense-Laser-Heated Clusters. Phys. Rev. Lett. 2003, 90, 023401. 14. Kumarappan, V.; Krishnamurthy, M.; Mathur, D. Asymmetric Emission of High-Energy Electrons in the Two-Dimensional Hydrodynamic Expansion of Large Xenon Clusters Irradiated by Intense Laser Fields. Phys. Rev. A 2003, 67, 043204. 15. Rusek, M.; Orłowski, A. Different Mechanisms of Cluster Explosion Within a Unified Smooth Particle Hydrodynamics Thomas-Fermi Approach: Optical and Short-Wavelength Regimes Compared. Phys. Rev. A 2005, 71, 043202. 16. Dorchies, F.; Blasco, F.; Bonte, C.; Caillaud, T.; Fourment, C.; Peyrusse, O. Observation of Subpicosecond X-Ray Emission From laser-Cluster Interaction. Phys. Rev. Lett. 2008, 100, 205002. 17. Milchberg, H. M.; McNaught, S. J.; Parra, E. Plasma Hydrodynamics of the Intense Laser-Cluster Interaction. Phys. Rev. E 2001, 64, 056402.

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18. Murphy, B. F.; Hoffmann, K.; Belolipetski, A.; Keto, J.; Ditmire, T. Explosion of Xenon Clusters Driven by Intense Femtosecond Pulses of Extreme Ultraviolet Light. Phys. Rev. Lett. 2008, 101, 203401. 19. Thomas, H.; Bostedt, C.; Hoener, M.; Eremina, E.; Wabnitz, H.; Laarmann, L.; Pl€ onjes, E.; Treusch, R.; de Castro, A. R. B.; M€ oller, T. Shell Explosion and Core Expansion of Xenon Clusters Irradiated With Intense Femtosecond Soft X-Ray Pulses. J. Phys. B 2009, 42, 134018. 20. Skopalova, E.; El-Taha, Y. C.; Zaı¨r, A.; Hohenberger, M.; Springate, E.; Tisch, J. W. G.; Smith, R. A.; Marangos, J. P. Pulse-Length Dependence of the Anisotropy of Laser-Driven Cluster Explosions: Transition to the Impulsive Regime for Pulses Approaching the Few-Cycle Limit. Phys. Rev. Lett. 2010, 104, 203401. 21. Peltz, C.; Varin, C.; Brabec, T.; Fennel, T. Time-Resolved X-Ray Imaging of Anisotropic Nanoplasma Expansion. Phys. Rev. Lett. 2014, 113, 133401. 22. Liu, C. S.; Tripathi, V. K.; Kumar, M. Interaction of High Intensity Laser With Non-Uniform Clusters and Enhanced X-Ray Emission. Phys. Plasmas 2014, 21, 103101. 23. Iwayama, H.; Harries, J. R.; Shigemasa, E. Transient Charge Dynamics in Argon-Cluster Nanoplasmas Created by Intense Extreme-Ultraviolet Free-ElectronLaser Irradiation. Phys. Rev. A 2015, 91, 021402(R). 24. I. Last and J. Jortner. Bridging between Microsopic Dynamics and Macroscopic Hydrodynamic Models for Kinetic Energy Transfer from Nanoplasma Electrons to Ions in Exploding Clusters. Elsevier Science B.V. (In preparation). 25. Last, I.; Jortner, J. Electron and Nuclear Dynamics of Molecular Clusters in Ultraintense Laser Fields I. Extreme Multielectron Ionization. J. Chem. Phys. 2004, 120, 1336. 26. Deiss, C.; Rohringer, N.; Burgd€ orfer, J. Laser-Cluster Interaction: X-Ray Production by Short Laser Pulses. Phys. Rev. Lett. 2006, 96, 013203. 27. Heidenreich, A.; Last, I.; Jortner, J. Nanoplasma Dynamics in Xe Clusters Driven by Ultraintense Laser Fields. Eur. Phys. J. D 2008, 46, 195. 28. Last, I.; Jortner, J. Scaling Procedure for Simulations of Extreme Ionization and Coulomb Explosion of Large Clusters. Phys. Rev. A 2007, 75, 042507. 29. Ron, S.; Last, I.; Jortner, J. Nuclear Fusion of Deuterons With Light Nuclei Driven by Coulomb Explosion of Nanodroplets. Phys. Plasmas 2012, 19, 112707. 30. Holkundkar, A. R. Parallel-Implementation of Three-Dimensional Molecular Dynamic Simulation for Laser-Cluster Interaction. Phys. Plasmas 2013, 20, 113110. 31. Holkundkar, A. R.; Mishra, G.; Gupta, N. K. Molecular-Dynamic Simulation for Laser-Cluster Interaction. Phys. Plasmas 2011, 18, 053102. 32. Last, I.; Jortner, J. Electron and Nuclear Dynamics of Molecular Clusters in Ultraintense Laser Fields. III. Coulomb Explosion of Deuterium Clusters. J. Chem. Phys. 2004, 121, 3030. 33. Rose-Petruck, C.; Schafer, K. J.; Wilson, K. R.; Barty, C. P. J. Ultrafast Electron Dynamics and Inner-Shell Ionization in Laser Driven Clusters. Phys. Rev. A 1997, 55, 1182. 34. Mikaberidze, A.; Saalmann, U.; Rost, J. M. Energy Absorption of Xenon Clusters in Helium Nanodroplets Under Strong Laser Pulses. Phys. Rev. A 2008, 77, 041201 (R). 35. Last, I.; Ron, Sh.; Jortner, J. Aneutronic H + 11B Nuclear Fusion Driven by Coulomb Explosion of Hydrogen Nanodroplets. Phys. Rev. A 2011, 83, 043202. 36. Moll, M.; Bornath, Th.; Schlanges, M.; Krainov, V. P. Inverse Bremsstrahlung Heating Rate in Atomic Clusters Irradiated by Femtosecond Laser Pulses. Phys. Plasmas 2012, 19, 033303. 37. Slater, J. C. Quantum Theory of Molecules and Solids. Electronic Structure of Molecules, Vol. 1; McGraw-Hill Book Comp: New York, 1963.

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38. Moshammer, R.; Feuerstein, B.; Schmitt, W.; Dorn, A.; Schr€ oter, C. D.; Ullrich, J.; Rottke, H.; Trump, C.; Wittmann, M.; Korn, G.; Hoffman, K.; Sandner, W. Momentum Distributions of Nen+ Ions Created by an Intense Ultrashort Laser Pulse. Phys. Rev. Lett. 2000, 84, 447. 39. Last, I.; Jortner, J. Theoretical Study of Multielectron Dissociative Ionization of Diatomics and Clusters in a Strong Laser Field. Phys. Rev. 1998, 58, 3826. 40. Ishikava, K.; Blenski, T. Explosion Dynamics of Rare-Gas Clusters in an Intense Laser Field. Phys. Rev. A 2000, 62, 063204. uller, J. P.; Przystawik, A.; G€ ode, S.; Tiggesb€aumker, J.; 41. D€ oppner, T.; M€ Meiwes-Broer, K.-H.; Varin, C.; Ramunno, L.; Brabec, T.; Fennel, T. Steplike Intensity Threshold Behavior of Extreme Ionization in Laser-Driven Xenon Clusters. Phys. Rev. Lett. 2010, 105, 053401. 42. Wabnitz, H.; et al. Multiple Ionization of Atom Clusters by Intense Soft X-Rays from a Free-Electron Laser. Nature (London) 2002, 420, 482. 43. Zamith, S.; Martchenko, T.; Ni, Y.; Aseyev, S. A.; Muller, H. G.; Vrakking, M. J. J. Control of the Production of Highly Charged Ions in Femtosecond-Laser Cluster Fragmentation. Phys. Rev. A 2004, 70, 011201 (R). 44. Sakabe, S.; Shirai, K.; Hashida, M.; Shimizu, S.; Masuno, S. Skinning of Argon Clusters by Coulomb Explosion Induced With an Intense Femtosecond Laser Pulse. Phys. Rev. A 2006, 74, 043205. 45. Rajeev, R.; Rishad, K. P. M.; Trivikzam, T. M.; Narayanan, V.; Brabec, T.; Krishnamurthy, M. Decrypting the Charge-Resolved Kinetic-Energy Spectrum in the Coulomb Explosion of Argon Clusters. Phys. Rev. A 2012, 85, 023201. 46. Krainov, V. P.; Roshchupkin, A. S. Dynamics of the Coulomb Explosion of Large Hydrogen Iodide Clusters Irradiated by Superintense Ultrashort Laser Pulses. Phys. Rev. A 2001, 64, 063204. 47. Siedschlag, C.; Rost, J. M. Electron Release of Rare-Gas Atomic Clusters under an Intense Laser Pulse. Phys. Rev. Lett. 2002, 89, 173401. 48. Megi, F.; Belkacem, M.; Bouchene, M. A.; Suraund, E.; Zwicknagel, G. On the Importance of Damping Phenomena in Clusters Irradiated by Intense Laser Fields. J. Phys. B 2003, 36, 273. 49. Jungreuthmayer, C.; Geissler, M.; Zanghellini, J.; Brabec, Th. Microscopic Analysis of Large-Cluster Explosion in Intense Laser Fields. Phys. Rev. Lett. 2004, 92, 133401. 50. Kundu, M. Asymmetric Explosion of Clusters in Intense Laser Fields. Phys. Plasmas 2012, 19, 083108. 51. Cheng, R.; Zhang, C.; Fu, L.-B.; Liu, J. Molecular Dynamics Simulations of Anisotropic Explosions of Small Hydrogen Clusters in Intense Laser Pulses. J. Phys. B 2015, 48, 035601.

CHAPTER THREE

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles Stine T. Olsen, Asbjørn Bols, Thorsten Hansen, Kurt V. Mikkelsen1 Department of Chemistry, University of Copenhagen, Copenhagen, Denmark 1 Corresponding author: e-mail address: [email protected]

Contents 1. 2. 3. 4.

Introduction A Quantum Mechanical Subsystem Coupled to Electrodes €dinger and Poisson Equations The Schro The Quantum Mechanical/Molecular Mechanics Method 4.1 Outline of the QM/MM Method 4.2 The Combined DFT and MM Method 5. Response Functions and DFT/MM 5.1 The Kohn–Sham Operators 5.2 Linear Response 6. Transport Theory 6.1 Fixed Energy Level Approach 6.2 Affected Energy Level Approach 7. The Conducting Magnitude 7.1 A Five-Level System 7.2 Three-Level With Access to First Excited State 7.3 Summary of Transport 8. Computational Section 9. Results 9.1 Investigations of Length of the Conjugated System: OPV2 vs OPV3 9.2 Investigation of Size of Gold Cluster; 170 Au vs 310 Au 9.3 Investigation of Redox State Optimization vs Frozen Redox Structures 9.4 Investigation of Rotational Effects 10. Summary and Conclusion Acknowledgments References

Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.03.002

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2017 Elsevier Inc. All rights reserved.

54 56 59 64 65 69 70 72 73 74 76 77 80 83 84 84 85 86 86 90 91 93 95 96 96

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Abstract We consider two different theoretical methods for investigating molecules sandwiched between electrodes and nanoparticles. One method is a heterogeneous and structured dielectric model for describing the physical situation of a molecule located between electrodes where the molecule is described by quantum mechanics and the electrodes as heterogeneous dielectric media. The interactions between the quantum subsystem and the dielectric media are given by polarization terms that are included in the quantum mechanical equations. The second method is a theoretical method that describes the effects of nanoparticles on molecular properties of molecules, and it is based on a quantum mechanical/molecular mechanics (QM/MM) response method. This method enables us to calculate frequency-dependent molecular properties of molecules interacting with nanoparticles having specific structures. Thereby, we are able to investigate how the specific structures of the nanoparticles affect the molecular properties of the molecules located next to or between nanoparticles. These methods enable us to perform calculations of different electronic and redox states of molecules and their molecular properties between nanoparticles or electrodes. The presented methods make it possible to investigate electron transport in molecular devices.

1. INTRODUCTION Modern methods for determining static and dynamic molecular properties have typically utilized response theory for molecules in the gas phase1–12 and in solution13–26 and the electronic structure of the investigated molecule is described by density functional theory or by uncorrelated/correlated electronic wave function. We obtain the molecular properties of the molecule from the poles and residues of the response functions, and we are able to calculate excitation energies and transition matrix elements, one- and two-photon transition moments, mixed electric–magnetic properties, polarizabilities, and first-order and second-order hyperpolarizabilities. One method for investigating the molecular system sandwiched between nanoparticles or electrodes utilizes a dielectric media for representing the environment around the molecular system. The surrounding environment around the molecular system can be represented as a homogeneous or a heterogeneous dielectric medium and the dielectric medium interacts with the molecule through induced polarization interactions.27–62 The other approach utilizes a quantum mechanics/molecular mechanics (QM/MM) model that allows for an atomistic description of the surrounding environment. In this method the molecule is treated fully quantum mechanically and the surrounding environment is described by molecular mechanics, furthermore, the electromagnetic response of the environment is included as a

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

55

dynamic electric field in the quantum mechanical equations. The quantum mechanical equations coupled to the classical environment are solved selfconsistently.18–26 We present methods for the study of linear and nonlinear frequencydependent responses of a molecule in the vicinity of two electrodes or two nanoparticles, and thereby we have three components—light, molecule, and nanoparticles–electrodes—that are in play resulting in a multitude of relevant length and energy scales. Our focus is on the molecule interacting with the electrodes or nanoparticles, but we are also interested in the properties of nanostructured gold, be that nanoparticles or rough surfaces with structure on a subwavelength length scale. As the nanoparticles are irradiated with light of certain wavelengths, plasmons (charge density waves) are typically excited and this causes rather dramatic local enhancements of the electric field. Based on previous work63,64 it is known that the location and magnitude of the enhancements are very sensitive to the details of the geometry of the nanoparticles. This provides an optimal environment for molecular spectroscopy since the local field enhancements will perturb the molecule such that its Raman cross section or two-photon absorption cross section is enlarged by orders of magnitude.65–70 Furthermore, surfaceenhanced spectroscopy can in some situations attain single molecule sensitivity.71,72 At certain distances the molecule–nanoparticle interactions are well described by electrodynamics using the approach of image charge effects. At closer distances between the molecule and the nanoparticles, the on-set of chemical bonding between the molecule and the nanoparticle will lead to further enhancement of the spectroscopical cross sections and these so-called chemical enhancements and are the least understood.73–76 Surface-enhanced Raman scattering was recently reported for molecules near a semiconductor nanoparticle, where plasmons are absent77 and in this case, the enhancement must be attributed fully to the chemical effect. Research groups have observed that for some structures where a molecule is located between two gold surfaces such as two counterposing triangular nanoparticles or two crossed nanorods, one observe strong field enhancements in the gap.78,79 Experimental efforts have been undertaken in order to investigate the molecular properties of a molecular transport junctions80–82 and these experiments will investigate the complex interplay between plasmons and the transport properties of the molecular junction. Our present aim is to present methods for investigating how energies and linear along with nonlinear molecular properties of sandwiched molecular systems are influenced by the presence of two electrodes or two

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nanoparticles. We show that the presented response methods for these physical systems will give us the possibility of calculating molecular properties up to fourth order and they include • one-, two-, and three-photon absorptions, • one-, two-, and three-photon matrix elements, • two-photon absorption between excited states, • frequency-dependent polarizabilities, • frequency-dependent polarizabilities of excited states, • frequency-dependent first-order hyperpolarizability tensors, • frequency-dependent second-order hyperpolarizabilities (γ).

2. A QUANTUM MECHANICAL SUBSYSTEM COUPLED TO ELECTRODES Many research groups have advanced the methods for performing calculations of solvated molecular systems coupled to a dielectric medium.27–58,83–86 For these models the molecular subsystem is described by quantum mechanics and the interactions between the molecular system and the surrounding media are given by an interaction operator. The interaction operator is included in the quantum mechanical equations, and the electronic structure of the sandwiched molecular system is obtained including the effects of the surrounding environment.27–38,40–42,44,49,51,53–58 Presently, we focus on the physical system where the molecular system is sandwiched by two metal electrodes, and we include the interactions with the electrodes in the molecular Hamiltonian. We described the molecular system, M, by the electronic wave function, Ψ and the charge distribution, ρM. The molecular charge distribution located between the two electrodes gives rise to polarization charges in the two surrounding dielectric media. Thereby, a reaction field and a polarization potential, Φind(r), are created and the induced reaction field interacts with the molecular charge distribution, ρM. The polarization, Epol, and interaction energies, Eint, are determined by: Z 1 Epol ¼ (1) dr ρM ðrÞΦind ðrÞ 2 and

Z Eint ¼

dr ρM ðrÞΦind ðrÞ

(2)

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

57

The quantum mechanical subsystem, the sandwiched molecular system, is described by the following Hamiltonian HM ðq; QÞΨðq; QÞ ¼ EðQÞΨðq; QÞ

(3)

0 HM ðq; QÞ ¼ HM ðq; QÞ + Wpol :

(4)

where

The coordinates of Nel electrons and M nuclei are termed q q1 ,…,qNel and Q Q1, …, QM, respectively. The Hamiltonian for the sandwiched molecular system is given by two parts: • the Hamiltonian of the molecular system in vacuum for a given nuclear 0 configuration, HM . • The interaction operator, Wpol, describes the interactions between the molecular system and the dielectric media. The operator, Wpol, depends on two terms – the induced potential, Φind(r), in the two surrounding dielectric media, – and the molecular charge distribution ρm(r). The molecular charge distribution ρm(r) is given by ρm ðrÞ ¼

M X Qi δðr Ri Þ

(5)

i¼1

where we denote Qi as the partial charge on the i-th nucleus at position Ri. ˚ strand et al.89 we are able to Based on the work by Cioslowski87,88 and A assign the molecular charge distribution. Based on the evaluated molecular charge distribution we need to solve the Poisson equation with appropriate boundary conditions. The boundary conditions are rather simple when we consider metallic electrodes and we wish to specify the potential applied to each electrode. Hereby, we impose Dirichlet boundary conditions on the potential and for a simple electrode configuration we obtain the corresponding Green’s function GD(r, r0 ), where GD(r, r0 ) ¼ 0 when r0 is on a surface. In the presence of grounded surfaces, we have the following potential due to a charge distribution ρ(r0 ) 1 ΦðrÞ ¼ 4πε0

Z

dr0 ρðr0 ÞGD ðr,r 0 Þ

(6)

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and the induced potential is given by Z 1 1 dr 0 ρðr0 Þ GD ðr, r0 Þ Φind ðrÞ ¼ : 4πε0 jr r0 j

(7)

As we keep the surfaces of the electrodes at finite voltages we have that the molecule is exposed to an external potential90: Z 1 @GD bias da0 Φbias ðr0 Þ 0 , Φ ðrÞ ¼ (8) @n 4π S where we define the normal derivative of the Green’s function as @GD ¼ rr0 GD ðr,r 0 Þ nðr0 Þ: @n0

(9)

The vector, n(r0 ), is the unit vector normal to the surface at r0 , and it points into the electrode. We let the electrode denoted i have a potential Φbias and i thereby we have that the total potential is Z 1 X bias @GD Φi da0 : Φbias ðrÞ ¼ (10) @n0 4π i Si All in all, we have determined the total external potential that the sandwiched molecule is subjected to ΦðrÞ ¼ Φind ðrÞ + Φbias ðrÞ

(11)

We determine the Green’s function using the method of image charges, and we have that a charge at the point r0 ¼ (x0 , y0 , z0 ) near an electrode surface in the x ¼ 0 plane leads to an image charge at r00 ¼ (x0 , y0 z0 ) and thereby the Green’s function is GD ðr,r 0 Þ ¼

1 1 jr r0 j jr r00 j

(12)

As we introduce the second electrode in the plane with x ¼ l, we obtain an infinity of point charges that are induced in each electrode. We denote the unit vector in the x-direction as ^ x and furthermore r00 ¼ r0 2x0 ^x . Based on this we write the Green’s function as ∞ X 1 1 0 GD ðr, r Þ ¼ (13) x Þj jr ðr 0 2x0 ^x + 2ln^x Þj jr ðr0 + 2ln^ n¼∞ and we determine the induced potential using Eq. (7).

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

59

Based on the above electrostatic considerations we are able to express the interaction operator between the dielectric media and the molecule as: XX X Wpol ¼ ΦðkÞ ðrj Þ hϕr j qj jϕs iErs (14) j

rs

k

where we let ϕr and ϕs represent the molecular orbitals r and s, respectively. The creation and annihilation operators for an electron in the spin orbital ϕrσ are a{rσ and arσ , respectively. We have introduced the excitation operator, Ers, X a{rσ asσ : Ers ¼ (15) σ

In order to calculate molecular properties of a molecule between surrounding media we must solve simultaneously a quantum mechanical problem—(the Schr€ odinger equation) and a classical electrostatic problem (the Poisson equation). These equations have to be solved subject to the boundary conditions defined by the geometry of the electrodes and that will be the focus of the next section.

€ 3. THE SCHRODINGER AND POISSON EQUATIONS In this section we outline how to solve simultaneously the Schr€ odinger and Poisson equations and how to obtain the response of a molecular system coupled to surrounding electrodes while being exposed to a time-dependent perturbation. We need to consider the time-dependent Schr€ odinger equation since the applied external electromagnetic field leads to a time-dependent perturbation of the system, and therefore, we invoke Frenkel’s variation principle in the form of the Ehrenfest’s equation.9 This leads to the following time-dependent expectation value for the operator A dhAi @A ¼ ih½A, Hi (16) dt @t and we have defined the total Hamiltonian for the sandwiched molecular system interacting with an external field as H ¼ H0 + Wpol + V ðtÞ where we assume that

(17)

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• •

•

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the time-independent Hamiltonian of the system is given by (H0 + Wpol). The time-dependent perturbation is given by V (t), and it corresponds to the interactions between the external electromagnetic field and the sandwiched molecular system. We determine at a time t the time-dependent expectation values using the time-dependent wave function, j0ti and therefore h i ¼ ht 0j j0t i

(18)

We express the time-dependent wave function j 0ti at time t by j0t i ¼ exp ½iκðtÞ exp ½iSðtÞj0i:

(19)

where the electronic wave function, j0i, for the sandwiched molecular system is given by an optimized multiconfigurational self-consistent reference wave function and is a solution to the following Schr€ odinger equation ðH0 + Wpol Þj0i ¼ E0 j0i,

(20)

and thereby it satisfies the generalized Brillouin condition h0j½λ, H0 + Wpol ji

(21)

for both the orbital and configurational variation parameters, λ. At this point we have solved simultaneously the Schr€ odinger and Poisson equations for the sandwiched molecular system coupled to the two electrodes, and thereby we have a wave function of the sandwiched molecular system that has been optimized in the presence of the surrounding environment. In order to solve the time-dependent evolution of the sandwiched molecular system as the time-dependent perturbation has been turned on we utilize the two transformation operators exp½iSðtÞ and exp ½iκðtÞ. These operators perform unitary transformations in the configuration and orbital space, respectively, and they are defined by • X SðtÞ ¼ ½Sn ðtÞRn{ + Sn* ðtÞRn (22) n

– where we have the following expression for the state transfer operator Rn Rm{ ¼ jnih0j where jni belongs to the orthogonal complement space to j0i.

(23)

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

• κðtÞ ¼

X ½κ k ðtÞck{ + κ *k ðtÞck :

61

(24)

k

– where we write the excitation operator, ck, as ck ¼ Epq p > q:

(25)

We describe the time evolution of the sandwiched molecular system utilizing a set of operators, T: T ¼ ðc{ ,R{ , c,RÞ

(26)

where we express the individual components as ck{, t ¼ exp½iκðtÞck{ exp ½iκðtÞ

(27)

ckt ¼exp ½iκðtÞck exp½iκðtÞ

(28)

Rn{, t ¼exp ½iκðtÞ exp ½iSðtÞRn{ exp ½iSðtÞexp ½iκðtÞ

(29)

Rnt ¼exp ½iκðtÞ exp ½iSðtÞRn exp ½iSðtÞexp ½iκðtÞ:

(30)

At this point we introduce the Ehrenfest equation for the time transformed operator Tt,† as: {, t d {, t @T ih½T{, t , H0 i ih½T{, t ,V ðtÞi ih½T{, t , Wpol i (31) hT i ¼ @t dt where we perform the expectation values from the time-dependent wave function, jOti. The time-dependent perturbation of the electromagnetic field is turned on adiabatically at t ¼ ∞, and we represent the perturbation as: Z ∞ dωV ω exp ½ðiω + EÞt V ðtÞ ¼ (32) ∞

and we have introduced a positive infinitesimal number E ensuring that the perturbation is zero at t ¼ ∞ and V ω is the Fourier transform of the V (t). From response theory, we obtain the following expression of the timedependent expectation value of a time-independent operator A: Z ∞ Z 1 ∞ t t ω1 dω1 exp ½ðiω1 + EÞthhA;V iiω1 + dω1 h0 jAj0 i ¼h0jAj0i + 2 ∞ ∞ Z ∞ dω2 exp ½ðiðω1 + ω2 Þ + 2EÞthhA;V ω1 ,V ω2 iiω1 , ω2 + ⋯ ∞

(33)

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where the functions hhA;V ω1 iiω1 and hhA;V ω1 ,V ω2 iiω1 , ω2 are the linear and quadratic response functions, respectively.9 The linear, quadratic, and cubic response functions are obtained when we solve the Ehrenfest equation for each order of the perturbation. Our focus is on the latter term in Eq. (31)9–12 since it describes the changes of the response function due to the presence of the induced charges in the surrounding environment, the two electrodes and we can by using Eq. (14) express the last term of Eq. (31) as: XX X ih½T{, t ,Wpol i ¼ i ΦðkÞ ðrj Þ hϕp j qj jϕq ih0t j½T{, t , Epq j0t i: j

pq

k

(34) Our next step involves an expansion of jOti and T†,t and thereby we isolate the linear, quadratic, and cubic contributions to the response equations due to the presence of the surrounding environment, the two electrodes described by dielectric media ih½T{, t , Wpol i ¼ Gð0Þ ðT{, t Þ + Gð1Þ ðT{, t Þ + Gð2Þ ðT{, t Þ + Gð3Þ ðT{, t Þ + ⋯ (35) The linear response equations are modified due to the first-order perturbation terms from Eq. (31) and we find when only considering the linear terms in κ(t) and S(t) the following modifications: • for the orbital operator, ckt , XX X Gð1Þ ðckt Þ ¼ ΦðkÞ ðrj Þ hϕp j qj jϕq i h0j½SðtÞ, ½ck , Epq j0i j

k

pq

+ h0j½ck ,½κðtÞ, Epq j0iÞ • •

(36)

and similar for the operator ck{, t where we replace ck by ck{ everywhere. For the configurational operator Rnt , XX X Gð1Þ ðRnt Þ ¼ ΦðkÞ ðrj Þ hϕp j qj jϕq i h0j½Rn , ½SðtÞ, Epq j0i j

k

pq

+ h0j½Rn , ½κðtÞ,Epq j0iÞ

(37)

• and similar for operator Rn{, t except all Rn are replaced by Rn{ . The quadratic response functions are modified due to the second-order perturbation terms from Eq. (31), and we find when only considering the quadratic terms in κ(t) and S(t):

63

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

•

For the operator ckt Gð2Þ ðckt Þ ¼

X i X X ðkÞ Φ ðrj Þ hϕp j qj jϕq iðh0j½SðtÞ,½SðtÞ, ½ck , Epq j0i 2! j k pq + h0j½ck , ½κðtÞ,½κðtÞ, Epq j0i + 2h0j½SðtÞ,½ck ,½κðtÞ, Epq j0iÞ:

•

•

(38)

The contributions to the second-order response function related to the operator ck{, t are the same but the operator ck is replaced with the operator ck{ . For the configurational operator Rnt we have the following extra terms Gð2Þ ðRnt Þ ¼

X i X X ðkÞ Φ ðrj Þ hϕp j qj jϕq iðh0j½Rn , ½SðtÞ,½SðtÞ, Epq j0i 2! j k pq + h0j½Rn , ½κðtÞ,½κðtÞ, Epq j0i + 2h0j½Rn , ½SðtÞ,½κðtÞ, Epq j0iÞ:

(39)

For the configurational operator Rn{, t we obtain a similar equation where the operator Rn is replaced by Rn{ . In order to calculate fourth-order molecular properties it is crucial to be able to determine cubic response functions and the additional terms for the sandwiched molecular system contributing to the cubic response function have the following expressions: • For the operator ckt

•

Gð3Þ ðclt Þ ¼

X 1 X X ðkÞ Φ ðr j Þ hϕp j qj jϕq i 3! j k pq fh0j½SðtÞ, ½SðtÞ,½SðtÞ, ½cl ,Epq j0i + h0j½cl , ½κðtÞ,½κðtÞ, ½κðtÞ,Epq j0i + 2h0j½SðtÞ, ½SðtÞ, ½cl , ½κðtÞ,Epq j0i + 2h0j½SðtÞ, ½cl ,½κðtÞ, ½κðtÞ, Epq j0ig

• •

(40)

We obtain the extra terms for the operator cl{, t by replacing cl with cl{ . For the configurational operator Rnt we have

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Stine T. Olsen et al.

Gð3Þ ðRnt Þ ¼

X 1 X X ðkÞ Φ ðrj Þ hϕp j qj jϕq i 3! j k pq fh0j½Rn , ½SðtÞ,½SðtÞ, ½SðtÞ,Epq j0i + h0j½Rn , ½κðtÞ, ½κðtÞ, ½κðtÞ,Epq j0i + 2h0j½Rn , ½SðtÞ,½SðtÞ, ½κðtÞ,Epq j0i + 2h0j½Rn , ½κðtÞ,½κðtÞ, ½SðtÞ, Epq j0ig:

(41)

In order to obtain an expression for the configurational operator Rn{, t we replace all Rn by Rn{ . Hereby, we have obtain a method for calculating frequency-dependent molecular properties of molecules sandwiched between two electrodes. We are able to determine energies and frequency-dependent electromagnetic molecular properties of ground states and excited states. •

4. THE QUANTUM MECHANICAL/MOLECULAR MECHANICS METHOD The next approach, the combined QM/MM method, is presented in this section, and we present the strategy for utilizing this approach for investigating a molecular system sandwiched between electrodes and nanoparticles. The molecular subsystem is described using quantum chemical calculations and includes the interactions between the electrodes and the molecule.18,19,91–100 In this section we illustrate the mathematical model for incorporating the molecular and environmental structures in a QM/MM model. We let the metal electrodes be described by an atomistic description.20–26 The molecular system that is sandwiched between metallic junctions is described by quantum mechanics at the level of density functional theory or coupled cluster wave functions. We represent the electrodes using classical electrodynamics and a structured atomistic environment where the properties of each atom in the metallic junction is given by an isotropic atomic polarizability. The electromagnetic response of the electrodes is included as a dynamic electric field in the electronic structure calculation, and the entire system is solved self-consistently.18–23,25,26 Our approaches focus on treating large-scale systems containing many electrons, and a fully quantum mechanical calculation of these systems requires a substantial computational effort in calculating the property of

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

65

interest. Our focus is to introduce methods that do not require substantial computational efforts and enable a method for investigating specific redox states of the organic molecule sandwiched between the metal electrodes and nanoparticles. We introduce the QM/MM model in order to model this situation, and we do it by combining the more flexible quantum mechanics description with the simpler molecular mechanics approach.101 The division into two regions or subsystems is performed by describing the chemical part of interest by quantum mechanics (QM) and the less interesting part by molecular mechanics (MM). Presently, we focus on the behavior of the organic molecular system and how the molecule is affected by the interactions with the metal clusters, and therefore, the metal clusters or nanoparticles clusters are treated with MM and the sandwiched molecular subsystem with QM.

4.1 Outline of the QM/MM Method We divide the total system into two subsystems, one subsystem that is treated with quantum mechanics and another region treated classically utilizing molecular mechanical force fields and electrostatic interactions. This division of subsystems enables us to write the total energy in terms of three contributions: E ¼ EQM + EMM + EQM=MM

(42)

The three terms represent the energy of the QM region (EQM), the energy of the MM regions (EMM) and the interaction energy between the two regions (EQM/MM). The corresponding Hamiltonian to this energy expression is given by: ^ MM + H ^ QM=MM ^ ¼H ^ QM + H H

(43)

^ QM ) describes the many electronic where the first part of the Hamiltonian (H ^ MM ) is Hamiltonian of the molecule in vacuum, the second term (H the Hamiltonian for the molecular mechanics area, and the final part ^ QM=MM ) takes care of the interactions between the of the Hamiltonian (H two regions. In order to describe in detail the two latter parts of the total Hamiltonian we utilize the following interaction tensors T¼

1 R

(44)

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Rα R3 3Rα Rβ R2 δαβ T αβ ¼ R5 2 15Rα Rβ Rγ 3R ðRα δβγ + Rβ δαγ + Rγ δαβ Þ T αβγ ¼ R7 1 T αβγδ ¼ 9 ð105Rα Rβ Rγ Rδ 15R2 ðRα Rβ δγδ + Rα Rγ δβδ R Tα ¼

(45) (46) (47)

+ Rα Rδ δβγ + Rβ Rγ δαδ + Rβ Rδ δαγ + Rγ Rδ δαβ Þ + 3R4 ðδαβ δγδ + δαγ δβδ + δαδ δβγ ÞÞ

(48)

and in these definitions we have that R is the distance between, e.g., two sites and we denote it with a subscript in order to indicate which distance is used, 1 and Rα is the α component of the distance vector, and hence Tss0 ¼ jrs rs0 j finally we let the symbol δαβ denote the Kronecker delta. 4.1.1 The Classical Region Is the Molecular Mechanics Region We divide the Hamiltonian for the classical region such that we have the intermolecular and the intramolecular parts ^ MM ¼ H ^ intra ^ inter H MM + H MM

(49)

intra inter EMM ¼ EMM + EMM

(50)

which gives us the energy

We follow the standard representation of the intramolecular contribution102 and we have intra str bend tors cross ¼ EMM + EMM + EMM + EMM EMM

(51)

where the first and second term give the energy contribution due to the stretching and bending of the molecules, respectively. The third term provides the energy contributions related to the torsional rotations within the the molecule. The final part of the energy expression is related to the coupling between stretching, bending, and torsional motions. For the situations where the atoms and molecules in the classical region are fixed with respect to their positions, we have the case where the intra energy will be constant. Since our focus is on energy differences and molecular

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

67

properties of the sandwiched organic molecule these terms will be constant in our approach. The contributions related to the intermolecular interactions are given by the sum of the electrostatic energy (EelMM), the polarization energy (Epol MM), vdw and the van der Waals energy (EMM) pol

inter el vdw ¼ EMM + EMM + EMM EMM

(52)

We write the electrostatic energy by summing over all the atomic sites in the MM region.103,104 We consider how a site in the MM region at a position rs interacts with another site at position rs0 , and by summing over all the sites we obtain the total electrostatic energy 0 1 1 s α s0 0 0 0 s s s α s s α s B q Tss0 q + q Tss0 μα μα Tss0 q + 3 q Tss0 Θαβ C C XXXB B C el s0 C B μs T αβ0 μs0 + 1 Θs T αβ0 qs0 + 1 Θs T αβγ ¼ EMM (53) αβ ss0 μγ C B α ss β 3 αβ ss 3 C s s0 >s αβγδ B @ 1 A 1 s αβγδ s0 s0 μsα Tssαβγ Θαβ Tss0 Θγδ 0 Θβγ + 3 9 In this expression we have utilized the previously defined tensor notation and we denote the charge at site s by qs, and the dipole moment at site s is represented as μsα which is the α component of the dipole moment. Furthermore, the quadrupole moment at the site s is given by Θsαβ with the α and β components. We obtain the polarization energy over the polarizable sites as a summation of the products of the total electric field components and the induced dipole moments, and we have utilized the linear dipole approximation for calculating the induced moments at the respective sites X ,s ¼ αsαβ Ftot μind α β ðrs Þ (54) β

and here we denote the β component of the total electric field at the polarizable site s as Ftot β (rs), and we have that α is the polarizability tensor. Therefore, we can write the polarization/induction energy as103,104

1 X X X ind, s α s0 ind, s αβ s0 ind, s0 1 ind, s αβγ s0 pol + μα Tss0 Θβγ EMM ¼ μα Tss0 q μα Tss0 μβ +μβ 2 s s0 >s αβγ 3 (55)

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105 We represent the van der Waals energy Evdw MM term by Lennard-potential and we have " # XX 0 0 A B ss ss vdw (56) EMM ¼ 12 jRs Rs0 j6 s s0 >s jR s R s0 j

In this expression we have the interaction coefficients Ass0 and Bss0 and the summations run over the classical sites. 4.1.2 The Interactions Between the Classical and the Quantum Regions We wish to focus on the interactions between the QM and MM regions, and we write the interaction Hamiltonian as ^ QM=MM ¼ H ^ elQM=MM + H ^ pol ^ vdw H QM=MM + H QM=MM

(57)

^ elQM=MM , In this expression the electrostatic interactions are described by H ^ pol the polarization interactions are given by the term H QM=MM the polarization energy, and the van der Waals interactions between the MM region and the ^ vdw QM region are represented by the term H QM=MM . The electrostatic interactions are separated into parts related to the nuclei and electrons in the QM region, respectively. el, m el , i ^ elQM=MM ¼ H ^ QM=MM ^ QM=MM H +H

(58)

For the nuclei in the QM region we have103,104 ,m ^ elQM=MM H

¼

X

Zm T^ ms q + s

X α

ms

α Zm T^ ms μsα

1X αβ + Zm T^ ms Θsαβ 3 αβ

and for the electrons in the QM region we have el, i ^ QM=MM H

¼

X

T^ is qs +

X α

is

α T^ is μsα

1 X ^ αβ s T Θ + 3 αβ is αβ

el, i EQM=MM ¼

ρðrÞ

X s

Tis qs +

X α

(59)

!

which gives the expectation value as103 Z

!

1 X αβ s Tisα μsα + T Θ 3 αβ is αβ

(60)

!! dr (61)

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

69

The polarization Hamiltonian is also represented by a nuclei part and an electron part, and we obtain the polarization energy as Z 1 X X ind, s α m 1 X X ind, s pol Tisα ρðrÞdr EQM=MM ¼ μ Tsm Z μ (62) 2 sm α α 2 s α α where the nuclear charge on the atom in the QM region is given by Zm. For the van der Waals interactions between the QM and MM regions we also use the Lennard-Jones potential with the interaction coefficients Ams and Bms. We perform the summations over the van der Waals interaction between the nuclei m in the QM region and the classical sites s in the MM region. " # S X X A B ms ms vdw EQM=MM (63) ¼ 12 jRm Rs j6 s¼1 m:center jR m R s j

4.2 The Combined DFT and MM Method For this method, we introduce the total energy in terms of three terms which are a functional of the electron density. E½ρ ¼ EDFT ½ρ + EMM ½ρ + EDFT=MM ½ρ

(64)

where EDFT[ρ] depends on the chosen functional. The energy of the molecular mechanics part is written as pol

el vdw EMM ½ρ ¼ EMM + EMM ½ρ + EMM

(65)

and of the three terms it is only the polarization energy that depends on the electron density. This is related to Eq. (55) as the induced dipole moment is defined as103 X μαind,s ¼ αsαβ Ftot β ðrs Þ β X (66) αsαβ Fβi ðrs Þ + Fβm ðrs Þ + Fβs ðrs Þ + Fβind ðrs Þ ¼ β

with the corresponding electric fields Z i Fβ ðrs Þ ¼ Tisβ ρðrÞdr X β Fβm ðrs Þ ¼ Tsm Zm m

(67) (68)

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Stine T. Olsen et al.

Fβs ðrs Þ ¼

X s6¼s0

0 Tssβ0 qs

+

Fβind ðrs Þ ¼

X γ

0 Tssβγ0 μsγ

XX s0 6¼s

γ

1 X βγδ s0 T 0 Θ 3 γδ ss γδ

,s Tssβγ0 μind γ

! (69)

0

(70)

From the interactions between the QM and MM regions we have the following energy contribution pol

el vdw EDFT=MM ½ρ ¼ EDFT=MM ½ρ + EDFT=MM ½ρ + EDFT=MM

(71)

and it contains the electrostatic energy (EelDFT/MM[ρ]), the polarization energy vdw (Epol DFT/MM[ρ]), and the van der Waals energy (EDFT/MM). It is only the first two terms in the energy contribution that depend on the electron density, and these two terms can be separated into a nuclei-dependent part and an electrondependent part. The DFT/MM interaction functional vDFT/MM is given by vDFT=MM

0 1 1 X X X ind, s α s0 ind , s0 ind , s αβ s ind , s αβγ s0 + μα Tss0 Θβγ ¼ μα Tss0 q μα Tss0 μβ + μβ 2 s s0 >s αβγ 3 ! ! Z X X 1 X αβ s dr ρðrÞ Tis qs + Tisα μsα + T Θ 3 αβ is αβ s α Z 1 X X ind, s Tisα ρðrÞdr μ 2 s α α (72) pol, i

r,s and μind is the reduced induced dipole moment μαindr , s ¼ αsαβ ðFβ ðrs Þ + α Fβi ðrs ÞÞ. The final step is to combine this result with the Kohn–Sham vacuum energy from vacuum DFT which gives

Etotal ½ρ ¼ Evacum ½ρ + vDFT=MM ½ρ

(73)

5. RESPONSE FUNCTIONS AND DFT/MM We consider response functions expressed within time-dependent density functional theory, TD-DFT. We represent the time-dependent Kohn–Sham reference determinant as jΦKS ðtÞi ¼ e^κ ðtÞ jΨKS ð0Þ i

(74)

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

and jΨKS ð0Þ i is the unperturbed Kohn–Sham determinant Y + i ¼ ^apσ jvaci jΨKS ð0Þ p, σ

71

(75)

+ where apσ is the creation operator, which adds electrons to the vacuum state in level p with spin σ. The time evolution operator is given by an exponential parametrization of the antihermitian operator ^κ ðtÞ in the following manner: X X X + ^κ ðtÞ ¼ κ pq ðtÞE^pq ¼ κpq ðtÞ ^apσ ^aqσ (76) pq

pq

σ

with ^aqσ being the annihilation operator, which remove electrons from an orbital p or q with spin σ. We have the following ^ + V^ ðtÞÞjΨKS ðtÞi ¼ iℏ @ jΨKS ðtÞi ðH @t

(77)

where the Hamiltonian in the Kohn–Sham model is represented by ^¼ H

X

f ðri ,tÞ

i

(78)

and depends implicitly on time due to the time-dependent electron density. The operator f(ri, t) is the Kohn–Sham operator. The time-dependent electron density is ^κ ðtÞ ^ ρðr, tÞ ¼ hΨKS ρ ðrÞe^κ ðtÞ jΨKS ð0Þ je ð0Þ i

by taking the expectation value of the operator X ^ ρ ðrÞ ¼ ϕ*p ðrÞϕq ðrÞE^pq pq

(79)

(80)

We expand the time evolution operator in powers of perturbation in order to obtain the parameters κ pq(t) ^κ ðtÞ ¼ ^κ ð1Þ ðtÞ + ^κ ð2Þ ðtÞ + ⋯ and the Fourier transformation is ð1Þ

^κ ðtÞ ¼

Z

^κ ω eiωt dω

(81)

(82)

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The perturbed density matrices up to first order are KS ^ KS Dð0Þ pq ¼ hΨð0Þ jE pq jΨð0Þ i h i KS Dð1Þ κ ð1Þ ðtÞ, E^pq jΨKS pq ¼ hΨð0Þ j ^ ð0Þ i

with the n-th order correction ρðnÞ ðr, tÞ ¼

X

ϕ*p ðrÞϕq ðrÞDðnÞ pq ðtÞ

pq

(83) (84)

(85)

5.1 The Kohn–Sham Operators The Kohn–Sham Hamiltonian is dependent on the electron density and we expand it as X ðnÞ X X ðnÞ ^¼ ^ ¼ H H f^pq E^pq (86) n

n

pq

with ðnÞ ðnÞ ðnÞ + vxc f^pq ¼ δn0 hpq + jpq , pq

(87)

The one-electron integral over the kinetic energy and the nuclearattraction is + * 1 X Za 2 (88) hpq ¼ ϕp r + ϕ 2 jr Ra j q a The n-th order Coulomb interaction is X ðnÞ jpq ¼ gpqrs DðnÞ rs rs

where gpqrs is

1 gpqrs ¼ ϕp ð1Þϕr ð2Þ ϕq ð1Þϕs ð2Þ r12

(89)

(90)

The n-th order exchange-correlation potentials are ðnÞ v ðnÞ vxc , pq ¼ hϕp j^ xc ðr,tÞjϕq i

which depend on the chosen DFT functional.

(91)

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

73

5.2 Linear Response We determine the evolution parameters κ pq using the Ehrenfest equation and we find

KS @ ^κ KS ^ κ ^ + V^ ðtÞ i Ψð0Þ q (92) ^,e H e Ψð0Þ ¼ 0 @t with q ^ representing the vector collecting the excitation operators E^pq 0 1 ⋮ (93) q ^ ¼ @ E^pq A ⋮ We expand Eq. (92) and we have h h h D i i E D i E ð1Þ KS KS ð1Þ ^ ð0Þ ð1Þ KS ^ ^ ΨKS + H + i Ψ q ^ , κ , H q ^ ,^ κ Ψ Ψ ð0Þ ð0Þ ð0Þ ð0Þ D E ^ , V^ ðtÞ ΨKS ¼ ΨKS ð0Þ q ð0Þ

(94)

and next we perform a Fourier transformation of Eq. (94) and we find h h h D i i E D i E ðωÞ KS ðωÞ ^ ð0Þ KS ðωÞ KS ^ ^ ΨKS + H + ω Ψ q ^ , κ , H q ^ ,^ κ Ψ Ψ ð0Þ ð0Þ ð0Þ ð0Þ D E ^ , V^ ω ðtÞ ΨKS (95) ¼ ΨKS ð0Þ q ð0Þ We write this in matrix form by defining 0 1 ⋮ ^κ ω ¼ ⋯E^pq ⋯ @ ^κ ωpq A ¼ q ^ { κω ⋮

(96)

and we obtain the following for the right-hand side of Eq. (95) D KS E ω q Ψ ^ Vω ¼ ΨKS ^ , V ðtÞ ð0Þ ð0Þ

(97)

and for the left side we obtain h h D i i E ð0Þ ðωÞ ðωÞ KS ^ ^ E κω ¼ ΨKS H q ^ , H ,^ κ Ψ ð0Þ ð0Þ

(98)

and the overlap matrix S

D E ^,q ^ { ΨKS S ¼ ΨKS ð0Þ q ð0Þ

(99)

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Stine T. Olsen et al.

Our final matrix equation is ðE ωSÞκω ¼ Vω and we are able to calculate the linear response functions D E ^ V^ ω ii ¼ ΨKS ^κ ω , Ω ^ ΨKS hhΩ, ð0Þ ð0Þ D { ω KS E ^ κ ,Ω ^ Ψ ¼ ΨKS ð0Þ q ð0Þ D E ^ {, Ω ^ ΨKS κω ¼ ΨKS ð0Þ q ð0Þ D E ^ {, Ω ^ ΨKS ðE ωSÞ1 Vω ¼ ΨKS ð0Þ q ð0Þ

with

(100)

(101) (102) (103) (104)

¼ Ω{ ðE ωSÞ1 Vω

(105)

D E q ^ ΨKS ^ , Ω Ω ¼ ΨKS ð0Þ ð0Þ

(106)

The two vectors Ω and Vω are denoted property gradient vectors. We have now obtained a method within DFT to calculate the desired properties. Hereby, we have reached the point where we are able to calculate the desired properties of molecules interacting with externally applied electromagnetic fields and metal electrodes or nanoparticles.

6. TRANSPORT THEORY We present briefly a previously published approach for describing electron transport for a molecular system sandwiched between two electrodes. Our present focus is for the situation where the interactions between the electrodes and the molecule are weak and that brings us into the Coulomb blockade regime. We will outline two approaches for constructing the Coulomb diamonds that are presented by areas that are either conducting or not conducting areas.106 We consider the transport mechanism of the total system as the electrons are able to travel from one electrode (the source), to the molecule and further out on the other electrode (the drain). A third electrode (the gate) gives rise to an external potential and this added external potential makes it possible to change the electronic levels of the molecular compound placed between the electrodes. Thereby, we can control the current going through the total system and it is important to note that the gate is capacitively

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

75

coupled, and therefore, the electrons are not able to pass through it. For the situations when the couplings between the sandwiched molecule and electrodes are weak we have that the discreteness of the molecular energy levels is maintained. Therefore, the current will be restricted to run when there is an energy match between the molecular levels and the levels of the electrodes and there is not a current outside of the energy match. This is different compared to the case of strong coupling between the molecular compound and the electrodes, as the current run continuously due to the broadening of the energy levels. One approach to take when considering electron transport in the Coulomb blockade regime is to consider three different cases depending on the ionization energy Δq+1, qE defined in Eq. (107). The three cases are A The ionization energy is above the bias window of the Fermi levels of the electrodes, i.e., Δq+1, qE > μS > μD, and the electron transport will be blocked. An electron is not able to be transferred from the source to the molecule due to the cost in energy. B Inside the bias window, i.e., μS > Δq+1,qE > μD where we will have transport of electrons since the energy level is within the bias window. Therefore, a change in the electronic population is energetically favorable. C Under the bias window, i.e., μS > μD > Δq+1,qE we have an energy mismatch and the electron transport will be blocked because the electron cannot be transferred from the molecule to the drain. We have defined the ionization energy as Δq + 1, q E ¼ Eq + 1 Eq

(107)

For the two cases A and C we have the situation that transport of electrons does not occur and the current will be blocked. This leads to the Coulomb blockade diamonds observed experimentally.107–110 It is also important to note that the current is not always completely blocked inside the diamonds which is due to the occurrence of different electronic phenomena related to spin and coherent correlation and given by the Kondo effect.111,112 We do not consider the Kondo effect in this work. We present our results in terms of stability diagrams, and these will be compared directly to the experimental diagrams containing the well-known harlequin pattern related to the Coulomb blockade diamonds. For these diagrams, we state the following two observations: 1 that the edges of the diamonds correspond to the conductance peaks, 2 and the cross point between two diamonds correspond to two degenerated redox states.

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We briefly outline our two approaches for calculating the stability diagrams. Generally, we construct the Coulomb diamonds by deriving the equations for the edges of the diamonds, and they represent the limit between conducting and nonconducting situations. The transport of electrons depends on the location of the ionization energy with respect to the electrochemical potentials of the source and drain. The two electrodes, source and drain, are of the same metal and therefore the electrochemical potentials of the source–drain at zero bias will be W where W is the work function of the metal (μ ¼ W ).

6.1 Fixed Energy Level Approach For the fixed energy level approach, we apply a bias VSD and the electrochemical potentials of the metal electrodes will change. We assume that shifts in the electrochemical potential can be divided equally between the two electrodes and we have that μ ¼ 1/2eVSD where μ is the electrochemical potential of the electrodes. We consider this to be a valid assumption since in our calculations we represent the electrodes as composed of the same metal and having the same geometry. We assume that the external potential from a gate (VG) electrode change the energy levels of the molecule in a linear fashion and we have Eq ðVG Þ ¼ Eq ð0Þ + qKVG

(108)

Here the subscript q is the charge of the redox state, Eq(0) (denoted as Eq in the following) is the energy of the molecule when the gate potential is zero. The factor K is an unknown constant describing the connection between the potential of the gate Φ and the potential, ϕ, that the molecule is exposed to, hence Φ ¼ K ϕ. We have that transport only occurs when the ionization energy is inside the bias window W ¼ Δq + 1, q ðVGcross Þ

(109)

and here VGcross is the crossing point between two diamonds. We can rewrite by using Eq. (108) and Eq. (107) this as W ¼ Eq + 1 ðVGcross Þ Eq ðVGcross Þ , Eq + 1 ðVGcross Þ W ¼ Eq ðVGcross Þ Eq: ð108Þ

, Eq + 1 + ðq + eÞKVGcross W ¼ Eq + qKVGcross , Eq + 1 + qKVGcross W ¼ Eq Eq Eq + 1 + W , VGcross ¼ eK

and this gives us the crossing point between the two diamonds.

(110)

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

77

We find the equations for the edges of diamonds using the requirement that they represent the limit between conducting and nonconducting areas and we have μðVSD Þ ¼ Δq + 1, q ðVG Þ

(111)

and using that the drop in the electrodes is the same for both electrodes, we have W + 1=2eVSD ¼ Eq ðVG Þ Eq + 1 ðVG Þ , W + 1=2eVSD ¼ Eq Eq + 1 eKVG

2 , VSD ¼ Eq + 1 Eq W + 2KVG e

(112)

where the two equations correspond, respectively, to the negative and the positive slope of the diamonds. We can express the height of the qth diamond as

2 2 Eq + 1 Eq W + 2KVG ¼ Eq Eq1 W + 2KVG e e 1 , 2KVG ¼ Eq1 Eq + 1 + 2W e (113) We get when inserting Eq. (113) in Eq. (112) VSD ¼

1 Eq1 + Eq + 1 2Eq e

(114)

and this corresponds to half the height (H) of a diamond. The expression in the bracket is referred to as the addition energy Uq. Finally we have H ¼ 2 Uq

(115)

and thereby we have established a very direct connection between theory and experiment, and we can compare directly the current–voltage spectra measured with the calculated results.

6.2 Affected Energy Level Approach In this section we consider an approach where the molecular levels can be affected by the electrodes, and this leads to a change in the molecular energy.113 We do this by higher order effects with respect to the applied

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electric field from the gate, and we assume that the dependence of energy with respect to the field F is114 1 Eq ðFÞ ¼ Eq μ0 F αF2 ⋯ 2

(116)

Expressing the field for a plate capacitor the energy of the molecular transistor can be expressed as: 1 V2 + qKVG Eq ðVSD , VG Þ ¼ Eq αq SD 2 d2

(117)

where the distance d between the two electrodes as well as the polarizability are new parameters compared to the first approach. Utilizing the same procedure as in the first approach, however, with the new energy expression, equations for the edges of the diamonds can be obtained as: W 1=2eVSD ¼ Δq + 1, q ¼ Eq Eq + 1

VSD 1 αq αq + 1 2 eKVG d 2

(118) (119)

In order to find VSD Eq. (119) is rewritten as: 0 ¼ Eq Eq + 1

VSD 1 αq αq + 1 2 eKVG + W 1=2eVSD d 2

(120)

where the solutions are given as: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1=2e 1=4e2 + 2ðαq αq + 1 Þ1=d2 ðEq Eq + 1 eKVG + W Þ VSD ¼ ðαq αq + 1 Þ1=d2 (121) However, it should be noted that Eq. (120) yields four solutions where only the two given in Eq. (121) are physical possible due to the validity compared with the solution of the first approach, hence to yield the same result as the first approach within the limit of zero polarizability. In other words, the solution of the second approach has to converged to the first approach for Δα ¼ αq αq+1!0. In the case of the two nonmentioned solutions to Eq. (120) it is observed that they diverged for Δα ¼ αq αq+1!0. To evaluated the limit of the solutions in Eq. (121) for Δα ¼ αq αq+1!0 we observed that it yields

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79

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 an expression, and hereby Eq. (121) is rewritten in the form of 1 + y and 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y the Taylor expansion up to second order 1 + y ¼ 1 + + ⋯ is considered. 2 The limit can easily be evaluated and we obtain

2 (122) Eq + 1 Eq W + 2KVG lim Eq: ð121Þ ¼ Δα!0 e which is recognized to be exactly the same expression as in the first approach. Hereby, the second approach can be considered to be valid in the context of the first approach. To obtain the height of the diamond the expression of Eq. (121) is rewritten as: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1=2e 1=2e 1 + 8Δα1 ðEq Eq + 1 + W eKVG Þ VSD ¼ e2 Δα1 " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# (123) 1 1=2 1=2 1 + 8Δα1 ðEq Eq + 1 + W eKVG Þ ¼ e Δα1 αq αq + 1 αq1 αq and Δα1 ¼ . The height can now be 2 2 ed e2 d2 obtained by finding the crossing point between the positive and the negative slopes: " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# 1 1=2 1=2 1 + 8Δα1 ðEq Eq + 1 + W eKVG Þ Δα1 e (124) " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# 1 1=2 + 1=2 1 + 8Δα1 ðEq Eq + 1 + W eKVG Þ ¼ Δα1 e

where Δα1 ¼

Here KVG is isolated yielding: 1=4 4Eq1 Δα21 + 4Eq Δα21 + 4W Δα21 + Δα1 KVG ¼ Δα1 eðΔα1 Δα1 Þ

Δα1 Δα1 Eq Δα1 Δα1 Eq + 1 + Δα1 Δα1 W + Δα1 eðΔα1 Δα1 Þ

(125)

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Substituting Eq. (125) into Eq. (121) we obtain half the height of the diamond: 1 VSD ¼ 2 ½1=2 + 2Δα1 Eq Eq + 1 + W Δα1 e +

1 Eq1 Δα21 + Eq Δα21 Δα1 ðΔα1 Δα1 Þ

(126)

W Δα21 + 1=4Δα1 + Δα1 Δα1 Eq Δα1 Δα1 Eq + 1 + Δα1 Δα1 W ÞÞ Again we have obtained a connection between theory and experiment and it has been done in a more sophisticated manner. However, this time it should be noted that only the unknown constant K has dropped out and not the work function of the metal W. This is not of concern since the work function of several metals is known.

7. THE CONDUCTING MAGNITUDE In this section we improve the previous theoretical approach of “The Conducting Limit” by presenting an approach for obtaining the magnitude of the current and thereby we derive an expression for the current I in the Coulomb blockade regime. We start out by following the classical expression for the current and since the current is conserved, the total current is expressed as the current into the left electrode: X L ITotal ¼ IL ¼ e Pn ðwnL+ 1, n wn1 , nÞ (127) n

where we have defined the following: (1) Pn denotes the probability of finding the system in a state with n electrons, and (2) wnL+ 1, n denotes the rate of transition from state n to n + 1. The term wnL+ 1, n describes an electron movL ing from the left electrode to the molecule, and wn1 , n describes an electron moving from the molecule to the left electrode, see Fig. 1. The rates are divided into contributions from the left and right electrode: wn, n + 1 ¼ wnL, n + 1 + wnR, n + 1

(128)

where wnL, n + 1 and wnR, n + 1 are the contributions from the left and right, respectively. We have changed the notation to yield the number of electrons and not the charge as in the previous theoretical approach.

W Ln+1,n W-2-1 P-2

W-10

P-1 W-1-2

W01 P0

W0-1

W12 P1

W10

Pn*

1

,n-

Pn

P2

W n*

Pn-1

W21

Wn,n-1

n+ 1,n *

n*, n+

-1 Wn

Wn-1,n

W W

,n*

1

Pn

Wn,n+1 Wn+1,n

Pn+1

W Ln-1,n

Fig. 1 A schematic representation of the current into the left electrode, where Pn represents the n-level system of the molecule, which could be one of the two presented; the five-level system with five redox states able to participate in the transport, or the three-level if the first excited states are able to interact as well.

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We express the probabilities of the given states as well as the rate of transitions between states and this is done by considering master equations with dP _ ¼ P: dt P_ ¼ WP

(129)

and here we denote P as the probability matrix, and W the transition matrix. This set of equations are known as Pauli master equations and are linear, Markovian rate equations. Thereby, we assume that the dynamics of the system does not depend on previous history,115 and we obtain for a steady state situation an eigenvalue problem: 0 ¼ WP

P

(130)

We solve this along with the normalization condition of n Pn ¼ 1 and then the probability can be expressed in form of rates. Our final task is to express the transition rates and we do that by using Fermi’s golden rule116 and the expressions for the removal (r) and addition (a) of an electron are given by: X n n1 Γnr μα Þ wn1, n ¼ ijα ½1 f ðEi Ej (131) α¼L , R X n n1 Γna μα Þ wn, n1 ¼ ijα f ðEj Ei (132) α¼L , R where f denotes the Fermi function of the right (R) and left (L) electrode, μα denotes the electrochemical potential, and Γα denotes the rate constants. The latter ones are given in terms of coupling elements and are expressed as: 2 Γnr ijα ¼ γ α jhn 1,ijcα jn, jij

(133)

2 { Γna ijα ¼ γ α jhn,ijcα jn 1, jij

(134)

with γα ¼

2π ρ ℏ

(135)

where ρ is the charge density of the electrodes. We have shown that the current of any n-level molecular system can been expressed in the context of physical constants, redox energies, and coupling elements. We obtain these properties from computational methods described previously, and we are able to express the derivative of the current and also the conductance and the Coulomb blockade diamonds for a chosen n-level system.

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7.1 A Five-Level System In this section we consider a system where the dynamics of the transport is mainly covered by five states, and the n master equations will be reduced to five equations, see Fig. 1. Furthermore, we assume that only neighbor states couple, and this gives the following master equations: P_ 2 ¼ w21 P1 w12 P2

(136)

P_ 1 ¼ w12 P2 + w10 P0 ðw01 + w21 ÞP1 P_ 0 ¼ w01 P1 + w01 P1 ðw10 + w10 ÞP0

(137)

P_ 1 ¼ w10 P0 + w12 P2 ðw01 + w21 ÞP1 P_ 2 ¼ w21 P1 w12 P2

(139)

(138) (140)

where the negative signs correspond to a decrease in population and vice versa. It should be noted that this description is optimal for the neutral systems, and less for the state of n ¼ 2 since the states of n ¼ 3 could be expected to influence the process. This issue could have been handled by considering the system piecewise, however, this will not be considered. Instead, the steady state solution of Eq. (140) nonfragment three-level system will be considered. We have the following solution:

1 w10 w21 w10 w21 + 1+ 1+ P0 ¼ 1 + w01 w12 w01 w12

w01 w10 w21 w10 w21 1 + 1+ + P1 ¼ 1 + w10 w01 w12 w01 w12

1 w12 w01 w10 w01 w21 w10 w01 P2 ¼ 1 + 1+ + + w21 w10 w01 w10 w12 w01 w10

1 w21 w01 w10 w21 w10 P1 ¼ 1 + + 1+ + w12 w10 w01 w12 w01

w12 w01 w10 w01 w21 w10 w01 1 P2 ¼ 1 + 1+ + + w21 w10 w01 w10 w12 w01 w10

(141)

(142) (143) (144) (145)

Finally, we obtain the current based on these populations, Eq.(127): L L L I ¼ e½P2 w12 + P1 ðw01 w21 Þ L L L L L w10 Þ + P1 ðw21 w01 Þ P2 w12 +P0 ðw10

(146)

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7.2 Three-Level With Access to First Excited State We consider in this section how to include excited states of the molecular system into the model. This reflects that transport in the Coulomb blockade regime implies the energy levels to be inside the bias window, thus with an increased bias voltage the excited states could be expected to be participating. Therefore, we consider a three-level system where the first excited state of the neutral state is able to interact with the other states, see Fig. 1. This will lead to parallel lines along the edges of the Coulomb diamonds. Once again we express the dynamics of the system as the Pauli master equations that are connected as in a cyclic graph: P_ n ¼ wn, n + 1 Pn + 1 + wn, n1 Pn1 ðwn1, n + wn + 1, n ÞPn P_ n1 ¼ wn1, n Pn + wn1, n Pn ðwn, n1 + wn , n1 ÞPn1 *

*

*

P_ n + 1 ¼ wn + 1, n Pn + wn + 1, n* Pn* ðwn, n + 1 + wn*, n + 1 ÞPn + 1 P_ n ¼ wn , n1 Pn1 + wn , n + 1 Pn + 1 ðwn1, n + wn + 1, n ÞPn *

*

*

*

*

(147) *

and this is solved under the steady state condition. That the solution exists is known from Markov theory since the fundamental theorem of regular Markov chains yields117: Fundamental Theorem of Regular Markov Chains Suppose M is the transition matrix of a regular Markov chain. Then there is a unique vector P whose components add up to 1 and that satisfies MP ¼ P. The entries of the columns of M n converge to the corresponding entries of P as n increases to infinity which corresponds to our initial problem of the master equations, see Eq. (129). With the existence of an unique solution the expression of the current can be expressed: L L L I ¼ e½P0 ðw10 w10 Þ + P1 ðw01 + w0L*1 Þ L L L +P1 ðw0L*1 w01 Þ + P0 *ðw10 w10 Þ * *

(148)

where the solution can be found numerically.

7.3 Summary of Transport Based on the two novel transport theories “The Conducting Limit” and “The Conducting Magnitude” constructed, we are able to theoretically present transport in the Coulomb blockade regime both in the simple form of conducting areas vs nonconducting areas as well as in a more refined manner

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85

including the magnitudes of the current in the conducting areas. Especially, the methods enable us to combine transport theory with quantum mechanics, and thus to study the effect of the chosen quantum chemical method in context of change in the transport picture.

8. COMPUTATIONAL SECTION We have considered the conductance of molecules sandwiched between two electrodes, and our investigations have focused on current– voltage diagrams showing Coulomb blockage diamonds of conjugated organic molecules placed between two metal electrodes. We have investigated the effect on the Coulomb blockade diamonds with respect to different chemical properties of the organic molecules as well as the approach that we have taken to model the total system, electrodes, and molecular compound. We have considered the effects of increasing the length of the conjugated molecular compound between the gold electrodes, and this has been done by investigated the theoretical results of the two organic compounds OPV2 and OPV3 located between two gold electrodes. We compare these results to our previous work on smaller systems in order to have more than two different molecular systems with different conjugation lengths. The compound OPV3 is a larger conjugated system compared to OPV2, and thereby we are able to investigate for these two compounds the effect of increasing the conjugation of the organic compound between the two electrodes. Furthermore, we have investigated the importance of the finite size of the gold clusters and the importance of geometry relaxation of the sandwiched molecular compound. We have done this by varying the size of the gold cluster and by performing geometry optimizations of the organic molecules for each redox state. Finally, the orientation of the molecule with respect to the electrodes has been investigated. We utilize a description of the transport118 mechanism where only the energy and the polarizability of a range of redox states as well as the coupling elements are required in the construction of the Coulomb transport picture. The energies as well as the polarizabilities are calculated directly from a combined QM/MM approach, where the chemical interesting part, i.e., the molecule is treated with the more flexible QM approach, whereas the less interesting gold clusters are treated with the simpler MM approach. The QM region was obtained from Gaussian09 vacuum optimized geometries of the neutral state of OPV2 and OPV3 utilizing DFT with the long-rangecorrected functional CAM-B3LYP together with a series of the correlation

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consistent basis sets. Further, the redox states ranging from 2e to 2e were geometry optimized for OPV2 in order to investigate the approximation of a fixed molecular geometry under electron transfer. Thus the additional electron, electrons added or subtracted to the QM part of the system represents a simple alternative to a more elaborated charge localization schemes such as constrained DFT.119 In the MM region the left gold cluster was designed to be half-spheres by choosing a point at the x-axis and including all gold atoms of a finite radius, whereas the right gold cluster was constructed identical but inverted and translated along the x-axis to the other side of the molecule ˚ between the sulfur atoms and the gold clusters. yielding a distance of 1.58 A The binding site on the gold cluster was chosen to be a fcc site on a (111) surface. For both OPV2 and OPV3 each gold cluster was assigned with 85 gold atoms, however, the effect of an increased cluster size of 155 atoms was investigated for OPV3 to see the effect of cluster size. The geometries of the electrodes were treated to be frozen in geometry, and thus are the respectively gold atoms attributed with no permanent electrical effects but assigned with a polarizability of 31.04 a.u.3 The junction calculations were run selfconsistently in Dalton and thereby taking the interactions between the molecule and clusters into account and solving the coupled electromagnetic and quantum mechanical equations iteratively.

9. RESULTS This section contains results where we have compared the effects of (1) investigating molecular systems having different conjugation lengths and thereby different transport properties, (2) changing the sizes of the gold clusters in order to investigate how much the transport properties change as the sizes of the gold clusters are increased, (3) relaxation of the nuclear degrees of freedom during the transport of electrons through the molecular system, and (4) the importance of orientation of the sandwiched molecule.

9.1 Investigations of Length of the Conjugated System: OPV2 vs OPV3 We consider two different oligo(phenylvinylene) molecular systems termed OPV2 and OPV3, and the molecular structures in junction are presented in Fig. 2. We note that the two molecular systems differ with respect to the conjugation length of the π system, and we wish to investigate how the transport properties in the Coulomb blockade regime change with the conjugation length.

Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

A

87

B

Fig. 2 Schematic representation of the two oligo(phenylvinylene) molecules. (A) OPV2 and (B) OPV3 sandwiched in a gold junction.

5 0 –5

5 0 –5 –5

5 0 –5

0 5 Gate voltage (V) Bias voltage (V)

Bias voltage (V)

–5

Polarized density

OPV3 Bias voltage (V)

Bias voltage (V)

Static density

OPV2

0 5 Gate voltage (V)

–5

0 5 Gate voltage (V)

–5

0 5 Gate voltage (V)

5 0 –5

Fig. 3 OPV2 vs OPV3. Calculation: CAM-B3LYP/aug-cc-pVDZ.

The Coulomb blockade diamonds are depicted in Fig. 3 in the limit of conducting area vs nonconducting areas, whereas Fig. 4 also expresses the magnitude of the conductance. As the length of the conjugated system has been extended from OPV2 to OPV3, we note that the sizes of the Coulomb blockade diamonds are decreased and thereby the transport of electrons is easier of the large system (OPV3). These results are related to the closer lying molecular levels for OPV3 compared to OPV2, which is leading

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Fig. 4 OPV2 vs OPV3. Calculation: CAM-B3LYP/aug-cc-pVDZ.

to easier transport for the larger system. On the other hand, both molecules expose a 2-1-3 coulomb pattern which indicates that the first diamond is medium sized, the second diamond is the largest, and the last diamond the smallest. Thus the neutral state is the most restrained toward charging. The same diamond pattern illustrates the chemical similarity of the two compounds. For the approach where we include higher order effects due to the applied electric field, we note that we obtain almost the exact same diamonds, but the edges of the diamonds are curved and not straight. The

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Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

larger polarizability of OPV3 leads to a more pronounced curvature for the Coulomb diamonds of OPV3 than for the ones of OPV2. Based on this study and our previous work on related and smaller systems,106,118 we can conclude for several molecules that the diamonds get smaller as the length of the conjugated molecular compound gets larger which is what one would expect but our method shows that the patterns for all the investigated OPV systems are the same. This result has only been obtained with our model, and it basically illustrates that the chemistry of the sandwiched molecules more or less remains the same for all the OPV compounds that we have investigated at this point. However, it is observed that the stabilization of OPV3 through the interactions with the metal clusters is smaller than the stabilization observed with the OPV2 system, see Table 1. The stabilization is due to the polarization interactions between the quantum mechanical systems and the polarizable metal clusters, and it is possible that the amount of gold atoms is too small to polarize a molecule having the molecular size corresponding to that of OPV3, see Section 9.2. Considering the magnitude of the transport the Coulomb staircase for the two molecular systems OPV2 and OPV3 is depicted in Fig. 4 by the use of master equations.106 Each molecular system has the possibility of being in five different redox states. All five redox states interact with the metal clusters and this enables the possibility of five steps in the Coulomb staircase. As the step length only depends on the energy of the considered state we note that the energy levels are well separated since all steps emerge clearly. Since the energy levels of OPV3 are separated by smaller energy differences that the energy levels of OPV2, we obtain step lengths of OPV3 Table 1 Addition Energies Eadd, Q21

vac

Q0

(eV) Q1

Eadd, Q21

junc

Q0

(eV) Q1

Diff. (%) Q21

Q0

Q1

OPV2 170Au/aug-ccpVDZ

3.55 4.61 3.78 2.45 3.50 2.71 30.9 24.0 28.3

OPV2 170Au-Redox/ aug-cc-pVDZ

3.11 4.32 3.39 2.02 3.19 2.32 35.0 26.3 31.3

OPV3 170Au/aug-ccpVDZ

2.82 3.51 2.95 2.09 2.77 2.23 25.9 21.0 24.4

OPV3 310Au/cc-pVDZ 2.94 3.56 3.02 2.15 2.75 2.21 27.0 22.7 26.8

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that are smaller compared to the step lengths of OPV2. Further, as expected, the current is higher for OPV3. Additionally, the conductivity patterns of OPV2 and OPV3 are similar since the two compounds have similar structures and molecular properties. We have observed a 1-3-4-2 pattern, which indicates the largest conductivity is given from the first peak, the second largest from the fourth peak, the second lowest from the second peak, and the lowest from the third peak. This relates to the fact that the couplings to the states 2 are larger than those to states 1. Chemically, this could be explained from the difference in donating an electron from a radical or from a lone pair. From the stability plot we note the exact same Coulomb blockade diamonds as for the previous approach, but these figures provide the additional information about the magnitude of the current. Based on the agreement of the patterns of the Coulomb blockade diamonds between the two approaches we will for the following presentations only consider the ones constructed from the Conducting Magnitude approach.

9.2 Investigation of Size of Gold Cluster; 170 Au vs 310 Au In this section we consider the importance of the sizes of the metal clusters connecting the molecular systems. As mentioned in the previous section the OPV3 system yields a lower stabilization than the OPV2 system with the same sized gold clusters, i.e., each cluster consisted of 85 gold atoms. Therefore, we increase the sizes of the gold clusters in order to investigate the importance of the sizes of the clusters. From the calculations depicted in Fig. 5 and in Table 1 it is observed that neither the conductance nor the current changes significantly with increased cluster size. The Coulomb blockade diamond pattern is also maintained with increased cluster size, however, it is observed that the Coulomb blockade diamonds are 20%–35% smaller compared to vacuum, whereas it was only 20%–25% for metal clusters with 155 atoms per cluster for the OPV3 system, hence a total cluster size of 310 Au atoms. Due to the increased computational cost for the 310-Au junction calculations have been performed with the basis set of cc-pVDZ, whereas the previous calculations were performed using the basis set aug-cc-pVDZ. This leads to decreases in the energy separations between the molecular states. Hence a greater stabilization has been obtained. Thereby to get a realistic approximation of the system, it is crucial to investigate the importance of cluster size vs the size of the molecule. We find that increasing molecular size must be followed by increasing the number of metal atoms in the metal clusters.

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Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

OPV3-170Au

OPV3-310Au 1500

−5

1500

−5

1000 VSD (V)

VSD (V)

1000 0

0

500

5 −5

0

500

5

0

5

−5

0 VG (V)

200

150

150

100

100

50

50 I (µA)

I (µA)

VG (V) 200

0

0

−50

−50

−100

−100

−150

−150

−200 −15

−10

−5

0

5

10

−200 −15

15

−10

−5

0 VSD (V)

5

10

15

−10

−5

0 VG (V)

5

10

15

2000

2000

1500

1500 G (µS)

G (µS)

VSD (V)

1000

500

0 −15

0

5

1000

500

−10

−5

0 VG (V)

5

10

15

0 −15

Fig. 5 OPV3 170 Au vs OPV3 310 Au. Calculation: CAM-B3LYP/cc-pVDZ.

9.3 Investigation of Redox State Optimization vs Frozen Redox Structures In this section we present investigations of the importance of geometry relaxation as the electrons are transferred onto the molecule. It is expected that the geometry of the molecule will change during that process depending on the redox states and the time that the electron stays on the molecular system. Thus, the five different redox states from the previous OPV2 calculations have been optimized as well as the coupling constants, see Fig. 6. Compared to the nonredox geometry optimized OPV2 system the same

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OPV2

OPV2-Redox 1500

−5

1500

−5

1000 VSD (V)

VSD (V)

1000 0

0

500

5 −5

0

5

200

150

150

100

100

50

50 I (µA)

200

0

−50

−100

−100

−150

−150 −10

−5

0 VSD (V)

5

10

−200 −15

15

2000

2000

1500

1500

1000

500

0 −15

0 VG (V)

0

5

0

−50

−200 −15

G (µS)

0 VG (V)

G (µS)

I (µA)

5 −5

500

−10

−5

0 VSD (V)

5

10

15

−10

−5

0 VG (V)

5

10

15

1000

500

−10

−5

0 VG (V)

5

10

15

0 −15

Fig. 6 OPV2 nonredox optimized vs OPV2 redox optimized. Calculation: CAM-B3LYP/ aug-cc-pVDZ.

tendency in the current is observed, i.e., the relative step size is the same. However, the redox optimized geometries yield higher current and conducting magnitudes, which can be explained by the larger coupling constants. Thus, the trend in the conductivity peaks is not dependent on the relaxation of the molecular structure upon charging, whereas the magnitude of the conductivity is. However, the difference is not significant. Likewise, the Coulomb blockade diamonds for the redox optimized geometry OPV2 states show the same pattern as the nonoptimized states, however, the redox

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93

optimized OPV2 yields slightly smaller diamonds. Thus for a general view of the trend in the Coulomb blockade diamonds nonredox optimized structures can be utilized. On the other hand for an accurate and detailed investigation, we recommend that one considers both calculation procedures, one for the geometry unrelaxed and one for the geometry relaxed molecular redox states.

9.4 Investigation of Rotational Effects Experimentally, the orientation of the molecule with respect to the electrodes can change, which can be reflected in the statistical data of the experiment. Thus, this section contains investigations of the effects of changes in the orientation of the molecular system with respect to the metal clusters for a dihydroazule derivative, DHA 4a, sandwiched in a silver junction, as depicted in Fig. 7. The molecular system has been placed in six different orientations ranging from a horizontal placement of 0 degree to a 150 degrees rotation in steps of 30 degrees. The Coulomb blockade diamonds for both approaches in the conducting limit are depicted in Fig. 8 and the corresponding addition energies in Table 2. Calculations for five different redox states have been performed for all orientations of the molecules, hence yielding 3 Coulomb blockade diamonds. For the horizontal placement of the molecule, a 3-2-1 diamond pattern is observed, indicating the redox state of charge 1 to be the most restrained towards charging. However, rotating the molecule in any other

Fig. 7 DHA 4a in six different orientations with respective to the silver clusters. Rotation angle from 0 to 150 degrees.

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0

Bias voltage (V)

Fixed energy 5

5

0

0

−5

−5

Bias voltage (V)

−5

30

0

0

−5

−5

Bias voltage (V) Bias voltage (V) Bias voltage (V) Bias voltage (V)

5

5

0

0

−5

−5 0

5

5

5

0

0

−5

−5 0

5

5

5

0

0

−5

−5 −5

150

0

5

−5

120

5

5

−5

90

0

5

−5

60

Affected energy

0

5

5

5

0

0

−5

−5 −5

0 Gate voltage (V)

5

−5

0

5

−5

0

5

−5

0

5

−5

0

5

−5

0

5

0

5

−5

Gate voltage (V)

Fig. 8 Coulomb blockade diamonds for the DHA 4a in the six different orientations with respective to the silver clusters.

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Molecular Properties of Sandwiched Molecules Between Electrodes and Nanoparticles

Table 2 DHA 4a Distance VDW Eadd,vac (eV)

Eadd, junc (eV)

Q21

Q0

Q1

Q21

Q0

Q1

0°

3.66

5.08

2.83

3.63

2.17

1.42

30°

3.66

5.08

2.83

3.41

3.97

60°

3.66

5.08

2.83

2.97

90°

3.66

5.08

2.83

120°

3.66

5.08

150°

3.66

5.08

Diff. (%) Q21

Q0

Q1

0.82

57.28

49.82

2.01

6.83

21.85

28.98

3.48

1.70

18.85

31.50

39.93

2.69

3.55

1.69

26.50

30.12

40.28

2.83

3.00

3.47

1.87

18.03

31.69

33.92

2.83

3.46

3.98

2.04

5.46

21.65

27.92

Addition energies. Calculation: CAM-B3LYP/cc-pVDZ.

orientation yields a 1-3-2 pattern, now indicating the neutral state to be the most restrained towards charging. Furthermore, the absolute height of the diamonds are observed to change with the orientation, where the neutral state is the state being affected the most. These results are also observed when the polarizability of the different redox states is included. Hence, a change in the transport properties is observed upon rotation, and must thus be taken into account in order to resemble experiment.

10. SUMMARY AND CONCLUSION We have presented some of our strategies for investigating how organic molecules sandwiched between metal clusters/electrodes are affected by the interactions between the molecule and the clusters. We have presented the theoretical and computational approaches for coupling the electromagnetic equations and the quantum mechanical equations for solving iteratively how the molecular system and the nanoparticles interact. We have illustrated how we can model the transport of electrons through organic molecules sandwiched metallic electrodes/nanoparticles. Our methods involve determination of the molecular and electronic structure of the molecules in vacuum and in junction between two metal electrodes. We have performed geometry optimizations utilizing different basis sets and functionals and we select from this investigation which molecular geometries that we will use for the detailed studies of sandwiched organic molecules. We have utilized these molecular geometries for calculating the five different redox states Q ¼ 0, 1, 2 in vacuum and in the junction. For all five different redox states, we obtain the total energy and molecular

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properties. Furthermore, we obtain for the calculations in the junction the polarization energy. We have obtained the molecular properties by the described QM/MM response method and the molecular properties could be the frequency-dependent polarizability α and the excitation energies of the organic molecules interacting with the nanoparticles/metal clusters. We have shown that we are able to utilize the calculated energies and molecular properties for calculating the conductivity as a function of bias and gate voltage. We have shown that we are able to obtain the harlequin pattern of Coulomb blockade diamonds when transporting electrons through sandwiched molecular compounds. Additionally, we have shown how the Coulomb blockade diamonds depend on conjugation lengths of the molecules, relaxation of the molecular structure as it gets charged, the sizes of the metal clusters along with the orientational effects when rotating the sandwiched molecule. We do have a large voltage difference between the source and drain but in some cases this is important when constructing the theoretical diamonds and comparing with the experimental data since we are able to consider aspects that are outside the presently available voltage window for the experimental methods.120

ACKNOWLEDGMENTS The authors thank the Danish Center for Scientific Computing for computational resources. K.V.M. thanks the Danish Natural Science Research Council/The Danish Councils for Independent Research, and the University of Copenhagen for financial support. T.H. thanks Lundbeckfonden for financial support. S.T.O., T.H., and K.V.M. thank Prof. Mark A. Ratner for his support, his vision, ambition and generosity as a scientist and as a supervisor.

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118. Olsen, S. T.; Arcisauskaite, V.; Hansen, T.; Kongsted, J.; Mikkelsen, K. V. Computational Assignment of Redox States to Coulomb Blockade Diamonds. Phys. Chem. Chem. Phys. 2014, 16, 17473–17478. 119. Kaduk, B.; Kowalczyk, T.; Voorhis, T. V. Constrained Density Functional Theory. Chem. Rev. 2012, 112, 321. 120. Olsen, S. T.; Hansen, T.; Mikkelsen, K. V. Predicting Transport Regime and Local Electrostatic Environment From Coulomb Blockade Diamond Sizes. J. Chem. Phys. 2017, 146, 104306.

CHAPTER FOUR

Criterion for the Validity of D’Alembert’s Equations of Motion John W. Perram1 School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, Australia 1 Corresponding author: e-mail address: [email protected]

Contents 1. 2. 3. 4. 5.

Introduction Constrained Particle Models of Mechanical Systems Forces of Constraint The Equations of Motion The Principle of Virtual Work 5.1 Examples of Systems With Time-Dependent Constraints 5.2 Validity of D’Alembert’s Equations 5.3 Systems of Rigid Bodies Connected by Universal, Revolute, and Telescopic Joints 5.4 D’Alembert’s Equations of Motion 6. Conclusion Acknowledgments References

104 104 106 106 106 108 110 113 114 116 116 116

Abstract For mechanical systems subject to time dependent, holonomic constraints, the principle of virtual work is apparently required to derive D’Alembert’s equations of motion. This is in contrast to situations where the constraints are time independent, where the equations of motion can be derived by standard arguments using vector calculus and linear algebra. Attempts to apply this method when some of the constraints have an explicit time dependence lead extra terms in the equations of motion. These apparent terms are removed by appealing to the principle of virtual work. In this work we show that, for the cases of universal, revolute, and telescopic joints between two rigid bodies of which one’s motion is specified externally, these terms apparently vanish identically when the computer algebra system Mathematica is used. This leads us to provide lengthy but elementary analytic proofs that the extra terms vanish identically for the three cases which, we believe, are exhaustive for real mechanical system.

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1. INTRODUCTION Analytical mechanics seeks to derive equations of motion for complicated mechanical systems in terms of generalized coordinates which satisfy any constraints between the mechanical variables.1,2 The derivation of these equations if all the constraints involve only the mechanical variables and are explicitly independent of time can be done using standard arguments using vector calculus and linear algebra. This derivation apparently fails when the constraints have an explicit time dependence. Examples of such systems include Foucault’s pendulum, where, because the pendulum is attached to a support point rotating with the earth, its plane of oscillation changes periodically over the course of the day. In fact, this model is only an approximation to reality, because, due to Newton’s third law, the motion of the pendulum will affect the motion of the earth, however, infinitesimally. Another example is provided by the blast wheel, a machine for cleaning metal surfaces, which bombards them with small metal or ceramic particles, which are accelerated while sliding along rotating blades before being ejected at high speed in the direction of the metal surface being cleaned. This system can be modeled by a particle sliding on a uniformly rotating rod. For this system, the additional terms vanish identically. We derive the solution to the system with time-independent constraints, when the mass of the blade is very heavy and show that it tends to the case when the angular velocity of the blade is constant. Important abstractions in mechanics are point particles and rigid bodies. We begin by showing that the latter can be modeled by assemblies of point particles connected by rigid constraints, of which triangles or tetrahedra are simple examples. For systems consisting of two rigid bodies, where the dynamics of one rigid body is controlled externally and is connected to the second one by universal, revolute, or telescopic constraints, the extra terms in the equation of motion vanish identically. Thus for such systems, we have shown by elementary means that D’Alembert’s equations of motion apply to them.

2. CONSTRAINED PARTICLE MODELS OF MECHANICAL SYSTEMS The subject of analytical mechanics has a long history in theoretical physics or applied mathematics, with a wide area of application in engineering

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and science. A key issue in the traditional theory, for a system modeled by N point particles with coordinates fr1 ½t , r2 ½t , …, r N ½tg

(1)

subject to M-independent holonomic constraints of the form Gα ½t ¼ gα ½r1 ½t , r2 ½t, …, r N ½t, t ¼ 0

(2)

is to find the equations of motion of the system in terms of 3N M generalized coordinates fq1 ½t, q2 ½t , …, q3N M ½tg

(3)

in terms of which the particle coordinates can be expressed as vector-valued functions r j ½t ¼ X j ½q1 ½t , q2 ½t, …, qM ½t, t

(4)

of the generalized coordinates and time which identically satisfy the constraints. Constrained particle models of rigid bodies can be made by growing tetrahedra on the faces of tetrahedra, whereas rotation axes connecting rigid bodies can be modeled by rigid constraints connecting vertices. In these situations, none of the constraints has an explicit time dependence and will generally be of the form ri ½t rj ½t ri ½t r j ½t L 2 ij

(5)

requiring that the distance between two particles remain constant. For a tetrahedron composed of four particles, there are six constraints of this form, so it has 12 6 ¼ 6 degrees of freedom, 3 translational and 3 rotational, consistent with its being a rigid body. Each extra tetrahedron added introduces three new coordinates and three new constraints, leaving the number of degrees of freedom fixed. A system consisting of two tetrahedra connected by a common rotation axis has seven degrees of freedom, which can be modeled by imposing five distance constraints of the form (5) between particles in the two tetrahedra.3

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3. FORCES OF CONSTRAINT With each constraint is associated a constraint force, which, for each particle, must be in the direction of the vector joining that particle to another other particles connected do it, and has the form μα ½t ri ½t r j ½t (6) where we will call the unknown functions μα[t] constraint force multipliers. For any problem, these have to be either calculated, which is the usual strategy when simulating the motion of complicated systems using a computer,4 or eliminated, which is the strategy of analytical mechanics.1

4. THE EQUATIONS OF MOTION The starting point is Newton’s equation of motion for each particle in the system mj r00j ½t F j

M X α¼1

μα ½t

@gα ½r1 ½t, r 2 ½t, …, r N ½t, t ¼0 @rj ½t

(7)

where mj are the masses associated with the particles and Fj are the external forces on them, which can depend on particle positions.

5. THE PRINCIPLE OF VIRTUAL WORK The process of eliminating the constraint forces is usually done through the principle of virtual work, by considering virtual displacements δrj of the particles forming the system, given in terms of the virtual displacements of the independent generalized coordinates by δr j ¼

M X @X j ½q1 ½t , q2 ½t, …, qM ½t, t k¼1

@qk ½t

δqk

(8)

For systems where the time-dependent constraints do no work in a virtual displacement, the result of taking the scalar product of the constraint forces with the vector of virtual displacements gives

Criterion for the Validity of D’Alembert’s Equations

M X α¼1

μα ½t

107

N X M X @gα ½r 1 ½t, r 2 ½t , …, r N ½t , t @X j ½q1 ½t , q2 ½t, …, qM ½t, t δqk ¼ 0 @qk ½t @r j ½t j¼1 k¼1 (9)

which, since the virtual displacements of the generalized coordinates are independent, gives the crucial identity N X @gα ½r1 ½t , r2 ½t , …, r N ½t, t @X j ½q1 ½t , q2 ½t , …, qM ½t , t ¼0 @qk ½t @rj ½t j¼1

(10)

for each constraint. This subterfuge is necessary, since the rate of change of the constraint, which must also be zero, can be found to be Gα0 ½t ¼

N X M X @gα ½r 1 ½t , r 2 ½t , …, r N ½t , t @X j ½q1 ½t , q2 ½t, …, qM ½t , t 0 qk ½t @qk ½t @r j ½t j¼1 k¼1

+

N @gα ½r1 ½t, r2 ½t, …, rN ½t, t X @gα ½r 1 ½t , r 2 ½t , …, r N ½t , t + @t @r j ½t j¼1

@X j ½q1 ½t, q2 ½t , …, qM ½t, t ¼0 @t (11)

which, since the generalized velocities are independent, will apparently only yield Eq. (9) if the constraints are time independent or the last two terms vanish identically. In these situations, straightforward applications of vector calculus and linear algebra yield D’Alambert’s equations of motion d @T ½t @T ½t ¼ + Qj ½t dt qj 0 ½t qj ½t

(12)

where T is the kinetic energy expressed as a quadratic form in the generalized velocities and Qj[t] are the generalized forces to be calculated presently. There a two worrying aspects of this theory. First, no clear criterion is given for determining whether the time-dependent constraints do no virtual work in a virtual displacement. Second, it is difficult to see how Eq. (9) can be consistent with Eq. (11) unless the last two terms of Eq. (11), namely

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N @gα ½r1 ½t , r2 ½t , …, r N ½t , t X @gα ½r1 ½t , r2 ½t, …, r N ½t, t + @t @rj ½t j¼1 @X j ½q1 ½t, q2 ½t, …, qM ½t, t ¼0 @t

(13)

vanish identically. To show that the problem is not academic, we discuss a couple of familiar systems with real, time-dependent constraints.

5.1 Examples of Systems With Time-Dependent Constraints 5.1.1 Foucault’s Pendulum Sometimes, one rigid body will be much more massive than the others, in which case it may be convenient to ignore the effect of the smaller subsystems on its motion. An example of such a system is Foucault’s spherical pendulum, where, because the support point is rotating about the earth’s axis, the influence of the motion of the pendulum on the rotation of the earth is infinitesimal and can be ignored. Thus, the position R[t] of the support point relative to an axis system fixed in space with origin at the center of the earth, will be a known function of time, so that the position r[t] of the pendulum mass must satisfy the time-dependent constraint g½r ½t, t ¼ ðr½t R½tÞ ðr ½t R½tÞ L 2 ¼ 0

(14)

where L is the radius of the earth. The position of the pendulum can be expressed in terms of spherical polar coordinates as r½t ¼ R½t + L fSin ½θ½tCos ½ϕ½t , Sin ½θ½t Sin ½ϕ½t,Cos ½θ½t g

(15)

where θ[t] and ϕ[t] are the latitude and longitude of the pendulum mass. Since this expression satisfies the constraint (14) identically, Eq. (15) are suitable expressions for the coordinates of the pendulum mass in terms of generalized coordinates θ[t] and ϕ[t]. 5.1.2 The Blast Wheel Another simple example is provided by a small bead sliding along a rotating horizontal rod, shown schematically in Fig. 1. In the figure, the black circle is at the origin of coordinates, the black line is the rod, whose inertial properties are modeled by the green mass M at its end. As the rod rotates with constant angular velocity, the red bead slides along it.

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Fig. 1 Showing the red bead of mass m sliding along the black wire, whose mass M is supposed to be concentrated at the green point. The system rotates about the black circle, which, in the case where the angular velocity of the rod is controlled to be constant, could be a motor.

Provided the ratio of the masses is small or the motion of the rod is controlled, the angular velocity of the rod will remain roughly constant. This situation is found in an industrial cleaning device called a blast wheel, where small metal or ceramic particles slide along the blades of a rapidly rotating wheel before being ejected at high speed when they reach the ends. The constraint that the bead slide on the rod is expressed through the timedependent constraint g½r ½t, t ¼ r½t r½t ðr½t u½tÞ2 ¼ 0

(16)

u½t ¼ fCos ½ωt , Sin ½ωtg

(17)

where

is a unit vector in the direction of the rod rotating with constant angular velocity ω. A suitable generalized coordinate is the distance q[t] of the bead from the rotation axis, so that its coordinates at time t are given by r½t ¼ fx½t, y½tg ¼ q½tfCos ½ωt, Sin ½ωtg

(18)

which clearly satisfies the constraint (16). In the next two sections, we show that, for the cases for Foucault’s pendulum and the bead sliding on a uniformly rotating rod, the identity (13) is satisfied by the constraints and the generalized coordinates. For this latter case, we present the analytic solution when the effect of the sliding bead on the rotating rod is taken into account, a system with time-independent constraints, and show that the solution tends to the solution of the original problem as the ratio of the particle and rod masses tends to zero. In the last

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section, we show that the identities (13) are satisfied by any system of rigid bodies connected by universal, revolute, or telescopic joints.

5.2 Validity of D’Alembert’s Equations 5.2.1 Foucault’s Pendulum In Foucault’s pendulum, the pendulum mass is connected to the support point by a wire of length L, so that its position r[t] satisfies the constraint (14), where, since R[t] is an explicit function of time t, has an explicit time dependence. Suitable generalized coordinates are the azimuthal and polar angles, so that the transformation to generalized coordinates, given by Eq. (15) is X ½θ½t, ϕ½t , t ¼ R½t + L fCos ½ϕ½tSin ½θ½t , Sin ½ϕ½tSin ½θ½t, Cos ½θ½t g (19) For this system, the first term in Eq. (13) is easily evaluated to give @g½r½t, t ¼ 2R0 ½t ðr ½t R½tÞ @t

(20)

The second term can be evaluated to give @g½r ½t, t @X ½θ½t, ϕ½t , t ¼ 2ðr½t R½tÞ R0 ½t @r½t @t

(21)

so that the sum of the second and third terms vanish identically. So, for this system, D’Alambert’s equations are valid. 5.2.2 The Particle Sliding on a Rotating Rod When the expression (18) is substituted into the constraint that the bead in Fig. 1 slides along the rod, it becomes G½t ¼ r ½t r ½t ðr½t u½t Þ2 ¼ ðx½t Sin ½ωt y½t Cos ½ωt Þ2 ¼ g½r ½t , t ¼ 0 (22)

which clearly has an explicit time dependence. For this constraint and the expression (18) for the coordinates of the bead in terms of the generalized coordinate q[t], we can evaluate the two terms in Eq. (13). The first term is @g½r ½t, t ¼ 2ωðx½tSin ½ωt y½t Cos ½ωtÞðx½tCos ½ωt + y½t Sin ½ωt Þ (23) @t

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111

which, when we insert the expression (18) is zero because the first term in the product vanishes. The second term can likewise be evaluated to obtain @g½r½t, t @X ½q½t , t ¼ 2q½tðx½tSin ½ωt y½tCos ½ωtÞ @r ½t @t fSin ½ωt, Cos ½ωtg fCos ½ωt, Sin ½ωtg (24) which also vanishes identically. Thus only the first term in Eq. (11) survives, and, since the generalized velocity is not generally zero, the identity (13) is obtained without recourse to the principle of virtual work. The kinetic energy of the system is easily found to be 1 T ¼ m ω2 q½t 2 + q0 ½t2 2

(25)

leading to D’Alembert’s equation q00 ½t ¼ ω2 q½t

(26)

whose solution is well known,1 namely q½t ¼ q½0Cosh½ωt

(27)

5.2.3 Time-Independent Constraints A second experiment could be to set the rod in motion with an angular velocity ω. This will cause the bead to move along the rod, but, as it does so, will increase the effective moment of inertia of the rod, and thus slow down its rotation. Of course, if the mass M at the end of the rod is much larger than that of the bead, this slowing down will be quite modest. In this event, where we need to consider both the motion of the green effective mass point and the sliding particle, the first constraint expressing the fact that the distance of the green mass from the center of rotation must remain fixed and equal to L, is expressed through the equation G1 ½t ¼ X ½t2 + Y ½t 2 L 2 ¼ 0

(28)

The second constraint that the perpendicular distance of the sliding particle from the rotating rod must remain zero is expressed through the equation G2 ½t ¼ ðX ½t y½t Y ½t x½tÞ2 ¼ 0

(29)

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In this case, we see that the two constraints are explicitly independent of time, so that Eq. (13) is satisfied. For this system, suitable generalized coordinates are the distance q[t] of the bead from the rotation axis and the angle θ[t] between the rod and a fixed axis. With an obvious notation fX ½t, Y ½t g ¼ fLCos ½θ½t, LSin ½θ½tg fx½t , y½tg ¼ fq½tCos ½θ½t, q½t Sin ½θ½tg

(30) (31)

for which the constraints are clearly satisfied. The kinetic energy can be expressed in generalized coordinates to give T¼

1 2 1 ML + mq½t 2 θ0 ½t2 + mq0 ½t 2 2 2

(32)

Because the kinetic energy does not involve the angular coordinate, D’Alembert’s equation for that variable is readily solved to give θ 0 ½t ¼ ω

ML 2 + mq½02 ML 2 + mq½t2

(33)

where q[0] is the initial distance of the bead from the rotation axis. The equation of motion for q[t] can then be obtained as

2 ML 2 + mq½02 q ½t ¼ ω 2 q½t ML 2 + mq½t2 00

2

(34)

The general implicit solution of this equation can be found to be "

# q½t mq½02 iE ArcSin , ML 2 q½0 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t¼ C !ﬃ u 2 u ½ mq 0 ωt 1 + ML 2

(35)

where E[ϕ, m] is the elliptic integral of the second kind and C is a constant which can be found from the initial conditions. This simplifies considerably in the limit as m/M tends to zero, when the modulus of the elliptic integral tends to zero and the solution tends to the solution (27) of the problem modeled by a time-dependent constraint.

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5.3 Systems of Rigid Bodies Connected by Universal, Revolute, and Telescopic Joints We now consider a system of constrained particle models of rigid bodies connected by a variety of axes of joints, in which the motion of one of the rigid bodies is prescribed or controlled. We believe that this case is generic for all conceivable real mechanical system. 5.3.1 Universal Joints If R[t] is the position of a point fixed in the controlled body and r[t] the position of a point in an attached body, then if these are connected by a universal joint, they satisfy the constraint (14) in the discussion of Foucault’s pendulum, for which we have shown that Eq. (13) is identically satisfied. 5.3.2 Revolute Joints Two bodies connected by a revolute joint can rotate about a common axis. If w[t] denotes a unit vector fixed in the controlled body passing through the point R[t] and given by w½t ¼ fCos ½ϕ½tSin ½θ½t , Sin ½ϕ½tSin ½θ½t , Cos ½θ½t g

(36)

where the angles are regarded as specified functions of time, then the unit vectors u½t ¼ fSin ½ϕ½t , Cos ½ϕ½t , 0g v½t ¼ w½t u½t

(37) (38)

are perpendicular to this vector and each other. In addition to a constraint of the universal type, another constraint is that the perpendicular distance of the point r[t] from the rotation axis is fixed, expressed by the equation G½t ¼ g½r½t , t ¼ ðr ½t R½t Þ ðr½t R½tÞ ððr½t R½tÞ w½t Þ2 ðLSin ½αÞ2 ¼ 0 (39)

remain fixed, where L is the length of the vector r[t] R[t] and α is the angle between this vector and the rotation axis w[t]. An expression for the position of this particle in terms of a generalized coordinate q[t] is X ½q½t, t ¼ R½t + LSin ½αðu½tCos ½q½t + v½tSin ½q½t Þ + LCos ½αw½t (40) When these are inserted into Eq. (13), some quite complicated expressions result. When its left-hand side is simplified using the computer algebra system Mathematica, the result is zero.

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5.3.3 Telescopic Joints In a telescopic joint, the bodies are allowed to slide along the axis connecting them. This is expressed by the constraint G½t ¼ g½r½t , t ¼ ðr½t R½tÞ ðr½t R½tÞ ððr ½t R½tÞ w½tÞ2 ¼ 0 (41) The position of the sliding particle can then be expressed in terms of a generalized coordinate q[t] X ½q½t , t ¼ R½t + q½t fCos ½ϕ½t Sin ½θ½t , Sin ½ϕ½t Sin ½θ½t , Cos ½θ½t g (42)

Initially, we inserted these expressions into the left-hand side of Eq. (13) and used Mathematica to simplify the complicated expressions and show that the left-hand side of Eq. (13) was indeed zero.

5.4 D’Alembert’s Equations of Motion In this section, we consider the general case of a system of rigid bodies connected by universal, revolute, or telescopic joints, where the motion of one body is specified. For these systems, we have shown that Eq. (13) is identically satisfied, so that Eq. (11) becomes N X M X @gα ½r 1 ½t, r 2 ½t, …, r N ½t, t @X j ½q1 ½t, q2 ½t, …, qM ½t, t 0 qk ½t ¼ 0 @qk ½t @r j ½t j¼1 k¼1 (43)

for each constraint, be it time dependent or not. Since the generalized velocities are linearly independent, their coefficients must vanish identically, leading to the identities N X @gα ½r1 ½t , r2 ½t, …, r N ½t, t @X j ½q1 ½t , q2 ½t , …, qM ½t, t ¼0 @qk ½t @rj ½t j¼1

(44)

for each generalized coordinate and constraint. The expression of the Cartesian velocities in terms of the generalized coordinates and the associated accelerations can be obtained by differentiating Eq. (4) to obtain r0j ½t ¼

M X @X j ½r1 ½t, r 2 ½t, …, r N ½t, t l¼1

@ql ½t

q0l ½t +

@X j ½q1 ½t, q2 ½t, …, qM ½t, t @t (45)

This can be differentiated to obtain expressions for the Cartesian accelerations

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Criterion for the Validity of D’Alembert’s Equations

r 00j ½t ¼

M X @X j ½r 1 ½t, r 2 ½t, …, rN ½t, t l¼1 M X M 2 X @ X

+

m¼1 l¼1

+2

@ql ½t

q00l ½t

j ½q1 ½t , q2 ½t , …, qM ½t , t 0 qm ½t q0l ½t @qm ½t @ql ½t

M X @ 2 X j ½q1 ½t, q2 ½t , …, qM ½t , t @ 2 X j ½q1 ½t , q2 ½t, …, qM ½t , t + q0l ½t @ql ½t @t @t2 l¼1

(46)

The constraint forces can be eliminated by taking the scalar product of Newton’s equations of motion with the vectors @X j ½q1 ½t , q2 ½t, …, qM ½t, t @qk ½t

(47)

and summing over the particles to give N N X @X j ½q1 ½t , q2 ½t , …, qM ½t , t X @X j ½q1 ½t , q2 ½t , …, qM ½t , t mj r00j ½t Fj @qk ½t @qk ½t j¼1 j¼1

M N X X @gα ½r 1 ½t , r2 ½t , …, rN ½t , t @X j ½q1 ½t , q2 ½t , …, qM ½t , t ¼0 μα ½t @qk ½t @r j ½t α¼1 j¼1

(48)

The third term vanishes because of the identities Eq. (13). The second terms are called the generalized forces, defined in terms of the actual forces by Qk ¼

N X j¼1

Fj

@X j ½q1 ½t , q2 ½t, …, qM ½t, t @qk ½t

(49)

Straightforward but tedious calculations now reveal that the remaining terms of Eq. (48) are exactly equal to d @T ½t @T ½t (50) 0 dt qk ½t qk ½t which leads us immediately to D’Alembert’s equations of motion ! d @T ½t @T ½t + Qj ½t ¼ 0 dt qj ½t qj ½t

(51)

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6. CONCLUSION We have considered the case of mechanical systems composed of an assembly of rigid bodies connected by universal, revolute, or telescopic joints when the motion of one or more of the bodies is externally controlled. In this situation, some of the constraints will have an explicit time dependence. For such systems, we have shown that D’Alembert’s equations of motion are never the less satisfied. We conjecture that a similar criterion can be found for nonholonomic systems such as wheeled vehicles where the constraints are of the Pfaffian type.

ACKNOWLEDGMENTS The author would like to acknowledge a long collaboration with Professor Mark Ratner and to congratulate him heartily on his 75th birthday. Part of this paper was presented at the CMCGS conference in 2016 and published in the conference proceedings. The author would like to thank the copyright holder GSTF for permission to publish it here.

REFERENCES 1. Goldstein, H. Classical Mechanics; Addison-Wesley: Berlin, Heidelberg, 1950. 2. Arnold, V. I.; Kozlov, V. V.; Neishtadt, A. I. Mathematical Aspects of Classical and Celestial Mechanics; Springer: Berlin, Heidelberg, 2006; p 29. 3. Overgaard, L.; Petersen, H. G.; Perram, J. W. A General Algorithm for Control of MultiLink Robots. Int. J. Robotics Res. 1995, 14, 281–294. 4. Edberg, R.; Evans, D. J.; Morriss, G. P. Constrained Molecular Dynamics: Simulation of Liquid Alkanes With a New Algorithm. J. Chem. Phys. 1986, 84, 6933–6939.

CHAPTER FIVE

A Time-Dependent Density Functional Theory Study of the Impact of Ligand Passivation on the Plasmonic Behavior of Ag Nanoclusters Adam P. Ashwell, Mark A. Ratner, George C. Schatz1 Northwestern University, Evanston, IL, United States 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Computational Methodology 3. Results and Discussion 3.1 Ag135+ 3.2 Ag25 (SH)18 and Ag25 (NH2)18 3.3 Ag3214+ and Ag44 (SH)304 4. Conclusion Acknowledgments References

118 122 122 122 126 134 139 140 140

Abstract We present a detailed study of the impact of ligand passivation on the electronic structures and optical properties of plasmonic Ag nanoclusters using density functional theory (DFT) and time-dependent density functional theory (TD-DFT). The clusters studied are Ag13 5 + , Ag25 ðSHÞ18 , Ag25 ðNH2 Þ18 , Ag32 14 + , and Ag44 ðSHÞ30 4 . We find that the highest occupied ligand orbitals from S (3p) and N (2p) appear just above the conduction band, and this leads to significant ligand-to-metal charge transfer transitions at high energies. Dielectric screening associated with ligand passivation results in reduced HOMO–LUMO gaps and in an increased gap between the HOMO and the valence band associated with the Ag 4d orbitals. Ligand field effects result in splitting of plasmonic peaks, leading to reduced mixing between nearby single-particle excitations. The magnitude of these effects is found to decrease when thiolate ligands are replaced

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with amine ligands. We also find that, in the case of the Ag44 ðSHÞ30 4 cluster, the ligands localize plasmonic excitations into the core of the cluster.

1. INTRODUCTION Noble metal nanoparticles with plasmonic properties have been the focus of extensive study in recent years.1,2 The localized surface plasmon resonances (LSPRs) of noble metal nanoparticles appear in optical spectra as strong absorption peaks that change in energy in response to changes in properties such as size, shape, composition, and environment,1,3 giving rise to numerous applications in index of refraction and biological sensing, photocatalysis, and optical devices.4–13 Plasmon resonance also results in the enhancement of the nanoparticle’s local electric field, and as such noble metal nanoparticles are used as substrates for surface-enhanced Raman spectroscopy.14–19 Other materials such as alkali metals, Group 13 metals, and intermetallic alloys have been shown to exhibit plasmonic performance similar to or even superior to that of noble metals.20–22 However, gold and silver remain the most commonly used plasmonic materials due to their low reactivity, making them easier to work with as well as particularly well suited to biomedical applications.23–26 Classically, an LSPR in a metal nanoparticle involves excitation of collective oscillations of the conduction electrons with respect to the positively charged background of the nuclei.1 The plasmonic behavior of a variety of nanoparticles of different shapes, sizes, and compositions can be described accurately using classical electrodynamics methods.1,27–30 The oldest of these methods is Mie theory, based on a solution to Maxwell’s equations for a sphere, as presented by Gustav Mie in 1908.31,32 Solutions to Maxwell’s equations have since been extended to accurately describe optical behavior in nonspherical and heterogeneous particles33–35 using such methods as the discrete dipole approximation (DDA)36 and the finite-difference time domain method.37 While classical electrodynamics can accurately predict LSPRs in a range of nanoparticles, they have been shown to fail in the small particle limit,38 where quantum effects and the influence of passivating ligands become significant. These limitations can be overcome, however, using quantum mechanical methods. Plasmonic excitations found using time-dependent density functional theory (TD-DFT) have been shown to be correlated with those found by well-established classical methods. Few-atom, bare Ag

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nanoclusters embedded in solid Ar have been shown experimentally to produce molecule-like absorption spectra (with multiple absorption bands), but these evolve into plasmon-like behavior for clusters 20 atoms in size and larger.39 The absorption spectra of small tetrahedral Ag clusters calculated by TD-DFT show similar molecule-like discrete features for small clusters, but these become more plasmon-like as cluster size increases, with many closely spaced excitations underlying the plasmon band. Extrapolation of the size-dependent energy of the primary peak in the spectra of tetrahedral nanoclusters to the large particle limit leads to excellent agreement with the LSPRs of tetrahedral nanoparticles calculated by the classical DDA method with empirical dielectric functions.40 A detailed analysis of the TD-DFT excited states for tetrahedral Ag clusters has shown that even at sizes as small as Ag20, the primary excitation in the calculated absorption spectrum is a collective excitation of several particle–hole states associated with the conduction electrons, which is a signature characteristic of plasmon excitation.41 Several models for identifying LSPRs in the calculated absorption spectra of molecules and clusters have been put forth, all of which rely on the idea that a plasmonic excitation must be collective. One such model, proposed in 2013 by Bernadotte et al.,42 defines electronic excitations as the poles of the external response function ð 0 (1) χ ext ðr, r , ωÞ ¼ ε1 ðr, r00 , ωÞχ irr ðr00 , r0 , ωÞd 3 r 00 where χ irr is the irreducible response function (i.e., the response to changes in the total potential, rather than just the external perturbation), ω is the angular frequency of the external perturbation, and the dielectric function ε is defined as ð 00 00 (2) εðr, r , ωÞ ¼ δðr r Þ χ irr ðr, r000 , ωÞfCoul ðr000 r00 Þd 3 r 000 where ƒCoul is the Coulomb kernel. Poles of χ ext can appear either at frequencies corresponding to poles of χ irr or at frequencies where the dielectric function ε has a zero mode. The former are classified as single-particle excitations and the latter as plasmons. A primary difference between these two classes of excitations is that the frequencies of plasmons depend on ƒCoul, while those of single-particle excitations do not. The authors performed TD-DFT calculations on small plasmonic systems with a modified definition of ε in which they multiply ƒCoul by a scaling factor λ. As they scanned λ from 0 to 1, they found that the frequencies of single-particle excitations

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were largely unchanged, while those of plasmons changed substantially. This approach has been used in a number of studies to investigate plasmonic excitations in metal clusters and nanowires.43–45 Guidez and Aikens presented in 2014 a quantum mechanical analysis of plasmonic resonances using configuration interaction (CI) in which they defined a plasmon as a coherent superposition of single-particle excitations in which the transition dipole moments of the excitations interfere constructively.46,47 They describe the wave function of a given excitation in an ideal plasmonic system with N single particle states in which all single-particle excitations have the same energies and transition dipoles as Ψex ¼ A1 Φ1 + A2 Φ2 + ⋯ + AN ΦN

(3)

where Φi are the singly excited determinants of each interacting state and Ai are weighting coefficients. After calculating the energies and oscillator strengths of the N excitations using the CI matrix, they find a single plasmonic state resulting from constructive addition of the contributing single-particle transitions and N 1 degenerate zero-intensity nonplasmonic states. The authors also used a variation of this method to describe the plasmon resonances found in acenes.48 Quantum mechanically, an LSPR can be defined in terms of a collective excitation of conduction electrons into the unoccupied portion of the conduction band. In practical terms, a collective excitation is one that is a linear combination of several single excitations whose dipole moments interfere constructively.46,47 Due to the computational expense of ab initio methods such as TD-DFT, quantum mechanical studies of plasmonic nanoparticles are typically limited to small nanoclusters, usually below 2 nm in diameter. It is possible to treat larger nanoclusters using TD-DFT by applying the time-dependent local density approximation (TD-LDA) with a jellium model, in which the positively charged nuclei of the cluster are treated as a positive charge spread out uniformly over the volume of the cluster.49–51 However, as this method does not treat molecular structure explicitly, it cannot properly handle ligands. An interesting recent development, however, has been the use of the semiempirical INDO/CI method to study plasmonic excitations in Ag nanoclusters, which may in the future allow quantum mechanical study of larger nanoclusters while explicitly treating molecular structure.52 Nanoclusters are interesting both as a means to better understand larger, computationally intractable systems and as intriguing systems in their own

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right. In the nanocluster size regime, the continuous density of states of larger nanoparticles is broken into discrete energy levels, giving rise to new chemical, optical, and electrical properties and therefore new applications.53–57 While both Au and Ag are plasmonic materials, at small sizes only Ag clusters consistently display plasmonic behavior. The plasmons observed in larger Au structures disappear in the small size limit, as the intraband transitions of Au clusters and nanowires are mixed with interband transitions, resulting in localized excitations that do not show the dependence on cluster size and shape that is characteristic of plasmons.44,58–61 In contrast, plasmons have been observed in Ag clusters as small as Ag10.40,41,52 Given the computational expense of ab initio methods like TD-DFT, it is far more practical to study plasmons in small Ag clusters than in Au clusters large enough for plasmons to emerge. Ligand passivation is an important factor to take into account in characterizing plasmon behavior. Experimentally, ligand-passivated clusters are far more readily available than bare clusters, and at small sizes the impact of ligands on the electronic structure and optical properties of clusters is significant. In a 2010 study, Peng et al.38 synthesized and measured the absorption spectra of a number of ligand-passivated Ag nanoparticles ranging in diameter from roughly 2 to 20 nm. They found that, while classical electrodynamics predicted a blue shift in the LSPR with decreasing size, the experimental spectra showed a reversal in size dependence of the LSPR at a turnover point of roughly 12 nm, below which the spectra are strongly red-shifted with decreasing size. Both the location of the turnover point and the degree to which the LSPR of the Ag nanoparticles red shifts below the turnover point were found to change significantly when the amine ligands were replaced with thiolate ligands. Because ligand effects become stronger with decreasing particle size, we would expect to see very substantial ligand effects in few-atom Ag clusters. Plasmons have been observed in sufficiently large thiolated Au and intermetallic Au–Ag clusters (that is, clusters with thiol-based ligands),61–69 with plasmons being found to emerge in Au clusters as size increases from smaller clusters to Au144(SH)60 and Au314(SH)96.70 In contrast, most studies of plasmonic behavior in Ag clusters have been focused on bare clusters, rather than the more experimentally relevant thiolated clusters.40,41,71–74 One reason for this is that crystallographically determined structures of thiolated Ag clusters have only recently become available. In this work, we study the electronic structures and optical properties of two bare Ag nanoclusters and their ligand-passivated variants using

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TD-DFT. In particular, we examine the impact of ligand passivation on the plasmonic excitations observed in bare clusters. We show that the addition of ligands, particularly thiolate ligands, to Ag clusters results in the presence of numerous ligand-to-metal charge transfer excitations in the absorption spectrum, and that these ligand-to-metal transitions are dominant at higher energies. We also show that dielectric screening and ligand field effects associated with ligand passivation can shift and split the Kohn–Sham orbitals of the bare Ag clusters, resulting in large red shifts in the energies of plasmonic excitations and the splitting plasmons into multiple excitations that are not as coherently coupled. Lastly, we demonstrate that ligand passivation results in core localization of the plasmon in small Ag clusters.

2. COMPUTATIONAL METHODOLOGY Ground-state electronic structure and geometry optimization were carried out using density functional theory (DFT). The Xα exchangecorrelation functional75 and a double-ζ (DZ) Slater-type basis set with frozen cores were used for geometry optimization. Linear response time-dependent density functional theory (LR-TD-DFT) was used for excited-state calculations with the SAOP76,77 functional and an all-electron triple-ζ (TZP) Slater-type basis set,78,79 a level of theory that has been used in several earlier studies of noble metal nanoclusters and nanowires.62,63,74 Only singlet– singlet excitations that were optically allowed by symmetry were calculated. To account for relativistic effects, the zeroth order regular approximation (ZORA) was used.80–82 Charges were assigned using a grid-based Bader analysis.83,84 All calculated spectra are shown convoluted with a Lorentzian with a full-width at half-maximum (fwhm) of 0.1 eV. Calculations were carried out using the Amsterdam Density Functional (ADF) 2013.01 and 2014.01 programs.85–88

3. RESULTS AND DISCUSSION 3.1 Ag13 5 +

3.1.1 Electronic Structure We first consider the electronic structure and absorption spectrum of Ag13 5 + , which comprises the bare core (with eight valence electrons) of the Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18 clusters which will be discussed in detail later. The structure of Ag13 5 + consists of a single central Ag atom

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Fig. 1 Optimized structures of (A) Ag13 5 + , (B) Ag25 ðSHÞ18 , and (C) Ag25 ðNH2 Þ18 .

surrounded by an icosahedral cage of 12 Ag atoms, as shown in Fig. 1A. Calculations on the Ag13 5 + cluster were performed using the D5d symmetry point group, as the Ih point group is not supported in ADF. In the Ag structures, Kohn–Sham orbitals composed primarily of orbitals up to and including 4d are fully occupied, while the set of molecular orbitals made up of 5s and 5p orbitals is partially occupied. The calculated electronic structure for the Ag13 5 + conduction band (Fig. 2A) corresponds well with the superatom model proposed by Walter et al.89 wherein the valence electrons of metal clusters are found in delocalized superatomic orbitals following the Aufbau rule 1S2 j 1P6 j 1D10 j 2S2 1F14 j …. The eight valence electrons of Ag13 5 + fit neatly into this model, resulting in occupied 1S and 1P superatomic orbitals and a closed shell. We can identify superatomic Kohn–Sham orbitals as those composed primarily of 5s and 5p atomic orbitals. Furthermore, we find in Fig. 2A that the Kohn–Sham orbitals are divided into two regions. All orbitals shown below about 29 eV are dominated by Ag 4d atomic orbital character (with one exception discussed below), while all those above, both occupied and virtual, are dominated

124

eV 1Fb

–22.9 –23.8

Energy (eV)

–24.7

1Fa

B

2S

–25.6

1D

–26.5 –27.4 –28.3 1P –29.2 –30.1 –31.0

Ag135+ calculated absorption spectrum 4

Intensity (arbitary units)

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3

Total absorption Intraband Interband

2 1

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Energy (ev) (1S)

–31.9

Fig. 2 (A) The calculated electronic structure of Ag13 5 + (note that the superatom 1S is embedded in a large number (65) of occupied orbitals associated with the atomic 4d state of Ag) and (B) the calculated absorption spectrum of Ag13 5 + . The total absorption spectrum is broken down by the characters of the underlying transitions.

by Ag 5s and 5p atomic orbital character. While made up of discrete orbitals, these regions above and below 29 eV correspond to the conduction and valence bands, respectively, found in larger metal particles. Note that the very negative values of the energies of these orbitals arise from the high positive charge of the species being studied. The occupied 1S superatomic orbital (also a conduction orbital) is found at roughly 31.3 eV in Fig. 2A, firmly in the middle of the valence band. The next set of superatomic orbitals is found in the triply degenerate HOMO, corresponding to the 1P orbitals. Beyond this, in the virtual portion of the conduction band, we find the five 1D orbitals at roughly 25.9 eV, the single 2S orbital at 24.8 eV, and the seven 1F orbitals split into groups of three and four orbitals at 23.6 and 22.8 eV, respectively. Due to the Laporte selection rule governing optical transitions for this centrosymmetric system, for optically allowed transitions electrons from the ungerade 1P orbitals can only be excited into the gerade 1D and 2S orbitals. 3.1.2 Optical Properties We find a number of discrete excitations in the absorption spectrum of Ag13 5 + (Fig. 2B). To characterize these excitations as plasmonic or nonplasmonic and make sense of the spectrum, we first analyze the atomic orbital character of the occupied and virtual orbitals involved in each

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excitation. Excitations in LR-TD-DFT are linear combinations of single-particle excitations between occupied and virtual Kohn–Sham orbitals. Kohn–Sham orbitals are, in turn, linear combinations of atomic orbitals. Therefore, excitations can be characterized by the atomic orbital characters of the Kohn–Sham orbitals associated with each single-particle excitation making up the excitation. In Ag13 5 + , and indeed in all systems studied in this work, all virtual orbitals that are accessible at relevant energies are conduction band orbitals of predominantly Ag 5s and 5p character. Because all excitations carry electrons into the conduction band, we need only to determine the characters of the occupied orbitals involved to characterize an excitation as either interband (d ! sp) or intraband (sp ! sp). In addition to the absorption spectrum of Ag13 5 + , Fig. 2B shows the spectrum decomposed into interband and intraband contributions. Each peak in the spectrum is actually representative of three nearly degenerate excited states with mutually orthogonal transition dipole moments, but for simplicity each group of three will be referred to as a single excitation. We see that the first two peaks, at 3.75 and 4.33 eV, are predominantly intraband, while the peaks at higher energies are interband. There are only a few possible single-particle excitations low enough in energy and with the correct symmetries to mix around 4 eV—those between three 1P orbitals and the five 1D orbitals, and between the 1P orbitals and the 2S orbital. Examination of the single-particle excitation contributions to the two intraband peaks shown in Table 1 reveals that the first peak is collective, involving a mixture of 1P to 1D and 1P to 2S single-particle excitations. For our purposes, an excitation is considered collective (i.e., plasmonic) if no contributing single-particle excitation has a weight greater than 0.5 and if there are at least three single-particle excitations with weights greater than 0.1. Here the highest contribution is from a 1P to 1D transition with a weight of 0.420. The second peak is similar to the first, but with 1P to 2S excitations making up over half of the total contributions to the transition (with a weight of 0.542). This means that the transition at 4.33 eV is less collective than that at 3.75 eV. Of course neither of these transitions is collective in the sense that all possible conduction electrons are involved in the excitations, but if we describe states as plasmonic that satisfy the 0.5 weight rule noted above, then the 4.33 eV state would be somewhere between a plasmon and a single-particle excitation. The presence of two peaks rather than one and the differences in contribution from 1P to 2S single-particle excitations may be due to the 1.1 eV energy gap between the 1D and 2S orbitals, preventing single-particle excitations to each of them from mixing

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Table 1 Transitions Contributing to Important Excited States in Ag13 5 + Energy (eV) Transition

3.75

4.33

6.38

Weight

1P ! 2S

0.420

1P ! 1D

0.323

1P ! 1D

0.215

Valence band ! 1D

0.014

Valence band ! 1D

0.002

1P ! 2S

0.542

1P ! 1D

0.180

1P ! 1D

0.119

Valence band ! 1D

0.065

Valence band ! 1D

0.023

Valence band ! 1D

0.735

Valence band ! 1D

0.183

Valence band ! 2S

0.018

Valence band ! 1D

0.014

Valence band ! 1D

0.009

An interband transition at 6.38 eV is also shown here for the sake of comparison. The five single-particle excitations that most contribute to each excited state are given.

effectively. Another cause may be the differences in superatomic character between the 1D and 2S orbitals.

3.2 Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18

3.2.1 Electronic Structure We next examine the effects of ligand passivation on the electronic structure and optical properties of the Ag13 5 + cluster. The Ag25 ðSPhMe2 Þ18 cluster has recently been synthesized,90 but the very closely related Ag25 ðSHÞ18 structure was first studied as a hypothetical Ag analog to the Au25 ðSHÞ18 cluster.63 The structure of Ag25 ðSHÞ18 is based around the icosahedral Ag13 5 + core, with six [SH-Ag-SH-Ag-SH] oligomeric ligands attached octahedrally as shown in Fig. 1B. These ligands are so-called staples,91 which are seen in many ligand-passivated silver and gold clusters.68,92–96 This results in a structure composed of an Ag13 core and then 12 additional Ag

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atoms incorporated in the oligomeric ligands. The other cluster we examine here, Ag25(NH2), is a purely hypothetical structure in which the thiolate ligands of Ag25 ðSHÞ18 are swapped out for amines, as shown in Fig. 1C. All calculations on these two clusters were performed using the Ci symmetry point group. The electronic structures of Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18 are shown in Fig. 3. In both cases, ligand passivation results in the appearance of a band of ligand orbitals between the valence band (occupied Ag 4d orbitals) and conduction band 1P HOMO, dominated by either S 3p or N 2p contributions. In Fig. 3 only the upper portions of these ligand bands are shown. In the thiolated cluster the ligand band spans roughly 2.1 eV, while in the aminated cluster it spans 1.7 eV. The ligand bands of both clusters also have modest contributions from Ag 5s and 5p orbitals. A second effect of ligand passivation in Fig. 3 is that the HOMO– LUMO gap of the Ag13 5 + cluster, 2.84 eV, is reduced to 1.65 eV in Ag25 ðNH2 Þ18 and 1.45 eV in Ag25 ðSHÞ18 , and spacings between virtual A

B

eV 1Fb

eV

–1.8

–3.5 –4.0

1Fa

–2.2

2S 1Db

–2.6

Energy (eV)

–4.5 –5.0

–3.0 1Da

1Fa 1Db 2S 1Da

–3.4

–5.5

–3.8

–6.0

–4.2 1P

–6.5

–4.6

–7.0

–5.0

–7.5

1Fc and 2P 1Fb

1P

–5.4 –5.8

–8.0 –6.2

Fig. 3 Calculated electronic structures of (A) Ag25 ðSHÞ18 and (B) Ag25 ðNH2 Þ18 . The orbitals shown here below the 1P orbitals are the upper portion of the ligand band. The valence band is further below in energy.

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orbitals are also significantly reduced. These reductions in the conduction band width and spacings can be considered to be a dielectric screening effect of the ligands, where polarization of the ligand electrons by electrons in the conduction and valence orbitals stabilizes those orbitals. Related to this is an increase in the conduction–valence band gap, defined here as the difference in energy between the HOMO and the highest-energy valence orbital (Ag 4d). As shown in Table 2, the conduction–valence gap of 2.01 eV in the bare Ag13 5 + cluster expands to 2.53 eV in the aminated cluster and 3.08 eV in the thiolated cluster. This can be understood in terms of preferential stabilization of the conduction band by dielectric screening compared to the valence band, as would make sense given that the valence band involves 4d orbitals that are less exposed to the ligands. However, another issue to consider is that the Bader charge analysis shows that ligand passivation also increases the number of electrons in the Ag13 core of both Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18 , as shown in Table 2. To distinguish between the effects of electron addition and those of screening, we examine the electronic structure of two additional charge states of Ag13, the neutral Ag13 cluster and Ag13 5 . We see that the simple addition of electrons to the Ag13 5 + cluster decreases the conduction–valence band gap, so apparently charge addition is less important than differential screening. The amine and thiolate ligands have a similar influence on the electronic structure, though the influence of the amines is smaller.

Table 2 Charge Analysis and Energy-Level Spacing Data for the Ag13 5 + “Family” of Clusters Total Bader Charge Ag13 (Core)

Ag12 (SH)18 (Ligands) (Ligands)

Gap Between Conduction and Valence Bands (eV)

Ag13 5 +

5.00

N/A

N/A

2.01

Ag13

0.00

N/A

N/A

1.53

5.00

N/A

N/A

1.37

2.10

4.26

7.36

2.53

1.19

2.57

4.75

3.08

Ag13

5

Ag25 ðNH2 Þ18 Ag25 ðSHÞ18

The conduction–valence band gap is the difference in energy between the 3P conduction orbital and the highest-energy valence orbital (orbitals with predominantly Ag 4d atomic orbital character).

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A third effect of ligand passivation is that the 1D and 2S virtual superatomic orbitals, which are the primary virtual orbitals involved in the plasmonic excitations in Ag13 5 + , are substantially modified by ligand field effects. In the thiolated cluster the 1D orbitals are split into 1Da and 1Db orbitals that are 0.59 eV apart, and the 2S orbital becomes mixed in with the lower group of 1F orbitals. Note that while the ligands here are arranged octahedrally, the splitting is more akin to that seen in the presence of a tetrahedral ligand field. For the sake of clarity, groups of orbitals that have been split are referred to by their superatomic orbital characters followed by a subscript, as seen in Fig. 3. In the aminated cluster, the 1D orbitals are split as in the thiolated case, but are separated by only 0.46 eV, and the 2S orbital is shifted down such that it lies between the two sets of 1D orbitals. The splitting observed here is consistent with that seen in previous studies of the Au25 ðSHÞ18 and Ag25 ðSHÞ18 clusters.62,63,65 3.2.2 Optical Properties Although superficially similar, the calculated spectra of Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18 (Fig. 4) are more complicated than that of the bare Ag13 5 + cluster in Fig. 2. The significant reduction in HOMO–LUMO gap due to ligand-induced dielectric screening pushes the intraband peaks of interest from around 4 eV down to around 2.5 eV, and we also see that a number of peaks arise at higher energies due to ligand-to-metal charge transfer. In the absorption spectrum of Ag25 ðSHÞ18 (Fig. 4A) we find a large number of sharp peaks. Even at low energies there is a substantial ligand-to-metal contribution in every excitation, which is due to mixing of S 3p orbitals with the Ag 5s and 5p orbitals in the triply degenerate 1P orbitals as well as some contribution by single-particle excitations originating at the top of the ligand band (at 7.3 eV in Fig. 4A). Regardless of this mixing, we consider peaks that are about evenly intraband and ligand-tometal to be, for our purposes, intraband. The intraband transitions of the Ag25 ðSHÞ18 cluster are located at 1.64, 2.33, and 2.65 eV. The component single-particle excitations are shown in Table 3. The first of these, at 1.64 eV, is a noncollective excitation primarily between the 1P and 1Da orbitals. The second, at 2.33 eV, is a collective excitation primarily between the 1P and 1Db orbitals. These two excitations are equivalent to the plasmonic transition seen at 3.75 eV in the Ag13 5 + cluster, but lowered in energy by the shrinking of the HOMO–LUMO gap and split

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Fig. 4 Calculated absorption spectra of (A) Ag25 ðSHÞ18 and (B) Ag25 ðNH2 Þ18 .

in two by the splitting of the 1D orbitals. In contrast to the peak seen in the bare cluster, however, neither of these two peaks in the thiolated cluster has any significant contribution from single-particle excitations to the 2S orbital. The third intraband transition, at 2.65 eV, is almost entirely composed of single-particle excitations from the 1P orbitals to the 2S orbital—each of the three transitions making up the peak seen in the spectrum is dominated by a transition from one of the three 1P orbitals to the 2S orbital, and the overall transition is not collective and, therefore, not plasmonic. This is equivalent to the peak at 4.33 eV in the bare cluster, but with virtually no mixing between transitions to the 2S orbital and transitions to the 1D orbitals. Every excitation higher in energy than those discussed above is

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Table 3 Transitions Contributing to Important Excited States in Ag25 ðSHÞ18 Energy (eV) Transition Weight

1.64

2.33

2.65

1P ! 1Da

0.550

1P ! 1Da

0.161

1P ! 1Da

0.125

1P ! 1Da

0.037

1P ! 1Da

0.030

1P ! 1Db

0.263

1P ! 1Db

0.196

1P ! 1Db

0.155

1P ! 1Db

0.052

1P ! 1Db

0.052

1P ! 2S

0.855

Ligands ! 1Da

0.020

1P ! 1Db

0.018

1P ! 1Db

0.016

Ligands ! 1Da

0.010

predominantly ligand-to-metal, and in fact the spectrum as a whole is dominated by ligand-to-metal transitions. The 1P to 2S single-particle excitations in the thiolated cluster do not mix with transitions between the 1P and 1Da or 1Db orbitals, even though the gap between the 2S and 1Da orbitals, 1.1 eV, is unchanged from the corresponding gap in Ag13 5 + . This may be an effect of dielectric screening. Although the energy gap is the same, the conduction band energies of the thiolated cluster are reduced by the dielectric screening from the ligands, meaning that the 2S may be more weakly coupled to the 1D orbitals than in the bare cluster even though the energy gap is unchanged. We next look at the absorption spectrum of Ag25 ðNH2 Þ18 (Fig. 4B), where a number of differences from the Ag25 ðSHÞ18 spectrum are readily apparent, and the spectrum is more similar to that of the bare cluster. We see far less ligand-to-metal character spread throughout the spectrum, and even at higher energies the ligand-to-metal character is far less dominant than it is

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in the thiolated cluster. This difference in the magnitude of the ligand-tometal contribution to the spectra of Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18 is due to the relative energies of the valence orbitals of the thiolate and amine ligands—the S 3p valence electrons are much closer in energy to the HOMO than are the N 2p electrons. As in the case of Ag25 ðSHÞ18 , there are three notable intraband transitions in the Ag25 ðNH2 Þ18 excitation spectrum (the peak just below 2 eV in Fig. 4B includes two separate transitions). The first intraband transition, at 1.84 eV, is a plasmonic excitation primarily from the three 1P orbitals to the two 1Da orbitals, but with some contribution from transitions to the 2S orbital, as seen in Table 4. The second transition, at 1.93 eV, is almost entirely made up of transitions from the 1P orbitals into the 2S orbital and is equivalent to the peaks at 4.33 eV in Ag13 5 + and 2.65 eV in Ag25 ðSHÞ18 . The final intraband transition is at 2.43 eV and is mostly from Table 4 Transitions Contributing to Important Excited States in Ag25 ðNH2 Þ18 Energy (eV) Transition Weight

1.84

1.93

2.43

1P ! 1Da

0.212

1P ! 2S

0.208

1P ! 1Da

0.199

1P ! 1Da

0.173

1P ! 1Da

0.053

1P ! 2S

0.767

1P ! 1Da

0.069

1P ! 1Db

0.026

1P ! 1Da

0.019

1P ! 1Da

0.018

1P ! 1Db

0.242

1P ! 1Db

0.185

1P ! 1Db

0.180

1P ! 1Db

0.148

1P ! 1Da

0.042

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the 1P orbitals to the 1Db orbitals. In this case, the first and third intraband transitions, at 1.84 and 2.43 eV, are equivalent to those at 3.75 eV in Ag13 5 + and at 1.64 and 3.22 eV in Ag25 ðSHÞ18 , which also primarily consists of transitions into the 1D orbitals. The relationship between the spectra of the aminated and bare clusters is similar to that between the thiolated and bare clusters. In both of the ligand-passivated clusters, the plasmonic peak of the bare cluster, at 3.75 eV, is split into two separate plasmonic excitations due to splitting of the 1D superatomic orbitals. The nearly plasmonic peak in the bare cluster, at 4.33 eV, ceases to be plasmonic at all upon addition of the ligands. While some of the shifting and splitting of peaks in the spectrum of Ag13 5 + upon ligand passivation can be explained by splitting and shifting of orbitals, the inability of single-particle excitations close in energy to mix effectively in the spectra of the ligand-passivated clusters cannot. In the bare Ag13 5 + cluster, single-particle excitations from the 1P orbitals to the five degenerate 1D orbitals mix to a significant extent with those from the 1P orbitals to the 2S orbital, which is about 1.1 eV higher in energy. However, in the thiolated cluster, the two groups of split 1D orbitals are separated by only about 0.60 eV, and yet there is little mixing between the single-particle excitations from the 1P orbitals into each of the two groups. Furthermore, the 2S orbital in the thiolated cluster is about 1.1 eV higher in energy than the 1Da and only about 0.55 eV higher in energy than the 1Db orbitals. And yet, there is essentially no mixing between single-particle excitations to the 1D orbitals and those to the 2S orbital. The aminated cluster is a similar case. In Ag25 ðNH2 Þ18 , we do see separate peaks for transitions from 1P to 1Da, to 2S, and to 1Db, but in the transition to the 1Da orbitals there is a nontrivial contribution from single-particle excitations into the 2S orbital. This may be due to the relative closeness of these energy levels, as the 2S orbital is only 0.23 eV higher in energy than 1Da, and the three 1Db orbitals are only 0.44 eV higher in energy than 1Da. In contrast, in the thiolated cluster, where the 2S and 1Da orbitals are separated by 1.1 eV, there is no significant mixing between 1P to 1D and 1P to 2S single-particle excitations. Importantly, nearly all the conduction electrons involved in the plasmonic peaks in the thiolated and aminated clusters come from the icosahedral Ag13 core, rather than from the 12 Ag atoms in the six [SH-AgSH-Ag-SH] ligands.

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3.3 Ag32 14 + and Ag44 ðSHÞ30 4

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3.3.1 Electronic Structure The last two clusters we examine are Ag32 14 + and its thiolated form, Ag44 ðSHÞ30 4 . Ag44 ðSHÞ30 4 was the first all-thiol-protected Ag cluster to be synthesized and have its structure fully crystallographically determined.97,98 Its structure as well as that of Ag32 14 + is shown in Fig. 5. The structure of Ag32 14 + consists of an icosahedral Ag12 core (unlike Ag13 5 + , there is no central Ag atom) surrounded by an Ag20 dodecahedral shell. It has 18 valence electrons, which gives it a closed shell in the superatom model. The thiolated structure, Ag44 ðSHÞ30 , has six Ag2 S5 3 staple ligands attached octahedrally. The two distinct regions in the structure of Ag32 14 + allow us to study how ligand passivation affects surface and core Ag conduction electrons differently. All calculations for both clusters were performed using the Ci symmetry point group. As with the Ag13 5 + cluster, the superatomic orbitals of Ag32 14 + can be found in its electronic structure (Fig. 6A). The 1S orbital is found in the middle of the valence band at 53.23 eV and is not shown. The three 1P orbitals are located on the upper edge of the valence band at 50.63 eV and have a mixture of conduction and valence band character. The HOMO is comprised of the five nearly degenerate 1D orbitals. The LUMO, at 47.54 eV, is the 2S superatomic orbital, and the seven 1F orbitals are split into a group of four at 47.34 eV (1Fa) and a group of three at 45.94 eV (1Fb). Lastly, the three 2P orbitals are found at 45.60 eV. It is important to note that both the five occupied 1D orbitals and the virtual 2S orbital are of gerade symmetry, meaning that transitions between them are forbidden.

Fig. 5 Structures of (A) Ag32 14 + and (B) Ag44 ðSHÞ30 4 .

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A

B eV –44.4

eV 1G and 2P 1.2

1G –45.0

1Fc

0.9 –45.6

1Fb

2P 1Fb

Energy (eV)

–46.2

0.3

–46.8 –47.4

1Fa 2S

0.0 1Fa 2S

–0.3 –0.6

–48.0

1Db

–0.9

–48.6 –49.2

0.6

1D

–1.2

1Da

–1.5 –49.8 –1.8 –50.4 –2.1 –51.0

Fig. 6 Calculated electronic structures of (A) Ag32 14 + and (B) Ag44 ðSHÞ30 4 .

The first virtual orbitals that are accessible to excitation from the 1D orbitals are the ungerade 1F orbitals, followed by the ungerade 2P orbitals. In the electronic structure of Ag44 ðSHÞ30 4 + (Fig. 6B) we see that, as was the case for Ag25 ðSHÞ18 and Ag25 ðNH2 Þ18 , the addition of ligands results in the addition of a ligand band between the conduction and valence bands as well as orbital splitting. The five occupied 1D orbitals which in the bare cluster were just barely split by 0.08 eV are now split by 0.31 eV. The seven virtual 1F orbitals are now split into three separate groups rather than the two in the bare cluster. And one of the nine virtual 1G orbitals is now relatively low enough in energy that it is below the three virtual 2P orbitals. As before, the addition of ligands increases the gap between the valence and conduction bands (here taken to be between the gap between 1Da HOMO–1 and the 4d valence band), this time from 1.6 eV to 3.0 eV, and lowers the spacings between virtual orbitals. 3.3.2 Optical Properties The calculated absorption spectra of Ag32 14 + and Ag44 ðSHÞ30 4 are shown in Fig. 7. In the bare Ag32 14 + cluster, there is a single plasmonic excitation at 2.45 eV, composed primarily of single-particle excitations from the five 1D orbitals to the four 1Fa orbitals, the lowest energy set of virtual orbitals with the appropriate symmetry. The single-particle excitations that make up this

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Fig. 7 Calculated absorption spectra of (A) Ag32 14 + and (B) Ag44 ðSHÞ30 4 .

plasmonic excitation (all 1D to 1Fa) are shown in Table 5. In this spectrum, intraband transitions are further broken down by region of origin within the cluster—an intraband transition can come from the Ag12 core, the Ag20 shell, or some mixture of the two. The plasmonic excitation in Ag32 14 + comes from a nearly even mixture of core and shell, as seen in Fig. 7A. One reason we see a single plasmonic peak in this spectrum, as opposed to the two seen in that of Ag13 5 + , may be because of the larger spacing between accessible groups of virtual orbitals. The two groups of 1F orbitals are separated by 1.4 eV, compared to the 1.1 separating the 1D and 2S orbitals in Ag13 5 + , and we see (Table 5) that the 1Fb orbitals are not involved in the plasmonic transition. At energies high enough for electrons from the 1D orbitals to be excited into 1Fb orbitals, electrons from the

A Time-Dependent Density Functional Theory Study

Table 5 Transitions Contributing to Important Excited States in Ag32 14 + Energy (eV) Transition

2.45

137

Weight

1D ! 1Fa

0.128

1D ! 1Fa

0.119

1D ! 1Fa

0.102

1D ! 1Fa

0.074

1D ! 1Fa

0.056

valence band can also be excited into the 2S and 1Fa, suppressing any potential plasmonic excitation. Ligand passivation of Ag32 14 + results in a more complex absorption spectrum (Fig. 7B). As in the case of the Ag25 ðSHÞ18 cluster, we find that ligand-to-metal character dominates the absorption spectrum of Ag44 ðSHÞ30 4 . While Ag32 14 + has a single plasmonic excitation, Ag44 ðSHÞ30 features four of them. The first is at 1.62 eV and is a mixture of transitions from 1Db to 1Fb and from 1Da to 1Fa, as shown in Table 6. The second, at 1.75 eV, is a mixture of 1Db to 1Fb and 1Db to 1Fc. The third plasmonic excitation is at 2.01 eV, made up mostly of single-particle excitations from 1Da and 1Db to 1Fc. And finally, the fourth and most intense plasmonic excitation is at 2.56 eV, a mixture of single-particle excitations between the upper part of the ligand band and the 2S LUMO orbital and between 1Da and 2P. Importantly, we see that the core Ag atoms contribute substantially more than the shell Ag atoms to the plasmonic excitations. In contrast, the two layers contributed about equally to the plasmonic excitation in the bare Ag32 14 + cluster. This suggests that the presence of ligands localizes some portion of the conduction electrons in the Ag shell, perhaps because of dielectric screening, resulting in a core-localized plasmon not seen in the bare cluster. As with the Ag13 5 + cluster and its ligand-passivated variants, the 12 Ag atoms in the six Ag2 S5 3 ligands do not participate in any plasmonic excitations. The presence of four plasmonic excitations in Ag44 ðSHÞ30 4 , compared with just one in Ag32 14 + , appears to have several causes. The first is simply orbital splitting due to ligand field effects. Because the thiolate ligands split the 1D into two groups and split the 1F orbitals into three groups, a number of single-particle excitations that were degenerate in Ag32 14 + have different

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Table 6 Transitions Contributing to Important Excited States in Ag44 ðSHÞ30 4 Energy (eV) Transition Weight

1.62

1.75

2.01

2.56

1Db ! 1Fb

0.225

1Da ! 1Fa

0.212

1Da ! 1Fa

0.193

1Da ! 1Fa

0.109

1Db ! 1Fb

0.101

1Db ! 1Fb

0.276

1Db ! 1Fb

0.109

1Db ! 1Fc

0.104

1Db ! 1Fb

0.090

1Db ! 1Fc

0.086

1Db ! 1Fc

0.135

1Db ! 1Fc

0.134

1Da ! 1Fc

0.133

1Da ! 1Fc

0.070

1Da ! 1Fa

0.065

Ligands ! 2S

0.235

1Da ! 2P

0.175

1Da ! 2P

0.155

Ligands ! 2S

0.068

1Da ! 2P

0.035

energies in the thiolated cluster. As a result, there are a number of new combinations of occupied and virtual energy levels available. The second is that the presence of thiolate ligands shifts the relative energies of orbitals, lowering transition energies due to dielectric screening. For example, the plasmonic peak in the thiolated cluster at 2.56 eV involves intraband transitions into the virtual 2P orbitals. These single-particle excitations have energies of about 2.2 eV, whereas in the bare cluster, the same single-particle excitations have energies of 3.6 eV. And finally, the increase in the gap between the conduction and valence bands upon ligand passivation, from 1.6 to 3.0 eV due to differential screening of valence and

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139

conduction orbitals, pushes the onset of interband transition to higher energies, allowing intraband transitions to higher-energy virtual orbitals to occur without convoluting with interband transitions.

4. CONCLUSION We have performed DFT and TD-DFT calculations on two sets of Ag nanoclusters, both with and without ligand passivation. The calculated electronic structures of these clusters show that ligand passivation of both the Ag13 5 + and Ag32 14 + clusters results in three distinct effects. (1) The highest occupied ligand orbitals show up at energies above the valence band, leading to substantial contribution from ligand-to-metal charge transfer excitations to the spectra, particularly at higher energies. (2) Dielectric screening reduces the splitting of the conduction band orbitals and differential screening increases the gap between the valence and conduction bands and in most cases reduces the spacings between virtual orbitals. (3) Ligand field effects lead to splitting of superatomic orbitals in the conduction band, in particular the sets of 1D and 1F superatomic orbitals, and this reduces plasmonic character of the lower energy transitions. These factors combine to create multiple plasmonic peaks in the calculated absorption spectra of the ligand-passivated clusters, rather than the single plasmonic peaks seen in the bare clusters. The extent to which the ligands split the superatomic orbitals and shift virtual orbital energies was found in the case of Ag13 5 + to depend on the identity of the ligand. Amine ligands were found to have a smaller impact on the electronic structure of Ag13 5 + than thiolate ligands due to the lower energy of nitrogen’s valence orbitals relative to those of sulfur. This is consistent with the findings of Peng et al.,38 whose experiments demonstrated a larger red shift for thiolated Ag nanoparticles than for aminated nanoparticles as diameter was reduced. The Ag atoms in the oligomeric ligands in each ligand-passivated cluster studied were found not to contribute significantly to any plasmonic excitations, nor did the addition of these Ag atoms to the Ag cores result in any increase in the occupancy of the superatomic orbitals. While the plasmonic excitation in Ag32 14 + is composed roughly evenly of single-particle excitations from the Ag12 core and Ag20 shell, contributions from the Ag12 core become dominant in the spectrum of the thiolated cluster. In the case of Ag32 14 + cluster and its thiolated counterpart, Ag44 ðSHÞ30 4 , ligand

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passivation was found to localize as well as split the plasmonic excitation seen in the bare cluster.

ACKNOWLEDGMENTS The authors thank Christine Aikens for providing an ADF input file for the TD-DFT calculation of Ag25 ðSHÞ18 shown in Ref. 63, as well as Rebecca Gieseking and Lindsey Madison for helpful discussions. This research was supported by the Department of Energy, Basic Energy Sciences, under grant DE-FG02-10ER16153.

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75. Vosko, S. H.; Wilk, L.; Nusair, M. Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis. Can. J. Phys. 1980, 58, 1200–1211. 76. Gritsenko, O. V.; Schipper, P. R. T.; Baerends, E. J. Approximation of the Exchange-Correlation Kohn–Sham Potential With a Statistical Average of Different Orbital Model Potentials. Chem. Phys. Lett. 1999, 302, 199–207. 77. Schipper, P. R. T.; Gritsenko, O. V.; Van Gisbergen, S. J. A.; Baerends, E. J. Molecular Calculations of Excitation Energies and (Hyper)Polarizabilities With a Statistical Average of Orbital Model Exchange-Correlation Potentials. J. Chem. Phys. 2000, 112, 1344–1352. 78. Van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J. Implementation of Time-Dependent Density Functional Response Equations. Comput. Phys. Commun. 1999, 118, 119–138. 79. Rosa, A.; Baerends, E. J.; van Gisbergen, S. J. A.; van Lenthe, E. Electronic Spectra of M(Co)6 (M ¼ Cr, Mo, W) Revisited by a Relativistic TDDFT Approach. J. Am. Chem. Soc. 1999, 121, 10356–10365. 80. van Lenthe, E.; Baerends, E. J.; Snijders, J. G. Relativistic Regular Two-Component Hamiltonians. J. Chem. Phys. 1993, 99, 4597–4610. 81. van Lenthe, E.; Baerends, E. J.; Snijders, J. G. Relativistic Total Energy Using Regular Approximations. J. Chem. Phys. 1994, 101, 9783–9792. 82. van Lenthe, E.; Ehlers, A.; Baerends, E. J. Geometry Optimizations in the Zero Order Regular Approximation for Relativistic Effects. J. Chem. Phys. 1999, 110, 8943–8953. 83. Rodrı´guez, J. I.; Bader, R. F. W.; Ayers, P. W.; Michel, C.; G€ otz, A. W.; Bo, C. A High Performance Grid-Based Algorithm for Computing QTAIM Properties. Chem. Phys. Lett. 2009, 472, 149–152. 84. Rodrı´guez, J. I. An Efficient Method for Computing the QTAIM Topology of a Scalar Field: The Electron Density Case. J. Comput. Chem. 2013, 34, 681–686. 85. te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T. Chemistry With ADF. J. Comput. Chem. 2001, 22, 931–967. 86. Fonseca Guerra, C.; Snijders, J. G.; te Velde, G.; Baerends, E. J. Towards an Order-N DFT Method. Theor. Chem. Acc. 1998, 99, 391–403. 87. ADF2013, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, http://www.scm.com. 88. ADF2014, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, http://www.scm.com. 89. Walter, M.; Akola, J.; Lopez-Acevedo, O.; Jadzinsky, P. D.; Calero, G.; Ackerson, C. J.; onbeck, H.; H€akkinen, H. A Unified View of Ligand-Protected Whetten, R. L.; Gr€ Gold Clusters as Superatom Complexes. Proc. Natl. Acad. Sci. U. S. A. 2008, 105, 9157–9162. 90. Joshi, C. P.; Bootharaju, M. S.; Alhilaly, M. J.; Bakr, O. M. [Ag25(SR)18](): The “Golden” Silver Nanoparticle. J. Am. Chem. Soc. 2015, 137, 11578–115781. 91. Heaven, M. W.; Dass, A.; White, P. S.; Holt, K. M.; Murray, R. W. Crystal Structure of the Gold Nanoparticle [N(C8H17)4][Au25(SCH2CH2PH)18]. J. Am. Chem. Soc. 2008, 130, 3754–3755. 92. Pei, Y.; Gao, Y.; Shao, N.; Zeng, X. C. Thiolate-Protected Au20(SR)16 Cluster: Prolate Au8 Core With New [Au3(SR)4] Staple Motif. J. Am. Chem. Soc. 2009, 131, 13619–13621. 93. Ning, C. G.; Xiong, X. G.; Wang, Y. L.; Li, J.; Wang, L. S. Probing the Electronic Structure and Chemical Bonding of the “Staple” Motifs of Thiolate Gold Nanoparticles: Au (Sch 3) 2 and Au 2 (Sch 3) 3. Phys. Chem. Chem. Phys. 2012, 14, 9323–9329.

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94. Jiang, D.; Walter, M. The Halogen Analogs of Thiolated Gold Nanoclusters. Nanoscale 2012, 4, 4234–4239. 95. MacDonald, M. A.; Zhang, P.; Qian, H.; Jin, R. Site-Specific and Size-Dependent Bonding of Compositionally Precise Gold–Thiolate Nanoparticles From X-Ray Spectroscopy. J. Phys. Chem. Lett. 2010, 1, 1821–1825. 96. Jiang, D. Staple Fitness: A Concept to Understand and Predict the Structures of Thiolated Gold Nanoclusters. Chem. Eur. J. 2011, 17, 12289–12293. 97. Yang, H.; Wang, Y.; Huang, H.; Gell, L.; Lehtovaara, L.; Malola, S.; H€akkinen, H.; Zheng, N. All-Thiol-Stabilized Ag44 and Au12Ag32 Nanoparticles With Single-Crystal Structures. Nat. Commun. 2013, 4, 1–8. 98. Desireddy, A.; Conn, B. E.; Guo, J.; Yoon, B.; Barnett, R. N.; Monahan, B. M.; Kirschbaum, K.; Griffith, W. P.; Whetten, R. L.; Landman, U.; Bigioni, T. P. Ultrastable Silver Nanoparticles. Nature 2013, 501, 399–402.

CHAPTER SIX

Switching Activity of Allosteric Modulators Controlled by a Cluster of Residues Forming a Pressure Point in the mGluR5 GPCR: A Computational Investigation☆ Michael Sabio*,†, Sid Topiol*,†,1 *Stevens Institute of Technology, Center for Healthcare Innovation, Hoboken, NJ, United States † 3D-2Drug, Fair Lawn, NJ, United States 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Methods 2.1 Molecular Mechanics Calculations 2.2 Hybrid Quantum Mechanics/Molecular Mechanics Calculations 2.3 Quantum-Chemical Calculations 2.4 Database Searches and Figures 3. Results and Discussion 3.1 X-ray Structural Analyses 3.2 Computational-Protocol Considerations and Validation 3.3 The Puckering and Interaction of Pro655 3.4 Overview of NAM Docking Simulations: MPEP 3.5 NAM Interactions: Mavoglurant and MPEP 3.6 NAM Interactions: NAMs 3, 4, and 5 3.7 PAM Interactions: PAMs 6, 7, and 8 4. Conclusions References

148 150 150 151 152 152 152 152 156 157 158 161 164 170 171 173

Abstract X-ray structures for ligand-modulated GPCRs were not available until 2007 and were limited to class A GPCRs. The recent availability of X-ray structures for mGluR5, a class C GPCR, provides a valuable tool for understanding drug action. For mGluR5, pairs of ☆

In honor of Mark Ratner’s 75th birthday.

Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.03.003

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2017 Elsevier Inc. All rights reserved.

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extremely closely related ligands have been shown to have opposite (activating PAM vs inactivating NAM) pharmacological switching effects, which have defied understanding in drug-discovery studies. Using MM and QM calculations, we have identified a cluster of mGluR5 residues that provide a pressure point that is sensitive to these small NAM/PAM differences. These residues reside in the extracellular side of the transmembrane region of the mGluR5 protein on helices 5 and 6, which for class A GPCRs are known to require conformational changes in the intracellular region for activation to occur. We also find that docking studies presented herein provide a clear explanation for the highly efficient mGluR5-NAM MPEP in terms of its interactions with the protein.

1. INTRODUCTION The functioning of systems of different scales utilizes conceptually similar devices. Even at an electronic scale, for instance, Aviram and Ratner described1 molecular rectifiers as early as 1974. Other larger systems that are still significantly smaller than our macroenvironments function through the use of various devices. Physiological processes such as the biochemical networks inherent in mammalians depend heavily on the communication of messages between proteins. Perhaps the most pervasive and important of these, and certainly the single-most targeted category of proteins for all available medications, are the so-called G-protein-coupled receptors (GPCRs). The GPCRs sit in the membrane surface and receive a signal, generally in the form of binding of a ligand to its outer, extracellular (EC) surface, which switches the GPCR between its inactive and active states. Intriguingly, one GPCR, rhodopsin, is an exception to this description and is of a hybrid scale, in that it is driven by an optical/electronic trigger in which incoming light initiates the flipping of the chromophore 11-cis retinal to all-trans retinal that then induces the activated state of rhodopsin. The active and inactive states of GPCRs have different protein conformations wherein only the active state is capable of binding to an effector protein (most commonly a G-protein) to complete its role in the signaling cascade. To be sure, this simple on/off switch model does not specify a number of additional known complexities, such as the ability of a given GPCR to interact with different effector proteins, the existence of varying levels of activation, and others (see, e.g., Ref. 2). In spite of the prominent role played by GPCR proteins in physiological processes and pharmaceuticals, and despite the extensive atomic-level protein structural information available for many other receptor and enzyme proteins, the

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first atomic-level structure for a GPCR, the X-ray structure of rhodopsin,3 appeared in 2000. Rhodopsin is a member of the largest class (i.e., class A) of GPCRs. The first X-ray structure of a ligand-mediated GPCR was reported4 for another class A GPCR, the β2-adrenergic receptor, in 2007. While several X-ray structures of class A GPCRs have been reported since then, representatives of other classes, including classes B, C, and F, have only started to appear more recently. Both the unique architecture of class C GPCRs and a profile of subtle variable ligand-mediated switching mechanisms raise challenges to our understanding of these proteins and, in turn, the optimal deployment of the emerging structural information toward drug discovery. The common architectural feature of GPCRs is a collection of seven α-helices (H1–H7) of amino acids that traverse the cellular transmembrane domain, starting from the EC side and continuing to the intracellular (IC) side and back in alternating directions (with odd- vs even-numbered helices) and connected at their EC and IC ends by generally loop-like amino acid sequences (EC1–EC3 and IC1–IC3). This constellation of linked, transmembrane-spanning helices forms a barrel-like structure. For class A GPCRs, natural, endogenous ligands bind in the inner region in the approximately top third of this barrel structure, as generally do synthetic ligands. Class C GPCRs also contain a seven-transmembrane domain; however, they contain two additional domains: a so-called “Venus Fly-Trap” (VFT) domain and a cysteine-rich (C-rich) domain, which links the VFT and seven transmembrane (7TM) domains. Unlike class A GPCRs, class C GPCRs utilize these additional domains for activation-state control. In the case of metabotropic glutamate receptors (mGluRs), for example, the orthosteric-binding site for the endogenous glutamate ligand is in the VFT domain. Numerous developed synthetic compounds bind at sites that are allosteric to the VFT domain and actually coincide with the classical (for class A) 7TM binding sites. Compounds binding at these sites can either act as positive allosteric modulators (PAMs), which enhance the activity of the protein in response to glutamate, or as negative allosteric modulators (NAMs), which inhibit glutamate’s activity. For one of these mGluRs, mGluR5, the small, highly efficient NAM MPEP has received much attention and effort in trying to understand its efficiency. Also, for mGluR5, researchers have identified5 pairs of compounds that have only minor differences in chemical constitution, yet have opposing effects at these allosteric sites with NAM vs PAM “switching” activities. The recently published mGluR5 X-ray structures6,7 present an opportunity to try to unravel

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this intriguing structure–activity relationship within the context of their protein-activation mechanism. Herein, we expand on our recent preliminary investigations of these relationships.8

2. METHODS 2.1 Molecular Mechanics Calculations Starting with the coordinates of the three mGluR5 7TM X-ray structures (with the Protein Data Bank9 accession codes 4OO9,6 5CGC,7 and 5CGD7), we prepared10 this study’s protein models in the Schr€ odinger software suite,11 without relaxation of heavy atoms. We deleted all surface ligands and all water molecules except for the water molecule featured and described in this study. We adjusted the models, as necessary, to provide appropriate local protein environments by changing the protonation states of selected His, Asp, and Glu residues, the orientation (flipping) of functionality at the end of specific side chains (i.e., Asn, Gln, and His), the rotational states of X-H groups (X ¼ O or S), and the orientation of the OdH bonds of the preserved water molecule. In a final step, we fully optimized the positions of all hydrogen atoms. We used the LMOD software12 in the Schr€ odinger suite11 to extensively explore possible binding modes of the ligands within the prepared protein models of this study. We permitted full ligand (i.e., conformational, translational, and rotational) flexibility, full side-chain flexibility of all residues in the ligand-binding region, and full flexibility for the one retained water molecule described in this study. For each ligand that we investigated, we searched for plausible binding poses by conducting multiple LMOD simulations that differ in the ligand’s starting location, orientation, and conformation (including, where relevant, varied puckered states of five- and six-membered rings and alternative amide-bond isomers). The ligandbinding region, for any ligand, was defined to include all residues having ˚ of any of that ligand’s single or multiple at least one atom within 7.0 A starting positions. We imposed geometric constraints, which we extracted from the Cambridge Crystallographic Database,13 on each ligand’s internal geometry, to enforce (as appropriate) near planarity or near linearity, without restricting dihedral-angle rotations. Our goal here was to avoid unrealistic ligand distortions that are normally observed in a simulation within a protein model with a tight binding cavity. To compare experimentally observed or computationally predicted ligand–protein complexes, we superimposed the corresponding backbone atoms, which remained frozen

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during the simulations. We used the AMBER* force field (Schr€ odinger’s 14 enhanced or modified version of the AMBER force field ), implemented with a distance-dependent dielectric function and dielectric constant of 3.0. An additional protocol was used to model the possible effects of each PAM reported in this study. Starting with the result of the LMOD simulation of the corresponding, closely related NAM, we structurally modified the ligand’s capping group, using standard geometry, to transform the NAM into the PAM, without relaxation or disturbance of the NAM’s LMOD coordinates. We subjected each PAM to an LMOD simulation, following the procedure specified earlier, except that we also (a) applied constraints ˚ tolerance) to the atoms of the PAM and (b) allowed full (with only a 0.1 A relaxation (side-chain and backbone atoms) of all residues with at least one ˚ of the PAM’s heavy atoms that are not present in the atom within 4.0 A corresponding NAM.

2.2 Hybrid Quantum Mechanics/Molecular Mechanics Calculations With the QSite module15 within the Schr€ odinger suite,11 we explored the binding of mavoglurant in the mGluR5 protein model derived from the PDB:4OO9 X-ray structure6 and assessed the reproducibility of the X-ray binding pose by a hybrid QM/MM (quantum mechanics/molecular mechanics) approach. For the QM region, we used the Hartree–Fock method16,17 with the 6-31G basis set18 (i.e., with no diffuse or polarization functions), which is expressed as HF/6-31G in standard notation. Within the QM region, we allowed complete flexibility of the ligand and the one water molecule, which is involved in a hydrogen-bond network with Tyr659, Thr781, and the carbonyl group of Ser809. Also, we allowed full flexibility of the following specified protein structural components in the QM region, with the indicated distances taken relative to the starting position of the ligand: (a) all side chains and all atoms of any proline, alanine, or ˚ and (b) all atoms of glycine residues having at least one atom within 5.0 A any residue with one or more backbone atoms (including hydrogen atoms) ˚ . However, positional constraints were applied to all backbone within 3.5 A atoms (except hydrogen atoms) within the QM region. We defined the MM ˚ of region as containing any residue having at least one atom within 10.0 A the starting position of the ligand minus any atoms belonging to the QM region. We allowed full flexibility in the MM region, except for positional constraints that we applied to the MM region’s backbone atoms.

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2.3 Quantum-Chemical Calculations With the Jaguar module (version 9.1, release 14)19 within the Schr€ odinger suite,11 we calculated interaction energies with the Hartree–Fock method16,17 and density functional theory (DFT)20,21 with the B3LYP method22,23; with both methods, we utilized the 6-31G** basis set24 (i.e., HF/6-31G** and DFT(B3LYP)/6-31G**). For these calculations, we used the coordinates of the QSite and LMOD simulation results without further relaxation. However, to avoid exceeding Jaguar’s 1000-atom limit, ˚ of the ligand MPEP or (b) 6.5 A˚ of we removed all residues beyond (a) 7.0 A the ligand mavoglurant or any of the other ligands excluding MPEP in this study; we then added hydrogen atoms with standard geometry to fill valence requirements. We performed these single-point energy calculations on each frozen-geometry ligand, truncated protein, and truncated complex, without a correction for the zero-point energy (as no equilibrium geometries were produced) and without a correction for the basis set superposition error.

2.4 Database Searches and Figures We used the Cambridge Structural Database (CSD) with ConQuest13 to explore the conformational preference of the Ød^dØ chemical moiety, allowing the optional replacement of a carbon atom by a nitrogen atom at any Ø (six-membered aromatic) position that is not attached to the acetylenic linker. ConQuest searches of the CSD were conducted with options selected to find “only organics” (i.e., no organometallic compounds) that were not disordered. For all figures, the graphics were generated within the Schr€ odinger soft11 13 ware or within the CSD suite.

3. RESULTS AND DISCUSSION 3.1 X-ray Structural Analyses In a previous study,8 we gained insights about structural, geometric, and binding features that may facilitate ligand-discovery efforts involving class C GPCRs, by computationally investigating the recently published, first X-ray structures of mGluR 7TM domains, namely those of mGluR125 and mGluR56,7 complexed with NAMs. We focused on the first of the three published mGluR5 X-ray structures, whose structure is available in the Protein Data Bank with accession code 4OO96 as the more recent X-ray structures7 became available only in the final stages of that study. We noted that

153

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N H O

1

O

N

2 O O

O S

S N

N

O O N S

S N

N

3

N

4

5 O

O S

S N

S

N

N

6

O O N S

7

N

N

8

Fig. 1 The chemical structures of the negative allosteric modulators (NAMs 1–5) and positive allosteric modulators (PAMs 6–8) investigated in this study. For each molecule, the compound name and assay data are provided in parentheses; the data for molecules 1–2 and 3–8 are reproduced from Refs. 6 and 5, respectively. 1 (mavoglurant, Ki ¼ 5.6 nM); 2 (MPEP, Ki ¼ 3.5 nM); 3 (compound 11b, IC50 ¼ 23 nM); 4 (compound 30b, IC50 ¼ 140 nM); 5 (compound 28f, IC50 ¼ 370 nM); 6 (compound 11a, EC50 ¼ 2.5 nM, Emax ¼ 150%); 7 compound 30g, EC50 ¼ 150 nM, Emax ¼ 110%); and 8 (compound 28g, EC50 ¼ 81 nM, Emax ¼ 120%).

the meta-methylated phenyl ring of the ligand, mavoglurant (1 of Fig. 1), is deeply buried in a mainly hydrophobic pocket delimited by Gly624, Ile625, Gly628, Ser654, Pro655, Ser658, Tyr659, Val806, Ser809’s backbone carbonyl moiety, Ala810, and Ala813 (Fig. 2). That pocket is stabilized and further spatially restricted by a water-mediated hydrogen-bond network involving Tyr659, Thr781, and Ser809’s backbone carbonyl moiety. Mavoglurant’s acetylenic portion, which is a common motif in mGluR5 ligands, traverses a narrow channel that leads to a wider pocket. Mavoglurant’s bicyclic system, which is accommodated in this wider pocket, gives the ligand an overall L-shape and allows the remainder of the ligand’s structure to veer by almost 90 degrees from the lower portion (in the view of Fig. 2) of the molecule and then pass under Phe788. A hydroxyl group on mavoglurant, which is involved in hydrogen bonds with the hydroxyl groups of Ser805 and Ser809, is attached to the chiral carbon located on mavoglurant’s cyclohexyl ring at the upper end of the acetylenic linker. At the ligand’s terminus in the upper compartment of the binding cavity, the methyl ester is directed into a hydrophobic pocket comprised of Ile651, Val740, Pro743, and Leu744, and the ligand’s ester carbonyl moiety participates in a hydrogen bond with Asn747’s side-chain amino functionality. The other two more recently published7 mGluR5 X-ray structures contain the closely related inhibitors 3-chloro-4-fluoro-5-[6-(1H-pyrazol-1-yl)

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Fig. 2 The binding topology of mavoglurant in the first published X-ray structure of the mGluR5 7TM domain. The ligand’s meta-methylated phenyl ring is deeply buried in a hydrophobic pocket, the acetylenic portion traverses a narrow channel that leads to a wider pocket that accommodates the ligand’s bicyclic system, the ligand’s L-shape allows it to pass under Phe788, and the methyl ester substituent at the ligand’s terminus is directed into a hydrophobic pocket. The ligand participates in three hydrogen bonds (i.e., with Ser805, Ser809, and Asn747).

˚ respyrimidin-4-yl] benzonitrile (“Heptares 14” in PDB:5CGC with 3.1 A olution) and 3-chloro-5-[6-(5-fluoropyridin-2-yl)pyrimidin-4-yl] ˚ resbenzonitrile (“Heptares 25” or HTL14242 in PDB:5CGD with 2.6 A olution). When these X-ray structures are compared to the mGluR5mavoglurant X-ray complex, one notices that an important difference found in both the two most recent mGluR5 X-ray complexes is the rotation of Trp785’s side chain toward the center of the helical bundle (Fig. 3). The consequences of this rotation include a narrowing of the binding pocket and greater stabilization for the two closely related ligands. The spatial restriction due to this rotation would no longer allow mavoglurant to fit,

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Fig. 3 A comparison of the three mGluR5 7TM X-ray complexes. The most important differences between the two Heptares X-ray structures (PDB:5CGC in brown and PDB:5CGD in green) and the mavoglurant X-ray structure (PDB:4OO9 in cyan) include the rotation of the side chains of Ser805 (away from the ligands) and Ser809 (to form a hydrogen bond with an aromatic nitrogen atom), the formation of a hydrogen bond with a water molecule (and the nitrile group), and only slight overlap of the two Heptares ligands with mavoglurant’s acetylenic linker.

in that Trp785’s phenyl ring would overlap with a portion of mavoglurant’s cyclohexyl ring. Other differences found in PDB:5CGC and PDB:5CGD include the rotation of the side chain of Ser805 away from the center of the binding cavity and the rotation of Ser809’s side chain to form a hydrogen bond with the available nitrogen atom of a ligand’s five- or six-membered aromatic ring. One also notices that the nitrile group of either Heptares ligand participates in a hydrogen bond with a water molecules, which occupies a portion of the binding cavity where the methyl group of mavoglurant’s ester moiety sits in the PDB:4OO9 X-ray structure. Also, there is very little overlap of the ligands in the PDB:5CGC and PDB:5CGD X-ray structures and the acetylenic linker of mavoglurant in the PDB:4OO9 X-ray structure. Of apparently less importance is that the orientation of the amide group of Asn747’s side chain is different in the PDB:5CGC and PDB:5CGD X-ray structures, with only one of the two X-ray structures having the same Asn747 side-chain orientation found in the PDB:4OO9 complex.

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3.2 Computational-Protocol Considerations and Validation We further investigated the binding of mavoglurant in the mGluR5 protein model derived from the PDB:4OO9 (i.e., Protein Data Bank9 entry 4OO9) X-ray structure6 and assessed the reproducibility of the X-ray binding pose and observed interactions, by performing LMOD simulations12 and a QM/MM geometry optimization,15 as summarized in Section 2. Both the LMOD and QM/MM refined structures of the mavoglurant–mGluR5 complex are quite superimposable on the PDB:4OO9 X-ray structure, with only negligible differences in atomic positions (Fig. 4). The high degree of similarity among the structures of the three complexes is due, at least in part, to the presence of a large ligand that is anchored with the help of three hydrogen bonds within a spatially restricted binding cavity; these conditions restrict the motion of the ligand and some of the nearby side chains. In the QM/MM calculation, we allowed only geometry relaxation to a local energy minimum, while the LMOD simulation sampled dihedral-angle rotations and relaxed the geometry of the complex, subject to the constraints

Fig. 4 A comparison of the PDB:4OO9 X-ray structure with the results of an LMOD simulation and a QM/MM geometry relaxation. The LMOD (yellow) and QM/MM (pink) refined structures of the mavoglurant–mGluR5 complex are quite superimposable on the PDB:4OO9 X-ray structure (cyan), with only negligible differences in atomic positions. For clarity, only selected residues are shown. The most noticeable differences occur in mavoglurant’s bicyclic system, where both rings in the X-ray structure are flatter than in the computational counterparts; the insets focus on the cyclohexyl ring.

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specified in Section 2. However, it was still possible that the ligand’s position in the LMOD and QM/MM calculations may shift and the rotational states of some of the nearby residues may be different from those found in the X-ray structure. In either the LMOD or QM/MM calculations, another consequence could have been the loss of at least one of the three intermolecular hydrogen bonds. One observes (Fig. 4) that the cyclohexyl ring of the ligand in the X-ray structure appears flatter than in the LMOD or QM/MM complex; a more chair-like conformation would be achieved in the X-ray structure if the methylene carbon atom attached to one of the cyclohexyl ring’s chiral centers would shift slightly in position. Also, the ligand’s pyrrolidine ring seems to be flatter in the X-ray structure than that in the LMOD or QM/MM complex. Nevertheless, there seems to be good agreement between the results of the computational protocols used herein and the first published mGluR5 7TM X-ray structure.

3.3 The Puckering and Interaction of Pro655 When comparing the results of our computational study with the X-ray structures, one of the differences we noticed is a variation in the ring conformation of Pro655. As described later, we monitored and examined this feature. In the LMOD simulations of mavoglurant/4OO9, MPEP/ 5CGC, and MPEP/5CGD, we observe that Pro655 may exist in two puckered states, differing in whether the residue’s Cγ atom (Fig. 5) is directed toward or away from the ligand. The two most common proline puckered conformations, as observed in the LMOD simulations, are described in the literature,26,27 and the proline puckering is referred to as Cγ-endo or Cγexo, respectively, when proline’s Cγ atom is directed toward or away from proline’s carbonyl carbon atom. In all three X-ray structures (4OO9, 5CGC, and 5CGD) from the Protein Data Bank,9 Pro655 adopts the Cγ-exo conformation, which places proline’s Cγ atom closer to the X-ray’s ligand. In contrast, the LMOD12 calculations prefer the alternative puckering, albeit by less than 1.3 kcal/mol (data not shown) at the molecular mechanics level in all three simulations. However, this small energy difference is well within the uncertainty (of 5–10 kcal/mol) that one expects from the use of a molecular mechanics force field and prompted our use of ab initio quantum-chemical methods, as described later. In LMOD, the generation of a puckered state that is contrary to that observed in all the three X-ray structures reflects several factors, such as the force field’s tolerance of the easy convertibility of a proline ring’s conformation, the strain of the

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Fig. 5 The predicted binding mode of MPEP in models derived from the PDB:5CGC and PDB:5CGD mGluR5 7TM X-ray structures. (A) The restricted motion of MPEP in a narrow region of the binding cavity, the hydrophobic or (nonparallel) π-stacking interactions with Trp785 and Phe788, and the formation of hydrogen bonds may explain MPEP’s remarkable efficiency. MPEP binding models based on the PDB:5CGC and PDB:5CGD X-ray structures are essentially identical (see inset). (B) MPEP would superimpose quite well on mavoglurant with respect to mavoglurant’s methyl-phenyl-acetylenic substructure, although there is a slight rotation. Also, MPEP’s phenyl group overlaps the chlorinated phenyl ring of either of the two Heptares X-ray ligands.

proline ring when in reasonably close contact with the ligand in a spatially restricted binding cavity, and the accommodation by nearby residues to permit the flipping of a proline ring. We attempted to address the relative stability of the proline ring’s puckered states, while accessing ligand interactions or strain in the context of interaction-energy calculations (Table 1), at higher levels of theory, with the conditions and restrictions described in Section 2. We used the Hartree–Fock method16,17 and DFT20,21 with the B3LYP method22,23; with both methods, we utilized the 6-31G** basis set24 (i.e., HF/6-31G** and DFT(B3LYP)/6-31G**, respectively).

3.4 Overview of NAM Docking Simulations: MPEP MPEP has become almost a gold standard of mGluR5 (and perhaps of all mGluR) NAMs because of its extraordinary efficiency. It is therefore of great interest to try to determine its binding characteristics. In our previous investigation,8 we generated and analyzed potential, alternative binding modes of MPEP (2 of Fig. 1) in a protein model based on the PDB:4OO9 mGluR5 7TM X-ray structure. In the preferred binding mode, MPEP

Table 1 Single-Point (SP) Interaction Energies, at the HF/6-31G** and DFT(B3LYP)/6-31G** Levels of Theory, of Mavoglurant and MPEP in Various Protein Modelsa Interaction Protein Pro655 SP Energy Complex Protein Energyb Ligand Energy Energy Row Model Ligand Puckered State Method Energyb (a.u.) (a.u.) (a.u.) (kcal/mol)

1

QSite/ 4OO9

Mavoglurant Toward ligand

HF

24,368.189730 23,356.229674 1011.942037 –11.3

2

LMOD/ 4OO9

Mavoglurant Toward ligand

HF

23,324.868435 22,312.917748 1011.934041 –10.4 [0.0] [0.0]

3

LMOD/ 4OO9

Mavoglurant Away from ligand

HF

23,324.859051 22,312.909686 1011.933399 –10.0 [5.9] [5.1]

4

LMOD/ 5CGC

MPEP

Toward ligand

HF

21,988.443527 21,397.468718 [0.0] [0.0]

590.979110 +2.7

5

LMOD/ 5CGC

MPEP

Away from ligand

HF

21,988.440076 21,397.464505 [2.2] [2.6]

590.979246 +2.3

6

LMOD/ 5CGD

MPEP

Toward ligand

HF

21,912.417561 21,321.443162 [0.0] [0.0]

590.979447 +3.2

7

LMOD/ 5CGD

MPEP

Away from ligand

HF

21,912.413981 21,321.438082 [2.2] [3.2]

590.979568 +2.3

8

QSite/ 4OO9

mavoglurant Toward ligand

DFT (B3LYP)

24,507.371892 23,489.038365 1018.286255 –29.7

9

LMOD/ 4OO9

mavoglurant Toward ligand

DFT (B3LYP)

23,459.500490 22,441.172110 1018.286124 –26.5 [0.0] [0.0] Continued

Table 1 Single-Point (SP) Interaction Energies, at the HF/6-31G** and DFT(B3LYP)/6-31G** Levels of Theory, of Mavoglurant and MPEP in Various Protein Models—cont’d Interaction Protein Pro655 SP Energy Complex Energy Protein Energy Ligand Energy Energy Row Model Ligand Puckered State Method (a.u.) (a.u.) (a.u.) (kcal/mol)

10

LMOD/ 4OO9

mavoglurant Away from ligand

DFT (B3LYP)

23,459.492960 22,441.165873 1018.285535 –26.1 [4.7] [3.9]

11

LMOD/ 5CGC

MPEP

Toward ligand

DFT (B3LYP)

22,113.586696 21,518.746262 [0.0] [0.0]

594.828299

–7.6

12

LMOD/ 5CGC

MPEP

Away from ligand

DFT (B3LYP)

22,113.583214 21,518.742903 [2.2] [2.1]

594.828401

–7.5

13

LMOD/ 5CGD

MPEP

Toward ligand

DFT (B3LYP)

22,037.164035 21,442.324157 [0.0] [0.0]

594.828560

–7.1

14

LMOD/ 5CGD

MPEP

Away from ligand

DFT (B3LYP)

22,037.160257 21,442.320044 [2.4] [2.6]

594.828620 7.3

a We use shorthand nomenclature to describe the various protein models that we subjected to HF/6-31G** and DFT(B3LYP)/6-31G** single-point energy calculations. The terms in the “Protein Model” column (QSite/4OO9, LMOD/4OO9, LMOD/5CGC, and LMOD/5CGD) represent a two-step process, as described in Section 2, of running a particular type of calculation (named before the slash, i.e., QSite or LMOD) on a computationally prepared model that is based on a chosen X-ray structure (named after the slash, i.e., 4OO9, 5CGC, or 5CGD). In the simulation (QSite or LMOD) step, the ligand that is named in the next column was used. b Inside square brackets in some table entries, we insert relative energies expressed in kcal/mol, when comparing complexes with the two alternative Pro655 puckered states in the same protein model with the same ligand.

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overlaps extremely well with mavoglurant in the “lower” (more sterically restricted) portion of the binding cavity. In another binding mode, MPEP overlaps with mavoglurant’s pyrrolidine ring and the attached methyl ester group in the “upper” branch of the binding cavity. The more recently reported X-ray structures of mGluR5 contain, as described earlier, a narrower ligand-binding pocket which could potentially enhance our understanding of MPEP binding. We therefore herein used the PDB:5CGC and PDB:5CGD X-ray structures to further develop mGluR5-MPEP binding models (Fig. 5). The now more rigorously docked structure of MPEP binding in these cavities confirm the findings of our preliminary examination of MPEP in the PDB:5CGC and PDB:5CGD X-ray structures. The smaller MPEP ligand, unlike mavoglurant, cannot adopt an L-shaped conformation to allow the ligand to veer away from Phe788, and MPEP’s methyl group will not allow MPEP to rotate or tilt in a tight channel. These restrictions allow MPEP’s phenyl group and Phe788’s phenyl ring to adopt a snugly fitting, perpendicular stabilizing orientation. When Trps785’s side chain is located near the center of the helical bundle, thereby narrowing the binding cavity (as in the PDB:5CGC and PDB:5CGD X-ray structures), additional stabilization is afforded to MPEP, whose phenyl group would participate in hydrophobic or (nonparallel) π-stacking interactions with Trp785. MPEP’s aromatic rings deviate significantly (about 34–40 degrees) from coplanarity to avoid a collision with Trp785. Also, the hydroxyl group of Ser809 interacts through hydrogen bonds with MPEP’s aromatic nitrogen atom and Trp785’s side chain (Fig. 5). The combination of the encapsulation of MPEP in a narrow region of the binding cavity, the hydrophobic or π-stacking interactions with Trp785 and Phe788, and the formation of hydrogen bonds may explain this small ligand’s remarkable efficiency. One also notices that the MPEP binding models based on the PDB:5CGC and PDB:5CGD X-ray structures are essentially identical, and MPEP would superimpose quite well on mavoglurant with respect to mavoglurant’s methyl-phenyl-acetylenic substructure, although there appears a slight tilting of MPEP relative to mavoglurant. Also, MPEP’s phenyl group overlaps the chlorinated phenyl ring of either of the two Heptares X-ray ligands.

3.5 NAM Interactions: Mavoglurant and MPEP In the only protein model subjected to a quantum-chemical relaxation of the ligand (i.e., mavoglurant), one water molecule, and the side chains of nearby residues including Pro655, the proline ring retained its initial

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(X-ray observed) puckered state. The interaction energy (Table 1, row 1) is 11.3 kcal/mol at the HF/6-31G** level of theory. When mavoglurant is subjected to an LMOD simulation in the 4OO9-based model (Table 1, rows 2 and 3), the complex containing the toward-ligand proline-ring puckering is lower in energy, at the HF/6-31G** level, by 5.9 kcal/mol and the interaction between ligand and protein is negligibly stronger by 0.4 kcal/mol than in the complex with Pro655 adopting the alternative puckered state; the interaction energies are 10.4 and 10.0 kcal/mol, respectively. The energy of the complex and the interaction energy are functions of much more than Pro655’s ring puckered state (e.g., differences in overall geometry throughout the protein, differences in the orientations of side chains within or near the binding cavity, as well as other geometric perturbations generated during the LMOD simulation). Thus, one cannot conclude whether a proline ring puckered state is preferred, especially when the difference in complex energies is very small. In general, one would expect that the proline ring may provide additional stabilization by its interaction with the ligand and by holding the ligand more firmly in a very confined portion of the binding cavity. A Cγ hydrogen atom of Pro655 interacts with (or near) the acetylenic carbon atom attached to the ligand’s aromatic ring. When Pro655 adopts a toward-ligand puckered state in the LMOD/ 5CGC simulation with MPEP (Table 1, rows 4 and 5), the complex is more stable by 2.2 kcal/mol, at the HF/6-31G** level, than the complex with the alternative Pro655 puckering, although one should apply the same caveats mentioned earlier in the interpretation of the mavoglurant results. The positive value of the HF/6-31G** interaction energy of MPEP in either complex (rows 4 and 5) indicates a net destabilization, which we attribute to an incomplete relaxation of the binding cavity to accommodate MPEP, at least at this level of theory; in every LMOD calculation, protein backbone atoms were not allowed to relax from their initial (X-ray observed) positions, which are obviously more suited for the respective, original X-ray ligand. To accommodate MEP, further relaxation or expansion of the binding cavity may be necessary. In the LMOD/5CGD simulation with MPEP (rows 6 and 7), the HF/6-31G** interaction energies are again both positive, indicating a destabilization probably caused by excessive strain in the complexes. The LMOD/5CGD complex containing the toward-ligand puckered state of Pro655 (row 6) is again the more favored conformation (rows 6 and 7), where the complex energy difference is also 2.2 kcal/mol. Here, too, the strain in the LMOD/5CGD complex, just as in the LMOD/5CGC

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complex, probably originates from an insufficient relaxation of the protein to allow sufficient space for the ligand, MPEP. One cannot conclude from the comparison of the HF/6-31G** interaction energies in Table 1 that the experimentally (X-ray) observed puckered state of Pro655 is more appropriate for modeling MPEP and similar molecules (rows 2–7). However, it is interesting that, in all cases, the energies of the complex and protein are lower in the 5CGC-based (vs the 5CGD-based) protein model, and therefore the 5CGC-based protein model may be more appropriate for modeling MPEP and similar molecules. However, without this data, one might have assumed that a 5CGD-based protein model should perform better due to the difference in resolution of the two ˚ ) and 5CGD essentially superimposable X-ray templates, 5CGC (3.1 A ˚ (2.6 A). In the case of restrictive (or even ligand-induced) binding cavities, even small differences in binding-site topologies are sufficient to account for the better performance of one X-ray structure rather than another, for ligands with particular structural features. We also performed interaction-energy calculations at the DFT(B3LYP)/ 6-31G** level of theory (Table 1, rows 8–14), using the same ligands and protein models, for comparison to the interaction energies at the HF/6-31G** level (rows 1–7). The interaction energy of the QSite/4OO9-based complex with mavoglurant is 29.7 kcal/mol (row 8), which is 18.4 kcal/mol lower (i.e., more favorable) than the value (11.3 kcal/mol) at the HF/ 6-31G** level (row 1). With the LMOD-4OO9-based models, the DFT(B3LYP)/6-31G** interaction energy of the protein with mavoglurant is 26.5 kcal/mol (row 9) and 26.1 kcal/mol (row 10), which are both 16.1 kcal/mol more favorable than their value at the HF/6-31G** level (rows 2 and 3). At the DFT(B3LYP)/6-31G** level of theory, the mavoglurant complex containing the toward-ligand proline-ring puckering is lower in energy by 4.7 kcal/mol, which is similar to the value of 5.9 kcal/mol at the HF/6-31G** level. At the DFT(B3LYP)/6-31G** level of theory, even the interaction energies of MPEP with the LMOD/5CGC-based and 5CGD-based protein models (rows 11–14) are all negative (indicating a stabilizing protein-ligand complex) and span a narrow range of values from 7.6 to 7.1 kcal/mol; all values are approximately 10 kcal/mol lower in energy than their HF/ 6-31G** counterparts. Moreover, at the DFT(B3LYP)/6-31G** level of theory, the complex containing the Pro655 toward-ligand puckering is favored over the complex with the away-from-ligand puckering by 2.2 and 2.4 kcal/mol in the LMOD/5CGC and LMOD/5CGD calculations,

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respectively, i.e., the differences are virtually identical to their values in the HF/6-31G** calculations. Overall, one observes that the HF/6-31G** complex-energy differences are quite similar to the DFT(B3LYP)/ 6-31G** results. In all cases, the complex with the toward-ligand puckered state of Pro655 is favored by values ranging from 2.2 to 5.9 kcal/mol, and the energies of the complex and protein are lower in the 5CGC-based (vs the 5CGD-based) protein model; therefore, the 5CGC-based protein model may be more appropriate for modeling MPEP and similar molecules.

3.6 NAM Interactions: NAMs 3, 4, and 5 In our previous study,8 we computationally predicted the binding poses of NAMs 3, 4, and 5 (Fig. 1) using a protein model derived from the PDB:4OO9 X-ray structure and proposed a structure-based explanation of the pharmacological switching for their corresponding, very closely related analogs 6, 7, and 8 (Fig. 1), which are PAMs. In the current study, by considering the results described earlier, we used the PDB:5CGC-based protein model and LMOD12 to create binding models for NAMs 3, 4, and 5 (Fig. 6). The NAMs are quite superimposable, especially with respect to

Fig. 6 The computationally predicted binding poses of NAMs 3, 4, and 5 in the PDB:5CGC-based protein model. The NAMs are quite superimposable, and the cyclopropyl (NAM 3), isopropyl-sulfonyl (NAM 4), and cyclobutyl-carbonyl (NAM 5) substituents are directed into a hydrophobic pocket flanked by Ile651, Val740, Pro743, Leu744, Phe788, and Met802. MPEP is rotated with respect to the orientation of its acetylenic linker, and MPEP’s methylpyridine substituent overlaps with the phenyl ring of each of the other NAMs.

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the common phenyl-acetylene-thiazole substructure. At the opposite end of each ligand, starting from the five- or six-membered ring fused to the thiazole moiety and continuing with the capping group, the superimposition is quite good, and the cyclopropyl (NAM 3), isopropyl-sulfonyl (NAM 4), and cyclobutyl-carbonyl (NAM 5) substituents are directed into a hydrophobic pocket flanked by Ile651, Val740, Pro743, Leu744, Phe788, and Met802. The side chain of Ile651 adopts a conformation that is different from that observed in the three mGluR5 X-ray structures to accommodate the capping groups of NAMs 3, 4, and 5. In the alternative rotational state of Ile651’s side chain, the methyl and ethyl branches of the isobutyl group swing away from the capping groups of the NAMs, such that the isobutyl group avoids collisions with the capping groups, and the ethyl branch bifurcates Ile651’s N–Cα–C backbone angle. This alternative Ile conformation is commonly observed in both the Protein Data Bank9 and the Cambridge Crystallographic Database.13 In a superimposition of the binding poses of NAMs 3, 4, and 5 and MPEP (Fig. 6, top inset, showing two views differing by a rotation of about 180 degrees), the orientation of MPEP’s acetylenic linker is different from that of any of the other NAMs, although all acetylenic linkers are in the same region of the binding cavity. The tilting allows NAMs 3, 4, and 5 to avoid colliding into Phe788 and to achieve hydrophobic interactions in a pocket that is not accessible to MPEP (also see Fig. 6, bottom left inset, for a rotated view focusing on the capping groups). Also, as we discussed in our previous study,8 MPEP’s methylpyridine substituent overlaps with the phenyl ring of each of the other NAMs (Fig. 6, bottom right inset). If we were to superimpose MPEP on the other NAMs (with respect to the acetylenic linker and the “lower” six-membered aromatic ring), MPEP would lose much of its interaction with Phe788, MPEP would not access the pocket occupied by the capping groups of the other NAMs, and MPEP’s methyl group would no longer be sterically tolerated in the binding cavity. It is interesting to note, based on the data in Table 1, that the energies of the protein model and complex derived from the PDB:5CGC X-ray structure are significantly and uniformly lower, by 76.0 a.u. at the HF/6-31G** level and 76.4 a.u. at the DFT(B3LYP)/6-31G** level of theory, than those of the corresponding PDB:5CGD-derived structures, which would suggest that the PDB:5CGC X-ray structure provides a more reliable model for these studies. This conclusion would contradict one that is based on the relative experimental refinement of the (highly superimposable) X-ray struc˚ for tures, as judged by their resolution of 3.1 A˚ for PDB:5CGC vs 2.6 A PDB:5CGD. Also, despite the large single-point energy differences in

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evaluating the results of LMOD simulations in both protein models, one observes that the LMOD simulations produced almost identical (highly superimposable) MPEP binding poses and binding cavity environments. The same may be said of the LMOD binding poses of NAMs 3, 4, and 5 (Fig. 7). As a further, preliminary assessment of the binding models for compounds 3–5 presented earlier, we determined the single-point DFT(B3LYP)/6-31G** interaction energies for NAMs 3, 4, and 5 in the LMOD PDB:5CGC-based protein models with the toward-ligand puckered state of Pro655. As indicated in Table 2, NAMs 3 and 5 show considerable destabilization of 10.2 and 11.2 kcal/mol, respectively, while NAM 4 is the only one of the three with a negative but modest interaction energy (of 3.0 kcal/mol). We believe these results reflect the strain of snugly fitting the capping groups into a spatially restricted hydrophobic pocket, which is not sufficiently relaxed or expanded to adequately allow ligand binding, while the rest of each ligand structure is held firmly in position in a narrow binding cavity. The ligands of the three X-ray structures do not have chemical functionality that probe this pocket. Nevertheless, the values of these interaction energies are within the errors that may be expected when using quantum-chemical methods on structures determined by molecular mechanics approaches. It is then reasonable to predict that within a quantum chemically relaxed structural framework, these NAMs would form stable complexes with mGluR5 and would conform well to the binding cavity, whereas the PAMs, which protrude further than their NAM counterparts,

Fig. 7 A comparison of the computationally predicted binding poses of NAMs 3, 4, and 5 in protein models derived from the PDB:5CGC and PDB:5CGD X-ray structures. The LMOD simulations produced almost identical (highly superimposable) binding poses and binding-cavity environments for NAMs 3, 4, and 5, in the two protein models, as shown in superimpositions in A, B, and C, respectively. In each panel, the carbon atoms are color-coded according to the legend (in the panel) indicating the protein model (“5CGC” or “5CGD”).

Table 2 Single-Point (SP) Interaction Energies, at the DFT(B3LYP)/6-31G** Level of Theory, of NAMs 3, 4, and 5a SP Energy Method

Complex Energy (a.u.)

1

LMOD/ 5CGC

NAM 3

Toward ligand

DFT(B3LYP)

22,681.221323 21,481.484317 1199.753273 +10.2

2

LMOD/ 5CGC

NAM 4

Toward ligand

DFT(B3LYP)

24,648.637712 22,973.109904 1675.522977

3

LMOD/ 5CGC

NAM 5

Toward ligand

DFT(B3LYP)

24,806.899254 23,489.217330 1317.699701 +11.2

a

See Table 1’s footnote regarding nomenclature.

Protein Energy (a.u.)

Ligand Energy (a.u.)

Interaction Energy (kcal/mol)

Pro655 Puckered Row Protein Model Ligand State

–3.0

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would result in the costly steric repulsions that we postulate later. Such further relaxation or expansion of this pocket or the entirety of the binding cavity (to allow more flexibility for each NAM or PAM ligand) is beyond the scope of this study, but we may explore this opportunity in future investigations. Our binding models of our previous study8 revealed that docked conformations of NAMs 3, 4, and 5 (Fig. 1) differ from that of MPEP (2; Fig. 1) in the relative rotation of the two aromatic moieties attached to the acetylenic linker. In that study, which featured protein models derived from the Protein Data Bank9 entry 4OO9,6 the phenyl and pyridine rings of MPEP are nearly coplanar, while the phenyl and thiazole rings of NAMs 3 and 5 are approximately orthogonal; NAM 4 seems to be distorted. The shape of the binding cavity, which reflects the fitting of the bulky ligand mavoglurant in the X-ray structure, is at least partially responsible for the observed orthogonal conformations. However, the orthogonal relationship is not as common as near coplanarity, as suggested by the results of a search of the Cambridge Crystallographic Database13 to determine the measure of the interplanar dihedral angle of the aromatic rings attached to the acetylenic linker. Unfortunately, we could not find database entries conforming to the thiazole-containing scaffold of NAMs 3, 4, and 5. The closest match is contained twice in the entry JAPMOD, which holds the coordinates of 2,5-bis(phenylethynyl)-1,3,4-thiadiazole, and the average measure of the interplanar dihedral angle is 18.6 degrees. Instead of relying on this single database entry, we performed a search, with the conditions specified in Section 2, with two six-membered aromatic rings optionally substituted with one or more nitrogen atoms wherever appropriate, as a surrogate for NAMs 3, 4, and 5 and as a reasonably applicable representative of MPEP; see Fig. 8 for the distribution of dihedral-angle values. We found 967 database entries with 1719 structural matches of the query. The absolute value of the measure of the interplanar dihedral angle is 21.1 23.2 degrees (i.e., mean standard deviation); the median is 11.2 degrees, and the range is 0–88.6 degrees. Although crystal packing forces may be a major determinant in the observed interplanar twist, we believe that the conformational relationship would tend toward near coplanarity rather than orthogonality. Moreover, the results of the current investigation, especially for NAMs 3, 4, and 5, seem more compelling when considering the conformations observed in the previous study.8 In the current study, the absolute value of this interplanar dihedral angle ranges from 2.3 degrees to 16.7 degrees across the set of three NAMs (3, 4, and 5) in the two (PDB:5CDC-based and PDB:5CGD-based) protein models. In contrast, the absolute value of

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500 X

X

300

1

4 2

X

400 Frequency

X

X

X

3

X

X X

X

X = C or N abs( 1-2-3-4)

200

100

0 0

20 40 60 Interplanar dihedral angle (degrees)

80

Fig. 8 The frequency distribution of the interplanar dihedral angle in the Cambridge Structural Database (CSD), using the depicted search query and the options specified in Section 2. The absolute value of the measure of the interplanar dihedral angle is 21.1 23.2 degrees (i.e., mean standard deviation), the median is 11.2 degrees, and the range is 0–88.6 degrees for 1719 structural matches of the query.

this interplanar dihedral angle ranges from 34.4 degrees to 39.9 degrees for MPEP in the two (PDB:5CDC-based and PDB:5CGD-based) protein models. The greater deviation from coplanarity of the aromatic rings in MPEP is directly attributable to the steric restraints of Trp785, which is in close contact with MPEP’s phenyl ring. If rotated in either direction from the state in any of its binding poses, MPEP’s phenyl ring would collide with Trp785. Thus, the computational results reported herein seem well supported by related ligand-structural information and the binding-site characteristics of the X-ray structures. Using the results of the same Cambridge database search, for general interest, we examined the values of the two acetylenic bond angles (CdC^C and C^CdC) of the 1719 structural matches of the query. For the 3438 angles, the measure is 176.2 3.5 degrees (i.e., mean standard deviation). This value is relevant to the analysis of MPEP’s conformation. It is interesting that one of the Cambridge search hits, BIDVOA, is M-MPEP [i.e., 2-((3-methoxyphenyl)ethynyl)-6-methylpyridine], which is a selective antagonist of mGluR5. Its acetylene linker bond angles measure 177.6 degrees and 179.6 degrees, and the interplanar dihedral-angle value is 4.6 degrees.

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3.7 PAM Interactions: PAMs 6, 7, and 8 Modulators of GPCRs such as mGluRs can serve as inhibitors (NAMs) or enhancers (PAMs) of activation by their endogenous ligands. While compound 3 is a NAM, the closely related compound 6 (Fig. 1) is an activator (and similarly for the pairs of compounds 4/7 and 5/8 [Fig. 1]). To further probe our previously proposed8 structure-based explanation of the pharmacological (NAM-to-PAM) switching when replacing the NAMs 3, 4, and 5 with their corresponding, very closely related analogs (PAMs 6, 7, and 8; Fig. 1), we ran additional LMOD calculations with the PAMs using the protocol summarized in Section 2. In this exercise, we are attempting to identify which residues are likely to be involved in a protein conformational change to accommodate the PAMs within an expanded binding cavity, without computationally simulating the transition pathway to such a global change or attempting to predict the PAM-relevant protein conformation. When PAM 6 (Fig. 1) replaces NAM 3 (Fig. 9, panels A1 an A2) in the protein

Fig. 9 A structure-based explanation of the pharmacological (NAM-to-PAM) switching when replacing the NAMs 3, 4, and 5 with their corresponding, very closely related analogs (PAMs 6, 7, and 8). Very small structural changes in a NAM that occupies the depicted spatially restricted hydrophobic pocket may cause strain, which would be relieved by movement of the affected residues and attached or nearby residues. The upper (A1, B1, and C1) and lower (A2, B2, and C2) panels are paired, and each pair provides two views of a superimposed NAM and its corresponding, structurally similar PAM; the views differ by a rotation of about 180 degrees.

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model derived from the PDB:5CGC X-ray structure, perturbations are observed in Val740 and Phe788, due to the structural change from a cyclopropyl ring (in NAM 3) to a cyclopentyl ring (in PAM 6). When NAM 4 (panels B1 and B2) is structurally transformed into PAM 7 (Fig. 1), evidence of strain is observed in the distortions of Pro743, Leu744 (including the positions of the backbone atoms between Pro743 and Leu744), and (to a lesser extent) Val740, due to the structural change from an isopropyl group (in NAM 4) to a cyclohexyl ring (in PAM 7). Replacing NAM 5 (panels C1 and C2) with PAM 8 (Fig. 1) causes strain mainly in Val740, Pro743, and Leu744 (including the positions of the backbone atoms between Pro743 and Leu744), due to a structural change from a cyclobutyl ring (in NAM 5) to a 4-methyl cyclobutyl group (in PAM 8). Thus, the cluster of residues Val740, Pro743, Leu744, and Phe788 undergo increased strain when the inhibitor NAM 3 is replaced by the enhancer PAM 6, or for the replacements in the other two pairs of NAMs and PAMs. At its simplest level, this suggests that these residues serve as indicators of the relative activation state. However, over the past few years, published research has produced considerable information regarding the activation of class A GPCRs. Specifically, a key pair of conformational changes in the IC regions of transmembrane helices 5 and 6 are critical to adopting the active state. Strikingly, residues Val740, Pro743, and Leu744 reside on helix 5, and Phe788 resides on helix 6. As the same characteristics are expected to drive activation in the class A and class C (mGluR) GPCRs, the differential ligand/protein interactions initiated when switching from the NAMs to PAMs can be responsible for the precipitation of structural changes involved in protein activation.

4. CONCLUSIONS The ability of the QM/MM and MM methods to reproduce the salient features of the three available mGluR5 X-ray structures supports their use in the selection of appropriate X-ray-based models as well as the analysis of the binding and activation of other ligands. We find that common choices made for selection of X-ray structural models may not be operative. For the very important mGluR5-NAM MPEP, its central feature of an acetylenic moiety would suggest that the mavoglurant-bound mGluR5 X-ray structure, where mavoglurant also contains a central acetylenic moiety, would provide a more reliable template. Instead, we find that

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either of the other two X-ray structures with aromatic central moieties (occupying the same region) provide better structural models. Furthermore, of these two X-ray structures, the one with the lower resolution, counterintuitively, seems more reliable. Also, we found that complexes with an inward (toward-ligand) puckering of Pro655 to be slightly but consistently more stable than complexes with the outward puckering, in accordance with the Pro655 puckering observed in the three available mGluR5 X-ray structures. Overall, the emerging docking model for MPEP provides a sound explanation for the exceptional efficiency of this small ligand by means of a combination of hydrophobic and hydrogen-bond interactions in near-optimal positions. The availability of the mGluR5 X-ray structures provides an opportunity to examine a subtle and intriguing series of structure–activity relationships. By docking three pairs of chemically closely related ligands that have opposite effects (NAM vs PAM switching), we have identified a cluster of protein residues that, for all three pairs of ligands, experiences greater steric repulsions from interactions with the PAM structures than with their NAM counterparts. Moreover, this cluster seems to serve as a pressure point for activation enhancement. This cluster of residues resides in the upper (EC) part of the 7TM domain and is located on helices 5 and 6. In the case of class A GPCRs, it has become well-established over the past few years that conformational changes in the IC regions of these helices are associated with GPCR activation by agonists. Furthermore, changes in the EC-orthosteric site at these same helices, induced by ligand binding, correlate with the IC-region activation state. This suggests that the allosteric modulation of mGluRs operates via protein characteristics that are analogous to those in the orthosteric activation of class A GPCRs. This could be brought about by the influence of the 7TM NAM/PAM binding on the VFT orthosteric binding at mGluR5, which would be analogous to a similar mechanism recently identified for a class F protein, based on its X-ray structure.28 Alternatively, the 7TM binding by these NAM/PAM compounds may bypass the orthosteric-binding site control mechanism and directly activate mGluR5 in a manner similar to the operation of class A GPCRs. This latter potential mechanism is supported by studies demonstrating29 that a truncated mGluR (mGluR5), lacking the entire EC domain, can be fully activated. Moreover, this structure-based approach can provide a tool for more effective NAM/ PAM compound design.

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REFERENCES 1. Aviram, A.; Ratner, M. A. Molecular Rectifiers. Chem. Phys. Lett. 1974, 29, 277. 2. Topiol, S.; Sabio, M. The Role of Experimental and Computational Structural Approaches in 7TM Drug Discovery. Expert Opin. Drug Discov. 2015, 10, 1071. 3. Palczewski, K.; Kumasaka, T.; Hori, T.; Behnke, C. A.; Motoshima, H.; Fox, B. A.; Le Trong, I.; Teller, D. C.; Okada, T.; Stenkamp, R. E.; Yamamoto, M.; Miyano, M. Crystal Structure of Rhodopsin: A G Protein-Coupled Receptor. Science 2000, 289, 739. 4. Rasmussen, S. G.; Choi, H.-J.; Rosenbaum, D. M.; Kobilka, T. S.; Thian, F. S.; Edwards, P. C.; Burghammer, M.; Ratnala, V. R. P.; Sanishvili, R.; Fischetti, R. F.; Schertler, G. F. X.; Weis, W. I.; Kobilka, B. K. Crystal Structure of the Human β2 Adrenergic G-Protein-Coupled Receptor. Nature 2007, 450, 383. 5. Packiarajan, M.; Grenon, M.; Zorn, S.; Hopper, A. T.; White, A. D.; Chandrasena, G.; Pu, X.; Brodbeck, R. M.; Robichaud, A. J. Fused Thiazolyl Alkynes as Potent mGlu5 Receptor Positive Allosteric Modulators. Bioorg. Med. Chem. Lett. 2013, 23, 4037. 6. Dore, A. S.; Okrasa, K.; Patel, J. C.; Serrano-Vega, M.; Bennett, K.; Cooke, R. M.; Errey, J. C.; Jazayeri, A.; Khan, S.; Tehan, B.; Weir, M.; Wiggin, G. R.; Marshall, F. H. Structure of Class C GPCR Metabotropic Glutamate Receptor 5 Transmembrane Domain. Nature 2014, 511, 557 (plus extended data). 7. Christopher, J. A.; Aves, S. J.; Bennett, K. A.; Dore, A. S.; Errey, J. C.; Jazayeri, A.; Marshall, F. H.; Okrasa, K.; Serrano-Vega, M. J.; Tehan, B. G.; Wiggin, G. R.; Congreve, M. Fragment and Structure-Based Drug Discovery for a Class C GPCR: Discovery of the mGlu5 Negative Allosteric Modulator HTL14242 (3-Chloro-5-[6-(5Fluoropyridin-2-yl)Pyrimidin-4-yl]Benzonitrile). J. Med. Chem. 2015, 58, 6653. 8. Topiol, S.; Sabio, M. 7TM X-ray Structures for Class C GPCRs as New DrugDiscovery Tools. 1. mGluR5. Bioorg. Med. Chem. Lett. 2016, 26, 484. 9. Berman, H. M.; Westbrook, J.; Feng, Z.; Gilliland, G.; Bhat, T. N.; Weissig, H.; Shindyalov, I. N.; Bourne, P. E. The Protein Data Bank. Nucleic Acids Res. 2000, 28, 235. the URL of the RCSB PDB is www.rcsb.org. 10. Sastry, G. M.; Adzhigirey, M.; Day, T.; Annabhimoju, R.; Sherman, W. Protein and Ligand Preparation: Parameters, Protocols, and Influence on Virtual Screening Enrichments. J. Comput. Aided Mol. Des. 2013, 27, 221. odinger, LLC: New York, 11. Schr€ odinger Release 2016–1; Maestro, version 10.5.014, Schr€ NY, 2016. 12. Kolossva´ry, I.; Guida, W. C. Low-Mode Conformational Search Elucidated: Application to C39H80 and Flexible Docking of 9-Deazaguanine Inhibitors into PNP. J. Comput. Chem. 1999, 20, 1671. 13. Allen, F. H. The Cambridge Structural Database: A Quarter of a Million Crystal Structures and Rising. Acta Crystallogr. B 2002, 58, 380. ConQuest version 1.18 (Build RC2), CSD version 5.37 with updates through May 2016. 14. Duan, Y.; Wu, C.; Chowdhury, S.; Lee, M. C.; Xiong, G.; Zhang, W.; Yang, R.; Cieplak, P.; Luo, R.; Lee, T.; Caldwell, J.; Wang, J.; Kollman, P. A Point-Charge Force Field for Molecular Mechanics Simulations of Proteins Based on Condensed-Phase Quantum Mechanical Calculations. J. Comput. Chem. 2003, 24, 1999. 15. Murphy, R. B.; Philipp, D. M.; Friesner, R. A. A Mixed Quantum Mechanics/Molecular Mechanics (QM/MM) Method for Large-Scale Modeling of Chemistry in Protein Environments. J. Comput. Chem. 2000, 21, 1442. 16. Hartree, D. R.; Hartree, W. Self-Consistent Field, with Exchange, for Beryllium. Proc. Roy. Soc. London 1935, A150, 9. osung des quantenmechanischen Mehrk€ orper17. Fock, V. N€aherungsmethode zur L€ problems. Z. Phys. 1930, 61, 126.

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18. Hehre, W. J.; Ditchfield, R.; Pople, J. A. Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular-Orbital Studies of Organic Molecules. J. Chem. Phys. 1972, 56, 2257. 19. Bochevarov, A. D.; Harder, E.; Hughes, T. F.; Greenwood, J. R.; Braden, D. A.; Philipp, D. M.; Rinaldo, D.; Halls, M. D.; Zhang, J.; Friesner, R. A. Jaguar: A High-Performance Quantum Chemistry Software Program with Strengths in Life and Materials Sciences. Int. J. Quantum Chem. 2013, 113, 2110. 20. Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864. 21. Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133. 22. Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648. 23. Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. J. Phys. Chem. 1994, 98, 11623. 24. Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; DeFrees, D. J.; Pople, J. A.; Gordon, M. S. Self-Consistent Molecular Orbital Methods. XXIII. A Polarization-Type Basis Set for Second-Row Elements. J. Chem. Phys. 1982, 77, 3654. 25. Wu, H.; Wang, C.; Gregory, K. J.; Han, G. W.; Cho, H. P.; Xia, Y.; Niswender, C. M.; Katritch, V.; Meiler, J.; Cherezov, V.; Conn, P. J.; Stevens, R. C. Structure of a Class C GPCR Metabotropic Glutamate Receptor 1 Bound to an Allosteric Modulator. Science 2014, 344, 58. 26. Nemethy, G.; Gibson, K. D.; Palmer, K. A.; Yoon, C. N.; Paterlini, G.; Zagari, A.; Rumsey, S.; Scheraga, H. A. Energy Parameters in Polypeptides. 10. Improved Geometrical Parameters and Nonbonded Interactions for Use in the ECEPP/3 Algorithm, with Application to Proline-Containing Peptides. J. Phys. Chem. 1992, 96, 6472. 27. Saha, I.; Shamala, N. Investigating Proline Puckering States in Diproline Segments in Proteins. Biopolymers 2013, 99, 605. 28. Byrne, E. F. X.; Sircar, R.; Miller, P. S.; Hedger, G.; Luchetti, G.; Nachtergaele, S.; Tully, M. D.; Mydock-McGrane, L.; Covey, D. F.; Rambo, R. P.; Sansom, M. S. P.; Newstead, S.; Rohatgi, R.; Siebold, C. Structural Basis of Smoothened Regulation by Its Extracellular Domains. Nature 2016, 535, 517. 29. Goudet, C.; Gaven, F.; Kniazeff, J.; Vol, C.; Liu, J.; Cohen-Gonsaud, M.; Acher, F.; Prezeau, L. Heptahelical Domain of Metabotropic Glutamate Receptor 5 Behaves Like Rhodopsin-Like Receptors. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 378.

CHAPTER SEVEN

Singlet Fission: Optimization of Chromophore Dimer Geometry Eric A. Buchanan*, Zdeněk Havlas*,†, Josef Michl*,†,1 *University of Colorado, Boulder, CO, United States † Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 1.1 Why Is Theoretical Work on SF Important? 1.2 A Closer Look at SF 2. The HOMO/LUMO Model 2.1 Definition 2.2 Expressions for TRP 2.3 A Simple Approximation for TRP 2.4 Local Maxima of jTRPj2 in the 6-D Space of Rigid Dimer Geometries 2.5 State Energies in the 6-D Space of Rigid Dimer Geometries 3. Applications 3.1 Two Ethylene Molecules 3.2 Two Clipped Cibalackrot Molecules 4. Outlook Acknowledgments Appendix References

176 176 179 186 186 189 195 199 202 205 205 206 211 212 212 223

Abstract After a brief review of electronic aspects of singlet fission, we describe a systematic simplification of the frontier orbital (HOMO/LUMO) model of singlet fission and Davydov splitting in a pair of rigid molecules. In both instances, the model includes electron configurations representing local singlet excitation on either chromophore, charge transfer in either direction, and triplet excitation in both chromophores (biexciton). The resulting equations are simple enough to permit complete searches for local extrema of the square of the electronic matrix element and to evaluate the effect of intermolecular interactions on the exoergicity of singlet fission and on the biexciton binding energy in the six-dimensional space of rigid dimer geometries. The procedure is illustrated on results for the six best geometries for dimers of ethylene and of an indigoid heterocycle with 24 carbon, nitrogen, and oxygen atoms.

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1. INTRODUCTION The present chapter represents a generalization of our earlier efforts1 and describes a simplified method for the complete approximate evaluation of the effect of rigid dimer geometry on the rate of singlet fission (SF). It is illustrated by application to two examples. The purpose of the procedure is not the prediction of absolute rates of SF, but the prediction of all approximate regions of dimer geometries at which the rate is locally maximized. Some references to other work on SF are provided, but a comprehensive review is not attempted.

1.1 Why Is Theoretical Work on SF Important? In recent years, SF has become a popular subject of investigation, since it provides an intriguing intellectual challenge and at the same time offers a possible path toward overcoming the Shockley–Queisser limit2 to the efficiency of a single-junction solar cell while not requiring any expensive current matching typical of multijunction cells. In the simplest description, SF3,4 is a process in which a singlet excited molecule transfers some of its energy to a ground-state neighbor and both end up in their triplet states, which then separate and become independent. Since it occurs in the excited singlet state (usually the lowest one), whose lifetime is typically measured in ns, SF is easily detectable only if it occurs on a time scale of ns as well, and dominant only if it occurs on a time scale that is much shorter. Initially, the two triplets formed in SF are coupled into an overall singlet, making the process spin-allowed and potentially as fast as other types of spinallowed energy transfer (ET) between molecules in contact. It thus is the fastest way to produce triplet from singlet excited states without relying on very strong spin–orbit coupling (i.e., without flipping spins). In a few compounds, half a dozen years ago SF was found to outcompete all other decay channels and produce a nearly 200% yield of triplet states.5–8 Used in a solar cell, such ideal material could in principle produce twice the current at half the voltage, which in itself does not offer any advantage. However, when lowenergy photons transmitted by the SF-capable material are captured by a subsequent layer of an ordinary solar cell material, the theoretical maximum efficiency rises from the Shockley limit of roughly 1/3 to a value close to 1/2.9 Unfortunately, only a handful of materials are known to perform SF with a triplet yield anywhere near 200% and even fewer are practical, primarily because most are too sensitive to the combined action of light and

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atmospheric oxygen. A useful SF material must not only have a triplet yield very close to 200% but also needs to satisfy many additional requirements. Some of the most obvious ones are a long-lived and mobile lowest triplet state, preferably at 1–1.1 eV above the singlet ground state, and a lowest excited singlet state at somewhat more than twice the triplet excitation energy (2–2.3 eV), with strong absorption everywhere above this threshold. Redox properties must match the junction material used for charge separation in order to incur minimal losses. There should be no charge separation from the excited singlet, only from the triplet. The material must have desirable adhesive and/or mechanical properties and must be inexpensive. Poisonous elements are to be avoided. The list could be continued. Clearly, it would be best if a large selection of SF materials were available. The desire to deduce structural guidelines for the design of new efficient SF materials thus adds a practical motivation to the interesting intellectual challenge posed by the presently rather limited understanding of the mechanistic details of the SF process. Useful guidelines for finding superior SF materials need to address at least two types of principal problems: (i) selection of an optimal chromophore and (ii) selection of an optimal mutual disposition of the chromophores in space. Facile separation of the two triplets could be added to the list. (i) A search for optimal chromophores is based on the requirement that SF be isoergic or preferably10–12 slightly exoergic in order to be fast and competitive, and this implies a rarely satisfied relation between the singlet (S1) and triplet (T1) excitation energies, E(S1) 2 E(T1). It was recognized early on13 that members of two overlapping groups of chromophores can be theoretically expected to meet this condition: biradicaloids (derived from a perfect biradical by a relatively weak covalent perturbation14) and large benzenoid hydrocarbons. The case of biradicaloids has been elaborated in considerable detail.15,16 It is unfortunate that biradicaloid structures frequently have high chemical reactivity and sensitivity to traces of oxygen. The classical SF chromophores, tetracene and pentacene, as well as the first successful chromophore derived from theoretical considerations,13 1,3-diphenylisobenzofuran,6,17–19 have biradicaloid character and are highly reactive. Additional biradicaloids identified by computations as potentially interesting for SF20–22 have so far not been tested experimentally. Most of them were captodatively stabilized and carry an acceptor and a donor group on each radical center.

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In recent years the family of highly efficient SF materials with a triplet yield above an arbitrarily set limit of 170% has grown beyond the carotenoid, all-trans-3R,30 R-zeaxanthin,7 the biradicaloid, 1,3-diphenylisobenzofuran,6 and the classical parent and variously substituted or doubled polyacenes, tetracene8 and pentacene,23–28 to shorter (triisopropylsilylethynylanthracenes29) and longer (hexacene12) members of the series, and to derivatives of terrylene,30,31 diketopyrrolopyrrole,32,33 and a quinoidal bithiophene.34 (ii) The optimal arrangement of the chromophores in space has been of considerable interest and is well recognized as critical for high triplet yields.33,35–40 Theoretical guidance clearly is very valuable. For example, general consideration of the simplified diabatic HOMO/LUMO model3,4 described in more detail later and numerical calculations in adiabatic framework41 suggested that an exactly stacked pair of chromophores would be less favorable than a slip-stacked arrangement with the slip in the direction of the HOMO–LUMO transition moment. (this slip removes a plane of symmetry relative to which the S1 electronic function is antisymmetric and the 1(TT) electronic function is symmetric, making it impossible for a totally symmetric electronic Hamiltonian to perturb the former into the latter not only in the HOMO/LUMO model but at any level of approximation unless an antisymmetric vibration is present in the total wave function of the initial or the final state; this vibronic effect is likely to be weak). An approximate general formula based on the HOMO/LUMO model has since appeared4,11 for the electronic matrix element TRP (R ¼ S1 is the reactant and P ¼ 1(TT) is the product) as a function of the mutual disposition of two chromophores in space. The formula was subsequently elaborated in a paper that also formulated a simple explicit rule for the optimal geometries in the model system of two ethylenes,1 cf. Section 3.1. In the present chapter, we first provide a qualitative description of the overall SF process as it is presently understood. Subsequently, we focus on two aspects of SF related to electronic structure, leaving aside many important issues such as dynamics.10,11,42 Specifically, we consider the optimization of the mutual disposition of two SF chromophores in space, both with respect to maximizing the electronic matrix element for SF and to minimizing the potentially deleterious effect of Davydov splitting. We do this within the framework of the HOMO/LUMO approximation and in an Appendix provide full exact expressions for the matrix elements of the interaction Hamiltonian within this model, with inclusion of intermolecular overlap. We summarize the derivation of the approximate formula1 for the SF

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electronic matrix element TRP in a dimer. Its simplicity permits complete searches of the relevant parts of the six-dimensional (6-D) space defined by the location and orientation of two adjacent rigid bodies, even though the identification of all local maxima of jTRPj2 required its evaluation at more than half a billion geometries. We next derive a similar formula for the effect of intermolecular interactions on the exothermicity of the SF process and show how these interactions can discriminate against some of the local maxima of jTRPj2. We also note that an analogous approximate formula can be written for the binding energy of the biexciton. Finally, we provide examples of the application of the results to SF in a model system of ethylene dimer and in a realistic system of a dimer of a heterocycle related to indigo. Although in most cases the present limitation to only two chromophore molecules does not describe reality, it seems to us that there is considerable merit in attempts to understand dimers, which could provide a useful qualitative guide to structures that are optimal for SF in general.

1.2 A Closer Look at SF There are two distinct ways of viewing SF. In the adiabatic description,43 one starts with electronic states of the dimer or a larger conglomerate, and the transition to the final states is caused by the non-Born-Oppenheimer terms in the Hamiltonian that induce nonradiative transitions from one surface to another. In the diabatic framework,3,10,44 which we are using for the present description of the SF process, one starts with electronic states of the individual partners, in the simplest case obtained by diagonalization of the Hamiltonian of a single molecule. Then, a transition from the initial to the final state is induced by the intermolecular terms in the electronic Hamiltonian of the total system. The diabatic description is more commonly used for the discussion of experimental results and its merits have been discussed in detail.10 It has the practical advantage that calculations of the starting and final states are performed for a system that is only half the size of the total system in the case of a dimer, and less in the case of a larger conglomerate. Also, it permits the use of qualitative concepts that are familiar from theories of charge45,46 and energy47–50 transfer, such as Marcus theory and F€ orster theory. A disadvantage of the diabatic description is that the definition of the initial state for SF usually is not as simple as in a solution, where diffusion brings two initially truly isolated partners together. In most instances, e.g., in a crystal, the partners participating in SF interact from the very start. A diabatic treatment then often begins with an adiabatic calculation of a dimer or higher aggregate followed by a diabatization to produce the desired diabatic basis.51–53

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In the diabatic treatment, the expression for the rate of SF W(RP) is the Fermi golden rule, W ðRPÞ ¼ h1 TRP 2 ρðE Þ

(1)

where the electronic matrix element TRP reflects the intermolecular electronic interaction between the initial reactant state R and the product double triplet biexciton state P and ρ(E) is the density of states in the product state at the energy E at which it is generated. Both terms are critically important, but in the present text, we primarily deal with TRP and only at the end consider ρ(E). In most cases the description of SF in terms of just two interacting molecules and a single-step process is grossly oversimplified. A more realistic schematic picture is provided in Fig. 1. It is still incomplete in that it shows only two chromophores, and this limitation applies throughout the present text. The effects of delocalization of the initial singlet excited state over a larger number of molecules in larger aggregates and in crystals by excitonic and charge-transfer interactions have been investigated by many authors and we provide only a few leading references.42,54–61 D+ A

B

A

B

D+ S1 D– T1 S0

D+ D+ S1 1.2.1 S1 D– D– hv T1 T1 S0 S0

D+ S1 D– T1 S0

A D+ S1 D– T1 S0

S1

D–

T1

S0

A B

D+ D+ S1 S1 A B A 1.2.1 A B D– D– D+ D+ D+ D+ D+ T1 T1 S1 S1 1.2.2 S0 S0 1.2.2 S1 S1 1.2.4 S1 D– D– D– D– D– T1 T1 T1 T 1 T1 D+ D+ S0 S0 S1 S1 S0 S0 S0 B 1.2.1 D– D– D+ 1.2.3 T1 T1 S1 S0 S0 hv D– T1 1.2.5 S0

B D+ S1 D– T1 S0

Fig. 1 Top right: symbolic representation of electronic states of partners A and B in SF. Center: Sequence of events (competing decay paths not shown). Possible (black) and actually occupied (red) electronic states in frames (blue, real species; red, a species that can be virtual or real). Narrow frames: separated partners, wide frames: partners in contact. Top path: SF in solution; bottom path: SF in crystal, aggregate, or dimer (in covalent dimers, the last step on the right is impossible). All steps are reversible and the sections in which they are discussed are stated.

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Fig. 1 indicates that each of the chromophores could be in the ground state S0, in the lowest triplet state T1, in the lowest excited singlet state S1, in the lowest doublet state of the radical cation D+, or in the lowest doublet state of the radical anion D. For chromophores of interest, the excitation energy of the S1 state is about twice that of the T1 state. To conserve the total electron count, if one of the partners is in its D+ state, the other one must be in its D state. Fig. 1 shows the sequence of events during SF and refers the reader to the sections where they are discussed, and it also shows a schematic pictorial representation of the dominant electronic configuration in each state (other contributing configurations are not shown). It applies in the limit of complete localization of initial electronic excitation on one of the partners. It is easily modified to describe the other limit, in which electronic excitation energy is delocalized equally over the two partners. They can then be in the lower or the upper singlet excimer state (S1S0 S0S1), in the lower or the upper triplet excimer state (T1S0 S0T1), in one of the biexcitonic states 1,3,5 (T1T1), in the charge-separated states D+D DD+, or in a more or less complicated superposition of these states. 1.2.1 Initial Excitation The initial excitation normally occurs by absorption of a photon, which converts S0 to S1 (or a higher excited singlet, which then usually rapidly decays to S1). Fig. 1 shows that this process may excite a single chromophore, e.g., in solution, where diffusion may later provide an opportunity for an encounter and subsequent SF. Most SF experiments deal with solids, where the initial excitation is normally delocalized over several chromophores. In a pair with delocalized states, the allowed excimer state (S1S0 + S0S1) is reached initially. It is the higher of the two excimer states if the two partners are in a geometry typical of H aggregates (transition dipole moment vectors stacked) and the lower one if they are in a geometry typical of J aggregates (transition dipole moment vectors approximately parallel to a straight line). In dimers and small aggregates, excitation may be localized or delocalized, depending on geometry and the strength of interchromophore coupling.62 Coherent excitation of two or more states (not shown in Fig. 1) has been proposed in an effort to interpret the results of time-resolved two-photon photoionization experiments on crystals of polyacenes.54 An ultrashort laser pulse excites a coherent superposition of the initial S1S0, final 1(TT), and mediating D+D states. After a short time interaction with the phonon bath destroys the quantum coherence to produce the 1(TT) biexciton state or the S1S0 state, which then behaves as if it were produced in an ordinary

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absorption process by itself and proceeds to perform SF to yield an additional portion of the 1(TT) state. 1.2.2 The First SF Step (or Two?) The conversion of the excited singlet state S1 or the lower excimer state, (S1S0 + S0S1) at the J geometry or (S1S0 S0S1) at the H geometry, and at times even a higher electronic excited state, into the 1(TT) biexciton is the key process in SF. Depending on its rate, it may proceed before the excess vibrational energy usually present in the initially prepared electronically excited state is lost to the surroundings, concurrently with this loss, or only after vibrational equilibration. This provides ample opportunity for multiexponential kinetics. A simplified schematic representation is provided in Fig. 2. Fig. 2 is only symbolic in that it shows the dominant electron configurations, while in fact others contribute as well. The bottom shows frontier orbital occupancies and spins in the localized singlet configurations with one or the other chromophore excited, and in the biexciton configuration in the center. On the top, the charge-separated configurations are shown. D+

D–

D–

9

1

S1

2

5

3

S0

4

6

7

D+

T1

T1

8

S0

S1

›

›

›

›

›

›

›

›

10 1 = 〈S0S1|H|D+D–〉, 3 = 〈S1S0|H|D+D–〉, 5 = 〈D+D–|H|T1T1〉, 7 = 〈S1S0|H|T1T1〉

›

›

2 = 〈S1S0|H|D–D+〉, 4 = 〈S0S1|H|D–D+〉, 6 = 〈D–D+|H|T1T1〉, 8 = 〈S0S1|H|T1T1〉, 9 = 〈D+D–|H|D–D+〉, 10 = 〈S1S0|H|S0S1〉

Fig. 2 Electron configurations of a dimer important for SF, and interactions between them. Solid lines: potentially strong interactions; broken lines: weak interactions, dependent on repulsions between charge densities defined by products of orbitals located on different molecules.

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Potentially strong interactions between configurations are shown as solid lines and weak interactions as broken lines. The associated interaction Hamiltonian matrix elements are shown at the bottom. There are two ways (mechanisms) in which formation of the biexciton can proceed. (i) The usual process involves a single step and no intermediate. In the diabatic description and starting in a localized excited singlet state, its rate is proportional to TRP 2 (cf. Eq. 1), where in the usual simple approximation TRP contains three contributions. A usually small one is due to a direct interaction of the initial with the final state and is indicated by a red arrow in Fig. 2. Two often large ones, which can be of mutually opposite signs, are due to mediated interactions indicated by cyan and blue arrows. The two mediating states that provide the interaction (superexchange) paths are the charge-transfer configurations D+D and DD+ shown in Fig. 2 at the top. In these, an electron has been removed from one or the other chromophore and added to its partner. To the best of our knowledge, the mediated path was first proposed half a century ago.63 Although some authors still argue that it is not clear whether the direct or the mediated contribution to the matrix element is more important, there is only one case that we are aware of in which the two have been carefully separated in a computation and the direct contribution was found to dominate over the algebraic sum of the mediated ones (HOMO/LUMO approximation for a stacked pair of tetracenes slipped significantly along the long axis and only 0.2 A˚ along the short axis; the total value of the electronic matrix element was only 10 meV); we have not been able to reproduce this result and find that the mediated term exceeds the direct term by three orders of magnitude).64 The interference between the two mediated paths has interesting implications for the dependence of TRP on the choice of the mutual disposition (distance and orientation) of the two chromophores; at the best geometries the interference is constructive. A way to suppress one of the paths and thus minimize a destructive interference is to make the two partners inequivalent and the D+D and DD+ states thus very different in energy (cf. Section 1.2.5). (ii) In rare instances the energy of one or both charge-transfer configurations can be so favorable that they no longer represent virtual states but become observable real states with finite lifetimes (minima in the lowest excited singlet potential energy hypersurface). Then, the formation of a biexciton can proceed by a two-step mechanism with one of the charge-transfer species as an observable intermediate.65,66 This

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intermediate appears to have deactivation channels such as intersystem crossing to yield one triplet molecule and internal conversion to yield the ground state. It is therefore unlikely that two-step SF conversions will be optimal when one aims for a high triplet yield. 1.2.3 The Biexciton Since SF is not complete until the two resulting triplets have become independent, the initially formed triplet pair (biexciton) generally needs to be viewed as a possibly detectable intermediate 1(TT) that can proceed in several directions. If its binding energy is sufficient or if it occurs in a covalent dimer, it may be separately observable and kinetically significant, making the SF process a two-step event. First of all, the initially reached 1(TT) state of the biexciton is only one of the nine levels that result from the coupling of two triplets. They can be thought of as a singlet 1(TT), the three components of a triplet 3(TT), and the five components of a quintet 5(TT). They are mixed by a small tensor term in the spin Hamiltonian, magnetic dipole–magnetic dipole interaction, familiar from electron spin resonance of triplet states and jointly with spin–orbit coupling responsible for their zero-field splitting (D and E terms in EPR spectroscopy). Quantum beats in delayed fluorescence are observed67 and the quintet state has been observed directly by time-resolved EPR spectroscopy.68 The description of the S1S0–1(TT)–3(TT)–5(TT) interconversion cannot be provided by standard kinetic expressions but requires a density matrix approach. When the two partners are equivalent by symmetry, the situation is simplified in that 1(TT) does not directly couple to 3(TT) but only to 5(TT). A recent summary of the coupling of spins of four electrons is available.69 In the presence of an outside magnetic field, a Zeeman term also contributes to the mixing of the nine sublevels. This provides an opportunity for significant magnetic field effects on SF, and their study played an important role in investigations of SF when it was first discovered.70,71 The biexciton can return to the initial S1S0 state but it can also complete the SF process by overcoming its binding energy and dissociating to two independent triplets T + T (Fig. 1). Their spins may remain coherent for some time and they can encounter each other again either before or after complete loss of spin coherence by spin–lattice interaction. Little is known about structural effects on the biexciton binding energy, which is related to the energy difference between the three substates, singlet, triplet, and quintet. Simple expressions for the usually only slightly different energies of the 1 (TT), 3(TT), and 5(TT) configurations are available,3 but the effect of interaction with higher energy configurations of the three multiplicities, which

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would presumably stabilize the singlet the most and the quintet the least by energy arguments alone, is difficult to guess without a detailed computation. The importance of inclusion of usually unknown biexciton binding energy in the evaluation of the overall yield of free triplets in SF was emphasized in a simple kinetic model proposed a few years ago.72 Internal conversion of the biexciton to the only lower singlet, S0S0, looks improbable because of a large energy gap, but it sometimes occurs quite rapidly and reduces the yield of free triplets in a presently unpredictable way. Its mechanism is not well understood and it is possible that it occurs via a specific “photochemical” decay path through a conical intersection (e.g., those used in 2 + 2 or 4 + 4 dimerization). Moreover, to the degree that the biexciton has developed 3(TT) character, fast internal conversion to the lower energy TS0 state is always a possibility. Acquisition of 5(TT) character normally does not open a similar additional decay channel via locally excited Q1S0 states, which are typically too high in energy (agreement with this statement is not universal68). In general, it seems to us fair to say that the structural factors that dictate the outcome of the competition between dissociation of the biexciton state into two independent triplets and other decay paths are very important for the overall yield of free triplets in SF but are not well understood. 1.2.4 Separated Triplet Pair The 1(TT) to T + T separation (Fig. 1) is usually competitive in crystals, aggregates, or polymers, due to facile triplet hopping from chromophore to chromophore, or even in solution, due to diffusion. It is generally favored by entropy.72,73 In isolated covalent dimers, the separation cannot occur and the SF process stops at the stage of a biexciton, which ultimately follows one of its other decay channels. Among the properties of the free triplets, long lifetime and high mobility are essential for any practical applications in solar cells. 1.2.5 Heterofission So far, we have tacitly assumed that the two chromophores involved in SF are the same, and in most cases studied so far that was indeed so (homofission). The two partners can also be different. Even if they are the same chemical species, the geometry of the pair may make them inequivalent. They can however also be different chemical species altogether (heterofission). Unless the excitation energies of the partners are matched, the SF process will then involve some conversion of electronic into vibrational energy and finally into heat, and cannot be fully efficient. If the partners do have the same singlet as well as triplet excitation energies, there will be no efficiency penalty, and this is the case shown in Figs. 1 and 2.

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The most interesting case is encountered when one of the partners acts as an electron donor and the other as an electron acceptor, such that the D+D and DD+ states have very different energies. This can result in a significant advantage for the rate of SF, since the possibly destructive interference between the two paths mediated by these charge-transfer states discussed in Section 1.2.2 can be greatly diminished (cf. “charge-transfer polymers”74,75).

2. THE HOMO/LUMO MODEL 2.1 Definition The HOMO/LUMO model of SF3,4 treats explicitly only electrons in the frontier orbitals on each partner (HOMO is the highest occupied and LUMO is the lowest unoccupied MO in the ground configuration). It is assumed that the initial singlet state on each partner is well described as a HOMO–LUMO excitation, and that electrons in molecular orbitals of lower energy than the HOMO on each partner can be treated as a rigid core. This assumption is most easily fulfilled when the HOMO to LUMO excited state is S1, but it could also be met when this state is S2 (or an even higher excited singlet), if the absorbed photon has sufficient energy and if SF is faster than internal conversion to S1, as is the case in certain carotenoids.76 The HOMO/LUMO model has seen much use over the past decades for many purposes; see for instance Refs.63,77–82 Even under the best of circumstances, the description of the S1 state of a chromophore as a HOMO–LUMO excitation and the limitation of the active space to only two orbitals on each partner are only approximate. Although the HOMO/LUMO model therefore cannot be exactly correct, it is appealing conceptually and it has seen wide use in SF studies. In particular, it was employed very successfully in the first microscopic dynamical calculations of SF in polyacenes.10–12 It is also supported by the results of a treatment of tetracene and pentacene dimers by active space decomposition,44 and we believe that it represents a good starting point for even simpler treatments that are needed if thorough searches of the sixdimensional space of relative geometries of two rigid bodies are to be made, as described later. It is assumed that the two interacting chromophores A and B have equal excitation energies, but they may differ in their reduction and oxidation potentials. The singlet ground state of the chromophore pair is represented by a Slater determinant constructed from doubly occupied orbitals, S0 A S0 B ¼ S0 S0 . In this notation the state of chromophore A is on the left and that of chromophore B is on the right (cf. Fig. 1). We consider the ground

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state S0, the lowest excited singlet state S1, and the lowest triplet state T1 of each partner, and the two lowest-energy charge-separated states 1D+D and 1DD+, in which an electron is transferred from A to B or from B to A, respectively (Fig. 1). The singlet excited states of A and B result from electron promotion from the HOMO hA or hB into the LUMO lA or lB, respectively. When we need to refer to an unspecified general state of partner A or B, we use UA and UB. The six-dimensional singlet state subspace treated explicitly in the model allows local excitations in either chromophores, a simultaneous triplet excitation in both chromophores, and the charge-transfer (CT) 1D+D and 1 + D D excitations: S0 S0 ¼ NS0S0 jhA α hA β hB α hB βj

(2)

S1 S0 ¼ NS1S0 21=2 ðjhA α lA β hB α hB βj jhA β lA α hB α hB βjÞ

(3)

S0 S1 ¼ NS0S1 21=2 ðjhA α hA β hB α lB βj jhA α hA β hB β lB αjÞ

(4)

D + D ¼ N + 21=2 ðjhA α lB β hB α hB βj jhA β lB α hB α hB βjÞ

(5)

D D + ¼ N + 21=2 ðjhA α hA β lA α hB βj jhA α hA β lA β hB αjÞ

(6)

1 1

1

T1 T1 ¼ NT1T1 31=2 ½jhA α lA α hB β lB βj + jhA β lA β hB α lB αj 1⁄2ðjhA α lA β hB α lB βj + jhA α lA β hB β lB αj + jhA β lA α hB α lB βj + jhA β lA α hB β lB αjÞ

(7)

The normalization factor 1/√(N!) for a Slater determinant is included implicitly (N ¼ 4 is the number of electrons in the active space). In the wave functions (2)–(7), core electrons are not shown. Because orbitals hA and lA are not orthogonal to orbitals hB and lB, the normalization factors Nx depend on intermolecular overlap integrals: ⁄ NS0S0 ¼ ðShAhB 1Þ2 (8) ⁄ NS1S0 ¼ 2ShAhB 2 SlAhB 2 ShAhB 2 SlAhB 2 + 1 (9) ⁄ (10) NS0S1 ¼ 2ShAhB 2 ShAlB 2 ShAhB 2 SlAhB 2 + 1 ⁄ N + ¼ ShAhB 2 + ShAlB 2 + 1 (11) ⁄ N + ¼ ShAhB 2 + SlAhB 2 + 1 (12) 1

2

1

1

1

1

2

2

2

2

1⁄2 NTT ¼ ðShAhB SlAlB ShAlB SlAhB Þ2 + 1⁄2 ShAhB 2 + ShAlB 2 + SlAhB 2 + SlAlB 2 + 1

ð where Sab ¼ aðr1 Þbðr1 Þdr1 .

(13)

The Hamiltonian matrix is D D D E 3 1 + E 1 + E 1 ^ S0 S0 ^ S1 S0 ^ S0 S1 ^ D D ^ D D ^ ðT 1 T 1 Þ S0 S0 H S0 S0 H S0 S0 H S0 S0 H S0 S0 H S0 S0 H 6 D D D E 7 7 6 1 + E 1 + E 1 7 6 S S H ^ S1 S0 ^ S0 S1 ^ D D ^ D D ^ ðT 1 T 1 Þ S1 S0 H S1 S0 H S1 S0 H S1 S0 H S1 S0 H 0 0 ^ S1 S0 7 6 6 D D D E 7 7 6 1 + E 1 + E 1 7 6 S0 S0 H ^ S0 S1 ^ S0 S1 ^ S0 S1 ^ D D ^ D D ^ ðT 1 T 1 Þ S0 S1 H S0 S1 H S1 S0 H S0 S1 H S0 S1 H 7 6 6D E7 1 + E D 1 + E D 1 + E D1 + 1 + E D1 + 1 + E D1 + 1 7 6 6 S0 S0 H ^ D D ^ D D ^ D D ^ D D ^ D D ^ ðT1 T1 Þ 7 S1 S0 H S0 S1 H D D H D D H D D H 7 6 6D E7 1 + E D 1 + E D 1 + E D1 + 1 + E D1 + 1 + E D1 + 1 7 6 ^ D D ^ D D ^ D D ^ D D ^ D D ^ ðT1 T1 Þ 7 6 S0 S0 H S1 S0 H S0 S1 H D D H D D H D D H 7 6 4D E D E D E D E D E D E5 1 1 1 1 1 1 1 + ^ 1 + ^ 1 ^ ðT 1 T 1 Þ ^ ðT1 T1 Þ ^ ðT1 T1 Þ ^ ðT 1 T 1 Þ S1 S0 H S0 S1 H D D H ðT1 T1 Þ D D H ðT1 T1 Þ ðT1 T1 ÞH S0 S0 H 2

(14)

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and the model permits an approximate description of processes such as ET, CT, and SF. This requires an evaluation of matrix elements in terms of integrals over the one-electron and two-electron parts of the Hamiltonian and of overlap integrals ShAhB, SlAlB, ShAlB, and SlAhB (note that ShAlA ¼ ShBlB ¼ 0). The complete results, obtained using the program MAPLE,83 are shown in Appendix. Those expressions that are needed for a description of SF were published previously.1

2.2 Expressions for TRP In the first approximation,3 the initial state in SF is represented by the configuration S1S0 if it is localized, and either S1S0 + S0S1 or S0S1 S1S0 if it is fully delocalized. The final state is represented by the configuration 1(T1T1). Very similar formulas result in both cases.3 Here we state results for TRP applicable for a localized initial state. Those for the delocalized initial state are given in Section 2.5, Eq. (51). The matrix element TRP ¼ HRP SRPE becomes48 TRP ¼ TS1S0/T1T1¼ ^ 1S0i, HS1S0/T1T1 SS1S0/T1T1ES1S0, where HS1S0/T1T1 ¼ h1(T1T1)jHjS 1 ^ 1S0i. The matrix eleSS1S0/T1T1 ¼ h (T1T1)jS1S0i, and ES1S0 ¼ hS1S0jHjS ment TS1S0/T1T1 that approximates TRP in this treatment is commonly referred to as the “direct” term. In a better approximation, the initial state is assumed to be a linear combination of S1S0 with a small admixture of 1D+D and 1DD+, and the final state is a similar linear combination of 1(T1T1) with 1D+D and 1DD+. If the initial and final states are degenerate and the 1D+D and 1DD+ states are also degenerate and higher in energy by ΔE, and if we use first-order perturbation theory and neglect terms containing products of two small numbers, the matrix element TRP becomes4,11 TRP ¼ TS1S0=T1T1 TS1S0= + T +=T1T1 + TS1S0= + T + =T1T1 =ΔE (15) where the subscript + stands for 1D+D and the subscript + stands for DD+. The expression for TRP now consists of the direct term TS1S0/T1T1 and a term mediated by the virtual states 1D+D and 1DD+, which contains division by the energy difference ΔE. Formula (15) is the standard first-order expression for TRP that is applicable when the energies of the chargeseparated states 1D+D and 1DD+ are the same. It has seen much use and it shall also be used in the following. If the energies of the charge-transfer states 1D+D and 1DD+ are different, the result changes to

1

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TRP ¼ TS1S0=T1T1 ½ TS1S0= + T +=T1T1 =ΔE 1 D + D + TS1S0= + T + =T1T1 =ΔE 1 D D + (16)

If the 1D+D and 1DD+ states are very close in energy to the S1S0 and (T1T1) states, the first-order expressions (15) and (16) will overestimate the magnitude of the mediated term. Then, an explicit diagonalization within the three-dimensional [1D+D, 1DD+, S1S0] and [1D+D, 1DD+, 1 (T1T1)] spaces is necessary and a more complicated formula for TRP results. Formulas (15) and (16) also assume that the coupling between the initial S1S0 and final 1(T1T1) states is weak relative to the effects of the phonon bath and that the initial excitation does not produce their coherent superposition. If this condition is not satisfied, a diagonalization in the full four-dimensional space [1D+D, 1DD+, S1S0, 1(T1T1)] is needed. This appears to be the case for the very fastest SF events, observed in crystalline pentacene54 (cf. Section 1.2.1). Only the two-electron part of the Hamiltonian contributes to the direct ^ 1S0i when interchromophore overlap is term TS1S0/T1T1 ¼ h1(T1T1)jHjS neglected and it is very small (typically on the order of meV), because the two-electron integrals involved represent electrostatic interactions between overlap densities at least one of which is very small (it originates in the multiplication of a molecular orbital located on A with a molecular orbital located on B). It is generally reasonable to state that the rate of SF is primarily determined by interactions of starting and final states mediated by virtual charge-transfer states.4,10,84 The mediated (indirect) term on the right-hand side of Eq. (13), (TS1S0/CATCA/T1T1 + TS1S0/ACTAC/T1T1)/ΔE, contains contributions both from the two-electron and the one-electron part of the Hamiltonian. It typically amounts to hundreds of meV at realistic geometries, even though it contains a division by a potentially large energy difference between the initial and the charge-separated states and also sometimes suffers from destructive interference of the two paths mediated by the virtual charge-transfer states ^ 1D+D + 1DD+i ¼ (Fig. 2). The interference is reflected in the term hS1S0jHj ^ 1D+Di + hS1S0jHj ^ 1DD+i, where the two matrix elements on the hS1S0jHj right could be comparable in size and opposite in sign. 1

2.2.1 Intermolecularly Nonorthogonal Orbitals When the four interchromophore overlap integrals ShAhB, SlAlB, ShAlB, and SlAhB are not neglected, the expressions for the terms that occur in Eq. (15)

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are quite lengthy and complicated. They have been published, partly in supporting information.1 Full results for all elements of the Hamiltonian matrix (14) and the associated overlap integrals are given presently in Appendix, and permit not only an evaluation of the expression (15) for SF but also a similar treatment of processes such as ET and CT. The closest analog are the previously reported48 expressions for ET matrix elements ^ 1S0i, derived after setting ShAlB ¼ SlAhB ¼ 0. We see no physical hS0S1jHjS justification for such selective neglect of these two of the four overlap integrals. For realistic dimer geometries, the interchromophore overlaps are usually smaller than 0.1. Terms that are higher than first order in overlap can therefore be safely neglected. To first order in overlap, the results are ^ S1 S0 ¼NS1S0 2 fFhAhA + FlAlA + 2FhBhB ðhA hA jhA hA Þ ðhB hB jhB hB Þ S1 S0 H 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + 2ðhA lA jhA lA Þ ðhA hA jlA lA Þ 2SlAhB ½FlAhB ðhA hA jlA hB Þ + 2 ðhA lA jhA hB Þ 2ShAhB ½FhAhB ðhA hA jhA hB Þ + ðhA hB jlA lA Þ + ðhA lA jlA hB Þg D

1 + E ^ D D ¼ NS1S0 N + fFlAlB ðhA hA jlA lB Þ + 2 ðhA lA jhA lB Þ S1 S0 H

(17)

+ ShAlB ½FhAlA ðhA hA jhA lA Þ + SlAlB ½FhAhA + 2FhBhB ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhB hB jhB hB Þ SlAhB ½FhBlB + 2ðhA hB jhA lB Þ ðhA hA jhB lB Þ ShAhB ½2ðhA hB jlA lB Þ + ðhA lA jhB lB Þ + ðhA lB jlA hB Þg D 1 + E ^ D D ¼ NS1S0 N + fFhAhB ðhA hB jlA lA Þ + 2ðhA lA jlA hB Þ S1 S0 H

(18)

+ 2SlAhB ½FhAlA ðhA lA jhB hB Þ ShAhB ½FhBhB + FlAlA + FhAhA 3ðhA hA jhB hB Þ ðhA hA jlA lA Þ ðhA hB jhA hB Þ + 2 ðhA lA jhA lA Þ ðhB hB jhB hB Þ + 2ðlA hB jlA hB Þ ðlA lA jhB hB Þ ðhA hA jhA hA Þg

(19)

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D E 1 ^ ðT1 T1 Þ ¼ NS1S0 NT1T1 ð3=2Þ1=2 f ðlA hB jlA lB Þ S1 S0 H ðhA hB jhA lB Þ + SlAlB ½FlAhB ðhA hA jlA hB Þ ShAlB ½FhAhB ðhA hA jhA hB Þ + ðhA hB jlA lA Þ ðhA lA jlA hB Þ + SlAhB ½FlAlB ðhA hA jlA lB Þ + 2ðhA lA jhA lB Þ ðlA hB jhB lB Þ ðlA lB jhB hB Þ ShAhB ½FhAlB ðhA hB jhB lB Þ + ðhA lA jlA lB Þ ðhA lB jhB hB Þ + ðhA lB jlA lA Þ ðhA hA jhA lB Þg D

E 1 + ^ 1 D D H ðT1 T1 Þ ¼ N + NT1T1 ð3=2Þ1=2 fFlAhB ðhA hA jlA hB Þ

(20)

+ ðlA hB jlB lB Þ SlAlB ðhA hB jhA lB Þ + SlAhB ½FhAhA + FhBhB + FlBlB ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhA hA jlB lB Þ + 2ðhA lB jhA lB Þ ðhB hB jlB lB Þ ðhB hB jhB hB Þ + ShAlB ½2ðhA lB jlA hB Þ ðhA hB jlA lB Þ ShAhB ½FhAlA ðhA hA jhA lA Þ + ðhA hB jlA hB Þ ðhA lA jhB hB Þ + ðhA lA jlB lB Þ + ðhA lB jlA lB Þg D

1

(21)

E

1 ^ ðT1 T1 Þ ¼ N + NT1T1 ð3=2Þ1=2 fFhAlB ðhA lB jhB hB Þ + ðhA lB jlA lA Þ D D + H SlAlB ðhA hB jlA hB Þ + ShAlB ½FhAhA + FhBhB + FlAlA ðhB hB jhB hB Þ 3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðlA lA jhB hB Þ + 2ðlA hB jlA hB Þ ðhA hA jlA lA Þ ðhA hA jhA hA Þ + SlAhB ½2ðhA lB jlA hB Þ ðhA hB jlA lB Þ ShAhB ½FhBlB ðhB hB jhB lB Þ + ðhA hB jhA lB Þ ðhA hA jhB lB Þ + ðlA hB jlA lB Þ + ðlA lA jhB lB Þg

(22) S1 S0 j1 D + D ¼ NS1S0 N + ShAhB 2 SlAlB ShAhB ShAlB SlAhB + SlAlB (23) (24) S1 S0 j1 D D + ¼ NS1S0 N + ShAhB ShAhB 2 SlAhB 2 1 (25) S1 S0 j1 ðT1 T1 Þ ¼ NS1S0 NTT ð3=2Þ1=2 ðSlAhB SlAlB ShAhB ShAlB Þ 1 + 1 D D j ðT1 T1 Þ ¼ N + NTT ð3=2Þ1=2 ShAhB ShAlB SlAlB + ShAlB 2 SlAhB + SlAhB (26)

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1 + 1 D D j ðT1 T1 Þ ¼ N + NTT ð3=2Þ1=2 ShAhB SlAhB SlAlB + SlAhB 2 ShAlB + ShAlB (27)

where F^ is the Fock operator for the ground state configuration S0S0. This operator includes the mutual interactions of electrons in the active space but also their interaction with those in the inactive core. The symbols ðabjcdÞ ¼ ð aðr1 Þbðr1 Þð1=r12 Þc ðr2 Þd ðr2 Þdr1 dr2 represent the two-electron (electron– electron repulsion energy) integrals in the basis of molecular orbitals of the partners. Three comments can be made: (i) The direct term in Eq. (15) contains off-diagonal elements of the Fock operator such as FlAhB and FhAhB. They are generally small relative to the diagonal elements such as FhAhA, because they contain one molecular orbital on each partner. Moreover, they enter multiplied by an overlap integral. Nevertheless, their contribution to the direct term might still exceed the contribution provided by the minute twoelectron integrals, and this may lead to situations in which the direct term need not be entirely negligible in Eq. (15) relative to the mediated term. (ii) The mediated term in Eq. (15) contains not only these off-diagonal one-electron integrals but also the much larger diagonal ones, albeit multiplied by overlap integrals. The latter contribution could be comparable to those provided by off-diagonal one-electron integrals and could have a significant effect on the structural dependence of the mediated term. (iii) The third comment does not refer to SF itself, but to the possible decay of the real (not virtual) 1D+D or 1DD+ intermediate that intervenes when the conversion of the initial S1S0 state to the biexciton 1T1T1 state occurs in two steps (Section 1.2.2). We have noted under (ii) that the presence of overlap influences the mediated term in Eq. (15) through its effect on the indirect coupling of the S1S0 state and the 1T1T1 state via the virtual 1D+D and 1DD+ charge-transfer states. It introduces similar diagonal one-electron integrals into the coupling of the 1D+D and 1DD+ states with the S0S0 ground state and thus might play a role in expressions for back-electron transfer through which these charge-transfer states are deactivated to the ground state. Because of the potential importance of overlap-containing

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terms, in exact solutions of the HOMO/LUMO model we do not neglect intermolecular overlap even though it is small. In the development of simple formulas, we neglect it. The expressions (17)–(27) that include overlap are useful for qualitative considerations such as the above points (i)–(iii), but for actual numerical computations of exact solutions of the HOMO/LUMO model, we prefer to perform a L€ owdin orthogonalization of the orbitals located on different partners, and subsequently use the simple formulas for orthogonal orbitals. 2.2.2 Intermolecularly Orthogonalized Orbitals The neglect of intermolecular overlap in the simplified procedure or the use of intermolecularly L€ owdin orthogonalized molecular orbitals in the exact solution of the HOMO/LUMO model simplifies expressions (17)–(27) considerably. The normalization factors Nx in (8)–(13) become equal to unity, the overlap integrals in (23)–(27) vanish, and the expressions (17)– (22) for the Hamiltonian matrix elements simplify to the previously published3 simple formulas (28)–(32), written in terms of the Fock operator ^ which includes interactions with core electrons: for the ground state F, ^ 1 ðT1 T1 Þ ¼ ð3=2Þ1=2 ½ðlA hB jlA lB Þ ðhA hB jhA lB Þ (28) S1 S0 jHj ^ 1 D + D ¼ lA jFjl ^ B + 2 ðhA lA jhA lB Þ ðhA hA jlA lB Þ (29) S1 S0 jHj ^ 1 D D + ¼ hA jFjh ^ B + 2 ðhA lA jlA hB Þ ðhA hB jlA lA Þ (30) S1 S0 jHj 1 + 1 ^ ðT1 T1 Þ ¼ ð3=2Þ1=2 lA jFjh ^ B + ðlA hB jlB lB Þ ðhA hA jlA hB Þ D D jHj 1

(31) 1=2 ^ ðT1 T1 Þ ¼ ð3=2Þ ^ B + ðlB hA jlA lA Þ ðhA lB jhB hB Þ D D jHj hA jFjl

+

1

(32) The Fermi golden rule formula for the SF rate then is ^ 1 D + D ^ 1 ðT1 T1 Þi ½ S1 S0 jHj W ðSFÞ ¼ h1 jfhS1 S0 jHj ^ 1 T1 T1 =ΔE 1 D + D 1 D + D jHj ^ 1 D D + 1 D D + jHj ^ 1 T1 T1 =ΔE 1 D D + gj2 ρðE Þ + S1 S0 jHj (33)

where the matrix elements of the Hamiltonian are given by expressions (28)–(32).

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2.3 A Simple Approximation for TRP For a rapid search for local maxima of jTRPj as a function of the relative disposition of two chromophores, it is desirable to simplify the expressions derived so far. The approximations to be introduced need to be fairly accurate at dimer geometries at which the absolute value of TRP is large, but they do not need to be valid at all at geometries at which jTRPj is small. We shall assume the energy differences between the charge-separated states and the initial and final states are large enough for the charge-separated state to be virtual and not a separate minimum on the S1 surface and for permitting the use of the first-order perturbation approximation and Eq. (15). If needed, first-order perturbation can be replaced by exact diagonalization, which makes the resulting formulas more complicated but does not involve much penalty in computation time. Starting with the exact solution of the HOMO/LUMO model for a pair of chromophores A and B, the following approximations were introduced1,85: 2.3.1 Neglect of Intermolecular Overlap This simplifies the expressions for matrix elements that are needed for the rate Eq. (33) to expressions (28)–(32). As is common in semiempirical theories, atomic orbitals are considered to be intramolecularly L€ owdin orthogonalized and yet retain their atomic properties. 2.3.2 Zero Differential Overlap Neglect of all electron repulsion integrals that involve charge densities resulting from products of orbitals located on different partners makes the direct term vanish and simplifies the mediated term greatly. The matrix elements needed for Eq. (33) now are: ^ 1 ðT1 T1 Þ ¼ 0 S1 S0 jHj (34) ^ 1 D + D ¼ lA jFjl ^ B (35) S1 S0 jHj ^ 1 D D + ¼ hA jFjh ^ B (36) S1 S0 jHj 1 + 1 ^ ðT1 T1 Þ ¼ ð3=2Þ1=2 lA jFjh ^ B (37) D D jHj 1 + 1 ^ ðT1 T1 Þ ¼ ð3=2Þ1=2 hA jFjl ^ B (38) D D jHj The validity of the zero differential overlap (ZDO) approximation has been verified numerically for many different geometries of a pair of ethylenes.1 As an example, Fig. 3 shows the three contributions to the matrix

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^ 1(T1T1)i for two Fig. 3 Eq. (31): the three contributions to the matrix element h1D+DjHj parallel ethylene molecules separated by z ¼ 3.5 Å, as a function of slip along directions x and y. A similar example was published previously.1

^ 1(T1T1)i in Eq. (31) for two slip-stacked ethylenes. It element h1D+DjHj demonstrates that in the region of geometries where this matrix element is large, the two contributions neglected in the ZDO approximation (lAhBjlBlB) and (hAhAjlAhB) are indeed negligible relative to the contribution ^ B that is kept in Eq. (37). (3/2)1/2 lA jFjh The formula for the SF rate now simplifies to W ðSFÞ ¼ ð3=2Þ h1 lA F^lB lA F^hB =ΔE 1 D + D (39) hA F^hB hA F^lB =ΔE ð1D D + Þj2 ρðE Þ

2.3.3 A Minimum Valence Basis Set of Natural Atomic Orbitals In the next step, we express the HOMO and LUMO in a minimum basis sets of natural AOs (μ or κ on partner A and ν or λ on partner B). On each partner, this basis set is orthonormal (L€ owdin orthogonalized AOs), but the AOs on partner A may have a nonzero overlap with those on partner B. Fock ^ elements between AOs on different partners are equated to resoperator (F) onance (hopping) integrals βμν, which are then related to AO overlaps through the Mulliken approximation86,87:

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μF^ν ¼ βμν ¼ kSμν

(40)

where the proportionality constant k is a function of the nature of atoms μ uckel and ν via the Wolfsberg–Helmholtz formula86 used in Extended H€ 87 theory (EHT), (41) k ¼ K Hμμ + Hνν =2 and Hμμ is the standard EHT parameter describing the electron binding energy of orbital μ. In EHT calculations the value of K is usually set to 1.75 and this works well when the atoms μ and ν are separated by the usual intramolecular bonding distances. However, for distances close to or exceeding the sum of van der Waals radii, the value needs to be reduced. For ˚ , the Linderberg formula88 yields (2p–2p)π and (2p–2p)σ interactions at 3–5 A K ¼ 1.0. We adopt K ¼ 1, for which the jTRPj values obtained from the simplified and the exact solution of the HOMO/LUMO model for ethylene dimer at a variety of geometries are nearly identical. The final expression for the rate of SF now is W ðSFÞ ¼ 3k4 =2h ρðE Þ Σ μν clμ chν Sμν ðΣ κλ clκ clλ Sκλ Þ=ΔE 1 D + D (42) Σ μν chμ clν Sμν ðΣ κλ chκ chλ Sκλ Þ=ΔE 1 D D + 2 This general result can sometimes be simplified further. For qualitative considerations, it is often possible to neglect the dependence of ΔE(1D+D) and ΔE(1DD+) on the geometry of the A,B pair, since A and B need to be in contact. Typical values are 30–50 kcal/mol, but values outside this range are easily possible. If CT is equally likely in both directions, for instance, if the two partners are symmetry-related identical chromophores, it is possible to assume ΔE(1D+D) ¼ ΔE(1DD+) ¼ ΔE, and (42) then simplifies to W ðSFÞ ¼ 3k4 =2hΔE2 ρðEÞ 2 Σ μν clμ chν Sμν ðΣ κλ clκ clλ Sκλ Þ Σ μν chμ clν Sμν ðΣ κλ chκ chλ Sκλ Þ (43) If the two partners are different and electron transfer from A to B is much easier than from B to A, ΔE ¼ ΔE(1D+D) ≪ ΔE(1DD+), the second term in the brackets in Eq. (42) can be neglected and the expression simplifies to 2 (44) W ðSFÞ ¼ 3k4 =2hΔE 2 ρðE Þ Σ μν clμ chν Sμν ðΣ κλ clκ clλ Sκλ Þ

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The simple expressions (43) and (44) are easily programmed but in many cases they can be understood qualitatively upon visual inspection. After all, even for large molecules any one AO on partner A normally overlaps strongly with only very few AOs on partner B (often, none, but sometimes up to three or four), and most of the terms in the sums in Eqs. (42)–(44) are negligible. Although our treatment deals with cases in which the two partner molecules are distinct, it can be used for qualitative insight even if they are covalently bound. E.g., when a donor A and an acceptor B are connected through a single bond that links atom 1 on A with atom 10 on B, the only significant overlap integral is S110 and the brackets in Eq. (44) equal S110 2 cl1 2 ch10 cl10 . Then, to maximize jTRPj, the LUMO of A should have a large coefficient at its link atom 1, both the LUMO and the HOMO of B should have a large coefficient at its link atom 10 , and the linking bond should not be twisted too much. Our search for a mutual disposition of partners A and B that optimizes the rate of the S1S0 to 1TT conversion might thus appear to have been reduced to the maximization of the square in the brackets of Eqs. (42), (43), or (44), requiring only the knowledge of the expansion coefficients on HOMO and LUMO and of the overlap integrals between AOs on one and the other partner. However, only half of the work has been done, since the density of states factor ρ(E) in expressions (42)–(44) also depends on the choice of geometrical disposition of the partners. The chief reason for that is that the dimer geometry affects the energetics of the SF process by Davydov interaction that frequently stabilizes the lowest excited singlet state and leaves the energy of the lowest triplet nearly intact. Its effect on relative SF rates at various partner dispositions may be unimportant in practice if SF is sufficiently exothermic and rapid at all geometries, but it may be essential if SF is isoergic or endoergic at some of them. The factor ρ(E) has been treated by various authors, for example via microscopic dynamics,10–12 in the Marcus theory approximation,40 or using a simple kinetic model.72 In Section 2.5, we address the evaluation of the energetics as a function of geometrical arrangement of the partners in a fashion that resembles our treatment of jTRPj2. We take into account both factors that determine the magnitude of the Davydov splitting of the lowest singlet state, the direct interaction between transition densities and the contribution mediated by virtual CT states.89,90

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2.4 Local Maxima of |TRP|2 in the 6-D Space of Rigid Dimer Geometries The mutual disposition of two rigid molecules is described by six degrees of freedom, three translations (Tx, Ty, Tz) and three rotations (Rx, Ry, Rz). The value of the TRP matrix element is a function of these six variables, TRP ¼ TRP (Tx, Ty, Tz, Rx, Ry, Rz). Since in the rate expressions TRP appears in the second power, our search for the mutual dispositions of two chromophores that maximize the rate constant of the SF process starts with a search for the largest local maxima of jTRPj2 in the 6-D space. Afterward, we will discard unphysical maxima and possibly also those for which the ρ(E) term is unfavorable. Locating the maxima of jTRPj2 is an arduous task that requires a systematic search of that part of the 6-D space in which the partners A and B are close to each other. Techniques such as the genetic algorithm91 that have been developed for such searches do not guarantee that all the local extrema will be found. Our preferred procedure is to combine preselection of extrema on a 6-D grid with subsequent gradient optimization starting at the preselected points. To create a relatively sparse grid in a 6-D space, with ˚ steps for translations and 10° for rotations, one has to evaluate jTRPj2 at 0.2 A 107–109 points or even more, depending on the size of chromophores. A systematic search for maxima on the jTRPj2 surface is therefore presently limited to the use of simple formulas such as (42)–(44). These do not provide any information about intermolecular repulsions and the energetic accessibility of the geometries at which the maxima of jTRPj2 are located. By itself, the function jTRPj2 typically shows local maxima at geometries at which the HOMO and LUMO overlap strongly but the molecules interpenetrate to a ridiculous extent. This is illustrated in Fig. 4, which shows a perspective ˚, view of a plot of jTRPj2 as a function of Tx and Ty at Tz ¼ 3 A Rx ¼ Ry ¼ Rz ¼ 0, and thus displays the results for a two-dimensional subspace of the total six-dimensional space. The values of jTRPj2 are quite large, ˚ the molecules are pressed closer together than they would since at Tz ¼ 3 A ever come under ordinary conditions, making the intermolecular overlap integrals fairly large. The value at the maxima drops as Tz is increased and grows as it is decreased. It reaches a peak at Tz ¼ 0, a completely unphysical situation with the molecules interleaved in the same plane. This unphysical maximum could however be viewed as the parent of the various realizable maxima that can be arrived at by moving the two molecules further apart along one or another direction.

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Fig. 4 jTRPj2 in units of eV2 for two parallel ethylene molecules separated by 3.0 Å as a function of slip along in-plane directions x and y. The maxima are located at x ¼ 1.0 and y ¼ 0.0 Å.

The inaccessible maxima of jTRPj2 thus provide insight into the origin of maxima that can actually be accessed, and our present task is to find the latter. For this purpose, we use a “search function,” which combines information about jTRPj2 with information about the part of our 6-D space that is excluded when atoms in the two partner molecules are modeled as hard spheres. The search function is defined as F ¼ αEREP 2 jTRP j2

(45)

where EREP is a repulsion term and α is a weighting coefficient. The function F has no real physical significance and only helps us to find structures at which jTRPj2 is large, yet the two partners are not unrealistically close. We have tested the use of several van der Waals potentials for EREP and in the end concluded that for our purpose a hard sphere potential seems to be the best and use X EREP2 ¼ 10106 μν exp ςdμν = rμ vdW + rν vdW , (46) where the summation runs over all atoms of molecule A (μ) and molecule B (ν), ς ¼ 244.0, dμν is the distance between atoms μ and ν, and rμ vdW is the

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Fig. 5 The search function F2 in units of eV2 for two parallel ethylene molecules separated by 3.0 Å as a function of slip along in-plane directions x and y.

van der Waals radius92 of atom μ. The results are quite independent of the choice of the weighting coefficient α, and we set α ¼ 1. Fig. 5 shows the value of the search function F in the same two˚ , Rx ¼ Ry ¼ Rz ¼ 0) that was used dimensional subspace Tx,Ty (Tz ¼ 3 A for Fig. 4. Now, the unphysical region of interpenetrated molecules is excluded and the four minima of the search function lie at its circumference. As Tz is increased, the excluded region of the two-dimensional subspace shrinks and ultimately disappears, and the four minima of F coalesce pairwise into two, identical with those of jTRPj2. This is a typical result; the optimal geometries that can be realistically accessed surround a much higher maximum that cannot be accessed because of intermolecular repulsions. Clearly, increased hydrostatic pressure would in general be favorable for reaching higher values of jTRPj2. Using the search function F(Tx, Ty, Tz, Rx, Ry, Rz) and the simple approximation (43) for the evaluation of jTRPj2, numerical determination of the location of minima on a 6-D grid of 109 points takes several hours of CPU time on a modern Intel processor for a small molecule such as ethylene and several days for a large one, such as 1,3-diphenylisobenzofuran.

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Subsequent refinement of the best dimer geometries is performed by an optimization procedure based on numerical evaluation of gradients and the Hessian.

2.5 State Energies in the 6-D Space of Rigid Dimer Geometries Up to this point, we have ignored the ρ(E) term in Eqs. (43) and (44). When it is approximated similarly as in Marcus theory of CT, its effect on the expected rate of SF is reflected in two primary energy-related terms, the reorganization energy λ and the exoergicity of the SF process, ΔESF. While λ can be reasonably considered independent of the mutual disposition of the partners A and B, the exoergicity ΔESF cannot. For some chromophores, such as pentacene, the resulting variation of the SF rate may be of little practical consequence since E(S1) is sufficiently larger than 2E(T1) that SF is exothermic for any realistic mutual disposition of the chromophores A and B and will always prevail over competing decay channels. For chromophores in which E(S1) and 2E(T1) are less favorable, such as tetracene, the variation of ΔESF as a function of the mutual chromophore disposition may be critically important for the outcome of the competition between SF and other decay processes. In this section, we use the HOMO/LUMO model and the same approximations that were applied in the search for the local maxima of jTRPj2 in Section 2.4 to evaluate trends in the possibly detrimental effect of chromophore interaction on ΔESF. The results can then be used to eliminate some of the otherwise favorable local maxima of jTRPj2 from consideration. To estimate roughly the reduction of the exoergicity ΔESF of the conversion of the lowest excited singlet energy in a dimer into the biexciton 1(TT) relative to expectations based on the properties of the monomeric chromophore, we start with expressions given in Appendix for the elements of the Hamiltonian matrix (14). After neglect of intermolecular overlap and introduction of the ZDO approximation, the Hamiltonian matrix for the dimer system becomes

ð47Þ where

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^ AB UA UB EðUA UB Þ ¼ EðUA Þ + E ðUB Þ + UA UB H

(48)

^ AB describes the interaction between partners A and B. and H The elements of the Fock operator F^ in the HOMO/LUMO basis are already familiar from Section 2.4 and (hAlAjhBlB) is the Coulomb interaction between the HOMO–LUMO transition densities on chromophores A and B. In direct analogy to the procedure used in Section 2.2 to derive expression (15) for the matrix element TRP, we assume that the singlet excited states localized on one of the partners A and B and the biexciton state are described by mixtures of electron configurations dominated by S1S0, S0S1, or 1(T1T1), respectively, but also containing small admixtures of the higher energy configurations 1D+D and 1DD+. Next, the contribution of the charge-separated configurations is approximated by first-order perturbation theory and terms containing products of two small numbers are neglected. As before, we now assume that the partners A and B are the same chemical species, the S1S0, S0S1, and 1(T1T1) configurations have the same energy (i.e., SF would be isoergic in the absence of interactions between partners A and B), and the 1D+D and 1DD+ states are also degenerate and higher in energy by ΔE than S1S0 or S0S1. Strictly speaking, in this approximation, even if A and B are the same chemical species, at most geometries E(S1S0) and E(S0S1) will differ slightly, by 2[(lAlA|hBhB) (hAhA|lBlB)], while 1D+D and 1DD+ will differ by half as much, (lAlA|hBhB) (hAhA|lBlB). Also E(T1T1) and E(S1S0) will differ slightly, by (lAlA|lBlB) + (hAhA|lBlB) (lAlA|hBhB) (hAhA|hBhB). We neglect these differences presently, but they could be easily taken into account if necessary. Note that in alternant hydrocarbons such as tetracene, the charge distributions hAhA and lAlA are the same, as are hBhB and lBlB, and (lAlA|hBhB) (hAhA|lBlB) vanishes. The resulting approximations for the (positive) Davydov splitting ΔEDS between the in-phase S1+ ¼ ðS1 S0 + S0 S1 Þ=21=2 and the out-of-phase S1 ¼ ðS1 S0 S0 S1 Þ=21=2 combination of the localized singlet excited states, for the endoergicity ΔESF , and for the matrix elements TRP for SF from S1 to the biexciton TT are ΔEDS ¼ jE ðS1 + Þ E ðS1 Þj ¼ 4 ðhA lA jhB lB Þ + lA F^lB hA F^hB =ΔE

(49)

¼ E ð TT Þ E ΔE SF h S1 ¼ 2ðhA lAjhB lB Þ i 2 + lA F^lB hA F^hB ð3=2Þ j lA F^hB 2 + hA F^lB j2 =ΔE (50)

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^ ^ B > < lA jFjh ^ B> TRP ¼ ¼ 31=2 < hA jFjl ^ B > < lA jFjl ^ B > =2ΔE < hA jFjh

(51)

Introduction of a minimum valence basis set of natural atomic orbitals (NAOs) and conversion of Fock matrix elements between AOs on different partners into resonance integrals followed by the use of the Mulliken approximation for conversion to overlap integrals similarly as in Section 2.3.3 yields the formulas ΔEDS ¼ 4 ðhA lA jhB lB Þ + k2 Σ μν clμ clν Sμν ðΣ κλ chκ chλ Sκλ Þ=ΔE (52) n 2 ΔESF ¼ 2ðhA lA jhB lB Þ + k2 Σ μν clμ clν chμ chν Sμν (53) 2 2 o ð3=2Þ Σ μν clμ chν Sμν + Σ μν chμ clν Sμν =ΔE ^ > TRP ¼ < S1 jHjTT 1=2 2 ¼ 3 k Σμν chμ chν clμ clν Sμν ½Σκλ ðchκ clλ clκ chλ ÞSκλ =2ΔE

(54) where μ and κ are located on partner A, and ν and λ are on partner B. If we now approximate the ρ(E) term in the Fermi Golden rule using Marcus theory, the SF rates from the S1 + and S1 combinations of the localized excited singlet states are h i 2 2 W SF ¼ ð2π=ℏÞTRP ð4πλkB T Þ1=2 exp ΔESF + λ =4πλkB T (55) If the Davydov splitting ΔEDS is large enough, only the more stable combination of locally excited singlet states is populated significantly and needs to be considered in the evaluation of SF rate. If ΔEDS is small, both combinations will be populated according to Boltzmann statistics and will contribute to SF. Then, the overall rate will be W ðSFÞ ¼ 1 ½ exp ðΔEDS =kB T Þ + 11 W ðSF + Þ (56) + ½ exp ðΔEDS =kB T Þ + 11 W ðSF Þ It is seen that excitation delocalization and the resulting Davydov splitting make SF less exoergic or more endoergic and in that sense usually are detrimental to SF. Although the formulas (53) and (54) are only

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approximate, they are likely to reflect trends correctly and will permit a rejection of some of the optimal dimer geometries obtained in Section 2.4. Finally, for the biexciton binding energy, in this approximation we obtain jE(1TT) E(T + T)j ¼ j(hAhAjhBhB) + (hAhAjlBlB) + (lAlAjhBhB) + (lAlAjlBlB) (3/2) k2(jΣ μνclμchνSμνj2 + jΣ μνchμclνSμνj2)/ΔEj. In this case the entropy change associated with the dissociation into free triplets will not be negligible, as was assumed for processes treated in the rest of our derivations.

3. APPLICATIONS 3.1 Two Ethylene Molecules Our first illustration of the simplified HOMO/LUMO model is its application to the simplest π-electron chromophore, ethylene. Although the energies of its S0, T1, and S1 states fulfil the requirements for isoergic SF, it is obviously not a practical molecule for solar cells. We take advantage of its simplicity to derive intuitive understanding and also use it for method testing. Each ethylene molecule has two identical natural atomic orbitals, one on each atom. We have expanded the NAOs in terms of a contracted Gaussian basis set, taken from Pople’s 6-311+G basis set.93,94 The expansion coefficients of NAO in terms of contracted Gaussians were calculated by Weinhold’s NBO analysis95,96 using the SCF wave function of ethylene. Construction of the orbitals HOMO and LUMO is simple, as they are L€ owdin orthonormalized in-phase and out-of-phase combinations of 2pπ NAOs on each atom, respectively. For the evaluation of jTRPj we used ΔE ¼ 1 eV. For preselection of best structures in the 6-D space, we used a grid size of ˚ for translations and 10˚ for rotations and performed calculations at a 0.2 A total of 7.12 108 dimer geometries. We found about 13 103 local minima of the search function F. These were optimized, sorted, and duplicates (mostly due to symmetry) were removed. Only 43 structures remained, and only a dozen of these have jTRPj values larger than 0.01% of the largest jTRPj value found. The six best structures are shown in Fig. 7, along with their jTRPj2 values in units of eV2. They all obey the simple rule1 deduced from inspection of formula (43): to maximize jTRPj2, position the ethylene molecules A and B in such a way that one of the NAOs of A overlaps both NAOs of B, and the other NAO of A has as little overlap with the NAOs of B as possible. The resulting local maxima of jTRPj2 are relatively narrow peaks with a

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width of 1–1.3 A˚ at half height and if this result is general, it will easily account for the sensitivity of the SF rate to the details of molecular packing. The qualitative rule applies to all non-polar chromophores if one (i) replaces the two NAOs of ethylene by two semilocalized LMOs of a general chromophore, defined as the sum (LMO+) and the difference (LMO) of its HOMO and LUMO, with LMO+ located mostly on one side and LMO mostly on the other side of the nodal surface that is introduced upon going from HOMO to LUMO, and (ii) looks for geometries in which LMO+ on partner A overlaps both LMO+ and LMO of partner B, while LMO on partner A has as little overlap as possible with both LMO+ and LMO of partner B. Polarity introduces complications and even a perfectly stacked pair of polarized ethylenes has a non-vanishing jTRPj if it is arranged head-to-tail. In Table 1 we show the results obtained for the effects of intermolecular interaction at the six ethylene dimer geometries shown in Fig. 6. They are the excitonic coupling matrix element (hAlAjhBlB) and, with the choices ΔE ¼ 1 and 2 eV, the Davydov splitting ΔEDS and the singlet fission electronic matrix element TRP and endoergicity ΔESF starting at either one of the two exciton states (a positive value of ΔESF implies a process that is more endoergic than would be expected from the properties of isolated molecules). The dependence of the magnitude of excitonic splitting on the relative orientation of the HOMO–LUMO transition moments in the two molecules is apparent and it is clear that trends are not sensitive to the details of the calculation. It is seen that structure 1, which has the largest jTRPj2 value, also has the largest Davydov splitting and suffers the most from the endoergicity induced by intermolecular interactions. The next best structure 2, with a somewhat smaller jTRPj2 value, leads to no Davydov splitting and actually gains a little exoergicity from intermolecular interactions. It would most likely be the best dimer geometry choice if one wished to perform SF on an ethylene dimer.

3.2 Two Clipped Cibalackrot Molecules Our second illustration of the simplified HOMO/LUMO model is its application to a much larger chromophore, diindolo[3,2,1-de;3ʹ,2ʹ,1ʹ-ij] [1,5]naphthyridine-6,13-dione (Fig. 7). Its structure is related to indigo and to another industrial dye, cibalackrot, which differs only by the presence of a phenyl substituent on each of the carbon atoms adjacent to a carbonyl.

Table 1 Energetic Effects of Intermolecular Interaction in Ethylene Dimer (eV)a Structure 1 2 3

4

5

6

(hAlA|hBlB)

0.11

0

0

0.01

0.06

0

ΔEDS

1.58, 1.01

0, 0

0, 0

0.49, 0.27

0.69, 0.47

0, 0

ΔESF+

0.42, 0.32

0.05, 0.02

0.25, 0.12

0.42, 0.22

0.17, 0.15

0.04, 0.02

ΔESF

1.16, 0.69

0.05, 0.02

0.25, 0.12

0.08, 0.05

0.52, 0.32

0.04, 0.02

|TRP+|

0.13, 0.07

0.46, 0.23

0.14, 0.07

0.04, 0.02

0.08, 0.04

0.05, 0.03

|TRP|

0.86, 0.43

0.14, 0.07

0.08, 0.04

0.04, 0.02

0.05, 0.03

a

0.46, 0.23

For the first entry in a column, ΔE = ΔE( D D ) = ΔE( D D ) = 1 eV; for the second, ΔE = 2 eV. 1

+

1

+

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Fig. 6 The six highest local maxima of jTRPj2 for a pair of ethylene molecules. The four NAOs are each colored differently, with a different shade in each of their two lobes. Projections of the two molecules onto the interior walls of the front octant are also shown. jTRPj2 values are given in eV2.

The geometry of this molecule was optimized with the B3LYP/6-311 +G* method and its frontier orbitals, HOMO and LUMO, were expressed in the basis of NAOs of π-symmetry by performing a HF/6-311+G calculation followed by Weinhold’s NBO analysis with the Gaussian program suite in a single run. Each NAO was expressed as fixed combination of four pz contracted Gaussian functions, identical to the basis set used in the SCF

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Fig. 7 Chemical structure of diindolo[3,2,1-de;3ʹ,2ʹ,1ʹ-ij][1,5]naphthyridine-6,13-dione.

procedure. The NAOs on each molecule were L€ owdin orthogonalized and used as a representation of HOMO and LUMO. For the evaluation of jTRPj we used ΔE ¼ 1 eV. We calculated the values of the search function F defined in Eq. (45) on ˚ a 6-D grid of 3.13 x 108 points with steps Δ (20° for rotations and 0.75 A for translations). We used them for preselection of physically realizable dimer structures with the largest jTRPj and found 58 103 structures for which F(xi Δ) > F(xi) < F(xi + Δ) in all six dimensions. Subsequently, all 10,000 structures of local minima of F for which jFj exceeded 0.01% of its value in the deepest minimum were optimized starting from the preselected structures. We then removed all duplicates (either random, or resulting from symmetry of the system). Finally, we obtained 2500 structures of dimers with search function values in the range from 0.3 to 0.3 1014 eV2. Positive search function values reflect situations in which repulsion exceeds jTRPj2 and are of no interest. They were eliminated, leaving 1800 structures. At most of these, jTRPj is very small. Elimination of points with jTRPj smaller than 0.01% of the maximum jTRPj value found completed the search and provided about 350 dimer structures. Only a few of them are really significant and about a dozen have a jTRPj value that exceeds 10% of the value less than one order of magnitude lower than the best structure (i.e., a difference of two orders in expected rate constant). The first six best structures are shown in Fig. 8. They are all approximately slip-stacked and mostly also rotated to some degree. The jTRPj2 values in eV2 are shown. The first structure among those that do not contain two nearly coplanar stacked chromophores is 250th on the list of optimized structures and has a jTRPj value of 15 meV. In Table 2, we show the results obtained for the excitonic coupling matrix element (hAlAjhBlB) evaluated in the point charge approximation, Davydov splitting ΔEDS, the endoergicity ΔESF , and the electronic matrix element TRP for SF from the two exciton states at the six dimer geometries

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Fig. 8 The first six best structures of diindolo[3,2,1-de;3ʹ,2ʹ,1ʹ-ij][1,5]naphthyridine-6,13dione dimers in side and top projections and their jTRPj2 values in eV2, numbered 1–6 from left to right in the order of decreasing jTRPj2 value.

Table 2 Energetic Effects of Intermolecular Interaction in a Dimer of the Heterocycle of Fig. 7 (eV)a Structure 1 2 3 4 5 6

(hAlA| hBlB)

0.00

ΔEDS

3.38, 1.70 0.56, 0.28

0.36, 0.14 0.14, 0.06 1.15, 0.57 1.14, 0.57

ΔESF+

3.13, 1.57 0.16, 0.08

0.02, 0.01

0.03, 0.01

ΔESF

0.25, 0.13

0.34, 0.15

0.11, 0.05 0.97, 0.48 0.95, 0.47

|TRP+|

0.93, 0.46 0.45, 0.23

0.60, 0.30 0.38, 0.19 0.00, 0.00 0.00, 0.00

|TRP|

0.00, 0.00 0.17, 0.08

0.00, 0.00 0.15, 0.07 0.51, 0.25 0.51, 0.25

a

0.00

0.39, 0.19

0.02

0.00

0.00

0.18, 0.09

0.00

0.19, 0.10

For the first entry in a column, ΔE = ΔE(1D+D) = ΔE(1DD+) = 1 eV; for the second, ΔE = 2 eV.

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shown in Fig. 8 (a positive value of ΔESF implies a process endoergic relative to expectations based on properties of the monomer). As in Table 1, results are shown for two choices of ΔE, 1 and 2 eV. The excitonic coupling is very small when evaluated as the electrostatic interaction of transition charge densities. When approximated as a dipole–dipole interaction, it is about four times smaller than was the case for two ethylenes in Table 1, because the transition dipole is only about half of that in ethylene. The Davydov splitting is totally dominated by the term mediated by the charge-transfer states and seems unrealistically large in this approximation, but the trend within the six structures shown is probably reliable. Similarly as in the case of ethylene, structure 1, which has the largest jTRPj2 value, also has by far the largest Davydov splitting and as a result suffers the most from the endoergicity induced by intermolecular interactions. It would be a very good choice for H-type excimer formation, but a very poor choice for SF. Structures 5 and 6 are somewhat better, but still very unfavorable energetically. In the present approximation, in structures 2 and 4, intermolecular interactions also make the formation of the biexciton less favorable. However, in structure 3 we find an ideal combination. It has one of the largest jTRP+j2 values and at the same time, its energy remains essentially unaffected by intermolecular interactions. It is thus expected to be the best dimer geometry choice for SF in a dimer of this heterocycle.

4. OUTLOOK The ability to search through a six-dimensional space of dimer geometries in pursuit of approximate positions of local extrema of the SF electronic matrix element (jTRPj), the Davydov splitting (EDS), the endoergicity of SF (ΔESF), and the biexciton binding energy offers interesting possibilities for design of structures optimized for SF, either by crystal engineering of monomers or by synthesis and crystallization of covalent dimers. We plan to release a computer program that will allow any interested party to perform such a search for a chromophore of interest. First, however, it will be necessary to build confidence in results obtained by the ruthlessly simplified procedure outlined here. The agreement of the results of the full and the simplified HOMO/LUMO model is encouraging, but obtaining a comparison with trends in both experimental data and high-level computational results is critically important. Our hope is that the structures of the local extrema obtained by the

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simple and fast procedure will indicate the approximate locations in the six-dimensional search space in which focused searches by high-level methods should be performed. We are acutely aware of the inadequacy of all treatments that are limited to the treatment of dimers. If testing results are encouraging and provide some assurance that the simple method is valuable, one can imagine extending it to more general treatments of larger aggregates or single crystals.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Biosciences, and Geosciences, under award number DE-SC0007004. Work in Prague was supported by the Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic (RVO: grant 15-19143S. We are grateful to Alexandr Zaykov and Petr 61388963) and GACR Felkel for assistance with computations and computer graphics, respectively.

APPENDIX The complete expressions for the matrix elements of the 6 6 Hamiltonian matrix of the HOMO/LUMO model without neglect of intermolecular overlap are listed below. ^ 0S0i: hS0S0jHjS ^ S0 S0 ¼NS0S0 2 f4ShAhB 3 ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ S0 S0 H +½2FhAhA + 2ðhA hA jhA hA Þ + 6ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ 2FhBhB + 2ðhB hB jhB hB ÞShAhB 2 4FhAhB ShAhB + 2FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhB hB jhB hB Þ + 2FhBhB g

^ 0S1i: hS0S0jHjS ^ S0 S1 ¼NS0S0 NS0S1 ð21=2 f½FhAlB ðhA hA jhA lB Þ 2ðhA lB jhB hB Þ S0 S0 H + ðhA hB jhB lB ÞShAhB 3 + f3ShAlB ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ + FhBlB ðhA hA jhB lB Þ ðhB hB jhB lB Þ 2ðhA hB jhA lB ÞgShAhB 2 + f½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB Þ ShAlB + 3ðhA hB jhB lB Þ ðhA lB jhB hB Þ + FhAlB gShAhB + FhAhB ShAlB FhBlB gÞ

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^ 1S0i: hS0S0jHjS ^ S1 S0 ¼NS0S0 NS1S0 21=2 ð½FlAhB 2ðhA hA jlA hB Þ ðlA hB jhB hB Þ S0 S0 H + ðhA lA jhA hB ÞShAhB 3 + f3SlAhB ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ FhAlA + ðhA hA jhA lA Þ + ðhA lA jhB hB Þ + 2ðhA hB jlA hB ÞgShAhB 2 f½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB Þ SlAhB ðhA hA jlA hB Þ + 3ðhA lA jhA hB Þ + FlAhB gShAhB FhAhB SlAhB + FhAlA Þ ^ +Di: hS0S0jHjD + ^ D D ¼NS0S0 N + 21=2 ð½fFhBlB 2 ðhA hA jhB lB Þ ðhB hB jhB lB Þ S0 S0 H + ðhA hB jhA lB ÞgShAhB 3 + fShAlB ½FhBhB ðhB hB jhB hB Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ FhAlB + ðhA hA jhA lB Þ + ðhA lB jhB hB Þ + 2ðhA hB jhB lB ÞgShAhB 2 + f½2FhAhB + 2ðhA hA jhA hB ÞShAlB + ðhA hA jhB lB Þ 3ðhA hB jhA lB Þ FhBlB gShAhB + ½FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhB hB jhB hB Þ + 2FhBhB ShAlB + FhAlB Þ

^ D+i: hS0S0jHjD + ^ D D ¼ NS0S0 N + 21=2 f½FhAlA ðhA hA jhA lA Þ 2ðhA lA jhB hB Þ S0 S0 H + ðhA hB jlA hB ÞShAhB 3 + fSlAhB ½FhAhA + ðhA hA jhA hA Þ + 2ðhA hA jhB hB Þ ðhA hB jhA hB Þ FlAhB + ðhA hA jlA hB Þ + ðlA hB jhB hB Þ + 2ðhA lA jhA hB ÞgShAhB 2 f½2FhAhB 2ðhA hB jhB hB ÞSlAhB + 3ðhA hB jlA hB Þ ðhA lA jhB hB Þ + FhAlA gShAhB + ½2FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB ÞSlAhB + FlAhB g

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^ 1T1i: hS0S0jHjT

^ T1 T1 ¼NS0S0 NT1T1 31=2 ðf½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ S0 S0 H + 2ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB ÞSlAhB + ½ðhA lA jhB hB Þ FhAlA + ðhA hA jhA lA Þ ðhA hB jlA hB ÞShAhB ðhA hA jlA hB Þ + FlAhB gShAlB + f½ðhA hA jhB lB Þ FhBlB + ðhB hB jhB lB Þ ðhA hB jhA lB ÞShAhB ðhA lB jhB hB Þ + FhAlB gSlAhB + ShAhB 2 ðhA lA jhB lB Þ + ½ðhA lA jhA lB Þ ðlA hB jhB lB Þ ShAhB + ðhA lB jlA hB ÞÞ

^ 1S0i: hS1S0jHjS

^ S1 S0 ¼NS1S0 2 ðf½8ðhA hA jlA hB Þ + 4ðhA lA jhA hB Þ S1 S0 H 4ðlA hB jhB hB Þ + 4FlAhB SlAhB + 2ðhA hA jhB hB Þ + 2ðhA hA jlA lA Þ ðhA hB jhA hB Þ ðhA lA jhA lA Þ + ðhB hB jhB hB Þ + ðlA hB jlA hB Þ + ðlA lA jhB hB Þ FhBhB FlAlA gShAhB 2 + f½4ðhA hA jhA hB Þ 4ðhA hB jhB hB Þ + 4FhAhB SlAhB 2 + ½6ðhA hB jlA hB Þ + 2ðhA lA jhB hB Þ 2FhAlA + 2ðhA hA jhA lA Þ SlAhB + 2ðhA hA jhA hB Þ 2ðhA hB jlA lA Þ 2FhAhB 2ðhA lA jlA hB ÞShAhB +½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ FhBhB + ðhB hB jhB hB ÞSlAhB 2 + ½2ðhA hA jlA hB Þ 4ðhA lA jhA hB Þ 2FlAhB SlAhB + FhAhA 4ðhA hA jhB hB Þ ðhA hA jlA lA Þ + 2ðhA hB jhA hB Þ + 2ðhA lA jhA lA Þ ðhB hB jhB hB Þ + 2FhBhB + FlAlA ðhA hA jhA hA ÞgÞ

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^ 0S1i: hS1S0jHjS ^ S0 S1 ¼NS1S0 NS0S1 ðfShAhB 3 ½FlAlB 2ðhA hA jlA lB Þ 2ðlA lB jhB hB Þ S1 S0 H + ðlA hB jhB lB Þ + ðhA lA jhA lB Þ + fShAlB ½FlAhB 2ðhA hA jlA hB Þ ðlA hB jhB hB Þ + ðhA lA jhA hB Þ +SlAhB ½FhAlB ðhA hA jhA lB Þ 2ðhA lB jhB hB Þ + ðhA hB jhB lB Þ +3SlAlB ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ + 3ðhA hB jlA lB Þ ðhA lA jhB lB Þ + ðhA lB jlA hB Þg ShAhB 2 + f½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ + ðhA hB jhA hB Þ FhBhB + ðhB hB jhB hB ÞSlAlB + f2SlAhB + ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ FhAlA + ðhA hA jhA lA Þ + ðhA lA jhB hB Þ + ðhA hB jlA hB ÞgShAlB + ½ðhA hA jhB lB Þ + ðhA hB jhA lB Þ FhBlB + ðhB hB jhB lB ÞSlAhB + ðhA hA jlA lB Þ 2ðhA lA jhA lB Þ 2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ FlAlB Þg ShAhB FhAhB SlAlB + ½SlAhB ðhA hB jhA hB Þ ðhA lA jhA hB Þ ShAlB SlAhB ðhA hB jhB lB Þ ðhA hB jlA lB Þ + 2 ðhA lA jhB lB ÞÞ

^ +Di: hS1S0jHjD

+ ^ D D ¼NS1S0 N + ðf½4ðhA hA jhB lB Þ + 2ðhA hB jhA lB Þ 2ðhB hB jhB lB Þ S1 S0 H + 2FhBlB ÞSlAhB + ð2ðhA hA jhB hB Þ ðhA hB jhA hB Þ + ðhB hB jhB hB Þ FhBhB SlAlB + 2ðhA hA jlA lB Þ ðhA lA jhA lB Þ + ðlA hB jhB lB Þ + ðlA lB jhB hB Þ FlAlB gShAhB 2 + ðf½2ðhA hA jhB hB Þ ðhA hB jhA hB Þ + ðhB hB jhB hB Þ FhBhB ShAlB + ðhA hA jhA lB Þ + 3ðhA hB jhB lB Þ + ðhA lB jhB hB Þ FhAlB gSlAhB + ½2FhAhB + 2ðhA hA jhA hB ÞSlAlB + ½FlAhB + 2ðhA hA jlA hB Þ ðhA lA jhA hB ÞShAlB 2ðhA hB jlA lB Þ ðhA lA jhB lB Þ ðhA lB jlA hB ÞÞShAhB + f½FhAhB + ðhA hA jhA hB Þ ShAlB + ðhA hA jhB lB Þ 2ðhA hB jhA lB Þ FhBlB gSlAhB + ½FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhB hB jhB hB Þ + 2FhBhB SlAlB +½FhAlA ðhA hA jhA lA ÞShAlB ðhA hA jlA lB Þ + 2ðhA lA jhA lB Þ + FlAlB Þ

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^ D+i: hS1S0jHjD

+ ^ D D ¼NS1S0 N + fShAhB 3 ½FlAlA 2ðhA hA jlA lA Þ 2ðlA lA jhB hB Þ S1 S0 H + ðhA lA jhA lA Þ + ðlA hB jlA hB Þ + ½3ðhA hB jlA lA Þ + 3FhAhB 3ðhA hA jhA hB Þ 3ðhA hB jhB hB ÞShAhB 2 + f½FhAhA + 2ðhA hA jhB hB Þ ðhA hB jhA hB Þ + ðhA hA jhA hA ÞSlAhB 2 + ½2ðhA hA jlA hB Þ 2FlAhB + 2ðlA hB jhB hB Þ 2ðhA lA jhA hB Þ SlAhB FhAhA + 3ðhA hA jhB hB Þ + ðhA hA jlA lA Þ + ðhA hB jhA hB Þ 2ðhA lA jhA lA Þ + ðhB hB jhB hB Þ 2ðlA hB jlA hB Þ + ðlA lA jhB hB Þ FhBhB FlAlA + ðhA hA jhA hA ÞgShAhB + ½FhAhB + ðhA hB jhB hB ÞSlAhB 2 +½2ðhA lA jhB hB Þ + 2FhAlA ÞSlAhB ðhA hB jlA lA Þ FhAhB + 2ðhA lA jlA hB Þg

^ 1T1i: hS1S0jHjT ^ T1 T1 ¼NS1S0 NT1T1 ð3=2Þ1=2 f½ðlA lA jhB lB Þ + FhBlB 2ðhA hA jhB lB Þ S1 S0 H ðhB hB jhB lB Þ + ðhA hB jhA lB ÞShAhB 2 + f½ðlA lA jhB hB Þ FhBhB + ðhB hB jhB hB Þ + 2ðhA hA jhB hB Þ ðhA hB jhB hB Þ FlAlA + 2ðhA hA jlA lA Þ ðhA lA jhA lA Þ ðlA hB jlA hB ÞShAlB + ½ðhA lA jhB hB Þ FhAlA + ðhA hA jhA lA Þ hA hB jlA hB ÞSlAlB + ðhA hB jhB lB Þ ðhA lA jlA lB Þ + ðhA lB jhB hB Þ ðhA lB jlA lA Þ FhAlB + ðhA hA jhA lB ÞgShAhB + ½ðhA hA jhB lB Þ FhBlB + ðhB hB jhB lB Þ ðhA hB jhA lB ÞSlAhB 2 + f½ðhA lA jhB hB Þ + FhAlA ðhA hA jhA lA Þ + ðhA hB jlA hB ÞShAlB + ½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB ÞSlAlB ðhA hA jlA lB Þ + 2ðhA lA jhA lB Þ ðlA hB jhB lB Þ ðlA lB jhB hB Þ + FlAlB gSlAhB + ½FhAhB + ðhA hA jhA hB Þ ðhA hB jlA lA Þ + ðhA lA jlA hB ÞShAlB + ½ðhA hA jlA hB Þ + FlAhB SlAlB ðhA hB jhA lB Þ + ðlA hB jlA lB Þg

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^ 0S1i: hS0S1jHjS ^ S0 S1 ¼ NS0S1 2 ðf4 ShAlB ½FhAlB ðhA hA jhA lB Þ 2ðhA lB jhB hB Þ S0 S1 H + ðhA hB jhB lB Þ FhAhA + ðhA hA jhA hA Þ + 2ðhA hA jhB hB Þ ðhA hB jhA hB Þ FlBlB + ðhA hA jlB lB Þ + 2ðhB hB jlB lB Þ + ðhA lB jhA lB Þ ðhB lB jhB lB ÞgShAhB 2 + f4ShAlB 2 ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ + ½2ðhA hA jhB lB Þ + 6ðhA hB jhA lB Þ 2FhBlB + 2ðhB hB jhB lB ÞShAlB 2ðhA hB jlB lB Þ 2FhAhB + 2ðhA hB jhB hB Þ 2ðhA lB jhB lB ÞgShAhB + ½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ FhBhB + ðhB hB jhB hB ÞShAlB 2 + ½4ðhA hB jhB lB Þ + 2ðhA lB jhB hB Þ 2FhAlB ShAlB + 2FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhB hB jlB lB Þ + 2ðhB lB jhB lB Þ + FhBhB ðhB hB jhB hB Þ + FlBlB Þ

^ +Di: hS0S1jHjD + ^ D D ¼ NS0S1 N + fShAhB 3 ½FlBlB 2ðhA hA jlB lB Þ 2ðhB hB jlB lB Þ S0 S1 H + ðhA lB jhA lB Þ + ðhB lB jhB lB Þ + ½3FhAhB 3ðhA hA jhA hB Þ 3ðhA hB jhB hB Þ + 3ðhA hB jlB lB ÞShAhB 2 + f½FhBhB ðhB hB jhB hB Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ ShAlB 2 + ½2ðhA lB jhB hB Þ 2FhAlB + 2ðhA hA jhA lB Þ 2ðhA hB jhB lB ÞShAlB FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + ðhA hA jlB lB Þ 2ðhA lB jhA lB Þ + ðhB hB jlB lB Þ 2ðhB lB jhB lB Þ FhBhB + ðhB hB jhB hB Þ FlBlB gShAhB + ½FhAhB + ðhA hA jhA hB ÞShAlB 2 + ½2ðhA hA jhB lB Þ + 2FhBlB ShAlB ðhA hB jlB lB Þ + 2 ðhA lB jhB lB Þ FhAhB g

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+ ^ D D ¼NS0S1 N + ðf2ShAlB ½FhAlA ðhA hA jhA lA Þ S0 S1 H 2ðhA lA jhB hB + hA hB jlA hB Þ SlAlB ½FhAhA ðhA hA jhA hA Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ FlAlB + ðhA hA jlA lB Þ + 2ðlA lB jhB hB Þ ðlA hB jhB lB Þ + ðhA lA jhA lB gShAhB 2 + ðfSlAhB ½FhAhA ðhA hA jhA hA Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ FlAhB + ðhA hA jlA hB Þ + ðlA hB jhB hB Þ + 3ðhA lA jhA hB ÞgShAlB + ½FhAlB + 2ðhA lB jhB hB Þ ðhA hB jhB lB ÞSlAhB + ½2FhAhB + 2ðhA hB jhB hB ÞSlAlB 2ðhA hB jlA lB Þ ðhA lA jhB lB Þ ðhA lB jlA hB ÞÞShAhB + f½FhAhB + ðhA hB jhB hB ÞSlAhB 2ðhA hB jlA hB Þ + ðhA lA jhB hB Þ FhAlA gShAlB + ½FhBlB ðhB hB jhB lB ÞSlAhB + ½2FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB ÞSlAlB + 2ðlA hB jhB lB Þ ðlA lB jhB hB Þ + FlAlB Þ

^ 1T1i: hS0S1jHjT ^ T1 T1 ¼NS0S1 NT1T1 ð3=2Þ1=2 f½FhAlA ðhA hA jhA lA Þ 2ðhA lA jhB hB Þ S0 S1 H + ðhA hB jlA hB Þ + ðhA lA jlB lB ÞShAhB 2 f½FhAhA ðhA hA jhA hA Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ ðhA hA jlB lB Þ + FlBlB 2ðhB hB jlB lB Þ + ðhA lB jhA lB Þ + ðhB lB jhB lB ÞSlAhB + ½ðhA hA jhB lB Þ + FhBlB ðhB hB jhB lB Þ + ðhA hB jhA lB ÞSlAlB ðhA hA jlA hB Þ ðhA lA jhA hB Þ + ðlA hB jlB lB Þ + ðlA lB jhB lB Þ + FlAhB ðlA hB jhB hB ÞgShAhB + ½ðhA lA jhB hB Þ FhAlA + ðhA hA jhA lA Þ ðhA hB jlA hB ÞShAlB 2 f½ðhA hA jhB lB Þ FhBlB + ðhB hB jhB lB Þ ðhA hB jhA lB ÞSlAhB + ½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ 2ðhA hB jhA hB Þ FhBhB + ðhB hB jhB hB ÞSlAlB + ðhA hA jlA lB Þ + ðhA lA jhA lB Þ 2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ FlAlB gShAlB ½FhAhB ðhA hB jhB hB Þ + ðhA hB jlB lB Þ ðhA lB jhB lB ÞSlAhB +½ðhA lB jhB hB Þ + FhAlB SlAlB ðhA hB jlA hB Þ + ðhA lB jlA lB Þg

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^ +D-i: hD+D-jHjD + + ^ D D ¼ N + 2 f½ðhB hB jlB lB Þ + ðhB lB jhB lB Þ FhBhB + ðhB hB jhB hB Þ D D H + 2ðhA hA jhB hB Þ ðhA hB jhA hB Þ FlBlB + 2ðhA hA jlB lB Þ ðhA lB jhA lB ÞShAhB 2 + f½2FhBlB + 4 ðhA hA jhB lB Þ 2ðhA hB jhA lB ÞShAlB 2ðhA hB jlB lB Þ 2ðhA lB jhB lB Þ 2FhAhB + 2ðhA hA jhA hB ÞgShAhB + ½ðhB hB jhB hB Þ + 2FhBhB 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB ÞShAlB 2 + ½2FhAlB 2ðhA hA jhA lB ÞShAlB + FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhA hA jlB lB Þ + 2ðhA lB jhA lB Þ ðhB hB jhB hB Þ + 2FhBhB + FlBlB g

^ D+i: hD+D-jHjD + + ^ D D ¼ N + N + fShAhB 3 ½FlAlB 2ðhA hA jlA lB Þ 2ðlA lB jhB hB Þ D D H + ðlA hB jhB lB Þ + ðhA lA jhA lB Þ + fShAlB ½FlAhB 2ðhA hA jlA hB Þ ðlA hB jhB hB Þ + ðhA lA jhA hB Þ SlAhB ½FhAlB ðhA hA jhA lB Þ 2ðhA lB jhB hB Þ + ðhA hB jhB lB Þ + 3SlAlB ½FhAhB ðhA hA jhA hB Þ ðhA hB jhB hB Þ + 3 ðhA hB jlA lB Þ + ðhA lA jhB lB Þ ðhA lB jlA hB Þg ShAhB 2 + ðf2SlAhB ½FhAhB ðhA hA jhA hB Þ hA hB jhB hB Þ FhAlA + ðhA hA jhA lA Þ + ðhA lA jhB hB Þ 3ðhA hB jlA hB gShAlB + ½ðhA hA jhB lB Þ 3ðhA hB jhA lB Þ FhBlB + ðhB hB jhB lB ÞSlAhB + ½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ + ðhA hB jhA hB Þ FhBhB + ðhB hB jhB hB ÞSlAlB + ðhA hA jlA lB Þ 2ðhA lA jhA lB Þ 2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ FlAlB ÞShAhB + f½2FhAhA 2ðhA hA jhA hA Þ 6ðhA hA jhB hB Þ + 3ðhA hB jhA hB Þ + 2FhBhB 2ðhB hB jhB hB ÞSlAhB 2ðhA hA jlA hB Þ + ðhA lA jhA hB Þ + 2FlAhB ÞgShAlB + ½ðhA hB jhB lB Þ 2ðhA lB jhB hB Þ + 2FhAlB SlAhB FhAhB SlAlB ðhA hB jlA lB Þ + 2ðhA lB jlA hB Þg

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^ 1T1i: hD+DjHjT

^ T1 T1 ¼N + NT1T1 ð3=2Þ1=2 ðfSlAlB ½FhBlB 2ðhA hA jhB lB Þ ðhB hB jhB lB Þ D + D H + ðhA hB jhA lB Þ + ðlA lB jhB lB ÞgShAhB 2 + ðf½2ðhA hA jhB lB Þ ðhA hB jhA lB Þ + ðhB hB jhB lB Þ FhBlB SlAhB + ½2ðhA hA jhB hB Þ ðhA hB jhA hB Þ + ðhB hB jhB hB Þ FhBhB SlAlB + 2ðhA hA jlA lB Þ ðhA lA jhA lB Þ 2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ FlAlB gShAlB ðhA lB jhB lB ÞSlAhB + ½ðhA hB jhB lB Þ + ðhA lB jhB hB Þ FhAlB + ðhA hA jhA lB ÞSlAlB + ðhA hA jhA lA Þ ðhA hB jlA hB Þ FhAlA + ðhA lA jhB hB Þ ðhA lA jlB lB Þ ðhA lB jlA lB ÞÞShAhB + fSlAhB ½FhBhB ðhB hB jhB hB Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FlAhB 2ðhA hA jlA hB Þ + ðhA lA jhA hB ÞgShAlB 2 + f½ðhA hB jhB lB Þ 2ðhA lB jhB hB Þ + 2FhAlB 2ðhA hA jhA lB ÞSlAhB + ½FhAhB + ðhA hA jhA hB ÞSlAlB ðhA hB jlA lB Þ + 2ðhA lB jlA hB ÞgShAlB + ½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhA hA jlB lB Þ + 2ðhA lB jhA lB Þ ðhB hB jlB lB Þ + FhBhB ðhB hB jhB hB Þ + FlBlB SlAhB SlAlB ðhA hB jhA lB Þ ðhA hA jlA hB Þ + ðlA hB jlB lB Þ + FlAhB gÞ

^ D+i: hDD+jHjD + + ^ D D ¼N + 2 f½FhAhA + ðhA hA jhA hA Þ + 2ðhA hA jhB hB Þ D D H ðhA hB jhA hB Þ + ðhA hA jlA lA Þ + ðhA lA jhA lA Þ FlAlA + 2ðlA lA jhB hB Þ ðlA hB jlA hB ÞShAhB 2 f½2FhAlA + 4ðhA lA jhB hB Þ 2ðhA hB jlA hB ÞSlAhB 2ðhA hB jlA lA Þ 2ðhA lA jlA hB Þ 2FhAhB + 2ðhA hB jhB hB ÞgShAhB + ½2FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB ÞSlAhB 2 + ½2 + FlAhB 2ðlA hB jhB hB Þ SlAhB + 2FhAhA ðhA hA jhA hA Þ 4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + 2ðlA hB jlA hB Þ ðlA lA jhB hB Þ + FhBhB ðhB hB jhB hB Þ + FlAlA g

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^ 1T1i: hDD+jHjT + ^ T1 T1 ¼ N + NT1T1 ð3=2Þ1=2 ½ðSlAlB fFlAlB ½FhAlA ðhA hA jhA lA Þ D D H 2ðhA lA jhB hB Þ + ðhA hB jlA hB Þ + ðhA lA jlA lB ÞgShAhB 2 ðfShAlB ½FhAlA ðhA hA jhA lA Þ 2ðhA lA jhB hB Þ + ðhA hB jlA hB Þ + SlAlB ½FhAhA ðhA hA jhA hA Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FlAlB ðhA hA jlA lB Þ 2ðlA lB jhB hB Þ + ðlA hB jhB lB Þ + 2 ðhA lA jhA lB ÞgSlAhB + ShAlB ðhA lA jlA hB Þ + ½ðhA hA jlA hB Þ ðhA lA jhA hB Þ + FlAhB ðlA hB jhB hB ÞSlAlB ðhA hA jhB lB Þ + ðlA hB jlA lB Þ + ðlA lA jhB lB Þ + FhBlB ðhB hB jhB lB Þ + ðhA hB jhA lB ÞÞ ShAhB fShAlB ½FhAhA ðhA hA jhA hA Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FhAlB 2ðhA lB jhB hB Þ + ðhA hB jhB lB Þg SlAhB 2 + f½2ðhA hA jlA hB Þ ðhA lA jhA hB Þ 2FlAhB + 2ðlA hB jhB hB ÞShAlB ½FhAhB ðhA hB jhB hB ÞSlAlB + ðhA hB jlA lB Þ 2ðhA lB jlA hB ÞgSlAhB + ½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ ðhA hA jlA lA Þ + 2ðlA hB jlA hB Þ ðlA lA jhB hB Þ + FhBhB ðhB hB jhB hB Þ + FlAlA ShAlB + SlAlB ðhA hB jlA hB Þ + ðhA lB jhB hB Þ ðhA lB jlA lA Þ FhAlB

^ 1T1i: hT1T1jHjT

^ T1 T1 ¼ NT1T1 2 ð1=2Þðf½ 8ðhA hA jlA lB Þ + 4ðhA lA jhA lB Þ + 4ðlA hB jhB lB Þ 8ðlA lB jhB hB Þ T1 T1 H + 4FlAlB SlAlB + FlBlB 2ðhA hA jlA lA Þ 2ðhA hA jlB lB Þ + ðhA lA jhA lA Þ + ðhA lB jhA lB Þ 2ðhB hB jlB lB Þ + ðhB lB jhB lB Þ + ðlA hB jlA hB Þ 2ðlA lA jhB hB Þ + ðlA lA jlB lB Þ + 2ðlA lB jlA lB Þ + FlAlA gShAhB 2 + ðf½8 ðhA hA jlA lB Þ 4ðhA lA jhA lB Þ 4ðlA hB jhB lB Þ + 8ðlA lB jhB hB Þ 4ðFlAlB ÞSlAhB + ½8ðhA hA jlA hB Þ 4ðhA lA jhA hB Þ + 4ðlA hB jhB hB Þ 4FlAhB SlAlB + 4ðhA hA jhB lB Þ 2ðhA hB jhA lB Þ + 2ðhB hB jhB lB Þ 4ðlA hB jlA lB Þ 2ðlA lA jhB lB Þ 2FhBlB gShAlB + f½4ðhA hA jhA lB Þ 4ðhA hB jhB lB Þ 4FhAlB + 8ðhA lB jhB hB ÞSlAlB + 2ðhA hA jhA lA Þ 2ðhA hB jlA hB Þ 2FhAlA + 4ðhA lA jhB hB Þ 2ðhA lA jlB lB Þ 4ðhA lB jlA lB ÞgSlAhB + ½4ðhA hA jhA hB Þ + 4FhAhB 4ðhA hB jhB hB ÞSlAlB 2 +½8ðhA hB jlA lB Þ + 2ðhA lA jhB lB Þ 4ðhA lB jlA hB ÞSlAlB 2ðhA hA jhA hB Þ + 2FhAhB 2ðhA hB jhB hB Þ + 2ðhA hB jlA lA Þ + 2ðhA hB jlB lB Þ 2ðhA lA jlA hB Þ 2ðhA lB jhB lB ÞÞShAhB + f½8ðhA hA jlA hB Þ + 4ðhA lA jhA hB Þ 4ðlA hB jhB hB Þ + 4FlAhB SlAhB 2ðhA hA jhB hB Þ 2ðhA hA jlA lA Þ + ðhA hB jhA hB Þ + ðhA lA jhA lA Þ ðhB hB jhB hB Þ + FhBhB + 3ðlA hB jlA hB Þ ðlA lA jhB hB Þ + FlAlA gShAlB 2 + f½4ðhA hA jhA lB Þ + 4ðhA hB jhB lB Þ + 4FhAlB 8ðhA lB jhB hB ÞSlAhB 2 + f½4ðhA hA jhA hB Þ 4FhAhB + 4ðhA hB jhB hB ÞSlAlB 4ðhA hB jlA lB Þ + 2ðhA lA jhB lB Þ + 8ðhA lB jlA hB ÞgSlAhB + ½6ðhA hB jlA hB Þ 2FhAlA + 2ðhA hA jhA lA Þ + 2ðhA lA jhB hB ÞSlAlB 2ðhA hA jhA lB Þ 2ðhA lA jlA lB Þ + 2FhAlB 2ðhA lB jhB hB Þ + 2ðhA lB jlA lA ÞgShAlB + ½FhAhA ðhA hA jhA hA Þ 2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FlBlB ðhA hA jlB lB Þ 2ðhB hB jlB lB Þ + 3ðhA lB jhA lB Þ + ðhB lB jhB lB ÞSlAhB 2 + f½2ðhA hA jhB lB Þ 6ðhA hB jhA lB Þ 2FhBlB + 2ðhB hB jhB lB ÞSlAlB 2ðhA hA jlA hB Þ + 2ðlA hB jlB lB Þ 2ðlA lB jhB lB Þ + 2FlAhB 2ðlA hB jhB hB ÞgSlAhB + ½FhAhA ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ + 4ðhA hB jhA hB Þ + FhBhB ðhB hB jhB hB ÞSlAlB 2 + ½2ðhA hA jlA lB Þ 2ðlA lB jhB hB Þ + 2FlAlB SlAlB + 2FhAhA + 2FlBlB 2ðhA hA jhA hA Þ 6ðhA hA jhB hB Þ 2ðhA hA jlA lA Þ 2ðhA hA jlB lB Þ + 5ðhA hB jhA hB Þ + 3ðhA lB jhA lB Þ 2ðhB hB jhB hB Þ 2ðhB hB jlB lB Þ + 2FhBhB + 3ðlA hB jlA hB Þ 2ðlA lA jhB hB Þ + 2ðlA lA jlB lB Þ + ðlA lB jlA lB Þ + 2FlAlA Þ

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38. Dillon, R. J.; Piland, G. B.; Bardeen, C. J. Different Rates of Singlet Fission in Monoclinic Versus Orthorhombic Crystal Forms of Diphenylhexatriene. J. Am. Chem. Soc. 2013, 135, 17278–17281. 39. Roberts, S. T.; McAnally, R. E.; Mastron, J. N.; Webber, D. H.; Whited, M. T.; Brutchey, R. L.; Thompson, M. E.; Bradforth, S. E. Efficient Singlet Fission Discovered in a Disordered Acene Film. J. Am. Chem. Soc. 2012, 134, 6388–6400. 40. Renaud, N.; Sherratt, P. A.; Ratner, M. A. Mapping the Relation Between Stacking Geometries and Singlet Fission Yield in a Class of Organic Crystals. J. Phys. Chem. Lett. 2013, 4, 1065–1069. 41. Matsika, S.; Feng, X.; Luzanov, A. V.; Krylov, A. I. What We Can Learn From the Norms of One-Particle Density Matrices, and What We Can’t: Some Results for Interstate Properties in Model Singlet Fission Systems. J. Phys. Chem. A 2014, 118, 11943–11955. 42. Teichen, P. E.; Eaves, J. D. Collective Aspects of Singlet Fission in Molecular Crystals. J. Chem. Phys. 2015, 143, 044118. 43. Feng, X.; Luzanov, A. V.; Krylov, A. I. Fission of Entangled Spins: An Electronic Structure Perspective. J. Phys. Chem. Lett. 2013, 4, 3845–3852. 44. Parker, S. M.; Seideman, T.; Ratner, M. A.; Shiozaki, T. Model Hamiltonian Analysis of Singlet Fission From First Principles. J. Phys. Chem. C 2014, 118, 12700–12705. 45. Marcus, R. A. On the Theory of Chemiluminescent Electron Transfer Reactions. J. Chem. Phys. 1965, 43, 2654–2657. 46. May, V.; K€ uhn, O. Charge and Energy Transfer Dynamics in Molecular Systems. John Wiley & Sons: Weinheim, Germany, 2008. 2004. 47. Scholes, G. D.; Ghiggino, K. P. Rate Expressions for Excitation Transfer. I. Radiationless Transition Theory Perspective. J. Chem. Phys. 1994, 101, 1251–1261. 48. Harcourt, R. D.; Scholes, G. D.; Ghiggino, K. P. Rate Expressions for Excitation Transfer. II. Electronic Considerations of Direct and Through-Configuration Exciton Resonance Interactions. J. Chem. Phys. 1994, 101, 10521–10525. 49. Scholes, G. D.; Harcourt, R. D.; Ghiggino, K. P. Rate Expressions for Excitation Transfer. III. An Ab Initio Study of Electronic Factors in Excitation Transfer and Exciton Resonance Interactions. J. Chem. Phys. 1995, 102, 9574–9581. 50. Scholes, G. D.; Ghiggino, K. P. Rate Expressions for Excitation Transfer. IV. Energy Migration and Superexchange Phenomena. J. Chem. Phys. 1995, 103, 8873–8883. 51. Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. Constructing Diabatic States From Adiabatic States: Extending Generalized Mulliken–Hush to Multiple Charge Centers With Boys Localization. J. Chem. Phys. 2008, 129, 244101. 52. Subotnik, J. E.; Cave, R. J.; Steele, R. P.; Shenvi, N. The Initial and Final States of Electron and Energy Transfer Processes: Diabatization as Motivated by System-Solvent Interactions. J. Chem. Phys. 2009, 130, 234102. 53. Subotnik, J. E.; Vura-Weis, J.; Sodt, A. J.; Ratner, M. A. Predicting Accurate Electronic Excitation Transfer Rates Via Marcus Theory With Boys or Edmiston-Ruedenberg Localized Diabatization. J. Phys. Chem. A 2010, 114, 8665–8675. 54. Monahan, N.; Zhu, X.-Y. Charge Transfer-Mediated Singlet Fission. Annu. Rev. Phys. Chem. 2015, 66, 601–618. 55. Chan, W. L.; Berkelbach, T. C.; Provorse, M. R.; Monahan, N. R.; Tritsch, J. R.; Hybertsen, M. S.; Reichman, D. R.; Gao, J.; Zhu, X. Y. The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 1321–1329. 56. Aryanpour, K.; Shukla, A.; Mazumdar, S. Theory of Singlet Fission in Polyenes, Acene Crystals, and Covalently Linked Acene Dimers. J. Phys. Chem. C 2015, 119, 6966–6979. 57. Petelenz, P.; Snamina, M. Charge-Transfer Coupling of an Embedded Pentacene Dimer With the Surrounding Crystal Matrix. J. Phys. Chem. C 2015, 119, 28570–28576.

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CHAPTER EIGHT

Continuum Contributions to Dipole Oscillator-Strength Sum Rules for Hydrogen in Finite Basis Sets Jens Oddershede*,†,1, John F. Ogilvie*,‡,§, Stephan P.A. Sauer¶, John R. Sabin*,† *University of Southern Denmark, Odense, Denmark † University of Florida, Gainesville, FL, United States ‡ Simon Fraser University, Burnaby, BC, Canada § Universidad de Costa Rica, San Jose, Costa Rica ¶ University of Copenhagen, Copenhagen, Denmark 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 1.1 Definition of Sum Rules 2. Computational Aspects 3. Results and Discussion 4. Conclusion Acknowledgments References

229 230 232 232 240 240 240

Abstract Calculations of the continuum contributions to dipole oscillator sum rules for hydrogen are performed using both exact and basis-set representations of the stick spectra of the continuum wave function. We show that the same results are obtained for the sum rules in both cases, but that the convergence toward the final results with increasing excitation energies included in the sum over states is slower in the basis-set cases when we use the best basis. We argue also that this conclusion most likely holds also for larger atoms or molecules.

1. INTRODUCTION Many molecular properties are expressed as sums over states; the list includes a range of electric and magnetic properties.1,2 Also, the only essential material constant in the simple Bethe theory of stopping of swift Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.02.001

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2017 Elsevier Inc. All rights reserved.

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charged particles in matter, the mean excitation energy, is a sum over states.3,4 In all cases, the sums run over all excited states of the molecules, discrete, and continuum. Contemporary calculations of electronic structure are nearly all performed using large, but finite, basis sets. Applying sufficiently large basis sets, one may obtain a good description of bound and excited states of an atom or molecule, but, no matter how large and flexible the finite basis set is, one can obtain an only approximate description of the continuum. The continuum contributions to the sum-overstates part of the molecular property hence rely upon a stick-spectrum representation of the true continuum contribution. All experience shows that this approximation works well for a range of molecular sum rules and other properties.5,6 In this paper, we assess the background for this experience by investigating how well the stick-spectrum representation of the continuum contributions works for a number of dipole oscillator-strength sum rules in the simple case of hydrogen; this atom is the only one for which we know the exact continuum wave functions.7 By comparing the computed sum rules using both the exact wave functions and a range of finite basis sets, we can thus acquire further insight into the background for the usefulness of the stickspectrum representation of continuum contributions to sum-over-states properties of atoms and molecules. The purpose of this paper is thus to give an improved understanding of how the convergence toward the exact result for the sum rules—which are known for hydrogen8,9—is obtained when we apply various basis sets as compared to the convergence when using the correct continuum wave function. We first give a brief summary of the sum rules that we tested. Then follow computational details and we end with the results and a discussion of the implications of our findings.

1.1 Definition of Sum Rules We consider electric-dipole oscillator-strength sum rules of two kinds, Z X df p ðEn En0 Þ fnn0 + Ep dE Sp ¼ (1) dE n6¼n 0

Z dSp X df p Lp ¼ ðEn En0 Þ fnn0 lnðEn En0 Þ + Ep lnðE Þ dE ¼ dp n6¼n dE

and

0

(2)

Dipole Oscillator-Strength Sum Rules for Hydrogen

231

Here, n0 and n index the ground and excited states, respectively; f is the dipole oscillator strength. In both equations, the summation extends over all bound states and the integration is over the continuum. In a calculation with a finite basis set, the integration over the continuum is approximated with a numerical integration in which the integration points are the stick-spectrum representation of the continuum. The integration points are always finite in number but vary with the choice of basis set. For large negative values of p in Eqs. (1) and (2), the sum rule in essence depends upon only the bound-state spectrum, but, for positive values of p, the continuum contributions to the sum rules dominate; these are the sum rules with which we are primarily concerned in this paper. To be able to monitor the convergence with the number of excited states in the sum over states, we report test calculations for hydrogen using both the exact continuum functions and calculations in a number of finite basis sets. The Thomas–Reiche–Kuhn (TRK) sum rule10 is probably the best known Sp sum rule; it states that S0 ¼ N

(3)

in which N is the number of electrons in the system. The TRK sum rule holds for exact wave functions and in the random-phase approximation (RPA)11; the fulfillment of this sum rule is commonly used as a measure of the completeness of basis sets in basis-set calculations of dipole oscillator-strength sum rules.12 Many other sum rules are related to properties of matter. S2 is the static dipole polarizability; S2i, i ¼ 2, 3, 4 … can be used to calculate the frequency dependence of the dipole polarizability at low frequencies.13 For an atom, S1 is related to the quadrupole moment of the ground state and S1 is related to the ground-state kinetic energy.8 Also, the ratio between the Lp and Sp sum rules Lp (4) Ip ¼ exp Sp is of interest in several connections, most prominently in the theory of the stopping power as I0, the mean excitation energy, is the only material constant in the simple Bethe theory of stopping.4 The broadening of a beam after its passage through a target, in the theory of stopping power referred to as straggling, may be related to the I1 sum rule.14

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2. COMPUTATIONAL ASPECTS Exact calculations of sum rules for hydrogen are readily available in the literature. As we seek to study the convergence of the sum rules as a function of the magnitude of excited states included in the sums and integrations in Eqs. (1) and (2), we repeated these calculations using exact wave functions for both bound states and the continuum,7 making use of advanced mathematical software (Maple).15 All those calculations were made with either exact algebraic formulae and arithmetic or floating-point arithmetic (precision at least 13 decimal digits) when exact formulae were impracticable. Basis-set calculations were performed with the DALTON program package.16 The full excitation spectrum was generated from the RPA method that for a one-electron system provides the exact solutions within the given basis set. A series of basis sets were tested. We began with the Dunning aug-cc-pVXZ, X ¼ 4, 5, and 6, which turned out to be unable to produce correct results for p ¼ 1 and 2 sum rules. We had to add both more tight and more diffuse functions to obtain agreement for all sum rules. The final basis set (25s29p) consists of the s- and p-type functions of Dunning’s d-aug-cc-pV6Z basis set augmented with extra tight and diffuse functions: 1s-function (α ¼ 18718.77), 11 sets of p-functions (α ¼ 8.649, 21.805, 54.962, 138.51, 348.99, 879.17, 2214.3, 0.6437, 0.2015, 0.006567, 0.002056), and a set of 12s and 12p continuum-like basis functions as suggested by Kaufman et al.17 with initial quantum number 1 and terminal quantum number 12. We refer to this basis as the 13s17p+1-1-12 basis set.

3. RESULTS AND DISCUSSION In Table 1, we report some sum rules for hydrogen calculated using both the exact discrete and continuum wave functions, labeled ExactMaple, and finite basis-set calculations of both bound and continuum states. In the latter calculation, we used the tailored 13s17p+1-1-12 basis set in both the length and the velocity approximation as it gave the best sum rules in all tested cases. We compare also with the value reported by Inokuti,9 which we assume to be correct to the quoted decimal places.

233

Dipole Oscillator-Strength Sum Rules for Hydrogen

Table 1 Calculated Sum Rules for Hydrogen in Atomic Units, Except for I0 Which Is in eV Length Velocity Exact-Maple Inokuti9a

S6

172.19

172.19

172.19

172.19

S2

4.500

4.500

4.500

4.500

S1

2.000

2.000

2.000

2.000

S0

1.000

1.000

1.000

1.000

S1

0.667

0.667

0.666

0.667

S2

1.333

1.319

1.333

1.333

L0

0.596

0.596

0.596

0.596

L1

0.081

0.081

0.082

0.082

I0

14.991

14.991

14.991

14.990

a

Inokuti’s results in rydberg are converted into atomic units.

From the agreement between the length and velocity results, we conclude that the basis-set results have converged; from a comparison between the last two columns in Table 1, we conclude also that they have converged toward the correct results. Thus, in this as in many previous applications, the basis-set calculations of the continuum contributions to the dipole oscillator sum rules work well for all values of p in Eqs. (1) and (2). The larger the continuum contributions to the sum rules are, the more care one must take with the choice of basis sets. To obtain a correct value of S2 for H, it was thus necessary to include both tight basis functions and basis functions tailored to describe Rydberg states and the continuum.17 We proceed to consider the convergence of the sum rule as a function of the number of excited states included in the summation and integrations in Eqs. (1) and (2); we see whether this convergence differs when we use the basis-set representation of the continuum states or the exact continuum states. This point is illustrated in Figs. 1–8 using our best basis set. All figures illustrate that there is no perceptible difference in the exact and basis-set calculations of the contributions to the sum rules from the bound states. For the continuum contributions, however, the convergence toward the correct result is slower in the basis-set cases than in the exact case. We thus have a slightly different representation of the continuum in the two

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4.5

exact 13s17p+1–1–12

S–2

4.0

3.5

3.0

2.5 0

1

2 3 4 Excitation energy (au)

5

6

Fig. 1 The S2 sum rule as a function of the excitation energy included in the sum overstates in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). The vertical dotted line marks the onset of the continuum.

2.0

1.8

exact 13s17p+1–1–12

S–1

1.6

1.4

1.2

1.0 0

1

2 3 4 Excitation energy (au)

5

6

Fig. 2 The S1 sum rule as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). The vertical dotted line marks the onset of the continuum.

235

Dipole Oscillator-Strength Sum Rules for Hydrogen

1.0 0.9 exact 13s17p+1–1–12

S0

0.8 0.7 0.6 0.5 0.4 1

0

2 3 4 Excitation energy (au)

5

6

Fig. 3 The S0 sum rule as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). The vertical dotted line marks the onset of the continuum.

0.6 exact 13s17p+1–1–12

S1

0.5 0.4 0.3 0.2 0.1 0

2

4

6 8 10 12 Excitation energy (au)

14

Fig. 4 The S1 sum rule as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p + 1-1-12). The vertical dotted line marks the onset of the continuum.

cases, but, eventually, when all excitations are included, the full space is spanned in both cases. Another way of expressing the same conclusion is to state that the intensities of the dipole transitions placed in the continuum are blue-shifted in the basis-set representation, which in turn implies a

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S2

1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

exact 13s17p+1–1–12

1

10 100 Excitation energy (au)

1000

10000

Fig. 5 The S2 sum rule as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). Notice the logarithmic scale on the energy axis. The vertical dotted line marks the onset of the continuum.

–0.4

L0

exact 13s17p+1–1–12

–0.6

0

2

4

6 8 10 Excitation energy (au)

12

14

Fig. 6 The L0 sum rule in atomic units as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). The vertical dotted line marks the onset of the continuum.

237

Dipole Oscillator-Strength Sum Rules for Hydrogen

–0.05

–0.10

–0.15 L1

exact 13s17p+1–1–12

–0.20

–0.25

–0.30 0.1

1

10

100

1000

Excitation energy (au)

Fig. 7 The L1 sum rule in atomic units as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). Notice the logarithmic scale on the energy axis. The vertical dotted line marks the onset of the continuum.

0.6

exact 13s17p+1–1–12 I0

0.5

0.4

0

2

4

6

8

10

12

14

Excitation energy (au)

Fig. 8 The I0 sum rule in atomic units as a function of the excitation energy included in the sum over states in the exact case (exact) and calculated in dipole length using the best basis set (13s17p+1-1-12). The vertical dotted line marks the onset of the continuum.

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slower convergence of a stick-spectrum representation of the continuum contributions. Only when we include all excitation energies in the stickspectrum representation do we obtain the correct result; that is, only then does the basis set span the same space as the exact continuum wave functions for hydrogen. One consequence of this behavior of the sum rule as a function of the number of states included in the sum over states is that truncated sum-overstates calculations of properties are not advisable, an issue that has been recognized for many years.18 Since the convergence of L0 and S0 (see Figs. 3 and 6) appears similar for medium continuum excitation energies, one might expect that the ratio of the two sum rule, that is the mean excitation energy (see Eq. 4), might show a faster convergence as a function of the states included in the sum over states. However, Fig. 8 shows that this does not appear to be the case. The Sp sum rules must increase monotonically with the inclusion of more states in the sum over states as all individual contributions in the sum in Eq. (1) are positive. However, the introduction of the logarithm in the definition of the Lp sum rule (Eq. 2) makes the behavior of Lp as a function of the excitation energy more unpredictable. We see in Figs. 6 and 7 that both L0 and L1 go through a minimum a bit above the ionization limit before they attain the monotonic behavior as a function of excitation energy that we saw for Sp sum rules. We have tested also how the convergence of sum rules are affected by the choice of the basis set. Only in the tailored 13s17p+1-1-12 basis set do we obtain all sum rules correct. We have, however, considered the TRK sum rules for which it is also possible to fulfill Eq. (3) in more modest basis sets. The first test is illustrated in Fig. 9. Here, we see that the Dunning basis sets show the opposite behavior relative to the exact solution as a function of the excitation energy as does the 13s17p+1-1-12 basis set. For low-lying continuum states the basis set results lie above the exact curve. The lack of agreement for the discrete part of the spectrum indicates, however, that these basis sets give a somewhat random representation of the discrete excitation spectrum and must be considered less reliable for more general sum-over-states calculations of properties. The same conclusion holds when we work with the pure sp-basis sets as we see from the comparisons in Fig. 10. Only when we include the basis functions tailored by Kaufmann et al.17 to describe the continuum and Rydberg states do we obtain an effective description both of the bound states and of all sum rules.

239

Dipole Oscillator-Strength Sum Rules for Hydrogen

1.0 0.9

exact 13s17p+1–1–12 aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z

S0

0.8 0.7 0.6 0.5 0.4 0.5

1.0

1.5 2.0 2.5 Excitation energy (au)

3.0

3.5

Fig. 9 The S0 sum rule for the aug-cc-pVXZ, X ¼ 4, 5, and 6 basis sets as a function of the excitation energy included in the sum over states. The vertical dotted line marks the onset of the continuum.

1.0 0.9

12s12p 12s13p 12s14p 12s15p 12s16p 12s17p 13s17p+1–1–12 exact

S0

0.8 0.7 0.6 0.5 0.4

0.5

1.0

1.5 2.0 Excitation energy (eV)

2.5

3.0

Fig. 10 The S0 sum rule for a series of tailored basis sets (see text) as a function of the excitation energy included in the sum over states. The vertical dotted line marks the onset of the continuum.

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4. CONCLUSION Using both the stick-spectrum representation of the continuum and the exact continuum wave functions, we show how the convergence with excitation toward the correct results for the Sp and Lp sum rules proceeds for hydrogen, the only example for which it is possible to do both. Both procedures converge toward the same result, but the stick spectrum in the best basis-set representation shows a slower convergence pattern than does the one using the exact wave functions. Other convergence patterns toward the correct results are found in less accurate basis sets. There is no reason to believe that the specific result for hydrogen cannot be generalized to larger and more complicated atoms and molecules. The main reason for that belief is that, for the high-lying continuum states that give the dominant contributions to the high-p sum rules, the continuum is little affected by the actual form of the molecular and atomic structure. We thus conclude that the test calculations on hydrogen yield adequate reason to believe that the widely applied procedure of approximating the continuum contributions with a stick-spectrum representation is able to produce as accurate results for dipole oscillator sum rules as if we were using a more correct continuum wave function, provided that the basis set is carefully chosen and balanced and that we do not truncate in the sum over states.

ACKNOWLEDGMENTS J.O. and J.R.S. thank Mark Ratner for suggesting the subject of this paper and for being a scientific collaborator and very good personal friend since we all started in science many years ago. J.O. acknowledges helpful discussion with Hans Jørgen Aagaard Jensen.

REFERENCES 1. Hirschfelder, J. O.; Brown, W. B.; Epstein, S. T. Recent development in perturbation theory. Adv. Quantum Chem. 1964, 1, 255–374. 2. Sauer, S. P. A. Molecular Electromagnetism. A Computational Chemistry Approach; Oxford University Press: Oxford UK, 2011. 3. Bethe, H. Zur Theorie des Durchgangs Schneller Korpuskularstrahlen durch Materie. Ann. Phys. 1930, 397, 325–400. 4. Inokuti, M. Inelastic Collisions of Fast Charged Particles With Atoms and Molecules— The Bethe Theory Revisited. Rev. Mod. Phys. 1971, 43, 297–347. 5. Sauer, S. P. A.; Oddershede, J.; Sabin, J. R. Directional Dependence of the Mean Excitation Energy and Spectral Moments of the Dipole Oscillator Strength Distribution of Glycine and its Zwitterion. J. Phys. Chem. A 2006, 110, 8811–8817. 6. Zarycz, M. N. C.; Provasi, P. F.; Sauer, S. P. A. On the Truncation of the Number of Excited States in Density Functional Theory Sum-Over-States Calculations of Indirect Spin-spin Coupling Constants. J. Chem. Phys. 2015, 143, 244107.

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7. Bethe, H. A.; Salpeter, E. E. Quantum Mechanics of One- and Two-Electron Atoms; Springer: Berlin Germany, 1957. 8. Bethe, H. A.; Jackiw, R. Intermediate Quantum Mechanics; Benjamin/Cummings: New York, USA, 1986. 9. Inokuti, M. Moments of the Oscillator Strength Distribution and Some Associated Quantities for the Hydrogen Atom: Argonne National Laboratory Report ANL 6769; 1963;pp 7–19. € die Zahl der Dispersionelektronen die einem station€aren Zustande 10. Thomas, W. Uber € Zugeordnet sind. Naturwissenschaften 1925, 13, 627–628; Reiche, F.; Thomas, W. Uber die Zahl der Dispersionelektronen die einem station€aren Zustande Zugeordnet sind. € Z. Phys. 1925, 34, 510–525; Kuhn, W. Uber die Gesamtstarke der von einem Zustande aus-gehenden Absorptionslinien. Z. Phys. 1925, 33, 408–412. 11. Jørgensen, P.; Oddershede, J. Equivalence Between Perturbative Calculated Transition Moments. J. Chem. Phys. 1983, 78, 1898–1904. 12. Jensen, P. W. K.; Sauer, S. P. A.; Oddershede, J.; Sabin, J. R. Mean Excitation Energies for Molecular Ions. Nucl. Instr. Meth. Phys. Res. B 2017, 394, 73–80. 13. see e.g. Packer, M. J.; Sauer, S. P. A.; Oddershede, J. Correlated Dipole Oscillator Sum Rules. J. Chem. Phys. 1994, 100, 8969–8975; Paidarova´, I.; Sauer, S. P. A. A Comparison of Møller-Plesset and Coupled Cluster Linear Response Theory Methods for the Calculation of Dipole Oscillator Strength Sum Rules and C6 Dispersion Coefficients. Collect. Czech. Chem. Commun. 2008, 73, 1415–1436. 14. Fano, U. Penetration of Protons, Alpha Particles, and Mesons. Annu. Rev. Nucl. Sci. 1963, 13, 1–66. 15. Maple 2016 (Waterloo Maple Inc., Waterloo, Ontario, Canada), https://www. maplesoft.com. 16. Aidas, K.; Angeli, C.; Bak, K. L.; Bakken, V.; Bast, R.; Boman, L.; Christiansen, O.; om, U.; Enevoldsen, T.; Cimiraglia, R.; Coriani, S.; Dahle, P.; Dalskov, E. K.; Ekstr€ Eriksen, J. J.; Ettenhuber, P.; Ferna´ndez, B.; Ferrighi, L.; Fliegl, H.; Frediani, L.; Hald, K.; Halkier, A.; H€attig, C.; Heiberg, H.; Helgaker, T.; Hennum, A. C.; Hettema, H.; Hjertenæs, E.; Høst, S.; Høyvik, I.-M.; Iozzi, M. F.; Jansik, B.; Jensen, H. J.; Jonsson, D.; Jørgensen, P.; Kauczor, J.; Kirpekar, S.; Kjærgaard, T.; Klopper, W.; Knecht, S.; Kobayashi, R.; Koch, H.; Kongsted, J.; Krapp, A.; Kristensen, K.; Ligabue, A.; Lutnæs, O. B.; Melo, J. I.; Mikkelsen, K. V.; Myhre, R. H.; Neiss, C.; Nielsen, C. B.; Norman, P.; Olsen, J.; Olsen, J. M. H.; Osted, A.; Packer, M. J.; Pawlowski, F.; Pedersen, T. B.; Provasi, P. F.; Reine, S.; Rinkevicius, Z.; Ruden, T. A.; Ruud, K.; Rybkin, V.; Salek, P.; Samson, C. C. M.; de Mera´s, A. S.; Saue, T.; Sauer, S. P. A.; Schimmelpfennig, B.; Sneskov, K.; Steindal, A. H.; Sylvester-Hvid, K. O.; Taylor, P. R.; Teale, A. M.; Tellgren, E. I.; Tew, D. P.; Thorvaldsen, A. J.; Thøgersen, L.; Vahtras, O.; Watson, M. A.; ˚ gren, H. The DALTON Quantum Chemistry Wilson, D. J. D.; Ziolkowski, M.; A Program System. WIREs Comput. Mol. Sci. 2014, 4, 269–284. 17. Kaufmann, K.; Baumeister, W.; Jungen, M. Universal Gaussian Basis Sets for an Optimum Representation of Rydberg and Continuum Wavefunctions. J. Phys. B: At. Mol. Opt. Phys. 1989, 22, 2223–2240. 18. Oddershede, J.; Jørgensen, P.; Beebe, N. H. F. Coupled Hartree-Fock and Second Order Polarization Propagator Calculations of Indirect Nuclear Spin-Spin Coupling Constants for Diatomic Molecules. Chem. Phys. 1977, 25, 451–458.

CHAPTER NINE

Features of Nearly Spherical Electronic Systems Jan Linderberg*,†,‡,1 *Aarhus University, Aarhus C, Denmark † Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, UT, United States ‡ Quantum Theory Project, University of Florida, Gainesville, FL, United States 1 Corresponding author: e-mail address: [email protected]

Contents 1. Preamble 2. An Electron Propagator 3. Using the Electron Propagator 4. An Energy Functional 5. Turning Points 6. A Dirac Density Matrix 7. Yet an Energy Functional 8. Accomplishments 9. Memories: Mark Ratner and the Late Osvaldo Goscinski References

243 244 249 250 253 257 262 264 265 265

Abstract Properties of electronic systems are a principal concern for quantum chemical theory and calculations and are succinctly expressed in terms of various Green functions, also termed propagators or response functions. Spherical symmetry offers simplifications and analytical options that will be explored in this chapter. A proper many-electron theory in the relativistic form is lacking, but the use of the Dirac equation and fourcomponent functions provides certain advantages in the formulation of the electron propagator. Energy functionals are developed for an inhomogeneous system with spherical symmetry. It is also shown how approximate response functions can be obtained for electric and magnetic perturbations.

1. PREAMBLE Spherical symmetry has been of paramount importance in the development of quantum theory. The electronic structure theory of atoms has generated a large body of literature, and this has carried over into molecular Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.02.003

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2017 Elsevier Inc. All rights reserved.

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theory where the atomic constituents brought with them features characterized by their free space states. A less extensive, but still significant, series of papers have used molecular models derived from spherical approximations. The results are useful, but the methods have had only marginal influence on the mainstream development of molecular theory. The homogeneous free electron gas remains a basis for studies of crystalline materials and for the development of the density functional approach for applications to general systems that are neither periodic nor particularly spherical. An attempt is made in this work to develop techniques for inhomogeneous systems from the study of a general electronic system where a reasonable starting model possesses spherical symmetry so that angular momentum algebra can be used. There is no complete many-electron theory in the relativistic form, but there are certain analytical advantages offered by the Dirac equation in a centrally symmetric situation. Thus, it will be shown in the following section how to reformulate a previously given approximation to the electron propagator. Subsequent sections are used to illustrate the use of the propagator to estimate the corrections to the spherical model, the exchange and correlation energy, the Breit energy, and photoelectron spectra. Speculations about future applications form the concluding section.

2. AN ELECTRON PROPAGATOR Condon and Shortley1 supply a comprehensive and well-presented sketch of the relativistic theory according to Dirac and its application to atomic theory. Their notations will be used in what follows. Thus we start with the basic equation, ! ! ! ! W + αμc 2 U + c β p ψ ¼ e φ + β A ψ where W represents an energy parameter and μ the electron mass. The central potential is denoted U, the velocity of light c, and the charge of the elec!

tron e. Further scalar and vector potentials are given by φand A , while α and !

β are the four-by-four Dirac matrices. Planck’s constant divided by 2π equals ℏ. The momentum operator involves the gradient vector operator, ! p ¼ iℏr, and ψ is the four-component Dirac state vector. The central field is conventionally given by an effective nuclear charge, U ¼ e2Z(r)/r. Additional terms that are not spherically symmetric enter the

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right-hand side of the equation as parts of the scalar potential φ and will be considered through perturbation theory. Two-component spinors are defined in terms of spherical harmonics fY‘, m ðθ, ϕÞg coupled with spin to form total angular momentum states. The options are gathered in the form, using Wigner’s 3j symbols, 2 0 1 3 1 6 B ‘ 2 j C 7 6 Y 1 ðθ, ϕÞB C 7 6 ‘, m 2 @ A 7 1 1 6 7 m m 6 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 6 7 jm χ k, m ðθ, ϕÞ ¼ 6 2j + 1 0 1 7ðÞ 6 7 1 6 j C7 B ‘ 6 7 2 C7 6 Y 1 ðθ, ϕÞB ‘ , m @ A5 4 2 1 1 m + m 2 2 1 1 1 1 k ¼ ‘ð‘ + 1Þ jðj + 1Þ , j ¼ jkj , ‘ ¼ k + 4 2 2 2

+

1 Each term gives a total angular momentum j ¼ jkj , where k is a pos2 itive or a negative integer and m can take values from –j to j in integer steps. 1 The orbital angular momentum quantum number ‘ ¼ j is the positive 2 root of the relation above. These spinors serve to express the state vector with its four components as j ∞ X 1 X χ k, m ðθ, ϕÞFk, m ðr Þ ψ¼ r k¼∞ m¼j iχ k, m ðθ, ϕÞGk, m ðr Þ The left-hand side of the Dirac equation becomes ! ! W + αμc 2 U + c β p ψ

9 8 k d > > > > > > χ k, m ðθ, ϕÞ ðhðr Þ + σ ÞFk, m ðr Þ Gk, m ðr Þ + Gk, m ðr Þ = r dr ℏc X < ¼

> > r k, m > k d > > ; : iχ k, m ðθ, ϕÞ Fk, m ðr Þ + Fk, m ðr Þ + ðhðr Þ σ ÞGk, m ðr Þ > r dr with the definitions W =ℏc + e2 Z ðr Þ=ℏcr hðr Þ; μc=ℏ ¼ σ

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for two entities with dimension of wave number, the latter, σ, is the inverse of the reduced Compton wavelength for the electron and has the value of the velocity of light in Hartree atomic units. The right-hand side of the Dirac equation has a corresponding expression ! ! ℏc X χ k, m ðθ, ϕÞdF k, m ðr Þ e φ + β A ψ ¼ r k, m iχ k, m ðθ, ϕÞdGk, m ðr Þ and the detailed form of the dFs and dGs depends on the particular potentials. There remain equations for the radial amplitudes, in the simplified form, as d k d k ðh + σ ÞF G G ¼ dF; F F + ðh σ ÞG ¼ dG dr r dr r An integral equation can be constructed by standard procedures to replace eis f F has the derivative the differential system. The expression X ¼ is e g G

eis f F eis f dF=dr is df =dr F d X ¼ ids=dr is + +e dg=dr G e g G eis g dG=dr dr d d or, after substitutions of the derivatives G, F , dr dr 0 0 F G F G is is d hσ k=r + ids=dr + e f 0 d lng=dr X ¼ e f dr eis g k=r ids=dr h + σ eis g d ln f =dr 0 0 dF dG + eis f 1 0 eis g 0 1 The first term is eliminated by putting the cofactors of F and G to zero: f k=r + ids=dr f h σ ¼ ¼0 g h+σ g k=r ids=dr Thus, it holds that

1 k=r + ids=dr f and ðds=dr Þ2 ¼ h2 σ 2 ðk=r Þ2 ¼ h+σ g N The proportionality factor N is to be chosen.

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A general, complex energy parameter W allows the specification of a qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ complex root p ¼ h2 σ 2 ðk=r Þ2 in the upper half plane, Ip > 0: Two options obtain for ds/dr and for s. The choice ds/dr ¼ p implies that lim r!∞ eis ¼ 0: This is used to define an “outer” function X∞ ¼

Z ∞ eis k=r + ip F d dr X∞ ¼ N h + σ G dr r

Alternatively, it follows that an “inner” function is Z r eis k=r ip F d dr X0 X0 ¼ ¼ N h + σ G dr 0 since lim r!0 eis ¼ 0: A more detailed examination of the functions near the qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ origin shows that lim r!0 rp ¼ i k2 ðe2 Z ð0Þ=ℏc Þ2 : The argument of the square root should always be positive. Consider further the forms

1 k=r + ip X∞ eis 2ipðh + σ Þ ¼F is k=r ip X e N N2 0

1 h + σ X∞ eis 2ipðh + σ Þ ¼G N h + σ X0 eis N2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ that suggest a useful choice for the factor N: N ¼ ð1 iÞ pðh + σ Þ. The results can be put in the form

is

Z r

Z ∞ e k=r + ip d eis k=r ip d F ¼ dr X0 + dr X∞ G h+σ N h+σ N dr dr 0 r

that completes the transformation of the system of differential equations to a matrix integral equation. The two functions X0 and X∞ depend on the logarithmic derivatives of the amplitudes (k/r ip)/N and (h + σ)/N that are regular in the complex W-plane except for parts of the real axis. Isolated essential singularities occur when p ¼ 0. The relevant values of r are identified with classical turning points in the Wentzel–Kramers–Brillouin theory and require special attention. Thus, it appears to be satisfactory to simplify so that

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d k=r ip 1 dp k=r ip k 1 dh 1 ln ¼ 2 dr N 2p dr k=r ip r ðk=r ipÞ 2 dr h + σ 1 dp k=r ip ﬃ 2p dr k=r ip d h+σ d pﬃﬃﬃ 1 dp ln ﬃ ln p ¼ dr N dr 2p dr and further analysis is deferred to later where the poles of the electron propagator on the real energy axis are identified. A matrix integral kernel,

0 eisðr Þ k=r + ipðr Þ r r 0 eisðr Þ 0 ðr, r ; W Þ ¼ ½k=r 0 ipðr 0 Þ hðr 0 Þ + σ Θ N ðr Þ hðr Þ + σ jr r 0 j N ðr 0 Þ

0 0 eisðr Þ k=r ipðr Þ r r eisðr Þ ½ k=r 0 + ipðr 0 Þ hðr 0 Þ + σ , Θ + N ðr Þ hðr Þ + σ jr r 0 j N ðr 0 Þ is defined with the use of the Heaviside step function Θ so that the integrals above can be compactly expressed as

Z ∞

1 dpðr 0 Þ 0 ^t F ðr 0 Þ F ðr Þ dF ðr 0 Þ 0 0 ¼ dr ðr, r ; W Þ Gðr Þ Gðr 0 Þ dGðr 0 Þ 2pðr 0 Þ dr 0 1 0 0 The only approximation stems from the simplifications in the forms of the logarithmic derivatives of the amplitudes f and g. An operator ^t with the property 1 1 ðk=r ipðr ÞÞ^t ¼ ðk=r ipðr ÞÞ N N serves to transform the f-amplitudes with the factor from the derivatives. There remains to include angular factors to form a full space electron propagator. Four-component column and row vectors will be used in the forms

eisk ðr Þ χ k, m ðθ, ϕÞðk=r ipk ðr ÞÞ ! ψ k, m r ¼ rNk ðr Þ iχ k, m ðθ, ϕÞðhðr Þ + σ Þ h i is ð r Þ k ! e { { ψ^ ¼ r ð θ, ϕ Þ ð k=r ip ð r Þ Þ iχ ð θ, ϕ Þ ð h ð r Þ + σ Þ χ k k, m k, m rNk ðr Þ k, m and the four-by-four matrix kernel or, equivalently, the electron propagator appears as

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0 ! ! ! 0 X r r ! 1 ψ^ k+, m r 0 0 r , r ; W ¼ ψ k, m r Θ 0 ℏc jr r j k, m 1 r r0 !0 + ! + ψ k, m r r ψ^ Θ ℏc jr r 0 j k, m † The row vectors ψ^ k, m differ from the conjugates (ψ k,m) when the pks are complex.

3. USING THE ELECTRON PROPAGATOR Properties that can be derived from the electron propagator2 will be estimated here by means of the approximate form 0 . Thus, it follows from the functional

I ! Z ! ! ! ! 1 ! S φ, A ¼ d r dW Tr e φ + β A 0 r , r ; W 2πi that the charge density equals the functional derivative of S with respect to φ: I ! ! ! e q r ¼ dW Tr 0 r , r ; W 2πi The current density is similarly obtained as I ! ! ! ! ! e I r ¼ dW Tr β 0 r , r ; W 2πic The contour integral circumscribes the real axis between 0 and μc2, the rest energy of the electron, to ensure that all bound states are included. A full propagator exhibits isolated poles on the real axis for bound states, while the approximate 0 gives a continuous spectrum. There is no current density in a spherical system and the charge density is spherically symmetric. It has the value I Z I X 2jkjhðr Þ ! ! e e qðr Þ ¼ dW dϕ sin θ dθ Tr 0 r , r ; W ¼ dW 2πi 2πi iℏcpk ðr; W Þ k and it is observed that hðr Þ @pk ðr; W Þ ¼ ℏcpk ðr; W Þ @W

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and consequently, it obtains qðr Þ ¼

eX jkj lim pk r; μc 2 + iη pk r; μc 2 iη η! + 0 π k

An approximate form of the summation over angular quantum numbers, and taking account of the symmetry between k and –k leads to the result 4er 2 qðr Þ ¼ 3π

Z

kmax kmin

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 @ 4e ^Z ðr Þ=r Þ2 ðk=r Þ2 ¼ λðr Þ3 dk 2σ^ αZ ðr Þ=r + ðα @k 3πr

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ^Z ðr ÞÞ2 ; kmin ¼ α ^Z ð0Þ λðr Þ ¼ kmax kmin ; kmax ¼ 2σ^ αrZ ðr Þ + ðα

The fine-structure constant is given a hat in order to distinguish it from the α-matrix in the original equation, and it is recognized that the product σ^ α equals the inverse Bohr radius a0. It was noticed earlier that the quantum number k should be larger than kmin in order to get a proper analytical behavior of the propagator near the center of the system.

4. AN ENERGY FUNCTIONAL A quantity of interest is the energy density ε(r) that is defined as 1 εðr Þ ¼ 2πi

I

Z WdW

! ! X jkj dϕ sinθ dθ Tr 0 r , r ; W ¼ π k

I W dpk

The energy parameter W has the form W ¼ ℏc

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p2k ðr; W Þ + σ 2 + ðk=r Þ2 e2 Z ðr Þ=r

It follows that the second term gives the standard potential energy term and thus the remaining contributions are εðr Þ + eqðr ÞZ ðr Þ=r ¼

X ℏc jkj I qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 p2k ðr; W Þ + σ 2 + ðk=r Þ dpk π k

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Nearly Spherical Systems

Integration yields I qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^Z ðr Þ=r pk ðr; μc 2 Þ σ 2 + ðk=r Þ2 σ + α p2k ðr; W Þ + σ 2 + ðk=r Þ2 dpk ¼ ln ^Z ðr Þ=r + pk ðr; μc 2 Þ 2 2 σ + α ^Z ðr Þ=r pk r; μc ½σ + α pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ^Z ðr Þ k2max k2 σr + α σ r +k pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ln ¼ 2r 2 ^Z ðr Þ + k2max k2 σr + α ﬃ ^Z ðr Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σr + α 2 k2 k max r2 and the “summation” gives the result μc 2 qðr Þ + Φðr; Z Þ e 2 1 ηðr Þ ℏcκ 4 ðr Þ 10 3 λðr Þ 2 η ; ηðr Þ ¼ Φðr; Z Þ ¼ 2η ð r Þ ð r Þ + 1 η ð r Þ ln 4πr 2 3 1 + ηðr Þ κ ðr Þ εðr Þ ¼

The functional Φ(r; Z) represents the radial energy density with excep^Z ðr Þ, is tion of the rest mass contribution. An auxiliary form, κ ðr Þ ¼ σr + α always larger than λ(r), and a Taylor series in powers of η(r) is considered for the functional Φ: μc 2 qðr Þ 4ℏcκ4 ðr Þη5 ðr Þ η2 ðr Þ 1 + … e 15πr 2 7 2 μc 2 qðr Þ 2eqðr ÞZ ðr Þ ^ +O α ¼ e 5r

εðr Þ ¼

Reduction to the limit of a vanishing fine-structure constant gives the original result of Fermi–Thomas theory. The balance between the potential and kinetic energies is coming out as a consequence of the propagator approximation and demonstrates that the statistical arguments underpinning the approach by Fermi3 and Thomas4 are replaced by an analytic approach. Fenyes5 showed how the use of the Wentzel–Kramers–Brillouin approximation method could be used to derive the Fermi–Thomas result from Hartree’s orbital theory. This was reintroduced by Plaskett6 and also by Broyles.7 An estimate of the total electronic energy obtains from the integral of ε(r) and subtraction of the self-interaction of the electronic density, including possible exchange and correlation adjustments:

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Z

∞

Ee ¼ 0

( 1 dr εðr Þ 2

Z

∞ r

qðr 0 Þ 2 dr 0 0 r

) ðterms from exchange and correlationÞ

Nearly spherical systems will be defined such that the nuclear potential is represented by a central point charge and a set of spherical charged shells. Then it holds that Z ∞ X Rj + r Rj r Z r qðr 0 Þ eZ ðr Þ ¼ Q0 + Qj dr 0 qðr 0 Þ r dr 0 0 2Rj r 0 r j in the Hartree approximation when exchange and correlation do not contribute to the central field. There are shells with charges and radii given by the set of values {Qj , Rj j j ¼ 1 , …}, which implies that the derivative of the effective charge Z(r) exhibits finite steps at these radii:

dZ Rj η dZ Rj + η Qj ; j ¼ 1,… lim ¼ η! + 0 eRj dr dr Otherwise it holds that e

Z ∞ dZ ðr Þ X Qj qðr 0 Þ dr 0 0 ¼ R dr r r Rj >r j

The total energy can now be further developed from the Hartree form. Thus " #2 ) Z ∞ ( μc 2 qðr Þ 1 X Qj dZ ðr Þ EH ¼ dr Φðr, Z Þ + e e 2 Rj >r Rj dr 0 The last form establishes the total energy as a functional of the effective central charge and implicitly of the charge distribution q(r). A variational formulation8 is to be based on a functional ) " #2 Z ∞ ( 1 X Qj dZ ðr Þ εF JH ðZ Þ ¼ dr Φðr, Z Þ e qðr Þ e 2 Rj >r Rj dr 0 with an appropriate Lagrangian multiplier that is written εF in order to account for a fixed number of electrons. The chemical potential, εF, is termed the Fermi energy and defines the energy of the least bound electronic state. It is chosen to be zero in the most simple propagator approximation.

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Thus it obtains that JH(Z) is stationary when e2

d2 Z ðr Þ eqðr Þ εF @qðr Þ eqðr Þ 4^ ακ ðr Þλðr Þ + ¼ + εF 2 ¼ e r @Z r πr dr

and the boundary conditions at the radii at the spherical shells of charge are satisfied. The original Fermi–Thomas equation is recovered for a vanishing finestructure constant and zero Fermi energy from d2 Z ðr Þ 4λ3 ðr Þ 8Z ðr Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2rZ ðr Þ=a0 ) 2 ¼ 2 3πr α^!0 3πra0 dr A functional such as JH(Z) lends itself to numerical estimates from the finite-element method, to be demonstrated below.

5. TURNING POINTS This section will show the effect of the possible singularities of the 1 dpðr Þ hitherto neglected terms with . They occur in the forms pðr Þ dr

Z ∞ 1 dpðr 0 Þ 0 ^t 0 0 ðr 0 , r 00 ; W Þ dr ðr, r ; W Þ 2pðr 0 Þ dr 0 1 0 0 in the iterated integral equation. No contribution occurs from points between the external ones since

0 ^t 0 ðr 0 , r 00 ; W Þ ðr, r ; W Þ 1 0

00 0 eisðr Þ k=r + ipðr Þ r r 0 2isðr 0 Þ r r ¼ Θ 00 0 Θ e 0 0 0 h r ð Þ + σ N ðr Þ jr r j jr r j 00

eisðr Þ ½ k=r 00 + ipðr 00 Þ hðr 00 Þ + σ N ðr 00 Þ

0 0 00 eisðr Þ k=r ipðr Þ r r 2isðr 0 Þ r r Θ 0 00 e Θ 0 0 0 N ðr Þ hðr Þ + σ jr r j jr r j 00

eisðr Þ ½ k=r 00 ipðr 00 Þ hðr 00 Þ + σ N ðr 00 Þ

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These forms show that a singularity in 1/p(r) will contribute to the integral and “scatter” the solution back where it came from, hence a turning point. The “scattering amplitude” will be evaluated through the saddle point method as the integral Z Z r 2isðr Þ 1 dpðr Þ T ¼ dr e dr 0 pðr 0 Þ ; sðr Þ ¼ 2pðr Þ dr a over an interval around a point r ¼ a, p(a; W ) ¼ 0 for some real energy parameter W. The vicinity of a admits an approximate representation of the wave number as pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pðr; W + iηÞ∝ r a _ a r where the choice indicates whether one passes from a “forbidden” to an “allowed” region or the opposite. Both options give the same integral. The saddle point technique utilizes a representation, Z Z 1 1 dr euðr Þ T ¼ dr e2isðr Þ 2 ¼ 4p ðr Þ 4 where an auxiliary function is defined as Z r 4ip3 ðr Þ dr 0 pðr 0 Þ 2 ln pðr Þ ¼ uðr Þ ¼ 2i 2 lnpðr Þ 3 a and a saddle point, in the complex r-plane, is located at r∗ where

duðr Þ 2 dpðr Þ 2 ¼ 4ip ðr Þ + dr pðr Þ dr vanishes and 2ip3 r∗ + 1 ¼ 0 Moreover, it holds that

d 2 u r∗ ¼ 6p2 r∗ 2 dr

so that the Taylor series for u around r∗ comes out to be 2 2 uðr Þ 2 ln p r∗ 3 r r∗ p r∗ 3

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There are three roots for p r∗ and two of them have positive imaginary parts. A new variable for the integration is used: rﬃﬃﬃ 3 z¼ r r∗ p r∗ π It is convenient to choose the root in the first quadrant. A slight change in the procedure gives the analogous result for the other integral above and it is concluded that rﬃﬃﬃﬃﬃ π 2 e3 ﬃ 0:99658i T ¼ i 12 The result demonstrates that within the realm of the saddle point method the kernel in the integral form is equivalent to a pointwise interaction expressed in terms of the Dirac δ-function. This implies in turn that the coupling in the integral equation for the propagator ðr, r 0 ; W Þ can be represented by a separable integral kernel and thus allow for the use of Fredholm’s theory. It is deduced that

1 dpðr 0 Þ 0 ^t ðr 0 < aÞða > r 00 Þðr 0 > aÞða < r 00 Þ 2pðr 0 Þ dr 0 1 0 and pﬃﬃﬃﬃ

T hða; W Þ + σ ðr < aÞ ¼ δðr aÞ N ða; W Þ k=a ipða; W Þ 0

0

pﬃﬃﬃﬃ T ða > r Þ ¼ ½ hða; W Þ + σ k=a ipða; W Þ δða r 00 Þ N ða; W Þ pﬃﬃﬃﬃ

T hða; W Þ + σ 0 0 ðr > aÞ ¼ δðr aÞ N ða; W Þ k=a + ipða; W Þ pﬃﬃﬃﬃ T ½ hða; W Þ + σ k=a + ipða; W Þ δða r 00 Þ ða < r 00 Þ ¼ N ða; W Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N ða; W Þ ¼ ð1 iÞ pða; W Þ½hða; W Þ + σ 00

A numerical factor of nearly unit modulus occurs pﬃﬃﬃﬃ iπ=40:00171 T ¼e

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Each turning point contributes a separable integral kernel of the type above and of rank unity. Considering a case with two turning points, 0 < a < a0 < ∞, the coupling is represented by ¼ ðr 0 > aÞða < r 00 Þðr 0 < a0 Þða0 > r 00 Þ ¼ a a + a0 a0 with a shorthand notation. The iterated propagator obtains from the symbolic equation

¼ + ¼ + a a + a0 a0 from which it follows that

a ¼ a + a a0 a0 ; a0 ¼ a0 + a0 a a since a a ¼ a0 a0 ¼ 0 The nonvanishing elements are equal, 0

a0 a ¼ a a0 ¼ Teisða ÞisðaÞ and the solution for the iterated propagator can be written as

¼ + 0

a a + a0 a0 + ½a a0 + a0 a Teisða ÞisðaÞ ¼ 1 T 2 e2isða0 Þ2isðaÞ

Singularities occur for the propagator when the denominator in vanishes, 0

0

1 T 2 e2isða Þ2isðaÞ ¼ 1 e2isða Þ2isðaÞiπ0:00684 This cannot happen due to the small number in the exponent, but when the Bohr–Sommerfeld quantization condition is met, Z a0 1 dr pðr; W Þ ¼ n + π; n ¼ 0,1,2, … 2 a there will be a near singularity.

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Pure Coulomb potentials with a constant effective nuclear charge Z lead to the stationary levels at vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 12ﬃ u 0 u u ^Z α B C W ¼ μc 2 u A ; n ¼ 0,1, … t1 + @ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ^ Z n+ + k α 2 and differ from the exact result by the one-half addition to the principal quantum number n. This may be compensated for by the Kramers modifi1 cation, k ! k , in the zeroth-order propagator at the expense of a differ2 ent behavior of the propagator near the origin and a modification of the amplitude derivative terms. A Taylor expansion of the energy above is 0 12 0 14 2 2 2 ^ e B e B Z Z 4n + 2 C α C 2 +… W ¼ μc @ A @ A 1+ 1 1 2a0 8a0 jkj n + + jkj n + + jkj 2 2 and the degeneracy between states for k and –k is explicit in the approximation that has been applied.

6. A DIRAC DENSITY MATRIX

! ! The propagator 0 r , r 0 ; W determines the one-electron density matrix through the contour integral I ! ! ! ! 1 dW 0 r , r 0 ; W ¼ Γ r , r 0 2πi It is rotationally invariant and depends only on the radii {r, r 0 }, the scalar ! ! ! ! ! ! product r r 0 rr 0 cosω, and Pauli matrices r τ r τr , r 0 τ r 0 τr 0 . These latter ones effect the transformation among spinors such that ! !

r τ χ k, m ðθ, ϕÞ ¼ r τr χ k, m ðθ, ϕÞ ¼ rχ k, m ðθ, ϕÞ

Four basic two-by-two matrices define the angular dependencies of the density matrix: kn, n0 ðθ, φ; θ0 , φ0 Þ ¼ ðτr Þn

j X m¼j

{

0

χ k, m ðθ, ϕÞχ k, m ðθ0 , ϕ0 Þ ðτr 0 Þn ; n, n0 2 f0, 1g

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Jan Linderberg

Another form is more compact and reads iðj‘Þ 2j + 1 k0, 0 ðθ, φ; θ0, φ0 Þ ¼ P‘ ð cos ωÞ 8π

! ! τ r r 0 0 P‘ ð cosωÞ 2πrr 0

!

It follows that kn, n ðθ, φ; θ, φÞ ¼ k0, 0 ð0, 0; 0, 0Þ ¼

jkj 1 0 4π 0 1

A partitioned form for the density matrix is written as " # ! ! 0 X Γ k1, 1 ðr, r 0 Þk1, 1 ðθ, φ; θ0, φ0 Þ Γ k1, 0 ðr, r 0 Þk1, 0 ðθ, φ; θ0, φ0 Þ Γ r,r ¼ Γ k0, 1 ðr, r 0 Þk0, 1 ðθ, φ; θ0, φ0 Þ Γ k0, 0 ðr, r 0 Þk0, 0 ðθ, φ; θ0, φ0 Þ k The radial charge density is then recovered in the form ! ! er 2 X k qðr Þ ¼ er 2 TrΓ r , r ¼ jkj Γ 1, 1 ðr, r Þ + Γ k0, 0 ðr, r Þ 2π k Spin features of the density matrix are absent in the charge density but need to be considered in relation to the contribution of the exchange energy among other quantities of interest. Exchange energy contributions arise from the off-diagonal elements of the density matrix in the form ! ! ! ! ! ! γ r , r 0 ¼ TrΓ r , r 0 Γ r 0, r It reduces to

! ! X γ r, r 0 ¼ γ k, k0 ðr, r 0 ÞΞ k, k0 ð cos ωÞ 0 k, k

where

1 γ k, k0 ðr, r Þ ¼ 2πiℏcrr 0 0

2 I

dWdW 0 Trk ðr, r 0 ; W Þk0 ðr 0, r; W 0 Þ

and 0

Ξk, k0 ð cos ωÞ ¼ Tr k0, 0 ðθ, φ; θ0, φ0 Þk0, 0 ðθ0, φ0 ; θ, φÞ ð2j + 1Þð2j0 + 1Þ P‘ ð cos ωÞP‘0 ð cos ωÞ ¼ 32π 2 ðj‘Þðj0 ‘0 Þ 2 0 sin ωP‘ ð cosωÞP‘0 0 ð cos ωÞ + 2π 2

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Nearly Spherical Systems

The second term on the right-hand side introduces a feature dependent on the explicit use of spinors in the formulation. Further simplification occurs from the expansion X 1 0 P‘ ð cosωÞP‘ ð cos ωÞ ¼ λ + Pλ ð cosωÞCλll 0 2 λ given by Condon and Shortley and its consequence sin 2 ωP‘0 ð cos ωÞP‘0 0 ð cosωÞ 1X 1 λ + Pλ ð cosωÞ½‘ð‘ + 1Þ + ‘0 ð‘0 + 1Þ λðλ + 1ÞCλll 0 ¼ 2 λ 2 The combined result has the form 1 X 1 ^ λkk0 0 Ξ k, k ð cos ωÞ ¼ 2 λ + Pλ ð cosωÞC 8π λ 2 1 1 1 with coefficients ( j ¼ jkj ; ‘ ¼ k + ; …) 2 2 2 (

λ j j0 0 0 ^ λkk0 ¼ ð2‘ + 1Þð2j + 1Þð2‘ + 1Þð2j + 1Þ 1 0 C ‘ ‘ 2

)2

λ ‘ ‘0 0 0 0

2

since

λ ‘ ‘0 Cλll 0 ¼ 2 0 0 0

2

An exchange energy contribution is estimated as the integral Z 2 ! !0 ! !0 e γ r ; r Ex ¼ d r d r ! ! 2 r 0 r The angular integrations lead to the result Z

Ξ k, k0 ð cosωÞ 1 X r< λ ^ sin θ dθ dφ sinθ dθ dφ ! ! ¼ C λkk0 λ r r 0 r r> > 0

0

0

and, with an approximation based on the Wentzel–Kramers–Brillouin tenet, it follows that

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Jan Linderberg

∞

r

2 I 0 1 0 γ k, k0 ðr, r Þ dr ¼ 2πiℏcrr 0 ðr 0 Þλ1 ﬃ r 3λ

1 2π

I dW

2 I

Trk ðr, r; W Þk0 ðr, r; W 0 Þ

dW 0

ðr Þλ1 ½ipk ðr; W Þ ipk0 ðr; W 0 Þ I dpk0 ðr; W 0 Þ idpk ðr; W Þ pk ðr; W Þ + pk0 ðr; W 0 Þ

This reduction is based on the approximation that the amplitude factors are close to unity since sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 h ¼ 1 + p0 ﬃ 1 σ σ along the integration paths. It follows that Z

∞

dr 0

γ k, k0 ðr, r 0 Þ

r

ðr 0 Þλ + 3

¼

1 2 2 0 r; μc , p min p r; μc k k 2πr 3 + λ k, k0

and finally Z 2e

2

∞

r 0

λ+2

Z dr 0

∞

Z ∞ 0 pκ 0 γ k, k0 ðr, r Þ 2 dr ¼e dr λ1 0 ðr Þ 0

r; μc 2 Rκ ; κ ¼ max0 ðjkj, jk0 jÞ k, k πr

The total exchange energy functional is then Ex ¼

∞ X κ¼1

Rκ

∞ X κ X ^ λκk0 ^ λκk0 + C C 2 λ¼0 k0 ¼κ

It holds that the summations over the angular momentum quantum numbers result, to a good approximation, in the form ∞ X κ X ^ λκk0 ^ λκk0 + C C ﬃ 1:3214κ2 + 1:5891κ ¼ ξ2 κ 2 + ξ1 κ 2 0 λ¼0 k ¼κ

C̿ ðκÞ; κ 1 There remains to evaluate the summation over the angular quantum number and again it is replaced by an integration: X

1 p r; μc C̿ ðκ Þ ﬃ κ κ r 2

Z

kmax

dκ kmin

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k2max κ 2 C̿ ðκÞ

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Nearly Spherical Systems

that gives the form

ξ2 2 2 2 π kmin 2 3 3 λ + kmin + kmin λ kmin λ p r; μc C̿ ðκÞ ﬃ arctan κ κ 8r λ 2 ξ πξ ξ + 1 λ3 ! 2 λ4 + 1 λ3 3r kmin ≪λ 16r 3r

X

and the exchange energy functional Z Ex ¼ e

∞

2 0

Z ∞ dr ξ2 4 ξ1 3 dr 2 0:082587λ4 + 0:16861 λ3 λ + λ ¼e 2 2 3π r 16 0 r

The original Dirac functional reads, in the present notation, as Z ExDirac

¼e

∞

2 0

dr λ4 ¼ e2 r 2 π2

Z

∞ 0

dr 0:10132 λ4 r2

A comparison between the two integrands is shown in the figure below: 10 8 6 4 2 s 0

l

1 4

2 3

Blue: s = 0.082587 l + 0.16861 l , Red: s = 0.10132 l4

The present function is somewhat larger than the original Dirac one for λ-values in the range for normal atomic fields and also for a field such as the one obtained for Buckminsterfullerene by Clougherty and Zhu.9

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Jan Linderberg

7. YET AN ENERGY FUNCTIONAL Inclusion of an exchange energy contribution to the energy functional gives the form

Z ∞ Z ∞ ( 0 2 eqðr ÞZ ðr Þ eqðr ÞZn ðr Þ 1 0 qðr Þ JX ðZ Þ ¼ dr Φðr, Z Þ + dr 0 + r r 2 r r 0

e2 ξ2 ξ1 εF 4 3 λðr, Z Þ + λðr, Z Þ qðr Þ 2 r 16 3π e The second term in the integrand eliminates the potential term included in the function Φ(r, Z), while the third one introduces the attraction from the nuclear configuration through the function Zn(r). Repulsion within the electronic distribution is added as the fourth term and the exchange energy follows, ideally eliminating self-interaction in the former one. A Lagrangian parameter, identified as the Fermi energy, completes the integrand. Optimization of the functional is feasible within the finite-element method and to illustrate this the functional is reduced to the limit of a ^ ! 0: Then vanishing fine-structure constant, α 8 2 s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z∞ < 2 2 qðr Þ 4 2 e ½Zn ðr Þ + ξ1 =4 3e a0 3 3πqðr Þr 2 JX ðqÞ ¼ dr μc εF + : e 10r 2 r 4e 0 9 3 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Z ∞ 0 2= 3πξ2 e2 3 3πqðr Þr 5 1 0 qðr Þ + dr 0 ; 64r 4e 2 r r where the integrand is a functional of the radial electron density. Jensen’s functional is retrieved when the Fermi energy is given relative to the rest mass, the exchange term is replaced by Dirac’s form, and the number density is replaced by the radial density: q(r) ¼ 4πr2en(r). An implementation the finite-element method is based on the representation of the potential integral as a piecewise linear function Z ∞ a r 0 r b 0 qðr Þ 0 dr 0 ¼ eZ a + eZb0 ; a