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 9781619427785, 1619427788, 44-2011-515-7

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PHYSICS RESEARCH AND TECHNOLOGY

THEORETICAL AND COMPUTATIONAL DEVELOPMENTS IN MODERN DENSITY FUNCTIONAL THEORY

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PHYSICS RESEARCH AND TECHNOLOGY

THEORETICAL AND COMPUTATIONAL DEVELOPMENTS IN MODERN DENSITY FUNCTIONAL THEORY

AMLAN K. ROY EDITOR

New York

Copyright © 2012 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Theoretical and computational developments in modern density functional theory / [edited by] Amlan K. Roy. p. cm. Includes bibliographical references and index. ISBN:  (eBook) 1. Density functionals. 2. Functional analysis. I. Roy, Amlan. QC20.7.D43T44 2011 515'.7--dc23 2011052310

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface

vii

Chapter 1

Density Functional Theory: From Fundamental Precepts to Nonlocal Exchange- Correlation Functionals Rogelio Cuevas-Saavedra and Paul W. Ayers

1

Chapter 2

Recent Progress towards Improved Exchange-Correlation Density-Functionals Pietro Cortona

41

Chapter 3

Constrained Optimized Effective Potential Approach for Excited States V. N. Glushkov and X. Assfeld

61

Chapter 4

Time Dependent Density Functional Theory of Core Electron Excitations: From Implementation to Applications Mauro Stener, Giovanna Fronzoni and Renato De Francesco

103

Chapter 5

Time Dependent Density Functional Theory Calculations of Core Excited States Nicholas A. Besley

149

Chapter 6

Density Functional Approach to Many-Electron Systems: The Local-Scaling-Transformation Formulation Eugene S. Kryachko

169

Chapter 7

Electron Density Scaling - An Extension to Multi-component Density Functional Theory Á. Nagy

189

Chapter 8

A Symmetry Preserving Kohn-Sham Theory Andreas K. Theophilou

201

Chapter 9

Self-Interaction Correction in the Kohn-Sham Framework T. Körzdörfer and S. Kümmel

211

vi

Editors

Chapter 10

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional Theories in the Presence of a Magnetostatic Field Xiao-Yin Pan and Viraht Sahni

223

Chapter 11

The Construction of Kinetic Energy Functionals and the Linear Response Function David García-Aldea and J. E. Alvarellos

255

Chapter 12

Variational Fitting in Auxiliary Density Functional Theory Víctor Daniel Domínguez Soria, Patrizia Calaminici and Andreas M. Köster

281

Chapter 13

Wavelets for Density-Functional Theory and Post-DensityFunctional-Theory Calculationss Bhaarathi Natarajan, Mark E. Casida, Luigi Genovese and Thierry Deutsch

313

Chapter 14

Time-Dependent Density Functional Theoretical Methods for Treatment of Many-Electron Molecular Systems in Intense Laser Fields Dmitry A. Telnov, John T. Heslar and Shih-I Chu

357

Chapter 15

A Hierarchical Approach for the Dynamics of Na Clusters in Contact with an Ar Substrate P. M. Dinh, J. Douady, F. Fehrer, B. Gervais, E. Giglio, A. Ipatov, P. G. Reinhard and E. Suraud

391

Chapter 16

Atoms and Molecules in Strong Magnetic Fields M. Sadhukhan and B. M. Deb

425

Chapter 17

Chemical Reactivity and Biological Activity Criteria from DFT Parabolic Dependency E=E(N) Mihai V. Putz

449

Chapter 18

Effect of a Uniform Electric Field on Atomic and Molecular Systems Santanu Sengupta, Munmun Khatua and Pratim Kumar Chattaraj

485

Chapter 19

A Quantum Potential Based Density Functional Treatment of the Quantum Signature aof Classical Nonintegrability Arup Banerjee, Aparna Chakrabarti, C. Kamal and Tapan K. Ghanty

505

Chapter 20

Properties of Nanomaterials from First Principles Study Arup Banerjee, Aparna Chakrabarti, C. Kamal and Tapan K. Ghanty

527

Preface

vii

Chapter 21

The Role of Metastable Anions in the Computation of the Acceptor Fukui Function Nelly González-Rivas, Mariano Méndez and Andrés Cedillo

549

Chapter 22

Kinetic-Energy/Fisher-Information Indicators of Chemical Bonds Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

561

Index

589

P REFACE

Today, our theoretical understanding of many-electron systems is largely dictated and dominated by Density functional theory (DFT). It plays a unique pivotal role for realistic and faithful treatment of materials in diverse fields such as chemistry, physics and biology. In many important research areas dealing with atoms, molecules, solids, clusters, nanomaterials including organic molecules, biomolecules, organometallic compounds, etc., DFT has become an indispensable and invaluable tool for nearly three and a half decade. Range of application is updated almost on a regular basis. Numerous exciting developments have been made in recent years which render quantum mechanical calculation of larger and larger systems more accurate and computationally approachable, which were otherwise impossible earlier. Scope of the method is extended for an overwhelmingly large array of systems; very well surpassing the limit and range of any other existing method available today. This book makes an attempt to present some of the important and interesting developments that took place lately, which have helped us in extending our knowledge on the electronic structure of materials. Fundamental and conceptual issues, formulation and methodology development, computational advancements including algorithm, as well as applications are considered. However, a topic as broad as DFT can not be covered in a single volume such as this. The chapters are mostly focused on theoretical, computational, conceptual issues, as the title implies. Therefore, purely application-oriented works are not included; applications scattered here and there in the book are mainly to assess the quality of the theory and feasibility of the method in question. The choice of the topics is far from complete and comprehensive; omissions are inevitable. Many important issues could not be taken up in this volume due to the space and time constraint (several authors expressed interest, but could not contribute finally because of lack of time). The first two chapters deal with one of the major issues in DFT, viz., the exchangecorrelation (XC) functionals. Its exact form remains unknown as yet and must be approximated for practical calculations. The authors start with a brief introduction to DFT, with special emphasis on XC functionals and a small review of the commonly used functionals. Chapter 1 discusses the inadequacy of conventional XC functionals, supposedly rooted in their inability to recover appropriate behavior for fractional charges as well as fractional spins. This arises primarily due to the neglect of dispersion interactions and

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Amlan K. Roy

strong correlation between non-spatially-separated electrons. This leads to the development of nonlocal 2-point weighted density approximated (2-WDA) functionals which are rigorously self-interaction free, closely mimics the proper fractional charge and spin behavior. This also produces dispersion interactions with the correct R−6 form and appears to hold great promise for the future advancements in XC functionals. Chapter 2 focuses on a local (SRC) and two generalized-gradient-type functionals (TCA and RevTCA). The local one offers very similar results as the LDA functional for equilibrium bond lengths while for atomization energies and barrier heights surpasses the LDA results. The TCA functional provides good results for thermochemistry, geometry and excellent-quality results for hydrogen bonded systems. The last one is found to be quite good for atomization energies and barrier heights. Although DFT has witnessed remarkable success for ground states, the same for excited states has come much later and somehow rather less conspicuous. Chapter 3 presents a constrained variational approach based on the asymptotic projection method along with its applications to the optimized effective potential problem. This facilitates the solution of relevant Kohn-Sham (KS)-type equation to handle appropriate local potential for excited states within the framework of both variational and non-variational approaches. The usefulness and efficiency of the method is illustrated by presenting results on various excitations in atoms and molecules. Chapters 4, 5 use the time-dependent (TD) DFT method to treat the core excitations which are notoriously difficult due to the presence of delicate correlation effects. High accuracy results are obtained in Chapter 4 for fine spectral features of small molecules in the gas phase, correctly taking into account the crystal field effect, configuration mixing and spin-orbit coupling. Chapter 5 sketches the current progress towards the NEXAFS spectra of relatively large systems including biologically significant molecules through TDDFT and development of suitable XC functionals in this regard. Chapter 6 reviews the so-called local-scaling-transformation of DFT for many-electron systems by introducing the concept of an orbit. Through a “variational mapping” procedure, it exploits the topological features of one-electron densities of atoms and molecules. The N − and v−representability criteria on the energy functional are satisfied. This is applicable to both Hartree-Fock and KS Hamiltonians, yielding corresponding orbitals and energies. Chapter 7 presents a generalization of DFT to a multi-component theory having relevance in non-adiabatic processes. Here, both the electrons and nuclei can be treated completely quantum mechanically without the use of Born-Oppenheimer approximation. This gives rise to two fundamental quantities: the electron density and nuclear N-body density. A density scaling route is advocated for the former via a new KS scheme. A value of the scaling factor exists for which the correlation energy disappears. Interestingly then one has to calculate exchange energy instead of the XC energy, which can be obtained very accurately in terms of the KS orbitals. The correlation energy, on the other hand, is not easily expressible in terms of the orbitals. A simple method to incorporate a major portion of correlation is also given. Chapter 8 considers the problem of a KS-type theory for the lower state belonging to an irreducible representation of a symmetry group of the exact Hamiltonian. The KS state reproducing an exact density does not have the transformation properties of an exact state. It is possible to develop a theory of the exact state properties in terms of approximate density or KS many-particle state. This relies on the availability of suitable functionals.

Preface

xi

Self-interaction remains one of the serious and nagging problems in DFT, and is presumably responsible for many qualitative defects of today’s XC functionals. Apparently the reason lies in its connection with the (semi-)local modeling of non-dynamic correlation. Chapter 9 gives KS self-interaction correction as a viable alternative to the traditional selfinteraction correction that employs orbital-specific potentials. Different KS self-interaction approaches are possible by means of different choices of the unitary transformation. Several such schemes are compared and contrasted with the traditional self-interaction approaches by taking the static electric polarizability of hydrogen chains as a reference problem. Chapter 10 summarizes the Hohenberg-Kohn, KS and Quantal DFT in presence of a magnetostatic field, B = ∇ × A(r). In presence of an external field, v(r), the basic variables in all these theories are the ground-state density ρ(r) and physical current density, j(r). This is achieved by proving the relationship between densities {ρ(r), j(r)} and external potentials {v(r), A(r)} to be one-to-one. Besides being a unique functional of {ρ(r), j(r)}, the ground-state wave function, however, must also be a functional of gauge function to ensure that the wave function expressed as a functional, is gauge variant. Extension of these to other Hamiltonians such as those in spin DFT or in which magnetic field interacts with both orbital and spin angular momentum, etc., is also considered. Chapter 11 reviews some of the most important nonlocal kinetic energy density functionals available today, all of which reproduce the linear response function of a free electron gas. General strategies behind the construction of these functionals are discussed, that make them suitable for use in both extended and localized electron systems. It is stressed that the local behavior of kinetic energy densities should be used as the guiding factor for designing new functionals as the latter is closely related to the potential. These ideas may also have relevance for XC functionals as well. In Chapter 12, variational fitting of auxiliary densities in DFT is discussed in detail. Through an iterative solution of the fitting equations, auxiliary DFT allows accurate and efficient first principles all-electron calculations of complex systems containing 5001000 atoms. A combination of singular value decomposition and preconditioned conjugate method offers a viable compromise between numerical stability and accuracy. By using diffuse auxiliary functions, calculation of structure, response property of large systems such as giant fullerenes and zeolites are possible relatively easily. Chapter 13, as the authors put succinctly, attempts to “make some waves about wavelets for wave functions”. Wavelets are essentially Fourier-transform like approaches and have been routinely used by engineers for several decades. Their advantages compared to standard Fourier-transform techniques are well known for multi-resolution problems with complicated boundary conditions. However, in the context of quantum chemistry or chemical physics, their usefulness and applicability has remained largely unexplored until very recently. At first, an elementary review of the subject is given. Then the authors discuss the theory behind the wavelet-based BIGDFT code for ground-state DFT and application of the same in the linear-response TDDFT. The possibility of making high-performance computing order-N wavelet-based TDDFT program for practical calculation of larger systems is also mentioned briefly. Chapter 14 reviews the latest developments in TDDFT front for studying the dynamical behavior of many-electron atoms/molecules interacting with a strong laser field. Use of optimized effective potential plus self-interaction correction facilitates the use of orbital-

xii

Amlan K. Roy

independent, one-electron local potential reproducing correct asymptotic behavior. Structure and dynamics are followed by solving the relevant KS equations quite accurately efficiently in a non-uniform, optimal spatial grid by means of the generalized pseudospectral method. Illustrative results are presented for multi-photon processes in diatomic and triatomic molecules through multi-photon ionization, high-order harmonic generation, etc. A hierarchical method is presented in Chapter 15 in order to study the dynamics of small metal clusters in contact with moderately active environments. The cluster can be treated at the fully quantum mechanical level, while the substrate at a classical level. Results on structural properties, dynamics in the linear-response regime as well as non-linear dynamics induced by strong femto-second laser pulses are given for small Na clusters in contact with Ar matrices. The interesting topic of atoms, molecules interacting with a strong magnetic field is taken up in Chapter 16. A density-based single equation approach within DFT is used to unravel the complex behavior of such systems. Some recent works on many-electron systems in presence of strong but static magnetic field are presented. Nature of molecular bonding is found to change under such fields, and a hydrogen atom in a strong oscillating magnetic field leads to the possibility of emission of coherent radiation. Chapter 17 reviews the general parabolic dependency of energy-number of electrons, E=E(N), within the context of DFT, in terms of electronegativity (negative of chemical potential) and chemical hardness, the two first- and second-order parameters respectively. Numerous applications ranging from atomic to molecular systems, as well as chemically reactive to biologically active environments are considered. Feasibility of DFT parabolic recipe for describing reactivity-activity principle in open systems (chemical, biological) by means of electronegativity and chemical hardness, viewed as “velocity/slope” and “acceleration/curvature”, in an abstract way, is explored. Effect of a uniform electric field on atomic, molecular systems is investigated in Chapter 18 within the KS approach. Writing the total energy in terms of a Taylor series expansion, interesting results are obtained for neutral atoms and ions. Explicit expressions are given for ionization potential and electron affinity changes in atoms. Consequently, it reveals that electronegativity of an atom exhibits an increment, when immersed in a uniform electric field. And hence the chemical reactivity of a system in such a field will be different from that in absence of the field. Molecules in excited states exhibit pronounced geometrical changes; excitation energies are decremented under the influence of such a field. Effect of basis set and XC potential on TDDFT excitation energies is also monitored. A combined quantum fluid dynamics and DFT-approach is employed to investigate the quantum-domain behavior of classically non-integrable systems in Chapter 19. Quantum signature of classical Kolmogorov-Arnold-Moser-type transitions in different anharmonic oscillators is probed starting from a toroidal to chaotic motion. Field-induced barrier crossing as well as the chaotic ionization in Rydberg atoms is also analyzed through such a quantum potential-based method. In the zero quantum-potential limit, a classical-like scenario is restored for a couple of quantum anharmonic oscillators. Theoretical investigation on the structural and optical response properties is made for various nano-clusters, nano-tubes and nano-cages (C20 , C60 , C80 , C100 ) through DFT and TDDFT in Chapter 20. Nano-clusters made of alkali metal atoms (N an , Kn ), noble metal gold atom doped with alkali and other coinage atoms (Au19 X, X =

Preface

xiii

Li, N a, K, Rb, Cs, Cu, Ag) as well as mixture of Ga and P atoms (Gan Pn ) have been considered, whereas carbon nano-tubes of various lengths and diameters are employed. Properties like binding energy, HOMO-LUMO gap, ionization potential, electron affinity, linear polarizability are used to follow the size-to-property relationship in these systems. Also van der Waals coefficient is calculated by means of Casimir-Polder relation, connecting this to the frequency-dependent dipole polarizability at imaginary frequencies. Chapter 21 concerns with the computation of vertical electron affinity and acceptor Fukui function. The role of metastable anions in the latter case is also examined. Chemical reactivity descriptors (local as well as global) for neutral molecules are severely restricted for unstable anions. The bound state electronic structure of such an unstable anion is satisfactorily obtained through an orbital swapping method. Applications of the methodology are given for small molecules, carbonyl organic compounds and inorganic Lewis acids. The density of non-additive Fisher information in atomic orbital resolution, related to the kinetic energy (contragradience) criterion, is demonstrated to provide a sensitive, viable probe for characterizing chemical bonds in Chapter 22. Regions of negative values mark the location of bonds in a molecule. The interference, non-additive contribution to the molecular Fisher-information density is used to determine bonding regions in molecules. Representative calculation on selective diatomics and polyatomics justifies the applicability and validity of the contragradience probe in exploring bonding patterns in molecules. Finally, I sincerely thank all the authors for agreeing to contribute in this edition, taking their valuable time off and adhering to the general time schedule. I am deeply indebted to Professor B. M. Deb, who initiated me into this wonderful and mysterious land of DFT while I was working as a graduate student in his laboratory. Fruitful and valuable discussion with professors Daniel Neuhauser, S. I. Chu, A. J. Thakkar, Emil Proynov, Z. Zhou, K. D. Sen, D. A. Telnov, is also gratefully acknowledged. Numerous valuable discussion with the IISER-K colleagues and students has helped me gain a deeper understanding of the subject with time. This book could not have been possible without the generous support from the Editorial and Publication staffs of NOVA Science Publishers in many ways, especially in extending the deadline several times; it is a pleasure to work with them.

In: Theoretical and Computational Developments… Editor: Amlan K. Roy, pp. 1-40

ISBN: 978-1-61942-779-2 © 2012 Nova Science Publishers, Inc.

Chapter 1

DENSITY FUNCTIONAL THEORY: FROM FUNDAMENTAL PRECEPTS TO NONLOCAL EXCHANGE- CORRELATION FUNCTIONALS Rogelio Cuevas-Saavedra and Paul W. Ayers∗ Department of Chemistry and Chemical Biology; McMaster University Hamilton, Ontario, Canada

Abstract Due to its favorable cost per unit accuracy, density functional theory (DFT) is the most popular quantum mechanical method for modeling the electronic structure of large molecules and complex materials. In DFT, the exchange-correlation functional has to be approximated since it’s exact from is unknown. While commonly used functionals are often successful, they have large and systematic failures for certain types of molecules are properties. In this chapter, we review the fundamentals of DFT, with particular emphasis on exchange-correlation functionals and the role of the exchange-correlation hole in developing new functionals. After reviewing the failures of conventional approaches for developing functionals, we review our recent work to develop fully nonlocal functionals based on the uniform electron gas. Keywords: Density-Functional Theory, Uniform Electron Gas, Nonlocal Functionals, Weighted Density Approximation.

Keywords: Density-Functional Theory, Uniform Electron Gas, Nonlocal Functionals, Weighted Density Approximation

I. The Electronic Structure Problem The majority of matter in universe consists of protons, neutrons and electrons. Under terrestrial conditions, protons and nucleons clump together to form positively charged atomic ∗

E-mail address: [email protected], [email protected]

2

Rogelio Cuevas-Saavedra and Paul W. Ayers

nuclei. Electrons, due to their negative charge, are attracted to and bound by the resulting nuclei, forming atoms. Molecules arise when atoms come close together, so that the electrons are attracted to more than one atomic nucleus. Nuclei and electrons behave very differently under ordinary conditions. Nuclei do not change significantly when atoms and molecules condense to form liquids and solids. The clouds of electrons surrounding the nuclei, on the other hand, dramatically deform. Electrons “pair” to form chemical bonds; they migrate from less electronegative to more electronegative regions; they correlate their motion to minimize their mutual repulsion, which leads (among other effects) to dispersion forces. Therefore, most of the problems in biology, chemistry, and condensed-matter and atomic/molecular physics are, at a fundamental level, manifestations of the electronic structure problem. The electronic structure problem—the problem of understanding, predicting, and modelling the behaviour of electrons in different atoms, molecules, and materials—is of undoubted importance. Obtaining quantitative results for the electronic structure problem usually entails approximately solving the electronic Schrödinger equation with a complicated form for the wave function. Once an accurate wave function is known, of course, then any molecular property can be computed. However, the daunting dimensionality of the wave function (3N spatial dimensions plus N spin dimensions) hinders progress. It would be desirable to have an alternative descriptor for the system, something much simpler than the wave function that nonetheless suffices to determine all molecular properties. Ideally, we would like the resulting descriptor to have a simple and direct physical interpretation (unlike the wave function) and we would like the corresponding theory to preserve the conceptual utility of the Hartree-Fock orbitals and orbital energies. One such theory is density functional theory (DFT).

II. Density Functional Theory (DFT) A. Overview The main idea in DFT is to change the descriptor of the system from the wave function to the ground-state electron density. To prove that this can be done, we must first prove that, just like in the wave function-based approach, all the information about an electronic system can be extracted from its ground state electron density. The key insight is that the form of the kinetic energy and electron-electron repulsion energy operators are universal: they do not depend on the particular system of interest, but only on the number of electrons, N.

 1  Tˆ = ∑  − ∇i2   2  i=1

(1)

e2 j=i+1 ri − rj

(2)

N

N

N

Vˆee = ∑ ∑ i=1

The only part of the electronic Hamiltonian that depends on the system of interest is the potential the electrons feel due to the nuclei in the system. Since electrons are not responsible

3

Density Functional Theory

for this potential, we will refer to it as the external potential. The electronic Hamiltonian now reads N

Hˆ = Tˆ + Vˆee + ∑ v ( ri ) i=1 N

= Fˆ + ∑ v ( ri ) (3)

i=1

where we have denoted the external potential by v(r) and grouped the kinetic and electronelectron repulsion energies in one term, denoted Fˆ . We say that the operator is universal because, no matter which electronic system we are dealing with, its form is always the same. Since the Hamiltonian determines the ground-state electronic wave functions of the system (from the variational principle), then any ground-state electronic property of the system can be expressed in terms of the number of electrons in the system, N, and the external potential, v(r). That is, every ground-state electronic property is a function of N and a functional of v(r). In order to motivate the subsequent development, we remind the reader that the wave function of a system does not possess any physical meaning per se. The most informative quantity that follows directly from the wave function is its squared modulus,

Ψ (r1 , σ1 ; r2 , σ 2 ;...; rN , σ N ) , which represents the probability than an electron located at 2

has spin

σ 1, another electron located at has spin σ 2 , etc.

Using this probabilistic interpretation of the wave function, the probability of observing an electron of either spin at position is given by N

ρ (r ) = ∑ ∫ δ (ri − r )∑ Ψ (r1 , σ 1; r2 , σ 2 ;...; rN , σ N σk

i =1

= Ψ

)

2

dr1 ⋅ ⋅⋅ drN

N

∑δ (r − r ) Ψ i

i =1

= N ∑ ∫ Ψ (r, σ 1 ; r2 , σ 2 ;...; rN , σ N ) dr2 ⋅⋅ ⋅ drN 2

σk

(4)

ρ (r ) is called the electron density of the system. Since it represents the probability of observing an electron at certain position, it is a nonnegative quantity. Since the operator in Eq. (4) is Hermitian, the electron density is an experimental observable. From the definition (4), it follows that the electron density is normalized to the number of electrons,

N [ρ ] = ∫ ρ (r )dr

(5)

4

Rogelio Cuevas-Saavedra and Paul W. Ayers

The square-bracket notation in Eq. (5) indicates that the number of electrons is a functional of the electron density.

B. The Ground-State Electron Density as the Descriptor of Electronic Systems: The First Hohenberg-Kohn Theorem The main attraction of DFT is that all the information about the system can be obtained from the ground-state electronic density, which generally depends on many fewer variables than, and is much simpler to interpret than, the electronic wave function. In the previous subsection, we showed that the number of electrons could be determined from the density. We also mentioned that if we know the number of electrons and the external potential of a system, then the Hamiltonian of the system is known and, by solving the Schrödinger equation, all properties of the system may be determined. The first Hohenberg-Kohn theorem states that the external potential is a functional of the ground-state electron density and implies that all properties of an electronic system are functionals of the ground-state electron density.[10,11] First Hohenberg-Kohn Theorem. For any system of interacting electrons in an external potential v(r), the external potential is uniquely determined, up to an arbitrary constant, by the ground-state electronic density ρ(r). The proof is simple, but not constructive. Consider two different N-electron systems with

{Ψ1 , v1}and {Ψ 2 , v2 }.

different ground-state wave functions and external potentials

Because the ground-state wave functions are different, the external potentials differ by more than a constant shift, v1 (r ) ≠ v2 (r ) + k . Accordingly, to the variational principle we have

Ψ1 Hˆ 1 Ψ1 < Ψ 2 Hˆ 1 Ψ 2 Ψ 2 Hˆ 2 Ψ 2 < Ψ1 Hˆ 2 Ψ1

(6)

Substituting the form of the electronic Hamiltonian in Eq. (3)and adding the two inequalities gives

 ˆ  Ψ 1 F Ψ1 + Ψ 1 

N

∑ v (r ) Ψ 1

i =1

i

1

  

 +  Ψ 2 Fˆ Ψ 2 + Ψ 2 

  

N

∑ v (r ) Ψ i

2

2

i =1

 <  Ψ 1 Fˆ Ψ1 + Ψ 1 

∑ v (r ) Ψ

 +  Ψ 2 Fˆ Ψ 2 + Ψ 2 

 Ψ v r ( )  ∑ 1 2 i i =1 

N

2

i =1

i

1

  

N

(7)

5

Density Functional Theory which simplifies to

Ψ1

N

∑ (v (r ) − v (r )) Ψ 1

i =1

i

2

i

1

− Ψ2

N

∑ (v (r ) − v (r )) Ψ 1

i =1

i

2

i

2

rs 

(58)

The possibility that three electrons could be located inside the sphere of radius rs is neglected in this model. For inter-electronic distances r rs the Overhauser model is not expected to be very reliable. It is also expected that the results become more accurate as the density decreases since the probability of having three electrons in the same sphere of radius rs becomes smaller as the density decreases. Finally, at high and intermediate densities the results for the exchangecorrelation function are expected to be closer to the actual hole for anti-parallel spin correlations that for parallel ones. Indeed, when two electrons of opposite spins are in a sphere of radius rs, a third electron tends to be excluded from the sphere because of both the Coulomb repulsion and the Pauli exclusion principle. This is not the case for parallel spin electrons, where only the Coulomb repulsion prevents a third electron of opposite spin from entering the Wigner-Seitz sphere.

D. An Approximation to the UEG’s the Exchange-Correlation Hole: The Model of Gori-Giorgi and Perdew The Overhauser potential makes it possible to study short-range correlation effects in the UEG. Now the Schrodinger equation and its solutions take the following form

 d 2 l ( l +1) 2   dr 2 − r 2 + k − V ( r, rs )  ul = 0  

(59)

18

Rogelio Cuevas-Saavedra and Paul W. Ayers

ul ( r ) = krRl ( r )

(60)

Equation (59) can be solved analytically, but the solution cannot be expressed in a closed form. Fortunately, we are not interested in the wave function itself, but the pair correlation function that comes from it. An analytical model for the exchange-correlation hole of the UEG, based on the Overhauser potential and various exact properties of the UEG, was developed by Gori-Giorgi and Perdew. From now on we will refer to it as the GGPxc model for the UEG hole.[30] The GGPxc model is actually not a model for the hole itself, but for its coupling-constant average. This coupling constant average corresponds to the adiabatic connection mentioned in previous sections of this chapter 1

g xc ( r, r ') = ∫ g xcλ ( r, r ') d λ 0

(61)

λ where g xc corresponds to the pair distribution function when the electron-electron interaction

is

λ r − r ' . In this case, the coupling constant is equivalent to an average over r

1 s gxc ( kF u,rs ,ζ ) = ∫ gxc ( kF u, Rs ,ζ ) dRs rs 0 r

=

1 s  gx ( kF u,ζ ) + gc ( kF u, Rs ,ζ )  dRs rs ∫0 

(62)

where we have indicated the usual separation of (and consequently g xc ) into its exchange and correlation contributions. Notice that rs is not part of the arguments for the exchange contribution in (62). This is because the explicit dependence on rs occurs only when the Coulomb repulsion is taken into account. Both gx and gc have long-range oscillations. These oscillations are unimportant from an energetic point of view in the following sense: one can design a model that describes shortrange correlation correctly but averages over the long-range oscillations, but still gives correct results for the total exchange-correlation energy. Since the long-range oscillations are not transferable between systems, it seems better to use non-oscillatory models. GGPxc is a nonoscillatory model for the pair correlation function that is designed to reproduce many of the known exact properties of the exchange-correlation pair distribution function. Perhaps the most important properties are the normalization of the exchange and correlation holes,

19

Density Functional Theory ∞



0

0

2 2 ∫ 4π u ρ hx (u )du = ∫ 4π u ρ  g x (u ) −1 du = −1



63) ∞

∫ 4π u ρ g (u )du = ∫ 4π u ρ g (u )du = 0 2

2

c

c

0

0

(64)

These constraints are also called the particle-conservation sum rules. They ensure that the model for the pair correlation function is self-interaction free. That is, the normalization of the exchange-correlation hole guarantees that an electron in a N-electron system interacts only with the remaining N-1 electrons in the system, and not with itself. The specific form of the integrals in (63) and (64) holds only for the UEG, but the hole normalization constraints are general requirements for any electronic system. The GGPxc model is designed to recover correct results for the exchange energy, ∞  g x (u ) − 1 1 2 u du = ε x (rs , ζ 4 π ρ u 2 ∫0

) (65)

the correlation energy ∞ g c (u ) 1 du = ε c (rs , ζ 4π u 2 ρ ∫ u 20

) (66)

and the potential energy of correlation, ∞ g c (u ) 1 2 π ρ u du = vc (rs , ζ 4 u 2 ∫0

) (67)

in the UEG. The short-range behaviour of the pair correlation distribution function is governed by the so-called cusp condition[31]

dg xc du

u =0

= g xc u =0 (68)

This cusp condition is modified when imposing it on the coupling constant average of the pair distribution function[27,32]. The cusp condition holds for all systems, not just the UEG. The long-range behaviour of the pair correlation function is also known, but it is more convenient to formulate in reciprocal (i.e., Fourier) space than in real space. The static structure factor[30,33] is related to the Fourier transform of the pair correlation function by the equation,

20

Rogelio Cuevas-Saavedra and Paul W. Ayers ∞

4  g xc (k F u, rs , ζ ) − 1 S xc  k , rs , ζ  = 1 + k  F  3π ∫0  × (k F u )

2

sin (k F u ) kF u

d (k F u ) (69)

The corresponding coupling constant average S xc can be obtained by replacing with g xc . Like the pair correlation function, the static structure factor can also be separated into exchange and correlation contributions. The long-range decay of the non-oscillatory gxc corresponds to the short-range behaviour of the static structure factor, which is constrained by the plasmon sum rule[30,33]

S xc (k → 0, rs , ζ ) =

k2 2 3

+ ϑ (k 4 ) rs3

(70)

Notice that there is no k3term in the small-k expansion of the static structure factor. Taking the Fourier transform of the exchange hole reveals that the small-k behaviour of the exchange-only static structure factor has the form,

3 k k3 2/3 2/3 S x (k → 0, rs , ζ ) = (1 + ζ ) + (1 − ζ )  −  kF 16kF3 8

(71)

In order to reconcile (71) and(70), there must be a linear and cubic term in the small-k expansion of the correlation static structure factor that exactly compensates the corresponding terms in the exchange structure factor. In real space, the k2 asymptotic decay of Sxc implies that the exchange-correlation hole decays as u–5 when u is large; this gives rise to the R–6 dispersion interaction between widely-separated systems. We can now present the functional form of the GGPxc model for the pair correlation function of the uniform electron gas. The notation

g xc refers to the non-oscillatory pair

distribution function. The derivation of the GGPxc model is quite complicated. The reader is referred to the original paper by Gori-Giorgi and Perdew for details.[30] The exchange piece of the non-oscillatory pair function is given by

g x ( k F u, ζ ) = 1 +

{

1 2 1/3 (1 + ζ ) J (1 + ζ ) kF u  2

+ (1 − ζ )

2

J (1 − ζ ) k F u    1/3

}

(72)

Where

J (y ) = −

9 4 y4

 Ax2 y 4 Ax3 y 6   − Ax y 2  2 − + + + 1 1 e A y    x 2 3!    

(

+e − Dx y Bx + C x y 2 + Ex y 4 + Fx y 6 2

)

(73)

21

Density Functional Theory The correlation piece is fit to the form

g c ( k F u , rs , ζ ) =

φ ( ζ ) rs f1 ( v ) ⋅ 2 κ ( kF u )

+e

− d (ζ ) x 2

 1 2 4  − d (ζ ) x 2  2  1 + d ( ζ ) x + d (ζ ) x   1 − e 2   

6

∑ c ( r ,ζ ) x n

n =1

n −1

s

(74)

Where

f1 ( v ) =

a0 + b2 v + a1v 2 + a2 v 4 + a3v 6

(v

φ (ζ ) = (1 + ζ )

2/3



2

)

+ b2

4

(75)

2/3 + (1 − ζ )  / 2 

(76)

x = k F u / φ (ζ )

(77)

v = φ (ζ ) κ rs kF u

(78)

d (ζ ) = 0.131707 (1 + ζ 

)

2/3

+ (1 − ζ )

2/3

− 1 

(79)

The constants Ax, Bx, Cx, Dx, Ex and Fx in equation (73) and a0, a1, a2, a3, b2and b in equation (75) are chosen to satisfy the aforementioned constraints on the pair correlation function. In equation(74), the coefficients depend on the Wigner-Seitz radius and the spin polarization. c1, c2, and c3 stipulate the short-range form of the pair correlation function and are determined by the Overhauser model to be[27]

(1 − ζ ) a c ( r ,ζ ) = 2

1

s



2

1 3

(

 4  1−ζ c2 ( rs , ζ ) = φ (ζ )   2  9π 

2

)

↑↓ 0

( r ) −1 ↑↓ s

(80)

(1 + ζ ) 13 + (1 − ζ ) 13    ↑↓ ↑↓ a1 rs 2

( )

8 8  4 23 1 + ζ ) 3 + (1 − ζ ) 3  (     + c1 ( rs , ζ ) d (ζ ) c3 ( rs , ζ ) = φ (ζ )   a2 rs , ζ − 20  9π     2

(

(81)

)

(82)

22

Rogelio Cuevas-Saavedra and Paul W. Ayers

where

rs↑↓ =

( )

a0↑↓ rs =

( )

a1↑↓ rs =

2rs (1 + ζ ) 3 + (1 − ζ ) 13    1

(83)

1 − A + Brs + Crs2 + Drs3 + Ers4 ) e − d0 rs + A  (  rs

(

)

1 − A1 + B1rs + C1rs2 + D1rs3 + E1rs4 + F1rs5 e− d0 rs + A1  rs 

(84)

(85)

2 −1   1+ ζ  3 ↑↑  3 a2 ( rs , ζ ) =   (1 + ζ ) a2  rs (1 + ζ )  2   2

2 −1   1− ζ  3 ↑↑  3 +  (1 − ζ ) a2  rs (1 − ζ )  2   2

2

1 1     2rs  1 − ζ   (1 + ζ ) 3 + (1 − ζ ) 3  ↑↓   + a 2  1 1 2   (1 + ζ ) 3 + (1 − ζ ) 3   2       2

(86)

and

1  9π  a ( rs ) =   5 4  ↑↑ 2

a2↑↓ ( rs ) =

2

3

(

)

1 − d p rs 2 − A + B r + C r e + Ap  p p s p s rs 

(

)

1 − Aa + Ba rs + Ca rs2 + Da rs3 + Ea rs4 e − da rs + Aa   rs

(87)

(88)

c4, c5, and c6are constrained by the plasmon sum rule, the particle conservation sum rule, and the expression for the correlation energy of the UEG. These considerations lead to the following forms

{

cɶ4 ( rs ,ζ ) = 100 π ( 3π − 8) cɶ1 ( rs ,ζ ) + ( 690π − 2048 ) cɶ2 ( rs ,ζ ) + π ( 225π

−672 ) cɶ3 ( rs ,ζ ) + (8192 − 2100π ) A ( rs ,ζ ) S (α ) + A( rs ,ζ ) P (α ) ( 600π −2048 ) + 960 π  A( rs ,ζ ) R (α ) − E ( rs ,ζ ) 

}

 4 ( 512 − 165π )   

(89)

23

Density Functional Theory

{

cɶ5 ( rs ,ζ ) = 2 ( 30π − 128) cɶ1 ( rs ,ζ ) + 8 π cɶ2 ( rs ,ζ ) + ( 39π −128 ) cɶ3 ( rs ,ζ ) −144 π A( rs ,ζ ) S (α ) + 16 π A( rs ,ζ ) P (α )

} (512 −165π )

−256  A( rs ,ζ ) R (α ) − E ( rs ,ζ )  cɶ6 ( rs ,ζ ) =

{

(90)

π (180π − 624 ) cɶ1 ( rs ,ζ ) + (150π − 512 ) cɶ2 ( rs ,ζ ) + π (135π

−432 ) cɶ3 ( rs ,ζ ) + ( 3072 − 1260π ) A( rs ,ζ ) S (α ) + A( rs ,ζ ) P (α ) ( 360π −1024 ) − 480 π  A ( rs ,ζ ) R (α ) − E ( rs ,ζ ) 

where

}

 6 (165π − 512 )   

c ( r ,ζ ) cɶn ( rs ,ζ ) = ( nn−1) s2 d (ζ )

t=

(91)

(92)

d (ζ )k F u

A ( rs , ζ ) =

φ (ζ )

(93)

φ (ζ ) rs d (ζ ) κ

(94)

α = φ 2 (ζ ) κ

rs

d (ζ )

(95)

and ∞

2  1  S (α ) = ∫ f1 (α t ) e− t 1 + t 2 + t 4  dt 2   0

(96)



2 1   P (α ) = ∫ f1 (α t ) e− t t 2 1 + t 2 + t 4  dt 2   0



R (α ) = ∫ 0

f1 (α t )  2  1  1 − e − t 1 + t 2 + t 4   dt  t  2  

(97)

(98)

24

Rogelio Cuevas-Saavedra and Paul W. Ayers

2rs d (ζ )  9π 2 3 E ( rs , ζ ) = 2   ε c ( rs , ζ ) 3φ (ζ )  4 

(99)

The correlation energy of the UEG is not known exactly. The highly accurate parameterisation of by Perdew and Wang is used.[34] Namely

) 1− ζ 4 ( ) f '' (0 ) 4 + ε c (rs ,1) − ε c (rs , 0 ) f (ζ )ζ

ε c (rs , ζ ) = ε c (rs , 0 ) + α c (rs )

f (ζ

(100)

where

(1 + ζ f (ζ ) =  All three terms— ε c (rs , 0 ), α c (rs ) and

(

)

+ (1 − ζ ) − 2   43 2 −2

43

(

43

)

(101)

ε c (rs ,1)—have the same functional form

)

(

G rs , AJ ,α 1,J , β1,J , β 2,J , β 3,J , β 4,J , p J = −2 AJ 1+ α 1,J rs

)

    1 × ln 1+  1 3  2 AJ  β1,J rs 2 + β 2,J rs + β3,J rs 2 + β 4,J rspJ +1       The label

(102)

J = 0, α , 1in (102) refers to the functions and ε c (rs ,1)respectively.

Even for the UEG, modeling the exchange-correlation hole is very challenging. However, many properties of the exchange-correlation hole of the UEG are transferable to real systems, and we may hope that the structure of the exchange-correlation hole is semi-universal. This is the motivation for many DFT approximations. The UEG already exhibits many of the most problematic characteristics of inhomogeneous electronic systems and, in fact, some of the failures of approximate exchange-correlation functionals can be understood based on their failure to capture certain features of the UEG.

IV. Approximations and Challenges in DFT A. Jacob’s Ladder The exchange-correlation hole is the key element in DFT. The more accurate the model for the exchange-correlation hole is, the more accurate the functional will be. More generally,

25

Density Functional Theory

the more information one uses to determine the hole, the more exact constraints on the hole one can satisfy and the more accurate the functional will be. This motivates a hierarchy of approximations, each more complicated—but also more accurate—than the previous one. Surprisingly, some of the simpler approximations frequently reproduce chemical phenomena with useful accuracy. Most approximate functionals are based not on the expression for the exchangecorrelation energy in terms of the exchange-correlation hole (Eq.(39)) but on the expression for the exchange-correlation energy in terms of the (usually adiabatically-connected and spherically-averaged; Eq.(40)) exchange-correlation charge,

∞  1 Exc [ρ ] = ∫ ρ (r )  ∫ ρ xc (r , u )4π udu  d r 2 0 

(103)

Many functionals are in fact based on an even simpler quantity, the exchange-correlation energy per electron per unit volume, ∞

ε xc  ρ;r  = ∫ ρ xc ( r,u ) 4π u du

(104)

0

ε xc  ρ;r  is then usually approximated as a function (not a functional) of the electron density and other quantities at the point r. This gives rise to the most common form for approximate exchange-correlation functionals,[35]

(

)

Exc  ρ  = ∫ ρ ( r ) ε xc ρ ( r ) ,∇ρ ( r ) ,∇ 2 ρ ( r ) ,...

(105)

As the number of pieces of information in the functional increases, the complexity (and, on average, the accuracy) of the functional also increases. In order to categorize the complexity and achievements of the different types of approximate functionals proposed, John Perdew[35] introduced the idea of a “Jacob’s ladder” of approximations. Jacob’s ladder ascends from the Hartree world (where both exchange and correlation is neglected) to the “Heaven of Chemical Accuracy” (where total energies are accurate to one milliHartree) via five rungs. The first rung is the Local Density Approximation (LDA)[19], where εxc is a function of the electron density alone. The LDA functional is usually based on the UEG model. The Generalized Gradient Approximation (GGA) is the second rung of the ladder; in a GGA,

ε x c (ρ (r ) , ∇ ρ (r ))[36-40]. The gradient

of the electron density provides some information about the inhomogeneity of the electron density in real systems. Third-rung functionals are called meta-Generalized Gradient Approximations (mGGAs)[41-45]. By allowing the exchange-correlation energy density to be a function of the kinetic-energy density (alternatively, the Laplacian of the electron density), exact information about the behavior of close-together same-spin electrons is included. Functionals on the fourth rung use the exact exchange energy, the exchange-energy density

26

Rogelio Cuevas-Saavedra and Paul W. Ayers

or, more generally, any information about the occupied Kohn-Sham orbitals. Such functionals are called hybrid-GGAs. The most popular and accurate functionals are semi-empirical hybrid-GGAs like B3LYP, B3PW91, and PBE0[46,47]. These functionals mix a fixed fraction of the exact exchange energy with GGA exchange. More recent functionals are usually local hybrids, where the exchange-energy density is used instead.[48] The topmost rung of the ladder adds the last remaining component: information about the unoccupied Kohn-Sham orbitals. Such functionals are called hyper-GGAs. Functionals like B2PLYP, the random phase approximation, and Görling-Levy perturbation theory reside on this rung. It is important to try to work on the lower rungs of the ladder whenever possible because calculations on the fourth rung (hybrid functionals), and especially the fifth rung (hyperGGAs), can be very computationally demanding. Nevertheless, the hybrid functionals should not be neglected; they are often very accurate, which is unsurprising because their expression for the energy as a sum of an "exact" exchange (Hartree-Fock) term and a GGA term is directly motivated by the adiabatic connection. Even after one has decided which rung(s) of the ladder one wants to step upon on the way to the “Heaven of Chemical Accuracy,” developing functionals is very difficult. Functionals are only reliable if they satisfy certain restrictions that are imposed by the exact exchangecorrelation hole. Some of these restrictions are, in perceived order of importance: (1) the normalization of the hole (avoiding self-interaction error), (2) the hole’s depth (the probability of finding two electrons with specified spins at the same position; this is also called the on-top value), (3) the hole’s near-coalescence behaviour (the probability of finding two electrons with specified spins very close together, but not quite at the same place; this is where the local kinetic energy becomes important), (4) the hole’s asymptotics (related to the probability of finding two electrons far apart; this is important for dispersion forces), and (5) reproduction of the uniform electron gas limit. Although capturing the UEG limit is perceived to be less important than the reproduction of other properties of the exact functional, it is especially important in the solid state, because functionals that give poor results UEG limit have serious failures for the bulk and surface properties of simple metals. DFT calculations are now the standard computational method for quantum chemistry and condensed-matter physics. In most cases, DFT calculations give highly accurate results, suitable for experimental comparison and interpretation. Specifically, for systems where the electron hole is localized, and for properties whose values are dominated by the short-distance portion of the hole, existing exchange-correlation energy functionals are already close to the “Heaven of Chemical Accuracy.” By contrast, for systems where the exchange-correlation hole is delocalized (e.g., systems with “non-classical” chemical bonding and transition states of chemical reactions) or where the long-range behaviour of the exchange-correlation hole is decisive (e.g., dispersion interactions),[49,50] existing exchange-correlation functionals fail catastrophically. For example, commonly used exchange-correlation energy functionals often overestimate chemical reaction rates by orders of magnitude and give a qualitatively incorrect description of dispersion interactions. The functionals on Jacob’s ladder are all local functionals in the sense that depends only on the properties of the system at the point r. Sometimes post-LDA functionals are called “semi-local” and “nonlocal” because the density gradient and other higher-rung quantities contain information about the electron density in the vicinity of r. In our nomenclature,

Density Functional Theory

27

however, a functional is only nonlocal if it depends explicitly on properties of the system at more than one point. Our research is focussed on nonlocal functionals. The historical development of DFT functionals has consisted of scaling Jacob’s ladder. Early on, DFT was mainly used in solid-state physics because molecular binding energies are overestimated (though molecular geometries were excellent) with LDA. The introduction of the density gradient in GGA functionals gave molecular binding energies far superior to Hartree-Fock theory and led to the adoption of DFT in the chemistry community. As yet, meta-GGA’s have not shown significant improvements upon simple GGAs. The next major step was the inclusion of Hartree-Fock exact exchange; this led to the most widely used of all functionals, B3LYP[51-54]. B3LYP gives remarkably accurate results for a wide range of molecules and properties; in many cases, B3LYP is more accurate than computationally expensive post-Hartree-Fock wave function-based methods like second-order many-body perturbation theory. Developing functionals that systematically improve upon B3LYP is a challenge. Now there is great interest in 100% exact exchange hybrids and hyper-GGAs, especially hyper-GGAs based on the random phase approximation.

B. Challenges for DFT Even simple approximations to Exc[ρ] perform extremely well for a wide range of problems in physics and chemistry. In particular, DFT accurately predicts the thermodynamic properties of molecules and solids. These successes often lead researchers to assume that DFT is reliably accurate for other properties, however, which is often not true. We now enumerate some of the greatest challenges faced in DFT.[55] 1. Further development of simple and accurate functionals. One of the main attractions of DFT is its simplicity compared to traditional many-body theory electronic structure methods. If DFT functionals become as numerically demanding and mathematically challenging as full configuration interaction (FCI),[6] much of the appeal of the DFT approach would be lost. While one does not wish to have a fully rigorous (FCI-like) DFT approximation, one does not wish to degenerate into an entirely empirical method, devoid of physical insight and constraints, either. DFT methods must balance simplicity and accuracy. 2. Description of reaction barriers and dispersion interactions.LDA and GGA functionals tend to systematically over-stabilize transition states, which leads to an underestimate of chemical reaction barriers and an overestimate of chemical reaction rates. All local functionals on the first four rungs give imperfect treatments of London dispersion and tend to systematically underestimate the importance of weak, non-covalent, interactions.[56]In order to describe dispersion, there should be an attractive part of the energy that asymptotically decays as R-6 when the distance, R, between the interacting units increases. Such behaviour is automatically recovered by any model for the exchange-correlation hole that decays asymptotically as r − r ′ but when one decides to develop functionals based on the exchange-correlation charge, this characteristic asymptotic decay is obscured. Since the Hartree-Fock

−5

,

28

Rogelio Cuevas-Saavedra and Paul W. Ayers approximation also omits dispersion, traditional hybrid GGAs also fail to capture dispersion. 3. Strong Correlation. DFT is based on a Slater determinant model and, as such, is often inaccurate when this model is a very poor starting point for approximations. Systems where this is the case are said to be strongly correlated. The difficulty of DFT in capturing strong correlation leads to problems with stretched chemical bonds, transition-metal systems, and materials with several unpaired electrons. 4. Self-Interaction. Even though the Schrödinger equation for a one-electron system can be solved almost exactly, approximate exchange-correlation functionals give poor results if they have a self-interaction error.[57-60]. The self-interaction error arises because in DFT the exchange-correlation energy is written as a smooth and continuous function of the electron density, and systems with just one electron are not treated in a special way. A more general problem, called many-electron selfinteraction error, arises because the density (and therefore density functionals) do not have a special form for integer electron number. This leads to approximations that underestimate the band gap in solids and overestimate the extent of charge delocalization in molecules. The delocalization error is associated with underestimates for charge-transfer excitation energies, overestimation of chemical binding in charge-transfer metal complexes, and overestimation of the response of a molecule to an electric field.[55]

C. Systematic Errors in Exchange-Correlation Functionals Addressing the shortcomings of DFT approximations is easier if we can identify certain systematic errors in approximate exchange-correlation functionals that induce these problems. We can then try to develop new approximations that systematically correct these errors. Most of the problems identified in the previous section can be attributed to just two problems: (1) delocalization error and fractional-charge behaviour and (2) static correlation error and fractional spin behaviour. Fortunately, these errors can be explained and understood using simple molecular systems. Consider the molecule. Most DFT functionals describe this molecule well near equilibrium geometries but severely underestimate the energy when the molecule is stretched. In particular, at the dissociation limit, most DFT functionals describe as two hydrogen atoms with half an electron each and predict that the energy of this system is much lower than the energy of a hydrogen atom in the electric field of a proton, even though both descriptions of dissociated should have the same energy. This tendency of most functionals to incorrectly predict that it is favourable to delocalize the electron between the two centers is a consequence of self-interaction error.[61,62] This also reflects the discrete nature of electrons. The exact electronic energy, as a function of the molecular charge, is a straight-line interpolation between integer charges with discontinuities at the integers.[63] Few methods predict the correct straight-line behaviour; most DFT methods are convex between the integers, predicting that the “fractional charge” is more stable than the linear combination of integer charges. One consequence of the delocalization error is that transition states are overstabilized because it is too easy for electrons to delocalize across stretched bonds. (In more

Density Functional Theory

29

conventional chemical language, there are too many favourable “resonance structures” for the transition state.) Another consequence is that, when a molecule is place in an electric field, approximate functionals predict a polarizability that is too high because it is too easy for fractional charges to appear at the edges of the molecule[64]. Finally, when removing or adding an electron from a system, approximate functionals overestimate the delocalization of the added electron or hole[55]. This effect can be very dramatic in large systems since the delocalization error increases with the size of the system under consideration. This leads to problems describing charge defects in crystals. One reason for the success of hybrid-GGAs is that, for fractional charges, Hartree-Fock has concave energy vs. charge behaviour while LDA, GGA, and meta-GGA functionals have concave energy vs. charge behaviour. A suitable mixing between the two theories can give a better description of fractional charges. Unfortunately, this is not a general solution because the correct fraction of Hartree-Fock exchange varies not only from system to system, but between different regions in the same molecule. The static correlation and fractional spin errors can be described by stretching the H2 molecule. Approximate functionals describe the covalent bond well but they overestimate the dissociation energy. This problem is an example of static correlation: the type of electron correlation that arises when the Hartree-Fock (or Kohn-Sham) Slater determinant is very close in energy to several other determinants.[65,66] (In stretched H2, the degeneracy arises from the near degeneracy of the σg and σu molecular orbitals.) Consider just the atomic subsystems in the closed-shell H2 molecule at the dissociation limit; this is a hydrogen atom with one-half of a spin-up electron and one-half of a spin-down electron. The exact energy for this system should be the same as that of the normal H atom in an integer-spin state. In fact, the exact functional requires that systems with fractional spins have an energy equal to that of the normal spin states. We refer to this as the constancy condition[67]. Most approximate functionals do not accurately describe the interpolation between the degenerate spin-up and spin-down states. In the case of the H atom, the overestimation of this energy exactly matches the error of the stretched H2. The violation of the constancy condition explains many types of static correlation error and explains the difficulty in using the electron density as a descriptor of degenerate states. The previous behaviour for fractional charges and fractional spins has been combined into one unified condition called the flat-planes condition[68]. This is an exact condition for the energy from any electronic structure method and many wave function-based methods also fail to satisfy the flat-planes condition. It is useful that most of the essential problems in DFT can be assessed using atomic systems with fractional electron number and fractional electron spin because this provides a simple way to assess and diagnose the weaknesses of approximate functionals. It also provides guidance for new developments: ideally, new functionals will satisfy (at least approximately) the flat-planes condition. New functionals that reduce the delocalization error (fractional electron number) and the static correlation error (fractional spin) have in fact been designed; these functionals are among the most accurate functionals available, but they are more computationally expensive than traditional functionals like PBE and B3LYP.

30

Rogelio Cuevas-Saavedra and Paul W. Ayers

IV. A Way Forward? Nonlocal Two-Point Functionals A. Overview Our research focuses on building nonlocal exchange-correlation functionals based directly on models for the exchange-correlation hole,

Exc [ρ ] =

(

)

1 ρ (r ) ρ (r ' ) f hxc (r, r ') drdr ' 2 ∫∫ r −r'

(106)

For simplicity, we usually choose f to be the identity function and we usually choose an exchange-correlation hole based on the uniform electron gas. Our hope is that the inherent non-locality of this functional form will facilitate modelling strong correlation and long-range correlation. Because we are interested in inhomogeneous electronic systems like atoms and molecules, there is an ambiguity in how the exchange-correlation hole in Eq. (106)is determined. Should we choose based on the electron density at r or based on the electron density at rʹʹ? Either choice breaks the symmetry of the exchange-correlation hole,

hxc (r , r ') = hxc (r ', r )

(107)

Preserving the symmetry of the exchange-correlation hole means that the exchangecorrelation hole must depend on the properties of the system (e.g., the electron density) at both r and rʹʹ. For this reason we call Eq. (106)a two-point functional. Should the functional depend on the electron density at r and rʹʹ, or on some other property of the system? Remember that the exchange-correlation hole for the UEG has the form and depends parametrically on the density (which is constant) through the Fermi vector

k F (r ) = (3π 2 ρ (r ))

1/ 3

(108)

and the Wigner-Seitz radius

4  rs (r ) =  πρ (r ) 3 

−1/3

(109)

These variables seem to be more natural for describing the exchange-correlation hole than the density itself. For each pair of points r and rʹʹ, we need to rewrite so that it depends on kF and rs at both points. One way to do this is to define symmetric versions of these quantities using the generalized p-mean,

31

Density Functional Theory 1

 k pk (r ) + kFpk (r ' ) pk kF → k F (r, r ') =  F  2  

(110)

1

 rspr (r ) + rspr (r ') pr rs → rs (r, r ' ) =   2  

(111)

The spin polarization ζ(r) can be symmetrized in a similar way, but in this chapter we will mostly focus on closed-shell systems, where ζ(r) = 0. The p-parameter in the mean can be adjusted to improve the accuracy of the functional. Equations (108) and (109) are local-density-like approximations that are exact for the uniform electron gas, but which may not be ideal for inhomogeneous systems. We can define, for example, an “effective Fermi momentum”, so that the exchange-correlation hole satisfies exact constraints in the inhomogeneous system. For example, we can define and rseff (r ) so that the normalization constraints on the exchange-, correlation-, and/or exchange-correlation holes are satisfied:[69]

∫ ρ (r ')h (r ,r ')dr ' = −1

(112)

∫ ρ (r ')h (r,r ' )d r ' = 0

(113)

∫ ρ (r ')h (r ,r ' )d r ' = −1

(114)

x

c

xc

This type of approach is called a 2-point weighted density approximation. Other constraints, like the flat-planes condition, could also be used to determine the parameters entering into the exchange-correlation hole model. The 2-point weighted density approximations have several appealing features. 1. If the exchange-correlation hole satisfies the plasmon sum rule, then this model has a dispersion interaction that has the correct leading-order terms, C6/R6 + C8/R8. This is in stark contrast to traditional exchange-correlation functionals, which entirely omit long-range dispersion. 2. For systems with integer electron number, the normalization conditions of the exchange-correlation hole ensure that the exchange-correlation functional is selfinteraction free. 3. For systems with fractional electron number, information about the flat-planes conditions enters directly into the normalization conditions for the exchangecorrelation hole. For example, for fractional number of electrons, the normalization condition on the exchange-correlation hole becomes

32

Rogelio Cuevas-Saavedra and Paul W. Ayers 2

∫ ρ (r ) h N +x

2

N

N +x

2

1− x ) ρ ( r ) + x ρ ( r ) ( r ,r dr = − ( ) 1

2

N +1

1

ρ N + x ( r1 )

2

1

(1− x ) ρ ( r ) + x ρ (r ) =− 2

2

N

1

N

1

N +1

1

(1 − x ) ρ (r ) + xρ ( r ) N +1

1

(115)

here N is an integer and 0 ≤ x≤ 1.Satisfying this condition helps ensure that the delocalization error and the many-electron self-interaction error are small.



∫ ρN

For systems with fractional spin, the normalization conditions on the spin-resolved exchange-correlation hole contain information relevant for the flat-planes conditions. Specifically,

β

∫ρ

α

∫ρ

α

α + x;N β + y

Nα + x;N β + y

Nα + x;N β + y

(r ) h 2

αβ

N α +x;N β + y

(r ) h

βα

(r ) h

αα

2

2

N α +x;N β + y

N α +x;N β + y

(ρ (r ,r ) dr = y 1

2

Nα ;N β +1

2

2

(

(r ) − ρ 1

ρN

α Nα +x;N β + y

α

2

(ρ (r ,r ) dr = x 1

α

β Nα +1;N β

α + x;N β + y

(r ) − ρ

(r ) 1

β

1

Nα +x;N β + y

)

( )

ρ Nα + x;N β + y ( r1 ) β

∫ρ

Nα + x;N β + y

(r ) h 2

ββ

N α +x;N β + y

1

( r )) 1

( ) ( ) (r ) + yρ (r )

( )

(

 1− x − y ρ α r1 + y ρ αN ;N +1 r1  + x ρ αN +x;N + y r1 N α ;N β α β α β   r1 ,r2 dr2 = − α ρ N +x;N + y r1

(

 1− x − y ρ β r1 + x ρ Nβ +1;N N α ;N β α β  r1 ,r2 dr2 = − β

)

α

β

( r ))

)

(

)

β

( )

ρ Nα +x;N β + y

1

1

β Nα +x;N β + y

(r ) 1

(116)

This helps ensure that the fractional-spin error is small, and that the types of static electron correlation that are primarily associated with molecular dissociation (left-right correlation) or open shells (as in transition-metal complexes) will be treated accurately. The form of the 2-point weighted density approximation therefore automatically captures the key features that any good exchange-correlation density functional should possess.

B. Initial Assessment of the 2-Point Weighted Density Approximation (2-WDA) As an initial test, we studied the performance of the 2-point weighted density approximation (2-WDA) for exchange-energies. The advantage of testing exchange energies is that there is an abundance of reference data on exchange energies and the exact exchange hole of the UEG is known and does not depend on the Wigner-Seitz radius. As such, this is the simplest, and most reliable, way to test the basic 2-WDA ansatz. There is only one parameter, the value of pk in the p-mean that one uses to average the effective Fermi momenta. We choose this parameter to minimize the error in the exchange energies of the first thirty-six atoms in the periodic table (from H to Kr); we find pk = 5.

Density Functional Theory

33

To assess the quality of the 2-WDA functional, we computed its average and rms errors in the exchange energies of the atoms from H to Kr. The average error is 0.006 a.u. and the rms error is 0.329 a.u. This can be compared to the average and rms errors of conventional exchange functionals: 2.415 a.u. and 2.866 a.u. for LDA; -0.018 a.u. and 0.086 a.u. for B88; 0.922 a.u. and 1.135 a.u. for PBE; -0.186 a.u. and 0.348 a.u. for OPTx. The 2-WDA functional has a smaller average error than any of these functionals and a far smaller rms error than the LDA and PBE functionals. The rms error of 2-WDA is comparable to OPTx (which was partially parameterized based on these atoms) but inferior to B88 (which was parameterized for precisely these atoms). Encouraged by these results, we extended the results to exchange-correlation energies by using the exchange-correlation hole of Gori-Giorgi and Perdew in the 2-WDA. Although the analysis is quite similar to the exchange-only case, the presence of electron correlation means that now the Wigner-Seitz radius enters into the formula. So there are now two parameters, pk and pr, controlling the p-means for kF and rs, respectively that must be determined. We determined these parameters by minimizing the error in the 2-WDA for the exchangecorrelation energies of a set of small molecules, chosen based solely on the fact their exchange-correlation energies are accurately known.[70] We found the value pk = 10 and pr = 20. The average error of the 2-WDA was -0.001 a. u.; the rms error was .043 a.u. This is better than the PBE functional (average error = .046 a.u.; rms error = .104 a.u.) and comparable to the OPTCS1 functional (average error = .006 a.u.; rms error = .043 a.u.). Note that the OPTCS1 functional was partly parameterized using the data set we used for assessment. The performance of the 2-WDA functionals is remarkable insofar as they are based only on information from the uniform electron gas (like LDA) and do not contain any dependence on the gradient of the electron density. The choice of a constant for p is very naïve, and certainly incorrect: we observe, for example, that heavier atoms prefer larger p, and lighter atoms smaller p. By allowing p to depend on (1) the electron density, (2) the electron density and its gradient, (3) the electron density, its gradient, and the kinetic-energy density, (4) the occupied Kohn-Sham orbitals, or (5) the occupied and unoccupied Kohn-Sham orbitals, we can define a new “Jacob’s ladder” of exchange-correlation functional approximations. The fact that the zero-rung (p = constant) functional on this new ladder already performs as well as the best second rung (density and gradient dependent) functionals on the conventional Jacob’s ladder suggests that the 2-WDA functional form should be further pursued.

C. Implementation of Weighted Density Approximations After the exchange-correlation hole has been determined, the exchange-correlation energy is evaluated by performing the integral in Eq.(106). Practical difficulties arise from the singularity present in the Coulomb interaction: when electrons are close together, the value of the denominator in (106) tends to zero and the value of the integrand diverges. This causes naïve numerical integration methods to either diverge or converge slowly to the large-grid limit.[71]We have developed several new approaches to this integral.[72]The best of the new methods rely on basis-set expansions of the integrand. In fact, the basis-set approach can be applied to any kind six-dimensional integral of the form

34

Rogelio Cuevas-Saavedra and Paul W. Ayers

I [ρ ] =

1 ρ (r ) K (r, r ') ρ (r ') drdr ' 2∫∫ r −r'

The behaviour of the numerator,

(117)

ρ (r ) K (r , r ' ) ρ (r ' ), is density-like since this

function lies in the same Banach space as the electron pair density.[73] Therefore, we expand the numeratorin a (truncated) basis set for the space of densities,

{ηi }i=1 , obtaining Nb

Nb Nb

ρ (r ) K (r, r ') ρ (r ') = ∑∑ Lijηi (r )η j (r ') i =1 j =1

(118)

Substitution of (118) in (117) gives

I [ρ] = where

Sij = ∫∫

1 Nb Nb ∑ ∑ Lij S ji 2 i=1 j=1

ηi ( r )η j ( r') r − r'

(119)

drdr'

= ∫ ηi ( r )ξ j ( r ) dr

(120)

is the “overlap” between the density-like and potential-like basis functions

ξ j ( r) = ∫

η j ( r') r − r'

dr' (121)

Indeed, since has been identified with a density-like basis function then eq. (121) naturally defines a potential-like basis set

{ξi }i=1. Nb

To evaluate Eq. (119), we must determine the coefficients of the expansion for Eq. (118). Project the kernel onto the potential-like basis set,

K mn = ∫∫ ξm ( r ) ρ ( r ) K ( r, r') ρ ( r') ξn ( r )drdr'

(122)

The calculation of this matrix is performed numerically on the integration grid. Then, using Eqs. (120), (121) and (122), the integral in Eq. (117) can be calculated using

1 I [ ρ ] = Tr  KS −1  2

(123)

35

Density Functional Theory

We remind the reader that the whole intention of proposing this methodology was to avoid the singularity associated with the Coulomb potential. Notice that such singularity is still present in the potential-like basis set and so there is still a singularity in the overlap matrix,(120). We need to choose a density-like basis set for which the potential-like basis can be simply computed, and for which the overlap integrals can be computed analytically. Choosing a Gaussian functions for the density-like basis [73]

(

∂ N ∂ L ∂ M −α P r−P 2 η NLM ;α P ( r, P ) = N L M e ∂Px ∂Py ∂Pz

)

(124)

meets these goals. In Eq. (124), the vector represents the position of the nucleus at which the Gaussian is centered. The structure of the exchange-correlation hole is dominated on the distance between the electrons. This suggests that it might be more efficient to replace the atom-centered basis set in Eq. with an explicitly correlated basis set. This is achieved by augmenting the form in Eq. (124) with a Gaussian that depends on the distance between the electrons

η NLM ;α

P ,βP

( r;r', P ) = e − β

P

r−r'

2

(

∂ N ∂ L ∂ M −α P r−P 2 e ∂PxN ∂PyL ∂PzM

)

(125)

To test these methods, we computed the exchange energy of LiH using the UEG exchange hole. As shown in Figure 1, the explicitly correlated basis function requires many fewer basis functions; the main difference is that basis functions with high angular momentum are needed in the non-correlated case. In these calculations we did not force the normalization of the exchange hole to be correct, so they only demonstrate basis-set convergence; they are not indicative of the quality of 2-WDA functionals.

D. An Alternative Based on the Direct Correlation Function (DCF) Even though the short-range behaviour of the exchange-correlation hole in an inhomogeneous system probably strongly resembles its behaviour in the uniform electron gas, the exchange-correlation hole has much more long-range structure (associated with static correlation, for example) in inhomogeneous systems. By using a direct correlation function hole (DCF), we can model long-range contributions of the exchange-correlation hole as a convolution of DCFs, which are short-ranged and, therefore, hopefully more universal. That is, we hope that the DCF of the UEG will be more transferable than the UEG’s exchangecorrelation hole. This idea is motivated by the theory of classical liquids, where the hole correlation function is determined from the DCF by solving Ornstein-Zernike equation

h ( r, r') = c ( r, r') + ∫ c ( r, x1 ) ρ ( x1 ) c ( x1, r') dx1

(126)

36

Rogelio Cuevas-Saavedra and Paul W. Ayers

The long-range structure in the hole correlation function (e.g., due to solvation shells) is built up from a simple, short-ranged, and semi-universal form for the DCF. The model we actually use is derived by substituting the Ornstein-Zernike equation into itself, and accounting for the indistinguishable nature of electrons, obtaining

1 1 1 h ( r, r') = c ( r, r') + ∫ c ( r, x1 ) ρ ( x1 ) c ( x1, r') dx1 2! 2! 3! 1 + ∫∫ c ( r, x1 ) ρ ( x1 ) c ( x1, x 2 ) ρ ( x 2 ) c ( x 2 , r' ) dx1 dx 2 + .... 4!

(127)

Figure 1. The exchange energy for lithium hydride, approximated with the exchange hole of the UEG, versus the number of basis functions. The non-correlated basis set consists of atom-centered Gaussian functions; see Eq. (124). The explicitly correlated basis set consists of explicitly correlated atomcentered Gaussian functions; see Eq. (125).

Equation (127) is also a 2-point functional, but in this case f (see Eq. (106)) is not the identity function. While other researchers have explored the idea of using an electronic direct correlation function[74] to approximate the exchange-(correlation) hole, they did not appropriately account for the fact electrons are indistinguishable particles and, consequently, used a model, based on Eq. (126), which is incompatible with the normalization constraint on the exchange-correlation hole. Equation (127) has a simple interpretation. The first term on the right-hand side represents the “direct” interaction between an electron located at r with another that is at r ; the term is divided by 2 because the electrons at r and r are indistinguishable. The second term on the right-hand side represents a situation in which the electrons at r and r interact thought an intermediary at x1. The probability of observing an electron to “bridge” the interaction is

ρ (x1 ).This term is divided by 3! Because the electrons at r, x1, and r

are

indistinguishable. Subsequent terms represent interactions conveyed through additional “bridging” electrons. For our initial test of Eq. (127), we started with the spin-resolved Gori-Giorgi[33] model for the exchange-correlation hole of the spin-unpolarized uniform electron gas then inverted

37

Density Functional Theory

Eq. (127) to determine the spin-resolved DCF. When spin-resolution is considered, Eq. (127) is modified and takes the form 1 1 1 hσσ ' ( r, r') = cσσ ' ( r, r') + ∑ cσσ ( r, x1 ) ρσ1 ( x1 ) cσ 2σ ' ( x1, r') dx1 2! 2! 3! σ 1 =α , β ∫ 1 +

1 ∑ ∑ cσσ ( r, x1 ) ρσ1 ( x1 ) cσ1σ 2 ( x1, x2 ) ρσ 2 ( x 2 ) cσ 2σ ' ( x2 , r') dx1 dx 2 + .... 4! σ1 =α , β σ 2 =α , β ∫∫ 1

(128) This equation seems dauntingly complicated, but it has a simple form in reciprocal space.

(

)

Hɶ xc ( k, rs , ζ ) = 2 exp Cɶ xc ( k, rs , ζ ) − Cɶ xc ( k, rs , ζ ) − I 2  Cɶ xc−1 ( k, rs , ζ )

(129)

whereI2 is the 2x2 identity matrix and

  Hɶ xc k,rs ,ζ =      ɶ C xc k,rs ,ζ =   

(

(

)

)

ρα hɶαα ( k,rs ,ζ )

ρα ρβ hɶαβ ( k,rs ,ζ ) 

ρα cɶαα ( k,rs ,ζ )

ρα ρ β cɶαβ ( k,rs ,ζ )

ρα ρ β hɶαβ ( k,rs ,ζ )

ρα ρβ cɶαβ ( k,rs ,ζ )

ρ β hɶββ ( k,rs ,ζ )

ρβ cɶββ ( k,rs ,ζ )

       

(130)

Because of the form of Eq. (130), we can extend the Gori-Giorgi model to spin-polarized electron gasses. If the exchange-correlation energy computed from Eq. (130) is accurate even for spin-polarized electron gasses, then this suggests that the physics contained in Eq. (130) is relevant for electronic systems. For comparison, we also extended the Gori-Giorgi exchangecorrelation hole to spin-polarized systems by performing a suitable scaling of coordinates,

 k  GG , rs  hɶαα ( k, rs, ζ ) = hɶααGG  3  1+ζ   k  GG hɶββ , rs  ( k, rs, ζ ) = hɶββGG  3  1+ζ   k  GG , r hɶαβ ( k, rs, ζ ) = hɶαβGG  6 s 2  1−ζ 

(131)

Further details can be found in reference 75. The extended Gori-Giorgi model gives excellent results, but the results from the DCFbased model are even better, especially for strongly spin-polarized ( ζ = 0.6 ) low-density

38

Rogelio Cuevas-Saavedra and Paul W. Ayers

electron gases (rs> 5). In these cases the long-range correlations are important and the direct correlation function captures a significant fraction of the error in the Gori-Giorgi model, even though the model was based on the direct correlation function was parametrized using only the spin-unpolarized Gori-Giorgi model. This supports the assertion that the direct-correlation function is highly transferable between systems. These results suggest that Eq. (129) should be used to approximating the exchangecorrelation hole in atoms and molecules. Since the implementation is rather difficult, we started by studying the exchange energy for asmall set of closed shell molecules using the exchange hole of the UEG, but without adjusting kF to impose the normalization constraint on the exchange-correlation hole. The results from Eq. (127) are only 3% better than the results obtained by using the exchange hole itself. This suggests that perhaps the DCF of the uniform electron gas is not transferable to molecules. However, before discarding this direction of research, we should impose the normalization constraint on the exchange-correlation hole.

VI. Summary This chapter reviews the fundamentals of density-functional theory (sections I-III), with emphasis on recent advances that have improved our ability to approximate the exchangecorrelation energy functional (sections IV-V). These advances are of two main types: on the one hand, mainly due to the work of Weitao Yang and his collaborators, we have recently attained a deeper understanding of the problems that afflict exchange-correlation density functionals. In particular, almost all of the problems that afflict conventional density functional approximations can be traced to their inability to recover appropriate behavior for fractional charges (leading to poor reaction barriers, exaggerated polarizability, diminished charge-transfer, and over-delocalization of electrons) and fractional spins (leading to poor molecular dissociation curves, incorrect treatment of biradicals, poor treatment of open-shell molecules, and over-localization of electron spins). The main issues that escape this treatment are dispersion interactions (very-long-range electron correlations) and strong correlations between non-spatially-separated electrons (e.g., the Be isoelectronic series). The second advance is the development of 2-point weighted density approximations (2WDA functionals). These inherently nonlocal functionals are rigorously self-interaction free, can be constrained to recover reasonable fractional-charge and fractional-spin behavior, and give dispersion interactions with the correct R–6 form. (But 2-WDA functionals are not expected to solve the problem of strong correlations between nearby electrons.) We believe that the 2-WDA form holds great promise but attempts to improve the form, as by using the direct correlation function, have been discouraging.

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39

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Rogelio Cuevas-Saavedra and Paul W. Ayers

[43] Becke, A. D. and Roussel, M. R. Phys. Rev. A39, 3761-3767 (1989). [44] Perdew, J. P., Kurth, S., Zupan, A., and Blaha, P. Phys. Rev. Lett.82, 2544-2547 (1999). [45] Tao, J. M., Perdew, J. P., Staroverov, V. N., and Scuseria, G. E. Phys. Rev. Lett.91, 146401 (2003). [46] Ernzerhof, M. and Scuseria, G. E. J. Chem. Phys.110, 5029 (1999). [47] Adamo, C. and Barone, V. J. Chem. Phys.110, 6158 (1999). [48] Jaramillo, J., Scuseria, G. E. and Ernzerhof, M. J. Chem. Phys.118, 1068 (2003). [49] Becke, A. D. and Johnson, E. R. J. Chem. Phys.123, 154101 (2005). [50] Angyan, J. G. J. Chem. Phys.127, 024108 (2007). [51] Becke, A. D. J. Chem. Phys.98, 5648-5652 (1993). [52] Lee, C., Yang, W. and Parr, R. G. Phys. Rev. B37, 758-789 (1988). [53] Miehlich, B., Savin, A., Stoll, H. and Preuss, H. Chem. Phys. Lett.157, 200-206 (1989). [54] Vosko, S. H., Wilk, L. and Nusair, M. Can. J. Phys.58, 1200-11 (1980). [55] Cohen, A. J., Mori-Sánchez, P. and Yang, W. Science 321, 792-794 (2008). [56] F, L. Z. Phys. Chem. 11, 222-251 (1930). [57] Perdew, J. P. and Zunger, A. Phys. Rev. B23, 5048-79 (1981). [58] Vydrov, O. A., Scuseria, G. E., Perdew, J. P., Ruzsinsky, A., and Csonka, G. I. J. Chem. Phys.124, 094108 (2006). [59] Ruzsinszky, A., Perdew J. P., Csonka, G. I., Vydrov, O. A., and Scuseria, G. E. J. Chem. Phys.126, 104102 (2007). [60] Mori-Sánchez, P., Cohen, A. J. and Yang, W. J. Chem. Phys.125, 201102 (2006). [61] Perdew, J. P. and Zunger, A. Phys. Rev. B23, 5084-5079 (1981). [62] Zhang, Y. and Yang, W. J. Chem. Phys.109, 2604 (1998). [63] Perdew, J. P., Parr, R. G., Levy, M., and Balduz, J. L., Jr. Phys. Rev. Lett.49, 1691-1694 (1982). [64] Mori-Sanchez, P., Wu, Q. and Yang, W. T. J. Chem. Phys.119, 11001-11004 (2003). [65] Baerends, E. J. Phys. Rev. Lett.87, 133004 (2001). [66] Becke, A. D. J. Chem. Phys.119, 2972-2977 (2003). [67] Cohen, A. J., Mori-Sánchez, P. and Yang, W. http://arxiv. org/abs/0805. 1724 (2008). [68] Mori-Sánchez, P., Cohen, A. J. and Yang, W. Phys. Rev. Lett.102, 066403 (2009). [69] Parr, R. G. PHYS-400. 1998. Conference Proceeding [70] Tozer, D. J. and Handy, N. C. J. Chem. Phys. 88, 2547-2553 (1998). [71] Becke, A. D. J. Chem. Phys. 88, 2547-2553 (1998). [72] Cuevas-Saavedra, R. and Ayers, P. W. Int. J. Mod. Phys. B24, 5115 (2010) [73] McMurchie, L. E. and Davidson, E. R. One- and Two-electron Integrals Over Cartesian Gaussian Functions. Journal of Computational Physics.26, 218-231 (1978). [74] C. Amovilli and N. H. Norman, Phys. Rev. B 76, 195104 (2007) [75] Cuevas-Saavedra, R. and Ayers, P. W. Journal of Physics and Chemistry of Solids, submitted PACS:31.15.A-

In: Theoretical and Computational Developments … Editor: Amlan K. Roy, pp. 41-60

ISBN: 978-1-61942-779-2 © 2012 Nova Science Publishers, Inc.

Chapter 2

RECENT PROGRESS TOWARDS IMPROVED EXCHANGE-CORRELATION DENSITY-FUNCTIONALS Pietro Cortona Laboratoire Structures, Propriétés et Modélisation des Solides, École Centrale Paris, Grande Voie des Vignes, Châtenay-Malabry, France

Abstract The accuracy of electronic structure calculations performed in the framework of the densityfunctional theory (DFT) depends in a crucial way on the choice of the exchange-correlation functional. The latter is the only quantity that requires some approximations in DFT, thus the search for approximate expressions giving accurate results is a major research subject. In this chapter, I will account for the progress we have done in this field. I will start by recalling some DFT basic concepts and by describing the various kinds of exchange-correlation functionals that one can meet. Then, I will present our own contributions, which consist in three functionals: one is local and the other two belong to the generalized-gradient approximation class. The performances of these functionals will be assessed by comparing their results with those of some of the most frequently used functionals. At the end of the chapter, I will briefly mention the research axes we are presently following in order to get further improvements.

Keywords: Density functional theory, Exchange-correlation functional, Generalized gradient approximation, Benchmark calculations

Introduction Most modern electronic structure calculations are performed in the framework of the density-functional theory (DFT) [1,2]. This is mainly due to the important role of the electronic correlation in determining many material properties.



E-mail address: [email protected]

42

Pietro Cortona

Of course, rigorous methods in Quantum Mechanics exist, which allow one to account for the correlation effects. But these methods are very demanding in terms of computer resources and can only be applied to relatively small systems. DFT is a useful alternative to these standard Quantum Mechanics approaches. It should not be confused, however, with an approximated method, since the general theory is exact. It can be considered as a reformulation of the standard Quantum Mechanics in terms of electron densities instead of wave functions. The major result of this reformulation is that the properties of each given system can be deduced from the solutions of the one-electron equation called the Kohn-Sham equation [2]. Of course, the difficulties of the “many-body problem” have not disappeared: the Kohn-Sham equation contains a term, the so-called exchange-correlation potential, which sometimes is said to be “unknown”. Strictly speaking, this is incorrect. The exchange-correlation potential is the functional derivative of the exchange-correlation energy density-functional, which, in turn, is well-defined from a mathematical point of view. But the definition of this latter functional is useless in practical calculations. For this reason, approximate expressions of the exchange-correlation functional are necessary. The key point is that, once these approximations are available, one has just to solve a one-electron equation. This fact, combined with the good accuracy of the approximate exchange-correlation functionals, is the reason for the success of the DFT.

Summary of DFT’s Basic Concepts DFT is based on Hohenberg and Kohn’s two theorems (HK) [1]. The first of these theorems states that all the properties of a system can be deduced from the knowledge of its ground state electron density. Quantities such as the total energy are thus functionals of the electron density. Among these functionals, the electron “internal energy” F[] (the sum of the electron kinetic energy plus the electron-electron interaction of an N-electrons system having as ground-state density) is particularly important. This functional is “universal” in the sense that it is independent from the particular external potential acting on the electrons. Its importance is due to the second HK theorem, which states that a given system’s ground state electron density minimizes the functional:

E v   F   where

 vext r rdr

(1)

vext r is the potential acting on the electrons in that particular system. Thus, taking

into account the normalization condition:

 rdr  N

(2)

the basis quantity of the DFT, that is the ground state electron density, can be found by solving the Euler equation:

Recent Progress towards Improved Exchange-Correlation Density-Functionals

43

F  v ext r    

(3)

In this equation, the first term is the functional derivative of F[] and  is a Lagrange multiplier. The HK theorems were originally proved for systems having a non-degenerate ground state1. Furthermore, the class of functions for which the functionals were defined was quite peculiar: it was the class of functions satisfying the condition of V-representability, that is the functions which are the ground state densities of an N-electrons system in an arbitrary external potential vext r. These limitations of the original theory were bypassed by Levy [4]. The Levy reformulation of the HK theory is based on the following definition of the F[] functional:

F   Min  Tˆ  Vˆee   

(4)

where Tˆ and Vˆee are the kinetic energy and the electron-electron interaction operators, respectively, and the minimum is searched among all the anti-symmetric N-particles wave functions, which give rise to the particle density . Levy showed that this functional is an extension of the one originally defined by HK and that, using this definition in eq. (1), the minimum property established by the second HK theorem is still valid. As wave functions , which generate any arbitrary  that is a good candidate to be an N-electrons density, always exist [5], the Levy formulation does not suffer from the original drawbacks2. The main obstacle to using eq. (3) in actual calculations is given by the difficulty of finding accurate approximations of the F[] functional. In particular, the available approximations for the kinetic contribution do not give functional derivatives accurate enough to be used in eq. (3). The usual way of bypassing this difficulty is to use the Kohn-Sham theory [2]. In the Kohn and Sham theory, the kinetic energy is split in two contributions: the greater one is treated exactly, and approximations are only required for the smaller contribution. This was achieved by introducing a fictitious system of non-interacting “electrons”. For such system, the functional F[] contains just the kinetic contribution. For this reason, it is usually indicated by Ts :

Ts   Min  Tˆ   

(5)

If the system of N non-interacting electrons is submitted to an effective potential v eff r , its ground state density is a solution of the Euler equation:

1 2

The extension to degenerate ground states was provided by Kohn [3]. A detailed mathematical treatment of all these aspects of the theory can be found in ref. [6].

44

Pietro Cortona

 Ts  v eff r    

(6)

Considering now the real system of interacting electrons, the F[] functional can be rewritten as:

F   Ts   J  E xc where

(7)

J is the classical Coulomb energy of a charge density : J   

1 2



 r  r r  r

drdr (8)

Eq. (7) is actually the definition of E xc , the so-called exchange-correlation energy functional. Using eq. (7), eq. (3) becomes:

 Ts  J  E xc    v ext r      

(9)

The comparison between eq. (6) and eq. (9) shows that, if the fictitious system of noninteracting electrons is submitted to the effective potential:

v eff r   v ext r  

 J  E xc   

(10)

the real and the fictitious system have the same ground state electron density. This common electron density can be derived more easily for the fictitious system by solving the oneelectron (Kohn-Sham) equation:

 1 2  J  E xc     vext r    r  i i r    i  2

(11)

and then calculating  by:

r    ni i r  i

where ni are fermionic occupation numbers.

2

(12)

Recent Progress towards Improved Exchange-Correlation Density-Functionals

45

The exchange-correlation functional E xc  is the key quantity of the Kohn-Sham theory: if its exact expression could be used, the results would be exact. As this is not possible, approximations are required, and the result accuracy depends on the quality of these

approximations. It should be noticed that E xc  contains a kinetic contribution: the difference between the kinetic energies of the real and fictitious systems having the same electron density.

Approximate Exchange-Correlation Functionals There are many approximate expressions of E xc  . These approximate functionals have been classed by Perdew and coworkers using the Jacob ladder metaphor [7]. Climbing the ladder, i.e. elaborating more and more sophisticated functionals, one moves from the “Hartree world” towards the “heaven of the chemical accuracy”. On the first rung of the ladder, one finds the local functionals, which have the general expression:

E xc   

 f  dr

(13)

where f is an ordinary function of  (not a functional). Only few local functionals exist. The most important ones are the various expressions of the local-density approximation (LDA). In the LDA, f() is assumed to be the exchange-correlation energy per unit volume of a homogeneous electron gas having density . The exchange energy of such a gas has the wellknown expression derived by Dirac [8]: 1

3 3 3 E x      4  



4 3

dr

(14)

while for the correlation contribution various expressions exist. Those which are used nowadays [9, 10, 11] are parametrizations of the results of the quantum Monte Carlo calculations performed by Ceperley and Alder [12]. The approximations of the second rung depend on the density as well as on the gradient of the density. More precisely, they are usually expressed in terms of the density and of the reduced density gradient. The latter can be defined in different ways. The one, which will be used in the present chapter, is the following:

s



 

2 3 2

13 4 3



(15)

These approximations are usually called generalized-gradient approximations (GGA). The natural way of correcting the LDA for gradient dependent effects would be to perform a

46

Pietro Cortona

systematic expansion in terms of the density gradient. But such a gradient expansion approximation (GEA) does not work in practice. The reason is that some known properties of the exact exchange-correlation functional, which are preserved at the LDA level, are violated by the GEA. GGA approximations, such as those proposed by Perdew and Wang (PW86 [13, 14]; PW91 [15]), were obtained by enforcing the respect of these properties. On the contrary of the local functional case, there are many functionals belonging to the GGA class. In addition to the Perdew and Wang ones, we mention those developed by Becke, Lee, Yang and Parr (BLYP) [16, 17] and by Perdew, Burke and Ernzerhof (PBE) [18]. The last one is probably the most widely used, in particular by the solid-state physicist community. The functionals that will be discussed in the present chapter are local or GGA-like. However, I would like to mention that the third rung functionals have an additional dependence on the laplacian of the density or on the kinetic energy density. A typical example is the TPSS functional [19]. Very important are the fourth rung functionals, characterized by an additional dependence on the occupied Kohn-Sham orbitals. This class includes the hybrid functionals which, after the pioneering paper by Becke [20], have known a very quick development in terms of applications as well as theoretical refinements. The founding idea of the hybrid functionals is to mix standard density-functionals with the Hartree-Fock exchange. The simplest way of doing that is to use a fixed percentage of Hartree-Fock and densityfunctional exchange. This gives rise to the one-parameter “global” hybrids. An alternative procedure consists in splitting the Coulomb electron interaction in a long-range and a shortrange contribution. Hartree-Fock is then used for the long-range part, while a densityfunctional is used for the second one. These are the so-called “range-separated” hybrids [21, 22]. There are many different hybrids, in both these classes. Recently, the two approaches have been combined and some global hybrids with range-separation have been proposed [23, 24]. I will not describe in detail the various techniques, which have been used in order to construct hybrid functionals. I just mention that the most widely used hybrid, B3LYP [25, 26] is a three parameters global hybrid, with the parameter values determined by fitting reference data for a large set of molecules. Another very popular hybrid is PBE0 [27, 28], a oneparameter global hybrid, where the parameter value is fixed on a theoretical ground [29].

Construction and Tests of New Exchange-Correlation Functionals There are two strategies, which are currently used in order to construct new functionals. These strategies concern the determination of the parameters contained in the analytical expression of the functional. In the “fundamental” strategy, the parameter values are fixed by requiring that some known properties of the exact functional be verified by the approximate one. In such a case, the number of parameters contained in the analytical expression must be quite small. In the “pragmatic” strategy, the parameter values are fixed by fitting a reference dataset. Functionals constructed according to this second strategy can contain a very large number of parameters and the reference dataset can possibly concern many systems and a great variety of properties. Once a new functional has been constructed, it must be tested and its performances must be compared with those of the previously existing functionals. A lot of work has been done in order to make these comparisons systematic. Nowadays, many reference datasets can be used to this aim. The G2 dataset, for example, gathers the atomization energies of 148 molecules

Recent Progress towards Improved Exchange-Correlation Density-Functionals

47

[30]. Several databases have been conceived in order to assess the ability of the functionals to describe the thermochemical kinetics. The Database/3 [31] contains 44 barrier heights for hydrogen-transfer (HT) reactions. Among these reactions, 6 have been identified as representative, giving rise to the HTBH6 set [32]. Non-hydrogen-transfer barriers are considered in the NHTBH38/04 dataset [33], which includes 38 barrier heights for heavyatom transfer (HAT), nucleophilic substitution (NS), and unimolecular or association reactions (UA). Among these reactions, 18 have been considered as representative of the overall set. Combined with those of the HTBH6 set, these 18 barrier heights constitute the DBH24 database [34]. The functionals, which will be discussed in this chapter, will be tested by means of the G2, HTBH6, and NHTBH38/04 datasets. Furthermore, some additional classes of chemical reactions will be considered as well as bond lengths, hydrogen bonds, and atomic properties. The corresponding reference data will be introduced later.

The Local Correlation Functional The first functional considered in this chapter is the local correlation functional proposed by Ragot and Cortona (RC) [35]. It is given by:

cRC 

0.655868 arctan 4.888270  3.177037rs   0.897889 rs

(16)

where rs is the Seitz radius, related to the electron density by:

 3  rs     4 

1/ 3

(17)

More precisely, eq. (16) is the expression of the correlation energy per electron. The complete expression of the functional, including the spin-polarization factor, is the following:

E cRC 

 A(rs (r))C((r))dr

(18)

where A is the correlation energy per volume unit:

A( rs (r )) 

3 RC  c ( rs (r )) 4rs3

(19)

and

1 3 C((r))   (1 (r)) 2 / 3  (1  (r)) 2 / 3  2 





is the polarization factor introduced by Wang and Perdew [36]. As usual,  is given by:

(20)

48

Pietro Cortona

 

   

(21)

The deduction of eq. (16) is quite long. Let me give here just the outline, as the details can be found in the original paper [35]. The starting point was a Colle-Salvetti-like (CS) correlated wave function, given by the product of the HF wave function times a correlation factor [37]:

  

 x1,..., x N    HF x1,..., x N  1  f ri ,r j i j

(22)

where xi indicates the space and spin coordinates of the electron i, and the correlation function f is given by:

  r  2 2 f r1,r2   1  R12 1 12 e   c R12 r12  2  

(23)

here R12 and r12 are the center-of-mass and the relative coordinates of the electrons 1 and 2, respectively:

R12  r1  r2 /2

;

r12  r1  r2

(24)

The Gaussian function appearing in the f expression determines the range of the correlation interactions, which are taken into account. This range is fixed by c , which can be roughly interpreted as the reciprocal of a correlation hole radius. CS assumed that c is related to the electron density as follows:





c R12   q R12 

13

(25)

where q is a parameter. After some theoretical developments, they wrote the function  in terms of c as follows:



 c 1  c





(26)

In order to derive the RC correlation energy expression, an approximate one-electron reduced density matrix was determined from the wave function given in eq. (22). Then, applying this reduced density matrix to the homogeneous electron gas, RC obtained an

Recent Progress towards Improved Exchange-Correlation Density-Functionals

49

analytical expression of the kinetic correlation energy per electron tc. The total correlation energy was deduced inverting the well-known relation [38]:

tc  

 rs  c  rs

(28)

The general expressions of both c and tc are quite complicate functions of rs and q. However, they become remarkably simple, once the value of q is determined. This value can be obtained by considering the high and low electron density limit cases. In order to have the right sign of tc when rs tends to zero, q must be greater or equal to 1.57. Furthermore, the smallest value must be chosen in order to obtain the best description of the homogeneous electron gas when rs tends to infinity. Assuming q=1.57, the expressions of tc and c become:

t c rs  

1 11.9475 14.9062 rs  4.8440 rs2

(27)

and the one given in eq. (16), respectively. It is worth to notice that the usual LDA functionals are obtained by fitting an analytical expression to the results of Quantum Monte Carlo calculations for the homogeneous electron gas. In the present case, the q value has been chosen in the same way. Thus, the functional described in this section is parameter-free exactly in the same sense as the usual local-density approximation is parameter-free.

The First GGA Correlation Functional The simplest way of obtaining a gradient dependent functional from the one introduced in the previous section, is to include a third term in the integral given in eq. (18) [39]:

E cGGA 

 A(rs (r))Bs(r)C((r))dr

(28)

where s is the reduced density gradient defined in eq. (15). In order to recover the local functional for a homogeneous system, the factor B must be equal to 1 when s=0. Furthermore, it is known that in the opposite limit (s   ), the correlation must vanish [18]. A very simple function respecting both conditions is the following:

B( s(r )) 

1 1  s(r )

(29)

where  and  are two parameters to be determined. This can be done by an approach developed by Zupan et al [40, 41]. They showed that (for light atoms) the total exchangecorrelation energy can be estimated from the knowledge of the average values of rs, , and s. They also gave suitable definitions of these average values.

50

Pietro Cortona In analogy to their work, the average value of B(s) can be defined as follows:

 B 



E cGGA A(rs (r))C((r))dr

(30)

In order to approximately compute the right hand side of this equation for a given system, the numerator can be replaced by the exact correlation energy, while the denominator can be evaluated by means of the density resulting from a local calculation:

 B 

Ecexact Eclocal

(31)

Nevertheless, as this is not sufficient to determine the parameters and , a further approximation is required. According to Zupan et al, it can be then assumed that:

 B(s)   B( s )

(32)

So, after combining the equations (29), (31) and (32), we obtain:

E clocal 1 E cexact

  s  

The values of

(33)

exact , and E clocal are and  can be determined by eq. (33), if , E c exact

local

known for some reference systems. E c is known for light atoms. E c can be calculated for any system. Furthermore, according to Zupan et al, an effective value of can be defined for any atomic or molecular system in terms solely of the exchange energy. Considering a GGA exchange functional given by:

E GGA  x GGA

and proceeding for Fx obtains:

(rs (r))FxGGA (s(r))dr  e uniform x

(34)

as it has been done for B, for a system having a density  one

 

FxGGA s

eff



E GGA  x uniform Ex 

(35)

This equation can be solved, analytically or numerically, for any exchange functional having an analytical form, like PBE or PW91. A priori, eff depends on the GGA and choices. It is clear, however, that eff can be a useful quantity only if its value is

Recent Progress towards Improved Exchange-Correlation Density-Functionals

51

relatively independent from these choices. This is indeed the case: eff was calculated using the PBE and PW91 exchanges, and several densities, obtained using different DFT, HF and post-HF methods. The resulting values did not display significant differences. Assuming some light atoms as reference systems, and replacing by eff in eq. (33), the  and  values were determined:

  1.43 ,   2.30

(36)

Once these two values have been adopted, the functional expression is complete.

The Second GGA Correlation Functional There are several known properties of the exact exchange-correlation functional. One of these properties is that the correlation energy of a one-electron system is zero. This is a very intuitive property of the exact functional, but neither the most widely used GGA functionals, nor those discussed in the previous sections do have this feature. A GGA correlation functional, which almost exactly satisfies this condition, can be obtained by including a further factor in the integral given in eq. (28) [42]:

E cGGA 

 Ars BsC 1  Drs,s,dr

(37)

and requiring that the function D satisfies the following constraints:

 Drs ,0,  0 (i)    D  1 (ii)   D rs Hyd ,s Hyd ,1  1 (iii)  D r ,s,0  0 (iv) s  





(38)

Condition (i) enables to recover the local functional for a homogeneous system. Condition (ii) assures that the correlation energy density will remain negative everywhere and vanishes for infinite reduced gradient values. Condition (iii) imposes that the correlation density vanishes for the exact densities of any hydrogenoid atoms. Condition (iv) enables to recover the first GGA functional for unpolarized systems. Considering condition (iii), as the electron density of hydrogenoid atoms in the ground state is exactly known, one has (Z being the atomic number): 2Zr / 3   rs (r) Hyd  Ae  2Zr / 3  s(r) Hyd  aAZe

so that (in atomic units):

with

 4 1/ 3 a    9 

(39)

52

Pietro Cortona

s(r) Hyd ars (r) Hyd

Z (40)

In order to have a vanishing correlation for these atoms, we can assume: 2   s   Drs ,s,   1  Sinc     ars     4 

where Sinc is the sine cardinal function:

(41)

Sinc(u)  sin(u) / u , that equals 1 at the origin and

0 for every integer  multiple. Such a choice enables to fulfil conditions (i) and (iii). Condition (ii) and (iv) are also verified thanks to the square power in eq. (41) and to the factor  , respectively. The fourth power in this latter contribution was chosen because the Perdew and Wang interpolation (factor C) is valid at the second order of . It can also be noticed that the chosen expression is obviously not unique. For instance, all integer multiples of the argument in the Sinc could be used, but this would have the effect of introducing points where the correlation vanishes without any physical reason. 4

The Exchange-Correlation Functionals In order to perform actual calculations, the functionals described in the previous sections must be combined with an exchange functional. In the case of the RC local correlation, it is quite natural to choose the local Slater exchange:

E xLDA





1

eLDA x dr

4 3 3  3       3 dr 4  

(42)

The exchange functional for the spin-polarized case is then obtained by the spin-scaling relationship:

E xLDA ,   

E x 2   E x 2   2

(43)

In the following, the resulting approximation will be referred to as SRC. The choice is more difficult in the GGA case, because several GGA exchange functionals exist. We have selected the PBE exchange, because our functionals were deduced by an approach similar in spirit to the PBE one. The PBE exchange functional is given by:

Recent Progress towards Improved Exchange-Correlation Density-Functionals

E xPBE 

rs FxPBE sdr  e LDA x

where

FxPBE s  1  

PBE

The complete expression of E x

  1 s 2 

53

(44)

(45)

,  is then obtained by means of eq. (43). The

values of the two parameters  and  were fixed by Perdew et al. in the following way. The first one determines the behavior of the functional for small s values. It was chosen in order to recover the LDA linear response: =0.21951. The value of the second one was fixed on the basis of the Lieb-Oxford bound [43]:

E x  ,     1.679   3 dr 4

(46)

A simple way of satisfying this condition is to require that the exchange energy density is never greater, in absolute value, than 1.679 

4

3.

In other words, the integral condition given

in eq. (46), is transformed in a more restrictive local condition. Adopting this criterion, the  value was found to be 0.804. In our calculations, we have always combined the TCA correlation with the PBE exchange. In the following, the resulting approximation will be referred to simply as TCA. Of course, also the RevTCA correlation can be used together with the PBE exchange, giving a functional which was called nTCA in ref. [42]. However, the results obtained by this functional were deceiving. The mean absolute error (MAE) for the G2 dataset, for example, is 20.4 kcal/mol, while the PBE one is 17.0 kcal/mol. In order to obtain better performances, the PBE exchange was modified by changing the  parameter value. As I have mentioned above, the local Lieb-Oxford condition is more restrictive than the original integral condition. Zhang and Yang pointed out [44] that it is possible to choose a value of  greater than 0.804 without violating the relation (46). We decided to determine  by requiring that the local Lied-Oxford condition be satisfied only for

s  0,3.3 Furthermore, instead of eq. (46), we used the improved relation proposed by

Chan and Handy [45]. In such a way, we obtained =1.227. The combination of the RevTCA correlation with this modified PBE exchange will be referred to as RevTCA.

3

This interval is sometimes called the “physical interval” [40, 41] because for many systems (in particular for covalent systems) these values of s are the most relevant ones.

54

Pietro Cortona

Tests and Performance Assessment As the functionals discussed in the previous sections are based on the RC one, I start by analyzing the performances of the latter, in particular in comparison with those of the LDA. A natural starting point is to look at the correlation energies. Those for the rare gas atoms are reported in table 1 and compared with the LDA ones and with reference values. The latter are taken from refs. [46, 47]. Table 1. Absolute values of the rare gas atom correlation energies (in Hartrees) [35]

He Ne Ar Kr Xe

SRC 0.094 0.594 1.097 2.397 3.683

LDA 0.111 0.740 1.423 3.268 5.178

Reference values 0.042 0.39 0.73 2.07 3.43

Table 2. First ionization potentials (in eV) [35]. The experimental values are taken from ref. [48]. The values in parenthesis are the percentage errors with respect to the experimental data. The mean absolute error (MAE) is reported in the last line

Li Be B C N O F Ne Na Mg Al Si P S Cl Ar Kr Xe MAE

SRC 5.26 (-2.4) 8.81 (-5.5) 8.36 (0.7) 11.55 (2.6) 14.77 (1.7) 13.66 (0.3) 17.81 (2.2) 21.92 (1.7) 5.16 (0.4) 7.51 (-1.8) 5.79 (-3.3) 8.06 (-1.1) 10.32 (-1.6) 10.33 (-0.3) 13.03 (0.5) 15.71 (-0.3) 14.05 (0.4) 12.35 (1.8) 0.17

LDA 5.47 (1.5) 9.03 (-3.1) 8.58 (3.4) 11.76 (4.4) 14.99 (3.2) 13.90 (2.1) 18.06 (3.7) 22.18 (2.9) 5.37 (4.5) 7.73 (1.0) 6.00 (0.2) 8.27 (1.5) 10.53 (0.4) 10.55 (1.8) 13.25 (2.2) 15.94 (1.1) 14.27 (1.9) 12.57 (3.6) 0.28

Exp. 5.39 9.32 8.30 11.26 14.53 13.62 17.42 21.56 5.14 7.65 5.99 8.15 10.49 10.36 12.97 15.76 14.00 12.13

LDA largely overestimates the correlation energy. SRC strongly reduces this drawback and becomes progressively more accurate with the increase of the atomic number. Nevertheless, it is well-known that, in spite of the large errors for the correlation and total energies, the LDA energy differences are sufficiently accurate for many purposes.

Recent Progress towards Improved Exchange-Correlation Density-Functionals

55

Furthermore, physical properties are generally related to total energy differences instead that to the total energies themselves. An example of total energy differences is given by the ionization potentials (IPs). The first IPs of light atoms and of some heavy rare gas atoms are reported in Table 2. Table 3. Adiabatic and vertical ionization potentials of some molecules (in eV) [49]. Experimental data are taken from ref. [50]

H2O NH3 SiH3 Thiophene Furane Pyrrole MAE

B3LYP Adiab. Vert. 12.39 12.50 10.02 10.80 7.94 8.89 8.70 8.70 8.71 8.89 8.03 8.21 0.14 0.08

LDA Adiab. Vert. 13.23 13.35 10.77 11.52 8.66 9.33 9.51 9.68 9.54 9.68 8.86 9.01 0.63 0.73

SRC Adiab. Vert. 12.27 12.40 9.84 10.57 8.18 8.85 8.71 8.88 8.73 8.88 8.06 8.21 0.18 0.10

Exp. Adiab. Vert. 12.62 12.62 10.07 10.84 8.14 8.74 8.87 8.87 8.88 8.88 8.21 8.21 -

SRC generally reduces the errors of LDA. This is quantified by the mean absolute error: 0.17 eV for SRC as against 0.28 eV for LDA. Looking at the relative errors, those of SRC are always smaller than 3% with only two exceptions: Be and Al. In the LDA case, this value is exceeded for seven atoms. A similar behavior is observed considering the 3d transition metal atoms. In such a case, the errors on the IPs are almost systematically reduced by 0.2 eV [49]. IPs of molecules were studied in some extension by Tognetti, Adamo, and Cortona [49]. They considered three different datasets. Their results for the first one are reported in table 3. Those for the other two datasets (containing 10 and 7 molecules, respectively) give rise to similar remarks. In the case of molecules too, SRC reduces considerably the errors of LDA. The MAE is divided for more than a factor of 3 in the case of adiabatic IPs, while for vertical transition this factor is greater than 7. It is interesting to remark that, in spite of its local character, the performances of SRC in calculating the IPs are close to those of a hybrid functional such as B3LYP. The tests we have just discussed suggest that SRC is a good starting point in order to construct GGA functionals. Additional tests were actually made [51]. They concern standard, but very important, molecular properties, such as atomization energies, barrier heights, and equilibrium bond lengths. I will discuss the SRC results for these quantities below, together with the TCA and RevTCA ones. The results for the atomization energies are summarized in table 4, where are reported the MAEs of various functionals. Two sets of molecules have been considered: the already mentioned G2 dataset and the G2-1 set [52]. The latter is actually a subset of the former. The reason for considering both sets is that the molecules of the G2-1 set are small, while G2 includes also larger molecules. Thus, comparing the results for the two sets, one obtains some insights about how the functional performances change with the molecular size. Three different comparisons are proposed in table 4. The first one is with the LDA results. This is the natural comparison for SRC, and the conclusion is that the errors are remarkably reduced, the MAE being divided by a factor of about 3.

56

Pietro Cortona

TCA and RevTCA should be compared with a functional belonging to the same class, i.e. a GGA functional such as PBE. Two comments are in order. The first one concerns the MAE value, which is considerably reduced in passing from PBE to TCA and successively to RevTCA. The second one concerns the trend. The PBE MAE for the G2 set is more than the double of the one for the G2-1 set. In the case of TCA and RevTCA, the MAE increase is much smaller: 34% in both cases. The results of two hybrid functionals, PBE0 and B3LYP, are also reported in the table as a reference. It can be seen that the performances of RevTCA are not far from those of PBE0. Both TCA and RevTCA considerably reduce the performance gap between a standard GGA functional such as PBE and standard hybrid functionals. A similar comment can be done about SRC. Its performances are closer to the PBE ones than to those of LDA. Similar comments apply to the barrier heights reported in table 5. The reactions considered are those of the HTBH6 dataset [32] for hydrogen transfer reactions and those of the NHTBH38/04 [33] for heavy-atom, nucleophilic substitution, unimolecular or association reactions. In addition, in the case of the GGA functionals, we have also considered proton transfer (PT) reactions [39]. The SRC MAE is smaller than the LDA one by 16% for HT and HAT reactions, and by 24% in the UA case. Only for NS reactions the LDA results are slightly better. In the GGA case, both TCA and RevTCA systematically improve on PBE. Table 4. Atomization energies [42, 51]: mean absolute errors (MAE) for the G2-1 [52] and the G2 [30] datasets (in kcal/mol)

LDA SRC

G2-1 (55 molecules) 39.7 14.6

G2 (148 molecules) 91.4 26.3

PBE TCA RevTCA

8.1 6.7 4.4

17.0 9.0 5.9

PBE0 B3LYP

3.1 2.3

5.0 3.1

Table 5. Mean absolute errors and mean signed errors (in parenthesis) for the activation barriers of some chemical reactions (in kcal/mol) [42, 51] HT 18.1 (-18.1) 15.2 (-15.2)

HAT 23.3 (-23.3) 19.7 (-19.7)

NS 8.5 (-8.4) 8.7 (-8.7)

UA 5.8 (-5.0) 4.4 (-4.2)

PT

PBE TCA RevTCA

9.5 (-9.5) 8.1 (-8.1) 6.6 (-6.6)

15.0 (-15.0) 13.7 (-13.7) 12.6 (-12.6)

6.9 (-6.9) 6.6 (-6.6) 5.6 (-5.6)

3.4 (-3.0) 2.9 (-2.7) 2.9 (-2.3)

4.1 (-4.1) 3.7 (-3.7) 2.9 (-2.9)

B3LYP

4.9 (-4.9)

8.5 (-8.5)

3.4 (-3.4)

2.0 (-1.5)

1.8 (-1.8)

LDA SRC

Recent Progress towards Improved Exchange-Correlation Density-Functionals

57

The TCA improvements are between 4% and 15%, depending on the reaction class. In the RevTCA case, they are greater: between 14.7% and 30.5% for UA and HT reactions, respectively. The equilibrium bond lengths are an example of property for which the new functionals give essentially the same results as the old ones. This can be seen in table 6, where are reported the results for three different datasets: G2-32, G2-M, and C2. G2-32 is a subset of G2 containing 32 organic molecules [53]. G2-M comprises 10 metal complexes4, which are representative of a larger set proposed in ref. [54]. Finally, C2 is a set of 12 diatomic molecules containing a monochalcogenic atom5. The most interesting remark concerns B3LYP. The molecules belonging to G2-32 are part of the set used for the optimization of the B3LYP parameters. As a consequence, the results for this dataset are excellent. Those for the G2-M dataset are less accurate: they are not better than the GGA ones. The accuracy is still worse for the C2 dataset: in this case the B3LYP MAE is the double of its GGA counterpart. Actually this is not the only case where the GGA results are similar or better than the B3LYP ones6. Another example is given by the activation enthalpies of the 1,3-dipolar cycloadditions. We have recently calculated [56] these enthalpies for a set of 9 reactions, and we have found that the MAEs are 4.3, 2.0, 2.7, and 3.1 kcal/mol for PBE, TCA, RevTCA, and B3LYP, respectively. For a second set of other 9 reactions, the corresponding MAEs are 4.3, 2.4, 2.1, and 2.2 kcal/mol. For these reactions, both TCA and RevTCA reduce remarkably the errors with respect to PBE, and give results more accurate than the B3LYP ones. I conclude this section by discussing some results for hydrogen bonds. We have considered a set of 10 representative hydrogen-bonded complexes [57], and we have calculated the equilibrium bond lengths and the dissociation energies [39, 58]. The results are summarized in Table 7. The TCA results are especially good, having an accuracy similar to that of PBE0. On the contrary, RevTCA is deceiving. Its errors are greater than the PBE ones, particularly for the bond lengths. Thus it does not seem to be a well-suited functional for calculating properties related to hydrogen bonds. Table 6. Mean absolute errors (in Å) for the bond lengths of the G2-32, G2-M, and C2 datasets [42, 51]

4

G2-32 0.015 0.017

G2-M 0.036 0.032

C2

LDA SRC PBE TCA RevTCA

0.013 0.012 0.014

0.014 0.015 0.019

0.012 0.012 0.010

B3LYP

0.007

0.016

0.024

The list of these complexes can be found in ref. [51]. See ref. [42] for the list of these molecules. 6 For a more complete discussion of this item see ref. [55]. 5

58

Pietro Cortona Table 7. Mean absolute errors for the bond lengths and the dissociation energies of 10 representative hydrogen-bonded complexes [39, 58]

Bond lengths (Å) Diss. energies (kcal/mol)

PBE 0.070 0.98

TCA 0.044 0.72

RevTCA 0.125 1.09

PBE0 0.047 0.67

Conclusion In this chapter I have described our progress towards obtaining improved exchangecorrelation functionals. Three functionals have been presented and their performances have been discussed. The results obtained by the first one, the SRC local functional, display a welldefined trend. In some cases - for the equilibrium bond lengths, for example - it gives almost the same results as LDA. In other cases (atomization energies, barrier heights for some class of chemical reactions), it greatly improves with respect to LDA. This trend is confirmed by the study of other properties and/or systems not discussed here. Preliminary results for simple solids, for example, indicate that the cohesive energies are greatly improved, while the equilibrium lattice parameters are almost identical to the LDA ones. TCA is a quite reliable functional. It gives good results for thermochemistry, geometries similar to the PBE ones, and excellent results for hydrogen-bonded complexes. Thus, it seems well-suited for a wide range of applications, in particular for systems which present hydrogen bonds in addition to the iono-covalent interactions. The RevTCA performances display a less well-defined trend. They are indeed excellent for properties such as atomization energies and barrier heights. But this functional gives a quite poor description of the hydrogen bond and fails also in other cases, for example for agostic bonds [59] and for radicals [60]. These failures seem to be mainly due to the exchange part of the functional. Research with the objective of obtaining a functional giving the good performances of RevTCA without having the same drawbacks is in progress. Work is also in progress along another axis, i.e. the construction of hybrid versions of these functionals. As I have explained in the previous sections, different kind of hybrids can be constructed. Unfortunately, the performances of the resulting hybrid functionals are almost unpredictable from those of their pure counterpart, thus a systematic study of all the possibilities must be performed. For example, our preliminary results indicate that the best global hybrid is obtained from the nTCA functional, i.e. the one that, in its pure version, gives the worse results.

Acknowledgments This work has been supported by a grant of the Agence Nationale de la Recherche (Programme Blanc 2010 – Dinf DFT project n. 0425).

Recent Progress towards Improved Exchange-Correlation Density-Functionals

59

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

Hohenberg, P., Kohn, W. Phys. Rev.1964, 136, B864-B871. Kohn, W., Sham, L. J. Phys. Rev.1965, 140, A1133-A1138. Kohn, W. In: Highlights of Condensed Matter Theory; Bassani, F., Fumi, F. G., Tosi, M. P., Eds.; North Holland: Amsterdam 1985; 1. Levy, M. Proc Natl Acad. Sci. USA 1979, 76, 6062-6065. Harriman, J. E. Phys. Rev. A 1981, 24, 680-682. Lieb, E. H. Int. J. Quantum Chem. 1983, 24, 243-277. Perdew, J. P., Schmidt, K. AIP Conf. Proc. 2001, 577, 1-20. Dirac, P. A. M. Proc. Cambridge Phil. Soc. 1930, 26, 376-385. Perdew, J. P., Zunger, A. Phys. Rev. B 1981, 23, 5048-5079. Vosko, S. H., Wilk, L., Nusair, M. Can. J. Phys. 1980, 58, 1200-1211. Perdew, J. P., Wang, Y. Phys. Rev. B 1992, 45, 13244-13249. Ceperley, D. M., Alder, B. J. Phys. Rev. Lett. 1980, 45, 566-569. Perdew, J. P., Wang, Y. Phys. Rev. B 1986, 33, 8800-8802. Perdew, J. P. Phys. Rev. B 1986, 33, 8822-8824. ibid. 34, 7406. Perdew, J. P., Chevary, J. A., Vosko, S. H., Jackson, K. A., Pederson, M. R., Singh, D. J., Fiolhais, C. Phys. Rev. B 1992, 46, 6671-6687. Becke, A. D. Phys. Rev. A 1988, 38, 3098-3100. Lee, C., Yang, W., Parr, R. G. Phys. Rev. B 1988, 37, 785-789. Perdew, J. P., Burke, K., Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868. ibid. 1997, 78, 1396. Tao, J., Perdew, J. P., Staroverov, V. N., Scuseria, G. E. Phys. Rev. Lett. 2003, 91, 146401. Becke, A. D. J. Chem. Phys. 1993, 98, 1372-1377. Iikura, H., Tsuneda, T., Yanai, T., Hirao, K. J. Chem. Phys. 2001, 115, 3540-3544. Vydrov, O. A., Scuseria, G. E. J. Chem. Phys. 2006, 125, 234109. Rohrdanz, M. A., Herbert, J. M. J. Chem. Phys. 2008, 129, 034107. Rohrdanz, M. A., Martins, K. M., Herbert, J. M. J. Chem. Phys. 2009, 130, 054112. Becke, A. D. J. Chem. Phys. 1993, 98, 5648-5652. Stephens, P. J., Devlin, J. F., Chabalowski, C. F., Frisch, M. J. J. Phys. Chem. 1994, 98, 11623-11627. Adamo, C., Barone, V. J. Chem. Phys. 1999, 110, 6158-6170. Ernzerhof, M., Scuseria, G. E. J. Chem. Phys. 1999, 110, 5029-5036. Perdew, J. P., Ernzerhof, M., Burke, K. J. Chem. Phys. 1996, 105, 9982-9985. Curtiss, L. A., Raghavachari, K., Redfern, P. C., Pople, J. A. J. Chem. Phys. 1997, 106, 1063-1079. Lynch, B. J., Truhlar, D. G. J. Phys. Chem. A 2003, 107, 3898-3906. Lynch, B. J., Truhlar, D. G. J. Phys. Chem. A 2003, 107, 8996-8999. Y. Zhao, N. Gonzalez-Garcia and D. G. Truhlar, J. Phys. Chem. A 109 (2005) 20122018. Zheng, J., Zhao, J., Truhlar, D. G. J. Chem. Theory Comput. 2007, 3, 569-582. ibid. 2009, 5, 808-821. Ragot, S., Cortona, P. J. Chem. Phys. 2004, 121, 7671-7680.

60 [36] [37] [38] [39] [40]

Pietro Cortona

Wang, Y., Perdew, J. P. Phys. Rev. B 1991, 43, 8911-8916. Colle, R., Salvetti, O. Theor. Chim. Acta 1975, 37, 329-334. Levy, M., Perdew, J. P. Phys. Rev. A 1985, 32, 2010-2021. Tognetti, V., Cortona, P., Adamo, C. J. Chem. Phys. 2008, 128, 034101. Zupan, A., Perdew, J. P., Burke, K., Causà, M. Int. J. Quantum Chem.1977, 61, 835845. [41] Zupan, A., Burke, K., Ernzerhof, M., Perdew, J. P. J. Chem. Phys. 1977, 106, 1018410193. [42] Tognetti, V., Cortona, P., Adamo, C. Chem. Phys. Lett. 2008, 460, 536-539. [43] Lieb, E. H., Oxford, S. Int. J. Quantum Chem. 1981, 19, 427-439. [44] Zhang, Y., Yang, W. Phys. Rev. Lett. 1998, 80, 890. [45] Chan, G. K.-L., Handy, N. C. Phys. Rev. A 1999, 59, 3075-3077. [46] Davidson, R. E., Chakravorty, S. J. J. Phys. Chem. 1996, 100, 6167-6172. [47] Clementi, E., Hofmann, D. W. M. THEOCHEM 1995, 330, 17-31. [48] Handbook of Chemistry and Physics (1989-1990) 70th ed., R. C. Weast , Ed.; Chemical Rubber Company: Boca Raton, FL, 1990. [49] Tognetti, V., Adamo, C., Cortona, P. Chem. Phys. 2007, 337, 161-167. [50] Nyulászi, L., Veszprémi, T. Int. J. Quantum Chem. 1997, 61, 399-403. [51] Tognetti, V., Cortona, P., Adamo, C. Chem. Phys. Lett. 2007, 439, 381-385. [52] Curtiss, L. A., Raghavachari, K., Trucks, G. W., Pople, J. A. J. Chem. Phys. 1991, 94, 7221-7230. [53] Adamo, C., Barone, V. J. Chem. Phys. 1998, 108, 664-675. [54] Bühl, M., Kabrede, H. J. Chem. Theory Comput. 2006, 2, 1282-1290. [55] Tognetti, V., Adamo, C., Cortona, P. Interdiscip. Sci. Comput. Life Sci. 2010, 2, 163168. [56] Tognetti, V., Cortona, P., Adamo, C. Theor. Chem. Account 2009, 122, 257-264. [57] Boese, A. D., Martin, J. M. L. J. Chem. Phys. 2004, 121, 3405-3416. [58] Tognetti, V. Théorie de la Fonctionnelle de la Densité et Complexes Métaux-Hydrures: Un dialogue entre développements et applications; PhD thesis, Université Pierre et Marie Curie, Paris, FRANCE, 2009; pp. 1-282. [59] Tognetti, V., Joubert, L., Cortona, P., Adamo, C. J. Phys. Chem. A 2009, 113, 1232212327. [60] Tognetti, V., Cortona, P., Adamo, C. Int. J. Quantum Chem. 2010, 110, 2320-2329. PACS: 31.15.eg, 71.15.Mb.

In: Theoretical and Computational Developments … Editor: Amlan K. Roy, pp. 61-102

ISBN: 978-1-61942-779-2 © 2012 Nova Science Publishers, Inc.

Chapter 3

CONSTRAINED OPTIMIZED EFFECTIVE POTENTIAL APPROACH FOR EXCITED STATES V. N. Glushkov1, and X. Assfeld2, 1

Department of Physics, Electronics and Computer Systems Dnepropetrovsk National University, Ukraine 2 Equipe de Chimie et Biochimie Théoriques, Nancy-Universite, Vandoeuvre-les-Nancy, Cedex, France

Abstract Constrained variational approach based on an asymptotic projection (AP) method, and its applications to the optimized effective potential (OEP) problem for excited states is reviewed. The basic tenets of a simple-to-implement AP method in order to take into account the necessary orthogonality constraints are presented. Variational properties of excited-state energies and specific features of SCF excited-state calculations are discussed. Amended oneparticle Schrödinger equations with a multiplicative potential are derived to determine orbitals of singly and doubly excited states having same symmetry as the ground state. The Slater determinant for a given excited state, constructed from the orbitals, is orthogonal to monodeterminantal functions of low-lying states. OEP equations determining parameters of a local potential expressed in terms of the external potential are obtained and analyzed. It is shown that the AP methodology makes it possible to apply both the OEP procedure and orbitaldependent correlation functionals developed for the ground state, to excited-state problem with same computational effort. A general procedure for application of the constrained OEP method to excited-state problems is demonstrated by means of practical calculations for simple atoms and molecules including excited-state energies, potential energy curves and excitation energies at different levels of approximation. In addition, we also focus on the development of alternative computational strategies capable of optimizing relatively small distributed basis sets with respect to the positions and exponents of basis functions.

Keywords: Density functional theory, Optimized effective potential, Orthogonality constraints, Excited states

 

E-mail address: [email protected] E-mail address: [email protected]

62

V. N. Glushkov and X. Assfeld

1. Introduction Methods originating from density functional theory (DFT) [1, 2] have become very popular in electronic structure calculations of atoms, molecules, solids, and there are already many special textbooks dealing with the various aspects of this theory (e.g., [3-6]). As well known, the success of DFT mostly depends on the form of exchange-correlation potential,

  Vxc (r ) . In principle, it is an exact theory; however Vxc (r ) must be approximated, since,

unfortunately, DFT itself provides neither theoretical nor practical prescriptions on how to find such potentials. It is also worth noting the importance of constraints imposed by the

 Vext (r ) [7-9]. In the course of development of DFT, several  distinct levels of approximations for Vxc (r ) are worth mentioning [10]. The local spin symmetry of external potential

density approximation, where the potentials are explicit functionals of electronic density, i.e.,

  Vxc (r ) = Vxc [  (r )] , provides a reference point within DFT. The second level introduces the

generalized gradient approximations and different hybrid methods. Such methods are capable of providing impressive results but do not provide systematic converging series of approximations to the exact results. The third generation of DFT uses orbital-dependent representations, i.e., these potentials depend on the Kohn-Sham (KS) orbitals and thus implicitly, they are electron density functionals. Such a representation has been proved to be useful for deriving ab initio functionals relying on the experience accumulated from the wave-function-based methods (see, e.g., [11]). For example, in some approaches, a part of Exc , named as the exchange energy, is derived by the relation

Ex =  H int  − EH (  ) , where  is the KS

ground-state determinantal function,

H int is the electron-electron interaction operator and

EH (  ) is the Hartree energy due to the electron density   . The problem of determination of local potentials may be resolved by the optimized effective potential (OEP) method [12,13]. Ground-state OEP approach plays, to a certain extent, a role similar to that of the Hartree-Fock (HF) theory in wave function formalism. Indeed, the OEP methodology not only has proven to be a useful technique to calculate the ground-state properties of manyelectron systems, but also has served as a starting point for constructing local exchangecorrelation potentials of DFT in its orbital-dependent implementation. Unlike the HF approximation, however, in the OEP method, an additional constraint is imposed on the



single-particle orbitals, i (r ) : they must satisfy a single-particle Schrödinger equation with



the same effective potential Veff (r ) for all electrons (with the same spin    ,  ). In particular, in the exchange-only implementation (xOEP), one seeks for a local potential

 Veff (r ) such that the eigenfunctions i (r ) of the one-particle Schrödinger equation (the KS

equation),

 1 2          2   V eff r   i ( r )   i  i r ,

  ,  ;

Constrained Optimized Effective Potential Approach for Excited States

63

i  1, 2,..., N ,

(1)

minimize the same energy functional as in HF approach, i.e.,

E

OEP

n

1    1    {}    i   2  i    (r )Vext (r )dr    (r )VH (r )dr  E x {} 2 2  i 1

(2)



n  *    here  (r )   i (r )  i (r ) is the electron density, VH is the Hartree (Coulomb)



i

potential and E x {} is the orbital-dependent HF exchange energy expressed in terms of orbitals: *  * '    '    i (r ) j (r ) i (r ) j (r )  ' 1 n E x {}     dr  dr   2  i, j r r'

(3)

and n is the number of   spin electrons (    ,  ). A combination of DFT and OEP methodology can now be considered as the undisputed workhorse of modern electronic structure ground-state calculations. Nevertheless, as Singh and Deb pointed out in ref. [14] that, abandoning the concept of a state function within DFT is advantageous for ground states but disadvantageous for excited states. Indeed, as a result, for the excited-state calculations, the DFT-OEP experience is less satisfactory. This is a research area in which much is as yet not clearly understood. At present, there are scarce DFT calculations for exited states, and, in particular, based on the OEP methodology, especially for excited states having same symmetry as ground state. DFT has been extended to excited states by several methods including time-dependent DFT [15-18]. Most of the practical calculations of excited states are carried out via timedependent DFT, in which transition energies are obtained from the poles of dynamic linear response properties. Despite its tremendous success, the present time-dependent DFT-based schemes suffer from a number of shortcomings which restrict their applicability (see, e.g., [19] and references therein). Over the years, a vast number of DFT papers have appeared in the literature; here only the scantiest of selections will be given. We shall focus on the time-independent DFT-OEPbased method whose extension to excited states is neither unique nor straightforward. There exist a number of distinct formulations (see e.g., [20-27] and references therein) and their number continues to grow at a rapid pace. Some of them rely on a variational treatment focusing either on ensembles [20,28-33] or on an individual excited-state approach [34-36]. Others use non-variational approaches (e.g., [37-39]). The first rigorous DFT approach to treat excited states was given by Theophilou [20] and was reformulated as a subspace DFT [28] (later as equi-ensemble theory). It was later generalized into the theory of unequally weighted ensembles of excited states by Gross, Oliveria and Kohn [30-32]. An important step towards practical calculations within the

64

V. N. Glushkov and X. Assfeld

ensemble theory was made by Nágy who generalized the OEP idea for ensemble of excited states [33]. Later, incorporation of a ghost-interaction correction in this scheme [40] has shown that the ensemble KS theory with the exact ensemble-exchange potential can be as accurate as ground-state calculations for atoms [40,41]. However, partly due to the lack of accurate correlation energy functional, there exist very few reported applications on molecules. It is known that for the latter, an orbital finite basis set choice is important to achieve reasonable results. Some preliminary reports on the choice of an optimal basis set for ensemble of states can be found in refs. [42,43]. The second approach uses a “pure state” formulation [34-36]. Nágy et al., remark in ref. [27] that, at present, there exists no standard Hohenberg–Kohn theorem for a single excitedstate density. The universal variational functionals for kinetic and electron–electron repulsion energies in this excited-state variational theory are bifunctionals. On the other hand, the bifunctional approach is appealing in a sense that it actually generates the desired excitedstate density. This can be confirmed, to some extent, by calculations based on an OEP algorithm, which is a subject of this contribution and has been presented in refs. [44, 45] A useful analysis of the “pure state” formulation can be found in [26,27] and references therein. A physical interpretation of the exchange-correlation potential has led to an interesting non-variational approach named as the work-function DFT formalism [37-39]. According to this concept,

  Vxc (r ) is defined as the work done in bringing an electron to the point r against

the electric field produced by its Fermi-Coulomb hole charge density. This formalism was first developed for the ground state [37]. It gives a universal prescription, independent of any state [25]. Its applications to DFT excited-state problems have shown considerable success for singly, doubly, and triply excited states, low and moderately high states, valence and core excitations, as well as the autoionizing and satellite states (see e.g., [46-50]). For a recent review of the work-function approach, see ref. [39]. The fundamental DFT characteristic is the electronic density and we deal with an auxiliary non-interacting system whose density is equal to that of the interacting system of interest. Therefore, it remains unclear how to avoid the collapsing of excited-state functionals when they are expressed in terms of densities. Nevertheless, it is useful to stress that such a mapping of an interacting system to a model non-interacting one, has also to preserve other important properties of the exact eigenstates, such as symmetry (spin and spatial) and orthogonality of states. It is worth noting that the problem of ensuring orthogonalities is considered as one of most important reasons responsible for DFT's discomfiture with excited states [25]. In this chapter, we review the so-called constrained OEP (COEP) method, which in contrast to the known DFT- and OEP-based excited-state approaches, explicitly introduces orthogonality of KS excited-state function to the lower states of same symmetry. This makes it possible to solve the KS equation by avoiding the “collapsing” on the one hand and also allows us to handle any appropriate local potential (including the ground-state potentials based on explicit functionals of electronic density) for excited-state DFT calculations within both variational and non-variational approaches. In the COEP method, the only additional computation required, beyond that arising in a standard ground-state KS scheme, is the evaluation of overlap matrix elements. This remark is important, especially, when different orbital basis sets are used for different states.

Constrained Optimized Effective Potential Approach for Excited States

65

To be self-contained and to clarify some features of this work, the next section gives a brief general overview of the AP method, which has been proved to be a useful tool for solving a wide class of problems in quantum physics and chemistry. This is formulated as an eigenvalue problem for a self-conjugate operator with some orthogonality constraints imposed on its eigenvectors. We discuss the variational properties of excited-state energies and some specific features of SCF excited-state calculations (Section 3). In Section 4, we present how the HF- and KS-type equations can be obtained to generate a single excited-state Slater determinant, which is orthogonal to determinantal functions of lower-lying states. Parametric form of the OEP, and equations determining the exchange-only OEP for excited states, are derived in Section 5. The possibility of handling correlation energy for excited states is considered in Section 6. We discuss the second-order orbital-dependent correlation energy functionals using the KS excited determinant as a reference. At last, in Section 7, we present some applications of the developed methodology for excited-state calculations including single and double excitations in simple atoms, as well as to the potential energy curves of diatomics. Our concluding remarks are given in Section 8.

2. Outline of the Asymptotic Projection Method 2.1. Constrained Eigenvalue Problem and Rayleigh-Ritz Variational Principle The increasing significance of eigenvalue theory in applied sciences stimulates interest in various methods for approximate solutions of eigenvalue problems. Indeed, it is well known that many tasks in physics and chemistry, in particular, in quantum chemistry, can be formulated as an eigenvalue problem for a self-conjugate operator H with some orthogonality constraint imposed on its eigenfunctions, i.e.,

( H  Ei )  i  0, u s  i  0,

i  1,2,..., n s  1,2..., q  n

(4)

(5)

It should be noted that, in general, the constraint vectors us are arbitrary with respect to the spectra of the operator H . It is implied that H is a non-relativistic Hamiltonian for an N -electron system (atomic units used unless stated otherwise):

H 

1 N 2 N nuclei Z k 1 N 1  i          2 i i k r  Rk 2 i  j r  r j

(6)

There has been a growing interest in quantum chemical problems of this type in recent years (see e.g., [51-53]) and references therein). There are several techniques for solving

66

V. N. Glushkov and X. Assfeld

equations (4), (5) or their equivalent variational implementations (e.g., [54-57]). A natural solution of this problem constitutes the derivation of equations for the constrained function in a manner such that the orthogonality constraint is built into the variational procedure as an auxiliary condition. Traditionally, this can be done by either the Lagrange multiplier method (also called the elimination of off-diagonal Lagrange multiplier method [58]) or the projection operator technique [53,59]. Other existing methods, in most circumstances, can be reduced to either of these techniques. Below we shall show how this problem in Eqs. (4), (5) could be solved in a way, which is simpler than the traditional approaches employed in quantum chemical calculations. We review an easily implementable approach, termed the asymptotic projection (AP) method and apply this technique to DFT excited-state problems by using the OEP method. The AP methodology was earlier proposed by Glushkov and Tsaune [60], and then developed further in refs. [61-66] (see also references therein). It is based on the properties of self-conjugate operators. It is general and applicable to any problem that can be cast in the form of an eigenvalue equation with some orthogonality constraints imposed on the eigenvectors. Variational techniques, such as the Rayleigh-Ritz method, are well adapted for the approximate solution of an eigenvalue problem. In such cases, the desired vectors are represented by the linear combination of a finite number of basis vectors  k ( ) , usually depending on variational parameters a , a  1, 2,..., t , so that,

i 

n

 Cik  k ( )  P  i

(7)

k 0

here the vectors  k , in general, belong to a many-body space of states; P is the orthoprojector defined within a chosen basis set, so that the eigenvalue problem (4) is reduced to that for the operator PHP in a finite-dimensional subspace M  PX , of a Hilbert space of states X . Then the eigenvalues, for example, the lowest one,

E0 , in Eqs. (4) and (5) are found

from the requirement of minimum of the functional (the Rayleigh quotient):

i.e.,

E ( )   H  /   ,   M

(8)

E0  E ( 0 )   0 H  0  min E    0  0  1, M ,

(9)

subject to the constraints,

u s   0.

(10)

In other words, we are dealing with a constrained minimization problem. Equations (9) and (10) define the constrained variational method.

Constrained Optimized Effective Potential Approach for Excited States

67

2.2. The Basic Theorem of the Asymptotic Projection Method For the sake of simplicity, initially we limit ourselves to problems involving one constraint vector u . In this case, the constrained variational problem is to minimize the Rayleigh quotient (8), subject to the constraint,

u   0.

(11)

We can identify three possible situations for the constrained vector u with respect to a subspace M spanned by the finite basis set (see [60] for more details). However, in general, this vector can be divided into two parts (I stands for the identity operator),

u  P u  ( I  P) u .

(12)

Therefore, we should consider only the constraint vector P u , which, without loss of

~ u~  1. Then, generality, can be taken as normalized, u  P u / ( u P u )1/2 , i.e., u the condition (11) may be rewritten in a symmetrized form, convenient to perform the variations:

 Pu   0,

Pu  u~ u~

(13)

Multiplying Eq. (13) by an arbitrary real multiplier  and adding it to Eq. (8) we get the functional,

L()   ( H  Pu )  /   ,

  M.

(14)

We start from the stationary condition for L, i.e.,

 L()  0.

(15)

Using Eq. (7), the variations can be written in following form, t

  P    ( a P)  a . a 1

(16)

Hereafter, we use  a P  P / a , for simplicity. The first term in Eq. (16) corresponds to variations within the finite-dimensional subspace M , whereas the second term allows this subspace to be rotated within the Hilbert

68

V. N. Glushkov and X. Assfeld

space X . Substituting Eq. (16) in Eq. (15), taking into account the independence and arbitrariness of variations in Eq. (16), we arrive at the following equations,

P( H  Pu  E ) P   0,

(17)

 ( a P)( H  Pu ) P   0.

(18)

Equation (17) is an eigenvalue problem on the subspace M  PX for the modified operator H mod  P( H  Pu ) P . Equation (18) allows the basis set parameters to be determined via the variational principle and thus to find the optimal position M in X . However, the Lagrange multiplier  is yet undetermined and condition (11) is not satisfied. We now introduce the key theorem of AP method. As such, it states [60,61]: 

~ is not an eigenstate of the operator PHP, then If it is assumed that the vector u

~ tends to be an eigenvector of the modified operator H , if the constraint vector u mod and only if   . This theorem ensures that the constraint condition (11) will be automatically fulfilled because of the orthogonality of the eigenvectors corresponding to different eigenvalues of a self-conjugate operator. To prove this theorem, let us consider the action of

H mod on the vector u . Let ek , k =

0,1,… ,n, be the basis set vectors in the subspace M . Without loss of generality, we may

~ and e e   . The matrix corresponding to assume e1  u i j ij ~ , which is represented by a column multiplied by the vector u

H mod in the chosen basis is

[1,0,..., 0]T in the same basis

set. Then the action takes the following matrix form [60]:

 H 111   H 21   ...   H n1

 H 11/  1 H 12 .. H 1n   1  1      H 21/1  H 22 ... H 2n   0  .     1  ... ... ...   ... ...       H /  H n 2 ... H nn   0  n1 1  

(19)

here H ik  ei H e k , 1  E    u P u , and u P u > 0. It is easily seen that Eq. (19) becomes the eigenvalue problem for the modified operator, if and only if   . Q.E.D.

Constrained Optimized Effective Potential Approach for Excited States

69

~ tends to an eigenvector of As one can also see from Eq. (19), the constraint vector u the operator

H mod as 1/ . It means that u   0 as 1/  , and therefore, the limit,

lim    u  exists. Since  cannot be infinity in practical calculations, one has to settle on some large finite values. On one hand, the greater the value of  , more accurately the respective constraint is satisfied. On the other hand, large values of  may make it more difficult to converge the self-consistent field (SCF) procedure which arises in the HF and KS equations. A reasonable compromise is achieved when  is of the order of 102 – 103 hartrees

~ (see subsection 4.1). It is also clear that if u will be an eigenstate of

is already an eigenvector of PHP, then it

H mod for all  .

It should be noted that, for an operator bounded from below,

E must be positive and

   . The eigenvalues are found during minimization. If, instead, we deal with an upper bounded operator, then   , and the eigenvalues are determined by maximization procedure. The result obtained above for a single constraint vector can easily be extended to cases involving a number of constraints. In such cases,

Pu in the above discussion is replaced

by an orthoprojector on the subspace determined by all the constraint vectors.

2.3. Equivalence of the Original and Modified Problem The Eqs. (4), (5) or the corresponding constrained minimization, viz., Eqs. (9), (10) imply that the solutions are sought in the subspace

( P  Pu ) X , i.e., this is an eigenvalue problem

for the operator H eff  ( P  Pu ) H ( P  Pu ) (cf. with the projection operator technique). In order to clarify the question of validity of replacing the constrained minimization (9), (10) by unconstrained one in (14) , we shall show that the spectra of the original operator

PHP and the modified one H mod are same on the subspace ( P  Pu ) X , when   . ~ is an eigenvector of H , the subspace M can be presented by a direct Indeed, since u mod

sum of subspaces [67]:

M  ( P  Pu ) X  Pu X , which are invariant for the operator eigenvalue problem on the subspace

(20)

H mod and, therefore, Eq. (17) is equivalent to an

Pu X , for which the solution is known (the eigenvector

u~ and the corresponding eigenvalue E   u P u  ) , i.e.,

Pu ( H  E ) Pu   0 ,

E  

(21)

70

V. N. Glushkov and X. Assfeld

and an eigenvalue equation on the subspace

( P  Pu ) X ,

( P  Pu )( H  E )( P  Pu )   0. here we have assumed that,

 ( P  Pu ) Pu ( P  Pu )  0,

(22)

for any value of .

Equation (22) means that the spectra of the operators ( P  Pu )( H  Pu )( P  Pu ) and

( P  Pu ) H ( P  Pu ) are same. But, from a practical point of view, Eq. (17) is much easier to solve than Eq. (22). Indeed, Eq. (22) requires the additional calculations of  H u

and

u H u , whereas Eq. (17) requires only the overlap element  u . Such a difference in computational effort is especially important in the minimization over nonlinear basis set parameters. For example, for SCF excited-state calculations, the conventional techniques 4 2 require  m elements of the two-electron operator whereas we have only  m overlap elements (here m denotes the dimension of one-particle basis set).

2.4. Comparison with Other Methods A comparison of the present formalism with other existing methods, such as the elimination of off-diagonal Lagrange multipliers method [58] and projection operator technique [59] can be found in more details in ref. [68]. Here we comment briefly on these methods from a perspective of handling the orthogonality constraints in SCF theory. One of these was developed by Cole et al., [69,70]. The solution of the problem employed by these authors is interesting but specific to the HF case. In contrast, the AP method described in the present work is a universal method which can be applied to many systems and in many approximation schemes, including SCF technique (see Sections 3 and 4). It is also worth noting that the localized SCF (LSCF) approach [52], which was first successfully applied to quantum mechanical/molecular mechanical methodology for the description of bond links, was expressed in terms of some orthogonality conditions imposed on orbitals. Later this method was used for the determination of core-ionized states of large molecules (see [71] and references therein). One can show that it is closely related to the projection operator technique [59]; also could be considered as a benchmark for implementation of orthogonality constraint problems formulated within the framework of SCF theory. In the limit of   , both the AP and LSCF methods yield same results. However, if we use different basis sets for orbitals and constraint vectors, then the LSCF method leads to the additional calculation of  m elements of the two-electron operator. The AP method as described above, may seem similar to the shift operator technique introduced by Huzinaga [72]. This technique was found to be useful for shifting positions of eigenvalues in the spectrum of Fock operator in HF theory, thereby accelerating the convergence of the iterative process. However, as is known [72], this technique operates only with exact eigenvectors of the operator PHP, whereas in the AP method, arbitrary constraint 4

vectors are considered.

Constrained Optimized Effective Potential Approach for Excited States

71

We also note the similarity of AP method with the other techniques for incorporating orthogonality constraints by means of an energy penalty term. However, we recognize that this penalty term method requires several calculations of a functional (e.g., the energy functional) in order to determine an optimal value of the multiplier . There have been some preliminary attempts to apply the penalty term method to excited-state calculations (e.g., [73]). However, the target accuracy was not achieved in this work. For more details, the reader is referred to the work of [73], where different penalty functions are considered. In concluding this section it is worth noting that, from a practical point of view, the solution of the eigenvalue equation for operator

H mod in the AP method has a stable

character, although the condition number (i.e., ratio

Emax / Emin ) of the corresponding

matrix can be rather large. The problem is that the eigenvalue sensitivity is measured by the

H mod matrix [74].

condition number of the diagonalizing matrix, but not the

3. Excited-State Theory Based on the Asymptotic Projection Method 3.1. Constrained Variational Problem for Excited States It is well known that the ab initio study of excited electronic states of atoms and molecules contains elements that are not contained in the treatment of ground states. In particular, excited- state wave functions must be orthogonal to states of lower energy. For the lowest eigenstate in a given symmetry class, a trial wave function of that symmetry is automatically orthogonal to all lower eigenstates. For higher eigenstates, however, the imposition of orthogonality constraints is often difficult and cumbersome. The AP method described above can be directly applied to the calculation of approximate wave functions and energies of excited states having same spin and spatial symmetry as lower

u  0 and E0 are the approximate ground-state (GS) wave function and corresponding GS energy; this is defined as the lowest root of Eq. (17) at   0 . Then, the states. In this case,

first excited state or the second eigenvalue is defined by the relation,

E1  E (1)  1 H 1 

min



{ 0 }

1 1  1,

H 

 ,

(23)

where the minimum is taken over all vectors  belonging to the orthogonal complement

{  0 }  to the vector  0 , i.e., the vectors   {  0 }  satisfy the condition

72

V. N. Glushkov and X. Assfeld

 0   0. It should be stressed that, generally {  0 }   M i.e., a subspace different from M can be used for E1. Higher eigenvalues are determined in a similar fashion. Thus, following the AP methodology, an eigenvalue problem, for example, for the first excited state takes the following form (cf. Eq. (17)):

P( H  P0 ) P 1  E1 P 1 , with P0   0

 

(24)

 0 . In other words, we should use a modified operator H  H   P0 ,

instead of H . In matrix form suitable for practical programmable calculations, the excited-state problem can thus be written as follows, n

n ~ H C  E  ik ki 1  Sik Cki ,

k 1

where S ik 

i  k

i  1,2,..., n

(25)

k 1

is the overlap matrix and

~ H ik  i H k   i  0  0 k , with   

(26)

3.2. Variational Theorems and Bounding Properties for Excited-State Energies Löwdin [75] gives a clear discussion on the general problem of variation in a restricted subspace, and has emphasized some of the technical difficulties that arise in using Eq. (23). In particular, we must require orthogonality to the exact lower eigenfunctions of H , if the minimum given by Eq. (23) is to be an upper bound to the exact eigenstate energy. A more useful practical formulation, that avoids introducing the exact lower eigenfunctions, has been given by Hylleraas and Undheim [76], and MacDonald [77]. These authors have shown that convenient upper bounds are provided by Rayleigh-Ritz method, if the functions  i determined in the same basis set

k ,

are

k=0,1,…,n, in accordance with the expansion in

Eq. (7). If we use different basis sets for different states, then only imposition of orthogonality constraint (10) on an approximate lower-state wave function, does not, in general, yield an excited-state energy, which is an upper bound to the exact excited-state energy. “The desirability of using different basis sets for different states” was pointed out by Shull and Löwdin [78] long times ago in 1958. This is because, calculations on excited states can be meaningless without including sufficiently diffuse basis functions, whereas for ground states, such functions may not be needed at all. Such a presentation is preferable and can provide a more compact representation of the accurate eigenvectors and eigenvalues.

Constrained Optimized Effective Potential Approach for Excited States

73

Although the only imposition of orthogonality constraint (10) on an approximate lower state wave function does not, in general, provide an upper bound, there exists the so-called “weak bound” [79] with respect to exact energies Єj . However, it is based on the exact wave functions  j , which are not known. An upper bound to the excited-state energy is obtained if we impose the additional constraint,

i H  j  0.

(27)

It should be noted that the AP method can also be applied to deal with Eq. (27). In fact, in practical calculations, excited-state energies are expected to lie above exact energies if

 0 is a good approximation to the true ground-state eigenfunction. Therefore, we

shall now investigate the bounding properties when only the orthogonality constraint (10) is imposed on an excited function. In this case, it is our opinion that a max-min principle [67], not directly based on eigenvectors, is useful. For example, for the first excited state, we have,

E1  max min  E    . {0 }

0

In this equation, the maximum is attained when

(28)

 0 is equal to the exact wave function

0 of the ground state. Equation (28) suggests that the complete basis set is used for

E1

value is affected by two factors. On one hand, if a

 0 , then a finite-dimensional approximation to the excited

state leads to an upper bound for

E1 , i.e., E1 

is carried out in the subspace ( I   0 made for

Є1 . On the other hand, if the variation of 

 0 ) X , and a finite basis set approximation is

 0 , then the max-min principle (28) gives Є1  E1 ; that is E1 is a lower bound to

the true energy with the equality

0  0 .

In contrast to the “weak bound”, defined via the exact eigenvalues and wave functions, we are able to give a relation between

E1

upper

and an upper bound E1

[66], which proves

useful from the viewpoint of practical calculations. Indeed, for the approximate wave functions

 0 and 1 , the secular determinant may be written as, det

0 H 0  

 0 H 1

1 H  0

1 H 1  

 0.

This equation has two solutions which could be written as below,

(29)

74

V. N. Glushkov and X. Assfeld

1 1 2 E0upper  [ E0  E1 ]  [( E0  E1 )2  4 H 01 ]1/2 , 2 2

(30)

1 1 2 E1upper  [ E0  E1 ]  [( E0  E1 )2  4 H 01 ]1/2 , 2 2

(31)

where H 01   0 H 1 . By the Hylleras-Undheim-MacDonald theorem [76, 77], we know that the solutions of these secular equations are upper bounds and, in particular, upper

E1

 Є1.

(32)

Furthermore, we note that,

E0upper  E0  E1upper  E1 . Hence, if the ground-state energy is determined to a given accuracy, then comparable accuracy, and the coupling matrix elements,

(33)

E1

will have a

H 01 , may be neglected at this level

of approximation.

3.3. Some Problems and Specific Features of SCF Excited-States Calculations Below we shall consider excited electronic states which can be adequately described by a single open-shell Slater determinant. Existing one-particle eigenvalue problems such as the KS equation, or the SCF equation in HF and OEP theory for ground state, cannot be directly applied to the excited states of same symmetry as a lower state without “variational collapse”; that is, the approximation to excited-state wave function is contaminated by components of a lower state. Quantum mechanics requires exact wave functions to be orthogonal but it makes no such demand on SCF functions. Indeed, consider the orthogonality condition for exact manyelectron wave functions describing the ground state,

0 , and first excited state 1 , i.e., (see

also [80]),

 0 1  0. The exact ground-state wave function,

(34)

0 , can be written as,

 0  0   0 ,

(35)

Constrained Optimized Effective Potential Approach for Excited States where

75

 0 is the many-electron ground-state SCF wave function and 0 is the correlation

correction. Without loss of generality, we may assume,

 0  0  0. Similarly, the exact excited-state wave function,

(36)

1 , can be written as,

1  1  1 , where

1

is the many-electron excited-state SCF wave function and

(37)

1

is the corresponding

correlation correction. Again, without loss of generality, we may require,

1 1  0.

(38)

Substituting Eq. (37) into Eq. (34) we get,

 0 1   0 1   0 1  0.

(39)

 0 1  0,

(40)

 0 1  0,

(41)

Now, if we require that,

which, in turn, implies that,

then it can be easily shown that,

1 H 1 1 1

 Є1,

(42)

where Є1 is the exact energy of the excited state. However, Eqs. (40) and (41) cannot be used directly because, 0 , the exact wave function for ground state, is unknown. Substituting Eqs. (35), (37) into Eq. (34) we have,

 0 1   0 1   0 1   0 1   0 1  0,

(43)

 0 1  [  0 1   0 1   0 1 ].

(44)

or,

76

V. N. Glushkov and X. Assfeld

Thus we see that the SCF wave functions, do not, in general, satisfy orthogonality constraints analogous to those obeyed by exact wave functions. Within the framework of wave function formalism, several useful methods have been put forth to overcome the difficult “variational collapse” problem, and a number of different schemes have been proposed for obtaining SCF wave functions for excited states [66,69,70,81,82-88]. Some of these approaches, e.g., [66,69,70,81-83], explicitly introduce orthogonality constraints to lower states. Other methods such as [84-88], use this restriction implicitly. In both of these schemes, however, the excited-state SCF wave function of interest is orthogonal to the wave function of a lower state, or states of the same symmetry. However, these lower state or states are not necessarily the best self-consistent field functions for these states (see comments in [69]). It is clear that the experience accumulated from HF excited-state calculations, could be useful to derive the corresponding equations for DFT or OEP excited-state approaches (see Section 4). For example, the HF excited-state method based on an AP methodology [66] has been recently employed to generate effective local potentials for excited states of atoms [89]. Another example is a spin-dependent localized HF (SLHF) technique which has been successfully applied to DFT-based excited-state calculations of atoms (see Section 7 and references therein). Thus, we may impose a constraint upon the SCF excited-state function so that,

 0 1  0 i.e., we explicitly introduce the orthogonality constraint on function 1.

(45)

1

and the best SCF ground-state

 0 . Imposition of the above constraint (45) is important for following reasons: Any lack of orthogonality of the SCF wave functions may lead to excited-state energies lying below the corresponding exact energies. For example, Cohen and Kelly [90], found for He atom, the first singlet excited-state energy hartree, whereas the observed energy E1

2. 3. 4.

exact

E1 =−2.16984

= −2.14598 hartree.

It preserves the important orthogonality property of exact eigenstates. It facilitates the development of a simple perturbation theory expansion for correlation effects in excited states [66]. It allows the study of properties which depend on the wave functions of different states, e.g., in the evaluation of transition properties (see also [69,70]).

3.4. Orthogonality Constraints for Single-Determinantal Wave Functions In this subsection, we are concerned with the ground and excited electronic states which can be adequately described by a single determinantal wave function, i.e., doublet, triplet states, etc., with spin

S  0.

Let

 0 be the ground-state unrestricted Slater determinant

constructed from a set of spin-orbitals consisting of spatial parts

0i  , (i   1,2,..., n )

Constrained Optimized Effective Potential Approach for Excited States associated with

77

 spin functions and orbitals 0i  , (i   1,2,..., n  ) associated with 

spin functions, i.e.,

0   N !

1/2

det 01 ,..., 0 n ;01  ,..., 0 n   .

(46)

Without loss of generality, we define n > n  , n + n  = N, where N is the number of electrons and

S  S z  (n  n ) / 2 is the total spin. Similarly, 1 is a single determinantal

wave function for the first excited state,

1   N !

1/2

det 11 ,..., 1n ;11  ,..., 1n   .

(47)

It is well known that the orthogonality constraint (45) for functions (46) and (47) can be written in terms of spatial orbitals in a form as given below,

 0 1  det 01 11 ... 0 n 1n  det 01 11 ... 0 n  1n   0.

(48)

The annihilation of either of the two determinants in (48) leads to fulfillment of the orthogonality condition (45). From energy consideration and previous computational experiences, we impose orthogonality restrictions only via the first determinant which is associated with the  set and involves the occupied orbital, highest in energy. As is well known, the condition, 





det 01 11 ... 0 n 1n



0

(49)

is fulfilled if either the rows or columns in the first overlap determinant are linearly dependent. Therefore, two physically different schemes are possible which could satisfy Eq. (49): either n b1j 0i 1j  0, i = 1,2,…, n (50) j or n bi0 0i 1j  0 , j = 1,2,…, n (51) i

 

Equation (50) requires that all occupied ground-state orbitals be orthogonal to a linear combination of excited-state orbitals

n

j

b1j  j1 , which describes an excited electronic

state. Equation (51) requires the orthogonality of all occupied excited-state orbitals associated

78

V. N. Glushkov and X. Assfeld

with

 spin functions, to the arbitrary vector

ground-state orbitals associated with

n

i

bi0  0i , from the subspace of occupied

 spin functions. In general, the coefficients b j may be

determined by minimizing the excited-state HF energy. However, calculations show that the choice, n

 bi0 0i

 0n ,

(52)

i

where

 0 n  is the orbital from ground-state Slater determinant with highest energy (i.e.,

HOMO orbital), leads to a minimum energy for the excited state. In the complete basis set limit or for a common basis set for ground and excited state, the schemes defined by Eqs. (50) and (51), yield same energy values. In this work, we use the second scheme, i.e., that defined by equation (51), which upon using (52) becomes,

0n 1j  0, j = 1,2,…., n,

(53)

in order to impose the orthogonality constraint (45). Equation (53) could be rewritten in a symmetrical form as follows, which is useful when deriving the HF equations,

1j 0n 0n 1j  0,

(54)

or since the left-hand side of (54) is not negative, n

  1j Pn 1j  0

(55)

j 

here Pn is the projection operator,

Pn  0n 0n .

(56)

Thus the constraint (45) imposed on the first excited-state determinantal function can be expressed in terms of orthogonality conditions for one-particle orbitals and, therefore, may be considered when using the AP methodology.

4. Amended Hartree-Fock and the Exchange-Only Kohn-Sham Equations for Singly and Doubly Excited States To distinguish between HF and KS orbitals, we will use the notation  i for former and

i for latter. To keep our notation simple, here and below we use state-number subscripts

Constrained Optimized Effective Potential Approach for Excited States

79

only for orbitals of the lower-lying states, but omit those for orbitals of the target (highest) state. Some results of this section may be found in a recent paper [89].

4.1. Amended Hartree-Fock Equations for Singly Excited States To derive the HF equations for excited states (for simplicity we restrict ourselves to the first excited state), we start from a minimization of the total excited-state energy, expressed in terms of the unrestricted HF (UHF) orbitals (cf. with Eq.(2)): n

E

UHF

1 1 { }    i   2  i    (r )Vext (r )dr    (r )VH (r )dr  Ex{ }. 2 2  i 1

(57)

  E x is given by Eq. (3) with  i in lieu of  i .

To get the restricted open-shell HF (ROHF) function, we shall minimize Eq. (57) subject to the following constraints: 1.

Orbitals must satisfy restrictions (55) to ensure the orthogonality of Slater determinants for ground and first excited state.

2.

The excited-state Slater determinant must be an eigenfunction of impose the spin purity condition in the form (see [65]): n

   Q   k

k

Sˆ 2 operator, i.e., we

 0,

k

where

(58)

Q  I  P is an orthoprojector on the subspace of virtual  spin orbitals and n

P  i  i  i . It is useful to note that Eq. (58) means that the set of orbitals associated with the  spin functions lies completely within the space defined by the set associated with the  spin functions. The HF equation for excited-state orbitals can now be obtained by constructing a functional consisting of the UHF energy expression in conjunction with the orthogonality constraints (55), (58), by the method of Lagrange undetermined multipliers. In particular, the constraints (55), (58), multiplied by Lagrange multipliers to the energy

E

UHF

o

and

s , respectively, are added

{ } , so as to give the following functional,

LE

n

UHF

n

{ }   o  k Pn  k  s   k Q  k . k







k

(59)

80

V. N. Glushkov and X. Assfeld

Using the stationary condition L  0 , we arrive at the equations determining the ROHF determinantal function (see [65, 66] for more details): 



( F   s P   o Pn   i )  i  0,

(60)

( F   s Q   i  )  i  0. 

(61)

here F , F are the standard UHF operators constructed from excited-state orbitals  i , i.e., 

1 F     2  Vext  VH  VxHF , 2

  , 

 HF here V x is a non-local exchange potential of the HF method and P 

(62)



n i

 i  i .

Equations (60) and (61) represent the constrained variational SCF method in its general form. The orthogonality constraint of Eq. (55) is satisfied in the limit of

s  0,

o   . By setting

we can relax the spin-purity constraint and go back from ROHF to UHF solutions.

By setting

o  0, one falls back to the ground state.

In practical applications, we invariably invoke the algebraic approximation by parametrizing the orbitals in a finite one-particle basis set. This approximation may be written as, M

 i   cpi  p

(63)

p 1

It should be stressed that, in general, the basis set for excited state is distinct from that for the ground state. This is because calculations on excited states can be meaningless without including sufficiently diffuse basis functions, whereas for the ground states such functions may not be necessary at all. Once the basis set { p } is introduced, the integro-differential equations (60) and (61) become generalized matrix eigenvalue problem: M

~

M

 Fpq cqi  i  S pq cqi

q 1

with S pq  elements,

(64)

q 1

~  p q , being the overlap matrix and Fpq , the modifyed Fock matrix with

Constrained Optimized Effective Potential Approach for Excited States

81



n ~ Fpq   p F   q  s   p  i  i  q  o  p  0n  0n  q

(65)

i

 pq

F  p F



n

 q  s [ S pq    p  i  i  q ]. i

(66)

This result can be easily extended to the higher energy levels. For example, for the 

second excited state, the operator Pn should be substituted by the orthoprojector, 

Pn   0n  0n   1n  1n

(67)

etc. Hence, the problem of the choice of a determinantal wave function for higher excitations does not appear. Since neither

s nor o

can be infinity in practical calculations, one has to settle on some

large finite values. As mentioned earlier, a reasonable compromise is achieved when both and

o

are of the order of 102 – 103 hartrees (see P.8). The recommended values are

hartrees for the spin-purity constraint and

s

s =100

o =1000 hartrees for the orthogonality constraint.

4.2. Kohn-Sham-Type Equations for Excited States In the exchange-only OEP scheme for ground states, the total energy functional has the   form (58) with  i replaced by  i . This functional is minimized with respect to  i , subject

to the constraint that these orbitals be eigenfunctions of an effective one-electron Hamiltonian having a form as follows,

1 h   2  Vext (r )  VH (r )  Vx (r ), 2

(68)



 where Vx (r ) is a local exchange potential. In other words, a non-local exchange potential in





the Fock operator is replaced by a local potential. In addition, Veff  Vext  VH  Vx . The condition for an extremum of

EUHF {} then leads to the following KS equation for the

ground state:

h  i   i  i ,

  ,  ,

(69)

82

V. N. Glushkov and X. Assfeld

 0 constructed from the solutions i is referred to as a KS

A Slater determinant

determinant. By analogy with the excited-state HF scheme, we impose the orthogonality constraint on KS determinants, viz.,   0  0 via Lagrange multipliers. Using arguments similar to those leading to Eqs. (60), (61), we arrive at the spin-restricted exchange-only KS equation for first excited state: 



(h  s P   o Pn   i )  i  0, 

(h   s Q   i ) i  0,

(70)

(71)

where unlike the HF scheme, the orthoprojectors are built upon orbitals obtained with a local 

potential, i.e., P  KS orbitals, while



n k 1

k k is the orthopojector onto the space of occupied  -spin

Q  I 

n



k k is the orthoprojector onto the space of virtual

k 1

 -spin KS orbitals, and Pn  0n 0n . We have termed the optimized effective potential method based on the eigenvalue Eqs. (70) and (71) as the constrained OEP (COEP) method. Introducing, as before, a finite basis set { p }, we expand the KS orbitals,

i 

M

 api  p

(72)

p 1

and write Eqs. (70) and (71) in the following matrix form, M

M ~  h a    pq qi i  S pq aqi ,

q 1

  , 

(73)

q 1

with 

n ~ h pq   p h  q  s   p  i  i  q  o  p  0n  0n  q

(74)

i 

n ~ h pq   p h   q  s [ S pq  

 p i i  q ]

(75)

i

In general, modified KS equations are solved self-consistently in the same manner as the excited-state HF equations. Certainly, it makes sense if we know a local exchange potential for a given excited state. It was shown [45] that using AP methodology, the exchange-only

Constrained Optimized Effective Potential Approach for Excited States

83

OEP equations for excited state can be obtained. They have a structure similar to the groundstate OEP equations and existing approximate methods developed originally for ground states can be applied to solve them. In particular, practical solutions of the KS and OEP equations are based on a finite basis set implementation. An orbital basis set is used to represent the  orbitals  i and an auxiliary basis set is needed to expand the local effective potential. For



 example, Vx (r ) can be expanded in a set of auxiliary basis functions,

 f m (r ) , m =1,2,…,

Maux, as, 

Vx (r ) 

M aux

 b f m 1

m

m

(r ), (76)

 where the coefficients {bm } are determined by solving the OEP equations.

4.3. Doubly Excited States Based on the AP Methodology We will stick to our recent paper [93] to obtain the results in this subsection. Let

 0 be

a Slater determinant describing the ground state of an N-electron system (see Eq. (46)),

 0  ( N!) 1 / 2 det  01  ,...,  0n   ;  01   ,...,  0 n  

(77)

  In particular, for a closed shell-ground state, n  n  and { 0 i }  { 0 i }.

Consider now, for simplicity, a doubly excited state obtained by the excitation of an   electron associated, e.g., with the  0 k orbital and the other electron from the orbital  0 l . Let

 be the Slater determinant which describes this doubly excited state,

  ( N ! ) 1 / 2 det 1  ,...,  n  ;1  ,...,  n  

(78)

As in the case of ground state, the Slater determinant arises from the unrestricted scheme, where orbitals in Eq. (78) are different from those in Eq. (77), and these will be obtained from the amended KS-like equations for doubly excited states (see below). In contrast to the existing DFT-based methods, we achieve the effect of excitation of an electron by using some orthogonality constraint imposed on the orbitals of doubly excited state Slater determinant (78). In other words, we require a fulfillment of conditions,

 0k  j  0 ,

j = 1,2,…, nα,

(79)

 0l  j  0 ,

j = 1,2,…, n

(80)

84

V. N. Glushkov and X. Assfeld

which provides a correct description of the doubly excited state by means of Slater determinant (78). It should be noted that such an idea was successfully applied by us to generate ionized hole states and to carry out calculations of core and valence ionization potentials for molecules [71,91,92]. The constraints (79) and (80) for doubly excited states can be rewritten in the form below,

Pk  i  0 ,

i = 1,2,…, nα

(81)

Pl   i  0 , i = 1,2,…, nβ

(82)

and

with the orthoprojectors: Pk





  0k  0k and Pl   0l  0l .

Then again, applying AP methodology to the eigenvalue problem (69), subject to the constraints (81) and (82), we arrive at the following KS-type equation for orbitals determining a doubly- excited Slater determinant (86): 



(h  o Pk   i )  i  0, 

(h   o Pl    i )  i  0,

(83)

(84)

with h defined by Eq. (68). It is clear that a matrix form of the above equations can be obtained using analogy with singly excited equations. This result can be easily extended to triply, etc., higher multiply excited states in a similar manner. For example, a triply excited state, dealing with an excitation of one more electron from the

 0 j orbital can be obtained by imposing the constraint: 0j i  0 , i = 1,2,…,

nα. More details can be found in ref. [93].

5. Effective DFT Potential Expressed as a Direct Mapping of the External Potential and Its Optimization for Excited States 5.1. Parametrized Effective Potential for Molecules Although a large number of practical calculations have been carried out based on the OEP methodology, solution of the OEP equations is not straightforward. It was recently demonstrated [94] that, finite basis set implementation of the OEP method suffers from numerical instabilities. In particular, this may lead to: (i) existence of infinitely many multiplicative exchange potentials and (ii) collapse of the unconstrained exchange-only search to the corresponding HF solution. In addition, the basis set OEPs are characterized by unphysical oscillations around the nuclei. A lot of important works have been published on

Constrained Optimized Effective Potential Approach for Excited States

85

various strategies to “balance” the orbital and auxiliary basis sets (e.g., [95-98]). Without going into a discussion of this problem (see e.g., [98]), we shall consider another concept  proposed by Theophilou [99] to represent Veff (r ) . In such a scheme, the effective DFT potential is expressed in terms of the external potential,

 Vext (r ) . Unlike the traditional

implementation of DFT, where the potential is an explicit or implicit functional of the  electronic density, this method deals with Veff (r ) as a functional of the external potential, i.e.,

  Veff (r ) = Veff [Vext ( r )] . A direct mapping does not contradict the Hohenberg-Kohn theorem

[1], which establishes a one-to-one correspondence between ground-state density and external potential in absence of any degeneracy. Indeed, in the KS theory, there exists a one-to-one correspondence between the density and KS potential. The one-to-one correspondence between

  Vext (r ) and Veff (r ) , which plays the role of KS potential, follows from transitivity

property of the correspondence relation. It is important to note that such a representation makes it possible to obtain many-electron wave functions having same transformation properties as the eigenstates of an exact Hamiltonian. For example, spin is the case where, in order to get correct magnetic properties, many-particle states must be eigenstates not only of,

Sˆ z

but also of

Sˆ 2 operators.

The derivation of explicit form of such a mapping is not an easy task, as is the case with exchange-correlation potential. Using the requirement of correct asymptotic behavior of  Veff (r ) and covariance condition (see [99,100] for more details), however, it has been found that the lower-order approximation of the effective ground-state potential for molecules can be written in following form [101,102]:

Veff (r )   k

where Z 



k

Zk r  Rk



1  exp(d k r  Rk ) N 1 C  Zk , Z k r  Rk

(85)

Z k , and summations are over the nuclei. The adjusted parameters C and d k

can be determined by using different criteria; in particular, by minimizing the energy

 E UHF {}   H  of a noninteracting N-particle state. In this implementation, Veff (r )

is closely related to the OEP approximation. The potential (85) preserves symmetry properties of the exact eigenstates, and has been found to be successful for ground-state calculations of different atoms and molecules [101, 102].

5.2. Equations for Optimization of Excited-State Parametric Effective Potential 

A parametrized form of the effective local potential (e.g., representation of Vx via the auxiliary basis functions, as in Eq. (76) or parametrization, as in Eq. (85)) makes it possible to

86

V. N. Glushkov and X. Assfeld

simplify greatly, not only a solution of the OEP integral equation, but also allows derivation of the OEP equation in a simple manner. This also avoids the use of functional derivatives, whose existence has been subject of many discussions [104]. Below, for simplicity, instead of Eqs. (70), (71), their spin-unrestricted versions (by setting

s  0 ) for excited-state orbitals

will be considered, i.e., the corresponding orbitals have to satisfy the following one-particle equations: 



(h  o Pn   i )  i  0,

(86)



(h    i )  i  0, with h



(87)

1     2  Veff . Such a simplification is justified here as the spin-symmetry is 2

preserved by using a direct mapping of DFT potential (85) itself. We start from the requirement of minimum for the orbital-dependent functional,

EUHF {}   H  subject to the constraint that orbitals

,

   1,

(88)

i must satisfy orthogonality condition (see also Eq.

(56)), n



 j  Pn  j   0

(89)

j

By analogy with the previous sections, we use the stationary condition, n

 [  H      k  Pn  k  ] = 0

(90)

k

Variation of the orbitals

j can be divided into two parts as follows,

 j  PN  j  ( I  PN )  j ,

where

PN

  , 

(91)

n

   i  i is orthoprojector on a subspace of  -occupied excited-state i

orbitals. Energetically significant variations are described by the second term because the first does not lead to any change in total energy as it is invariant to any orthogonal transformations

Constrained Optimized Effective Potential Approach for Excited States

87

of the occupied orbitals among themselves. After some manipulations, the relation (90) can be written in terms of one-particle orbitals, n













i ( I  PN )( F  Pn )  i



n



i ( I  PN ) F   i  c.c.  0

i

i

(92) Variations



i



i

and

are independent; therefore Eq. (92) leads to different

relations for each spin set: n

 i ( I  PN )(F   Pn )  i  0

(93)

i

and n

 i ( I  PN ) F   i  0

(94)

i

Unlike the HF case, however, the variations

i , i are not arbitrary because

they are restricted by Eqs. (86), (87) with a local potential Veff and, therefore, have to be determined by these equations. For example, for  orbitals we have,   [hmod   i ] i   [ i  Veff ] i 

where hmod  h

 

Now, let Ri

(95)

  0 Pn . be the reduced resolvent operator (Green's function in coordinate 

representation) for the modified operator, hmod      ( i  hmod ) Ri = Ri ( i  hmod ) = I  PN .

(96)

Then, a solution of Eq. (95) takes the form,

( I  PN ) i  Ri [Veff ]  i

(97) 

Substituting Eq. (97) in Eq. (93) and using the following spectral representation for Ri ,



Ri 

virt

i i

a  n  1

 i   a



(98)

88

V. N. Glushkov and X. Assfeld

with the summation over virtual  -orbitals, we get, n  virt

 i

   i [Veff ]  a  a (VHF  Veff )  i

0

 i   i

a

(99)



here we have taken into consideration, the explicit form for Fock operator and hmod , i.e.,   F   0 Pn  hmod  VHF  Veff ,



(100)

  a hmod  i  0

where VHF  Vext  VH  Vx . Besides, HF

due to Eq. (86), and

Ri i  0 . The only difference in Eq. (99) compared to the ground-state OEP equation is that the virtual summation does not include the orbital

 0n .

By analogy, one can obtain an equation for optimization of Veff : n  virt

 i

    i [Veff ]  a  a (VHF  Veff )  i

 i   i

a

0

(101)

A parametrized form of the effective potential implies that Veff depends on a set of 

variational parameters, i.e., Veff  Veff {  } (e.g., { }  {bm } or { }  {C , d } ). 

Then,



 Veff   ( Veff )  , 

(102)

and the OEP equations (99), (101) can be reduced to a system of algebraic equations with respect to variational parameters n virt

 i

a

 :

  i [ Veff ]  a  a (VHF  Veff ) i

 i   i

 0 ,   , .

(103)

The left-hand side of Eq. (103) represents a gradient of the excited-state energy with respect to the potential parameters, and can be used to design a cost-effective computational algorithm for minimization. It is also worth noting that the representation (85) does not use splitting of Veff to the Hartree and exchange potential. This helps us in avoiding a selfconsistent iterative procedure for solving the KS-type equations for excited states.

Constrained Optimized Effective Potential Approach for Excited States

89

6. Second-Order Correlation Functionals Based on Many-Body Perturbation Theory for Excited States 6.1. Correlation Functionals Compatible with the Exchange OEP Consider now one of the possibilities to include correlation effects within the framework of DFT-OEP methodology. At present, it is unclear which approximation for the correlation energy should be used in combination with exchange potential of the OEP based methods. For atoms, for example, good accuracy can be obtained using the ColleSalvetti semi-empirical correlation density functional [104]. It is known that the combination of OEP exchange energy with traditional explicit correlation functionals expressed in terms of electronic density, leads to incorrect correlation contributions (e.g., [105]). For molecules, the most direct way for incorporating correlation effects in a systematic form is offered by many-body perturbation theory (PT), based on the KS determinant as a reference (see e.g., [11], [105-107] and references therein). Particularly for ground states, encouraging results have been obtained with orbital- dependent second-order correlation functionals. In addition, the authors of ref. [105] observed from their numerical computations that, inclusion of the correlation potential,

 Vc (r ) , based on a second-order

PT, in total effective potential, does not lead to an improvement compared to a complete neglect of correlation potential. This gives the possibility of using a compromised version: the total energy may be computed via second order PT based on exchange-only orbitals. To employ PT, one writes the Hamiltonian of an interacting system in following form,

H (k )  H (0)  kV . here H

(0)

(104)

( 0) is the zeroth-order Hamiltonian, and V  H  H is a perturbation operator.

Then, the PT series expansions for energy

Ei and wave function i take the form:

Ei (k )  Ei(0)  kEi(1)  k 2 Ei(2)  ...,  i (k )   i(0)  k i(1)  k 2  i( 2)  ... ,

(105)

i  0,1,2...

(106)

It is well known that, the success of any PT is crucially dependent on the choice of (0)

zeroth-order Hamiltonian H . Within DFT, there exist different choices for the separation of the Hamiltonian (see discussion in [11]). The Møller-Plesset (MP) version of many-body PT is one of the most popular computational tools in quantum chemistry for incorporating (0)

correlation effects. Its effectiveness is due to the successful choice of H and the effectiveness of HF approximation. Using the same analogy in an AP method between ground- and excited-state HF calculations, the MP like PT (MPPT) for excited states was

90

V. N. Glushkov and X. Assfeld

developed [66]. A single Slater determinant based on excited-state HF orbitals was used as a reference function.

6.2. Second-Order Correlation Energy Functionals of Excited States The good agreement between total energies computed by OEP and HF methods, and the high overlap between single-determinantal functions built from the KS and HF orbitals (see [102]) naturally incites us to use an MP-like perturbation theory for incorporating correlation effects for excited states. In this case, by analogy with MPPT, the zeroth-order Hamiltonian

H ( 0 ) for the first excited state can be written in following form [66]: H

( 0)

n

=

n

 H eff (i)   H eff (i) 

i

(107)

i

with  H eff 

M 1



 H eff 

i  i i ,

M 1

i

  i

 i  i .

(108)

i

here M is the dimension of orbital basis set employed for KS excited-state equations, and we use the spectral representation of H eff based on excited-state orbitals and energies of Eqs. (86), (87). M-1 means that one orbital

 0n

should be excluded from the virtual subspace.

It is important that singly (  i ), doubly (  ijab ), etc., excited configurations based on a a

reference KS excited-state function 

(0)

(0) are eigenfunctions of H . Also, these are, due to

AP technique, orthogonal to the ground-state KS function  0

(0)

and among themselves.

Therefore, these functions form an orthonormal basis in a many-body space, and can be used to develop a PT for construction of orbital-dependent correlation functionals for excited states. It was shown [66, 108] that the Rayleigh-Schrödinger PT adapted to our problem leads to the following expression for first-order correction to the KS reference excited-state function: (0) (1)  RV (0)  (0) (1) . 0 0 

where R  Q( E

( 0)

(109)

 H (0) )1Q is the reduced resolvent operator, Q is the orthoprojector

onto the complementary space, i.e., Q  I  

( 0)

 ( 0) .

The total energy including a second-order orbital-dependent correlation energy for first excited state, then takes the form as below,

E  E OEP  E ( 2)   H   E ( 2) , and

Constrained Optimized Effective Potential Approach for Excited States

E ( 2)   (0) H  (1)

91 (110)

or, taking into account Eq. (109), we arrive at an expression in terms of spin-orbitals which is suitable for programming:

occ virt

E ( 2)    i

a

i F  a i  a

 

occ virt

( ai b j   a j bi 

i  j a b

i   j   a  b

2



2

  ( 0) H  (00)  (01)  (0) (111)

here also, the summations are over spin-orbitals; a, b are virtual orbitals while i , j are occupied orbitals; and F is the Fock operator built from KS excited-state orbitals. In Eq. (111), we use the standard notation:

i j kl    dr1 i (r1)k (r2 ) r1  j (r2 )l (r2 )dr2

(112)

12

The first and second terms in Eq. (111) are readily recognized from the second-order KS PT expression for ground state. It is worth noting that unlike the PT based on HF orbitals [66], single excitations enter at the second-order corrections, because the KS orbitals, in general, do not satisfy Brillouin’s theorem, i.e.,  i H  a

(0)

 0 . The third term in Eq.

(111) appears because the KS ground- and excited-state functions are not eigenfunctions of the Hamiltonian. In usual cases, the coupling element

(00) H ( 0)  1 , and

(01) (0)  1 . It means that the contribution of last term in Eq. (111) may be neglected at the first stage of calculation. Thus, in terms of computational cost, the present PT for excited states is similar to a genuine Møller-Plesset PT, since the first term in Eq. (111) can be easily calculated. In practice, (as our calculations show) the contribution of first-order term is much less than that from double excitations.

7. Illustrative Applications to Atoms and Molecules The aim of this section is to present a brief survey of the applications of constrained optimized effective potential method for small atomic and molecular systems, so that we can calibrate the accuracy of our calculations from both theoretical as well as computational point of view. At present, there are only scarce finite basis set exchange-only calculations for excited states having same symmetry as the ground state, which are based on the exisiting DFT-OEP methods. In this respect, the numerical Hartree-Fock (NHF) [109] and other accurate numerical DFT calculations present excellent database to examine the performance of novel methods, in particular, an exchange-only COEP method. As far as the molecular

92

V. N. Glushkov and X. Assfeld

excited-state calculations are concerned, we could not find suitable literature data obtained from the DFT-based methods for direct comparison with either X-only or exchange+correlation (XC) results. However, there are numerous papers available on theoretical study of excited states of molecules using the configuration interaction (CI) approximation. Therefore, while comparing with these results, we pay attention on a balanced description of ground and excited states. All our calculations for the systems studied in this chapter were carried out in finite basis sets consisting of s-type distributed Gaussian functions:

 p ( x, y, z )  exp{ p [( x  X p ) 2  ( y  Y p ) 2  ( z  Z p ) 2 ]} .

(113)

The corresponding basis sets were extended to include some p-functions in order to compute second-order correlation energies. In addition, each p function (

px , p y , pz )

was

represented by a linear combination of two s-functions, i.e., the so-called lobe representation was used. The total energy of ground and excited states were minimized to determine nonlinear basis set parameters, viz., orbital exponents for atoms and orbital exponents + positions ( Z p ) for diatomic molecules, i.e., basis sets were optimized for each individual state. This allowed us to minimize the error associated with truncation of one-particle basis sets and, thus, to observe more clearly the errors of the method itself. This procedure due to the AP method for excited states, takes practically the same computational time as for a ground states. Our experience shows that the even-tempered prescription is a good choice to generate a zero approximation for orbital exponents. Some details of basis set optimization can be found in [110] and references therein. It should also be stressed that we employed the same effective potential for both  spin electrons and  spin electrons, i.e., Veff  Veff  Veff . Then we deal with

i  i ,

i  1,2,..., n  (doubly occupied orbitals ) and n  n  singly occupied orbitals, so that a 2 Slater determinant  constructed from these orbitals is an eigenfunction of Sˆ z and Sˆ operators, i.e.,

Sˆ z   S  ,

Sˆ 2   S ( S  1)  .

(114)

7.1. Singly and Doubly Excited States of Simple Atoms In Table I, we report the results [45, 93] for energies of some singly excited states of representative simple atoms He, Li and Be at the exchange-only level, by employing a potential (85), which for atoms takes the simple form,

Veff (r )  

Z 1  exp( dr )  ( N  1)C r r

(115)

Constrained Optimized Effective Potential Approach for Excited States

93

The variational parameters C and d for the systems studied in this chapter, can be found in refs. [45,108]. Here, we used small basis sets consisting of 8s functions for He atom and 12s for Li, Be both. The exchange-only results are compared with those of HF – AP-based method and with other accurate calculations based on the work-function DFT formalism [46] (column WF-X-only in Table I). We then compare our OEP results for He atom at different levels of approximation (Tables II), with those obtained from the HF + second-order PT method (HF-PT2), and also with the exact energies. For this purpose, the basis set was extended to 12s6p functions with respect to Table I. As results of Table II show, the accuracy of our proposed scheme in obtaining excitedstate energies is practically the same as that for ground-state energies. Tables III and IV, where doubly excited-state energies for He (Table III) and corresponding excitation energies (Table IV) are given along with the SLHF DFT-based method, facilitate to gauge the performance of present COEP method for doubly excited states. Our total XC energies, as well as the double excitations are in good agreement with literature data (the one considered as the best theoretical result, taken from Table IV of ref. [46]). Basis sets of 8s Gaussians and 8s3p functions were used for He at the X-only and XC levels, respectively. Table I. Calculated singly excited-state energies (hartree) of atomic systems with various X-only methods System He

State 3

S 1s2s S 1s3s 3 S 1s4s 3 S 1s5s 2 S 1s23s 2 S 1s24s 2 S 1s25s 3 S 1s22s3s 3 S 1s22s4s 3

Li Be

a

−(COEPa) (X-only) 2.171 68 2.067 24 2.035 17 2.019 92 7.305 84 7.271 95 7.257 09 14.368 23 14.311 60

−(HF-APa) 2.173 73 2.067 93 2.035 44 2.019 96 7.309 13 7.273 69 7.258 90 14.372 84 14.319 15

−(WFb) (X-only) 2.174 20 2.067 93 2.036 06 2.022 42 7.309 66 7.274 66 7.259 96 14.377 98 14.325 06

Current results. HF-AP denotes HF calculations based on an AP method. Work-function method ( ref. [46]).

b

Table II. The ground- and excited-state energies as well as excitation energies, (hartrees) for Li atom using different levels of approximation State (2S) 1s 2s 1s23s 2

E (2s  3s )

HF-AP −7.4327 −7.3091 −0.1236

COEP −7.4318 −7.3058 −0.1260

HF-PT2 −7.4703 −7.3452 −0.1251

COEP-PT2 −7.4779 −7.3520 −0.1259

Eexact −7.478 −7.354 −0.124

E

94

V. N. Glushkov and X. Assfeld Table III. Total energies (hartree) for doubly excited states, ns2 (n=2–5) of He at different levels of approximation −COEPa

−(HF-APa)

1

S 2s2 1 S 3s2 1 S 4s2 1 S 5s2

0.763 44 0.337 45 0.188 52 0.121 09

0.763 44 0.337 47 0.188 52 0.121 10

1

−COEPa

−SLHFb

0.772 27 0.342 20 0.191 19 0.122 74

0.734 73 0.330 61 0.188 14 0.121 29

State

S 2s2 S 3s2 1 S 4s2 1 S 5s2 1

−SLHFb

−WFc

X-only 0.719 68 0.319 96 0.179 95 0.115 11

0.7197 0.3200 0.1800 0.1152

−WFc

−“Exact”d

0.766 37 0.345 78 0.196 59 0.127 54

0.777 87 0.353 54 0.200 99 0.130 30

XC

a

Current results. Spin-dependent localized HF method [111]. c Work-function method [46]. d Taken from ref. [46]. b

Table IV. Double excitations of He (in hartree). Numbers in the parentheses denote absolute percentage errors with respect to the best theoretical data (“Exact”) State 1 S 2s2 1 S 3s2 1 S 4s2 1 S 5s2

COEP (XC)a 2.131 57 (0.28) 2.561 64 (0.46) 2.712 65 (0.38) 2.781 10 (0.29)

SLHF (XC)b 2.169 11 (2.05) 2.573 23 (0.92) 2.715 70 (0.49) 2.782 55 (0.34)

“Exact”c 2.125 53 2.549 86 2.702 41 2.773 10

a

All excitation energies are calculated with respect to the ground-state energy of E = −2.903 84 hartree. b Spin-dependent localized HF method [111]. c Taken from ref. [46].

7.2. Potential Energy Curves for Singly and Doubly Excited States of Molecules In this subsection, at first, we test the performance of our formalism on diatomic molecules such as, HeH and LiHe at fixed nuclear separations, R = 1.5 bohr and R = 11.36 bohr respectively, which are close to the equilibrium distances in excited states. The different levels of approximations, such as the exchange-only and exchange+second-order correlation correction were used (see the COEP and COEP-PT2 notations respectively in Tables V and VI).

Constrained Optimized Effective Potential Approach for Excited States

95

Then, we also observe the performance of COEP method for description of the potential energy curves of singly (1 3g +, 2 3g +) and lowest

g

1

+

doubly excited state for H 2

molecule. The HeH and LiHe excited states are of considerable experimental interest and have been studied using accurate ab initio CI calculations (see, e.g., refs. [112,113]). Here, our attention is focused on excited states having same spatial and spin symmetry as ground state. We used basis sets consisting of 12s6p gaussians for HeH and 18s6p functions for LiHe molecule respectively [108]. We compare our results with the accurate energies computed by the configuration interaction methods using extended basis sets [112,113]. One can see that for the systems under consideration, our proposed technique provides comparable accuracy for energies of both ground and excited states at different levels of approximation (Table V). In addition, our COEP-PT2 results are in agreement with experimental values (excitation energies) and benchmark energies computed with ab initio wave function methods (Table VI). Slight difference in X-only results for HeH (Table V) with respect to that obtained in ref. [45] is explained on the ground of different basis sets: 18s functions in [45] and 12s6p in the present chapter, as also employed in [108]. Table VII presents the computed X-only COEP energies of H2 at selected internuclear separations of the 1 3g + and 2 3g + excited states. The basis set consists of 12s=6sx2 functions, i.e., 6s Gaussians per nucleus. Later, for the study of lowest g doubly excited state at the exchange-correlation level, this basis set was extended to 12s4p, i.e., 6s2p functions for each atom. We calculated singly excited-state energies at various values of R ranging from 0.8 to 10 bohr, and it is found that the energy minima are obtained for R=1.9 bohr (1 3g +) and 2.0 bohr (2 3g +), which are in reasonably good agreement with those of the “exact” CI calculations of Corongiu and Clementi [114]. A comparable accuracy of our calculations for different electronic states leads to fairly accurate results for vertical excitation energies. 1

+

Table V. Total energies (hartrees) of HeH and LiHe molecules using Different levels of approximation State

HF-AP

COEP

HF-PT2

COEPPT2

HeH, R=1.5 bohr A 2+ C 2+ D 2+

−3.0678 −3.0149 −2.9876

−3.0672 −3.0144 −2.9871

−3.1008 −3.0462 −3.0183

−3.1124 −3.0559 −3.0275

LiHe, R = 11.36 bohr X 2+ B 2 2  2

−10.2860 −10.2182 0.0678

−10.2835 −10.2134 0.0701

−10.3539 −10.2824 0.0715

−10.3674 −10.2998 0.0676

E ( X 

 B  )

CI [112] −3.1127 −3.0558 −3.0300 [113] −10.3804 −10.3122 0.0682

Indeed, at R=2.0 bohr, we have E (2 3g+) − E (1 3g+) = 2.04 eV, which is in very good agreement with Corongiu and Clementi data: E (2 3g+) − E (1 3g+) = 2.06 eV. Another quantity of interest is the deviation E  ЕCOEP – ЕCI (4th and 7th columns of Table VII) as a

96

V. N. Glushkov and X. Assfeld

function of R. One can see that E near the equilibrium distance depends weakly on R, i.e., our curves show an equi-distant behavior with respect to the accurate potential curves. On the other hand, certainly, we cannot expect a correct behavior of computed curves for large R using the X-only approximation. Table VI. Vertical excitation energies (eV) from A 2+ of HeH at R = 1.5 bohr State

COEP

COEP–PT2

CI [112]

Expt. [112]

A  C 2+ D 2+

0 1.44 2.18

0 1.54 2.31

0 1.53 2.26

0 1.55 2.26

2 +

Table VII. Energy values (hartree) at internuclear distances R (bohr) for triplet 1 and 2

g+ excited states of H2 molecule. Entries in the bold correspond to energies at

3

the respective minimum internuclear distances

1 3g+ R 0.8 1.0 1.2 1.4 1.7 1.9 2.0 2.1 2.2 2.4 2.7 3.0 3.6 4.4 5.2 5.6 6.2 7.0 8.0 9.0 10.0 *

g+

3

−COEP 0.45697 0.59907 0.67149 0.70815 0.72933 0.73159 0.73068 0.72859 0.72572 0.71913 0.70668 0.69312 0.66728 0.63954 0.62047 0.61389 0.60702 0.60213 0.60323 0.60234 0.60019

E  E COEP  E CI .

−CI[114] 0.46142 0.60391 0.67667 0.71363 0.73491 0.73708 0.73610 0.73415 0.73147 0.72461 0.71228 0.69926 0.67500 0.65071 0.63665 0.63273 0.62935 0.62734 0.62636 0.62596 0.62576

2 3g+

E 0.00445 0.00484 0.00518 0.00548 0.00558 0.00549 0.00542 0.00556 0.00575 0.00548 0.00560 0.00614 0.00772 0.01117 0.01618 0.01884 0.02233 0.02521 0.02313 0.02362 0.02557

−COEP 0.36502 0.51042 0.58590 0.62548 0.65063 0.65540 0.65573 0.65507 0.65206 0.64882 0.64036 0.61381 0.59765 0.57562 0.56008 0.55444 0.54811 0.54263 0.57610 0.57188 0.56861

−CI [114] 0.36843 0.51444 0.59042 0.63030 0.65549 0.66010 0.66056 0.66024 0.65921 0.65556 0.64790 0.63945 0.62459 0.61393 0.61219 0.61303 0.61513 0.61827 0.62136 0.62326 0.62426

E 0.00341 0.00402 0.00452 0.00482 0.00486 0.00470 0.00483 0.00517 0.00715 0.00674 0.00754 0.02564 0.02694 0.03831 0.05211 0.05859 0.06702 0.07564 0.04526 0.05138 0.05565

Constrained Optimized Effective Potential Approach for Excited States

97

Table VIII. Comparison of doubly excited potential (calculated at the XC level) energy curves (hartree) of H2 ( 1 u2

1



 g

)

R (bohr)

HF-PT2

COEP+PT2

Ref.[115]

Ref.[116]

1.4 2.0 2.5 3.0 4.0

−0.099 77 −0.393 80 −0.540 06 −0.632 90 −0.730 06

−0.106 05 −0.402 97 −0.549 20 −0.641 63 −0.738 20

−0.1046 −0.3975 −0.5404 −0.6308 −0.7251

−0.101 62 −0.396 88 −0.540 35 −0.630 74 −0.725 69 21

Finally, in Table VIII, we show total energies for the lowest 1 u

 g doubly excited

state of H2, calculated for some points of the potential energy curve. Second and third columns of the table signify energies calculated using a second-order PT based on the HF excited determinant function (HF-PT2 column), and the Slater determinantal function constructed from orbitals according to the prescription in Eqs. (83) and (84) (COEP-PT2 column). One can see that our results are close to the values obtained from the CI method, as employed in refs. [115] and [116], although for some of the data in Table VIII, these are slightly more negative than those of the CI method. This situation is often encountered, because the PT approach is not variational. On the other hand, a comparison of COEP-PT2 and HF-PT2 energies shows that the COEP method tends to overestimate correlation energy compared to the HF-PT2 scheme. An analysis of the orbital energies

k

shows that the

lowest COEP virtual orbitals are lower in energy than those of the HF method. Therefore, the COEP implementation with the effective potential in Eq. (85) leads to smaller energy denominators in the perturbation series than HF one. This fact explains the tendency of overestimating correlation energy in the COEP method. It is worth noting that for ground state, the similar differences between PT based on OEP and HF orbitals has been observed by a number of authors (see, e.g., refs. [1,105,107]).

Conclusion In this chapter, we have reviewed the constrained variational approach based on an asymptotic projection method and demonstrated its applications to the optimized effective potential problem for excited states. Existing variational and non-variational approaches to DFT excited-state calculations have also been briefly discussed. In contrast to the known DFT-OEP-based methodologies for excited states, the constrained OEP method explicitly introduces orthogonality of the KS excited determinantal function to the lower states of same symmetry. Probably, a concept of explicit orthogonality of states does not sound comfortable from the view point of traditional DFT, as it is well known that this theory originally operates with electronic density, and not with the state functions playing an auxiliary role. Nevertheless, this concept as well as the time-independent bifunctional approach is able to generate reasonably good excited-state densities. In our implementation, a bifunctional

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character of the excited-state calculations implies dependence of excited-state energies not only on the orbital basis set and OEP employed directly for the given excited state, but also on the ground-state KS determinantal function. The COEP method works along the lines of time-independent DFT for a “pure” excited state and makes it possible to solve the KS-type equations. On one hand, it helps one to avoid the “collapsing” and, on the other hand, allows one to handle any appropriate local potential for excited-state DFT calculations within the framework of both variational and non-variational approaches. Some specific features of SCF excited-state calculations that implement the asymptotic projection methodology have been discussed. In particular, we have shown the similarity of HF and KS-type equations generating singly and doubly excited-state orbitals. In addition, the only additional computation required, beyond that arising in the standard ground-state SCF schemes, is the evaluation of overlap matrix elements. The parametric form of OEP, expressed as a direct mapping of external potential has been considered and corresponding equations determining an exchange-only OEP for excited states have been obtained. It has been shown that they have a structure similar to the groundstate OEP equations. We have presented one of the possibilities to construct orbital-dependent correlation functionals based on the many-body perturbation theory for excited states. Computational cost of the present PT for excited states is similar to a genuine Møller-Plesset PT. Effectiveness of the COEP method and its performance has been demonstrated by excited- state calculations of simple atoms (singly, doubly excited-state energies) and molecules (potential excited-state energy curves and vertical excitation energies) at different levels of approximation including both X-only results and X-only+correlation studies. Practical calculations have shown that a simple-to-implement COEP methodology is capable of delivering fairly accurate results on the energies of singly and doubly excited states and can be easily applied to both atoms and molecules. Unlike the major traditional methods, where an improvement in accuracy is achieved by using extensive ways and means (for instance, the considerable extension of atom-centered one-particle basis sets), here, we have focused on the development of alternative approaches capable of optimizing relatively small distributed basis sets.

Acknowledgment One of us (VNG) wishes to thank Professor A. Theophilou, Professor M.Levy and Professor N. Gidopoulos for stimulating discussions at different stages of this work and Professor A. K. Roy for inviting us to contribute a chapter in this book.

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PACS: 31.15.-p, 31.15.E-

In: Theoretical and Computational Developments… Editor: Amlan K. Roy, pp. 103-147

ISBN: 978-1-61942-779-2 © 2012 Nova Science Publishers, Inc.

Chapter 4

TIME DEPENDENT DENSITY FUNCTIONAL THEORY OF CORE ELECTRON EXCITATIONS: FROM IMPLEMENTATION TO APPLICATIONS Mauro Stener*, Giovanna Fronzoni and Renato De Francesco Dipartimento di Scienze Chimiche e Farmaceutiche, Università di Trieste Via L. Giorgieri 1, I-34127 Trieste - Italy

Abstract The time-dependent density functional theory (TDDFT) method to treat coreelectron excitations is outlined, according to its implementation in the ADF code. The most relevant applications of the method for a description of X-ray absorption at the L edges (metal 2p core orbitals) have been reviewed. A study of small molecules in the gas phase shows high accuracy of the method. It is able to describe correctly fine spectral features taking into account the crystal field effect, configuration mixing and spin-orbit coupling. For extended systems (solids and surfaces), the method has been applied by taking into consideration suitable cluster models, accurately chosen in order to simulate the electronic structure of the condensed phase at a very high level of accuracy.

PACS: 31.15.E-, 71.15.Mb, 78.70.Dm. Keywords: TDDFT, Core-electron excitations, NEXAFS.

1. INTRODUCTION The validity of Density Functional Theory (DFT) as well as its formulation based on the Kohn-Sham (KS) approach [1], is limited to the electronic ground state. Although this is not *

E-mail address: [email protected]

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an important obstacle for many studies, it hampers DFT applications to electron spectroscopy; a typical field where excited states are heavily involved. However, it has been recognized for many years that Time Dependent (TD) DFT is a rigorous extension of the formalism, which can be applied to excited electronic states. Early applications of TDDFT to optical absorption of atoms and molecules were reported in the literature [2,3]; a general demonstration of the basic theorem has been reported successively [4]. Many molecular quantum chemical implementations of TDDFT are now available [5-9] especially within the standard basis sets and taking advantages of the electron density fitting procedure. This makes DFT implementations computationally economic because calculation of the difficult bi-electronic integrals is avoided. Standard TDDFT basis-set implementations allow the calculation of outer valence excitation spectra in terms of excitation energies and oscillator strengths; however, in principle, cannot deal with the core-electron excitations lying above the first ionization potential. The latter, however, are only weakly coupled to the corresponding background valence continuum, which is responsible for the Auger decay and lifetime width of the core hole. For this reason, it is reasonable to explicitly suppress, in the calculation, the excited configurations, where valence electrons are promoted to discretized continuum states, keeping only those departing from the core shell under study. Within this procedure, validity of the finite basis set is extended up to the ionization limit of the current core shell. Therefore, the complete core excitation spectrum can be properly described. Moreover, the suppression of all valence excitations makes the calculation even more economic than in the valence shell, where all excitations must be included. Consequently, the dimensionality of the matrices involved is drastically reduced. Finally, it is worth noting that, with the implementation described in this chapter, the core excited energies are still extracted from the lowest eigenvalues of a matrix, and such a task is efficiently pursued by an iterative algorithm such as the Davidson diagonalization. The approach considered in this chapter has been implemented in the ADF code [10-12] and has been employed to calculate core excitations for various molecules and extended systems modelled by finite-size clusters. Core electron excitations occur when X-ray radiation is absorbed by matter, in particular, the X-Ray absorption structures near the ionization threshold (NEXAFS region) are closely related to the local electronic structure around an atom in which the excitation takes place. They contain the most significant information on low-lying unoccupied states of the system. The spectral structures are associated with transitions from a core orbital, strongly localized on a particular atom to the unoccupied electronic states, which lie near the ionization limit of this specific core state. Theoretical calculations relate each excitation energy to a virtual state whose nature (valence or Rydberg) can be defined. It also yields oscillator strengths which are connected with the composition of unoccupied orbitals; mapping in particular, the dipole-allowed atomic site component of the virtual states. The fine structure around the K absorption edge has been extensively studied for structural determination, presumably due to a simple interpretation of the spectra, which are governed by sp dipole transitions. The L-edge structures, in particular, of transition metals are potentially richer, since they are dominated by the pd dipole transitions which directly probe the metal d content of the different unoccupied levels, and therefore details of the d involvement into bonding.

Time Dependent Density Functional Theory of Core Electron Excitations

105

From a theoretical point of view, it has been demonstrated [12,13] that, a proper description of the L edge requires inclusion of configuration mixing in the calculations. This is brought into the picture by considering degeneracy of core hole, like 2p orbitals, as well as of the final atomic nd contributions. In this respect, the TDDFT approach represents an essential improvement with respect to the single-particle approaches, like the KS method, which encounters serious difficulties when applied to L2,3 or similar edges. Moreover the spin-orbit (SO) splitting of core orbitals leads to the existence of SO partner states, which give rise to distinct structures converging to different ionization thresholds. In particular, if a 2p core electron is excited, the XAS spectra shows two closely related manifolds of excited states; viz., those associated with the 2p3/2 (L3) and those associated with the 2p1/2 (L2) spin-orbit coupled ion core respectively. The SO coupling redistributes the spectral intensity over the two series of transitions; therefore its inclusion is of fundamental importance for a correct assignment of the spectral features, especially when the structures deriving from two SO components tend to overlap [14]. Therefore the availability of a general method which includes both configuration mixing as well as SO coupling is widely considered of fundamental importance for a quantitative description of XAS spectra of transition-metal compounds. In this respect, the formalism based on the relativistic two-component zeroth-order regular approximation (ZORA) along with TDDFT and its implementation in the ADF code [15], has proven to satisfy the desired requirements with competitive computational efficiency. An extension of the original nonrelativistic TDDFT computational scheme to Relativistic two component ZORA TDDFT version has, therefore, offered a decisive further improvement for the description of XAS core spectra [16], in particular, in case of the L2,3 edges of transition-metal compounds. For 3d systems, the L2 and L3 edges are in fact quite different; their branching ratios are far from the statistical weight expected within a single-particle model. The origin for these large differences is principally rooted in the coupling of 2p core wave function and the 3d valence wave function. These effects (so called “multiplet effects”) have same order of magnitude of the 2p SO coupling and they mix and reorder the L2,3 edges. The 3d SO coupling is, instead, a second origin of difference which remains much weaker [17]. Apart from the present extensions of TDDFT method to core excitations, other approaches are available to study the X-ray absorption in molecules and solids, in particular, the real-space-multiple-scattering code FEFF [18] and the pseudo-potential band structure WIEN2K code [19]. In a single-particle model for excitations, all other electrons do not participate in the spectroscopic transition. However, ignoring many-body effects other than the relaxation effect does not satisfactorily reproduce the experimental behavior of L3 and L2 edge spectra of 3d metals [20]. Fairly good agreement with experiment can be achieved by taking into account the 2p-3d two-particle interactions within a model, based on atomic multiplet theory [21]. Multiplet theory, including charge transfer and covalency effects, has become a de facto standard for the interpretation of L3 and L2 edge spectra in solid state [2123]. In this approach, however, the treatment of orbital mixing due to molecular bond formation is simplified; furthermore a number of adjustable parameters are used. Approaches that go beyond the ground state and are significantly more time consuming are based on TDDFT or on the Bethe Salpeter equations [24]. It is worth mentioning here the molecular TDDFT implementations of the Q-Chem program [25], that takes advantages of the TammDancoff Approximation (TDA). In this context, efforts to design new exchange-correlation functionals suitable for core excitations [26] have proven very promising, especially to obtain

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very accurate absolute excitation energies, which is one of the problems of TDDFT approach. A somewhat similar approach to that developed by Zangwill and Soven for the simulation of optical absorption of atoms and molecules [2,3], has been recently presented [27], with an aim to describe the X-ray absorption in solids. In addition, this scheme has been formulated in a fully relativistic way in order to describe the SO interaction. Its application to the L2,3 absorption spectra of 3d transition metals demonstrates the importance of core-hole interaction on the L3/L2 branching ratio. In summary, the ADF implementation of TDDFT method for core-electron excitations is particularly suitable and competitive for the description of transition-metal 2p excitations and, therefore, we will focus in the following, mainly on applications to L3 and L2 edges.

2. THEORETICAL METHOD The TDDFT approach for valence electron excitations and its implementation in the ADF code has been described in the literature [5,6]; so here we just recall the salient features briefly. We will describe in detail the new features regarding the extension of this method to core-electron excitations. The general problem is cast in the following eigenvalue equation:

Fi  i2Fi ,

(1)

where  is a four-index matrix with elements  ia , jb ; the indices consist of products of occupied-virtual (ia and jb) KS orbitals, while  and  refer to the spin variable. The eigenvalues

 i2 correspond to squared excitation energies while the oscillator strengths are

extracted from the eigenvectors Fi [5]. The -matrix elements can be expressed in terms of KS eigenvalues () and the coupling matrix K:

ia , jb    ij ab   a   i   2 ( a   i ) K ia , jb ( b   j ). 2

(2)

The elements of the coupling matrix K are given by:

 1  K ia , jb   dr  dr 'i  r  a  r    f xc  r, r ',     j  r '  b  r '  ,  r r' 

(3)

where ’s arethe KS orbitals and f xc r,r' ,   is the exchange-correlation kernel. In this approach, the kernel is approximated according to the Adiabatic Local Density Approximation (ALDA) [28]. The space spanned by the solutions of this eigenvalue equation (1) corresponds to the 1h1p excited configurations. So it is possible to approximate this space operating on a selection over the configurations, keeping only those necessary for an accurate description of the phenomenon, as it is customary in ab-initio CI calculations. Thus, in practice, the indices which span the occupied orbitals space (i and j) are limited to run only over the core shell. This selection of active core orbitals can be profitably described following Scheme 1. As a result of this selection, the complete -matrix is reduced to a submatrix which is much smaller than the original one. At this point, two important advantages are observed; (i) the core-electron excitations now correspond to the lowest eigenvalues of a reduced matrix (ii)

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the dimension of this reduced matrix is much smaller than the original one, which makes core-electron excitation calculations much cheaper than the valence excitation calculation on the same molecule.

Scheme 1. Reduction of Ω matrix after selection of only core orbitals in the active space.

For comparison, also KS results may be considered in order to assess the difference with respect to TDDFT, which may throw some light on the importance of response effects. In a KS scheme, the oscillator strength is calculated directly as dipole transition moments between one-electron KS eigenfunctions, and excitation energies as eigenvalue differences. The core-electron TDDFT method has been further extended in order to include SO coupling [16] through a combination of relativistic two-component ZORA and a modified form of TDDFT formalism [15]. In this section, at first, we briefly summarize the essential ideas involved in this method. These are extensively described in refs. [15,16] and further details could be found there. Then we outline the modifications which have been implemented in order to apply the formalism to core excitations. The relativistic twocomponent ZORA TDDFT formalism exploits the full use of symmetry and is designed for the calculation of valence excitation energies as well as intensities for closed-shell molecules including SO coupling. The equation which is actually solved is the same eigenvalue equation (1) as in the non-relativistic formalism, but now the -matrix is defined according to the following expression:

ia , jb    ij ab   a   i   2 ( a   i ) 2

Fia ( b   j ). Pjb

(4)

In Eq. (4), indices i,j indicate occupied spinors and a,b denote virtual spinors; i,a are molecular spinor energies, while F,P are the Fock matrix and density matrix respectively. The explicit expression for the coupling matrix

Fia is given in ref. [16] for closed-shell Pjb

systems. It assumes an ALDA approximation for exchange correlation kernel and a noncollinear scheme for the exchange correlation (XC) potential [29]. In keeping with the demonstrated possibility, it neglects molecular contributions to the KS potential in ZORA kinetic energy beyond a sum-of-atomic-potentials-approximation [30]. It also neglects response contributions to the potential in this term. The use of non collinear XC potential in TDDFT relativistic calculations on closed-shell molecules allows one to treat SO coupling and ensures correct non-relativistic limit [15]. This completes the definition of the elements of  matrix in expression (4), which is diagonalized with Davidson's iterative algorithm. The implementation takes advantages of auxiliary basis functions, which are used to fit the induced density and to calculate efficiently the product between  matrix and a trial vector [6]. Another fundamental aspect of the implementation lies in its full use of molecular

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symmetry, which produces important computational economy if the molecule is highly symmetric. All the details of relativistic double-group symmetry implementation with various advantages, such as block diagonal  matrix and reduction of numerical integrals to symmetry-unique part of the integration grid, by virtue of the Wigner-Eckart theorem, are addressed in ref. [15]. The modifications needed to extend the relativistic two-component ZORA-TDDFT method to treat core-electron excitations remains essentially the same as needed in a nonrelativistic case. The only difference, however, is that, now the selection of active space is done on core spinors rather than on molecular orbitals. So, Scheme 1 remains valid with the understanding that i, j and a, b indices run on occupied and virtual spinors respectively. Scaled ZORA spinor eigenvalues are used in the calculation of core excitation energies. The spectra calculated at the TDDFT level are always obtained as a discrete set of excitation energies and oscillator strengths. However, for an easier comparison with experimental data, it is convenient to convolute the discrete lines by a Gaussian function with suitable Full Width at Half Maximum (FWHM). For this reason, in the following, convoluted profiles are reported in figures, with the FWHM being chosen according to the resolution of the experimental spectra.

3. 2P CORE EXCITATIONS OF GAS PHASE MOLECULES The first application of the TDDFT method for core excitations of atomic 2p orbitals has been dedicated to gas phase molecules for many reasons. First, very accurate experimental results are available for gas phase molecules, which make them excellent candidates to validate the method and to assess the performance in terms of accuracy. Second, the present method is naturally suited to describe molecules, since in this case we do not need to introduce models to mimic the systems, like for bulk or surfaces. Third, gas phase molecules are small and symmetric systems; so these are not so much demanding in terms of computer time.

3.1. TiCl4 TiCl4 is a typical system suitable for the study of metal 2p excitations, due to its high symmetry coupled with the availability of accurate experimental data. And also previous abinitio calculations are available, which could be useful for comparison. The most intense excitations from the Ti 2p shell are expected to be towards valence virtual orbitals with mainly metal 3d character, which are split into 3e and 10t2 manifolds due to the crystal field. In Table 1, the excitation energies and oscillator strengths are reported for such transitions, calculated at both KS and TDDFT non-relativistic level, as well as with different potentials and with the QZ3P-3DIF basis set as included in ADF database [12]. Irrespective of the choice of potential, a very large difference is found between the KS and TDDFT approaches. For instance, at the LB94-GS level, the KS scheme gives an energy splitting of 0.91 eV between the 3e and 10t2 states, and a comparable intensity (f = 33.09 x 10-2 and f = 45.14 x 10-2 respectively). On the other hand, TDDFT calculations give a doubled energy splitting of 2.11 eV and a dramatic intensity redistribution; the 10t2 transition

Time Dependent Density Functional Theory of Core Electron Excitations

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being 19.5 times more intense than 3e. This result is in excellent agreement with the experiment, as can been seen from the lower panel of Figure 1, where the measured absorption profile is reported in the energy region of Ti L2 and L3 edges. The two peaks around 457 and 459 eV correspond to the excitations related to the L3 edge, and correctly display the stronger intensity of second peak with respect to the first one. The only discrepancy is an overestimate of the calculated LB94-GS term value with respect to experiment, probably associated with the too much attractive character of LB94 potential. Table 1. Excitation Energies (eV) and oscillator strengths f for Ti 2p valence transitions in TiCl4, using various DFT schemes (QZ3P-3DIF basis set) 3e

10t2

E

f x 10

2

E

f x 102

SAOP - GS

KS TDDFT

439.24 439.60

33.03 1.553

439.89 441.35

40.81 41.46

LB94 - GS

KS TDDFT

453.41 453.88

33.09 2.343

454.32 455.99

45.14 45.69

VWN - TS

KS TDDFT

456.62 457.09

33.95 2.093

457.49 459.24

46.83 47.64

Therefore, we can state that the TDDFT scheme represents a definite improvement with respect to the KS scheme, the latter being inadequate to describe even the qualitative appearance of the spectrum. It is interesting to identify the actual reason for such an improvement. From a careful inspection of the eigenvectors of Eq. (1), it is found that the TDDFT excitations are actually described by strongly mixed configurations. The first excitation is characterized by a combination of 77% 2t2  3e and 22% 2t2  10t2 excited configurations, while the second transition, by 76% 2t2  10t2 and 21% 2t2  3e. In the KS scheme, such a mixing is not allowed and hence the results are deteriorated. The TDDFT results reflect similar findings of previous ab-initio Configuration Interaction (CI) calculations [13], where the CI 1h-1p results parallel the present TDDFT one, while the separate single channel 2p  10t2 and 2p  3e CI parallel the present KS. When comparing with the experiment, we have restricted our discussion to the Ti L3 edge, since we start with a non-relativistic analysis. We will consider the relativistic effects later. It is worth noting that the other schemes considered in Table 1 (SAOP-GS and VWNTS) give very similar results to that of LB94-GS, as far as it concerns the energy splitting and intensity distributions; for both KS and TDDFT. Therefore it appears that, the inclusion of response effects via TDDFT formalism is much more important than the choice of exchange correlation potential. On the other hand, absolute energies are quite sensitive to the potential. And VWN-TS gives the best results (see also Figure 1).

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Figure 1. TDDFT QZ3P-3DIF Ti 2p excitation spectra of TiCl4 calculated with various exchange correlation potentials. Convoluted profiles are obtained with a Gaussian broadening of 0.6 eV of Full Width Half Maximum (FWHM). Lower panel: experimental data from ref. [12]. The vertical dashed line shows the calculated ionization limit.

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Effect of the choice of the potential is considered in Figure 1, where the TDDFT QZ3P3DIF basis-set results are reported for various potentials, together with a recent experimental measurement [12]. Apart from the large shift in absolute scale, there is no apparent effect in the valence region, as already observed in the discussion of Table 1. However, important differences are recorded in the high energy part of the spectrum; the SAOP-GS and LB94-GS give rise to a weak but non-negligible Rydberg structure just below the calculated ionization limit (such limit is taken as opposite of the Ti 2p eigenvalue), while the VWN-TS result places such a Rydberg structure above the ionization limit. This effect could be ascribed to the correct asymptotic behavior of SAOP and LB94 exchange correlation potentials, which support Rydberg states due to the presence of Coulomb tail. The VWN potential does not behave correctly in the asymptotic region, and hence the Rydberg states are pushed above the threshold. Now let us consider the inclusion of relativistic effects, namely the SO coupling which splits the Ti 2p (2t2) initial level into 2p3/2 and 2p1/2 spinors. The effect of such coupling on Ti 2p core-excited spectrum is clearly apparent, as a major splitting of the spectrum features into two main bands corresponding to the L3 (2p3/2) and L2 (2p1/2) components (see the experimental spectrum reported in Figure 2). Each L3 and L2 band exhibits a similar but nonidentical fine structure with two major components having additional weaker structures. The two components derive from the splitting of “Ti 3d” Molecular Orbitals (MOs) dictated by the Td symmetry of the molecule, and are associated with the 2t23e and 2t210t2 transitions. A reliable and complete theoretical description of the Ti 2p spectrum requires both inclusion of configuration mixing among different excitation channels due to the degeneracy of 2p core hole, and an explicit treatment of the SO effect. The configuration mixing has already been pointed out before at the non-relativistic level (Table 1) [12]. Now, as concerns the relativistic effects, two different phenomena are expected to influence the calculated excitation energies and intensity distribution of core excited spectrum; the scalar relativistic corrections and SO coupling. In order to analyze the first effect in our calculated Ti 2p spectrum, we compare in Table 2 the scalar-ZORA TDDFT results [16] with the Non Relativistic (NR) TDDFT results [12], obtained using the same basis set (QZ3P-3DIF). It has been verified that this large basis set and the ZORA QZ4P basis set [31] give results very close to each other for the Ti 2p core excitation energies at the two-component TDDFT level. As one can see in Table 2, the effect of scalar relativistic correction is to increase the excitation energies by about 0.8 eV with respect to NR-TDDFT results, bringing them in a better accord with the experimental values (reported in Table 3). The fact that the excitation energies in both calculations are underestimated, is generally ascribed to the deficiencies in XC potential. We underline that a substantial agreement between the NR and scalar-ZORA TDDFT results is obtained considering the relative energy shifts between calculated transitions, instead of the absolute energy values, which actually represent the most significant data in comparing the calculated and experimental spectra. Also, the oscillator strength values appear to be very close in the two theoretical approaches. Therefore we can state that the substantial improvement on the Ti 2p spectral features is not ascribable to the inclusion of scalar relativistic correction in the TDDFT approach. When comparing the scalar-ZORA TDDFT results with experimental L3 edge features (see Figure 2), a correct reproduction of the first two peaks (around 457 and 459 eV) is easily seen, in

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particular, as regards the energy separation. Also, the intensity ratio is consistent with the experimental pattern, although the calculated 2t210t2 main transition appears overestimated. The latter is strongly associated with the neglect of SO effects in the calculation, which is expected to redistribute intensity among the two L3 and L2 manifolds of excited states.

Figure 2. TDDFT LB94-GS Ti 2p excitation spectra of TiCl4 from two-component ZORA TDDFT and scalar-ZORA TDDFT calculations. Convoluted profiles are obtained with a Gaussian broadening of 0.6 eV of Full Width Half Maximum (FWHM). Upper panel: experimental spectrum from ref. [12]. The vertical dashed lines show the calculated ionization limits.

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Table 2. Excitation energies (eV) and oscillator strengths f for Ti 2p excitation spectrum of TiCl4 from non-relativistic and scalar-ZORA TDDFT calculations (ET-QZ3P-3dif basis set) Non-relativistic resultsa E(eV) 2p (t2)

453.88 455.99 460.03 460.97 461.49 462.12 462.34 463.06 463.26

2b E (eV) f x 10

(0.000) 2.11 6.15 7.09 7.61 8.24 8.46 9.17 9.38

2.343 45.69 0.558 0.847 1.354 0.900 3.183 3.402 0.559

Scalar relativistic resultsa E(eV) 454.77 456.85 460.87 461.82 462.35 462.90 463.20 463.92 464.12

E (eV) (0.00) 2.08 6.10 7.05 7.58 8.13 8.43 9.15 9.35

f x 102 b 2.336 45.07 0.561 0.840 1.361 0.804 3.247 3.247 0.569

Final state 3e+10t2 10t2+3e 11t2 12t2 13t2 11a1 14t2 5e 15t2

a

Calculated DFT-KS 2p ionization limits are 463.53 eV and 464.38 eV for non-relativistic and scalar relativistic calculations, respectively. b Only below-edge calculated transitions with f × 102 > 0.100 are reported.

The effect of SO coupling on the calculated Ti 2p spectrum is included at the twocomponent ZORA TDDFT level of calculation [16]. The results are obtained employing an ET-QZ4P-2diff basis set and LB94-GS potential, which has proven to generate the best reproduction of experimental spectrum. The excitation energies and oscillator strengths are reported in Table 3 and Figure 2. These are compared with scalar-ZORA results as well as the experimental data [32,12]. Each excited state from the two-component calculation is characterized according to its double-group symmetry. As regards the comparison with experimental spectrum in Figure 2, the agreement is fairly nice. The main structures of L 3 (the two lines below 455 eV, in the calculated Excitation Energy scale) and L2 (the two lines below 460 eV) edges, associated with the 2p3d transitions, are correctly reproduced, both in terms of energy separation and intensity distribution. An assessment of the general quality of two-component results can be made comparing the following data: SO splitting is 6.1 eV and 5.9 eV in experiment and calculation respectively, while separations of the most intense L3 and L2 features are 5.6 eV and 5.4 eV respectively. Also, the energy separation of two components of L3 and L2 bands (1.52 eV and 1.20 eV respectively), deriving from the “crystal field splitting”, are in good accord with the experimental values. They however, appear to be slightly underestimated with an opposite trend with respect to the scalar-ZORA results, which overestimate the splitting of L3 main structure. Apart from some minor details, inclusion of SO effect in the calculation brings about a decisive improvement in the description of spectrum with respect to the scalar-ZORA results, as can be seen from Figure 2 clearly. The inclusion of L2 manifold of excited states not only allows to obtain a complete representation of all the observed spectral features but also gives rise to a redistribution of intensity among excited states relative to the L3 edge. In particular, we observe a strong reduction of intensity ratio of the two L3 components from 19.5 to 9.06, which is somehow in better agreement with

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Table 3. Excitation energies (eV) and oscillator strengths f for Ti 2p excitation spectrum of TiCl4 from two-component ZORA TDDFT and scalar-ZORA TDDFT calculations (ET-QZ4P-2diff basis set) Initial Edge a E(eV) E (eV) f x 102 b Final state state Two-components resultsd 2p3/2 (2u3/2) L3 452.73 (0.00) 2.439 14u3/2(83%)+15u3/2(14%) 454.25 1.52 22.12 (15u3/2+10e5/2)(83%)+14u3/2 (13%) 460.05 7.32 3.234 12e5/2+17u3/2 460.47 7.74 0.315 12e1/2 460.49 7.76 1.875 13e5/2+19u3/2 461.04 8.31 0.405 13e1/2 461.30 8.57 3.705 14e5/2+20u3/2 461.94 9.21 3.555 22u3/2 462.33 9.60 0.675 23u3/2+15e5/2 462.46 9.73 0.369 16e5/2+24u3/2 2p1/2 (2e5/2) L2 458.45 5.72 3.618 14u3/2(88%)+15u3/2(10%) 459.62 6.89 12.02 15u3/2(74%)+14u3/2(7%) 464.90 12.17 0.243 16u3/2 465.85 13.12 0.465 17u3/2 466.30 13.57 0.516 19u3/2 467.06 14.33 1.089 20u3/2 467.71 14.98 1.092 22u3/2 Scalar results 2p (2t2) L3

a

454.85 456.93 461.00 461.96 462.40 462.42 462.99 463.26 463.90 464.27 464.39

(0.00) 2.08 6.15 7.11 7.55 7.57 8.14 8.41 9.05 9.42 9.54

2.345 44.96 0.598 0.972 0.357 0.903 0.763 3.159 3.534 0.765 0.463

EXPc

(456.9) 1.80

5.60 7.40

3e + 10t2 10t2 + 3e 11t2 12t2 10a1 13t2 11a1 14t2 5e 15t2 16t2

Calculated two-components DFT-KS 2p ionization limits are 462.53 eV (L3 edge) and 468.38 eV (L2 edge) respectively. Calculated scalar DFT-KS 2p (2t2) ionization limit is 464.46 eV. b Only below-edge calculated transitions with f × 102 > 0.100 are reported. c Ref. [32]. d The initial and final states from the two-component calculations are characterized according to their double-group symmetry.

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the experimental trend. A more detailed comparison would require careful deconvolution and band area integration, which, at present, are not available even for the most recent Synchrotron Radiation measurements [12]. The two-component TDDFT excitations towards the virtual valence states (14u3/2, 15u3/2 and 10e5/2 in the double-group representation) are described by strongly mixed configurations for both L3 and L2 edges (see Table 3). The latter is apparently responsible for the puzzling intensity distribution between the two 2p3d channels (t2+e in the single-group representation), as found also in the scalar TDDFT calculation; although referred to the only L3 edge. This effect cannot be described with a pure one-electron model, such as KS scheme, as previously commented. There is no indication in the calculated spectra (both at two-component and scalar ZORA level) of a structure at about 460.2 eV, as experimentally suggested [32]. Since next virtual levels are Rydberg types, close to threshold, the identification of this feature could be doubtful. A further improvement of two-component results over the scalar-ZORA calculation concerns the description of Rydberg part of the spectrum. This region is quite complex due to the overlap between Rydberg transitions converging to L3 threshold and valence excitations of L2 edge. The L3 Rydberg transitions fall at higher energy than the main L2 components (at 458.45 eV and 459.62 eV) and their intensities are increased with respect to the scalar-ZORA results. This allows one to describe correctly the two peaks experimentally observed at high energy side of the L2 main structure (see Figure 2). The last calculated feature, at around 466 eV, is caused by the Rydberg transitions converging to L2 edge. A last comment concerns the effect of potential employed, which is considered in Figure 3. The two-component TDDFT results obtained with the ET-QZ4P-2diff basis set are reported [16], together with the experimental spectrum. Apart from a large shift in the absolute scale, as such, there is no significant difference in the valence region, although the splitting of two L3 valence components from SAOP is smaller than that found in the LB94 case, worsening the comparison with experiment. This recurs also in the two L2 valence components which appear closer in energy and less intense in the SAOP calculation. More significant differences are apparent in the high-energy part of the spectrum, in particular as regards the intensity distribution among Rydberg states of L3 series. The intensities generated by the SAOP potential are higher than those based on the LB94 potential, and this appears less consistent with the experimental trend. The results with these two potentials agree well in the region of Rydberg states converging to the L2 threshold. Although a quantitative comparison with experimental intensity is not possible at this moment, our investigation finds LB94 to be more satisfactory in agreement with the experimental spectrum compared to the SAOP result, both in terms of energy separation and intensity distribution.

3.2. SO2 The core excitations in SO2 provide a good test for the theory because the inner-shell spectra relative to the three core edges S 1s, S 2p and O 1s have been the subject of numerous experimental investigations, some of them at high level of resolution [33-39]. In particular, for the S 2p threshold, the experiment [38] has allowed to resolve a variety of Rydberg states in a region, where the SO splitting as well as ligand-field splitting are responsible for a high level of complexity in the attribution of excited states. Relatively little work has been done from a theoretical point of view, and only a few calculations existed [36,39] before the recent investigation based on TDDFT [40]. A direct comparison of relative energies and intensities of the transitions obtained at the same level of accuracy for different core edges appears

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instructive for a better characterization of the core-excited states. These latter are expected to be present in all the spectra with different oscillator strengths according to the dipole selection rules and symmetry properties. The SO2 molecule (C2V symmetry) has three unoccupied valence orbitals: 3b1, 9a1 and 6b2. According to the molecular electron configuration in ground state, the LUMO (3b1) is a * antibonding orbital while the next two orbitals 9a1 and 6b2 represent * type antibonding MOs. These three MOs are dominated by the S 3p component; a significant O 2p contribution is present in the LUMO as a result of the S-O covalent interaction, which decreases in the two following * orbitals while the other sulfur atomic components (in particular 3d) increase. The remaining virtual orbitals obtained in the extended basis set correspond to Rydberg orbitals mostly centered on the central S atom. Within this MO scheme, spectral features below the core ionization threshold in the XAS spectra can be interpreted in terms of transitions from the particular core level to the three virtual valence orbitals. In the higher energy region of the spectra, less intense structures are present which involve transitions towards Rydberg orbitals. The discrete calculated structures of the S 2p spectrum are discussed below. The two-component ZORA TDDFT S 2p core excitation spectrum is reported in Figure 4, along with a comparison with experimental data [38]. Figure 4 also shows the S 2p spectrum obtained at a scalar-ZORA TDDFT level, in order to analyze the effect of SO coupling on the calculated structures. The apparent complexity of the calculated spectrum is evident from this figure, where a large number of lines are visible. The main factors which contribute to the shape and structure of the spectrum are SO splitting and molecular-field splitting. In the experimental spectrum, vibrational excitations also contribute, which are however not included in the present calculation. The SO interaction splits the S 2p core hole in two levels, well described as 2p3/2 and 2p1/2. Furthermore, the molecular field splits the degenerate 2p orbitals into a1, b1 and b2 orbitals in C2v symmetry, removing the degeneracy of 2p3/2 level. It is to be noted that in the relativistic SO description, the 2p orbitals correspond to the 7e1/2, 6e1/2 (LIII components split by the crystal field) and 5e1/2 (LII component) spinors. The excited states therefore converge towards three ionization thresholds, namely, two S 2p3/2 thresholds (7e1/2 and 6e1/2, LIII components) and one S 2p1/2 threshold (5e1/2, LII component). The intensity distribution among the final excited states is also modulated by configuration mixing included in the TDDFT scheme. An assessment of the general quality of our present two-component results can be carried out comparing the following data: the calculated crystal-field splitting (energy separation between 7e1/2 and 6e1/2 spinors) is 0.13 eV, while the experimental value estimated in ref. [38] from the two LIII ionization thresholds is 0.113 eV. For the SO splitting of LIII edge and LII thresholds, present calculated value is 1.3 eV, using a mean value of the two LIII components. This is in excellent agreement with the experimental XPS measurement (1.3 eV) [41] and the value of 1.2 eV from ref. [38].

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Figure 3. Two-component ZORA TDDFT Ti 2p excitation spectra of TiCl4 calculated with the LB94 and SAOP exchange correlation potentials. Convoluted profiles are obtained with a Gaussian broadening of 0.6 eV of Full Width Half Maximum (FWHM). Upper panel: experimental spectrum from ref. [12]. The vertical dashed lines show the calculated ionization limits.

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Figure 4. TDDFT S 2p excitation spectra of SO2 from two-component ZORA TDDFT and scalarZORA TDDFT calculations. Convoluted profiles are obtained with a Gaussian broadening of 0.25 eV of FWHM. Upper panel: experimental spectrum from ref. [38]. The vertical dashed lines show the calculated ionization limits.

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When comparing the two-component ZORA TDDFT results with experimental spectrum (Figure 4), reproduction of certain features is apparent. Current calculated absolute energies are generally underestimated by about 1 eV; however, the energy separation between the main features is correctly described. Only the shift between last two calculated structures just before the threshold appears larger than the experimental pattern. The lowest energy region (from 164 to 166 eV) includes only the S 2p excitation to the lowest unoccupied * orbital (3b1), which is split by SO effects into two main components (labeled a and b) relative to the 2p3/2 and 2p1/2 edges. These structures appear very weak, although they are clearly discernible both in experiment and in calculations. The following intense valence peak (peak c at 168.3 eV) is assigned to the 2p3/2  9a1 excitation, while the highest valence 2p3/26b2 excitation is described by a series of less intense lines indicated as region d in Figure 4. The main line in this energy region corresponds to the 2p1/2 component (at 169.49 eV) of 9a1 excitation which gives rise to peak e; note that the SO separation between two 9a1 SO components is 1.2 eV, again in excellent agreement with the experimental data [36]. In the energy region above 170 eV, the core excitations to Rydberg states are calculated. As we can see in Figure 4, the energy separation between valence and Rydberg transitions is very small with a consequent partial overlap of the relative spectral features. The agreement between theory and experiment is so nice that a complete assignment of Rydberg features has been proposed. We refer to ref. [40] for a more complete analysis of the Rydberg features. We would like to make one last comment concerning the comparison between twocomponent and scalar relativistic ZORA results (panels b and c of Figure 4). It is apparent that the inclusion of SO effect in our calculations brings about a decisive improvement in the description of spectrum. The ordering of three valence transitions, and also the Rydberg series remains the same; however the inclusion of LII manifold of excited states gives rise to a redistribution of intensities among the excited states relative to the LIII edge. This strongly modifies the shape of spectral features, both in the valence and in Rydberg region.

3.3.Transition-metal compounds: VOCl3 and CrO2Cl2 In this section, we present a theoretical and experimental study of the XAS spectroscopy of oxychlorides, VOCl3 and CrO2Cl2 at the metal 2p thresholds. In the original work [42], MnO3Cl and ligand edges have also been considered as well. These molecules are isoelectronic with TiCl4, in which the metal 3d orbitals are significantly involved in bonding. Along the series, a halogen atom is substituted stepwise by the oxygen atom, as the number of d electrons increases in going from V to Cr, inducing variations in the L-edge spectra. A complete set of absorption spectra is available at the 2p and 3p edges of the 3d metal atoms [43]. There is a wealth of data for solid compounds such as oxides, but high-resolution spectra at the L2,3-edges of several classes of transition metal compounds are still rare. Some results for free molecules are available in [14,44]. Same for the powders of transition-metal complexes (with weak interaction between the metal centers) are discussed in [45,46]. Some lower resolution EELS spectra of organometallic compounds are provided in [32,47]. For the clusters of transition metals, ref. [48] presents some results. Also, a few MO calculations of transition-metal L2,3 edges are available including the SO contribution [16,49]. In particular, for the oxychlorides, the only previous calculation of the metal L-edges has been performed using a non-relativistic configuration-interaction (CI) approach [14]. Both SO as well as molecular field splitting are resolved in the present VOCl3 and CrO2Cl2 experimental

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measurements [42]. This results in a significant complexity of the spectral features, so much so that the attribution of core-excited states can only be achieved with adequate theoretical support. At the V L2,3-edge, an additional difficulty arises from overlap among structures relative to the V L2,3 and O K absorption edges, which fall in the same-energy region. The calculated metal 2p excitation energies and oscillator strengths are reported in Tables 4 and 5 together with the experimental data. Figure 5 and 6 display the experimental and calculated spectra of VOCl3 and CrO2Cl2 respectively. The calculated excitation energies appear to be underestimated for both molecules with respect to the experiment. This can be generally attributed to the deficiencies in XC potential and to the self-interaction error, typical of the KS scheme of DFT, which for core levels is much more pronounced than those in the valence region. This, however, has no consequence on the relative energy scale, which represents the most important data for comparison with experimental spectra; the experimental data have been accordingly shifted to the calculated ones in these figures. The MO scheme of virtual valence levels, which are characterized by strong 3d metal contribution, is reported in Scheme 2. This could be useful in correlating the spectral features to electronic structure of the molecules.

3.3.1. VOCl3 The experimental V L2,3-edge spectrum of VOCl3 (Table 4 and Figure 5) shows three distinct regions of absorption. The lowest energy region is characterized by three distinct peaks, the central one by a broad and less resolved band, while a two-peaked structure is present around 530 eV, in the region of O K-edge absorption. The first two structures are well separated in energy and correspond to two SO components which converge to the V L3 (2p3/2) and L2 (2p1/2) ionization thresholds. The last structure can be safely assigned to the O 1s excitation, since the V L2,3-edge absorption is very low in this energy region according to the theoretical results. When comparing the two-component ZORA results for V L2,3 structure with experimental spectrum (Figure 5), quite good agreement is observed overall. The energy separation between the main features is correctly described and intensity ratio is also consistent with the experiment. The lack of the above two-peak structure is clear in theoretical V L2,3 spectrum at higher energy; only weak absorption is present before the 2p1/2 threshold, therefore confirming the O K-edge origin of the experimental feature. The main factors which influence the shape and structure of theoretical spectrum are SO, multiplet and molecular field splitting. The SO coupling splits the V L2,3 core hole into 2p3/2 and 2p1/2 levels. The C3v symmetry of VOCl3 removes degeneracy of the 2p3/2 level, so that in a relativistic description, 2p orbitals of V correspond to the 2a3/2, 7e1/2 (L3 components, split by only 0.03 eV) and 6e1/2 (L2 component) spinors. The SO interaction leads to a series of SO partner states responsible for the appearance of two main features converging to L3 and L2 edges. As concerns the molecular field effect on VOCl3, the C3v symmetry splits the 3d shell into three virtual valence levels, viz., 13e (LUMO), 14e and 16a1 MOs, which are dominated by the V 3d component. The contributions from O 2p and Cl 3p components are also present as a result of the V-ligand covalent interaction. The molecular field is responsible for the splitting of L3 band into three major components. The latter are associated with the transitions from 2p core level to three virtual valence MOs, and are also present in the L2 band. The quality of present theoretical results can be assessed by a comparison of the calculated L2-L3 splitting (7.2 eV) with the experimental one (7.7 eV). Also, the energy separations between three components of the first band (1.42 and 1.00 eV) are in reasonably good agreement with

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the experimental values (1.20 and 1.10 eV). We now consider in detail the theoretical results and assignment of the spectral features. The V 2p excitations map the V nd and V ns character of the final MOs. In the lower energy region, virtual valence MOs have predominant V 3d character, so that a significant portion of intensity for these valence transitions is calculated quite well. Peak A is assigned to the 13e LUMO transition; the main transition of peak B is towards the second 14e valence level, and peak C derives from two excitations with similar intensity from the two 2a3/2 and 7e1/2 spinors to the third 16a1 valence level. In general, we are led to attribute the energy splitting, as well as different intensity distribution among states with dominant V 3d participation, to interference between energetically close excitations induced by configuration mixing. The calculated peaks A, B, and C correctly describe the shape of valence experimental features, in particular the energy separation. Only the intensity of second peak is slightly overestimated with respect to the experiment, as it is apparent from a comparison of the data in Table 4. The experimental spectrum shows a small peak at about 1 eV lower than peak A (at 514.5 eV of absolute experimental energy). This peak is similar to the small leading peak found in similar x-ray absorption spectra of d0 compounds, which is explained as an atomic multiplet line with a predominant triplet character [50,51]. The present SO relativistic TDDFT scheme includes all the electronic states deriving from SO coupling and also provides a series of weak lines (the total f value summed over the lines is 0.039) at the lower energy side of peak A, at around 510.7 eV. These are associated with transitions of same nature as the more intense lines contributing to peak A, namely the 2a3/2  13e LUMO transitions. These lower energy excited states are not present in the spectrum calculated at a scalar relativistic level, where the transitions to triplet states are forbidden; therefore they result from the SO coupling. The comparison of the Ao pre-peak in experimental spectrum is however qualitative. The calculated intensity for these transitions is very low and, it also appears closer to peak A than in the experiment. In addition to a possible deficiency of TDDFT in describing coupling strength of such states, other important factors could also contribute to the A0 experimental peak; e.g., transitions allowed by a quadrupole or vibronic coupling falling in this energy region. The second band in the theoretical spectrum is dominated by three peaks (D, E and F). The first two represent the L2 components of 13e and 14e valence transitions. Peak F has contributions from several lines. The first one is the 6e1/2 16a1 transition, corresponding to the third valence 2p1/2 component, while the following states are expressed as a mixing of configurations involving also L3 Rydberg transitions. Further, intensity associated with the 6e1/2 16a1 transition is therefore spread over several other excited states, as shown in Table 4. Other excitations contributing to the peak F are due to transitions from 2p3/2 core state to Rydberg levels converging to the L3 edge, with V nd and ns character. Therefore peak F consists of overlapping structures between the third valence excitation of L2 edge and Rydberg transitions converging to the L3 edge. We underline that the molecular field splittings between the three valence excitations in L2 band (1.3 and 0.9 eV) closely reflect the splittings calculated for L3 components (1.4 and 1.0 eV). The higher energy side of F peak (G and H structures in Figure 5) consists of a very large number of lines with low oscillator strengths associated with transitions from the 2p3/2 core level (2a3/2 and 7e1/2 spinors) to discrete states, most of which fall above the L3 edge. The D, E and F peaks are assigned to the experimental L2 band, whose intensity appears quite large, although three lesser resolved

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Mauro Stener, Giovanna Fronzoni and Renato De Francesco Table 4. Experimental and calculated excitation energies (eV) and oscillator strengths f for V L2,3 spectrum of VOCl3 a EXP

E (eV)

b

2

f x 10 Label

E (eV)b

f x 10

2c

Final stated main configuration/dominant character final MO

±10%

-0.8 (2) (515.3) 0.30 0.40 1.10

0.25 0.94 1.1 1.3 0.79

A0 A

B

1.60

6.9

2.70 3.20

11.3 2.4

C D

E

9.70

F

a

-(0.520.26) (511.16) 0.25

weak transitions 2a3/2, 7e1/2  13e 0.397 2a3/2  13e / 3d V, 3pCl 3.418 2a3/2  13e

0.88

0.060

1.62 1.67 1.92

1.714 7.308 0.161

2.46 2.89 7.03 7.10 7.23 8.52 8.53 8.74 8.89 9.38

6.213 6.696 0.496 0.189 3.662 1.770 4.340 0.579 0.409 1.404

9.44

0.640

9.63 9.67 9.87 9.99

3.714 0.901 1.712 3.287

2a3/2  14e / 3d V, 2p O, 3pCl 2a3/2  14e 7e1/2  14e 7e1/2  16a1 / 3d V, 2p O, 3pCl 2a3/2  16a1 7e1/2  16a1 6e1/2  13e 2a3/2  15e / 4s Cl 6e1/2  13e 6e1/2  14e 6e1/2  14e 7e1/2  21a1 / 4s V 2a3/2  18e / nd V + 3d Cl 6e1/2  16a1 + 2a3/2  19e /4p V + nd V 7e1/2  22a1 /ns V + 6e1/2  16a1 2a3/2  20e / 3d Cl + nd V 7e1/2  23a1 + 6e1/2  16a1 2a3/2  23a1 / np, ns V 7e1/2  23a1 + 6e1/2  16a1

Calculated two-component DFT-KS V 2p ionization limits are 521.78 eV (2a3/2), 521.75 eV (7e1/2) (L3 edge) and 528.94 eV (6e1/2, L2 edge) respectively. Experimental values are: 524.85 eV (2p3/2, L3 edge), 532.55 eV (2p1/2, L2 edge). b The absolute excitation energy is given for the first intense line. Excitation energies relative to the first line are reported for other transitions. c L3 edge: only calculated transitions with f × 102 > 0.100 below the 2p3/2 ionization threshold (521.75 eV) are reported. L2 edge: only transitions from 6e1/2 spinor (2p1/2) are reported with f × 102 > 0.0100 up to the energy region of F feature. d The initial MOs are characterized according to their double-group symmetry.

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peaks may possibly contribute to it as well. The G and H theoretical features describe the higher energy tail of L3 experimental band. The following region (I, L and M features) is characterized by a very low intensity, and is assigned to Rydberg transitions from the 2p1/2 level (6e1/2 spinor) which converge to the L2 edge. This result confirms that the two-peaked structure observed in this energy range in the experiment, does not derive most of its intensity from V 2p excitations, and is mainly composed of O 1s core excitations. VOCl3 (C3v)

CrO2Cl2 (C2v)

MO  (eV) 13e(LUMO) (10.79) [3dV(58%), 3pCl(23%), 2pO(5%) ] 14e 1.27 [3dV(58%), 2pO(19%), 3pCl(17%)] 16a1 2.37 [3dV(48%), 2pO(27%), 3pCl(11%)]

MO  (eV) 17a1(LUMO) (11.60) [3dCr(40%), 3pCl(18%), 2pO(17%)] 4a2 0.37 [3dCr(46%), 2pO(35%), 3pCl(6%)] 11b1 0.87 [(3dCr(45%), 2pCl(23%), 2pO(18%)] 18a1 1.92 [3dCr(45%), 2pO(33%), 3pCl(6%)] 9b2 2.54 [3dCr(44%), 2pO(38%), 3pCl(4%)]

Scheme 2. Molecular orbital scheme of the virtual valence levels of the DFT ground-state of VOCl3 and CrO2Cl2. For each MO, following quantities are reported: the main character in terms of atomic metal, chlorine and oxygen percentage contribution, and the energy separation,  from the LUMO KS energy value, which is taken as zero. (The absolute LUMO KS eigenvalues (-) are reported in parentheses.

Table 5. Experimental and calculated excitation energies (eV) and oscillator strengths f for Cr L2,3 spectrum of CrO2Cl2 a EXP E(eV)b f x 102 Label

E (eV)b

f x 10

2c

Final state d main configuration/dominant character final MO

±20% (577.2) (2)

1.2

0.4

1.0

A

-(0.660.15) weak transitions 7e1/2, 6e1/2  17a1, 4a2 (573.15) 0.914 6e1/2  17a1 / 3d Cr, 3pCl, 2pO 0.28 0.995 7e1/2  4a2 / 3d Cr, 2pO 0.31 0.187 7e1/2  11b1 / 3d Cr, 3p Cl, 2pO 0.34 0.740 6e1/2  11b1 0.41 0.462 6e1/2  11b1 +7e1/2  4a2

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Mauro Stener, Giovanna Fronzoni and Renato De Francesco Table 5. Continued

EXP E(eV)b f x 102 Label

E (eV)

b

f x 10

2c

Final state d main configuration/dominant character final MO

±20% 0.8 1.1 1.4

1.1 1.4 1.5

1.8

1.1

2.1

3.7

2.7 3.1

5.5 4.5

B

C

D

E

F

a

-(0.660.15) weak transitions 7e1/2, 6e1/2  17a1, 4a2 0.88 1.911 7e1/2  11b1 1.06 2.471 6e1/2  11b1 1.33 0.102 6e1/2  18a1 / 3d Cr, 2pO 1.74 0.739 6e1/2  18a1 + 7e1/2  18a1 1.89 1.196 6e1/2  9b2 / 3d Cr, 2pO 2.02 0.225 7e1/2  9b2 2.13 5.040 7e1/2  18a1 2.88 5.300 6e1/2  9b2 2.97 7.475 6e1/2  9b2 + 6e1/2  18a1 8.34 0.196 5e1/2  17a1 8.54 1.084 5e1/2  17a1 8.80 0.297 7e1/2  21a1 / nd Cr + 5e1/2  4a2 8.84 0.286 7e1/2  11b2 /nd Cr + 5e1/2  4a2 8.85 0.233 7e1/2  11b2 + 5e1/2  4a2 8.88 0.356 7e1/2  21a1 + 5e1/2  4a2 8.93 0.499 6e1/2  22a1 /nd Cr + 5e1/2  4a2 9.31 0.469 5e1/2  11b1 9.34 0.388 7e1/2  23a1 /4s Cr, 4s Cl + 5e1/2  11b1 9.44 0.291 6e1/2  12b2 / nd Cl, np Cr, nd Cr 9.51 1.560 5e1/2  11b1 + 7e1/2  23a1 9.52 1.140 6e1/2  12b2 + 5e1/2  11b1 9.66 0.305 6e1/2  24a1 /4s Cr + nd Cr 9.69 0.214 6e1/2  14b1 / 3d Cl, nd Cr 9.81 0.161 7e1/2  15b1 / 3d Cl, np Cr , nd Cr 9.83 0.128 6e1/2  15b1

Calculated two-component DFT-KS Cr 2p ionization limits are: 584.42 eV (6e1/2), 584.39 (7e1/2) (L3 edge) and 593.13 eV (5e1/2, L2 edge) respectively. b The absolute excitation energy is given for the first intense line. Excitation energies relative to the first line are reported for other transitions. c L3 edge: only calculated transitions with f × 102 > 0.100 below the 2p3/2 ionization thresholds are reported. L2 edge: only transitions from 5e1/2 spinor (2p1/2) are reported with f × 102 > 0.0100, up to the energy region of F feature. d The initial MOs are characterized according to their double-group symmetry.

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Figure 5. Vanadium L2,3 spectrum of VOCl3. Black lines: two-component ZORA TDDFT spectrum. Vertical dashed lines: calculated vanadium 2p3/2 and 2p1/2 ionization limits. Convoluted profiles are obtained with a Gaussian broadening of 0.5 eV of full width at half maximum. Red line: experimental spectrum. The experimental profile has been shifted on the calculated energy scale (by 4.1 eV).

Figure 6. Chromium L2,3 spectrum of CrO2Cl2. Black lines: two-component ZORA TDDFT spectrum. Vertical dashed lines: calculated chromium 2p3/2 and 2p1/2 ionization limits. Convoluted profiles are obtained with a Gaussian broadening of 0.5 eV of full width at half maximum. Red line: experimental spectrum. The experimental profile has been shifted on the calculated energy scale (by 4.0 eV).

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3.3.2. CrO2Cl2 The experimental Cr L2,3 spectrum (Figure 6) is characterized by two main features whose fine structure is less clearly resolved than the V 2p spectrum, in particular for the higher energy structure. The large energy separation of two main bands corresponding to the L3 and L2 SO components reflects the SO splitting of Cr 2p initial states. The latter splitting amounts to 8.72 eV in the present calculation, comparing well with the experimental value of [52,53]. The agreement of two-component ZORA Cr 2p results with experimental spectrum is fairly good, both for energy separation between two main features, and reproduction of main components of the lowest energy L3 band. The Gaussian profiles used for the convolution of theoretical results also describe the fine structure for higher energy band, and is useful to distinguish the excited states ascribed to the valence L2 from Rydberg L3 transitions. The intensity ratio among the calculated features is quite consistent with observed experimental pattern, as can be verified from the data in Table 5. It is again fruitful to consider both the molecular field splitting and SO splitting to interpret the Cr 2p spectrum. Chromyl chloride has an approximately tetrahedral configuration, but the decrease of symmetry to C2v causes a loss of degeneracy of the 3d shell, which splits into five virtual valence levels. The three lowest 17a1, 4a2 and 11b1 MOs, are roughly correlated to the three-fold degenerate t2 orbitals in Td symmetry, and the remaining two 18a1 and 9b2 MOs, could correspond to e orbitals. The SO coupling splits the Cr 2p core hole into 2p3/2 and 2p1/2 levels. In C2v symmetry, the 2p3/2 level is further split into two almost degenerate 7e1/2 and 6e1/2 spinors (L3 components, split by only 0.03 eV) and 5e1/2 (L2 component) spinors. The lowest energy L3 band is assigned to transitions starting from 2p3/2 core hole to virtual valence levels of mainly Cr 3d character. These virtual levels have contributions also from Cl 3p and O 2p components, and therefore represent the virtual counterparts of the Cr-ligand bonding MOs. The molecular field splitting is responsible for the fine structure of L3 band. Peaks A and B are due to transitions to the first three 17a1, 4a2 and 11b1 virtual valence MOs, while peaks C and D are associated with the two higher 18a1 and 9b2 MOs, respectively. A series of weak lines (the total f value summed over the lines is 0.258) are calculated at the lower energy side of peak A (around 573 eV), as in the previous V 2p spectrum. They describe lower lying excited states relative to the transitions from Cr 2p orbital to 17a1 and 4a2 virtual MOs. These calculated lines are hardly visible in Figure 6. Also in the experiment, it is quite difficult to verify the possible presence of a pre-peak due to lower resolution than in the V 2p experimental spectrum. The oscillator strength, which maps the Cr 3d participation in final MOs, increases on going from peak A to D, due to a decrease of p ligand components and of the parallel shift of Cr 3d contribution to the upper levels of virtual valence manifold. The four theoretical peaks describe the four features resolved in experiment correctly, from the perspective of both energy separation an intensity trend. Agreement between theoretical and experimental oscillator strength of the single feature is not quantitative (see Table 5); this gives a general overestimation of the calculated value with respect to experimental value. The convolution of theoretical lines calculated above 580 eV reveals four main peaks for the second feature of calculated spectrum. Peak E represents the 2p1/2 SO component of peak A. Analogously, the most intense excitations of peak F correspond to the SO partner states of peak B, although several transitions from 2p3/2 core hole to Rydberg states of Cr ns and nd character appear in this energy region. We emphasize that several excited states in this energy region are expressed as a mixing of configurations, which is responsible for a spread of intensity over a large number of lines. The overlap of Rydberg transitions converging to the 2p3/2 edge and valence

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transitions starting from the 2p1/2 core hole is more noticeable on going towards higher energy. Both the remaining SO partner virtual valence states (5e1/2 18a1 and 9b2 transitions) contribute to peak G and only to a lesser extent to peak H. This last peak mainly derives from higher Rydberg excitations converging to the 2p3/2 ionization threshold. Therefore the second experimental band can be assigned to 2p1/2 (L2) valence transitions overlapping with a large number of less intense Rydberg L3 transitions. In the region labelled I, one can see the presence of transitions from 2p3/2 spinors into virtual MOs falling above the L3 edge. The Rydberg excitations from the 2p1/2 core hole start at about 588 eV, and give rise to the featureless structure labelled L at the higher energy side of calculated spectrum.

4. 2P CORE EXCITATIONS OF BULK METAL OXIDES The ADF implementation of TDDFT method for core excitations follows a typical quantum chemistry approach, which consists of treating a finite system by employing a localized basis set (Slater functions centered on atoms). While this is optimal for molecules, its extension to periodic systems (bulk or surfaces) is not straightforward, since the latter must be modeled by finite systems whose size should be rather large in order to capture correctly the electronic structure of solid or surface under investigation. In practice, in order to employ molecular or cluster models to accurately mimic the periodic systems, two requisites must be considered. Firstly, the cluster model should be large enough to converge to the correct limit. Secondly, it must be properly embedded in order to keep the electronic structure adherent to that of the extended system. While the first requirement is, at least in principle, easily checked by simply enlarging the model cluster, the second one is much more difficult. In practice, the embedding should simulate periodic systems by properly joining the border of the finite cluster, and this can be done in different ways depending on the chemical properties of the solid to simulate. In practice, for ionic solids the embedding can be done properly using arrays of point charges on nuclear positions outside the model. For covalent systems, however, it is more effective to saturate the dangling bonds with hydrogen atoms. More elaborate embedding schemes can be designed for surfaces, e.g., employing pseudo-hydrogen atoms with fractional nuclear charge has led to one of the best possible simulation routes for electronic structure of a particular site [54]. In the following applications, the embedding has been chosen accordingly, in order to obtain the best simulation of extended systems by finite cluster.

4.1. Alkaline-earth oxides This series of oxides is representative of solids of ionic character, for which both the cluster shape and point charge embedding to represent the bulk, can be a common choice. It is worth noting that, although the natural choice for description of a solid would be in terms of crystal Bloch functions with proper treatment of periodic boundary conditions, there is a series of reasons which suggest that a finite size cluster model would be more suited to study core excitations. In fact, a core excitation is well localized on a specific atomic site; so it is reasonable to assume that the proper incorporation of a limited region would be sufficient to describe it accurately. On the other hand, periodic calculations based on plane wave expansions do not include core orbitals (in order to improve convergence by mean of

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pseudopotentials) and, therefore, cannot deal with multiplet effects which may be treated at the TDDFT level. Therefore, we have addressed the best possible choices in the cluster in order to suggest a general strategy for the design of TDDFT calculations of core electron excitations in bulk materials.

4.1.1. MgO MgO was the first solid system studied with the TDDFT method for core excitations, employing a series of cluster models to mimic the bulk [55]. MgO has been chosen because it is one of the simplest and most studied solid state systems (both theoretically [56-63] and experimentally [64-68]). So it is particularly suited to assess the performances of the method. The clusters have been built with the following rule: the central atom which lies on the origin is the atom which is excited; successive shells of Mg and O atoms are then added keeping an octahedral symmetry for computational economy, as shown in Figure 7. The cluster charge is chosen assuming a formal ionic configuration, so that, in general, the charge of the cluster MgmOn will be 2(m-n). For a better simulation of the bulk situation, the clusters have been embedded within an array of point charges, which are optimized for the clusters having an Mg atom in the center [69]. Point charges are mandatory to obtain the proper closed-shell electronic structures; in fact, clusters in absence of embedding, in general, give rise to open-shells structures, which are not suitable to model the bulk situation. In order to study the convergence with respect to cluster size and shape, we have considered the series [MgmOn]2(m-n); the pictures of these clusters are shown in Figure 7. In all these cases, the embedding procedure of [69] has been employed. In [MgO6]10-, the array consists of 336 point charges, while for larger clusters, successive shells of point charges are substituted by Mg and O atoms in the calculation in such a way that, usually, the sum of the cluster nuclearity and point charge number is constant and equals to 343. From this analysis, it was concluded [55] that the cluster [Mg55O38]34+ is suitable for a realistic description of the excitation of core electrons of the central atom. Basis set choice is a rather delicate issue. In fact, from a detailed analysis [55] it was found that the DZP basis set is appropriate for this kind of calculations and that basis sets beyond TZP should be avoided to prevent errors caused by numerical instabilities, which are ascribed to numerical linear dependency problems caused by the presence of too diffuse basis functions. In summary, for Mg 2p excitations, the Mg centered [Mg55O38]34+ cluster model with DZP basis set and optimized point charges have been employed. Non-relativistic TDDFT results [55] concerning the Mg 2p excitations are reported in Figure 8, panel (a), together with the experimental data [70]. The agreement between theory and experiment is remarkably good on the whole energy range. All the spectral features at low energy are properly described by the calculation; namely, the first shoulder at 54–55 eV, the first maximum at 58 eV, as well as the second maximum at 60 eV. The experimental features at higher energy are rather smooth; generally speaking, the agreement with the theoretical spectrum is rather satisfactory. The Mg 2p spectrum has also been calculated at the KS level, in order to identify the role of configuration mixing by comparison with the TDDFT results. However, the KS result is very close to the TDDFT one; hence we do not report the KS spectrum in the figure. This finding is somehow surprising; in fact, it is well known that the excitation of degenerate 2p core electrons is properly described by strongly

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mixed excited configurations [13]. For MgO, this does not seem to be necessary, probably because most of the intensity is gained by 3s virtual final states, in contrary to other transition-metal compounds, where the most prominent features regarding the 2p excitations are related to promotion to 3d valence virtual orbitals, as observed previously in TiCl4 and isoelectronic transition-metal molecules. So, we are led to attribute the reduced importance of configuration mixing for Mg 2p excitations to the weaker 2p-3s correlation, with respect to the 2p-3d one for transition metals. In panel (b), the theoretical results from ref. [71] are reported for comparison, calculated with a cluster model employing the ground-state electron configuration with a minimal basis set. The profile resembles closely the present TDDFT one; the main difference being the slight intensity redistribution towards the low energy region. Finally, in panel (c) of Figure 8, the MS-X results of ref. [70] are reported; the agreement seems to be remarkably good despite the crudity of the numerical approach. So we are led to consider the good performances of MS-X to the inclusion of continuum boundary conditions, which prevents the discretization of states above the ionization threshold. More accurate approaches than MS-X method to treat the continuum are now available, in particular at DFT level with B-spline basis set [72]. However they are computationally much more demanding than the present method.

Figure 7. Cluster models employed to simulate MgO bulk.

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Figure 8. Panel (a), solid line: TDDFT Mg 2p excitation spectra of bulk MgO, calculated by employing the cluster model [Mg55O38]34+. Convoluted profile is obtained with Gaussian broadening of 1.0 eV; dotted line: experimental data [70]. Panel (b), solid line: theory [71], dotted line: experimental data [70]. Panel (c), solid line: theory [70], dotted line: experimental data [70].

4.1.2. CaO, SrO, BaO After the initial systematic study on MgO, in this subsection, the analysis of core excitations has been extended to the following members of the series of alkaline-earth oxides, viz., CaO, SrO and BaO [73]. This study has proven interesting because the series of alkalineearth oxides was treated at the same level of accuracy allowing us to analyze spectral trends across a number of compounds with the same rock salt structure. Very appealing is the

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possibility to correlate the spectral features, in particular at the Metal L edge, with the presence of empty nd orbitals of the metal cations. The latter is expected to decrease in energy in going from MgO to BaO with a consequent increase in participation in bonding with the O atom [56,80]. The non-relativistic TDDFT results, relative to the metal 2p excitations are reported in Figure 9 together with the available experimental data [70,74,75]. The calculated spectra show noteworthy change in the near- edge structures on going along the series. In particular, in the pre-edge region, where the features strongly increase their intensity from MgO to CaO, then decrease in SrO (see the oscillator strength f scale in the figure) and become very low in BaO spectrum. Also in the case of L edge, the BaO spectrum differs significantly from other members of the series. The CaO experimental spectrum [75] shows two very sharp structures corresponding to the 2p3/2 (L3) and 2p1/2 (L2) edges, which are well separated in energy. It is important to underline that the non relativistic calculation employed in [73] does not allow the treatment of SO effect. So it is not possible to distinguish between the L2 and L3 edges; therefore the TDDFT results have to be compared with the experimental features converging to a single ionization threshold (generally Ca 2p3/2). At the moment, the two-component relativistic ZORA method would be suitable to re-analyze these systems including SO coupling to distinguish the edges. A good agreement with the experiment in L3 region is found, both for the energy separation and intensity distribution of the two low energy structures. Above the threshold region, calculations suggest very low intensity. Also, in the SrO theoretical spectrum, most of the intensity is concentrated in the structures around edge, while in BaO there is a drop of intensity at lower energy. The appearance of the most important features occurs at about 15 eV from threshold. The calculated PDOS can be useful for the attribution of calculated spectral features. The metal 2p electron has dipole-allowed transitions into s and d like final states. Therefore, both the contributions of metal ns and nd PDOS’s are considered in Figure 10, and compared with the theoretical spectra obtained at the KS level. A closer inspection of the results reported in Figure 10 (left panel) shows that the calculated KS oscillator strengths are not properly approximated by the ns and nd contributions (we do not report their sum in the left panel). This means that the contribution of the metal ns and np components to the oscillator strengths are different, in accordance with the conclusion in literature [76,77] regarding the different contribution of transitions towards nd and (n+1)s orbitals to the L-edge absorption spectra of heavier atoms. We have explicitly calculated the 2pns and 2pnd oscillator strengths of the metal dications M2+ (M= Mg, Ca, Sr and Ba). From there, we find that the transitions to ns states are much less intense compared to the transitions to nd states, and follow a well defined trend along the series from Mg2+ to Ba2+. In particular, the calculated ratio between the nd and ns oscillator strengths is the smallest for Mg2+ (3d/3s=11.4), while it increases significantly in Ca2+ (3d/4s=62.4), Sr2+ (4d/5s=51.2) and Ba2+ (5d/6s = 235.6). This behavior can be correlated with the more compact nature of an nd wave function with respect to an (n+1)s wave function, resulting in a stronger overlap with the 2p wave function. In case of Mg dication, the 3s and 3d orbitals have the same principal quantum number. So their spatial distributions are expected to be less different, resulting in a lower ratio between the intensity of the two p-s and p-d channels. This analysis points out that the L spectrum of these oxides is dominated by the d absorption channel. If each metal ns and nd PDOS’s of the oxides is weighted for (multiplied by) the corresponding 2pns and 2pnd oscillator strengths of metal dications computed, the total PDOS reported in the right panel of Figure 10 is obtained.

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As we can see, the calculated KS oscillator strength is now found to be well approximated by the “weighted” (ns+nd) PDOS’s. The previous investigation of the PDOS diagrams can be useful to analyze the trend of spectral features along the series (see Figure 9). In MgO, most of the intensity around the edge is gained by the 3s virtual states and therefore the spectral features are very low. On going towards higher energy, the spectral intensity increases with increase of Mg 3d character of the final states. The high intensity of the pre-edge features of CaO spectrum is associated with the presence of low-lying virtual states with strong 3d metal contribution; the main s character of the higher energy states is responsible for the drop of intensity calculated above the ionization limit. A quite similar behavior is found for SrO spectrum, although the intensity of transitions around the edge is lowered by about one order of magnitude with respect to the CaO spectrum. In fact, the maximum f value calculated for CaO is 0.1811, while that for SrO is 0.0119. Closely connected to this fact is the lower d metal (4d) character of low-lying virtual states with respect to that found in CaO, together with a less effective overlap between the 2p wave function and 4d one with respect to the 3d wave function. Finally, the different shape of BaO spectrum reflects the energy increase of virtual levels with 5d metal character, which gives rise to an intense structure far away from the threshold. It is interesting to compare the TDDFT and KS 2p metal spectra (Figure 9 and 10 respectively), in order to identify the role of configuration mixing, which is expected to be important in case of degenerate core hole, like the 2p core electrons [13]. The most significant difference between the two levels of calculations is observed in cases of CaO and SrO spectra. The TDDFT approach significantly redistributes the intensity of transitions near edge, bringing them in a better accord with the experiment in case of CaO. Here in particular, the mixing of configurations tends to concentrate the intensity on two main lines (at 347.6 and 348.6 eV in Figure 9), which are associated with transitions towards final states with 3d dominant character, and give rise to the two well distinct peaks of L3 edge structure. In contrast, less improvement with respect to the KS results is obtained in case of MgO and BaO TDDFT spectra, probably because the excited states at lower energies have a small d contribution. In fact, as shown before for TiCl4, it is known that the strong configuration mixing found in case of 2p excitations in transition-metal compounds is associated with the promotion of 2p electron to the d valence virtual orbitals, and requires an accurate theoretical treatment for a correct description of the intensity distribution [12].

4.2. Transition-metal oxides The application of a relativistic ZORA-TDDFT scheme has given very satisfactory results for L2,3 metal edge spectra in transition-metal compounds, as has been presented elsewhere [16,42]. Thanks to excellent performance of the scheme and considering its efficient parallel implementation, it is natural to extend the application of the method to calculations of 2p core excited states of larger systems. In particular, we are interested in solid systems for which the NEXAFS technique is particularly useful for characterization of local bonding and geometrical environment of a particular atomic species. A difficult class of solids to perform spectroscopic predictions through theoretical calculations are the transition-metal oxides; in fact the ionic picture does not describe their electronic structures well, due to the significant mixing between transition-metal d states and oxygen p states. Furthermore, many-body effects to describe the interaction between 2p core hole and 3d excited electron, as well as the SO coupling, have also to be accounted for. Solid state effects through the use of adequate cluster

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models should also be included in the calculations. The assessment of relativistic ZORATDDFT performances for L2,3-edge absorption in transition-metal oxides has been conducted on two systems: TiO2 and V2O5.

Figure 9. TDDFT metal 2p excitation spectra (solid line). Convoluted profiles are obtained with Gaussian broadening of 1.0 eV. Experimental data: MgO – dotted line [70], dashed line [74]; CaO – dotted line [75]. Vertical dashed lines: calculated Metal 2p DFT-KS ionization thresholds (opposite eigenvalues): MgO: 55.13 eV; CaO 350.02 eV; SrO 1927.12 eV; BaO 5201.80 eV.

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Figure 10. Left panel: KS metal 2p excitation spectra (solid line - convoluted profiles are obtained with Gaussian broadening of 1.0 eV) and metal s- (dotted line) and d- (dashed line) PDOS profiles. DOS profiles are amplified (for best fitting calculated spectra) using the following coefficients: MgO 50, CaO 25, SrO 20, BaO 10. Right panel: KS metal 2p excitation spectra (solid line) and sum of s- and dPDOS “weighted” profiles; weight coefficients are the respective 2p  ns and 2p  nd oscillator strengths relative to the calculated transitions in M2+ ions; DOS profiles are reduced (for best fitting calculated spectra) using the same coefficient (2) for all the profiles. Vertical dashed lines: calculated Metal 2p DFT-KS ionization thresholds (opposite eigenvalues): MgO: 55.13 eV; CaO 350.02 eV; SrO 1927.12 eV; BaO 5201.80 eV.

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Figure 11. Cluster models of TiO2 rutile employed in the calculations, and PDOS profile vs. energy levels in terms of KS eigenvalues (energy is rescaled with respect to the LUMO KS eigenvalue taken as zero). Panel a: bulk cluster and Ti 3d-PDOS profile. Panel b: (110) surface cluster and Ti 3d-PDOS profile. In the cluster figures, green balls represent the Ti atoms, red balls the O atoms and gray, small balls the pseudo-hydrogen atoms. For each cluster, the excited Ti atom is represented in black.

4.2.1. TiO2 TiO2 has been considered in its rutile phase, which is the most studied one due to the ready availability of high-quality single crystals [78]. The Ti L2,3 spectra has been calculated at the relativistic level, both for bulk rutile and its (110) surface [79], with an aim to analyze the sensitivity of its spectral features with respect to the different symmetry environments of Ti atom in these two conditions. The complexity of TiO2 spectra arises from both correlation effects and SO relativistic effects, which transfer spectral intensity from L3 to L2 edge structures. These are also split by a ligand field effect depending on the local symmetry of excited Ti atom. The cluster models for the simulation of bulk and (110) surface are shown in Figure 11. In both cases, the clusters have been terminated with pseudo-hydrogen atoms following a procedure described in ref. [79]. We refer to this work for a detailed description of the cluster building as well. Here we only recall that, a geometry optimization has been carried out on the Ti19O32 surface cluster, keeping in mind the well known importance of relaxation effects on the geometric and electronic structure of TiO2(110). For the simulation of Ti 2p core excitation spectra, only the titanium atom in central position has been excited; both for the surface and bulk cluster (as indicated in Fig. 11). Partial density of states (PDOS)

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calculations have been performed also in order to provide a pictorial representation of the unoccupied states of the clusters, as described at the ground state SCF-KS level. The Ti 3d component dominates the low-lying virtual MOs involved in the core excitations from Ti 2p core levels; therefore only the contribution of Ti 3d atomic orbitals are reported in Fig. 11. Figures 12 and 13 show the Ti L2,3-edge spectra of bulk and (110) surface clusters respectively, calculated both at the Scalar Relativistic ZORA-TDDFT level including 1h-1p interactions, and at the two- component ZORA-TDDFT level incorporating SO interaction as well. This allows us to analyze the effect of SO coupling on the calculated spectra. The experimental profiles are also reported in these figures for direct comparison, after a red-shift on the computed energy scale, to reach the best match with theoretical features.

Figure 12. TDDFT Ti L2,3-edge excitation spectra of bulk TiO2 from scalar-ZORA TDDFT (upper panel) and two-component ZORA TDDFT (lower panel) calculations. Convoluted profiles are obtained with a fixed Gaussian broadening (FWHM = 0.5 eV) up to 458 eV and with an increasing Gaussian broadening (FWHM = 0.5 to 1.5 eV) from 458 to 465 eV. Lower panel: dashed line shows the experimental spectrum from ref. [80]. The vertical lines show the calculated ionization limits.

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It is useful to recall that in an MO picture of TiO2 bulk, the crystal field of the O atoms around Ti atom, in a slightly distorted octahedron (D2h symmetry of the rutile), splits the Ti 3d levels into two groups, viz., the three-fold t2g and two-fold eg. The latter two levels correspond to two distinct energy levels in a perfect Oh symmetry, while apparently split into b1g, b3g, ag (t2g-like levels) and ag, b2g (eg-like levels) in the lower D2h symmetry (see also Fig. 11). Considering the experimental spectrum [80] reported in Fig 12, we observe that the SO interaction of Ti 2p core electrons, gives rise to two groups of peaks separated by about 5.5 eV which converge to the L3 and L2 edges. The calculated SO relativistic results show a very satisfactory matching with all the observed experimental features, both at L3 and L2 edges; only some differences in the intensity distribution are noticed. The calculated SO splitting of 5.57 eV is in excellent agreement with the experimental value. The importance of including SO coupling in the theoretical scheme is abundantly clear when comparing the theoretical results in upper and lower panels of Fig. 12. At the Scalar Relativistic TDDFT level of theory, only a qualitative reproduction of L3 experimental features is provided by calculations, while the SO interaction, completely including the L2 manifold of excited states, redistributes intensity. This also provides a good reproduction of the experimental pattern. The L3 edge of the SO relativistic spectrum (up to about 458 eV) is characterized by two low-intensity peaks (A3 and B3) which precede the t2g-related peak (C3) while the eg-like peak is split into two peaks (D3 and E3). The two weak pre-peaks A3 and B3 are assigned to transitions towards low-lying virtual orbitals of t2g-like symmetry, and are caused by the mixing of particle-hole configurations [79]. Their weak oscillator strengths also reflect the composition of the final MOs, which have a strong contribution from 3d orbitals of Ti neighbors, and only a minor participation from 3d components of the central Ti atom. The higher intensity of t2g-like transitions of C3 peak is related to the increase of contribution of 3d orbitals centered on the central Ti atom. The energy splitting of the e g-band into D3 and E3 features reflects the splitting value calculated for the Ti 3d orbital components in a D2h symmetry, as can be seen from the PDOS related to the Ti 3d band for the present TiO2 bulk cluster (Fig. 11). However, the strong participation of 3d components of neighboring Ti atoms to the eg-like virtual MOs, as well as the correlation effects and SO coupling completely redistribute intensities over the 3d features at L3 edge. This suggests that both local multiplet and long-range contribution from neighboring Ti atoms are responsible for the shape of this eg-like peak. This analysis is in substantial agreement with the conclusions drawn in ref. [81], which has shown that the L3-eg splitting is a long-range bandstructure effect that reflects the crystal structure of rutile on a length scale of about 1 nm (cluster sizes of about 60 atoms). The L2 calculated transitions start above 458 eV, and give rise to features more intense than the corresponding L3-edge features, in accordance with the experimental pattern. This trend shows that the inversion of L3/L2 branching ratio from an ideal statistical ratio of 2:1 is due to the interplay of configuration mixing and SO relativistic coupling. The calculations provide three main L2 peaks (C2, D2 and E2) in close correspondence with the L3 peaks. This is more so, as regards the nature of transitions involved; a splitting of the eg-related peak (into D2 and E2 peaks) occurs also for the L2 edge.

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Figure 13. TDDFT Ti L2,3-edge excitation spectra of TiO2 (110) surface from scalar-ZORA TDDFT (upper panel) and two-component ZORA TDDFT (lower panel) calculations. Convoluted profiles are obtained with an increasing Gaussian broadening (minimum FWHM = 0.8 eV, maximum FWHM = 2.0 eV). Lower panel: dashed line shows the experimental spectrum from ref. [82]. The vertical lines show the calculated ionization limits.

The theoretical results of Ti L-edge of rutile (110) surface are reported in Figure 13, together with the experimental spectrum [82]. The effect of the inclusion of SO coupling on our calculated TDDFT Ti 2p spectrum is well apparent in the figure. This confirms the importance of SO coupling in transferring spectral weight from L3- to L2-edge structures, as well as among the L3 3d components. The experimental spectrum is characterized by two double-peaked features separated by about 6 eV [82], which are correctly reproduced by theoretical calculations, with a calculated SO splitting of 5.67 eV. It is relevant to observe the great number of calculated L3 and L2 transitions, very close in energy over which the intensity

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spreads. This is a result of the symmetry lowering with respect to the rutile bulk structure. In fact, the arrangement of atoms in the surface cluster produces two distinctly different Ti positions concerning their oxygen environment: (i) a six-fold-coordinated Ti, as in the bulk, and (ii) a five-fold-coordinated Ti, which is missing one oxygen due to the surface formation. The symmetry of the surface cluster reduces to C2v from the D2h of the bulk. The distribution of the Ti 3d atomic components in C2v symmetry is shown in the plot of PDOS of the surface cluster (figure 11). It appears that the separation of Ti 3d components into two groups (t2gand eg-like) is less defined than in the bulk structure. A thorough analysis of the excitations [79] indicates that the transitions which give rise to peaks A3 and B3 start from the 2p3/2 spinors towards final orbitals, which are mainly contributed by 3d orbitals of Ti neighbors, with a smaller participation of 3d components of the central Ti atom. As in the bulk calculations, each excited state derives from a significant mixing of configurations. This factor and the significant 3d character of the Ti neighbors’ virtual states are responsible for the low intensity of these transitions, as well as its spread over a large number of lines. The two calculated A3 and B3 peaks are split by the crystal field by about 2.2 eV, which is in agreement with the experimental spectrum (about 2.3 eV from a graphical assessment) and also the intensity trend is correctly reproduced. The L2 calculated transitions appear around 459 eV, and give rise to a double-peaked feature. This is very similar to the L3 feature, apart from the stronger intensity of peaks A2 and B2, in keeping with the experimental pattern and the behavior previously observed in the bulk spectrum. The nature of these transitions is in close correspondence with the L3 transitions [79]. The underestimation of calculated B2 intensity could depend on the lack of Rydberg transitions converging to the L3 edge, which are expected to fall in this energy region. The absence of diffuse functions in the basis set of central Ti atom prevents the contributions from Rydberg transitions. Also the cluster size, which is smaller than that in a bulk simulation, could be responsible for some deficiencies of the calculated spectrum, if we consider the fact that clusters of almost 60 atoms have been found to be necessary to reach a convergence of TiO2 bulk spectra [81].

4.2.2. V2O5 Vanadium pentoxide V2O5 has an enormous importance as a catalyst and many microscopic details of its catalytic behaviour are still under debate. The spectroscopic techniques involving core excitations can give strong support in understanding its electronic structure, and thus it has been a subject of intense research activity, both from experimental and theoretical point of view [83-89]. The crystal structure of V2O5 can be represented as alternating layers containing either vanadium and bridging oxygen atoms, or alternatively, vanadyl oxygen atoms. There are three differently coordinated oxygen species and each vanadium atom is placed in a highly distorted environment; the distance between the two adjacent layers (about 4.4 Å) indicates that the inter-layer interactions are mainly electrostatic, and therefore rather weak [90]. The present ZORA-TDDFT calculations [91] employ a single-layer cluster V10O31 (Fig. 14) to simulate the bulk V2O5. The dangling bonds of the peripheral oxygen atoms in the cluster are saturated by hydrogen atoms, and this leads to a neutrally-charged cluster (V10O31H12). The strategy used to cut out the cluster from the bulk structure of V2O5 is described in ref. [92]. We have been forced to use a cluster with only one layer because of the computational demands required by the TDDFT relativistic SO approach for simulation of V 2p spectra. However, a previous analysis on the convergence of core non-relativistic spectra in terms of number of layers in the cluster [92] has demonstrated

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that a good representation of the spectroscopic features is already recovered by using the V10O31 cluster, although a convergence can be reached with a three-layered cluster (V30O93H36).

Figure 14. Cluster model of bulk V2O5 employed in the calculations, and PDOS profile vs. energy levels in terms of KS eigenvalues (energy is rescaled with respect to the LUMO KS eigenvalue taken as zero). Panel a: bulk cluster V10O31H12. In the cluster figures, blue balls represent the V atoms, red balls the O atoms and gray balls the hydrogen atoms.

The present calculations have been performed employing a DZ basis set for V, O and H atoms. Note that, the V10O31H12 cluster has two symmetrically equivalent central V atoms.

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Therefore, the calculated spectra and PDOS profiles have been evaluated for both atoms and finally averaged. For these two V atoms, a DZ all-electron basis set has been used while for the other V and O atoms, the Frozen Core (FC) computational scheme has been used (V FC 3p and O FC 1s) [91].

Figure 15. TDDFT V L2,3-edge excitation spectra of bulk V2O5 from scalar-ZORA TDDFT (upper panel) and two-component ZORA TDDFT (lower panel) calculations. Convoluted profiles are obtained with a fixed Gaussian broadening (FWHM = 0.5 eV). The vertical lines show the calculated L3 ionization limits. Lower panel: dashed line shows the experimental spectrum from ref. [85].

Figure 14 reports partial density of the unoccupied states (PDOS) of V10O31H12 cluster, in particular, the V nd PDOS and O np PDOS, which could be useful to analyze the nature of lower lying virtual levels. In a purely ionic representation of V2O5 ground state, O 2p orbitals would be completely filled (O2-), whereas V 3d states would be unoccupied (V5+, 3d0

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configuration). As a result of the V-O covalent interaction, a significant contribution from O 2p orbitals is present above the Fermi level, in an energy region where the V 3d orbitals dominate. In a perfect Oh symmetry, V 3d orbitals are split into two sub-groups: t2g at lower energy and eg at higher energy (crystal field splitting). Deviations from the octahedral environment are particularly strong in case of V atoms in the peculiar V2O5 crystal structure and further splitting of the 3d orbitals appears to be related to the C2v symmetry of the cluster. The t2g-like structure is split into two different peaks; the lowest one due to the 3dxy component, and the highest one, around 2 eV, relative to 3dxz and 3dyz components. The higher energy eg-like structure around 4 eV, on the other hand, is contributed by the 3dx²-y² and 3dz² components. The calculated V 2p spectra from TDDFT, both at scalar relativistic and SO relativistic levels, are reported in Fig. 15, together with the experimental spectrum [85], arbitrarily shifted along the excitation energy axis, in order to obtain the best match with theory. Two broad structures, assigned to the series of transitions converging to L3 and L2 edges, are present in experiment with the edge maxima separated by about 7 eV. The present calculated SO splitting between V 2p3/2 and V 2p1/2 initial levels is 6.95 eV, in excellent agreement with the experimental value. A decent matching between the SO relativistic spectral profile and experiment is apparent in Fig. 15 (lower panel). Considering also the results obtained at a Scalar Relativistic TDDFT level (upper panel of Fig. 15), we again observe that the inclusion of SO interaction in our calculation strongly modifies the spectral shape, not only for the appearance of L2 structure, but also for the significant redistribution of intensity among the excited states of the L3 edge. Now let us consider in detail the L3 structure of the spectrum. The peak labels reported in this figure are the same as used in the experiment [85], where the L3 pattern shows two main recognizable features: a first peak (P2) preceded by two weak, broader lower-energy peaks (P6 and P1), followed by a sharp, more intense peak (P4) with a shoulder on the lower energy side (P3). The set of L3 peaks mainly arises from the crystal field splitting of 3d-like final orbitals. However, certain features in the intensity distribution cannot be explained simply in terms of a single-particle picture, such as a representation provided by the V3d-PDOS profile of Fig. 14. In particular, the very weak lines in the lower energy range are described by a significant mixing of several configurations relative to transitions towards virtual orbitals. These are largely dominated by 3d contributions from both central V atom and neighboring V atoms and, this analysis accounts for the low oscillator strength calculated for the relative transitions. The excited states of P6 and P1 calculated peaks can be described in terms of the virtual MOs made up by both central V 3dxy component and 3d t2g*-like components of the neighboring V atoms. The other two t2g*-like components, namely, 3dxz and 3dyz of the central V atom characterize transitions contributing to the peak P2. The increased intensity of this peak is due to the larger number of transitions in this energy with higher oscillator strength with an increase in the 3d contribution from excited V atoms, compared to the peripheral V atoms in the final MOs. The comparison with the lower energy features of the L3 experimental band is satisfactory; in particular, the estimated splitting of the experimental P1 and P2 features (about 1.1 eV in NEXAFS experiment [85]) is correctly reproduced by the calculations (1.19 eV). The following P3 and P4 peaks, as calculated, are assigned to the eg*-like components (3dx²-y² and 3dz²) of the V 3d conduction band. The excited states always derive from a strong mixing of configurations. Therefore, transitions to the MOs originating from both 3dx²-y² and 3dz² components of central and peripheral V atoms are distributed in this whole energy range. However, the transitions contributing to the P3 peak, mainly correspond to the 3dz² component

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of central V atom, while the P4 peak derives from transitions mainly associated with the 3dx²* y² component of central V atom. The participation of 3d eg -like components from neighboring V atoms in the higher MOs is reduced with respect to the lower-lying eg*-like MOs, and this change of composition explains the higher oscillator strength calculated for P4 transitions with respect to P3 one. We assign the strongest calculated P4 peak to the P4 experimental feature. However, the calculations do not provide the higher energy shoulder identified as P5 in the experiment. Apart from the effects not explicitly included in calculation, for example, a possible coupling of electronic and vibrational states, this discrepancy could be ascribed to a limited cluster size employed (only one layer of V atoms). A three-layered structure could provide a more correct description of the bulk situation, with the excited V atoms placed in middle layer and completely surrounded by cluster atoms. Towards this end, this leads to a minimization of the surface effects, as pointed out in the non relativistic results in ref. [92]. The calculated L2 transitions start around 518 eV, and give rise to features in close correspondence with those of the L3 structure, apart from a reduction of global intensity. The nature of the L2 transitions is also similar to that of L3 transitions. Therefore, the three lower energy features of L2 edge represent t2g*-like 3d transitions starting from the V 2p1/2 initial orbital, while the higher-energy double-peaked feature derives from the eg*-like 3d transitions. The reduction in intensity of this last feature compared to the correlated L3 one reflects the decrease of 3d contribution from central V atom to virtual MOs which are involved in the higher-lying L2 transitions. These are characterized by increasing contributions from the V 4p atomic components. The comparison with experiment is not significant in this energy region, due to the lack of structured features in the broad L2 edge band.

5. CONCLUSIONS In this chapter, we have described the TDDFT method to treat core electron excitations, according to its implementation in the ADF code. Although the method is completely general and has been applied to K thresholds as well as to L edges, we have described the most important applications to L edges, since this is the field where this method is more competitive. The spectroscopic study of molecules in gas phase has allowed us to analyze the effects of crystal field splitting, configuration mixing and SO relativistic effects on spectral features, showing high accuracy of the method. Inclusion of these effects in the theoretical scheme appears essential to correctly describe the L3- and L2-edge transitions. This is especially true in case of transition-metal compounds. Because, in such a case, the large SO coupling gives rise to an intensity distribution between the series of excited states converging to L3 and L2 thresholds that deviate from the 2:1 statistical intensity ratio. The method can also be efficiently applied to the simulation of spectra of large solid systems despite the significant computational effort required, in particular, when the SO relativistic effects need to be included in such calculations. The computational study of solid state systems requires the design of suitable cluster models to mimic the electronic structures of bulk and surface condensed phase in a reasonably good way. In summary, the TDDFT method for core excitation is suitable for the description of accurate NEXAFS gas-phase experiment of

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molecules, and can as well be used as a practical tool to describe the X-ray absorption on condensed systems of interest in the field of materials science.

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In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 149-168

Chapter 5

T IME D EPENDENT D ENSITY F UNCTIONAL T HEORY C ALCULATIONS OF C ORE E XCITED S TATES Nicholas A. Besley∗ School of Chemistry, University of Nottingham, University Park, Nottingham, UK

Abstract Recent advances in X-ray sources have led to renaissance in spectroscopic techniques in the X-ray region. Theoretical calculations can often play an important role in the analysis and interpretation of experimental spectra. In this Chapter, recent progress on the development of time-dependent density functional theory to calculate core excited states and near-edge X-ray absorption fine structure (NEXAFS) spectra is described, including the development of new exchange-correlation functionals. These new methods are illustrated through calculations on important biological systems.

PACS: 78.70.Dm Keywords: Core-excited states, TDDFT, Short-range corrected functional

1.

Introduction

Since density functional theory (DFT) was adopted by the quantum chemistry community, its popularity has grown rapidly to the point where it now dominates electronic structure calculations. For many problems in chemistry and biology, electronically excited states are important. Excited states can be studied within DFT through time-dependent density functional theory (TDDFT), wherein the response of the electron density to a time varying electric field is considered. Early work using TDDFT simply applied exchangecorrelation functionals that had proved successful in calculations of ground state properties to study excited states. However, it was soon recognized that such functionals were deficient and provided qualitatively inaccurate predictions for certain classes of excited states, such as Rydberg or charge transfer states [1]. Currently, there is a considerable ongoing ∗

E-mail address: [email protected]

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research effort to develop exchange-correlation functionals that can correct these deficiencies, and this has led to the introduction of long-range corrected or Coulomb attenuated functionals [2, 3]. e-

e-

V

V

V

V

O

O

O

O

C

C

C

C

X-ray emission spectroscopy (XES)

Auger spectroscopy

X-ray photoelectron spectroscopy (XPS)

near-edge X-ray absorption fine structure (NEXAFS)

Figure 1. Spectroscopic techniques in the X-ray region.

Another less familiar failure of commonly used exchange-correlation functionals is in the calculation of core excited states using TDDFT. If molecules are irradiated with X-rays with sufficient energy to excite core electrons a number of different processes can occur, leading to a variety of different spectroscopic techniques, which are depicted in Figure 1. In X-ray photoelectron spectroscopy a core electron is ionized leaving a core-hole. Valence electrons can relax and fill the core-hole resulting in X-ray emission spectroscopy. Furthermore, this emission can be accompanied by the ionization of a valence electron in Auger spectroscopy. This Chapter is primarily concerned with near-edge X-ray absorption fine structure (NEXAFS) spectroscopy which corresponds to the excitation of a core electron to give a bound state below the ionization continuum. This technique provides information on the unoccupied orbitals and has the advantage of being element specific, i.e. it is possible to study excitations from the core orbitals of different elements separately. NEXAFS spectroscopy has a long history, and is used extensively in surface science, providing information on the structure and orientation of the adsorbed molecules, and the nature of the bonding to the surface [4, 5] and in bio-inorganic chemistry [6–8] where it can differentiate between different oxidation states. Recently, NEXAFS has also been central in high profile experiments probing the structure of water [9, 10]. Reliable theoretical calculations can be invaluable in the interpretation of experimental data, and in this Chapter we describe recent developments towards accurate calculations of core-excited states and NEXAFS spectra with TDDFT. This includes recent progress in the development of exchange-correlation functionals designed for core-excited states, and recent applications in biological chemistry.

2.

Calculations of NEXAFS Spectra

Early calculations of NEXAFS spectra used the multiple scattering Xα method [4]. In the NEXAFS region, the muffin-tin approximation at the heart of the method led to inaccuracies in the computed spectra, and stimulated the search for more accurate methods. One of the most successful and widely used methods that followed was the direct static exchange

TDDFT Calculations of Core Excited States

151

(STEX) method [11–13]. In this independent channel single electron approach the contribution to the molecular potential of the excited electron is neglected. The calculation of the absorption spectrum comprises a number of steps. A calculation of the core-hole state is performed with the valence orbitals frozen, followed by optimization of the valence orbitals with the core hole frozen. The STEX Hamiltonian is diagonalized and the excitation energies are obtained by summing the core ionization potential to the eigenvalues of the STEX Hamiltonian. The oscillator strengths are calculated from the dipole matrix elements between the ground and the final STEX states. The limitations of this approach are the neglect of electron correlation and the independent channel approximation. In an effort to improve the STEX approach, the transition potential method was introduced [14,15]. In this approach, the orbital binding energy is computed as the derivative of the total energy with respect to the orbital occupation number. To take into account the relaxation of the orbitals, the energy is approximated by calculating the derivative at the point corresponding to the occupation 0.5. Formally, this corresponds to a core orbital with half an electron removed which captures a balance between final and initial states. The transition potential method has proved successful and has been applied to a wide variety of problems. The excitation energies obtained from the transition potential method are about 1.5 - 2 eV too low, and this error has been attributed to higher order contributions to the core relaxation energies [15]. With advances in DFT, it is possible that DFT can provide a framework to compute accurate core-excitation energies and NEXAFS spectra. Within Kohn-Sham DFT, NEXAFS spectra can be computed using a ∆Kohn-Sham approach. In this approach the core excitation energy is the difference in the expectation values of the neutral and core-excited KohnSham Hamiltonians, where the orbitals have been variationally optimized for the different states. However, obtaining a core-excited state with a Kohn-Sham formalism is not straightforward, and usually some constraints, overlap criterion or intermediate optimization with a frozen core hole is used to prevent the collapse of the core hole during the self-consistent field procedure [16–19]. An advantage of the ∆Kohn-Sham approach is that the relaxation of the core hole is included, and a recent study showed that core excitation energies computed with the B3LYP functional were in good agreement with experiment provided uncontracted basis functions were used [19]. The principal disadvantage of ∆Kohn-Sham calculations is that a separate calculation is required for each core-excited state. Computing NEXAFS spectra for even relatively small molecular systems requires many different core-excited states to be computed, and the calculations can become expensive and tedious. Consequently, TDDFT in which the excited states are obtained within a single calculation becomes an attractive option for computing NEXAFS spectra.

3.

Calculation of Core Excited States with TDDFT

If the Tamm-Dancoff approximation [20] of TDDFT is adopted, excitation energies and oscillator strengths are determined as the solutions to the eigenvalue equation [21] AX = ωX

(1)

Aiaσ,jbτ = δij δab δστ (aσ − iτ ) + (iaσ|jbτ ) + (iaσ|fXC |jbτ )

(2)

where the matrix A is given by

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Nicholas A. Besley Table 1. Computed excitation energies (in eV) for a GGA and hybrid exchange-correlation functional with the overlap diagnostic Λ Excitation CO C(1s)→ π ∗ CO O(1s)→ π ∗ HF F(1s)→ π ∗ SiH4 Si(1s)→ π ∗ HCl Cl(1s)→ π ∗

Exp. 287.4 534.2 687.4 1842.5 2823.9

BLYP 271.1 512.0 659.7 1784.1 2749.8

B3LYP 276.1 519.6 669.3 1800.9 2769.1

Λ 0.18 0.14 0.08 0.03 0.02

and (iaσ|jbτ ) =

(iaσ|fXC |jbτ ) =

Z

Z Z

∗ ∗ ψiσ (r1 )ψaσ (r1 )

∗ ψiσ (r1 )ψaσ (r1 )

1 ψjτ (r2 )ψbτ (r2 )dr1 dr2 r12

∂ 2 EXC ∗ ψjτ (r2 )ψbτ (r2 )dr1 dr2 ∂ρσ (r1 )∂ρτ (r2 )

(3)

(4)

and i are the orbital energies and EXC is the exchange-correlation functional. However, standard implementations of TDDFT exploit the iterative subspace algorithm of Davidson [22] which is inefficient for core-excited states. By their nature, core excitations are high in energy, and consequently a very large number of roots are required to obtain coreexcited states making calculations prohibitively expensive even for small molecular systems. A number of groups have proposed different methods for overcoming this problem. The Sakurai-Sugiura projection method can be used to find excitation energies in a specified range, and this has been implemented within TDDFT and shown to be an efficient approach for core excitations [23], and a resonant converged complex polarization propagator has been implemented to study NEXAFS [24, 25]. Perhaps the simplest solution to this problem is to restrict the single excitation space to include only excitations from the relevant core orbital(s) [26–28]. This approximation is remarkably accurate, and it has been shown that for a range of core excitations from 1s orbitals, the largest error observed was 0.01 eV in the excitation energy and 0.01 in the oscillator strength [29]. This is a consequence of the large energy separation between core orbitals localized on nuclei with different atomic charge making the mixing between excitations included and those excluded negligible. The accuracy of these calculations depends primarily on the exchange-correlation functional. It has been shown that standard generalized gradient approximation (GGA) and hybrid functionals result in a large underestimation of core-excitation energies [30]. This is illustrated in Table 1, which shows some calculated core-excitation energies computed using TDDFT with BLYP and B3LYP exchange-correlation functionals and the 6-311(2+,2+)G∗∗ basis set. The results show that the amount the excitation energy is underestimated increases with the nuclear charge of the relevant atom and is slightly less with B3LYP compared to BLYP. Also shown are values for the Λ diagnostic, which was introduced to determine for which states GGA and hybrid functionals would fail [31]. Λ is a measure of the overlap

TDDFT Calculations of Core Excited States

153

between donating and accepting orbitals, and is given by Λ=

P

2 i,a κia Oia 2 i,a κia

P

(5)

where Oia is a measure of the spatial overlap between occupied orbital ψi and virtual orbital ψa Oia =

Z

|ψi(r)||ψa(r)|dr

(6)

and within the Tamm-Dancoff approximation κia = Xia.

(7)

The compactness of core orbitals make the values of Λ small, and comfortably in the regime where GGA and hybrid functionals fail [31]. For nuclei with higher nuclear charge, the core orbitals will be more compact and a greater failure of the functional would be expected. This relationship between the overlap and underestimation of the computed core-excitation energies suggests an analogy between the failure of the functionals for core-excited states and charge transfer states. The origin of the failure of exchange-correlation functionals for charge transfer states is now a well understood and is associated with the self-interaction error [32]. One successful remedy is the use of long-range corrected functionals, and it would be intuitive to suppose that a similar approach would be appropriate for core excitations.

4.

Exchange-Correlation Functionals for Core Excited States

Nakai and co-workers reported the first attempts to improve the description of coreexcited states within TDDFT. The CV-B3LYP [33] and CVR-B3LYP [34, 35] functionals were introduced, which were designed to be accurate for all types of excitation, including core excitations, and work by using an appropriate fraction of Hartree-Fock (HF) exchange depending on the type of excitation. These functionals were applied to core-excitations from first and second row nuclei and showed a substantial improvement in accuracy, yielding mean absolute errors below 1 eV. In other work, the fraction of HF exchange in a hybrid B3LYP type functional was optimized for core excitations [27]. Building on the analogy with charge transfer excitations, Coulomb attenuated or range separated functionals have been developed for core excitations. These functionals exploit a partitioning of the 1/r12 operator in the evaluation of the exchange energy. In traditional long-range corrected exchange functionals the 1/r12 operator is partitioned using the error function (erf) according to erf(µr12 ) erfc(µr12 ) 1 = + r12 r12 r12

(8)

where r12 =|r1 -r2 | and erfc=1-erf [2]. The first term of equation 8 is the long-range interaction term and the second term accounts for the short-range interaction. In standard longrange corrected functionals the short-range term is treated using DFT, while the long-range

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Nicholas A. Besley

Table 2. Computed excitation energies (in eV) for core→valence transitions with a range of exchange-correlation functionals. a Mean absolute deviation, † reference [38], ‡ reference [37] and ] reference [34] Excitation

Exp.

C2 H4 C(1s)→ π ∗ C2 H2 C(1s)→ π ∗ H2 CO C(1s)→ π ∗ CO C(1s)→ π ∗ N2 N(1s)→ π ∗ H2 CO O(1s)→ π ∗ CO O(1s)→ π ∗ HF F(1s)→ σ ∗ MADa

284.7 285.8 286.0 287.4 401.0 530.8 534.2 687.4 -

LCgaucore-BOP‡ 286.1 285.0 285.6 286.5 401.6 531.8 535.2 686.4 0.7

CVR-B3LYP]

SRC1†

SRC2†

286.1 285.1 286.0 286.9 401.3 531.4 534.5 686.0 0.5

285.1 286.1 285.5 286.1 400.6 530.8 534.4 686.7 0.5

285.3 286.3 286.0 286.7 400.7 530.8 534.2 686.9 0.3

term is evaluated using HF exchange [2]. However, the application of such functionals for core-excited states results in a negligible change in the computed excitation energy. This behavior can be rationalized. Long-range corrected functionals introduce HF exchange for terms in which r12 is large. For these terms either r1 or r2 must be large. Any of the exchange integrals that involve a core orbital will be vanishing small due to the fact that the core orbital is very short-ranged. Consequently, it is perhaps not surprising that such long-range corrected functionals have a negligible influence on core-excitation energies. The logical conclusion from this analysis is that it is necessary to introduce HF exchange at short-range, i.e. when r12 is small. This was achieved within the LCgau-DFT scheme, wherein the 1/r12 operator is partitioned as [36] erfc(µr12 ) 2µ 2 2 erf(µr12 ) 2µ 2 2 1 = − k √ exp−(1/a)µ r12 + + k √ exp−(1/a)µ r12 r12 r12 π r12 π

(9)

where the first two terms describe the short-range interaction and the remaining terms give the long-range interaction. The inclusion of the Gaussian correction provides an additional contribution to the short-range term that can be tailored to introduce HF exchange at shortrange [37]. This functional form was optimized, through the three parameters µ, k, and a, and tested on a set of core-excitation energies from first row nuclei, and the resulting functional was called LCgau-core-BOP [37]. The performance of the LCgau-core-BOP functional is illustrated in Table 2 for a range of core→valence transitions. The functional does demonstrate a large reduction in the error compared to functionals such as BLYP and B3LYP. However, the mean absolute deviation (MAD) remains relatively high compared to typical errors that would be expected for valence→valence transitions, and is greater than the MAD for the CVR-B3LYP functional. One way to improve the LCgau-core-BOP functional is to have a larger fraction of HF exchange at r12=0. To achieve this, a short-range corrected functional that is based on a reversal of the standard long-range partitioning scheme was introduced [38]. In this func-

TDDFT Calculations of Core Excited States

155

tional, the electron repulsion operator is partitioned in the evaluation of the exchange energy using the error function according to 1 r12

erfc(µSR r12) erfc(µSR r12 ) − CSHF r12 r12 erf(µLR r12 ) 1 erf(µLR r12 ) − CLHF + . + CLHF r12 r12 r12 = CSHF

(10)

Treating the first and third terms of equation 10 with HF exchange and the remaining terms with DFT exchange leads to the following functional SRC1 Exc = CSHF ExSR−HF(µSR ) − CSHF ExSR−DFT (µSR) (11) LR−HF LR−DFT DFT DFT + CLHF Ex (µLR ) − CLHF Ex (µLR ) + Ex + Ec

where occ

ExLR−HF

1 XX = − 2 σ i,j x

Z Z

erf(µLR r12 ) r12

(12)

erfc(µSRr12 ) r12

(13)

∗ ∗ ψiσ (r1 )ψjσ (r1 )

ψiσ (r2 )ψjσ (r2 )dr1 dr2

and ExSR−HF

occ 1 XX = − 2 σ i,j

x

Z Z

∗ ∗ ψiσ (r1 )ψjσ (r1 )

ψiσ (r2 )ψjσ (r2 )dr1 dr2

respectively. The long and short-range DFT exchange is computed from modifying the usual exchange energy [39] Ex = −

1X 2 σ

Z

ρ4/3 σ Kσ dr

(14)

to give ExLR−DFT = −

1X 2 σ

Z

8 √ ρ4/3 σ Kσ aσ [ π erf 3



1 2aσ



+ 2aσ (bσ − cσ )]dr

(15)

and ExSR−DFT

1X = − ρ4/3 σ Kσ 2 σ   1 8 √ + 2aσ (bσ − cσ )]}dr x {1 − aσ [ π erf 3 2aσ Z

(16)

where 

bσ = exp −

1 4a2σ



−1

(17)

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Table 3. Computed excitation energies (in eV) for second row nuclei. a Mean absolute deviation Excitation SiH4 Si(1s)→ σ ∗ PH3 P(1s)→ σ ∗ H2 S S(1s)→ σ ∗ SO2 S(1s)→ π ∗ HCl Cl(1s)→ σ ∗ Cl2 Cl(1s)→ σu∗ MADa

Exp. 1842.5 2145.8 2473.1 2473.8 2823.9 2821.3 -

SRC1 1843.1(+0.6) 2146.2 (+0.4) 2473.1 (0.0) 2473.4 (-0.4) 2824.6 (+0.7) 2821.8 (+0.5) 0.4

SRC2 1842.5 (0.0) 2145.8 (0.0) 2472.9 (-0.2) 2473.1 (-0.7) 2824.5 (+0.6) 2821.7 (+0.4) 0.3

and cσ = 2a2σ bσ +

1 2

(18)

For the short-range component µSR Kσ1/2 aσ = √ ρ−1/3 6 π σ

(19)

and for the long-range component µLR aσ = √ ρ−1/3 Kσ1/2 6 π σ

(20)

This functional is combined with the LYP correlation functional [40] to give the full exchange-correlation functional. There are four parameters that are introduced, CSHF , CLHF , µSR and µLR . These are optimized to minimize the mean absolute deviation for a set of core excitation energies with the 6-311(2+,2+)G∗∗, yielding values of 0.50, 0.17, −1 0.56a−1 0 and 2.45a0 for CSHF , CLHF , µSR and µLR , respectively, for the K-edge of first row nuclei. Calculated excitation energies using this functional are given in Table 2 and show an improved agreement with experiment. This functional is implemented within the Q-Chem software package [41]. One limitation of this form of the exchange-correlation functional is that it is not accurate for core excitations from the K-edge of heavier second row nuclei. For these elements it is necessary to have a greater fraction of HF exchange at r12 = 0. This can be achieved through appropriate optimization of the four parameters, and −1 values of 0.87, 2.20, 0.25a−1 0 and 1.80a0 for CSHF , CLHF , µSR and µLR provide accurate core excitation energies for these nuclei as shown in Table 3. A closely related short-range corrected functional form was also considered SRC2 Exc = CSHF ExSR−HF(µSR ) + (1 − CSHF )ExSR−DFT(µSR )

+

CLHF ExLR−HF (µLR )

+

(1 − CLHF )ExLR−DFT(µLR )

(21) +

EcDFT

When µSR =µLR , this functional is equivalent to the SRC1 functional (equation 11). However, if µSR 6= µLR , the two functionals differ, and the SRC2 functional no longer corresponds to a rigorous partitioning of the electron-repulsion operator (equation 10). Re−1 optimization of the parameters gives values of 0.55, 0.08, 0.59a−1 0 and 1.02a0 and 0.91,

TDDFT Calculations of Core Excited States

157

Figure 2. Experimental spectra (black line) and computed spectra (red line). Experimental spectra adapted from references [42–44].

2.20, 0.28a0−1 and 1.80a−1 0 for CSHF , CLHF , µSR and µLR , for first row and second row nuclei, respectively. For this form of the functional there is a further decrease in the error, and the resulting MAD is comparable to the level of accuracy that is achieved with TDDFT calculations of excitations in the ultraviolet/visible region. Figure 2 shows experimental and computed NEXAFS spectra for four molecules not included in the parametrization of the functionals [38]. The spectra have been computed with the SRC1 functional and generated by representing each excitation by a Gaussian function with a full-width at half maximum of 0.3 eV. For butadiene, the calculation reproduces the essential features of the experimental spectrum well. Two peaks at 284.7 eV and 285.4 eV that correspond to C(1s)→ π ∗ excitations are predicted. The peak at low energy arises from excitation from the core orbitals on the end carbon atoms, while the peak at 285.4 eV arises from the 1s orbitals of the central carbon atoms. The broader peak at higher energy has contributions from a number of excitations. Peaks at 288.4 eV correspond to excitations to the higher energy π ∗ orbital, while additional peaks arise from excitation to Rydberg 3p orbitals. The experimental spectrum of ethane has a sharp peak at 288.4 eV with a shoulder on the low energy side and a broad band at 291.4 eV, which lies above the ionization threshold. The calculations predict four excitations with non-zero intensity below 289 eV that correspond to excitations to 3s and 3p Rydberg states that lie broadly in agreement with the experimental spectrum. Above the ionization threshold the calculations show a ∗ ornumber of significant bands, the most intense of which arises from excitation to a σCH bital. The sulphur K-edge experimental spectrum for 1-propanethiol has a broad intense peak at 2472 eV, with a very broad band at higher energy above the ionization threshold. The calculations show two intense bands at 2472.8 eV and 2473.6 eV, in good agreement with the first band in the experiment. These excitations correspond to excitations from the 1s orbital of sulphur to virtual orbitals localized on the thiol group. Two distinct peaks are

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evident in the silicon K-edge spectrum of Si(CH3 )3 Cl. The first peak has been assigned to ∗ a Si(1s)→σSiCl excitation [44]. The calculations predict an intense band in good agreement with this. In the remainder of this Chapter some specific applications of these functionals to study NEXAFS spectra of biologically important systems are described.

5. 5.1.

Applications NEXAFS of Protein Backbone Secondary Structure Elements Carbon K-edge

Nitrogen K-edge

Oxygen K-edge glycl-glycine

280 290 300 310 Energy / eV

400 410 420 Energy / eV

530

540 550 Energy /eV

Figure 3. Experimental (black line) and computed (red line) NEXAFS spectra for glyclglycine. Spectroscopic methods are used extensively to probe the geometric and electronic structure of polypeptides and proteins. Commonly used techniques, such as electronic circular dichroism (CD) and infrared spectroscopy (IR), provide information on secondary structure combined with the advantage that they can be measured on a sufficiently fast timescale to provide dynamic structural information [45]. In recent years, there has been a growing interest in the application NEXAFS, since the application of such techniques have the potential to inform on protein structure. Figure 3 shows computed and experimental NEXAFS spectra for glycyl-glycine [46], which serves as a model fragment of protein backbone. Calculations are presented for the SRC1 functional in conjunction with the 6-311G∗ basis set using the structure optimized at the MP2/cc-pVTZ level of theory. In these spectra, the region above the ionization threshold has been generated using the Stieltjes imaging technique [47]. For the carbon K-edge the calculation reproduces the experiment well. The intense peak at 288.3 eV arises from a combination of ∗ ∗ ∗ C1s (COOH)→πCOOH and C1s (CONH)→πCONH (or πamide ) transitions, which are com∗ ∗ puted to lie at 287.5 eV and 287.8 eV. Where πCOOH is a π orbital localized on the acid ∗ end of the molecule and πCONH is a π ∗ orbital localized on the amide end. The calculations predict that these two transitions are of similar intensity, and once broadened by Gaussians cannot be distinguished in the spectrum. The nitrogen and oxygen K-edge spectra show two distinct peaks before the broad continuum feature. These features are also evident in the calculated spectra. At the nitrogen K-edge the lower energy band arises from ∗ ∗ N1s(NH)→πCONH and N1s(NH2 )→πCONH transitions which are computed to have similar excitation energies, while the higher energy band corresponds to excitations to Rydberg 3p ∗ states. In the oxygen spectrum, the lower energy feature arises from C1s (CONH)→πCONH

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159

∗ ∗ and C1s (COOH)→πCOOH excitations, while the C1s (OH)→πCOOH lies at higher energy and contributes to the higher energy feature at 536 eV which also has contributions from excitations to Rydberg 3p orbitals.

"-sheet

turn

agitoxin !-helix

Figure 4. Secondary structure fragments.

Carbon K-edge

280 285 290 295 300

Energy / eV

Nitrogen K-edge

400

405

410

Energy / eV

Oxygen K-edge

530 535 540 545

Energy /eV

Figure 5. Computed NEXAFS spectra for the helix (black line), β-sheet (red line) and turn (blue line) secondary structure elements from the agitoxin 2 protein. These calculations on the relatively small molecules in the gas phase show that TDDFT with the short-range corrected functional provides an accurate description of the NEXAFS spectra, particularly in the region below the ionization continuum. The calculations also illustrate a useful feature that can aid in the interpretation of the spectra of the larger systems, namely, that intense bands can occur when the core and virtual orbital are spatially close, otherwise excitations will be much weaker. Overall, the agreement with experiment is quantitative, and the excitation energies are predicted within 1 eV. This agreement with experiment may be improved by accounting for the different molecular conformations. Furthermore, the calculations are largely consistent with previous theoretical analysis of these systems [48, 49]. Consequently, this theoretical approach can now be applied to larger polypeptides with some confidence to explore the NEXAFS spectra of proteins. More specifically, the dependency of the predicted spectra to the underlying secondary structure. The protein agitoxin 2 (PDB code 1AGT) has three distinct secondary structure elements, an α-helix, β-sheet and turn, shown in Figure 4, and the calculated NEXAFS spectra

160

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for these structures extracted from the protein are shown in Figure 5. The largest fragment is the β-sheet which comprises 110 atoms, and the spectra have been normalized such that the maximum peak height is equal. The carbon K-edge spectrum for the α-helix has a distinct band at 287.6 eV followed by a broad band. There are two general types of carbon atom in the helix, the carbonyl carbon (C=O) and the CH2 carbon. The band at 287 eV arises ∗ excitations. Furthermore, the intense contribution to this band occurs from C1s(CO)→ πCO ∗ for excitation to the πCO orbital that is localized predominantly on the same residue as the ∗ core orbital. The higher energy feature corresponds to excitations to σCH orbitals. Typical ∗ and σ ∗ orbitals are shown in Figure 6. Excitations from the C (CH ) orbitals result πCO 1s 2 CH in a broad feature centered at about 290 eV. The lower energy part of this band arises from ∗ excitations, and the higher energy part to transitions to σ ∗ orbitals. C1s (CH2 )→ πCO CH

!*C=O

"*CH

∗ and σ ∗ orbitals on an α-helix. Figure 6. Typical πCO CH

The oxygen K-edge has distinct bands at 532.4 eV for the α-helix and 532.7 eV for the ∗ 310 -helix arising from O1s→πCO excitations. Similarly to the carbon spectra, this feature ∗ comes from excitation to πCO orbitals localized on the same residue as the core orbital. ∗ The feature at higher energy can be assigned to excitation to πCO orbitals on neighboring ∗ residues and excitation to σCH orbitals. The nitrogen spectrum shows more structure with distinct peaks at 402.7 eV, 404.7 eV and 406.5 eV. The lower two peaks can be assigned to ∗ excitations. Closer inspection shows that the 402.7 eV band arises from excitaN1s→πCO tion to the π ∗ orbital located predominantly on the same residue as the nitrogen core orbital, whereas the 404.7 eV band is associated with the π ∗ orbital on a neighboring residue. The higher energy band is more difficult to assign, but comprises predominantly contributions ∗ from N1s→σCH excitations. At the carbon and oxygen edges, there is relatively little difference between the com∗ puted spectra for the three different elements of secondary structure. The C1s (CO)→πCO band lies at 287.6 eV with no significant variation. For the turn, a small feature at 285.9 eV is evident. However, this arises from a carbonyl group at the end of the model turn that is not part of a peptide bond. The most significant difference is in the intensity of the broad band at higher energy, which is more intense for the helix than the β-sheet. The spectra for the oxygen K-edge are qualitatively similar and show the same trends observed for the

TDDFT Calculations of Core Excited States

161

Table 4. Computed core excitations energies (in eV) for plastocyanin.‡ reference [7], † uncontracted basis functions Excitation Cu(1s)→SOMO Cu(2p)→SOMO S(1s)→SOMO

Exp.‡ 8978 931 2469

TD-B3LYP 8863 929 2417

TD-SRC1 8978 947 2469

B3LYP 8986 946 2465

B3LYP† 8980 936 2466

carbon edge. The most variation between the different structural elements is observed at the nitrogen edge. The spectra for the helix is similar to that obtained for the model α-helix ∗ ∗ with three distinct peaks which were assigned to N1s →πCO (lower two) and N1s→σCH excitations. The predicted spectra for the turn and β-sheet are significantly different with a single broader band at 403.2 eV with a shoulder at higher energy. The underlying cause of this change is that while for the α-helix there is a significant energy change between differ∗ excitations, i.e. to π ∗ orbitals located on the same or neighboring ent types of N1s→πCO CO residue, while for the turn and β-sheet this change is much smaller. Furthermore, for the ∗ and σ ∗ orbitals and this turn and β-sheet there is considerable mixing between some πCO CH ∗ ∗ leads to some of the intensity for the σCH band being absorbed into the predominantly πCO band.

5.2.

Copper Centered Proteins

Blue copper proteins, such as plastocyanin are important in a number of biological processes, such as photosynthesis [50]. The active site of plastocyanin comprises a copper centre coordinated with two histidine ligands, a cysteine ligand and a methionine ligand. The structure of the active site is characterized by a short Cu-S bond to cysteine and a long Cu-S bond to methionine. The oxidized form of the active site has a single occupied molecular orbital (SOMO) that is a out-of-phase combination of the Cu 3dx2 −y2 and Scys3pπ orbitals. Excitation of core electrons to this singly occupied orbital appear as pre-edge features in the X-ray absorption spectrum. Table 4 shows computed core excitation energies to the SOMO for the copper K and L-edges and the sulphur K-edge with TDDFT and ∆Kohn-Sham approaches. The calculations use the geometry taken from the crystal structure (1PLC) and the 6-311(2+,2+)G∗∗ basis set except for copper where the 6-31(2+,2+)G∗∗ basis set was used. The ∆Kohn-Sham approach gives core excitation energies that are in reasonable agreement with experiment. However, the difference from experiment is many electron volts, particularly for the excitations from copper. In a recent study, it has been shown that using uncontracted basis functions leads to a significant improvement in core excitation energies computed within a ∆Kohn-Sham approach [19]. For the K-edge excitations in plastocyanin, uncontracting the basis functions does result in an improvement in the calculated excitation energies and the results are in quite good agreement with experiment. The discrepancy for the Cu Ledge remains high. This is a result of the neglect of the 2p core-hole spin-orbit interaction which leads to two components [51]. The experimental value quoted is for the lower energy

162

Nicholas A. Besley

component of this excitation, and the calculations are predicting correctly a value that lies between the two components. For the TDDFT calculations, excitation energies for the B3LYP functional are, as expected, much lower than experiment. Due to its relatively high nuclear charge, this is particularly true for the Cu(1s) excitation, where an error of over 80 eV is observed. In contrast, the error for the excitation from the Cu 2p orbital is much lower. The 2p orbital of copper will be larger than the 1s orbital and can be thought of as less “core-like”. Following the overlap arguments discussed earlier, there will be a greater overlap between the 2p orbital and the single occupied orbital, and consequently the failure of the B3LYP functional will be less dramatic. This presents a further problem to finding a universal exchange-correlation functional that can be applied to all core excitations since a functional designed for K-edge excitations is likely to fail for L-edge excitations. This is illustrated by calculations with the SRC1 short-range corrected functional. This functional predicts excitation energies for the Cu and S K-edges that are in excellent agreement with experiment. This is in some sense surprising for the Cu(1s) excitation, since this type of excitation was not present in the data that was used to parametrize the functional [38]. However, the excitation energy for the Cu-L edge is too high and is further from experiment than the calculation with the B3LYP functional. In addition to the excitation energy, the intensity of the transition is also of interest. From the intensity it is possible to extract information about the nature of the wave function of the SOMO. For blue copper proteins, the intensity of the pre-edge feature at the S Kedge can inform about the relative contribution of sulphur p orbital and copper d orbital character in the SOMO [7]. The SOMO in these proteins is a mixture of copper d and sulphur p orbitals. Since the s→p transition is allowed and s→d is forbidden, the more intense the pre-edge feature, the greater the p character in the orbital. These intensities have been computed to be 0.0015 and 0.0011 for plastocyanin and the closely related blue copper protein cucumber basis protein, respectively [52]. This indicates that there is a greater p orbital character in the SOMO for plastocyanin. 2000

Time / s

1500

1000

500

0

0

1

2

3

4

5

6

7

Number of CPU

Figure 7. Cost of the TDDFT calculation for alanine. Black line with default integral thresholds and integration grid, and red line with modified integral thresholds and integration grid.

TDDFT Calculations of Core Excited States

6.

163

Towards Larger Systems

In order to extend the application of TDDFT such that it can readily be applied to study the NEXAFS spectra of larger systems, it is necessary to reduce the cost of the calculations. One obvious strategy is to exploit parallel computing. For TDDFT calculations with a SRC functional, the vast majority of the computational effort is used in evaluating exchange integrals involving the error function and complementary error function, and integrals involving the exchange-correlation functional (equation 2). An efficient approach to parallelizing TDDFT is to exploit parallel architecture in the evaluation of these integrals. Further computational savings can be made by re-examining the two-electron pre-screening threshold and the numerical integration grid used for the integrals involving the exchangecorrelation functional. In standard quantum chemistry packages two-electron integrals are screened and only evaluated if they satisfy conditions such as κab κcd ≥ τ where κab =

(22)

q

(ab|ab)

(23)

and τ is a threshold parameter that is typically set to 1x10−11 . We have found that for the TDDFT part of the calculation this threshold can be increased significantly to values such as 1x10−5 without any effect on the calculated excitation energies to a precision of 0.01 eV. Similarly, much smaller numerical integration grids can be used within the TDDFT calculation. The default integration grid in Q-CHEM is the SG-1 grid [53] which is a pruned Euler-MacLaurin-Lebedev-(50,194) grid (i.e., 50 radial points, and 194 angular points per radial point). We find that reducing this grid to (30,38) also results in no loss of precision for excitation energies at the 0.01 eV level.

! !"

! !"

! !"

! !"

! !"

! !"

! !"

! !"













Figure 8. Low lying π ∗ orbitals of porphine and zinc porphine. Figure 7 shows the scaling of the time for the TDDFT calculation with number of processors for a calculation of the lowest 50 core excitations at the K-edge of alanine with the 6-311(2+,2+)G∗∗ basis set using a desktop computer with dual quad core Intel Xeon 2000 MHz processors and 8 MB of memory. In this example, the time for the calculation using seven processors is reduced by about a factor of four. This represents quite a modest scaling

164

Nicholas A. Besley

Table 5. Excitation energies (in eV) for the bands below the ionization threshold. † Reference [55] Excitation Porphine N(1s)→π1∗ NH(1s)→π2∗ N(1s)→π4∗ NH(1s)→π4∗

Exp.†

Calc.

398.2 400.3 402.3 403.9

398.3 400.5 401.3 403.3

Excitation Zinc Porphine N(1s)→π2∗ N(1s)→π1∗ N(1s)→π4∗

Calc. 399.1 399.1 402.0

with the number of processors. However, this is a reflection of the size of the system and a more linear scaling would be expected for larger systems. The modification of the integral thresholds and integration grid results in a further decrease in time by about 20%. Table 5 shows excitation energies for the intense pre-ionization threshold excitations at the nitrogen K-edge for porphine and zinc porphine calculated using the SRC2 functional and 6-311(2+,2+)G∗ basis set. The calculations show that these intense bands arise from excitations to π ∗ orbitals, which are shown in Figure 8. This is consistent with previous calculations on these systems [54–56]. The calculations show that intense bands arise when exciting to π ∗ orbitals that are spatially located on the relevant nitrogen atom(s). Thus the NH nitrogens have an intense band from excitations to π2∗ , the other nitrogens to π1∗ , all nitrogens to π4∗ and excitations to π3∗ are weak and not observed since this orbital is localized on the outer ring. For porphine the excitation energies for the two lowest bands are very close to the experimental values. For the higher transitions there is a larger difference with experiment, but overall the agreement remains reasonably good. For zinc porphine, fewer distinct bands are predicted. The lowest band arises from excitations to π1∗ and π2∗ orbitals, while the higher energy band corresponds to excitations to the π4∗ orbital. This is consistent with previous calculations using the static exchange (STEX) method [55].

Conclusion The application of spectroscopic techniques in the X-ray region is becoming increasingly common across a wide range of research areas. Accurate calculations of X-ray absorption spectra provide an extremely useful tool that can be used alongside experimental measurements. Quantum chemical excited state methods such as TDDFT are used routinely to study electronic excitations in the UV region, and the application of such methods to the problem of core excitations has the prospect to provide accurate predictions of NEXAFS spectra at a relatively low computational cost that can be used by non-specialist users. The calculation of core excitations highlights deficiencies in exchange-correlation functionals. New exchange-correlation functionals that contain a high fraction of Hartree-Fock exchange in the short-range can lead to an improved agreement with experiment. This provides a framework that can be applied to give accurate NEXAFS spectra for relatively large systems. This has been illustrated through a number of applications involving biologically significant systems. However, a number of challenges remain to be addressed. The most

TDDFT Calculations of Core Excited States

165

significant is to describe relativistic effects adequately within the calculations. These effects lead to a shift in the energies of the core orbitals which becomes very large for heavier elements. Furthermore, spin-orbit coupling effects become important for L-edge excitations. Currently, the SRC functionals require different parameters for the K-edge of first and second row elements, and further parameters would be required to describe L-edge excitations. Ideally it would be possible to describe all core excitations with a single functional without resorting to different parametrizations.

References [1] D. J. Tozer, R. D. Amos, N. C. Handy, B. J. Roos, L. Serrano-Andres, Mol. Phys. 1999, 97, 859. [2] Y. Tawada,T. Tsuneda, S. Yanagisawa, T. Yanai and K. Hirao, J. Chem. Phys. 2004, 120, 8425. [3] T. Yanai, D.P. Tew and N.C. Handy, Chem. Phys. Lett. 2004, 393, 51. [4] J. St¨ohr, NEXAFS Spectroscopy, Springer Series in Surface Science Springer: Heidelberg,1996. [5] A. Nilsson and L.G.M. Pettersson, Surf. Sci. Rep. 2004, 55, 49. [6] J.L. DuBois,P.M. Mukherjee,T.D.P. Stack, B. Hedman, E.I. Solomon and K.O.Hodgson, J. Am. Chem. Soc. 2000, 122, 5775. [7] E.I. Solomon, R.K. Szilagyi, S.D. George and L. Basumallick, Chem. Rev. 2004, 104, 419. [8] J.E. Penner-Hahn, Coord. Chem. Rev. 2005, 249, 161. [9] P. Wernat, D. Nordlund, U. Bergmann, M. Cavalleri, H. Ogasawara, L.A. Naslund, T.K. Hirsch, L. Ojamae, P. Glatzel,L.G.M. Pettersson and A. Nilsson, Science 2004, 304, 995 [10] T. Tokushima, Y. Harada, O. Takahashi, Y. Senba, H. Ohashi,L.G.M. Pettersson, A. Nilsson and S.Shin, Chem. Phys. Lett. 2008, 460, 387. [11] W.J. Hunt and W.A. Goddard III, Chem. Phys. Lett. 1969, 3, 414. ˚ [12] H. Agren, V. Carravetta, O. Vahtras and L.G.M. Pettersson, Chem. Phys. Lett. 1994, 222, 75. ˚ [13] H. Agren, V. Carravetta, O. Vahtras and L.G.M. Pettersson, Theor. Chem. Acta 1997, 97, 14. [14] M. Stener, A. Lisini and P.Decleva, Chem. Phys. 1995, 191, 141. ˚ [15] L. Triguero,L.G.M. Pettersson and H. Agren, Phys. Rev. B 1998, 58, 8097.

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[16] H. Hsu,E.R. Davidson and R.M. Pitzer, J. Chem. Phys. 1976, 65, 609. ˚ [17] A. Naves de Brito, N. Correla, S. Svensson and H. Agren J. Chem. Phys. 1991, 95, 2965. [18] A.T.B. Gilbert, N.A. Besley and P.M.W. Gill J. Phys. Chem. A 2008, 112, 13171. [19] N.A. Besley, A.T.B. Gilbert and P.M.W. Gill, J. Chem. Phys. 2009, 130, 124308. [20] S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 1999, 314, 291. [21] A. Dreuw and M. Head-Gordon, Chem. Rev. 2005, 105, 4009. [22] D.R. Davidson, J. Computat. Phys. 1975, 17, 87. [23] T. Tsuchimochi, M. Kobayashi, A. Nakata, Y. Imamura and H.Nakai, J. Comput. Chem. 2008, 29, 2311. [24] U. Ekstr¨om and P.Norman, Phys. Rev. A 2006, 74, 042722. ˚ [25] U. Ekstr¨om, P. Norman, V. Carravetta and H. Agren Phys. Rev. Lett. 2006, 97, 143001. [26] M. Stener, G. Fronzoni and M. de Simone, Chem. Phys. Lett. 2003, 373, 115. [27] N.A. Besley and A. Noble, J. Phys. Chem. C 2007, 111, 3333. [28] S.D. George, T. Petrenko and F. Neese, Inorg. Chim. Acta 2008, 361, 965. [29] F.A. Asmuruf and N.A. Besley, J. Chem. Phys. 2008, 129, 064705. [30] Y.: Imamura, T. Otsuka and H.Nakai, J. Comp. Chem. 2006, 28, 2067. [31] M.J.G. Peach, P. Benfield, T. Helgaker and D.J. Tozer, J. Chem. Phys. 2008, 128, 044118. [32] A. Dreuw, J. Weisman and M. Head-Gordon, J. Chem. Phys. 2003, 119, 2943. [33] A. Nakata, Y. Imamura, T. Ostuka and H. Nakai, J. Chem. Phys. 2006, 124, 094105. [34] A. Nakata, Y. Imamura and H. Nakai, J. Chem. Phys. 2006, 125, 064109. [35] A. Nakata, Y. Imamura and H.Nakai, J. Chem. Theory Comput. 2007, 3, 1295. [36] J.-W. Song, S. Tokura, T. Sato,M. A. Watson and K.Hirao, J. Chem. Phys. 2007, 127, 154109. [37] J.-W. Song, M. A. Watson, A. Nakata and K. Hirao, J. Chem. Phys. 2008, 129, 184113. [38] N.A. Besley, M.J.G. Peach and D.J. Tozer, Phys. Chem. Chem. Phys. 2009, 11, 10350.

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In: Theoretical and Computational Developments... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 169-187

Chapter 6

D ENSITY F UNCTIONAL A PPROACH TO M ANY-E LECTRON S YSTEMS : T HE L OCAL -S CALING -T RANSFORMATION F ORMULATION Eugene S. Kryachko∗ Bogolyubov Institute for Theoretical Physics Kiev, Ukraine

Abstract We are definitely witnessing an ever-increasing need to study quantum mechanically the 21-million molecular world that surrounds us. The unique approach that is suitable for this purpose is the density functional theory. This Chapter outlines the basic features of the so called local-scaling-transformation formulation of the density functional approach or shortly LSDFT. “As a grad student at Columbia around 1950, I had the rare opportunity of meeting Albert Einstein. We were instructed to sit on a bench that would intersect Einstein’s path to lunch at his Princeton home. A fellow student and I sprang up when Einstein came by, accompanied by his assistant who asked if he would like to meet some students. “Yah,” the professor said and addressed my colleague, “Vot are you studying?” “I’m doing a thesis on quantum theory.” “Ach!” said Einstein, “A vaste of time!” He turned to me: “And vot are you doing?” I was more confident: “Im studying experimentally the properties of pions.” “Pions, pions! Ach, vee don’t understand de electron! Vy bother mit pions? Vell, good luck boys!” Leon Lederman, “Life in Physics and the Crucial Sense of Wonder” (CERN Courier, September 30, 2009) ∗

E-mail address: [email protected]

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PACS: 65.40DE, 31.15.E-, 31.15.-p, 31.10.+z, 31.15.bt Keywords: many-electron system, Density functional theory, Local-scaling transformation, Orbit, Variational principle, Self-consistent field concept.

1.

Density Functional Entourage

The present molecular world consists of about 21 million molecules. Generally speaking, a molecule M is a stable quantum Coulomb system consisting of the following two subsystems: • The electronic - of N electrons of the mass me and the charge −e which positions in the spin-configurational space are determined by the corresponding radii vectors r1 , r2 , . . . , rN where each ri , i = 1, 2, . . . , N belongs to the real three-dimensional space R3 and the spins σ1 , σ2 , . . . , σN where each σi , i = 1, 2, . . . , N takes the value from Z2 = {±1/2}, the discrete two-dimensional spin space • The nuclear - of M nuclei carrying the nuclear charges {Zα }M α=1 and located at {Rα ∈ R3 }M . α=1 According to L¨owdin’s definition [1, 2]: “A system of electrons and atomic nuclei is said to form a molecule if the Coulombic Hamiltonian H ′ with the centre of mass motion removed has a discrete ground state energy Eo ” (see also [3–5] and references therein) where b =H b e + Tbnn + U bnn is, respectively, the sum of the electhe total Hamiltonian H := H tronic Hamiltonian operator, the nuclear kinetic energy operator, and the nuclear-nuclear Coulomb interaction energy operator. Consider, within the Born-Oppenheimer approximation, the electronic Hamiltonian operator (in the atomic units) of M: b e = Tbe + U bee + Vben = − 1 H 2

N X

∇2ri

i=1

+

N X

1=i −∞: (Fii) the boundness from below of the expectation value hΨ | H In fact, (Fii) results from the aforementioned definition of molecule which lowest energy is bee and Vben are of Coulomb type, (Fii) is equivalent to finite. If U Te [Ψ] = hΨ | Te |Ψi < ∞

(4)

implying that Ψ ∈ LN is a differentiable function of all spatial coordinates, together with each component of ∇ri Ψ ∈ LN . One can prove [6, 7] that the conditions (Fi) and (Fii) fully determine LN of “admissible” N -electron wave functions where the energy functional b e | Ψi. E[Ψ] ≡ hΨ | H

(5)

b e Ψo = Eo Ψo , H

(6)

is thus well defined. Its lowest energy, the infimum, equal to the ground-state electronic energy Eo as the lowest eigenenergy of the N -body Schr¨odinger equation

is attained at the ground-state electronic wave function Ψo , that is Eo ≡

inf Φ ∈ LN



E[Φ] = E[Φ]|Φ=Ψo ∈LN .

(7)

The stationary quantum mechanical variational principle then reads as δE[Φ]|Φ=Ψo = 0.

(8)

The basic postulate of the many-electron density functional theory [8–13] suggests, first, the existence of the so called functional  E[ρ(r)] spin-restricted functional (9) E[ρ(x)] = E[ρ↑ (r), ρ↓ (r)] spin-polarized functional that has the meaning of the energy and depends, in some functional manner, on one-electron density ρ(r), Z X Z d3 r2 . . . d3 rN |Ψ(r, s1 ; r2 , s2 ; . . . ; rN , sN )|2 , Ψ ∈ LN (10) ρΨ (r) := N s1 ,...,sN

or on its both spin components, ρΨ↑ (r) and ρΨ↓ (r), ρΨ s (r) := ρΨ (r, s) = Z X Z d3 r2 . . . d3 rN |Ψ(r, s; r2 , s2 ; . . . ; rN , sN )|2 , = N s2 ,...,sN

s =↑, ↓ .

(11)

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The latter yield together ρΨ (r) = ρΨ↑ (r) + ρΨ↓ (r). Each ρΨ s (r) is normalized to Ns so that N↑ + N↓ = N . The second suggestion is that the infimum of E[ρ(r)] does exist and Eo ≡ inf {E[Φ]} = E[Φ]|Φ=Ψo = inf {E[ρ(r)]} = E[ρΦ (r)]|Φ=Ψo . Φ∈LN

(12)

ρ∈PN

where PN is a set of one-electron densities associated with LN (see below). Formally, this postulate looks rather strong, however it is widely accepted that it is guaranteed by the Hohenberg-Kohn theorem [8] (for the new proof of the Hohenberg-Kohn theorem see [14, 15]). Equation (9) assumes the existence of the “Functional mapping” F : E[Ψ] 7→ E[ρΨ (r)]

(13)

that implicitly presumes the existence of the “Variable mapping” Ψ ←→ ρΨ (r).

(14)

Obviously, the mapping (14) is valid if, first, there are defined the sets of “variables” on its left- and right-hand sides. Second, the symbol ←→ does not mean at all that this is precisely a one-to-one correspondence. The sub-mapping of (14), V : Ψ → ρΨ (r), is given by the reduction mapping, either (10) or (11), that is, ρΨ (r) = V(Ψ) and PN ≡ VLN . Besides, the reduction mapping has another facet - this is a so called N representability: any one-electron density obtained via V possesses its own image in LN . Generally speaking, the inverse mapping V−1 is one-to-many, that is, a given one-electron density has many pre-images in LN . It is trivial to show this. Let us consider any stable two-electron system which ground-state wave function and one-electron density are Ψo (r1 , r2 )[α(s p 1 )β(s2 ) − β(s1 )α(s2 )] and ρo (r), respectively. The two-electron Slater determinant ρo (r1 )ρo (r2 )[α(s1 )β(s2 ) − β(s1 )α(s2 )]/2 possesses the same one-electrondensity ρo (r) as well. Q. E. D. The Hohenberg-Kohn theorem [8] states however that there exists a one-to-one correspondence between the ground-state wave functions and groundstate densities.

2.

Local-Scaling Transformations, One-Electron Densities, and Many-Electron Wave

In order to properly assess the local-scaling-transformation formulation of the density functional theory, we first consider the concept of local-scaling transformation and second, apply it to the topological features of atomic and molecular one-electron densities.

2.1.

Mathematical Preliminaries: Local-Scaling Transformations f

Define on the Euclidean R3 the following mapping: R3 − → R3 such that r ∈ R3 is mapped into f (r) := f (r)er = f (r; er )er (15)

Density Functional Approach to Many-Electron Systems . . .

173

where er ≡ r/r ≡ e(Ω) is a unit vector, specified in R3 and defined by the spherical angles Ω = (θ, φ), and r =| r |. For a given er (Ω), the transformation (1) that deforms R3 onto itself, non-uniformly in general, is referred to as a local-scaling transformation or LST for short [7, 16–24] and is the special class of point transformation [21, 25]. LSTs satisfy all axioms of group and hence form the group F of local-scaling transformations. A scalar function f (r) in Eq. (1) can be arbitrary, though often it belongs to C 1 or higher. In the former, f is a C 1 -diffeomorphism on R3 . (1) non-trivially generalizes the well-known scaling: fλ (r) := λr which Fock [26] used in 1930 to prove the virial theorem. λ 6= 0 is a constant that means that all vectors r ∈ R3 are scaled uniformly. Bearing in mind that an arbitrary vector r is uniquely determined by its Cartesian coordinates r = (x, y, z), the equivalent representation of (1) is the following   x f → f (r) ≡  r≡ y − z 

x r f (x, y, z) y r f (x, y, z) z r f (x, y, z)

   xσ(r) fx (r)  ≡  fy (r)  ≡  yσ(r)  zσ(r) fz (r) 



(16)

where f (r) = σ(r)r. The Jacobian of (1) is defined as J{f (r); r} ≡ J{f ; r}  1  f − x32 f + x ∂f r ∂x  r r  y ∂f =  − xy f + r ∂x r3   − xz3 f + z ∂f r ∂x r =

− xy f+ r3

x ∂f r ∂y 2 y y ∂f 1 f − f + r r ∂y r3 z ∂f − yz f + r ∂y r3

f 2 ∂f ∂f ∂f 1 (x +y + z ) = 3 r · ∇f 3 . 3 r ∂x ∂y ∂z 3r

x ∂f r ∂z y ∂f − yz f + r ∂z r3 1 z2 z ∂f f − f + r r ∂z r3

− xz f+ r3

      

(17)

In terms of σ(r), the Jacobian (3) has the form J{f (r); r} = σ(r)[1 + r · ln σ(r)]. For the uniform scaling fλ :=, the corresponding Jacobian is equal to λ 0 0 J{fλ ; r} = 0 λ 0 0 0 λ

= λ3 .

(18)

It is trivial to generalize a three-dimensional local-scaling transformation (1) on other dimensions, say RD , simply by considering a vector r as a D-dimensional one. If D = 1, f (r) is a function of a single variable r. The corresponding Jacobian J{f (r); r} = df (r)/dr. Let consider some examples of local-scaling transformations fLST : [1]: f [1] = [( 1r )m + ( √δr )m ]−1/m , where δ - constant [27]. ( r(1 + ar2 )1/3 if r ≤ R q [2] [2]: f = d−1 d−2 + r + do + d1 r + dL ln r otherwise. r2

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This form results ( from the asymptotes at small and large r [28]. rq if r ≤ a [3]: f [3] = 8a a2 r 8r a a − r + r2 − 12 ln( a ) otherwise

[29]. [4]: Let Ω := ] − L/2, L/2[3 ⊆ R3 be a cube with volume |Ω| = L3 . f [4] is defined

as a periodic deformation on the cube Ω if it is a C 1 -diffeomorphism on R3 that leaves Ω invariant: f [4] (Ω) = Ω and if f [4] (r + Lm) = f [4] (r) + Lm for any m ∈ Z3 [30]. [5] [5] [5] [5] q [5]: fp,q,r is defined by the inverse function r(fp,q,r ) = [fp,q,r ]p (1 + α[fp,q,r ] )r where [5]

α, p, q, and r are variational parameters. If p = q = r = 1, r(fp,q,r ) refers to the [5] Hall’s transformation [31]. The other r(fp,q,r ) with q = r = 1, p = r = 1, and p = q = 1 were defined in [32, 33]. The Hall’s local-scaling transformation is then [5] f1,1,1 = [(1 + 4αr)1/2 − 1]/(2α). Let φ(r) be an arbitrary function given on domain Σ ⊂ R3 . A local-scaling transformation (16) transforms φ(r), generally speaking, into another function ψ(r) := φ(f (r))

(19)

within the Jacobian (3), depending on the normalization of φ(r) on Σ if any. If φ(r) = exp(−r) is a simple exponential orbital, under the Hall’s local-scaling transformation it converts to ψ(r) =

(1 + 4αr)1/2 − 1 exp(−[(1 + 4αr)1/2 − 1]/(2α)) 2αr(1 + 4αr)1/4

(20)

[32] that was used in [34] to approximate the 1s orbital.

2.2.

One-Electron Densities: Definition

A function ρ(r) : R3 → R1+ is defined as a one-electron density associated with some system of N electrons if: (Di) ρ(r) is non-negative everywhere in R3 ; (Dii) ρ(r) is normalized to the total number N of electrons, Z

d3 rρ(r) = N.

(21)

R3

Here R1+ stands for the non-negative semi-axis of R1 . (21) merely implies that the square root of ρ(r) is a square-integrable function, i. e. [ρ(r)]1/2 ∈ L2 (R3 ); (Diii) ρ(r) is a continuously differentiable function of r almost everywhere in R3 . It is a well-behaveness of densities. Let DN be the class of the one-electron densities associated with a Coulomb system of N electrons and obeying the conditions (Di)-(Diii). Obviously, VLN ⊂ DN .The fact that the condition (Diii) is valid for VLN is the consequence of the following

Density Functional Approach to Many-Electron Systems . . .

175

Proposal 1: For any ρΨ (r) = VΨ where Ψ ∈ LN , ∇r ρΨ (r) ∈ L2 (R3 ). Proof [35]: According to the Schwarz inequality, it follows from Eq. (11) that 2

[∇r ρΨ (r)] ≤ 4N ρΨ (r)

X Z

s1 ,...,sN

3

d r2 . . .

Z

d3 rN |Ψ(r, s1 ; r2 , s2 ; . . . ; rN , sN )|2 . (22)

Q.E.D. Corollary 1.1: ∇r [ρΨ (r)]1/2 ∈ L2 (R3 ). Proof [10]: Z Z 1 [∇r ρΨ (r)]2 3 1/2 2 d r(∇r [ρΨ (r)] ) = d3 r ≤ Te [Ψ]. 4 ρΨ (r)

(23)

Q.E.D. The term [∇r ρΨ (r)]2 /ρΨ (r) is known as the von Weizs¨acker kinetic energy tW [ρΨ (r)] [36]. 1/2 Hence, in the other words, Corollary 1 tells that tW [ρΨ (r)] is square-integrable. Usually, the von Weizs¨acker term is only a part of the total many-electron kinetic energy [7]. The exception is the Hartree-Fock 2-electron model systems for which tW [ρΨ (r)] is the exact kinetic energy1 . We further have Corollary 1.2: Thomas-Fermi [37, 38] one-electron density ρT F (r) is not N −representable. 1/2 Proof: According to [39, 40], tW [ρT F (r)] is not square-integrable. Q.E.D. Furthermore, the Thomas-Fermi energy density functional cannot be inserted in the density functional philosophy presented by the mappings (13) and (14) for all ρ(r) ∈ DN since the ground-state energies of many Thomas-Fermi atoms and ions2 lie below the exact ones3 . Consider a N-electron atom or ion with the nucleus centered at the origin of the Cartesian coordinate system. Let ρ(r) ∈ DN be one-electron density associated with a given  atom and ρ(r) = ρ(r, er ) | r ∈ R1+ , Ω ≡ (θ, φ), 0 ≤ θ ≤ π, 0 ≤ φ ≤ π is merely a bundle of one-dimensional curves. Let two densities ρ1 (r) and ρ2 (r) from DN be given. Both of them are represented by the corresponding bundles of curves. Let choose the unit vector er and in these bundles, the projections of ρ1 (r) and ρ2 (r) onto er - the curves ρ˜1 (r) and ρ˜2 (r) which are, according to (Diii), are continuously differentiable functions of r =| r |. Hence, they are homo-topically equivalent, or equivalently, there does exist such topological deformation that maps or deforms ρ˜1 (r) into ρ˜2 (r). Formally, ρ˜2 (r) = J{f (r); r}˜ ρ1 (f (r, er )).

(24)

1 Except H− which is unstable within the Hartree-Fock picture since its Hartree-Fock ground-state energy is equal to -0.488 Hartree and placed above Eo [H] = −0.5 Hartree. Note that the exact ground-state energy of H− is -0.5278 Hartree. 2 Thomas-Fermi molecules are unstable (see e. g. [7] and references therein). 3 Some of the widely used density functionals predict the ground-state energies below the experimental ones. For example, the B3LYP density functional in conjunction with the 6-31+G(d, p) basis set yields the energy -0.500273 Hartree < Eo [H] [41]. The B3LYP and B3PW91 show a similar trend for the atoms Li, C, O, F, Na, and Mg, and diatomics O2 , F2 , and LiF [42]. This implies that the corresponding ground-state wave functions, if do exist, are not square-integrable.

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Eugene S. Kryachko

The Jacobian in (24) ensures the normalization (Dii) for both densities ρ1 and ρ2 . Generalizing Eq.(24) over all directions in R3 results in that [7] ρ2 (r) =

1 r · ∇f 3 ρ1 (f (r)). 3r3

(25)

To hold the electron-nuclear Kato cusp, the nuclear position is invariant of f . If f is a uniform scaling fλ , the latter equation takes the form ρλ (r) = λ3 ρ1 (λr).

(26)

Given er , combining Eqs.(24) and (25) yields df (r, er ) r2 ρ2 (r, er ) = 2 dr f ρ1 (f (r, er ), er )

(27)

or in spherical coordinates, along a chosen unit vector er determined by Ω = (θo , φo ), r2 ρ2 (r, θo , φo ) df (r, θo , φo ) = 2 . dr f ρ1 (f (r, θo , φo ), θo , φo )

(28)

Equation (27), or (28), is the first-order nonlinear differential equation for deformation f (r) for given densities ρ1 and ρ2 . Due to (Diii), its solution exists and it is unique (see e. g. [7] and references therein). Therefore, for any pair of well-behaved densities, one enables to determine the deformation that transforms one of them into another. This means that F acts on DN transitively, that is, in algebraic terminology, DN is a single orbit with respect to F. For a given and fixed density ρ1 (r), defined hereafter as the generator density ρg (r), Eq. (26) then implies Proposal 2: There exists the one-to-one correspondence between F and DN that is explic[g] itly expressed as f ∈ F ⇔ ρf (r) := J{f ; r}ρg (f (r)). In the integral form, Eq. (28) is as follows  Z f (r, θo , φo ) = 3

r

ro

r2 ρ2 (r, θo , φo ) dr ρ1 (f (r, θo , φo ), θo , φo )

1/3

.

(29)

Note that the rhs of (29) contains a cubic root that reflects that the group F of local-scaling transformations acts on R3 . It is shown in [43] that the dimensionality D of RD enters the corresponding Jacobian in the power D and, respectively, the corresponding integral form as 1/D. This is on the one hand. On the other there exists another remarkable facet of Eq. (28). This equation is well-known in mathematics as the “Jacobian problem” ( [44, 45], see also [46, 47]).

2.3.

Many-Electron Wavefunctions and Concept of Orbit

To build the “variable mapping” (14), let generalize the concept of the local-scaling transformations on LN . This is rather simply and straightforward. For this purpose, let us choose an arbitrary “reference” or generator wave function Φg ({ri , σi }i=N i=1 ) where σi is

Density Functional Approach to Many-Electron Systems . . .

177

spin of the ith electron and ρg (r) ∈ DN is the associated one-electron density. Then define a new wave function Φf ({ri , σi }) = Φρ ({ri , σi }) ≡

N Y i=1

1/2 J(f (ri ); ri ) Φg ({f (ri ), σi })

(30)

with the density ρ(r) ≡ ρg (f (r)) casting in Proposal 2. Φf is a locally scaled image of the “reference” wave function. Formally, Φf ≡ F Φg where F ∈ F ×N := [×]N F and F = (f, f, . . . , f ) := f ×N and (30) is nothing then else as the definition of the action of the group F ×N on LN . Arbitrariness in choosing Φg ensures the validity of the definition (30) on the entire LN . Due to the isomorphism of the groups F and F ×N , it is obvious that a local-scaling transformation that maps a given pair of N-electron wave functions into each other matches unambiguously the local scaling that transforms the corresponding oneelectron densities into each other. However, although any pair of densities are locally scaled, this property no longer holds for an arbitrary pair of N-electron wave functions. Hence, LN is non-trivially partitioned, with respect to the group F ×N of local-scaling transformations, into the orbits [ O[i] . (31) LN = i

In this sense, the group F entangles DN and LN . By construction, an arbitrary orbit O[i] is closed with respect to F ×N , that is, for any pair Φ1 and Φ2 in O[i] , there exists such localscaling transformation F1⇒2 that Φ2 = F1⇒2 Φ1 . In the other words, if Φ1 is the generator wave function of O[i] , for all F ∈ F ×N , F1⇒2 Φ1 ∈ O[i] . We thus prove Proposal 3: There exists a one-to-one map of variables on any orbit in LN . Corollary 3.1: Orbit O[i] is invariant relative to generator wave function. Corollary 3.2: On each orbit O[i] ⊂ LN , there exists one and only one N -electron wave function which one-electron density is ρ(r) ∈ DN . [k] Corollary 3.3: For any given orbit O[k] ⊂ LN generated by Φg and the latter one-electron [k] [k] density ρg , Fρg exhausts the whole DN . Remark 1: Corollary 3.3 implies that any density ρ(r) ∈ DN is N -representable. In the other words, the group F of local-scaling transformations and its actions on DN and LN defined above ensures the N-representability of DN . The uniqueness of the local-scaling transformation as the solution of Eq. (27) guaran[i] tees that the transformed wave function Φρ is also unique. Thus, for any ρ(r) ∈ DN there [i] exists a unique wave function Φρ generated by means of local-scaling transformation from [k] the arbitrary generator wave function Φg . The orbit in LN is actually the set of all the wave functions thus generated which yield one-electron densities ρ(r) in DN : [i] O[i] ≡ {Φ[i] r); ρ | Φρ −→ ρ(~

Φ[i] ρ ∈ LN ;

ρ(~r) ∈ DN }.

(32)

Therefore, the orbit patterns in LN predetermine the inverse “variable mapping” V that was the premise in (13) and (14) and that naturally generalizes the Hohenberg-Kohn theorem on the entire set DN . Note that LN includes N -electron Slater determinants which are structurally invariant with respect to F ×N . Define SN as the proper subset of LN consisting of Slater determi-

178

Eugene S. Kryachko S nants. Since F ×N SN ⊆ SN , then SN = i OS [i] over all Slater orbits. Thus, we have: Corollary 3.4: An arbitrary one-electron density ρ(r) ∈ DN is N -representable in SN .

3.

Energy Density Functional and Variational Principle

3.1.

Energy Density Functional: Definition

Proposal 3 definitely allows to propose the following rigorous definition of the energy density functional     (33) Ei ρ(r) ≡ Ei ρ(r); Φ[i] g := E[Φ]|Φ∈O[i] ⊂LN

and hence express the “functional mapping” (13) in the explicit way. It is evident that this mapping is one-to-many functionals as there are orbits in   and there exist as many density [i] LN . To derive Ei ρ(r) that is defined on the orbit O ⊂ LN explicitly, let first write [i]

down the explicit expression for the energy functional E[Φg ] of the orbit-generating wave 2[i] [i] 1[i] function Φg in terms of its 1- and 2-matrices, Dg (x1 , x′1 ) and Dg (x1 , x2 ; x1 , x2 ), respectively, and its one-electron density ρg (x): Z Z 1 [i] 4 1[i] ′ E[Φg ] = d x1 ∇x1 ∇x′1 Dg (x1 ; x1 ) x′ =x1 + d4 xρ(x)ˆ v (r) (34) 1 2 Z Z 2[i] Dg (x1 , x2 ; x1 , x2 ) 4 + d x1 d4 x2 . |r1 − r2 | R P R where d4 x ≡ s d3 r. Let us apply the local-scaling transformation that casts in Pro[i] posal 3 to the wave function Φg , precisely to its 1- and 2-matrices, and its density. This yields: h i1/2 1[i]  Dg (f (r1 ), s1 ; f (r′1 ), s′1 ), (35) Dρ1[i] (r1 , s1 ; r′1 , s′1 ) = J f (r1 ); r1 J f (r′1 ); r′1   Dρ2[i] (r1 , s1 , r2 , s2 ; r1 , s1 , r2 , s2 ) = J f (r1 ); r1 J f (r2 ); r2

(36)

Dg2[i] (f (r1 ), s1 , f (r2 ), s2 ; f (r1 ), s1 , f (r2 ), s2 ),

and   ρ(r, s) = J f (r); r ρg f (r), s .

(37)

Partitioning the 1- and 2matrices, appearing in the rhs of Eqs. (35) and (36), into their local and non-local components: Dg1[i] (f (r1 ), s1 ; f (r′1 ), s′1 ) = =

(38) h

 ρg f (r1 ), s1 ρg f (r′1 ), s′1

Dg2[i] (f (r1 ), s1 , f (r2 ), s2 ; f (r1 ), s1 , f (r2 ), s2 ) =

i1/2

e g1[i] (f (r1 ), s1 ; f (r′1 ), s′1 ), D

  1 ρg f (r1 ), s1 ρg f (r2 ), s2 2

(39)  [i] 1 + Fxc,g (f (r1 ), s1 , f (r2 ), s2 )

Density Functional Approach to Many-Electron Systems . . . 1[i]

179

[i]

e g is the non-local part of the 1-matrix and Fxc,g is the non-local exchangewhere D correlation factor. Therefore, the 1- and 2-matrices of (30) take the appearance:

and

h i1/2 e g1[i] (f (r1 ), s1 ; f (r′1 ), s′1 ), Dρ1[i] (r1 , s1 ; r′1 , s′1 ) = ρ(r1 , s1 )ρ(r′1 , s′1 ) D

(40)

2[i]

Dρ (r1 , s1 , r2 , s2 ; r1 , s1 , r2 , s2 ) =   [i] = 21 ρ(r1 , s1 )ρ(r2 , s2 ) 1 + Fxc,g (f (r1 ), s1 , f (r2 ), s2 ) .

(41)

Finally, we obtain [1]   [i] E[Φ[i] ρ ] ≡ E ρ(x); Φg  2 Z Z ∇r ρ(x) 1 1 4 e g1[i] (f (r), s; f (r′ ), s′ )|x′ =x = d x d4 xρ(x)∇r ∇r′ D + 8 ρ(x) 2 Z + d4 xρ(x)v(r)   [i] Z ρ(x )ρ(x ) 1 + F (f (r ), s , f (r ), s ) xc,g 1 2 1 1 2 2 1 d4 x1 d4 x2 . (42) + 2 |r1 − r2 | Few statements can be drawn from Eq. (42): (i) The kinetic energy density functional is composed of two components. The first, the von Weizs¨acker term, is local and orbit-invariant. The second is non-local, orbit-dependent, and due to the one-third power in Eq. (29), transformed to the modified Thomas-Fermi term within the local density approximation; (ii) The exchange-correlation energy density functional is explicitly expressed as   := (43) Exc [Φρ[i] ] ≡ Exc ρ(x); Φ[i] g Z [i] ρ(x1 )ρ(x2 )Fxc,g (f (r1 ), s1 , f (r2 ), s2 ) 1 d4 x1 d4 x2 ; := 2 |r1 − r2 |  [i]  (iii) In fact, each density functional E ρ(x); Φg depends upon two basic variables: the [i]

one-electron density ρ(x) and the generator wave function Φg . Equation (42) expresses the energy as a functional of the one-electron density ρ(x) within the orbit O[i] . True, Eq. (42) satisfies the condition of N -representability; (iv) One of the orbits in the decomposition (31) of LN is actually the orbit that contains the exact ground-state wave function. Refer it to as the Hohenberg-Kohn orbit O[HK] ⊂ LN . If a generator wave function is chosen to belong to O[HK] , Eq. (42) then determines the Hohenberg-Kohn energy density functional in the explicit manner.

3.2.

Orbit Variational Principle and Euler-Lagrange Equation

The variational principle of the energy density functional theory based on the definition (33) is a straightforward consequence of the quantum mechanical variational principle (8)

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Eugene S. Kryachko

and the “functional mapping” (13). It is clearly orbit-dependent or, equivalently, it is of the [i] intra-orbit type. Let consider the energy density functional E[ρ(r, s); Φg ] given by Eq. (42) and defined within the O[i] only. In this functional, ρ(r, s) stands for the density variable [i] resulted from the initial density ρg (r, s) associated with the generator wave function Φg . [i] [i] The extremum of E[ρ(r, s); Φg ] on DN is attained at the ith-optimal density ρopt (r, s) which is obtained by varying the following auxiliary functional  Z [i] [i] E[ρ(r, s); Φg ] − µ d4 x ρ(r, s) − N (44) where µ[i] is the Lagrange multiplier that accounts for the normalization of the density and that actually plays the role of a chemical potential on the orbit O[i] . Therefore, the stationary [i] ground-state variational principle for the energy density functional E[ρ(r, s); Φg ] is given by ) (  Z δ [i] = 0, ρ(r, s) ∈ DN , (45) E[ρ(r, s); Φ[i] d4 x ρ(r, s) − N g ]− µ δρ(r, s) we obtain the following integro-differential equation for the one-electron density [1]:    1 ∇2 ρ(r, s) 1 ∇ρ(r, s) 2 [i] − + vT,g [ρ(r, s)]; r, s + v(r) 8 ρ(r, s) 4 ρ(r, s)   [i] [ρ(r, s)]; r, s = µ[i] (46) + vH [ρ(r, s)]; r + vxc,g R where vH ([ρ(r, s)]; r) = d4 xρ(r, s)|r − r ′ |−1 is the Hartree potential, (    1 [i] e g1[i] f (r), s; f (r ′ ), s′ ′ vT,g [ρ(r, s)]; r, s = ∇r ∇r ′ D (47) r =r,s′ =s 2 )    δ ′ ′ 1[i] e g f (r), s; f (r ), s ∇r ∇r ′ D . + ρ (r, s) r ′ =r,s′ =s δρ(r, s) is the potential originated from the non-local component of the kinetic energy in (42), and  [i] [i]   δExc,g [ρ(r, s); Φg ]; r, s [i] [i] [i] vxc,g [ρ(r, s)]; r, s = Exc,g [ρ(r, s); Φg ]; r, s + ρ(r, s) (48) δρ(r, s) the exchange-correlation potential resulted from the non-local part of the electron-electron interaction where  Z [i]  1 ρ(r2 , s2 )Fxc,g f (r1 ), s1 ; f (r2 ), s2 [i] [i] 4 Exc,g [ρ(r1 , s1 ); Φg ]; r1 , s1 = d x2 . (49) 2 |r1 − r| [i]

Solving Eq.(46) for the given generator wave function Φg , we obtain the i-th optimal or [i] i-th approximate ground-state density ρo (r, s) ∈ DN and the i-th optimal or i-th groundstate energy   (50) Eo[i] ≡ Ei ρo (r)

Density Functional Approach to Many-Electron Systems . . .

181

that simply casts as the i-th orbit variational principle: Eo[i] ≡

inf

Φ∈O[i] ⊂LN

{E[Φ]} = E[Φ]|Φ=Ψ[i] ∈O[i] ⊂L = o

N

inf

ρΦ →Φ∈O[i]

{Ei [ρΦ ]}.

[i]

(51) [i]

The next step is to substitute the densities ρ1 (r) and ρ2 (r) by ρg (r) and ρo (r) in Eq.(25) correspondingly and to solve the latter. The solution is the i-th optimal local[i] [i] scaling transformation fo (r) ∈ F which is further applied to Φg to get, via Eq.(30), the [i] i-th optimal, ground-state wave function Φo ∈ LN . True, generally speaking, the latter is the approximate ground-state wave function that yields an upper bound to the exact groundstate energy Eo which is attained, by definition, only at the Hohenberg-Kohn orbit O[HK] , [HK] that is, Eo = Eo .

4.

Global Variational Principle: The Concept of Local-Scaling Self-Consistent Field

The orbit variational principle (44) deduced in Subsection 3.2 is solely defined on a particular orbit. The reason is trivial: this is precisely that orbit where the energy density functional is defined according to Eq. (33). In contrast, the global ground-state quantum mechanical variational principle (8) is carried out over the whole Hilbert space LN . Within the local-scaling formulation of the density functional approach is achieved due to the fact that the energy density functional in fact depends on two basic variables of theory: one the one-electron density - is the key variable of the density functional theory and the other is the generator wave function that determines an orbit. Hence, the orbit partitioning (31) of LN is governed by the orbit generators. Therefore, Eo =

inf

{ inf

over all orbits ρ(r)∈DN

{E[ρ(r); Φ[i] g ]}}.

(52)

O [i] ⊂LN

(52) implies that the search for the exact ground-state wave function must be carried out by a combined intra-orbit and inter-orbit minimization [7]. The former reflects the charge consistency variational principle, whereas the latter the inter-orbit one, the orbit consistency. The latter is actually the variational principle of the “inter-orbit” self-consistent field that resembles the Kohn-Sham self-consistent field approach and results in inter-orbit “jumps” that finally leads to the exact, Hohenberg-Kohn orbit.

4.1.

Computational Aspects of Energy-Density-Functional Variational Principle

One-electron density casts as the basic variable of the density functional theory.

4.2.

Trial One-Electron Densities

The key computational problem in applying the variational principle (45) to the density functional (42) is to properly describe the set(s) of trial one-electron densities. One may

182

Eugene S. Kryachko

define a set of trial densities as [48] n o [ n X ci rai exp(−bi rdi )|n, {ai }, {bi }, {ci }, {di } = A[r] Aρ := ρ(r) = ρ :

A1 ⊂ A3 ⊂ A5 ⊂ A6 ⊂ A7 ⊂ A8 ⊂ A9 ⊂ A10 ⊂ A11 ⊂

r i=1 [1] [2] Aρ : a1 = 0, d1 = 1; A2 ⊂ Aρ : d1 = 1; [3] A[2] ρ : a1 = 0; A4 ⊂ Aρ : none of above A[3] ρ : a1 = a2 = 0, d1 = d2 = 1, c = c2 /c1 ; A[4] ρ : a1 = a2 , d1 = d2 = 1, c = c2 /c1 ; [4] Aρ : a1 = a2 = 0, d1 = d2 , c = c2 /c1 ; A[4] (53) ρ : a1 = 0, d1 = d2 = 1, c = c2 /c1 ; ′ A[5] ρ : a1 = a2 = a3 = 0, d1 = d2 = d3 = 1, c = c2 /c1 , c = c3 /c1 ; ′ A[7] ρ : a1 = 0, d1 = d2 = d3 = 1, c = /c2 /c1 , c = c3 /c1 ; A[10] : a1 = 0, d1 = d2 = d3 = d4 = 1, c = c2 /c1 , c′ = c3 /c1 , c” = c4 /c1 ρ

and so on. r is a number of adjustable parameters. True, Aρ does not cover all trial densities for atoms and their ions, however, it shows pretty good variational results. Moreover, Remark 2: It is not guaranteed that Aρ is a set of N -representable densities for some N . All trial densities that belong to Aρ , excepting those from A3,4,7 , are precisely linear combinations of generalized Slater-type functions. This allows to handle the integrals in Eq.(28), determining the required local scaling, analytically by means of the formula [49] Z x dt tb exp(−ct) = c−(b+1) γ(b + 1, cx) (54) 0

for x > 0 and b > −1. Here γ(a, x) is incomplete gamma function. For a given set of parameters, the energy density functional EHF [ρ; parameters] can be evaluated numerically by means of the Romberg integration method. The parameters can be optimized to minimize the energy via the Powell method of conjugate directions.

4.3.

Final Act: A Purely Computational Theme

It would be a quite natural wish to finalize the present review of the local-scaling energy density functional approach with the words that definitely show the perspectives of this approach. Very often, the perspectives are perceived through the computational results, through the numbers. Such are undoubtedly the words which are expressed by the authors of the work [50]: “The LSDFT is a rigorous formulation of DFT, constructive in nature, and satisfies the N and v representability conditions on the energy functional. In principle, the method is applicable to Hartree-Fock or Kohn-Sham Hamiltonians and yields the corresponding orbitals and energies.” Tables 1-3 demonstrate those numbers obtained within the LSDTF [51–55] 4 . The subscript a is the Hartree-Fock limit, b the unrestricted Hartree-Fock limit, c is the exact level, and d is the Kohn-Sham self-consistent field method. 4

Density Functional Approach to Many-Electron Systems . . .

183

Table 1. The He-isoelectronic series by means of the local-scaling-transformation density functional approach

System H−

He

Li+

Be2+

B3+

Li

Be

Trial density A9 Kellner orbit: a = 0.8106, b = 8.2572, c = 1.3857, d = 13.704, e = 2.4380 Eckart orbit: a = 1.0425, b = 10.331, c = 2.0879, d = 0.0913, e = 0.4951, α = 1.0371, β = 0.2836 Kellner orbit: a = 2.8024, b = 1.4190, c = 3.5822, d = 1.5099, e = 5.2275 Eckart orbit: a = 2.7120, b = 1.5012, c = 3.5358, d = 2.1147, e = 4.9848, α = 2.1926, β = 1.1900 [1s2s] orbit SCF LSDFT intra-orbit: Eckart orbit intra-orbit: [1s2s] orbit inter-orbit: Eckart orbit inter-orbit: [1s2s] orbit LS to 2-matrix Kellner orbit: a = 4.7577, b = 0.9456, c = 5.4893, d = 0.9931, e = 7.8069 Eckart (∼Kohn-Sham) orbit: a = 4.6691, b = 1.1892, c = 5.3512, d = 1.4029, e = 7.5699, α = 3.3015, β = 2.0789 Kellner orbit: a = 6.7564, b = 0.6527, c = 7.5102, d = 0.6669, e = 10.448 Eckart (∼Kohn-Sham) orbit: a = 7.4391, b = 1.4854, c = 6.7092, d = 1.1684, e = 10.225, α = 4.3960, β = 2.9848 Kellner orbit: a = 8.7828, b = 0.3945, c = 9.6049, d = 0.4661, e = 13.079 Eckart (∼Kohn-Sham) orbit: a = 8.7622, b = 0.3273, c = 9.7153, d = 0.4800, e = 12.915, α = 5.4789, β = 3.9012 A11 co = 6.05500, a1 = 0.637707, b1 = 1.77484, c1 = 5.78600, a2 = 0.13263 × 10−3 , b2 = 3.66310, c2 = 1.50250, a3 = 0.80130 × 10−3 , b3 = 4.26890, c3 = 2.28769, ζ1 = 2.56539, ζ2 = 0.62120 Raffenetti orbit Spin-polarized LSDFT:ρα ∈ A11 , ρβ ∈ A9 co = 8.50132, a1 = −0.30266, b1 = 2.00550, c1 = 5.13355, a2 = 0.16893, b2 = 0.00000, c2 = 4.64942, a3 = 6.88132 × 10−3 , b3 = 1.21894, c3 = 1.69038, ζ1 = 3.65424, ζ2 = 0.95693 Clementi-Roetti orbit Raffenetti orbit [1s2 2s2 ] orbit CI [1s2 2s2 ] + [1s2 2p2 ] [2s2 2p2 ] + [1s2s2p2 ] orbit

Energy

−0.487926 [−0.4879297]a

−0.5134555 −2.8616799 [−2.861680]a

−2.8768084 −2.8768069 −2.8771522 −2.8768084 −2.8771907 −2.903002 [−2.903724]c −7.2364148 [−7.2364152]a

−7.2501004 −13.611299 [−13.611299]a

−13.624400 −21.986233 [−21.98623]a

−21.999018

−7.431670 [−7.4327269]a −7.432720 −7.431859 [−7.432751]b

−14.569644 [−14.573023]a −14.5725 −14.573014 −14.5730039 −14.612495

184

Eugene S. Kryachko Table 2. The ground-state calculations of the selected atoms via the local-scaling-transformation density functional approach System B N F Na Al P Cl

Energy −24.529049 [−24.529057]a −54.398488 [−54.400934]a −99.409204 [−99.409349]a −161.84417 [−161.85893]a −241.84121 [−241.87671]a −340.65832 [−340.71878]a −459.39162 [−459.48207]a

System C O Ne Mg Si S Ar

Energy −37.688695 [−37.688619]a −74.805922 [−74.809399]a −128.547096 [−128.54710]a −199.58802 [−199.61464]a −288.80725 [−288.85436]a −397.42942 [−397.50490]a −526.71369 [−526.81751]a

Table 3. The ground-state calculations of the some atomic clusters via the local-scaling-transformation density functional approach System LiH 1 Σ+ R = 3.015 bohr Li2 1 Σ+ g R = 5.051 bohr Na2 ,Na7 Al, Na20 ,Si14 , Al13

Method and Energy −7.9814 [−7.9874]a [−7.9870]d −14.8638 [−14.8716]a [−14.8708]d Pseudopotential LSDFT [50]

Acknowledgments The author gratefully thanks all friends and colleagues with whom he shared the ideas of the density functional theory during the last three decades, in particular Ivan Zh. Petkov, Mario V. Stoitsov, Eduardo V. Lude˜na, Toshi Koga, Jean-Lois Calais, Ingvar Lindgren, PerOlov L¨owdin, Julian Schwinger, B. M. Deb, Enrico Clementi, Bob Parr, and Erkki Br¨andas, and Brian Sutcliffe. Amlan K. Roy is gratefully acknowledged for his kind invitation.

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[7] E. S. Kryachko and E. V. Lude˜na, (1990) Energy Density Functional Theory of ManyElectron Systems, Kluwer Academic Publishers, Dordrecht. [8] P. Hohenberg and W. Kohn,Phys. Rev. 136B, (1964) 864. [9] W. Kohn and L.J. Sham,Phys. Rev. 140A (1965) 1133. [10] E.H. Lieb, Int. J. Quantum Chem. 24 (1983) 243. [11] R. G. Parr and W. Yang, (1989) Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford. [12] R. M. Dreizler and E. K. U. Gross, (1990) Density Functional Theory, SpringerVerlag, Berlin. [13] N. H. March, (1992) Electron Density Theory of Atoms and Molecules, Academic Press, New York. [14] E. S. Kryachko,Int. J. Quantum Chem. 103, 818 (2005). [15] E. S. Kryachko, Int. J. Quantum Chem. 106, 1795 (2006). [16] I.Zh. Petkov and M.V. Stoitsov, Theor. Math. Phys. 55, 584 (1983). [17] E. S. Kryachko, (1984) In J. P. Dahl and J. Avery (eds.), Local Density Approximations in Quantum Chemistry and Solid State Physics, Plenum, New York, pp. 207-227. [18] I. Zh. Petkov, M. V. Stoitsov and E. S. Kryachko, 149.(1986).

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[19] E. S. Kryachko, I. Zh. Petkov and M. V. Stoitsov, Int. J. Quantum Chem. 32, 467.(1987). [20] Kryachko, E. S., Petkov, I. Zh. and Stoitsov, M. V. (1987) Int. J. Quantum Chem. 32, 473; Ibid. 34, 305(E) (1988). [21] E. S. Kryachko and E. V. Lude˜na, Phys. Rev. A 35, 957.(1987). [22] M.V. Stoitsov and I.Zh. Petkov, Ann. Phys. (N.Y.) 184, 121 (1988). [23] I.Zh. Petkov and M.V. Stoitsov,Nuclear Density Functional Theory, Oxford Studies in Physics Clarendon, Oxford, 1991. [24] E. S. Kryachko and E. V. Lude˜na, Phys. Rev. A 43, 2179-2193.(1991). [25] F. M. Eger and E. P. Gross, Ann. Phys. (N.Y.) 24, 63 (1963). [26] V. A. Fock, Z. Phys. 63, 855 (1930). [27] A. M. Moro, J. M. Arias and J. G´omez-Camacho, Phys. Rev. C 80, 054605 (2010). [28] S. Pittel and M. V. Stoitsov,Phys. At. Nucl. 64, 1055 (2001).

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Eugene S. Kryachko

[29] M.V. Stoitsov, P. Ring, D. Vretenar and G.A. Lalazissis, Phys. Rev. C 58, 2086 (1998). [30] O. Bokanowski, B. Grebert and N. J. Mauser, J. Mol. Struct. (Theochem) 501-502, 47 (2000). [31] G. G. Hall, Proc.Phys. Soc. (London) 75, 575 (1960). [32] M. J. ten Hoor, Int. J. Quantum Chem. 33, 563 (1988). [33] M. J. ten Hoor, J. Phys. B: At. Mol. Opt. Phys. 22, L89 (1989). [34] G. H¨ojer, Int. J. Quantum Chem. 15, 389 (1979). [35] T. L. Gilbert, Phys. Rev. B12 (1975) 2111. [36] C. F. von Weizs¨acker, Z. Phys. 96 (1935) 431. [37] L.H. Thomas, Proc. Cambridge Phil. Soc. 23 (1927) 542. [38] E. Fermi, Rend. Accad. Naz. Lincei 6 (1927) 602. [39] P. Gomb´as, Die Statistische Theorie des Atoms und Ihre Anwedungen, Springer, Wien, 1949, [40] J. Katriel and M. R. Nyden, J. Chem. Phys. 74, 1221 (1981). [41] L. A. Eriksson, E. S. Kryachko and M. T. Nguyen, Int. J. Quantum Chem. 99, 841 (2004). [42] J. Paier, M. Marsman and G. Kresse, J. Chem. Phys. 127, 024103 (2007). [43] E. S. Kryachko, In Lecture Notes in Chemistry, Vol. 50, Springer, Berlin, 1989. pp. 503-522. [44] J. Moser, Trans. Amer. Math. Soc. 120, 286 (1965). [45] B. Dacorogna and J. Moser Ann. Inst. Henri Poincar?, Anal. non. lin. 7, 1 (1990). [46] O. Bokanowski and B. Grebert Int. J. Quantum Chem. 68, 221 (1998). [47] O. Bokanowski and B. Grebert J. Math. Phys.37, 1553 (1996). [48] E. S. Kryachko and E. V. Lude˜na, (1992) in S. Fraga (ed.), Computational Chemistry: Structure, Interactions and Reactivity, Elsevier, Amsterdam. pp. 136-165. [49] T. Koga, Phys. Rev. A 42, 3763 (1990). [50] D. G. Kanhere, A. Dhavale, E. V. Lude˜na, and V. Karasiev, Phys. Rev. A 62, 065201 (2000). [51] E. V. Lude˜na and E. S. Kryachko, Rev. Mex. Astr. Astrofis. 23, 95-106. (1992). [52] E. S. Kryachko and E. V. Lude˜na, New J. Chem. 16, 1089-1098.(1992).

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[53] E. S. Kryachko,E. V. Lude˜na and T. Koga, (1993) in E. Clementi (ed.), Methods and Techniques in Computational Chemistry, METECC-94, Vol. B: Medium Size Systems, STEF, Cagliari, pp. 23-56. [54] E. V. Lude˜na,R. L´opez-Boada,J. Maldonado,E. Valderrama, E. S. Kryachko,T. Koga and J. Hinze, (1995) Int. J. Quantum Chem. Special Issue ”Thirty Years of Density Functional Theory. [55] E. V. Lude˜na,E. S. Kryachko,T. Koga ,R. L´opez-Boada, J. Hinze, J. Maldonado and E. Valderrama, (1995) in P. Politzer and J. M. Seminario (eds.), Theoretical and Computational Chemistry: Density Functional Calculations, Elsevier, Amsterdam.

In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 189-199

Chapter 7

E LECTRON D ENSITY S CALING - A N E XTENSION TO M ULTI - COMPONENT D ENSITY F UNCTIONAL T HEORY ´ Nagy∗ A. Department of Theoretical Physics, University of Debrecen, Debrecen, Hungary

Abstract The theory of density scaling is presented with an extension to multi-component density functional theory. Via density scaling a new Kohn-Sham scheme is constructed. It can be shown that there exists a value of the scaling factor for which the correlation energy disappears. Therefore exchange energy has to be determined instead of the exchange-correlation energy and it can be calculated very accurately. ζKLI method incorporating correlation is proposed.

PACS: 31.15EKeywords: Electron density scaling

1.

Introduction

In density scaling, proposed by Chan and Handy [1], the density n(r) is changed to ζn(r). We have shown [2, 3] that there exists a value of the scaling factor for which the correlation energy disappears. It has the consequence that one has to calculate only the exchange energy instead of the exchange-correlation energy. There is a simple explicit expression of the Kohn-Sham orbitals for the exchange energy. The correlation energy, on the other hand, cannot easily be expressed with the orbitals. Therefore using the scaled exchange energy instead of the original exchange-correlation energy leads to a very simple method. Moreover, the optimized potential method (OPM) [4] and the Krieger-Li-Iafrate (KLI) [5, 6] approach can be also generalized [2, 3]. Here we emphasize the importance the ∗

E-mail address:[email protected]

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ζKLI method that is as simple as the original KLI method but it incorporates a large portion of correlation. Nowadays, there is a growing interest in non-adiabatic processes. Density functional theory has been generalized to a multi-component theory [8] in which both the electrons and the nuclei are treated completely quantum mechanically, without the use of the BornOppenheimer approximation. There are two fundamental quantities: the electron density and the nuclear N -body density. Density scaling is applied here to the electron density.

2.

Density Scaling in the Electron Density Functional Theory

The density scaling was applied to obtain a Kohn-Sham scheme with the scaled electron density [2, 3]. Here, we suppose a non-degenerate state. Extension to degenerate states is detailed in [7]. Consider a non-interacting system with the density nζ (r) = n(r)/ζ, where ζ = N/Nζ is a positive number. In the present theory we suppose that ζ is larger but close to 1. If the original real system has N -electrons Z n(r)dr = N, (1) our new Kohn-Sham system has Nζ -electrons: Z nζ (r)dr = Nζ .

(2)

N is always integer, but Nζ is generally non-integer. Therefore, the new Kohn-Sham equations will differ from the ones of the original N -electron Kohn-Sham system. To construct the new Kohn-Sham system we define the density nζ = (1 − q)n + qnion ,

(3)

q = N − Nζ = N (1 − 1/ζ).

(4)

where

We consider only the case for which q is a small positive number: q 0. ion

ee

ion

(20)

Consequently, there exists a value of ζc for which Eζc = 0. Note that Eq. (19) has two solutions, however, the other solution is not close to 1. Moreover it can even be negative and thus physically not acceptable.

3.

Electron Density Scaling in Multi-Component Density Functional Theory for Electrons and Nuclei

A multi-component density functional theory has been recently proposed for electrons and nuclei by Gross and collaborators [8]. Here we briefly summarize the theory. Consider a system of N electrons and Nn nuclei with masses M1 , ..., MNn and charges Z1 , ..., ZNn . The electron and the nuclei coordinates are denoted by r1 , ..., rN and R1 , ..., RNn , respectively. The total Hamiltonian has the form ˆ = Tˆe + W ˆ ee + U ˆext,e + Tˆn + W ˆ nn + U ˆext,n + W ˆ en , H

(21)

where 1 Tˆe = − 2

N X i=1

∇2i

(22)

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193

and Tˆn = −

Nn X

1 ∇2 2Mα α

α=1

(23)

are the kinetic energy operators of the electrons and nuclei, while ˆ ee = W

N −1 X

N X

k=1 j=k+1

Nn−1

ˆ nn = W

Nn X X

α=1 β=α+1

1 , |rk − rj |

(24)

Zα Zβ , |Rα − Rβ |

(25)

Zα |Rα − rj |

(26)

and ˆ en = − W

Nn X N X

α=1 j=1

ˆext,e and U ˆext,n are the external operators. are the inter-particle operators, respectively. U In the multi-component density functional theory the single-particle density is obtained by defining it with respect to a coordinate frame which is attached to the system. Therefore the electronic coordinates are transformed to a body-fixed frame according to r0j = R(α, β, γ)(rj − RCM N )

j = 1, .., N,

(27)

where RCM N =

Nn 1 X Mα Rα M

(28)

α=1

is the center of mass of the nuclei, while M=

Nn X



(29)

α=1

gives the total mass of the nuclei. R denotes the three-dimensional orthogonal matrix representing the Euler rotations. The densities are defined as Z Z 0 Nn n(r ) = N d R dN −1 r0 |Ψ(R1 ..., RNn , r01, ..., r0N )|2 (30) and Γ(R) =

Z

dN r0 |Ψ(R1 ..., RNn , r01 , ..., r0N )|2

(31)

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The multi-component Hohenberg-Kohn theorem tells that the ground-state densities n and Γ determine the ground-state wave function, therefore any observable is a functional of the ground-state densities n and Γ and the total energy functional takes its minimum at the exact ground-state densities. The multi-component Kohn-Sham scheme can also be constructed. The multi-component Kohn-Sham equations have the form   1 2 (32) − ∇ + vs,e (r) uj = εe,j uj , 2 and "

# 1 X ∇2α − + VS,n (R) χ(R1 , ..., RNn ) = n χ(R1 , ..., RNn ). 2 α Mα

(33)

The densities are given by n(r) =

X

|uj |2

(34)

j

and Γ(R1 , ..., RNn ) = |χ(R1 , ..., RNn)|2 .

(35)

The multi-component Kohn-Sham equations are coupled because of the mutual influence of electrons and nuclei on each other. Eqs. (32) have the same form as the Kohn-Sham equations in the electron density functional theory. In fact, however, they are different. The electron orbitals do not parametrically depend on the nuclear coordinates. Dependence on the nuclear configuration is included in the Kohn-Sham potential through the functional dependence on Γ. The total energy functional is given by Z Z E[n, Γ] = F [n, Γ] + drn(r)ve(r) + dNn RΓ(R1 , ..., RNn )Vn (R1 , ..., RNn ), (36) ve (r) and Vn (R1 , ..., RNn ) are ’auxiliary’ external potentials and ˆ ee + W ˆ en + W ˆ nn + U ˆext,e + U ˆext,n |Ψmin [n, Γ]i. (37) F [n, Γ] = hΨmin [n, Γ]|Tˆe + Tˆn + W The functional F [n, Γ] can be decomposed in the following way: F [n, Γ] = Ts,e [n] + Ts,n [Γ] + EU,Jxc [n, Γ],

(38)

where Ts,e [n] and Ts,n [Γ] are the non-interacting electron and nuclear kinetic energy functionals, respectively, while the functional EU,Jxc [n, Γ], defined by Eq. (38) contains the classical Coulomb and exchange-correlation effects: en EU,Jxc[n, Γ] = J[n] + Exc[n] + EJc [n, Γ] + TM P C [n, Γ] + UJxc [n, Γ].

(39)

The first two terms in Eq. (39) give the classical Coulomb and exchange-correlation energies of the electrons. It is important to emphasize that these functionals are, by construction,

Electron Density Scaling-An Extension to Multi-Component ...

195

en identical to the ones of the standard electronic density functional theory. EJc [n, Γ] contains the many-body effects due to the electron-nuclear interaction. TM P C [n, Γ] denotes the mass-polarization and Coriolis effects. UJxc [n, Γ] incorporates all effects arising from the ˆ . The Kohn-Sham potentials in Eqs. (32) and (33) can presence of true external potentials U en be given by the functional derivatives of EU,Jxc [n, Γ]: δEU,Jxc [n, Γ] (40) vs,e = δn [n0 ,Γ0 ]

and

vS,n = Wnn +

δEU,Jxc[n, Γ] , δΓ] [n0 ,Γ0 ]

(41)

where n0 and Γ0 are the ground-state densities. Now, we extend the theory of electron density scaling to multi-component density functional theory. We construct a Kohn-Sham scheme with the electron density nζ = n/ζ. The scaled Kohn-Sham equations can be obtained in the same way as in the electron density functional theory.   1 2 (42) − ∇ + vζ,s,e (r) uζj = εζ,e,j uζj . 2 The scaled electron density is given by nζ (r) =

X

λζj |uζj |2 .

(43)

j

Now, we utilize the fact that the electron energy part of the total energy is defined exactly as in the standard density functional theory. The electron density scaling changes the electron non-interacting kinetic energy, but the total electron energy ET Jxc[n] = Ts,e [n] + J[n] + Exc [n]

(44)

ET Jxc [n] = Tζs,e [n] + J[n] + Eζxc[n].

(45)

remains the same

Therefore, we are led to an equation similar to Eq. (14) Ts,e [n] + Exc [n] = Tζs,e [n] + Eζxc[n].

(46)

Obviously, Tζs,e = Ts,e if ζ = 1. The relationship between Tζs,e and Ts,e is the same as in the electron density functional theory: Tζs,e [n] = ζTs,e [nζ ].

(47)

The scaling of the exchange energy remains also the same: Eζx,e [n] = ζ 2 Ex,e [nζ ].

(48)

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The electron correlation energy takes the form ˜ Tˆe + W ˆ ee |Φi ˜ Eζc,e [n] = hΦ| ˜ 0 |ζ Tˆe + ζ 2 W ˆ ee |Φ ˜ 0 i + qhΦ ˜ 0 |ζ Tˆion + ζ 2 W ˆ ion |Φ ˜ 0 i]. (49) − [(1 − q)hΦ ion e ee ion ˜ 0 and Φ ˜ denote the non-interacting and the interacting electron wave functions, respecΦ ˜ ˜ Tˆe + W ˆ ee |Φi, ˜ while Φ ˜ 0 is the tively: Φ is the wave function that yields n and minimizes hΦ| 0 ˆ ˜0 ˜ wave function that yields n and minimizes hΦ |Te |Φ i. There is a theorem similar to the one in the electron density functional theory: Theorem 2 There exists a parameter ζ˜c for which the electron correlation energy disappears: Eζc,e = 0. The proof is the same as in the electron density functional theory, with the exception ˜ 0 and Φ ˜ instead of Φ0 and Φ. that we should write Φ Though the multi-component Kohn-Sham equations (42) with the scaled density have the same form as the scaled Kohn-Sham equations in the electron density functional theory, ˜ 0 and Φ ˜ are different from the wave functions Φ0 and Φ of the electron the wave functions Φ ˜ 0 and Φ ˜ do not parametrically depend on density functional theory. The wave functions Φ the nuclear coordinates. Dependence on the nuclear configuration is included in the KohnSham potential through the functional dependence on Γ. Consequently, that particular value ζ˜c for which the electron correlation energy disappears will somewhat differ from ζc .

4.

The ζKLI Method

Exchange can be very accurately treated in the density functional theory via the optimized potential method (OPM) [4] or the Krieger-Li-Iafrate method (KLI) [5]. However, the available correlation functionals are still not accurate enough. Moreover, it is a very delicate problem to find a correlation functional that performs well together with the OPM or KLI exchange. In the existing approximate functionals exchange and correlation are treated together and if we change only the exchange part (e. g. into KLI) the balance between the exchange and correlation is ruined. As we have seen it is possible to derive a Kohn-Sham system with scaled electron density, in which the correlation energy disappears. Therefore it is enough to calculate the exchange energy instead of the exchange-correlation energy and it can be done very accurately. In the electron density functional theory exchange can be treated exactly via the optimized potential method [4]. This method has been generalized using density scaling [3]. To find the optimized potential is very tedious even in the ground-state. However, Krieger, Li and Iafrate [5] introduced a very accurate approximation. An extension to the scaled density was presented [3] using an alternative derivation of the KLI approximation [6]. The generalized ζKLI approximation for the Kohn-Sham potential has the form: vζKS = v + vJ + vζx ,

(50)

where vζx = vζS +

ζ X huζj |λζj vζx − vζxj |uζj i|uζj |2 . n j

(51)

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197

The first term in Eq. (51) vζS (r) =

ζ X ∗ uζj (r)vζxj (r)uζj (r) n(r)

(52)

j

is the Slater potential and vζxj denotes the Hartree-Fock-like exchange potential vζxj (r)uζj (r) = −

Z

dr0

X

λζj

u∗ζj (r0 )uζj (r)

j

|r − r0 |

uζj (r0 ) ,

(53)

For ζ = 1 Eq. (50) gives the original KLI exchange potential. As we used Hartree-Fock like expression we obtained only the exchange. The results above are valid for any value of ζ. We search that ζc for which the correlation energy disappears. For that value of ζ the ζKLI method provides a very simple approximation that includes correlation.

5.

Discussion

We have already emphasized that the value ζ˜c for which the electron correlation energy disappears will somewhat differ from ζc . In the original electron density functional theory the electron correlation energy depends parametrically from the nuclear coordinates. Therefore ζc also depends parametrically from the nuclear coordinates. However, there is not parametrical dependence on the nuclear configuration in the electron correlation energy of the multi-component density functional theory. Instead, the information on the nuclear distribution is incorporated in the functional dependence of Eζc,e on Γ. Therefore ζ˜c will not depend parametrically from the nuclear coordinates. In this chapter only the electron density is scaled. It is in principle possible to introduce scaling into the nuclear density. Note that in the multi-component density functional theory the nuclear density is a many-body density. As it can be seen from Eq. (31) Γ depends on all nuclear coordinates. It is a very general and flexible formalism. In actual applications it might be useful to reduce the variables with the introduction of reduced density matrices. Introducing internal nuclear coordinates might be desirable. The choice of such coordinates is specific to the system under study. It has been shown [2, 3] that the Kohn-Sham scheme can be generalized so that the electron correlation energy disappears in it. The value of q = N (1 − 1/ζ) for which Ec = 0 is not universal. In the electron density functional theory it has been found [2] that ζc is close to a constant q0 ≈ 0.04. Finally, we present an approximate expression for the electron correlation Ec of the original Kohn-Sham scheme [2]. We have shown that [2] for a small q 1 Ts − Tζ ≈ − qhuN |∇2i |uN i. 2

(54)

It can further be approximated as Ts − Tζ ≈ q

Ts . N

(55)

´ Nagy A.

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The difference in exchange energy can be similarly calculated Z 1 2 |uN (r1 )|2 |uN (r2 )|2 q dr1 dr2 Ex − Eζx ≈ − 2 |r1 − r2 | N Z X u∗i (r1 )u∗N (r2 )ui (r2 )uN (r1 ) − q dr1 dr2 . |r1 − r2 |

(56)

i=1

A further approximation leads to Ex − Eζx ≈ −q 2

Ex J + 2q 2 N N

(57)

As q is selected so that Eζc = 0, using Eqs. (14) Ec = Tζ − Ts + Eζx − Ex

(58)

Eqs. (55), (57) and (58) lead to the approximate relation for the electron correlation energy:  q 2 q Ec ≈ J − (Ts + 2Ex). (59) N N

This relation can be useful as it provides an approximate expression of Ec if the kinetic, exchange and classical Coulomb energies are known supposing an approximate value for q. We can immediately see that the same approximation can be written in the multi-component density functional theory. To conclude it has been shown that the Kohn-Sham scheme can be modified so that the electron correlation energy disappears in it both in the original electron density functional and the multi-component density functional theories. Therefore exchange energy has to be determined instead of the exchange-correlation energy.

Acknowledgments The work is supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007 project. The project is co-financed by the European Union and the European Social Fund. Grant OTKA No. K 67923 is also gratefully acknowledged.

References [1] G. K.-L. Chan AND N. C. Handy, Phys. Rev. A 1999, 59, 2670. ´ Nagy, Chem. Phys. Lett. 2005 411, 492. [2] A. ´ Nagy, J. Chem. Phys. 2005 123, 044105. [3] A. [4] R. T. Sharp and G. K. Horton, Phys. Rev. 1953, A 30, 317; K. Aashamar, T. M. Luke and J. D. Talman, At. Data Nucl. Data Tables 1978 22 443. [5] J. B. Krieger,Y. Li and G. J. Iafrate, Phys. Rev. A 1992 45, 101; Phys. Rev. A 1992 46, 5453. ´ Nagy, Phys. Rev. A 1997 55, 3465. [6] A.

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´ Nagy, J. Phys. B 2011 44, 035001. [7] A. [8] T. Kreibich, E. K. U. Gross, Phys. Rev. Lett. 2001 86, 2984; T. Kreibich, R. van Leeuwen, E. K. U. Gross, Phys. Rev. A 2008 78, 022501.

In: Theoretical and Computational Developments... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 201-209

Chapter 8

A S YMMETRY P RESERVING KOHN -S HAM T HEORY Andreas K. Theophilou∗ DEMOKRITOS National Center for Scientific Research Athens, Greece

Abstract In this work we deal with the problem of the lowest energy KS state when the exact state belongs to an irreducible representation of the symmetry group of the exact Hamiltonian. As Fertig and Kohn have shown (H.A. Fertig, W. Kohn, Phys. Rev. A 62, 052511 (2000)) the KS state reproducing the exact density does not have the transformation properties of the exact state. We show that one can develop a theory with the exact state properties, at the expense of the accuracy of the density. This theory, however, demands new functionals for the exact state and therefore more appropriate is the subspace theory, where the symmetric part of the density can be reproduced exactly, but not the one of a single state.

PACS: 21.15.E-, 31.15.ec Keywords: Density functional, Subspace, Asymmetry

1.

Introduction

Although there is a lot of progress in density functional theory, DFT [1, 2], in its Kohn and Sham, KS, formulation, there are still some fundamental questions which expect their answer. One of them has to do with the qualitative features of the many particle KS wave function in relation to those of the exact one. Thus, e.g. if the external potential is spherically symmetric, then the energy eigenstates of the exact Hamiltonian can be chosen as eigenstates of the angular momentum operators L2 and Lz . Then, one could ask whether the many electron KS state having the same density as the exact one is also an eigenstate of these operators with the same l and m. The answer is no. Fertig and Kohn [3] have shown by an example, that the KS potential does not have the symmetry properties of the ∗

E-mail address: [email protected]

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Andreas K. Theophilou

external potential and therefore the density cannot be derived by a Slater determinant having the proper angular momenta. Although, they used a harmonic oscillator model, with harmonic interaction, one can extend these conclusions for real physical systems. Thus, there is a serious problem that one has to deal with, since symmetry breaking results not only to conceptual difficulties but also to serious computational problems. For example, it is much easier to solve the KS equations with a spherically symmetric potential than one with an angular dependence. The above finding is strange from the conceptual point of view, as it implies that in the case of degenerate states, we must have different KS potentials according to the state we consider from the space of degeneracy. Thus, e.g., the eigenstate |Ψ11 >, with l = 1 m = 1, will have a different KS potential from the |Ψ10 > state which has m = 0, whereas their energies must be the same. Further, to a linear combination of such states will also correspond a different KS potential. These, results are also in contradiction with the subspace theory, where symmetry is preserved [4, 5]. One would argue that in general, eigenstates with l ≻ 0 are not ground states. However, one one can formulate a DFT theory for the lower energy eigenstates with a definite l and m as there is one to one correspondence between such a state and the corresponding density [6]. In this work we shall show that it is possible to formulate a density functional theory where the many electron KS state has the same transformation properties as the exact eigenstate. This will be at the expense of some features such as the need of the equality of the KS density to the exact one. For this purpose we shall review some concepts of group theory, necessary for the understanding of this problem.

2.

Review of Group Theory in the Many Electron Problem Let us consider a Hamiltonian H, H = T + Hint + V

(1)

which is invariant under a group of transformations G, i.e., gHg −1 = H, f or all g ∈ G

(2)

If G is a group which involves only geometric transformations (rotations and translations) then the kinetic energy operator T is invariant under any rotation and translation. The same is true with the electron-electron interaction operator Hint . Then, for obtaining the symmetry of the Hamiltonian, one has to search for the symmetry group which maps the potential on itself, i.e. to find the group elements g for which V (gr) =V (r). Such an example is the case of an atom where the external potential V (r) due to the charge of the nucleus is invariant under rotations, i.e. the potential does not depend on the orientation of r. Let us consider a subspace S Γ of the Hilbert space which is invariant under the group G, i.e. if a state |ΨΓγ > belongs to S Γ , so does the state g|ΨΓγ > for all g ∈ G. Thus, by choosing an orthonormal basis from states in S Γ we find g|ΨΓγ >=

X β

Γ Dβγ (g)|ΨΓβ >

(3)

A Symmetry Preserving Kohn-Sham Theory

203

We shall show that the states H|ΨΓγ > belong to the same subspace S Γ i.e. they have the same transformation properties as |ΨΓγ > . Thus, gH|ΨΓγ >= gHg −1 g|ΨΓγ >= Hg|ΨΓγ >

(4)

For deriving the above property we used the relation gHg −1 = H. After taking into account Eq. 3, we have gH|ΨΓγ >=

X

Γ Dβγ (g)H|ΨΓβ >

(5)

β

The we conclude that the states H|ΨΓγ > have the same transformation properties as the |ΨΓγ > . From the above it follows that in searching for an eigenstate of H we can search it in an invariant subspace S Γ since an eigenstate by its definition maps a one dimensional state on itself. For a certain group G, the Hilbert space can be separated in mutually orthogonal subspaces invariant under G, which have different transformation properties. These transformation properties are specified by the matrices DΓ (g) which have matrix elements Γ (g). The upper index Γ indicates the irreducible representation (Irrep) of a group G, Dβγ i.e. Γ specifies the various types of transformation properties. Let us now see what happens with the electron density ρ(r) =< ΨΓγ | ρb(r)|ΨΓγ > of an eigenstate |ΨΓγ > of H. Since < ΨΓγ |ρb(r)|ΨΓγ > is a bilinear function of |ΨΓγ > it follows that it does not belong to any definite transformation with the exemption of the cases that the Irrep is one dimensional, i.e. the case the g|ΨΓγ >= ω(g)|ΨΓγ > . Take for example the case of spherical symmetry where the Irreps are labeled by l denoting the eigenvalues l(l + 1) of L2 . Then the density corresponding to the state |Ψ10 > is ρ(r) = ρ0 (r) + ρ1 (r)Yo2 (Ω)

(6)

where Yo2 (Ω) = Yo2 (θ, ϕ) is the spherical harmonic with l = 2. Thus, in general the density corresponding to an energy eigenstate is not invariant under the group of transformations under which the external potential is invariant. Then, the KS potential is not invariant under the symmetry group of the external potential. This is easy to see, since the Hartree potential, Vh . which obeys the equation ∇2 Vh (r) = 4πρ(r) has the same transformation properties as the density. The exchange and correlation potential Vxc is also asymmetric since it is a more complicated function of the density. As a result of the density asymmetries, the many particle wave functions as well as the spin orbitals do not transform according to the Irreps of the symmetry group of the exact Hamiltonian. Thus, they cannot be used for selection rules or other physical applications where symmetry properties are important. Some of the above difficulties can be overcome by using the subspace theory for degenerate states.

3.

Subspace Theory

In the subspace theory, one considers a subspace of states and for the case that symmetry is present, one has to consider the space of degeneracy S. The subspace density is defined

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Andreas K. Theophilou

as ρ(r, S) = T rS {ρb(r)},

(7)

b A(S) = T rS (A).

(8)

where by T rS we mean the trace restricted in the subspace S. In the same way one can b define subspace functions A(S) for any operator A, When an orthonormal set |Ψk > of S is considered, the above formula takes the form A(S) =

M X

< Φk |H|Φk >,

(9)

k=1

where M is the dimension of the subspace. In the case that S is invariant under g, ρ(gr, S) = T rS {g ρb(r)g −1 },

(10)

T rS {ρb(gr)} = T rS {ρb(r)}.

(11)

b = T rS (B b A), b we find Remembering now that T rS (AbB)

Therefore, if g is an element of a group G under which a Hamiltonian H is invariant and S is the subspace of degeneracy of a group of eigenstates, then the subspace density is invariant under G. Thus, if G is a rotation under which the external potential is invariant, then gHg −1 = H, since, as stated earlier, both the kinetic energy operator T and the electron-electron interaction operator, Hint , are invariant under any rotation or translation. For the case treated by Fertig and Kohn, where the exact eigenstate is an eigenstate of the angular momentum operator L2 with eigenvalue l = 1, the subspace density is 1 ρ(r) = (A + Br2 ) exp(−wr2 ), 3

(12)

where A, B and w have fixed values and since it does not depend on the angles θ, φ, it is spherically symmetric. We shall prove that this density corresponds to the subspace of a two-particle Slater determinants (SL) having the form |Φ1m >= |φ00 , φ1m >

(13)

where φ00 (r) = φ0 (r)Y00 (Ω) and φ1m (r) = φ1 (r)Ym1 (Ω) whereas, as Fertig and Kohn have proved, the exact single eigenstate density is not reproducible by a single KS eigenstate of L2 . The density corresponding to the subspace of |Φ1m >, m = −1, 0, 1, is ρ(r) = |φ0 (r)2 + |φ1 (r)2

(14)

Since the φlm (r) states must behave as rl in the neighborhood of 0, we consider the following expansions φ0 (r) =

k X

n=0

a2n r2n

(15)

A Symmetry Preserving Kohn-Sham Theory

205

and φ1 (r) = exp(−wr2 /2)

k+1 X

a2n+1 r2n+1

(16)

n=1

Then the subspace density is ρ(r) = exp(−wr2 ){

k X



a2n a2n′ r2(n+n ) +

k+1 X



a2n+1 a2n′ +1 r2(n+n +1) }

(17)

n,n′ =1

n,n′ =0

In order to get an approximation to the density of Eq. 12, we equate the coefficients of the zero and 2nd power of r to those of 12, while higher coefficients must be equal to 0. Thus, taking as example the case that k = 7, we have a20 = A (18) 2a0 a2 + a21 =

B 3

2a0 a4 + 2a1 a3 + a22 = 0

(19) (20)

2a0 a6 + 2a1 a5 + 2a2 a4 + a23 = 0 (21) √ From these equations we see that a0 is A and if we let a2 a4 and a6 as arbitrary parameters, a1 is determined by a2 , a3 by a2 and a4 , a5 by a2 , a4 and a6 . We still have the two normalization conditions since 2π

Z

drr2 |φi (r)|2 = 1.

(22)

We see that we still have a free parameter for the appropriate choice of the 23 φ00 (r), φ1m (r). For higher order corrections we get the general formula for the coefficient of the 2nth power of r, 2(a0 a2n + a1 a22n−1 + a2 a2n−2 + ... + am a2n−m ... + +an−1 an+1 ) + a2n = 0.

(23)

From this formula, each odd coefficient a2m+1 is determined by a2 , a4 and a2m . Thus, when the highest order approximation has the power 2k + 1, we have k free parameters, which become k − 2 when the normalization conditions are taken into account. Thus, we conclude that it is possible to reproduce the spherical part of the density by a subspace of determinants having the proper transformation properties although the exact individual eigenstate densities cannot be reproduced [3]

206

4.

Andreas K. Theophilou

The General Case

One could argue, that we restricted our proof to the harmonic oscillator wave functions. For the general case we can consider the minimization of the kinetic energy with respect to the spherical part of the density, i.e., Inf {

l X

< Φlm |T |Φlm > :

m=−l

l 1 X < Φlm |ρb(r)|Φlm >= ρ0 (r)} l m=−l

(24)

Then, by considering a ”complete system of potentials” vn we have a denumerable set of condition and the above equation becomes

min{

Pl

m=−l

< Φlm |T |Φlm > : =

R

1 l

Pl

m=−l

R

d3 r < Φlm |ρb(r)|Φlm > vn (r)

d3 rρo (r)vn0 (r), n = 1, 2...}

(25)

l (r) = where vn0 (r) are spherically symmetric potentials. The potentials of the form vnm R 3 l l l vn (r)Ym (Ω) do not enter into the constraint conditions because d rρo (r)vn (r)Yml (Ω) = 0. The equation resulting after the minimization is

T |Φ > +

X

λn

Z

d3 rvn (r)ρb(r)|Φ >= E|Φ > .

(26)

and the KS potential is spherically symmetric since it is a sum of potentials having this property. Let us consider now the case of a single eigenstate density like e.g. that of Eq.1. Then, we must take into account the additional density constraints 1 l

Pl

m=−l

R

d3 r

q

R

d3 r < Φlm |ρb(r)|Φlm > vn2 (r)Y02 (Ω) =

16π 1 2 2 2 15 3 Br exp(−wr )vn (r),

n = 1, 2...

(27)

By repeating the minimization procedure, we find that the KS potential depends on θ. Further, the additional constraints make the space in which the minimum is searched a subspace of the previous one, having thus higher kinetic energy. Since the above results can be generalized for any geometric symmetry group, where ρo (r) refers to the identity Irrep, one has to draw some conclusions about DFT: (i) One can use the subspace theory and get the subspace density by means of which all properties of the eigenstates can be derived, although their densities are not equal to those of the exact states or (ii) One can use the single state density and possibly get KS potentials which are not invariant under the symmetry group of the Hamiltonian. In any case, the general conclusion is that the KS potential is a complex mapping of the density and one can consider various approximations appropriate for special classes of systems than search for the general mapping of the density to the KS potential.

A Symmetry Preserving Kohn-Sham Theory

5.

207

The Single Eigenstate Equation

Since in practice many times we do not know the symmetry group of the Hamiltonian, the dimension of the subspace of degeneracy is not known and therefore we do not know how many states have to be included in order to get a density having the proper symmetry. In addition, we may have small perturbations of the external potential which break symmetry. For these reasons it is useful to have a single state KS theory. For this purpose the kinetic energy minimization must be over density constraints where the vn (r) have the symmetry of the external potential. Thus the variational principle is min{< ΦΓγ |T |ΦΓγ > : R

R

d3 r < ΦΓγ |ρb(r)|ΦΓγ > vn (r) =

d3 rρΓγ (r)vn (r), n = 1, 2...M }

(28)

and if V (gr) = V (r) for g ∈ G, we must also have vn (gr) = vn (r) for g ∈ G, where G is the symmetry group of the external potential. Then the vn (r) must be linear mappings of the external potential, otherwise the symmetry is lost. We shall give some examples of such mappings. v1 (r) = V (r), v2 (r) = ∇2 V (r), (29) It is not possible to have ∇4 V (r) since this would exclude the case of the coulomb potential due to a point charge since ∇2 (e/(r − R)) = −eδ(r − R) and higher order derivatives would involve unphysical potentials. The other linear mappings are nonlocal ones vn (r, V ) =

Z

d3 r′ Kn (|r − r′ |)V (r′ ), n ≻ 2,

where the label V in vn (r, V ) indicates the dependence on the external potential. The single particle equations resulting after minimization are M X 1 − ∇2 ϕi (r) + λn vn (r)ϕi (r) =∈i ϕi (r) 2 n=1

where the λn are Lagrange multipliers to be determined by the condition of the density constraints. The many particle KS equation is T |ΦΓγ > +

M X

λn

n=

Z

d3 rvn (r, V)ρb(r)|ΦΓγ >= E|ΦΓγ >

We shall show that to different potentials correspond different KS states since by considering the same state as being eigenstate of a potential V ′ , we get the equation T |ΦΓγ > +

X n

λ′n

Z

d3 rvn′ (r, V′ )ρb(r)|ΦΓγ >= E ′ |ΦΓγ >

By subtracting we find that XZ n

d3 r[λn vn (r, V) − λ′n vn′ (r, V′ )ρb(r)|ΦΓγ >= (E − E ′ )|ΦΓγ >

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Andreas K. Theophilou

But, multiplicative operators do not have physical eigenstates. Hence to different external potentials correspond different states. Thus, by keeping the same mappings of V and the same M, we can find unique |ΦΓγ > associate with the exact lower energy state |ΨΓγ > of the physical system under consideration. In increasing the number M of vn (r), we increase the accuracy of the density of |ΦΓγ > relative to that of |ΨΓγ > . However, after some M, the density constraints cannot be satisfied since the densities of the two states must have a finite deviation. For M = 1 we always have a solution, since we have an external potential differing by a multiplicative factor from the exact one, whereas Hint which is repulsive is missing from the KS Hamiltonian. Hitherto, we dealt with the existence of a KS type theory for the minimum energy states of certain Irrep, in the Levy formulation of DFT [7], i.e. determining the many particle determinant under certain density constraints. However, one has to specify how the energy of the exact |ΨΓγ > is determined, i.e., how the exact energy is expressed as a functional of |ΦΓγ > . As a first approximation, one could calculate the < ΦΓγ |H|ΦΓγ > and minimize it with respect to the coefficients λn . This approximation is equivalent to the effective potential approximation of Hartree-Fock. Applications were carried out in a version of effective potentials derived by a similar method. The energies found for various molecules compare well with those of the exact HF [8–10] although two terms v1 and v2 were used. A better approximation needs an accurate expression of the lowest energy of |ΨΓγ > as a functional of |ΦΓγ > . Unfortunately, most work on this matter concerns density functionals and one has to develop new functionals E(ΦΓγ ) in order to go beyond the optimized effective potential approximation. It is to be noted once again that the constraints introduced do not determine uniquely the exact density, as these are finite. The problem is: what is the best choice of linear mappings of the external potential, i.e. which and how many vn (r) has one to use. If we search for constraints defining the exact density of the physical system, we may not be able to find eigenstates having the exact density.

Conclusion One can develop a Kohn and Sham type theory for the lower state |ΦΓγ >, belonging to an Irreducible representation Γ of a symmetry group of the exact Hamiltonian, the density of such a state cannot be equal to that of the exact state |ΨΓγ >. Further, one has to develop new functionals giving the exact energy in terms of the approximate density or in terms of the Kohn and Sham many particle state |ΦΓγ > . A more convenient approximation is by using the subspace density functional theory, where one can reproduce the exact subspace density but not that of the single state.

References [1] Hohenberg and W. Kohn, Phys. Rev. B 136 , 864 (1964). [2] W. Kohn and L.J. Sham, Phys. Rev. A 140 , 1133 (1965). [3] H.A. Fertig, W. Kohn, Phys. Rev. A 62, 052511 (2000).

A Symmetry Preserving Kohn-Sham Theory [4] A.K. Theophilou, J. Phys. C 12 , 5419 (1979). [5] A.K. Theophilou and P. Papaconstantinou, Phys. Rev. A 61 022502 (2000). [6] U. Barth,Phys. Rev. A, 20, 1693 (1979). [7] M. Levy, Proc. Natl. Acad. Sci., USA, 76 6062 (1979). [8] A.K. Theophilou and V. Glushkov, Int. J. Quantum Chem. 104, 538 (2005). [9] A.K. Theophilou and V.N. Glushkov,J. Chem. Phys. 124, 034105 (2006). [10] V.N. Glushkov and S.I. Fesenko,J. Chem. Phys. 125, 234111 (2006).

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In: Theoretical and Computational Developments... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 211-222

Chapter 9

S ELF -I NTERACTION C ORRECTION IN THE KOHN -S HAM F RAMEWORK 1

1∗ T. K¨orzd¨orfer1 and S. Kummel ¨ Theoretical Physics IV, University of Bayreuth Bayreuth, Germany

Abstract Electronic self-interaction is one of Density Functional Theory’s most notorious problems. It is responsible for many of the qualitative failures of present day exchangecorrelation functionals. In this chapter we first review why the seemingly simple problem of self-interaction is very hard to solve in practice due to the close relation between self-interaction and the modeling of non-dynamic correlation in semi-local density functionals. We then discuss in detail the concept of the self-interaction correction and stress the differences between the traditional self-interaction correction using orbital-specific potentials and recently developed Kohn-Sham Optimized Effective Potential self-interaction approaches. Finally, we study how much the electric dipole polarizability of hydrogen chains, which is badly overestimated by local and semi-local exchange-correlation functionals, is improved by different self-interaction correction schemes.

PACS: 31.15.EKeywords: Density functional theory, Self-interaction correction, Polarizability

1.

Introduction: Self-Interaction in Density Functional Theory

Density functional theory (DFT) [1] in both its ground-state and its time-dependent formulation (TDDFT) [2] has seen enormous success in the past decades and has become a standard method for investigating the electronic structure of molecules and solids. Bond lengths and geometries of many systems are reliably predicted with semi-local functionals such as the Local (Spin) Density Approximation (LDA) or Generalized Gradient Approximations (GGA) at a computational cost that is significantly lower than the one of a HartreeFock calculation. When hybrid-functionals [3] are used, the computational cost of a DFT ∗

E-mail address: [email protected]

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T. K¨orzd¨orfer and S. K¨ummel

calculation increases to that of a Hartree-Fock one. However, typically also the accuracy for the structural properties of many systems increases and in many cases hybrid functional calculations reach a predictive power that is comparable to second-order perturbation theory or better. Thus, for a wide range of electronic structure problems, DFT appears almost as an “to good to be true” solution. However, there is a range of problems for which DFT and TDDFT with standard semilocal or hybrid functionals fail badly [4]. The proper description of systems with localized orbitals [5, 6, 7], of reaction-barriers [8, 9], ionization [10] and long-range charge-transfer [11, 12] are prominent examples. The source of these problems is well known – it is the self-interaction error. At first sight the problem appears as a rather trivial issue. Common sense tells us that in a system with just one electron there must not be any electron-electron interaction, as a single particle cannot interact with itself. In terms of the Kohn-Sham way of writing the energy functional as a sum of non-interacting kinetic energy, external potential energy, Hartree electrostatic interaction, and exchange-correlation energy, i.e., Z E[n] = Ts [n] + vext (r)n(r) d3 r + EH [n] + Exc [n], (1) there will be a contribution from EH [n] even for a one-electron density, and this contribution must be canceled exactly by Ex [n], while Ec [n] must vanish. However, LDA, GGA and hybrid functionals are not free from one-electron self-interaction, i.e., if such functionals are used to evaluate EH [n] + Exc [n] for, e.g., the hydrogen atom density, one obtains a spurious non-vanishing contribution to the energy. This seemingly simple issue is one of the most notorious and serious problems of DFT. It has far reaching consequences, and some of them can be understood quite intuitively. Properly describing situations in which localization of electrons plays a role obviously will be hindered by self-interaction, as an electron that “feels its own Hartree repulsion” will try to spread its spatial probability distribution as much as possible as a consequence of classical electrostatics. The fact that functionals such as LDA and GGA largely overestimate polarizabilities of extended systems and current densities in molecular electronics, and underestimate the energy of charge transfer excitations, i.e., make it “too easy” for electrons to be shifted around, can also be understood in terms of the self-interaction problem. Without self-interaction, the energetic difference between “being alone at a given site” and “being at a given site together with another electron” is very pronounced, as seen, e.g., in effects such as Coulomb blockade. With self-interaction, this sharp energetic difference is “smeared out”, because an electron’s self-interaction energy is present independent of the presence of other electrons. Consequently, blockade effects are less pronounced and transferring electrons becomes “too easy”. Whereas the above arguments invoke physical intuition to explain why self-interaction has many detrimental consequences, the same conclusions can also be reached on formal grounds. Self-interaction is closely tied to the famous derivative discontinuity [13, 14] of Exc . Lack of the latter can be seen as the self-interaction problem’s “Doppelganger” [15]. An Exc that is self-interaction free typically has a derivative discontinuity, leads to a vxc with proper asymptotic behavior and a particle-number discontinuity, and a vxc with a particle number discontinuity will typically show the potential “step-structures” that are

Self-Interaction Correction in the Kohn-Sham Framework

213

decisive for the description of charge transfer [11, 12, 4]. Quite some progress has been made in coping with these problems, e.g., by using rangeseparated hybrid functionals with a system-dependent choice of the range-separation parameter [16, 17]. However, eliminating self-interaction by a more universal approach is highly desirable. It remains a challenge, though, and this is so for a fundamental reason that can best be understood by an example. It may appear as one of the most natural ways of eliminating one-electron self-interaction to combine full exact exchange with a semilocal correlation functional of meta-GGA type that is self-correlation free [18, 19]. The full exact exchange would cancel the Hartree self-interaction exactly, and the meta-GGA correlation would vanish for any one-electron density. However, such a functional – though self-interaction free - would not be of much use as it would seriously underestimate electronic binding [20]. The explanation for this effect is that local and semi-local exchange functionals do not model exact exchange, but also effectively include electron interaction effects that from a wave function perspective appear as non-dynamic correlation [21]. Thus, the true challenge is not just removing the self-interaction, but removing it in such a way that the important non-dynamic correlation that (semi-) local exchange models is not removed at the same time.

2.

The Concept of the Self-Interaction Correction

There are different ways of trying to correct for self-interaction, and overviews with pertinent references can be found in [4, 22]. However, the so far most frequently used idea goes back to Perdew and Zunger [23]. They proposed to enforce the condition that the electron interaction effects must vanish for any one-electron density by explicitly subtracting the single-orbital contributions, SIC Exc [n↑ , n↓ ]

=

LDA Exc [n↑ , n↓ ]

Nσ X X   LDA EH [ni,σ ] + Exc [ni,σ , 0] . −

(2)

σ=↑,↓ i=1

We have written the self-interaction correction (SIC) for LDA (which is understood to refer to the spin-polarized functional), but obviously the scheme can be applied to any functional. We chose LDA because our results presented later in this chapter refer to LDA-SIC. Eq. (2) looks rather natural, but it is important to understand its implications. It enforces the previously discussed condition EH [ni,σ ] + Exc [ni,σ , 0] = 0

(3)

which must hold for any one-electron (spin) density ni,σ . This is a necessary condition for a self-interaction free functional. If the SIC is applied to a functional that already is selfinteraction free, the SIC will vanish, as it should. However, when applying Eq. (2) to some sort of condensed matter system – and after all, that is what DFT is about – we are transferring the one-electron condition of Eq. (3) to a many-electron system. In other words, we are taking Eq. (3) out of the context for which it was guaranteed to hold. As a consequence, there is no guarantee that the SIC will necessarily lead to an improvement in many electron systems. However, there is hope that it frequently will, and this hope is particularly large for certain observables because of the following observation. Using Eq. (2) in a

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T. K¨orzd¨orfer and S. K¨ummel

many-electron system corresponds to identifying orbitals with electrons. In doing so we are going beyond the traditional DFT point of view in which orbitals were introduced merely as auxiliary quantities to build up the density [1]. It is also not too hard to imagine situations in which identifying orbitals with electrons is not possible; e.g., the ionization of the He-atom is a well-studied example in this respect [10]. However, very frequently electrons can be identified with orbitals. This may be seen as a consequence of the Fermionic nature of electrons which explains their tendency to effectively behave “single-particle” like, and it is obvious for systems in which the wave function can be reasonably well approximated by a single Slater determinant, a situation encountered for many organic molecules and simple solids. Realizing that the notion of identifying orbitals with electrons is central to Eq. (2) helps us to sharpen our expectation about which observables are most likely to profit from the SIC: especially the ones that are single-particle like. Prominent ones are the single-particle density of states and related probes such as photoemission intensities. Indeed it has been shown that SIC is a powerful tool for helping in the interpretation of photoemission spectra [6, 7]. At the same time, we also get a feeling for where the dangers of the SIC may lie. By stressing the correspondence between electrons and orbitals the SIC may yield a density functional that stresses the single-particle character too much. In other words, by applying the correction of Eq. (2) we may take out important features of local and semi-local functionals that get their strength from a global, density-based description of exchange and correlation in terms of the total density. Over-correcting LDA and GGA by stressing the single-particle nature of orbitals may lead to a functional that is “too HartreeFock like”, i.e., we may lose important aspects of correlation. The traditional SIC of Ref. [23] minimizes the total energy with respect to orbital variations δ/δϕ∗i (r), i.e., it is close to Hartree-Fock theory in its variational ansatz. While this type of SIC has proved successful for certain classes of problems [23, 5], it is computationally tedious. More importantly yet, it does not offer general improvement for the description of electronic binding [24, 25] and has a tendency to underestimate bond strengths. One may interpret this as a consequence of being “too much Hartree-Fock like” [21]. Therefore, one may hope to achieve a more balanced combination of SIC and (semi-) local exchange-correlation functionals by bringing the SIC back under the umbrella of Kohn-Sham theory, i.e., instead of orbital variations δ/δϕ∗i (r) use the Kohn-Sham density variation δ/δn(r). This is the idea that we focus on in the next sections.

3.

Kohn-Sham Self-Interaction Corrections and Its Different Flavors

For any functional Exc [{ϕi }] that explicitly depends on the orbitals, taking the functional derivative δExc vxc (r) = (4) δn(r) is considerably more involved than for explicitly density dependent functionals such as LDA and GGA. Mathematically, however, the functional derivative with respect to the density is well defined in either case as long as the orbitals are themselves functionals of the density. Therefore, orbital functionals can be used in the Kohn-Sham framework, i.e., with

Self-Interaction Correction in the Kohn-Sham Framework

215

one multiplicative potential that is the same for all orbitals. For historical reasons the procedure of calculating vxc (r) for an orbital functional is called the “Optimized Effective Potential” (OEP) approach. It has been reviewed in detail, e.g., in [4, 26, 27], and we here focus only on the aspects of the OEP approach that are special [28] to the SIC energy expression. The Kohn-Sham approach defines the exchange-correlation potential by Eq. (4). Evaluating δExc /δn(r) for an orbital dependent Exc -expression proceeds by using the chain-rule for functional derivatives, schematically written as N

X δ = δn(r) i=1

Z

d3 r ′

δϕ∗i (r′ ) δ + c.c., δϕ∗i (r′ ) δn(r)

(5)

to turn the derivative with respect to n into derivatives with respect to the N occupied orbitals (c.c. stands for complex conjugate). A particular feature of the SIC energy functional is that Eq. (2) is not invariant under a unitary transformation U that leaves the density invariant but maps a given set or orbitals {ϕj } into another occupied set {ϕ˜i }, ϕ˜i =

N X

Uij ϕj .

(6)

j=1

(Spin indices are suppressed for notational clarity. In a spin-DFT calculation, all quantities in Eq. (6) would carry an additional spin-index σ.) Because of the unitary variance of Eq. (2) one therefore has an additional degree of freedom. The functional derivative with respect to the density requires an additional chain rule, Z N Z N X SIC [{ϕ SIC X δExc ˜j }] δ ϕ˜∗j (r′ ) ϕ∗i (r′′ ) δExc 3 ′ d r d3 r′′ = + c.c. (7) δn(r) δ ϕ˜∗j (r′ ) ϕ∗i (r′′ ) δn(r) i=1 j=1

SIC by which reflects this freedom. In other words, for a given, fixed density we can vary Exc choosing different unitary transformations between the occupied orbitals. The evaluation of the chain rules in Eq. (7) has been described in detail in Ref. [28], therefore we here only cite the result. The Kohn-Sham exchange-correlation potential for the SIC energy expression is defined as the solution of the integral equation N Z X  j=1

 SIC ′  3 ′ ′ ′ ′ ϕ∗j (r′ ) vxc (r ) − uSIC xc,j (r ) Gj (r , r)ϕj (r ) + c.c. d r = 0

(8)

where N

uxc,j

SIC [{ϕ ˜k }] 1 X δExc Uij = ∗ ′ ϕj (r ) δ ϕ˜i (r′ )

(9)

i=1

and Gj (r′ , r) =

N X ϕi (r′ )ϕ∗ (r) i

i=1

i6=j

εj − εi

(10)

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T. K¨orzd¨orfer and S. K¨ummel

is defined in terms of the Kohn-Sham orbitals and eigenvalues. Eq. (8) is called the Generalized Optimized Effective Potential (GOEP) equation. It yields the vxc (r) which leads via the Kohn-Sham equations to Kohn-Sham orbitals that, when subjected to the unitary transformation U, yield the lowest total energy. For a given unitary transformation the GOEP equation leads to a unique set of KohnSham orbitals {ϕi }. From these one obtains, via Eq. (6), a unique set of energy minimizing orbitals which typically are more localized than the Kohn-Sham orbitals and are therefore called localized orbitals. However, it is important to note that the GOEP equation does not specify the unitary transformation. Choosing U is an additional step, and different choices of U can be motivated. One natural choice is to make U the unitary transformation which yields the lowest possible energy. This U is not trivial to compute, but has been identified years ago [29, 30, 31]. The energy minimizing transformation appears as the best choice when one is thinking from a total energy perspective. On the other hand, one may also speculate that this choice makes the Kohn-Sham SIC in some sense closest to traditional Perdew-Zunger SIC and Hartree-Fock theory. Another natural choice is U = . Then, there is only one set of orbitals, the Kohn-Sham orbitals. Straightforward Kohn-Sham SIC has been shown to be a powerful method [32]. However, converging the SIC-OEP calculations iteratively [27, 33, 32] is very difficult. All variants of the SIC are harder to converge than LDA or GGA calculations, but the particularly tedious convergence of straightforward Kohn-Sham SIC may reflect that the balance between localization (driven by the Hartree orbital-correction) and de-localization (driven by the kinetic energy and the exchange-correlation correction) is a very subtle one when only one set of orbitals is used. One can also choose U to be the Foster-Boys localization [34, 35, 36], i.e., choose U such that the spatial spread of the orbitals is minimized. As the orbitals ϕ˜i that one finds from the energy minimizing U are typically localized, the Foster-Boys choice may be seen as a physically motivated approximation to the energy minimizing transformation. Its advantage is that it is numerically easier to evaluate than the energy minimizing criterion. These three choices are the ones that have been tested in some detail so far. However, in principle one may choose U according to other criteria. For example, one may try to choose U such that “the exchange-correlation hole stays localized”, hoping that in this way one may avoid the over-correction of taking out too much static correlation as discussed in section 2. Another idea is to choose U such that the total energy as a function of particle number consists of straight-line segments, a condition that has been associated with “manyelectron self-interaction freeness” [37, 38]. Using such conditions in practice, however, at the moment is not possible, because it is not known how the corresponding U-matrices would have to be constructed. In any case it is important to note that different variants of SIC can be defined, and the unitary variance, which may appear as a drawback on first sight, may actually be turned into a strength.

4.

SIC Static Electric Response

In order to develop an understanding for in how far the unitary transformation is important for the results that one obtains from a SIC calculation we here study a paradigm

Self-Interaction Correction in the Kohn-Sham Framework

217

problem for which LDA and GGA fail: The static electrical response of Hydrogen chains. Hydrogen chains are frequently employed [39, 40, 41, 42, 43, 44, 12, 45, 46, 47, 48, 32, 49] model systems that allow to test a functional’s ability to describe long-range charge transfer. The static R electric polarizability is defined as the derivative of the electrical dipole moment µ = − |e|n(r)r d3 r with respect to the electrical field F at vanishing field strength. Its longitudinal component ∂µz (F ) αzz = , (11) ∂Fz F =0

i.e., change of dipole moment along the chain’s backbone divided by the field applied along

LDA KS-Slater KS-KLI PZ-SIC KS-OEP CCSD(T)

3

longitudinal polarizability [a0 ]

250

200

150

100

50

4

6

8

10

12

number of hydrogen atoms in chain

Figure 1. Longitudinal polarizability of hydrogen chains in atomic units as a function of chain length. Here, Kohn-Sham SIC approaches without localizing transformation (i.e., U=1) are compared to LDA, Perdew-Zunger SIC with orbital specific potentials [47] and coupled-cluster (CCSD(T)) results [39]. Kohn-Sham OEP values are close to the coupledcluster benchmark. the chain’s backbone, is frequently used as a measure for an approaches’ ability to describe charge transfer. LDA and GGAs overestimate this quantity noticeably. Here, we calculated αzz using different density functionals, in particular different SIC approaches. The Hydrogen atoms were placed at alternating distances of 2 and 3 a0 . We employed a finite field approach as described in detail in Ref. [50], applying a linear electric field |e|rF and using the Bayreuth version [51, 28] of the PARSEC [52] electronic structure package. We used a field strength of 0.005 a.u., a Troullier-Martins pseudopotential with a cutoff radius of 1.39 a0 and a grid spacing of 0.25 a0 . In the energy minimizing OEP calculations the Pederson criterion was iterated to be fulfilled to at least 5 × 10−6 . In Fig. 1 we compare the longitudinal component of the polarizability as a function of chain length for different approaches that do not use a localizing transformation. For the Kohn-Sham approaches this corresponds to U = in the notation of the previous section. In addition to full OEP we also show the results that are obtained with two approximations to the OEP, the Krieger-Li-Iafrate (KLI) approximation [53] and the Slater approximation.

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T. K¨orzd¨orfer and S. K¨ummel

The latter two approximations have been described in detail in the literature, see, e.g., Refs. [26, 4, 28]. As a benchmark against which the DFT calculations are to be compared we included the Coupled-Cluster (singles,doubles and perturbative triples, CCSD(T)) results from [39], which can be regarded as quasi-exact for these systems. Kohn-Sham SIC is extremely close to the benchmark and thus proves itself as a powerful approach. However, as stated previously, these calculations are tedious to converge. One may therefore be tempted to resort to OEP approximations such as the KLI- or Slater scheme. However, Fig. 1 shows

LDA LOC-Slater LOC-KLI Garza-SIC LOC-OEP CCSD(T)

3

longitudinal polarizability [a0 ]

250

200

150

100

50

4

6

8

10

12

number of hydrogen atoms in chain

Figure 2. Longitudinal polarizability of hydrogen chains in atomic units as a function of chain length. Here, SIC approaches with an energy-minimizing localizing transformation are compared to LDA and coupled-cluster (CCSD(T)) results [39]. Also included are polarizabilities that were obtained with the approximate potential of Ref. [54]. that the Kohn-Sham KLI approximation is much less reliable than the OEP, and Kohn-Sham Slater behaves erratically. As an alternative GOEP calculations with an energy minimizing transformation are of great interest. These results are shown in Fig. 2. Again we also show the values that are obtained from using the KLI- and Slater approximations, and in addition we also constructed a local SIC potential following the procedure described in [54]. A number of interesting observations can be made. One immediately notes that again SIC considerably improves over LDA. However, the polarizabilities obtained from the generalized OEP equation with the energy minimizing transformation (denoted LOC-OEP in the plot) are not as close to the benchmark as the plain OEP. In fact they are relatively close to the Perdew-Zunger SIC values from Fig. 1. This finding is in line with the arguments given above. We further note that the KLI- and Slater approximations when used with the localized orbitals again lead to worse results than LOC-OEP, but the differences between the LOC-OEP and the LOC-KLI and LOC-

Self-Interaction Correction in the Kohn-Sham Framework

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Slater approximations are less pronounced than the differences between the corresponding approaches without localizing transformation. A somewhat puzzling result are the good polarizabilities that are obtained from the Garza-potential [54], which cannot be explained based on the GOEP formalism [28]. The success of orbital functional Kohn-Sham response calculations has previously been explained in terms of a field-counteracting term [41, 12, 32] stemming from the response part of vxc (r). The observation that LOC-KLI and LOC-Slater lead to noticeably better results than straightforward Kohn-Sham KLI and Kohn-Sham Slater suggests that the distribution between Slater- and response contributions to vxc (r) is considerably changed when an energy-minimizing unitary transformation is introduced. Thus, the concept of distinguishing between the Slater- and response-potential may only be of limited use in the GOEP context.

Conclusion The self-interaction error is one of the most fundamental problems in DFT. Curing it is not a trivial task due to the close relation between self-interaction and the (semi-) local modeling of non-dynamic correlation. We have discussed Kohn-Sham SIC as an alternative to the traditional SIC approach that used orbital specific potentials. The missing unitary invariance of the SIC energy functional can be exploited as a means of defining different SIC approaches via different choices for the unitary transformation. We tested different Kohn-Sham SIC approaches and SIC-approximations by calculating the static electric polarizability of hydrogen chains. SIC leads to polarizabilities that are noticeably closer to the benchmark wave function results than the polarizabilities from (semi-) local functionals. Yet, the specific definition of the SIC is important for the quality of the results.

Acknowledgments S.K. acknowledges support by Deutsche Forschungsgemeinschaft GRK 1640.

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[7] T. K¨orzd¨orfer, S. K¨ummel, N. Marom, and L. Kronik, Phys. Rev. B 82, 129903(E) (2010). [8] K. D. Dobbs and D. A. Dixon, J. Phys. Chem. 98, 12584 (1994). [9] T. N. Truong and W. Duncan, J. Chem. Phys. 101, 7408 (1994). [10] M. Lein and S. K¨ummel, Phys. Rev. Lett. 94, 143003 (2005). [11] D. Tozer, J. Chem. Phys. 119, 12697 (2003). [12] S. K¨ummel, L. Kronik, and J. P. Perdew, Phys. Rev. Lett. 93, 213002 (2004). [13] J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz Jr., Phys. Rev. Lett. 49, 1691 (1982). [14] M. Mundt and S. K¨ummel, Phys. Rev. Lett. 95, 203004 (2005). [15] J. P. Perdew, Adv. Quant. Chem. 21, 113 (1990). [16] T. Stein, L. Kronik, and R. Baer, J. Am. Chem. Soc. 131, 2818 (2009). [17] A. Karolewski, T. Stein, R. Baer, and S. K¨ummel, J. Chem. Phys. 134, 151101 (2011). [18] J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. 82, 2544 (1999). [19] J. Tao, J. P. Perdew, V. N. Staroverov and G. E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003). [20] J. P. Perdew and K. Schmidt, in Density Functional Theory and Its Application to Materials, Ed. V. Van Doren, C. Van Alsenoy and P. Geerlings, 1–20, American Institute of Physics, New York (2001). [21] D. Cremer, Mol. Phys. 99, 1899 (2001). [22] S. K¨ummel and J. P. Perdew, Mol. Phys. 101, 1363 (2002). [23] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [24] S. Goedecker and C. J. Umrigar, Phys. Rev. A 55, 1765 (1997). [25] O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 121, 8187 (2004). [26] T. Grabo, T. Kreibich, S. Kurth, and E. K. U. Gross, in Strong Coulomb Correlation in Electronic Structure: Beyond the Local Density Approximation, Ed. V. Anisimov, Gordon & Breach, Tokyo, (2000). [27] S. K¨ummel and J. P. Perdew, Phys. Rev. B 68, 035103 (2003). [28] T. K¨orzd¨orfer, S. K¨ummel, and M. Mundt, J. Chem. Phys. 129, 014110 (2008). [29] M. R. Pederson, R. A. Heaton, and C. C. Lin, J. Chem. Phys. 80, 1972 (1984).

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[30] M. R. Pederson, R. A. Heaton, and C. C. Lin, J. Chem. Phys. 82, 2688 (1985). [31] E. S. Fois, J. I. Penman, and P. A. Madden, J. Chem. Phys. 98, 6352 (1993). [32] T. K¨orzd¨orfer, M. Mundt, and S. K¨ummel, Phys. Rev. Lett. 100, 133004 (2008). [33] S. K¨ummel and J. P. Perdew, Phys. Rev. Lett. 90, 43004 (2003). [34] S. F. Boys, Rev. Mod. Phys. 32, 296 (1960). [35] J. M. Foster and S. F. Boys, Rev. Mod. Phys. 32, 300 (1960). [36] C. Edminston and K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963). [37] A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, J. Chem. Phys. 125, 194112 (2006). [38] P. Mori-S´anchez, A. J. Cohen, and W. Yang, J. Chem. Phys. 125, 201102 (2006). [39] B. Champagne, D. H. Mosley, M. Vra˘cko, and J.-M. Andr´e, Phys. Rev. A 52, 178 (1995). [40] B. Champagne, E. A. P. Stan, J. A. van Gisbergen, E.-J. Baerends, J. G. Snijders, C. Soubra-Ghaoui, K. A. Robins, and B. Kirtman, J. Chem. Phys. 109, 10489 (1998). [41] S. J. A. van Gisbergen, P. R. T. Schipper, O. V. Gritsenko, E. J. Baerends, J. G. Snijders, B. Champagne, and B. Kirtman, Phys. Rev. Lett. 83, 694 (1999). [42] M. van Faassen, P. L. de Boeij, R. van Leeuwen, J. A. Berger, and J. G. Snijders, Phys. Rev. Lett. 88, 186401 (2002). [43] N. Maitra and M. van Faassen, J. Chem. Phys. 126, 191106 (2007). [44] P. Mori-S´anchez, Q. Wu, and W. Yang, J. Chem. Phys. 119, 11001 (2003). [45] P. Umari, A. J. Willamson, G. Galli, and N. Marzari, Phys. Rev. Lett. 95, 207602 (2005). [46] C. D. Pemmaraju, S. Sanvito, and K. Burke, Phys. Rev. B 77, 121204(R) (2008). [47] A. Ruzsinszky, J. P. Perdew, G. I. Csonka, G. E. Scuseria, and O. A. Vydrov, Phys. Rev. A 77, 060502(R) (2008). [48] A. Ruzsinszky, J. P. Perdew and G. I.Csonka, Phys. Rev. A 78, 022513 (2008). [49] R. Armiento, S. K¨ummel, and T. K¨orzd¨orfer, Phys. Rev. B 77, 165106 (2008); in particular section IV. [50] S. K¨ummel and L. Kronik, Comput. Mater. Sci. 35, 321 (2006). [51] M. Mundt, S. K¨ummel, B. Huber, and M. Moseler, Phys. Rev. B 73, 205407 (2006).

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In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 223-253

Chapter 10

H OHENBERG -KOHN , K OHN -S HAM , AND Q UANTAL D ENSITY F UNCTIONAL T HEORIES IN THE P RESENCE OF A M AGNETOSTATIC F IELD Xiao-Yin Pan∗ Department of Physics and Institute of Modern Physics, Ningbo University, Ningbo, China Viraht Sahni† Brooklyn College and the Graduate School of the City University of New York, Brooklyn, NY, US

Abstract Traditional Hohenberg-Kohn (HK), Kohn-Sham (KS), and Quantal (QDFT) density functional theories are concerned with electronic structure in the presence of an external electrostatic field E(r) = −∇v(r). In this review, we describe recently developed understandings of these theories in the added presence of an external magnetostatic field B(r) = ∇ × A(r). We establish within the rigorous context of the original HK theorem that for the non-degenerate ground state the basic quantum-mechanical variables are the ground state density ρ(r) and physical current density j(r). This is achieved by proving, in a manner different from HK, that the relationship between the densities {ρ(r), j(r)} and the external potentials {v(r), A(r)} is one-to-one. As such the ground state wave function Ψ is a functional of {ρ(r), j(r)}. It is shown that Ψ must also be a functional of a gauge function. A {ρ(r), j(r)} functional theory is thereby constructed. The proof of bijectivity between {ρ(r), j(r)} and {v(r), A(r)}, valid for (v, A) - representable densities, is then extended via the constrained-search framework to N - representable densities and non-degenerate states. The corresponding Kohn-Sham and Quantal density functional theories are developed. An example of the mapping from the interacting to the non-interacting fermion model with the same {ρ(r), j(r)} is provided via QDFT employing the 2D Hooke’s atom. Finally, it is explained why the basic variables cannot be the ground state density ρ(r) and paramagnetic current density jp (r) as is thought to be the case. ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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PACS: 03.65.Ta, 31.15.E-, 31.15.ec, 71.15.Mb Keywords: Hohenberg-Kohn theorem in magnetostatic field, Hohenberg-Kohn density functional theory, Kohn-Sham density functional theory, Quantal density functional theory in magnetostatic field. Percus-Levy-Lieb constrained-search theorem.

1.

Introduction

Hohenberg-Kohn (HK) [1], Kohn-Sham (KS) [2], and Quantal density functional (QDFT) [3, 4] theories are concerned with the structure of electrons in the presence of an external electrostatic field E(r) = −∇v(r), with v(r) a scalar potential . HK theory proves that for a non-degenerate ground state, there is a one-to-one relationship between the ground state density ρ(r) and the external potential v(r) to within a constant. As such the ground state wave function Ψ is a functional of the density: Ψ = Ψ[ρ]. It has been proved [4, 5] that the wave function Ψ must also be a functional of a gauge function α(R) (with R = r1 , . . ., rN ) so as to ensure that the wave function written as a functional be gauge variant. Therefore, the wave function Ψ = Ψ[ρ, α]. As each choice of gauge function α corresponds to the same physical system, the choice of α = 0 is equally valid. Thus, knowledge of the ground state density ρ(r) uniquely determines all the properties of the system, and hence also the ground state energy E. The ground state density ρ(r) is thus said to be a basic variable of quantum mechanics. The KS and QDFT theories map the interacting system as described by Schr¨odinger theory to one of non-interacting fermions or bosons with the same ground state density ρ(r). The ground state energy E and the ionization potential or electron affinity are then subsequently obtained from these model systems [3, 4]. This article is concerned with our recent understandings [6, 7, 8] of these theories in the added presence of an external magnetostatic field B(r) = ∇ × A(r), where A(r) is a vector potential. Of significance are properties such as the Zeeman effect in atoms and molecules, and the de-Haas-van Alphen effect, the Hall effect, and Magneto-resistance in solids. The more recent applications have focused on electrons confined to two-dimensions: hetero-structures, the integer and fractional Quantum Hall effects, Quantum Dots, and other properties in the presence of high magnetic fields. As in HK theory, it is first proved that there exists a one-to-one relationship between the ground state density ρ(r) and physical current density j(r), and the external potentials {v(r), A(r)}. Consequently, the non-degenerate ground state wave function is a functional of {ρ(r), j(r)}. To ensure gauge variance, the wave function must also be a functional of a gauge function α(R). Therefore, the wave function Ψ = Ψ[ρ, j, α]. As α = 0 is a valid choice, knowledge of the ground state {ρ(r), j(r)} uniquely determines all the properties of the system including the ground state energy E. Thus, in the presence of a magnetic field B(r), the basic variables in quantum mechanics are {ρ(r), j(r)}. The proof of the bijectivity between {v(r), A(r)} and {ρ(r), j(r)} differs in a fundamental way from the original proof of HK. Having established this bijectivity, it is then possible to construct a (ρ, j) - functional theory. The proof of the bijectivity between the densities {ρ(r), j(r)} and the potentials {v(r), A(r)}, and the subsequent (ρ, j) - functional theory are valid for (v, A) - representable densities. Again, having first established that {ρ(r), j(r)} are the

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basic variables, it is then possible to prove via a Percus-Levy-Lieb [9] constrained-searchtype proof that the weaker requirement of N -representability suffices. The restriction to non-degenerate ground states is also removed. Next we map the interacting system to one of non-interacting fermions - the S system – with the same ground state {ρ(r), j(r)} within both a KS and QDFT frame work. The KS theory is in terms of the energy functional of {ρ(r), j(r)} and its functional derivatives. The QDFT description of the S system is in terms of “classical” fields and their quantal sources. We then provide an example of the mapping via QDFT for the exactly solvable 2D Hooke’s atom [10]. Finally, we address the work of Vignale and co-workers [11, 12, 13] which has been the basis of the understandings in this area thus far. According to this understanding, the basic variables are the ground state density ρ(r) and the paramagnetic current density jp (r). This conclusion does not fall within the rigorous rubric of the HK theorem definition of a basic variable because the relationship between {ρ(r), jp(r)} and {v(r), A(r)} is not one-to-one [14, 15]. We explain where the arguments and proof of these authors breaks down. For completeness, we note that the HK theorem [1] was generalized by Rajagopal and Callaway [16] to the relativistic case in which the variables are the four-current {ρ(rt), j(rt)} and the four-potential {v(rt), A(rt)}. For the stationary-state theory, the idea of employing {ρ(r), j(r)} within the context of KS theory [2] was due to Ghosh and Dhara [17] and Diener [18]. The former employ these variables without proving the oneto-one relationship between {ρ(r), j(r)} and {v(r), A(r)}. The latter does not account for the fact that in the presence of a magnetic field, the relationship between the potentials {v(r), A(r)} and the wave function Ψ can be many-to-one and not one-to-one. There is also a magnetic-field density functional theory [19, 20] in which the variables employed are the density ρ(r) and the magnetic field B(r). We begin with an explanation of the concept of a basic variable, the definition of which stems from the Hohenberg-Kohn theorem [1].

2.

Definition of a Basic Variable

For a system of N electrons in an external electrostatic field E(r) = −∇v(r), the Schr¨odinger equation in atomic units (e = ~ = m = 1) is ˆ ˆ + Vˆ )Ψ(X) = EΨ(X), H(R)Ψ(X) = (Tˆ + W

(1)

where {Ψ(X), E} are the eigenfunctions and eigenenergies, with R = r1 , . . . , rN ; X = x1 , . . . , xN ; x = rσ, {rσ} being the spatial and spin coordinates of the electron. The operators are the kinetic 1X 2 Tˆ = pk 2

;

k

electron-interaction

p ˆ k = −i∇rk ,

X 1 ˆ =1 W , 2 |rk − r` | k,`

(2)

(3)

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and external potential Vˆ =

X

v(rk ).

(4)

k

The density ρ(r) is the expectation ρ(r) =< Ψ(X)|ˆ ρ(r)|Ψ(X) >,

(5)

with the density operator ρˆ(r) =

X k

and

δ(rk − r),

(6)

R

ρ(r) dr = N . The total energy E of the system is the expectation ˆ E = < Ψ(X)|H(R)|Ψ(X) > = T + Eee + Eext

(7) (8)

where the kinetic energy T =< Ψ(X)|Tˆ|Ψ(X) >=< Ψ(X)|

1X 2 pk |Ψ(X) >, 2

(9)

P (rr0 ) dr dr0 , |r − r0 |

(10)

k

the electron-interaction energy Eee

ˆ |Ψ(X) >= 1 =< Ψ(X)|W 2

Z

with the pair-correlation function P (rr0 ) being the expectation P (rr0 ) =< Ψ(X)|Pˆ (rr0 )|Ψ(X) >,

(11)

where the pair-correlation operator Pˆ (rr0 ) =

X0 k,`

δ(rk − r)δ(r` − r),

(12)

Z

(13)

and the external energy Eext

=< Ψ(X)|Vˆ |Ψ(X) =

ρ(r)v(r) dr.

To arrive at the Hohenberg-Kohn theorem, it is first proved (Map C) that the relationship between the external scalar potential v(r) and the non-degenerate ground state wave function Ψ(X) is one-to-one. This fact is then employed to prove (Map D) that the relationship between the ground state wave function Ψ(X) and the ground state density ρ(r) is also one-to-one. (Map D is established via the contradiction E + E 0 < E + E 0 , where E 0 corresponds to v 0 (r) and Ψ0 (X).) Thus, Map C v(r)

←→

Map D Ψ(X)

←→

ρ(r).

(14)

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Knowledge of the ground state density ρ(r) then uniquely determines the external ˆ operators are aspotential v(r), and since the kinetic Tˆ and electron-interaction W ˆ sumed known, so is the system Hamiltonian H(R) determined uniquely. Solution of the Schr¨odinger equation Eq. (1) for the wave function Ψ(X) then leads to all the properties of the system. Equivalently, the ground state wave function Ψ(X) is a functional of the ˆ i.e. ground state density ρ(r), i.e. Ψ(X) = Ψ[ρ(r)]. Expectations of all operators O, ˆ O =< Ψ[ρ(r)]|O|Ψ[ρ(r)] > are unique functionals of the density: O = O[ρ(r)]. Prior to defining a basic variable, there is one additional point to note. A unitary or gauge transformation of the above Schr¨odinger equation that preserves the density ρ(r) can P P be performed [4, 5]. The unitary operator is U = exp[i j α(rj )]; j α(rj ) = α(R), where α(r) is a smooth function of position. The transformed wave function Ψ0 (X) and ˆ 0 (R) are, respectively Hamiltonian H Ψ0 (X) = U † Ψ(X),

(15)

and ˆ 0 (R) = U † H(R)U ˆ H 1X ˆ + Vˆ , = [ˆ pk + A(rk )]2 + W 2

(16)

k

ˆ 0 (R)Ψ0 (X) = where A(r) = ∇α(r). The transformed Schr¨odinger equation is then H E 0 Ψ0 (X) with E 0 = E. Thus, the wave function and Hamiltonian are gauge variant. All the properties of the system such as ρ(r), T, Eee, E etc., are gauge invariant. As one can perform the above unitary transformation for any arbitrary choice of gauge function α(R), one concludes that the wave function is a functional of the gauge function α(R) : Ψ(X) = Ψ[α(R)]. Additionally, the Hohenberg-Kohn theorem is then generalized to be valid for each choice of gauge function α(R). In other words, the relationship between the ground state density ρ(r) and the external potential v(r) is unique for each choice of gauge function α(R). Thus, a more general statement can be made with regard to the ground state wave function viz. that it is a functional of both the density ρ(r) and the gauge function α(R) : Ψ(X) = Ψ[ρ(r), α(R)]. As the wave function Ψ(X) is gauge variant [4, 5, 21] and the density ρ(r) gauge invariant, it is the presence of the gauge function α(R) which ensures that the wave function written as a functional Ψ[ρ(r), α(R)] is gauge variant. As the physical system remains unchanged for each gauge function α(R), the choice of α(R) = 0 is equally valid. Hence, the wave function can be considered as a functional of the ground state density ρ(r) : Ψ(X) = Ψ[ρ(r), α(R) = 0]. With the above understanding, the definition of a basic variable is as follows: A basic variable is a gauge invariant property knowledge of which uniquely determines the external potential, and thereby determines uniquely the Hamiltonian of the system. Solution of the corresponding Schr¨odinger equation then leads to all the properties of the system. It is only within the rigorous context of the bijective relationship between a basic variable and the external potential that the non-degenerate ground state wave function can be said to be a functional of the basic variable.

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Definitions in the Added Presence of a Magnetostatic Field

We next define the system of electrons in an external scalar v(r) and vector A(r) potential, discuss the gauge variance/invariances of properties, and via a unitary transformation show that the wave function is a functional of the gauge function α(R). When both an external electrostatic field E(r) = −∇v(r) and a magnetostatic field B(r) = ∇ × A(r) are present, the Schr¨odinger equation in units such that (e = ~ = m = c = 1; for atomic units replace A by A/c) is ˆ ˆ )Ψ(X) = EΨ(X), H(R)Ψ(X) = (TˆA + Vˆ + W

(17)

where TˆA is the physical kinetic energy operator 1X (ˆ pk + A(rk ))2 2 k Z Z 1 = Tˆ + ˆj(r) · A(r)dr − ρˆ(r)A2(r)dr, 2

TˆA =

(18) (19)

with ˆj(r) the physical current density operator ˆj(r) = ˆjp (r) + ˆjd (r), where ˆjp (r) is the paramagnetic current density operator  X ˆjp(r) = 1 ∇rk δ(rk − r) + δ(rk − r)∇rk , 2i

(20)

(21)

k

and ˆjd the diamagnetic current density operator ˆjd (r) = ρˆ(r)A(r).

(22)

The total energy E is the expectation ˆ E = < Ψ(X)|H(R)|Ψ(X) Z = TA + Eee + ρ(r)v(r) dr, where the physical kinetic energy TA is Z Z 1 TA = T + j(r) · A(r)dr − ρ(r)A2(r)dr, 2

(23) (24)

(25)

with T the “canonical” kinetic energy of Eq. (9), and j(r) the physical current density which is the expectation

j(r) = Ψ(X)|ˆj(r)|Ψ(X) (26) = jp (r) + jd (r),

(27)

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229

with the paramagnetic current density jp (r) the expectation

jp(r) = Ψ(X)|ˆjp(r)|Ψ(X)

(28)

and the diamagnetic current density jd (r) being jd (r) = ρ(r)A(r).

(29)

Thus, the total energy E may be written as (see also Eq. (8)) A E = T + Eee + Eext ,

(30)

A where the external component Eext of the energy is A Eext

=

Z

ρ(r)v(r) dr +

Z

1 j(r) · A(r) dr − 2

Z

ρ(r)A2(r) dr.

(31)

Performing the same density preserving unitary transformation on Eq. (17) as in the previous section, the transformed wave function is Ψ0 (X) = U † Ψ(X), and the transformed Hamiltonian is X 2 ˆ + Vˆ . ˆ 0 (R) = 1 ˆ k + A(rk ) + ∇α(rk ) + W p H 2

(32)

k

ˆ 0 (R)Ψ0 (X) = E 0Ψ0 (X) with E 0 = E. The transformed Schr¨odinger equation is H Once again the wave function is a functional of the gauge function α(R) : Ψ(X) = Ψ[α(R)](X). Equivalently, if one performs a gauge transformation of the vector potential A(r) [21] such that A0 (r) = A(r) + ∇α(r),

(33)

but let v 0 (r) = v(r), the Hamiltonian of Eq. (17) changes to that of Eq. (32). Thus, the Hamiltonian is gauge variant. Since the physical systemP remains the same, the wave function Ψ(X) must be multiplied by a phase factor exp[−i j α(rj )]. The system wave function is therefore also gauge variant. However, all the physical properties of the system such as the energy E and its individual components, the density ρ(r), and the physical current density j(r) which are all expectations of Hermitian operators remain the same and are gauge invariant. The component paramagnetic jp (r) and diamagnetic jd (r) current densities, on the other hand, are gauge variant. The choice of gauge function α(R) is arbitrary because the physical properties of the system remain unchanged. The choice of α(R) = 0 is thus an equally valid one. Finally, as the system is time-independent, the continuity equation for the physical current density j(r) is ∇ · j(r) = ∇ · jp (r) + ∇ · jd (r) = 0.

(34)

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Xiao-Yin Pan and Viraht Sahni

4.

Proof of Bijectivity between {v(r), A(r)} and {ρ(r), j(r)}

We next prove the bijectivity between {v(r), A(r)} and the non-degenerate ground state {ρ(r), j(r)}. As the expression for the energy of Eq. (30) is in terms of the properties {ρ(r), j(r)}, it would appear that one could prove a one-to-one relationship between {ρ(r), j(r)} and the external potentials {v(r), A(r)} along the lines of the original Hohenberg-Kohn proof. However, such a proof is not possible because the relationship between the external potentials {v(r), A(r)} and the non-degenerate ground state wave function Ψ(X) can be many-to-one:

{ v(r), A(r)} &

{ v 0 (r), A0(r)} →

{ v 00 (r), A00(r)} → ...

Ψ(X)

%

(35)

Hence, there can be no equivalent of Map C of Eq. (14). The many-to-one relationship is exhibited [10] by the example of the 2D Hooke’s atom in a magnetic field in which there exist an infinite number of {v(r), A(r)} that generate the same ground state Ψ(X). (The many-to-one relationship between external potentials and the wave function was noted originally by von Barth and Hedin [22] with regard to spin density functional theory. It has also been noted [14] for the Hamiltonian of Eq. (17) as well as the Hamiltonian for superconducting density functional theory.) The bijectivity can be proved as follows: Suppose now that there exists another {v 0 (r), A0(r)} (with corresponding ground state wave function Ψ0 ) that also lead to the same ground state {ρ(r), j(r)}. (The v 0 (r) differs from the v(r) by more than a constant and the A0 (r) differs from A(r) by more than a gauge transformation.) We prove that this cannot be the case. Let us assume that Ψ 6= Ψ0 . Then according to the variational principle for the energy

ˆ E = Ψ|H|Ψ

<

0 ˆ 0 . Ψ |H|Ψ

(36)

Now Z 02 ˆj(r) · A0 (r)dr − 1 ρ(r)A ˆ (r)dr|Ψ0 2 Z

0

Ψ |Vˆ − Vˆ 0 |Ψ0 + Ψ0 | ˆj(r) · [A(r) − A0 (r)]dr|Ψ0 Z

0 1 ρˆ(r)[A2(r) − A02 (r)]dr|Ψ0 (37) Ψ| 2     Z Z E 0 + ρ(r) v(r) − v 0 (r) dr + j(r) · A(r) − A0 (r) dr Z 1 ρ(r)[A2(r) − A02 (r)]dr. (38) 2

0

ˆ 0 = Ψ0 |Tˆ + Vˆ 0 + U ˆ+ Ψ |H|Ψ + − = −

Z

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional ... Thus, from Eqs. (36) and (38),     Z Z 0 0 0 E ≥ < Ψ|H|Ψ ˆ > = E, < Ψρ,j |H|Ψ

(60)

which on employing the expression for E of Eq. (30) is equivalent to ˆ |Ψρ,j > ≥ < Ψ|Tˆ + W ˆ |Ψ > . < Ψρ,j |Tˆ + W

(61)

Thus, of all antisymmetric functions Ψρ,j that lead to the ground state {ρ(r), j(r)}, the true ˆ >. ground state wave function is that which minimizes the expectation < Tˆ + W It follows from Eq. (61) that the universal functional F [ρ, j] for the ground state {ρ(r), j(r)} of Eq. (51) may be assigned the constrained-search definition: ˆ |Ψρ,j > F [ρ, j] = < Ψρ,j |Tˆ + W ˆ |Ψρ,j > . = min < Ψρ,j |Tˆ + W Ψρ,j →ρ,j

(62) (63)

Thus, searching over all N -representable functions Ψρ,j that lead to the ground state ˆ >. {ρ(r), j(r)}, the functional F [ρ, j] delivers the minimum of the expectation < Tˆ + W Furthermore, the functional F [ρ, j] as defined by Eq. (63) is equally valid for degenerate states as well as for {ρ(r), j(r)} that are not (v, A)-representable. We note that for a given {ρ(r), j(r)}, it is possible to construct Slater determinantal functions that will reproduce these properties but which are not necessarily eigenfunctions of a differential equation. With the definition of the functional F [ρ, j] of Eq. (63), the ground-state energy, minimization may be divided into two steps: Z Z ˆ ˆ E = min < Ψ|T + W + ρˆ(r)v(r)dr + ˆj(r) · A(r)dr Ψ Z 1 − ρˆ(r)A2(r)dr|Ψ > (64) 2 Z Z  ˆ + ρˆ(r)v(r)dr + ˆj(r) · A(r)dr = min min < Ψ|Tˆ + W ρ,j Ψ→ρ,j Z 1 ρˆ(r)A2(r)dr|Ψ > (65) − 2 Z Z   ˆ ˆ = min min < Ψ|T + W |Ψ > + ρ(r)v(r)dr + j(r) · A(r)dr ρ,j Ψ→ρ,j Z  1 − ρ(r)A2(r)dr . (66) 2 The inner minimization of Eq. (65) is constrained to all antisymmetric functions that lead to {ρ(r), j(r)}. The outer minimization releases this constraint by searching over all

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional ...

235

{ρ(r), j(r)}. As the first term of Eq. (66) is F [ρ, j] of Eq. (63), we have Z Z Z  1 E = min F [ρ, j] + ρ(r)v(r)dr + j(r) · A(r)dr − ρ(r)A2(r)dr , (67) ρ,j 2 = min E[ρ, j]. (68) ρ,j

The variations in Eq. (68) are over all N -representable {ρ(r), j(r)}. This then eliminates the stringent (v, A)-representability constraint.

7.

Construction of the S system within the Kohn-Sham Framework

We next assume that there exists a model system of non-interacting fermions – an S system – that can possess the same ground state density ρ(r) and physical current density j(r) as that of the interacting system described previously. This implies that one must determine an effective scalar potential vs (r) and an effective vector potential As (r) which on substitution into the corresponding Schr¨odinger equation for the model fermions will generate a Slater determinant that will reproduce the true ρ(r) and j(r). In essence, one thus assumes there exists an effective magnetic field Bs (r) = ∇ × As (r). The S system Hamiltonian is X ˆ s = TˆA,s + Vˆs = hs (rk ), (69) H k

TˆA,s

Vˆs

1X = (ˆ pk + As (rk ))2 2 k Z Z 1 ρˆ(r)A2s (r)dr = Tˆ + ˆjs (r) · As (r)dr − 2 X = vs (rk ),

(70) (71) (72)

k

1 hs (r) = (ˆ p + As (r))2 + vs (r), 2 ˆjs (r) = ˆjp (r) + ˆjd,s (r), ˆjd,s (r) = ρˆ(r)As(r). The corresponding S system Schr¨odinger equation for each model fermion is   1 2 (ˆ p + As (r)) + vs (r) φa (x) = a φa (x); a = 1, . . ., N, 2

(73) (74) (75)

(76)

the wave function being a Slater determinant Φ{φa} of the orbitals φa (x) which occupy the lowest states. Hence, the density ρ(r) and current density j(r) are the expectations

ρ(r) = Φ{φa }|ˆ ρ(r)|Φ{φa} , (77)

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Xiao-Yin Pan and Viraht Sahni

and

j(r) = Φ{φa}|ˆjs (r)|Φ{φa} = jp,s (r) + jd,s (r).

(78)

The proof of the bijectivity between {ρ(r), j(r)} and {vs (r), As(r)} is similar to that for the interacting system. Thus, the orbitals φa (x) are functionals of {ρ(r), j(r)} : φa = φa [ρ, j]. So are the components of the energy, the expression for which we determine P next. ? Multiplying Eq. (76) by φa (x), integrating over r, and performing the sum σ,a , we obtain the physical kinetic energy of the non-interacting fermions TA,s as

TA,s [ρ, j] = Φ{φa }|TˆA,s|Φ{φa} (79) Z Z 1 ρ(r)A2s (r)dr (80) = Ts [ρ, j] + j(r) · As (r)dr − 2 Z X = a − ρ(r)vs(r)dr, (81) a

where Ts [ρ, j] is the “canonical” kinetic energy of these fermions:

Ts [ρ, j] = Φ{φa}|Tˆ|Φ{φa} .

The interacting system ground state energy E of Eq. (30) is Z Z E[ρ, j] = ρ(r)v(r)dr + j(r) · A(r)dr Z 1 ρ(r)A2(r)dr + T [ρ, j] + Eee [ρ, j]. − 2 To this expression we add and subtract Z Z 1 Ts [ρ, j] + j(r) · As(r)dr − ρ(r)A2s (r)dr 2

(82)

(83)

(84)

to obtain (using Eqs. (80) and (81) Z Z P E[ρ, j] =  + ρ(r)[v(r) − v (r)]dr + j(r) · [A(r) − As (r)]dr a s a Z 1 − ρ(r)[A2(r) − A2s (r)]dr + T [ρ, j] − Ts [ρ, j] + Eee . (85) 2 Defining the Correlation-Kinetic energy Tc [ρ, j] as Tc [ρ, j] = T [ρ, j] − Ts [ρ, j],

(86)

KS [ρ, j] functional as and the Kohn-Sham (KS) electron interaction energy Eee KS Eee [ρ, j] = Tc [ρ, j] + Eee [ρ, j],

(87)

we have the expression for the ground state energy E[ρ, j] to be Z Z X E[ρ, j] = a + ρ(r)[v(r) − vs (r)]dr + j(r) · [A(r) − As (r)]dr a



1 2

Z

KS ρ(r)[A2(r) − A2s (r)]dr + Eee [ρ, j].

(88)

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional ...

237

To obtain the effective potentials {vs , As} within the KS framework, we take the functional derivatives of E[ρ, j] with respect to the density and physical current density: δE[ρ, j] = 0, (89) δρ(r) j(r) which yields

  1 2 2 vs (r) = v(r) + vee (r) − A (r) − As (r) , 2 where the local effective electron-interaction scalar potential vee (r) is defined as KS δEee [ρ, j] vee (r) = , δρ(r) j(r)

and

δE[ρ, j] = 0, δj(r) ρ(r)

which yields

As (r) = A(r) + Aee (r), where the effective electron-interaction vector potential Aee (r) is defined as KS δEee [ρ, j] Aee (r) = . δj(r) ρ(r)

(90)

(91)

(92)

(93)

(94)

KS (Notice that since the functional Eee [ρ, j] depends upon the gauge invariant properties {ρ, j}, the potentials {vee , Aee} are gauge invariant.) On substituting for vs (r) and As (r) from Eqs. (90) and (93) into Eq. (88), we obtain the expression for the ground state energy E[ρ, j] solely in terms of the S system properties: Z Z X KS E[ρ, j] = a − ρ(r)vee(r)dr − j(r) · Aee (r)dr + Eee [ρ, j]. (95) a

This expression for the energy is gauge invariant. Note that the S system scalar and vector potentials {vs , As} are comprised of the external potentials {v, A} and an effective electron-interaction potential component {vee , Aee }. Thus, with the same external potentials {v, A} also applied to the model system, it is the effective electron-interaction potentials {vee , Aee } that ensure the orbitals φa (x) generate the true density ρ(r) and current density j(r). The potentials {vee , Aee} are defined in terms KS of the functional derivatives of the KS electron-interaction energy functional Eee [ρ, j]. This energy functional represents contributions due to the Pauli exclusion principle and Coulomb repulsion between the electrons in the presence of the potentials {v, A}. It also incorporates Correlation-Kinetic effects which arise due to the difference between the kinetic energy of the interacting and model fermions possessing the same {ρ, j}. These are the correlation contributions to the kinetic energy. All these correlations are incorporated in this energy functional, and hence in the functional derivatives {vee , Aee} that generate the orbitals φa(x). Note that the expression for vs (r) of Eq. (90) also includes the difference

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Xiao-Yin Pan and Viraht Sahni

of the square of the vector potentials for the interacting and model S systems. Finally, since the potentials {vee , Aee } are gauge invariant, the S system Schr¨odinger equation is gauge covariant. Furthermore, the continuity equation is satisfied by the self-consistent solution of the S system differential equation.

8.

Construction of S System via Quantal Density Functional Theory

As in conventional QDFT [3, 4], the description of the mapping in the presence of an external magnetostatic field to an S system with the same {ρ(r), j(r)} is in terms of “classical” fields whose sources are quantal in that they are expectations of Hermitian operators. In addition to the fields representative of correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects, there now exists a field representative of correlation-magnetic effects. The new field is a consequence of the difference between the vector potentials for the interacting and model systems. The equations of the QDFT are derived from the generalized “quantal Newtonian” first law (or differential virial theorem) and the integral virial theorem as written in terms of fields [7] for both the interacting and S systems. As QDFT is based on the virial theorems, the theory is valid for both a nondegenerate ground or excited state of the interacting electrons. Additionally, the state of the model S system is arbitrary, in that it may be in a ground or excited state configuration. We begin with a description of the interacting system as described by the Hamiltonian of Eq. (17). The “quantal Newtonian” first law for the interacting system, according to which the sum of the external F ext(r) and internal F int(r) fields experienced by each electron vanishes, is F ext(r) + F int (r) = 0. (96) This law is valid for arbitrary gauge and derived employing the continuity condition of Eq. (34). (For two different derivations of the law see [7, 24].) The external field F ext(r) is the sum of the electrostatic E(r) and Lorentz L(r) fields: F ext (r) = E(r) − L(r),

(97)

where L(r) is defined in terms of the Lorentz “force” l(r) as L(r) =

l(r) , ρ(r)

(98)

and where l(r) = j(r) × B(r),

(99)

with its components given as lα(r) =

3 X 

β=1

 jβ (r)∇αAβ (r) − jβ (r)∇β Aα (r) .

(100)

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional ...

239

The internal field F int(r) is the sum of the electron-interaction E ee (r), kinetic Z(r), differential density D(r), and internal magnetic I(r) fields: F int(r) = E ee (r) − Z(r) − D(r) − I(r).

(101)

These fields are defined in terms of the corresponding “forces” eee (r), z(r; γ), d(r), i(r; jA), respectively, as E ee (r) =

eee (r) z(r; γ) d(r) i(r; jA) ; Z(r) = ; D(r) = ; I(r) = . ρ(r) ρ(r) ρ(r) ρ(r)

(102)

The electron-interaction “force” eee (r), representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, is obtained via Coulomb’s law via its quantal source, the pair-correlation function P (rr0 ) of Eq. (11): Z P (rr0 )(r − r0 ) 0 eee (r) = dr . (103) |r − r0 |3 The kinetic “force” z(r; γ), representative of kinetic effects, is obtained from its quantal source, the reduced single-particle density matrix γ(rr0 ) which is the expectation

γ(rr0 ) = Ψ(X)|ˆ γ (rr0 )|Ψ(X) , (104)

where the single-particle density matrix operator γˆ(rr0 ) is ˆ γˆ(rr0 ) = Aˆ + iB,

(105)

N

Aˆ =

 1 X δ(rj − r)Tj (a) + δ(rj − r0 )Tj (−a) , 2

(106)

j=1

ˆ = −i B 2

N X  j=1

 δ(rj − r)Tj (a) − δ(rj − r0 )Tj (−a) ,

(107)

Tj (a) is a translation operator such that Tj (a)Ψ(. . . rj , . . .) = Ψ(. . . rj + a, . . .), and a = r0 − r. The “force” z(r; γ) is defined in terms of its components as zα (r; γ) = 2

3 X

β=1

∇β tαβ (r; γ),

(108)

where the kinetic energy tensor tαβ (r) is   ∂2 ∂2 1 0 00 + γ(r r ) tαβ (r; γ) = 00 0 0 00 . 4 ∂rα0 ∂rβ ∂rβ ∂rα00 r =r =r

(109)

The differential density “force” d(r) whose quantal source is the density ρ(r), is defined as 1 d(r) = − ∇∇2 ρ(r). 4

(110)

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Xiao-Yin Pan and Viraht Sahni

Finally, the contribution of the magnetic field to the internal “force” i(r; jA) for which the quantal source is the physical current density j(r) is defined in terms of its components as iα(r; jA) =

3 X

β=1

∇β Iαβ (r; jA),

(111)

with   Iαβ (r; jA) = jα(r)Aβ (r) + jβ (r)Aα(r) − ρ(r)Aα(r)Aβ (r).

(112)

E = Eext + (T + Eee + I),

(113)

The fields L(r), E ee (r), D(r), and the sum [Z(r) + I(r)] are gauge invariant [24]. The energy E is then the sum of the kinetic T , external Eext, electron-interaction Eee , and internal magnetic contribution I energies:

where in integral virial form in terms of the respective fields Z 1 ρ(r)r · Z(r)dr T = − 2 Z Eext = ρ(r)r · F ext(r)dr Z Eee = ρ(r)r · E ee (r)dr Z I = ρ(r)r · I(r)dr. By operating on the first law by

R

(114) (115) (116) (117)

drρ(r)r· one obtains the integral virial theorem [25]:

2T + Eee − I = −Eext.

(118)

The “quantal Newtonian” first law [7] for the S system defined by Eqs. (69-78) derived employing the continuity condition of Eq. (34) is int F ext s (r) + F s (r) = 0,

(119)

F ext s (r) = E(r) − Ls (r).

(120)

where Here Ls (r) is the corresponding effective Lorentz field defined in terms of the Lorentz “force” as ls (r) , (121) Ls (r) = ρ(r) with ls (r) = j(r) × Bs (r),

(122)

and where the components of the “force” are  3  X ls,α (r) = jβ (r)∇αAs,β (r) − jβ (r)∇β As,α (r) . β=1

(123)

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional ...

241

The internal field F int s (r) of the S system is F int s (r) = −∇vee (r) − D(r) − Z s (r) − I s (r),

(124)

where the kinetic Z s (r) and internal magnetic I s (r) fields are defined in terms of the “forces” zs (r; γs) and is (r; jAs) as Z s (r) =

zs (r; γs) ρ(r)

and

I s (r) =

is (r; jAs) . ρ(r)

(125)

The S system kinetic “force” in turn is defined in terms of its quantal source, the Dirac density matrix γs (rr0 ) as zs,α (r; γs) = 2

3 X

β=1

∇β ts,αβ (r; γs)

where the kinetic energy tensor ts,αβ (r; γs) is   ∂2 ∂2 1 0 00 + γ (r r ) ts,αβ (r; γs) = s 0 00 , 4 ∂rα0 ∂rβ00 ∂rβ0 ∂rα00 r =r =r

(126)

(127)

and the source

N

XX γs (rr0 ) = Φ{φi }|ˆ γ (rr0 )|Φ{φi} = φ?i (rσ)φi(r0 σ). σ

(128)

i=1

The internal magnetic “force” is (r; jAs ) is defined as X is,α (r; jAs ) = ∇β Is,αβ (r; jAs ),

(129)

β

with   Is,αβ (r; jAs ) = jα (r)As,β (r) + jβ (r)As,α (r) − ρ(r)As,α(r)As,β (r).

(130)

The effective scalar and vector potentials {vs (r), As(r)} of the S system differential equation Eq. (76) are determined as follows: (a) The effective vector potential As (r) is obtained from the requirement that the physical current density j(r) of the S system Eq. (78) is the same as that of the interacting system Eq. (26). (b) We assume that the external electrostatic field E(r) is the same for the interacting and model systems. Thus, we write the scalar potential vs (r) as vs (r) = v(r) + vee (r),

(131)

where vee (r) is an effective scalar electron-interaction potential whose rigorous physical interpretation will be given below. We then equate the corresponding expressions for this field given by the “quantal Newtonian” first law for each system. Thus, we obtain the

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Xiao-Yin Pan and Viraht Sahni

effective electron-interaction potential energy vee (r) to be the work done to move a model fermion from some reference point at infinity to its position at r in the force of a conservative effective field F eff (r): Z r

vee (r) = −



F eff (r0 ) · d`0 ,

(132)

where F eff (r) is the sum of the electron-interaction E ee (r), correlation-kinetic Z tc (r), and correlation-magnetic Mc (r) fields: F eff (r) = E ee (r) + Z tc (r) + Mc (r),

(133)

and where Z tc (r) = Z s (r) − Z(r),

Mc (r) = Ls (r) − L(r) + I s (r) − I(r).

(134) (135)

As in the B = 0 case [3, 4], the field E ee (r) may be subdivided into its Hartree E H (r), Pauli E x (r), and Coulomb E c (r) field components. The quantal sources for these fields are the density ρ(r), the Fermi hole ρx (rr0 ), and the Coulomb hole ρc(rr0 ), respectively. Thus, the effective field may be expressed as F eff (r) = E H (r) + E x (r) + E c (r) + Z tc (r) + Mc (r),

(136)

with each field being representative of a specific electron correlation. Note that ∇ × F eff (r) = 0 so that the work done vee (r) is path-independent. The individual components of F eff (r) are separately curl free for systems with certain symmetry, as in the example of the following section which is one of cylindrical symmetry. The work done in each field is then path-independent. Writing the effective vector potential As (r) as As (r) = A(r) + Aee (r),

(137)

where Aee (r) is an effective electron-interaction vector potential, the expression for the ground state energy in terms of S system properties is Z Z X E= i − ρ(r)vee(r)dr − j(r) · Aee (r)dr + Eee + Tc , (138) i

where the correlation-kinetic energy Tc is Z 1 ρ(r)r · Z tc (r)dr. (139) Tc = 2 R On applying drρ(r)r· to Eq. (133) one obtains the corresponding integral virial theorem for the S system: Z Eee + 2Tc + Mc =

where Mc =

Z

ρ(r)r · F eff (r)dr,

ρ(r)r · Mc (r)dr,

(140)

(141)

The “quantal Newtonian” first law is of course valid for both ground and excited states. Hence, the mapping via QDFT is applicable to ground and excited states. Furthermore, as in the B = 0 case [3, 4, 26, 27], the mapping to the S system is arbitrary in that the model fermions may be in a ground or excited state.

Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional ...

9.

243

Application of QDFT to the 2D Hooke’s Atom

We next apply QDFT to map the interacting system of the 2D Hooke’s atom [10] in a ground state to one of non-interacting fermions with the same {ρ(r), j(r)} also in its ground state. The 2D Hooke’s atom is comprised of two electrons in a harmonic external potential of frequency ω0 in which the electrons are confined to the x−y plane by a magnetic field B applied in the z-direction. With the vector potential chosen such that A = 21 B × r, the Coulomb gauge ∇ · A = 0 is satisfied. The Hamiltonian for this system (in a.u. with e = ~ = m = 1) is ˆ = H

2   X 1 i=1

1 ˆ i + A(ri) p 2 c

2

 1 2 2 1 + ω0 ri + . 2 |r1 − r2 |

(142)

There exist analytical solutions q to the corresponding Schr¨odinger equation for effective ω02 + ωL2 belonging to certain denumerably infinite set of oscillator frequencies ω ˜ = values, where ωL = B/2c is the Larmor frequency. We employ ω ˜ = 1. Thus, there exist an infinite number of physical systems (with different ω and ω 0 L that satisfy the condition q

ω ˜ = ω02 + ωL2 = 1 and with different j(r) that have the same ground state wave function. The model is an example of the many-to-one relationship of {v(r), A(r)} to the ground state Ψ. For ω ˜ = 1, the ground state wave function (nodeless solution) is 1

2

2

(143) Ψ(r1 r2 ) = C(1 + r12 ) e− 2 (r1 +r2 ), √ where r12 = |r1 − r2 | and C 2 = 1/π 2(3 + 2π). The ground state density is        1 2 2 1 2 2 2 −r 2 √ − 12 r 2 2 √ r + (2 + r ) , e πe ρ(r) = (1 + r )I0 r + r I1 2 2 π(3 + 2π) (144) where I0 (x) and I1 (x) are the zeroth- and first-order modified Bessel functions [28]. The density has cylindrical symmetry: ρ(r) = ρ(r). The density ρ(r) and the radial probability density rρ(r) are plotted in Fig 1. As expected for this harmonic external potential, the density does not exhibit a cusp at the nucleus. As the wave function is real, the paramagnetic current density jp (r) = 0. Thus, the physical current density 1 j(r) = ρ(r)A(r), (145) c and satisfies the continuity condition ∇ · j(r) = 0. For the mapping of the above interacting system in its ground state to an S system also in its ground state, the corresponding S system orbitals φa (x) are of the general form r ρ(r) iθ(r) e ; a = 1, 2 (146) φa (r) = 2 where θ(r) is an arbitrary real phase factor. The S system paramagnetic current density jp,s (r) is then jp,s (r) = −ρ(r)∇θ(r). (147)

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Figure 1. Electron density ρ(r) and radial probability density rρ(r).

Figure 2. The Hartree potential energy WH (r).

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Figure 3. The Pauli-Coulomb potential energy Wxc (r). The function −1/r − 3/2r 2 is also plotted. The phase factor θ(r) may be incorporated into a gauge transformation so that for the resulting S system the effective vector and scalar potentials are A0s (r) = As (r) + ∇θ(r) and p 0 vs (r) = vs (r) with the orbitals being φi (r) = ρ(r)/2. Thus, for different gauge functions, the corresponding S systems differ only by a gauge transformation. We emphasize, however, that the mapping from the ground state of the interacting system to the model system in its ground state is unique. The unique S system reproduces the density and physical current density of the interacting system. As the phase factor is arbitrary, for convenience we set θ(r) = 0, so that jp,s (r) = 0. The requirement that the S system produce the same physical current density j(r) then leads to the effective vector potential As (r) to be As (r) = A(r).

(148)

The S system differential equation is then 

 p p 1 2 1 2 2 ρ(r) =  ρ(r). pˆ + ω ˜ r + vee (r) 2 2

(149)

Note that as a consequence of Eq. (148), the contribution of the correlation-magnetic field Mc(r) to the potential energy vee (r) vanishes, as does the contribution of Aee (r) to the total energy E. Thus, it is only the electron-interaction E ee (r) and correlation-kinetic Z tc (r) fields that contribute to vee (r) and E.

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Figure 4. The Pauli potential energy Wx (r). The function −1/r − 5/4r 2 is also plotted.

Figure 5. The Coulomb potential energy Wc (r). The function −1/4r 2 is also plotted.

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Figure 6. The correlation-kinetic potential energy Wtc (r). The function 3/2r 2 is also plotted. For the structure of the various quantal sources and their corresponding fields, we refer the reader to [7]. Here we present the structure of the effective electron-interaction potential vee (r) and of its components. Due to cylindrical symmetry, the electron-interaction field E ee (r) is conservative. Hence, the contribution of Pauli and Coulomb correlations Wee (r) to the effective electroninteraction potential energy vee (r) is the work done in this field: Z r E ee (r0 ) · d`0 . (150) Wee (r) = − ∞

This work done is path-independent. The electron-interaction potential Wee (r) may be further subdivided into its Hartree WH (r), Pauli-Coulomb Wxc (r), Pauli Wx (r) and Coulomb Wc (r) components, each being the work done in the conservative fields E H (r), E xc (r), E x (r), and E c (r), respectively. The potentials WH (r), Wxc(r), Wx(r), Wc(r), and Wee (r) are plotted in Figs 2-5, 7. The asymptotic structure of the potentials is Wee (r) Wx (r)



r→∞



r→∞

1 1 2 5 1 3 + , WH (r) ∼ + 2 , Wxc (r) ∼ − − 2 r→∞ r r→∞ r r2 2r r 2r 1 5 1 − − 2 , Wc (r) ∼ − 2 . (151) r→∞ r 4r 4r

Once again, as a consequence of cylindrical symmetry, the correlation-kinetic field Z tc (r) is conservative, and therefore the contribution of this effect to the effective electron-

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Figure 7. The electron-interaction Wee (r), correlation-kinetic Wtc (r), and effective electron-interaction vee (r) potential energies. The function 1/r is also plotted. interaction potential energy vee (r) is the work done in this field: Z r Wtc (r) = − Z tc (r0 ) · d`0 .

(152)



This work done is also path-independent. The potential energy Wtc (r) is plotted in Figs. 6, 7. Its asymptotic structure is 3 . (153) Wtc (r) ∼ r→∞ 2r 2 It is evident from Eqs. (151) and (153) (see also Fig. 7) that Wtc (r) decays asymptotically much faster than the electron-interaction potential Wee (r). The effective electron-interaction potential vee (r) is then the sum of the electroninteraction Wee (r) and correlation-kinetic Wtc (r) potentials: vee (r) = Wee (r) + Wtc (r).

(154)

The potential vee (r) is plotted in Fig. 7. Its structure near the nucleus and in the classically forbidden region are vee (r) ∼ 1.50 − 0.99r 2 , (155) r→0

5 1 + 2. (156) r 2r Observe (see Figs 5 and 6), that the Coulomb Wc (r) and correlation-kinetic Wtc (r) components of vee (r) are of the same order of magnitude but opposite in sign. Hence, there vee (r) ∼

r→∞

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is a substantial cancellation of these effects in the potential vee (r). There is also a significant cancellation between the Hartree WH (r) and Pauli Wx (r) potentials (see Figs. 2 and 3). It is due to this cancellation that the asymptotic structure of vee (r) is 1/r (see Eq. 151), and is due to the residual Hartree potential. The Pauli and Coulomb correlations, and correlation-kinetic effects, all contribute to the term of O(1/r 2 ) of vee (r). The eigenvalue of the p S system differential equation Eq. (149) can be obtained directly from it since the solution ρ(r) is known. Or it may be determined by writing vee (r) with ω ˜ = 1 as √ 1 ∇2 ρ 1 2 (157) vee (r) =  + √ − r . 2 ρ 2 Since vee (r) vanishes at infinity, and ∇2 = ∂ 2 /∂r 2 + (1/r)∂/∂r, we obtain  = 2 a.u. The potential vee (r) may also be obtained directly from Eq. (157). What is achieved via QDFT is an understanding of the individual components of the potential. The results for the components of the total energy and other properties of the S system in its ground state are given in Table I.

10. Why the Paramagnetic Current Density Cannot Be a Basic Variable Finally, we address the arguments and proof of Vignale-Rasolt et al [11, 12, 13, 14] (VR) for the claim that the basic variables are {ρ(r), jp(r)}. The reasons why this conclusion is erroneous are the following: 1. At the outset, VR reject the physical current density j(r) as a basic variable. Their reasoning [13], which is based on the understanding of the Hohenberg-Kohn theorem prior to our work of [5], is as follows: According to the Hohenberg-Kohn theorem proved for the B(r) = 0 case, there is a unique one-to-one relationship between the ground state density ρ(r) and the ground state wave function Ψ(X). However, in the case of B(r) 6= 0, because the wave function is gauge variant, there can be no such one-to-one relationship between j(r) which is gauge invariant and the wave function Ψ(X). This argument is fallacious because as discussed in Sect. II, density preserving gauge transformations can also be applied to the Hamiltonian and wave function of Eq. (1) for the B(r) = 0 case. The uniqueness of the one-to-one relationship between the ground state density ρ(r) and the ground state wave function Ψ(X) is for each choice of gauge function α(R). If the VR argument were applied to this case, their corresponding statement would be that there can be no one-to-one relationship between the density ρ(r) which is gauge invariant and the wave function Ψ(X) which is gauge variant. As a consequence there would be no density functional theory. The reason for rejecting j(r) as a basic variable is inconsistent with quantum mechanics and the generalization of the Hohenberg-Kohn theorem to density preserving unitary transformations. 2. In their proof, VR ignore the fundamental physical difference between the B(r) = 0 and the B(r) 6= 0 cases, viz. that the relationship between {v(r), A(r)} and Ψ is not one-toone but many-to-one. They claim this difference is of no significance and ignore it. As such, they then attempt to prove a one-to-one (Map D) relationship between the ground state wave function Ψ(X) and the properties {ρ(r), jp(r)}. The fact that the properties {ρ(r), jp(r)}

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Table 1. Quantal density functional theory properties of the ground state S system that reproduces the density, physical current density, and total energy of the Hooke’s atom in a magnetic field in a ground state with effective oscillator frequency ω ˜=1 Property

Value (a.u.)

E

Total energy

3.000 000

Eee

Electron-interaction energy

0.818 401

EH

Hartree energy

1.789 832

Exc

Pauli-Coulomb energy

−0.971 431

Ex

Pauli energy

−0.894 916

Ec

Coulomb energy

−0.076 515

Eext

External energy

1.295 400

Ts

Kinetic energy

0.780 987

Tc

Correlation-Kinetic energy

0.105 212



Eigenvalue

2.000 000

< δ(r) >

∝ Fermi contact term

0.436 132

Size of atom

2.037 89

∝ diamagnetic susceptibility

2.590 8

∝ nuclear magnetic constant

2.996 87

2




do not conform to the definition of a basic variable within the rigorous Hohenberg-Kohn context as described previously is acknowledged by VR. According to Capelle and Vignale [14] the “CDFT potentials [{v(r), A(r)}] are not uniquely determined by the densities [{ρ(r), jp(r)}]”. 3. We next address the VR proof which is along the lines of [22] for spin density functional theory. As noted previously for the B(r) = 0 case, the fact that Ψ(X) 6= Ψ0 (X) (for v(r) 6= v 0 (r)+ constant) as arrived at via Map C, is explicitly employed in the proof of the one-to-one relationship (Map D) between Ψ(X) and ρ(r). In their proof of the corresponding Map D for the B(r) 6= 0 case, VR still assumed Ψ(X) 6= Ψ0 (X) to arrive at the contradiction E+E 0 < E+E 0 in spite of the fact that there is no Map C in this case. This fact has also been noted by Taut et al [15] who state that “Vignale and Rasolt presupposed the existence of the generalization of (Map) C”. Thus, there is no mathematical rigor to the VR assumption which is critical to their proof of the one-to-one relationship between Ψ(X) and the properties {ρ(r), jp(r)}.

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4. The properties {ρ(r), jp(r)} cannot determine uniquely all the properties of a system as required by the definition of basic variables. Because of the many-to-one relationship between {v(r), A(r)} and Ψ(X), knowledge of {ρ(r), jp(r)} cannot uniquely determine j(r). This is because j(r) = jp (r) + jd (r); jd(r) = ρ(r)A(r), and there are an infinite number of A(r) that lead to the same Ψ(X) and thus jp (r), but not the same j(r). The model of [10] constitutes an explicit counter example to the claim of VR. Thus, Ψ(X) cannot be a unique functional of {ρ(r), jp(r)}. 5. Basic variables are fundamental properties of the system that are gauge invariant. As the wave function is a functional of these basic variables, the construction of a density or current density functional theory is based on these gauge invariant properties. The VR theory, on the other hand, is based on the paramagnetic current density jp (r) which is gauge variant. Recognizing this, VR employed instead the vorticity ν(r) = ∇ × [jp(r)/ρ(r)] as the gauge invariant property in addition to the density ρ(r). This adds a layer of complexity [29] to the theory that is unnecessary when the correct basic variable j(r) is employed instead. Furthermore, for systems for which the ground state wave function is real, the vorticity vanishes since the paramagnetic current density jp (r) is zero. 6. In their ‘proof’ VR consider Map D between the ground state wave function Ψ(X) and the variables {ρ(r), jp(r)}. The choice of the properties {ρ(r), jp(r)} on the right hand side of Map D, which is based on erroneous reasoning as explained in point 1 above, is entirely arbitrary. The same Map D type ‘proof’, once again ignoring the many-to-one relationship between {v(r), A(r)} and Ψ(X), has been given by Diener [18] for the relationship between Ψ(X) and the properties {ρ(r), j(r)}. Thus, it is evident that conclusions based solely on Map D are inadequate to determine whether a property is a basic variable or not. For a property to be a basic variable, one must prove a one-to-one relationship between the property and the corresponding external potential. 7. In time-dependent current density functional theory, it has been proved [28] that the basic variables are {ρ(rt), j(rt)}. For a theory to be self-consistent, the time-independent version ought to be a special case of the time- dependent theory, as is the case in Schr¨odinger theory. The time-independent theory of VR is not a special case of the time-dependent version because in the absence of the temporal variable, j(rt) reduces to j(r) and not to jp (r).

Conclusion We have provided here a brief summary of Hohenberg-Kohn, Kohn-Sham, and Quantal density functional theories in the presence of a magnetostatic field B(r) = ∇ × A(r). These theories are all based on the understanding that the basic variables in the presence of such an external field are the ground state density ρ(r) and physical current density j(r). This understanding in turn is achieved by proving the relationship between the densities {ρ(r), j(r)} and the external potentials {v(r), A(r)} to be one-to-one. The proof thus lies within the rigorous context of the original Hohenberg-Kohn theorem. It is only as a consequence of this bijectivity that one can then state that the ground state wave function Ψ is a unique functional of {ρ(r), j(r)}. The wave function is also a functional of a gauge function to ensure that the wave function written as a functional is gauge variant. Again, it is the knowledge that the basic variables are {ρ(r), j(r)} that allows for the

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extension of the original proof from (v, A) - representable to N - representable densities via a constrained-search-type proof. The latter proof also eliminates the restriction from non-degenerate ground states to include degenerate states. Once again, it is the knowledge that the basic variables are {ρ(r), j(r)} which then allows for a mapping from the interacting system to one of non-interacting fermions with the same {ρ(r), j(r)}. We have demonstrated this by example via Quantal density functional theory. (The construction of such a model system with the choice of any other property as a basic variable such as the paramagnetic current density jp (r) is of little value. The reason for this is because the relationship between the external potentials {v(r), A(r)} and the ground state wave function Ψ is many-to-one, and hence, a property such as the physical current density j(r) cannot then be determined uniquely.) The extension of these theories to other Hamiltonians such as those of spin density functional theory, or that in which the interaction of the magnetic field with both the orbital and spin angular momentum is considered, or superconductivity density functional theory, all first require a proof of the bijective relationship between the basic variables and external potentials. For each of these systems, the relationship between the external potentials and the ground state wave function is many-to-one [14]. It is the bijectivity proof that will then define what constitutes a basic variable.

Acknowledgments The work of XP was supported by the National Natural Science Foundation of China (Grant No. 10805029), ZheJiang NSF (Grant No.R6090717) and the K.C. Wong Magna Foundation of Ningbo University. The work of VS was supported in part by the Research Foundation of CUNY.

References [1] P. Hohenberg, W. Kohn, Phys. Rev. 1964, 136, B 864. [2] W. Kohn, L. Sham, J. Phys. Rev. 1965, 140, A 1133. [3] V. Sahni, Quantal Density Functional Theory, Springer-Verlag: Berlin, Heidelberg, 2004. [4] V. Sahni, Quantal Density Functional Theory II; Approximation Methods and Applications, Springer-Verlag: Berlin, Heidelberg, 2010. [5] X.-Y. Pan,V. Sahni, Int. J. Quantum Chem. 2008, 108, 2756. [6] X.-Y. Pan,V. Sahni, Int. J. Quantum Chem. 2010, 110, 2833. [7] X.-Y. Pan,V. Sahni, Phys. Rev. A 2011, 83, 042518. [8] X.-Y. Pan,V. Sahni, J. Phys. Chem. Solids (accepted for publication). [9] J. Percus, Int. J. Quantum Chem. 1978, 13, 89; M. Levy, Proc. Natl. Acad. Sci. USA 1979, 76, 6062; E. Lieb, Int. J. Quantum Chem. 1983, 24, 243; M. Levy, Int. J. Quantum Chem. 2010, 110, 3140.

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[10] M. Taut, J. Phys. A: Math. Gen. 1994, 27, 1045; Corrigenda ibid 1994, 27, 4723; M. Taut, H. Eschrig, Z. Phys. Chem. 2010, 224, 999. [11] G. Vignale and M. Rasolt, Phys. Rev. Lett. 1987, 59, 2360. [12] G. Vignale and M. Rasolt, Phys. Rev. B 1988, 37, 10685. [13] G. Vignale, M. Rasolt and D.J.W. Geldart, Adv. Quantum Chem. 1990, 21, 235. [14] K. Capelle and G. Vignale, Phys. Rev. B 2002, 65, 113106; H. Eschrig and W.E. Pickett, Solid State Commun. 2001, 118, 123; K. Capelle and G. Vignale, Phys. Rev. Lett. 2001, 86, 5546. [15] M. Taut, P. Machon and H. Eschrig, Phys. Rev. A 2009, 80, 022517. [16] A.K. Rajagopal and J. Callaway, Phys. Rev. B 1973, 7, 1912. [17] S.K. Ghosh and A.K. Dhara, Phys. Rev. A 1989, 40, 6103; A.K. Dhara and S.K. Ghosh, ibid 1990, 41, 4653. [18] G. Diener, J. Phys.: Condens. Matter 1991, 3, 9417. [19] C.J. Grace and R.A. Harris, Phys. Rev. A 1994, 50, 3089. [20] F.R. Salsbury and R.A. Harris, J. Chem. Phys, 1997, 107, 7350. [21] H.A. Kramers, Quantum Mechanics, North Holland: Amsterdam 1957. [22] U. von Barth and L. Hedin, J. Phys. C 1972, 5, 1629. [23] M. Levy, Phys. Rev. A 1982, 26, 1200. [24] A. Holas and N.H. March, Phys. Rev. A 1997, 56, 4595. [25] S. Erhard and E.K.U. Gross, Phys. Rev. A 1996, 53, R5. [26] M. Slamet, R. Singh,L. Massa and V. Sahni, Phys. Rev. A 2003, 68, 042504. [27] V. Sahni and M. Slamet, Int. J. Quantum Chem. 2004, 100, 858. [28] Handbook of Mathematical Functions, edited by Abramowitz, M.; Stegun, I.A.; Dover: New York, NY 1972. [29] See comments in W. Zhu and S. Trickey, J. Chem. Phys. 2006, 125, 094317. [30] S.K. Ghosh and A.K. Dhara,Phys. Rev. A 1988, 38, 1149; G. Vignale, Phys. Rev. B 2004, 70, 201102 (R).

In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 255-280

Chapter 11

T HE C ONSTRUCTION OF K INETIC E NERGY F UNCTIONALS AND THE L INEAR R ESPONSE F UNCTION David Garc´ıa-Aldea∗ Department of Physics and Astronomy. University of California Los Angeles, Los Angeles, CA, US and Departamento de F´ısica Fundamental, UNED, Madrid, Spain J. E. Alvarellos† Departamento de F´ısica Fundamental, UNED, Madrid, Spain

Abstract In this chapter we present a brief review of some of the most sophisticated nonlocal kinetic energy density functionals formulated to date. The different families of functionals we will introduce share an important property, as all of them reproduce the linear response function of the free electron gas. These functionals have given promising results yielding not only small relative errors but also a relatively accurate local behavior and they are able to show quantum effects when applied variationally. We will show the general strategies behind their construction that make them suitable to be used in both extended and localized electron systems. For extended systems an approximate implementation of some of the functionals will allow them to present a computational cost that scales quasi-linearly with the system size.

PACS: 31.15.E-, 31.15.ec, 71.15.Mb, 71.10.Ca Keywords: Density functional theory, kinetic energy density functionals, Linear response theory ∗ †

E-mail address: [email protected] E-mail address: [email protected]

256

1.

David Garc´ıa-Aldea and J. E. Alvarellos

Introduction

After the foundations of modern Density Functional Theory (DFT) by the seminal paper of Hohenberg and Kohn [1], and its opening for the practical applications by Kohn and Sham (KS) [2], DFT has been widely used for accurate electronic structure calculations (first in solid state physics and later in quantum chemistry) because it needs a smaller computational effort when compared with the methods available before. Nowadays, DFT has a privileged position in the field of electron structure calculations, due to its advantages over other methods in computational efficiency and precision of the results [3, 4, 5, 6, 7]. But KS formulation of DFT has also built the theoretical background for making a clear partition of the total energy that can be used to evaluate the different pieces of this energy by proposing functional approximations for each of them. The KS method allows the DFT total energy density functional E[n] of a system of N electrons to be splitted as E[n(r)] = TS [n(r)] + V [n(r)] + J[n(r)] + Exc [n(r)],

(1)

where V [n] is the energy due to the interaction of the electronic cloud with the external potential and J[n] is the repulsion energy of the electronic cloud (i. e. the Hartree energy). The two other terms, Exc [n], the exchange-correlation energy, and TS [n], the non-interacting kinetic energy (KE), can be both expressed as functionals in terms of the electron density, n(r). For an N electron system, the KS orbitals, ϕi (r), of the KS method can be used to exactly calculate the non-interacting kinetic energy density functional (KEDF) as TS [{ϕi (r)}N i ]

1 =− 2

Z

dr

N X

ϕ∗i (r)∇2 ϕi (r)

(2)

i=1

(atomic units are used in this chapter), while the electron density is given by n(r) =

N X

ϕ∗i (r)ϕi(r).

(3)

i=1

On the other hand, DFT give us a beautiful example of how the progress in most of science disciplines is highly non linear in time. While the formal foundations of DFT are dated in 1964, several energy density functionals were formulated several decades before: (i) In 1927 the first functional for the kinetic energy was independently proposed in 1927 by Thomas [8] and Fermi [9] Z Z TT F [n] = dr tT F (r) = CT F dr n(r)5/3, (4) where tT F (r) is the Thomas-Fermi (TF) kinetic energy density (KED) and CT F = 3 2 2/3 is the TF constant. This KED only uses the local values of the density at 10 (3π ) each point r. By construction, the TF functional yields the correct kinetic energy of an homogeneous non-interacting system, i. e. the free electron gas. (ii) A second form for the kinetic energy was proposed by von Weizs¨acker [10] (vW) in 1935, Z |∇n(r)|2 1 dr . (5) TvW [n] = 8 n(r)

The Construction of Kinetic Energy Functionals ...

257

In this case, the values of the density around each point r are needed for the evaluation of ∇n(r). The vW functional is known to be exact for one or two electron systems in their ground state (i. e., for those systems that can be described by a single spatial orbital). (iii) In 1957 Kirznits [11] developed another functional for the kinetic energy, called the second-order gradient expansion approximation (GEA2). It is a correction of the TF functional, that includes a term proportional to the von Weizs¨acker functional 1 TGEA2 [n(r)] = TT F [n(r)] + TvW [n(r)]. 9

(6)

All these functionals are actually older than the HK formulation of the DFT,1 and they are extremely important not only for historical reasons, indeed most of the approximate energy functionals proposed in the literature are based on them. Besides the functionals already introduced, there are other KEDFs that could be classified into three different branches (see a general study of these functionals, stressing the importance of their local behavior, in [14]). (a) Linear combinations of the Thomas-Fermi and von Weizs¨acker functionals. It is known that the TF functional is exact for homogeneous systems while the vW functional is exact for systems with one spatial orbital. These functionals are then proposed as a mix of these two functionals trying to describe systems that do not belong to any of those opposite limits, so they lack a strong theoretical background and are essentially proposed as heuristic ones. They do not represent a clear improvement on the performance of the GEA2 approximation. (b) Gradient Expansion Approximations (GEA), that systematically correct the TF functional by adding terms in a gradient expansion around the density of the homogeneous system ν Z X TGEA−ν [n(r)] = dr t(i) (8) s [n(r)], i=0

(i)

being ts [n(r)] the local contributions of each of the different terms of the expansion (only even terms appear). The zeroth-order expansion is the Thomas-Fermi functional, and when the 2nd-order one (that equals 91 of the von Weizs¨aker functional) is added we get the GEA2. When going beyond, the fourth-order expansion developed by Hodges [15] involves the calculation of the Laplacian of the electron density; unfortunately, this functional do not improve the results of the GEA2. The sixth-order expansion formulated by Murphy [16] involves the bilaplacian of the electron density and a pathology is found: for atomic densities it is divergent both near the nuclei and at large distances (the same happens for corrections of higher order). So, even when evaluating the density functionals with accurate densities 1

Note that for the exchange energy, the functional presented by Dirac in 1930 [12], Z x ED = CD n(r)4/3 dr,

(7)

` ´1/3 where CD is the Dirac constant, CD = − 34 π3 , corresponds to the local approximation, depending only on the local density at r. It is constructed following the same spirit as the TF functional and therefore provides the correct exchange energy for the free electron gas. A useful and widely used generalization of the Dirac functional for the exchange energy was proposed by Slater (this X-α approximation is also local, see [13]).

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(i.e., those obtained with accurate orbital-based methods such as the Hartree-Fock (HF) [19, 20] or KS ones) the strategy of systematically correct the TF functional with the gradient expansion does not appear to be successful. Note that these functionals are constructed to reproduce the linear response function for density perturbations with a large wavelength; their relation with the static response function will be discussed later in Section 1.2 (c) Generalized Gradient Approximations (GGA). These functionals are formulated following a different strategy. They have the form of the TF functional, but in this case at each r the KED is evaluated as the local KED of the TF functional, tT F (r), weighted by an enhancement factor, fenh (s(r)). So, the general form for these functionals can be written as Z Z TGGA [n(r)] = dr tT F (r) fenh (s(r)) = CT F dr n5/3 (r) fenh(s(r)). (9) where the dimensionless variable s(r) = |∇n(r)| /n4/3 (r) takes into account the deviations from the homogeneous density. This quantity has a clear physical interpretation because it controls the speed of the variation of the electron density. Large values of s(r) correspond to fast variations in the electron density and small values to slow ones; a zero value indicates a region of the space where the electron density has no variation (in this case the TF functional is the correct one, so the enhancement factor is enforced to have a value of 1 when s = 0). Many functionals have been formulated using this approach (see [14] and references therein) and their enhancement factor have been usually adjusted in a semi-empirical way. They usually yield results close to the GEA2 ones when the functionals are evaluated with accurate densities, giving relative errors usually below 1%.

1.1.

Nonlocal Functionals

All the functionals presented above can be labeled as local or either semi-local functionals. The Thomas-Fermi and Dirac functionals are local ones, because they only use the electron density at a specific point of the real space to calculate the contribution of that point to the kinetic and exchange energy, respectively. On the other hand, the von Weizs¨acker, the GEA and the GGA functionals are all semi-local functionals because they use not only the value of the electron density at that point but also the density at points around it, through the information about the variation of the density (i. e. the density gradient, Laplacian, ...). So, in a general sense, we could call all of these kinds of functionals semi-local (SL) functionals, being the local ones the simplest of them. By contrast, for evaluating the contribution to the functional from a given point of the real space we can gather information from all the other points in the system. These more complicated functionals should calculate the energy contribution coming from a point r by using the density of the whole space at every point r0 . They are the so-called nonlocal (NL) functionals. This procedure would surely improve the quality of the functional, but the computational cost of evaluating this NL functionals is much higher than when using the SL ones. In fact, while the SL functionals have a computational cost that scales almost linearly with the number of electrons of the system, the NL functionals scale with a higher power of the system size. So a natural question arises: should we make the effort to formulate NL functionals paying the price of their computational cost? We think there are several important reasons for doing that.

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First of all, from a quantitative point of view, when using accurate densities the relative errors of the more sophisticated SL kinetic energy functionals are much lower than those obtained with the simpler ones. As an example, using accurate atomic electron densities the TF functional usually yields relative errors around 10%, and the von Weizs¨acker functional only gives small relative errors when applied to very small atoms, its errors grow enormously for bigger ones. On the other hand, GEA2 and GGA functionals usually yields errors of the order of 1%, far away from the chemical precision usually obtained with the orbital-based methods, but that can be considered as enough accurate for some of the purposes of ground state energy calculations that usually are performed using orbital-free methods. But from a variational point of view, we need to get not only enough accurate total energies for accurate densities but also obtain accurate enough density profiles for the ground state, i. e. after the total energy functional E[n(r)] in Eq. (1) is completely minimized. This is somewhat related with the local behavior of the KED of the functionals and we have found that the SL functionals do not give good results for that behavior [14] and cannot either accurately reproduce total energies and density profiles after minimization. This minimization (equivalent to a complete self-consistent procedure) is also decisive for the correct evaluation of the forces in molecular mechanics calculations or to find the stationary points on the energy surface of the system. For that reason we think it is important to make a general discussion on the ability of KE functionals to describe the relevant characteristics of the quantum systems. This second reason is even more important than the first one from a physical point of view. When an approximate KE functional is used to calculate a system by minimizing the total energy, and obtaining both the ground state energy and its electron density, a number of quantum effects should arise — as the Friedel oscillations in a jellium surface and the shell structure of atomic systems. But none of those SL functionals gives these quantum oscillations after such a minimization, yielding instead density profiles with no structure and energies that are not generally very accurate: minimization of Eq. (1) for atoms using SL functionals gives density profiles with only one maximum [17] — instead of the the several maxima that reflect the shell structure of the atom — and Friedel oscillations are poorly described when the jellium surface is solved [18]. This pathology of the SL functionals comes from their semi-local nature because the quantum effects can be tracked to have their origins in the orthogonalization of the different spatial orbitals, a process that is extremely nonlocal. Electronic structure calculations with methods like the HF approximation — that includes the exchange exactly but neglects the Coulombic correlation — or the Kohn-Sham method [2] — that includes exchange and correlation in an approximate way — use spin-orbitals to individually describe each electron of the system. These spin-orbitals are orthogonal to each other, with the important consequence that the orthogonalization procedure is computationally expensive.2 The kinetic energy of the system is then evaluated through Eq. (2 ). As the SL functionals use only information of the density coming from the neighborhood of each point, and as they are not able to reproduce the quantum effects, we could expect that these effects would be better reproduced if adequate nonlocal functionals are used, trying to mimic in some way 2

Numerical techniques that use some properties of localized basis set functions are able to reduce the scaling to ∼ N x where x has a value between 3 and 4 for HF and between 2 and 3 for KS.

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the strong nonlocality introduced by the orthogonalization process of the spin-orbitals. Looking at the mathematical structure of the SL functionals, FSL [n(r)], it is easy to see that they essentially have a single integral of an approximate KED given by a function fSL (r), i. e. Z FSL [n(r)] =

dr fSL (r) .

(10)

This function fSL (r) can be though as a complicated function of the electron density and its derivatives, as the universality requirement for the functional — as given by the Hohenberg and Kohn principle — does not prescribe any specific spatial dependence but only an implicit one through the electron density. As a first step to construct a nonlocal functional, going clearly beyond any semi-local approximation, we can consider that at least one of the terms of the functional have a double integration on r and r0 . Thus, the integrand f (r, r0) should be a general function of the electron density and its derivatives, evaluated at two different points r and r0 , Z Z  dr dr0 f r, r0 .

For the sake of simplicity, we can allow a spatial dependence only through the relative position of the two points, i. e. through the distance |r − r0 |. So, we can propose a simple nonlocal (SNL) term with two integrals Z Z   FSN L [n(r)] = dr dr0 fSN L n (r) , n r0 , r − r0 , (11)

depending on the whole electron density distribution through n(r) and n(r0 ). In this case, the order in which integrals are evaluated could be exchanged. But when a more general form of a nonlocal expression with two integrals is used, exchangeable integrations cannot usually be done. As an example, assume that the nonlocal term of a functional is written as Z FN L [n(r)] = dr fN L (r) , (12) being its KED fN L (r) a function that includes another integral over r0 . The integrand of this second integration should also include powers of the density or differential operators acting on the density, enforcing the evaluation of the functional to be necessarily performed firstly on dr0 and then on dr. Note that the first kind of functionals (the SNL ones) is a special case of the second one. It can be easily seen that these nonlocal forms have a computation cost that scales as ∼ N 2 , overcoming the best scaling than can be obtained with orbital-based methods.

1.2.

Lindhard Function and nonlocal Kinetic Energy Functionals

This chapter has the aim of introducing some aspects of the general methodology behind the construction of nonlocal KE functionals that reproduce the linear response (LR) function of the free electron gas. Its is well known (see [1]) that the first functional derivative of the total energy functional plays the role of a potential while its second functional derivative

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becomes a response function. For a general system, the exact LR function can have a very complicate structure. But in the case of the free electron gas there is no interaction between electrons and the only nonzero part of the total energy functional is the kinetic energy. Consequently, the second functional derivative of the total energy (that coincides with the exact KEDF), when taking the limit for the homogeneous system with density n0 , is )  2 −1  2 −1 ( 2 δ T [n(r)] π 1 π =− = FLind (η), (13) F kF δn(r1 )δn(r2 ) n0 kF χ ˜Lind (q)

where F represents the Fourier transform and in this case the LR function is exactly known in momentum space. It is given by the Lindhard function, FLind (η) [21], that only depends 1/3 on the scaled momentum η = q/2kF , being kF the Fermi momentum kF = 3π 2 n0 . The mathematical form of the Lindhard function −1  1 1 − η 2 1 + η + ln , (14) FLind (η) = 2 4η 1 − η

shows a weak logarithmic singularity at η = 1, a non-analyticity due to the appearance of (1 − η) in the denominator of the logarithm function. At η = 1 the slope of the function changes and the first derivative becomes divergent. This singularity is related to the discontinuity in the occupation number at the Fermi energy (with non zero values for momenta smaller than Fermi momentum, and zero otherwise; it then arises from the exclusion principle). Despite of this singularity, the function has a well defined value at that point, FLind (η = 1) = 2. The reproduction of this non-analyticity is considered to be related with the ability of a functional to yield quantum effects. The exclusion principle also makes FLind (η) to have a rather different behavior when η < 1 than when η > 1. Specifically, the series expansions of the Lindhard function for low and high values of η are 1 8 FLind (η) ' 1 + η 2 + η 4 + O(η)6 η→0 3 45 and FLind (η) ' 3η 2 − η→∞

24 −2 3 − η + O(η)−4 . 5 175

(15)

(16)

We can see that for small values of η the function has a value close to 1, growing with 13 η 2 , whereas for large values of η the function diverges quadratically as 3η 2. It is easy to demonstrate that the second functional derivative of any semi-local KEDF, when taking the homogeneous system limit, is not able to reproduce the Lindhard function. Firstly, the Thomas-Fermi and the GEA2 functionals yield the following responses in that limit (see Fig. 1) )  2 −1 ( 2 π δ TT F [n(r)] F (η) = 1, kF δn(r1 )δn(r2 ) n0 )  2 −1 ( 2 δ TGEA2 [n(r)] 1 π F (η) = 1 + η 2 . (17) kF δn(r1 )δn(r2 ) n0 3

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David Garc´ıa-Aldea and J. E. Alvarellos

1.0 -1 FLind HΗL

0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0 Η

1.5

2.0

Figure 1. The thick solid line corresponds to the inverse of the exact Lindhard response function. The response corresponding to different approximations of the KEDF through their second functional derivative are also depicted. The dashed line corresponds to the TF functional yielding a constant response. The dotted line is given by the second functional derivative of the GEA2 functional, and the dotted-dashed line corresponds to the vW functional (that reproduces the correct behavior for large moments).

In general, the KEDF obtained through the gradient expansion approximation up to nth order yields a LR function that corresponds to the series expansion for small η of the Lindhard function up to that order. On the other hand, it is important to remark that the von Weizs¨acker functional produces a response in the homogeneous limit given by )  2 −1 ( 2 δ TvW [n(r)] π F (η) = 3η 2, (18) kF δn(r1 )δn(r2 ) n0

that equals the quadratic divergence of the large η behavior of the Lindhard function in Eq. (16). Fig. 1 shows how the GEA2 functional describes the small η limit whereas the vW functional reproduces the large η divergence of FLind (η). Finally, the response reproduced by the GGA functionals depends on their specific form, namely )  2 −1 ( 2 δ TGGA [n(r)] π 00 F (η) = fenh (0) + 4 CT F fenh (0)η 2, (19) kF δn(r1 )δn(r2) n0

showing that, in the best case, they are able to reproduce the LR up to the GEA2 level, but no better. It is easy to extract a clear conclusion from this analysis. If we look for a KEDF able to reproduce the LR function of the free electron gas (and this is the main objective of this chapter) we must construct an explicitly fully nonlocal functional.

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263

Nonlocal KEDFs Based in the Thomas-Fermi Functional

Before presenting a general procedure for constructing nonlocal functionals in subsection 2.4, let us review some of the nonlocal KEDFs that have been presented in the literature assuming that the contribution to the kinetic energy from each point of the space depends on the density of the whole system.

2.1.

Chac´on-Alvarellos-Tarazona Functionals

The Chac´on-Alvarellos-Tarazona (CAT) functional [22] was formulated in 1985 with a mathematical form previously used in the description of the XC energy of an electron system [23] and to approximate the free energy of classical fluids [24, 25, 26]. This functional has a nonlocal term with the structure of the TF functional given in Eq. (4), using an averaged density n e(r) in the expression for the KED per particle (instead of the local density itself), i. e.: Z Z nl TCAT [n] = dr n(r) eT F (˜ n(r)) = CT F dr n(r) ˜n(r)2/3, (20) where eT F (r) is the KED per particle corresponding to the TF model. Density n ˜ (r) is 0 obtained through a kernel Ω (n(r), |r − r |) that averages the density over the whole system Z  n ˜ (r) = dr0 n(r0 ) Ω n(r), r − r0 (21)

So, for this functional Ω can also be thought as a weight function and the CAT functional is an averaged density approximation that yields the TF energy when n e (r) equals the local electron density. The kernel Ω is the cornerstone of the functional, being its shape fixed by the LR through the relationship given in Eq. (13) between the Lindhard function and the second functional derivative δ 2 T [n (r)] /δn(r1 )δn(r2 ) n . Moreover, the kernel is dimensionless 0 and must be normalized to get the right energy in the homogeneous limit: Z  dr0 Ω n0 , r − r0 = 1 ∀r. (22)

Taking the Fourier transform of the second functional derivative of Eq. (20), going to the homogeneous limit, and making it equal to the Lindhard response function, we should obtain an equation that allows us to fix the kernel. But the Lindhard function only depends on η, being scaled on the Fermi momentum. So, we can rewrite the kernel in the following way3   Ω n(r), r − r0 = [2kF (r)]3 ω 2kF (r) r − r0 (23)

where we have introduced this natural scaling in position space, i. e. the local Fermi momentum at r, kF (r). In this way the relationship given in Eq. (13) becomes a first-order 3

Note that the scaling properties of any kernel Ω appears then to be fixed by the scaling of FLind . In general, any functional that reproduces the LR of the electron gas must explicitly include that scaling or reproduce it when the homogeneous limit is taken.

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David Garc´ıa-Aldea and J. E. Alvarellos

differential equation in the momentum space,  1 2 0 6 ω(η) − [ω(η)]2 − ω (η)η = FLind (η). 5 5 5

(24)

We know that the Lindhard function diverges as 3η 2 when η → ∞ (this would induce a really strong divergence if the kernel given by Eq. (24 ) were written in the position space) but we have also previously shown that the LR of the vW functional equals this quadratic divergence. So, we can introduce in the nonlocal KEDF we are constructing an explicit von Weizs¨acker term for sweeping away that divergence in the differential equation that we must solve. For a similar reason, we also add an explicit TF functional, weighted by a prefactor α (see a more detailed discussion in the original reference [22]). Then the final expression for the full kinetic functional is written as a sum of a vW term plus a term proportional to the TF functional, and then the nonlocal term: nl TCAT [n(r)] = TvW [n(r)] − αTT F [n(r)] + (1 + α)TCAT [n(r)] .

(25)

Finally, the equation for the kernel includes the contributions of the added TF and vW functionals and becomes  FLind (η) − 3η 2 + α 6 1 2 0 ω(η) − [ω(η)]2 − ω (η)η = , 5 5 5 1+α

(26)

similar to Eq. (24) but with no divergences. This differential equation can be numerically integrated using a standard method like the Runge-Kutta one. Choosing a value of α = 3/5 we make the real space kernel to go asymptotically to zero, a property that is desirable to allow real space calculations for localized systems. Note the important point we have found here: the shape of ω does not depend on the specific system under study and the density at each point r of the system only appears in the real space kernel ω via the scaling factor 2kF (r) introduced in Eq. (23). Putting the focus on getting a better description of the nonlocal nature of the KEDF, the simple scaling appearing in Eq. (23) can be generalized. As a first step, one can improve this nonlocality by introducing a scaling factor in ω that depends not only on the density at r but also on the density at any other point r0 we use in the evaluation of the NL functional. This is the key idea of a symmetrized CAT functional [17], that defines the averaged density n e (r) in a more general sense Z   3  n e (r) = n r0 2 ζγ r, r0 ω 2ζγ r, r0 r − r0 dr0 , (27)

where the two-point scaling factor ζγ (r, r0) depends on the density both at r and at r0 through the local Fermi wave vector at each point, 1/γ  γ  kF (r) + kFγ (r0 ) 0 . (28) ζγ r, r = 2 Note that ζγ can be interpreted as a mean of the local Fermi wave vector at both points. The proposed symmetrized CAT functional is then [17] 3 8 nl TCAT [n(r)] = TvW [n(r)] − TT F [n(r)] + TCAT [n(r)] , 5 5

(29)

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265

nl with TCAT [n] given by Eq. (20) and the new weight function ω is again obtained by imposing that for the homogeneous system the functional reproduce both the energy (giving a normalization condition on ω) and the LR function (giving the shape of ω). In this case, we get a second-order differential equation

FLind (η) − 3η 2 + 3/5 = 1 + 3/5

2 6 1 1  0 1 ω(η) − [ω(η)]2 − ω (η)η + ω(η)ω 0(η)η 5 5 180 15   1 00 1 1+γ 2 + ω (η)η + − 4 ω 0 (η)η. (30) 30 10 3

Note that both CAT functionals we have presented here include a TF and a vW contributions in a natural way, as the pathologies due to the behavior of the Lindhard function (when η → 0 and η → ∞) do not affect the solution of the differential equations and, as a consequence, the shape of the kernel when written in the position space is well behaved (see a more detailed explanation in [22]). When atoms and jellium surfaces are solved by a complete variational minimization for these functionals4 the results show that the symmetrization procedure makes an important difference, clearly improving the description of the quantum effects of the systems (e. g., for both the atomic shell-structure and the quantum density oscillations of a jellium surface [17, 27, 18]; see also the discussion on the real space LR function in [28]).

2.2.

Wang-Govind-Carter Functionals

After the CAT functional was proposed, many authors developed similar kinetic functionals suitable for being implemented in extended systems. Those of Wang-Teter (WT) [29], Perrot (P) and Smargiassi-Madden (SM) [30] are written in the simpler form given by Eq. (11) and follows the same physical principles of the CAT model. Note that, as commented, all these functionals must have an intrinsic dependence on the Fermi wave vector through the argument η of the Lindhard function. In position space this dependency must appear as a scaling factor in the argument of the kernel. For the CAT functionals the two-body nonlocal scaling ζγ (r, r0) was chosen as the quantity to be used in all scaled magnitudes that appear in the formalism. On the other hand, the WT, P and SM functionals scale through a constant Fermi wave vector, making them suitable for calculations of extended systems where an uniform reference density can be properly defined. The generalizations of those functionals by Wang, Govind and Carter (WGC) also follows the same procedure. The latest version of these functionals [31, 32] is written as Z Z    nl TW GC [n] ∝ dr dr0 nα (r) ωα,β ζγ r, r0 , r − r0 nβ r0 (31)

where the function ωα,β also depends on a two-point scaling factor5 ζγ (r, r0) and is also fixed by the LR in the homogeneous limit through the relationship given by Eq. (13). 4

In the symmetrized functional, two different values γ = 1/2 and γ = −1/2 were discussed in the original formulation of the functional [17]. The value γ = 1/2, for which ζγ can be somewhat seen as an intermediate case between the arithmetical and the geometrical means, has been proved to give the better results. 5 So, the idea of a two-point scaling factor introduced in the symmetrized CAT functional [17] is also used for this double density-dependent scaling factor. The formulation of these WGC functionals also uses a two-point scaling function, i. e. they have a double density-dependent kernel ωα,β .

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David Garc´ıa-Aldea and J. E. Alvarellos The full functional is given by nl TW GC [n(r)] = TT F [n(r)] + TvW [n(r)] + TW GC [n(r)],

(32)

with a normalized kernel, Z

 dr ωα,β n0 , r − r0 = 0,

(33)

and the corresponding differential equation for obtaining it.

  γ + 1 3(α + β) 9 0 αβωα,β (η) + − ηωα,β (η) FLind (η) − 3η − 1 = 5 20 10 1 00 + η 2 ωα,β (η). 20 2

(34)

These functionals perform better when they verify6 α + β = 5/3. As we commented before, as a consequence of how they are constructed, these functionals have been applied to extended systems, where a reference density can be defined. Only the WT functional has been solved for localized systems, for a one dimensional infinite well [28] as well as for a rectangular box and for a single aluminium atom [33]. Note that we will not discuss here density-independent kernel functionals, i. e. those with a kernel scaled through a constant Fermi wave vector (see [32] for an example). These functionals are much easier to construct than the nonlocal functionals with densitydependent kernels and they are reviewed in Refs. [5] and [34]. Thanks to the use of local pseudopotentials, the WGC functionals have been widely applied to extended systems, taking into account not the full electron density but the valence electron density instead [35, 36]. Applications of these functionals to extended systems have used a technique developed in [32] to simplify its evaluation by making an expansion of the scaling factor around a constant reference density. Total energy calculations of metallic materials [31], covalent materials [37] and semiconductors [38] have been performed obtaining results in good agreement with KS calculations.

2.3.

Generalized Chac´on-Alvarellos-Tarazona Functionals

The CAT functionals can be generalized to give a full family of KEDFs with a nonlocal term that has the mathematical structure of the TF functional (i. e. the KED per particle is calculated with the TF one but using an averaged density) and designed to be used in both localized and extended systems [34]. As previously, we impose on our approximated functionals two main conditions: they will give the correct kinetic energy for the free electron gas and they will reproduce the LR in the homogeneous limit. As an added condition, this family of CAT functionals will also comply with another requirement: long-range effects in the position-space kernel are avoided, in order to make them applicable to localized systems. 6

The optimal values chosen by the authors [31, 32] were α, β =

5 6

±



5 6

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So, the nonlocal term of this generalized CAT family of functionals will also have the structure of the TF functional, but using an averaged power of the density n eβ (r) in the expression for the KED (instead of the density itself), i. e.:7 Z nl TCAT [n] = CT F dr n(r) [e nβ (r)]2/3β . (35) Here n eβ (r) is an average of a power β of the electron density, and the nonlocality of the nl [n] through a kernel Ω that makes the averkinetic energy is supposedly included in TCAT aging, Z  n eβ (r) = dr0 nβ (r0) Ω ζγ (r, r0), r − r0 . (36)

In order to get a better description of the nonlocality, the scaling in Ω is introduced through a two-point wave vector ζγ (r, r0), as given in Eq. (28). Following the discussion in Ref. [22], this generalized CAT functionals [34] also includes the TF and the vW contributions, becoming nl TCAT [n] = TvW [n] − αTT F [n] + (1 + α) TCAT [n].

(37)

Note that now we use three parameters8 that need to be fixed in the formulation of the functional. But, in order to the functional be useful in localized systems, the kernel in momentum space must go asymptotically to 0 when q → ∞. On the other hand, the kernel must have a value of 1 when q → 0 as a consequence of the kernel normalization to get the correct energy for an homogeneous system. As a result, we get an explicit relation between α and β, namely −1  8 −1 , (38) α= 3β where the values of the Lindhard function, FLind (0) = 1 and FLind (∞) = −3/5, have been used. So, these CAT functionals depend on only two parameters: β, that appears as a power of the density, and γ, that gives the way the scaling factor ζγ (r, r0) is evaluated. For a fixed γ, each value of β specify the corresponding CAT kinetic functional. Being β the power of the electron density to be averaged in Eq. (36), we fix β and then get the corresponding value of α. As an example, for β = 1 we get the original CAT functional, a functional that includes the term TvW [n] − 35 TT F [n] that come from the correct expansion of the Lindhard function for large values of the scaled momentum η (see [22]). On the other hand, the value α = 1 (and its corresponding value β = 4/3) gives a full negative ThomasFermi contribution. Finally, for β = 2/3 we have the special case where the averaged density in Eq. (35) is powered to one and we get a simpler structure for the KEDFs, that will be discussed in the following subsection 2.4 When equating the LR and the second functional derivative of the functional, we get a complicated second-order differential equation in momentum space to fix the shape of the 7

Note the power of n eβ (r) we need to use in the KDE per particle in order to preserve the correct dimensional scaling of the KEDF, following the definition of n eβ in Eq. (36). 8 I. e. parameters α, β and the power γ in the scaling function ζγ given by Eq. (28), that makes the kernel double density-dependent through the two-point Fermi wave vector.

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David Garc´ıa-Aldea and J. E. Alvarellos Table 1. Values for the parameter β (and their corresponding values of α) [34] α β

1/7 1/3

1/3 2/3

3/5 1

1 4/3

5/3 5/3

kernel FLind (η) − 3η 2 + α 1+α

   2 6 1 1 1 2 2 = ω(η) + (2 − 3β) [ω(η)] + − 3 ω 0 (η)η 5 5 180 β β   1 2 3 1 00 − − 1 ω(η)ω 0(η)η + 6(β − 1) + ω (η)η 2 5 3β 5 30β   1 2 1+γ + − −2+ ω 0 (η)η. (39) 10 β 3β

Note that if we choose the value β = 2/3 three terms in the differential equation disappear. After integration of the former differential equation, we find that the shape of the kernel in momentum space depends strongly on the value of β but is almost independent of the value of γ. The values chosen in Ref. [34] for the parameter β and their corresponding values of α are listed in Table 1. As commented previously, two different choices for the parameter γ that gives how the scaling factor ζγ (r, r0) is evaluated have been used as with the original symmetrized functional (γ = 1/2 and γ = −1/2) [17, 27, 18, 28]). When using atomic densities (from He to Xe, calculated with the G AUSSIAN package [39], or from He to Ar with approximate orthogonalized Slater orbitals with exponents given by [40, 41]), the smaller relative errors of the total kinetic energies for these generalized CAT functionals are obtained for β = 2/3 and γ = −1/2, with similar errors (average values between 0.8% and 1.8%) to those obtained by usual semi-local functionals [42, 43, 14].

2.4.

Simple Nonlocal Thomas-Fermi Functionals

After the review of some of the NL functionals we focus now in presenting a general procedure for constructing SNL KEDFs that reproduce the LR of the free electron gas. The free electron gas is the simplest extended many electron system, and its energy is exactly given by the TF functional. So, the structure of this TF functional can then be used as a first proposal for constructing NL kinetic functionals able to reproduce the LR of the free electron gas. With the help of a Dirac δ function the TF functional can be written in two different ways, which appears to be practical to show how to construct a nonlocal one from it. Specifically, we introduce together a second integration in r0 and a Dirac δ function, and

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the TF functional can be written as9 Z Z  TT F [n] = CT F dr dr0 n5/3−β (r) δ r − r0 nβ (r0 ). TT F [n] = CT F

Z

dr n(r)

Z

0

β

0

0

dr n (r ) δ r − r



2/(3β)

(40) .

(41)

In the first expression the integrals could be evaluated in any order, but that cannot be done in the the second one because the integration over dr0 is affected by the 2/(3β) power. Note that in Eqs. (40) and (41) the Dirac δ function can be thought as a connection between the contributions from r and r0 . The general procedure we propose for constructing a fully nonlocal term for a KEDF is to substitute the Dirac δ (r − r0 ) function by a kernel that couples contributions from the two points r and r0 . We will then choose as a kernel a function with a general dependence on the electron density at each of these points and on the distance |r − r0 | between them. The WGC functionals have a mathematical structure similar to Eq. (40), following the simpler SNL scheme of Eq. (11). On the other hand, the generalized CAT functionals follow the spirit of Eq. (12) and have a mathematical structure that resembles Eq. (41). Besides, the kernel of the generalized CAT functionals have been constructed to allow the functionals to be applied to localized systems in real space, whereas the kernel of WGC functionals make them inapplicable to those systems. It would be interesting to generate KEDFs able to keep the main benefits of both types of functionals: the possibility of a simpler evaluation — like the WGC functionals — and a kernel suitable to work in position space for both localized and extended systems — like the CAT ones. To develop such a family of functionals with the mathematical structure of the TF functional, that we will call simple nonlocal TF-based (STF) [34], we are going to impose the two conditions we have always used to construct the previous NL functionals and the former requirement for the generalized CAT functionals, i. e. to avoid the long-range effects in the position-space kernel in order to make the functionals applicable to localized systems. So, we write the functional TST F [n] in the same way as we did in Eq. (37) nl TST F [n] = TvW [n] − αTT F [n] + (1 + α) TST F [n], nl but now the nonlocal term, TST F [n], is a double convolution of two different powers of the 0 density at points r and r and coupled by a kernel Ω (ζγ (r, r0), |r − r0 |) that is scaled with the two-point Fermi wave vector ζγ (r, r0), i. e. Z Z  nl TST F [n] = dr dr0 n5/3−β (r) Ω ζγ (r, r0) r − r0 nβ (r0 ). (42)

We now have a nonlocal term containing only powers of the real space densities at r and r0 ; the nonlocality is introduced by the convolution of the scaled kernel with the powered densities, instead of being generated by an averaged density. This nonlocal term of the KEDF is similar to the WGC one but in this case the kernel is normalized to one and, as The total power of the electron density should be equal to 53 , in order to have the correct dimensionality of a KEDF and the scaling of the TF functional. 9

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Table 2. Values for the parameter β, and their corresponding values of α. The generalized CAT functionals coincides with this simpler one when β = 2/3 (see [34]) α β

1 1/3

1/3 2/3

5 6

3/5√ ± 65

7/25 5/6

commented, can be directly constructed from the TF functional. In fact, changing the Dirac δ function in Eq. (40) by the kernel Ω (ζγ (r, r0) |r − r0 |) we get Eq. (42). So, the kernel can be then thought as a link between the contributions that come from r and r0 , These STF nonlocal functionals keep the essential properties of the CAT functionals, as the kernel Ω is fixed by enforcing the functional to reproduce the LR of the free electron gas: 1 FLind (η) − 3η 2 + α 9 = β (5/3 − β) ω(η) + ω 00 (η)η 2 1+α 5   20 9 1 γ+1 − 1 ω 0 (η)η + β 2 − 3β + 1. + 2 10 5

(43)

This is also a second-order differential equation but much simpler than the one obtained for the CAT family of functionals. For avoiding the long-range effects in the position-space kernel we must impose a relation between α and β 8/9 , (44) α = −1 − β (β − 5/3) and the values of the pairs α and β are presented in Table 2. Applying this relationship between α and β we finally obtain the differential equation FLind (η) − 3η 2 =

1 2 (9 − γ) 1 (8ω(η) − 3) + ω 0 (η)η + ω 00 (η)η 2. (45) 5 45 (β − 5/3) 45 (β − 5/3)

So, although the SFT functionals depend both on β (that appears as a power of the densities) and γ (that gives the way the scaling factor ζγ (r, r0) is evaluated), for a fixed value of γ only one parameter is needed to specify each of them. If we compare the total atomic kinetic energies obtained with the STF functional to the exact solutions of the KS method,10 we get very good results for all these functionals (the CAT and STF families) comparable to the best results of any GGA semi-local functional [14]. The results for the STF family of nonlocal functionals are better when γ = −1/2 and β equals 5/6 or 2/3 (as commented, for the later value, the STF functional coincides with the original CAT with the same value of β). 10

As explained before, we have calculated closed-shell atoms and used the accurate atomic densities obtained with the G AUSSIAN package [39] (from He to Xe), or with approximate orthogonalized Slater orbitals with exponents given by [40, 41] (from He to Ar).

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271

A Brief Summary

As a summary of this section, we must stress here that if we choose a given mathematical structure for the nonlocal term of the KEDF we want to construct, the details of the functional will be mainly fixed through the Lindhard function when the LR and the second functional derivative of the functional are equated (in a general sense, this will yield the shape of the kernel). As added rules when trying to make a choice between several nonlocal KEDFs, we must focus on the following points: the computational convenience of the mathematical structure (i. e., if the computational cost of the evaluation of the functional can be made much smaller by writing the nonlocal term in a simpler form for some systems, e. g. for extended ones); how the functionals reproduce the correct results for some given benchmark system (e. g., a specific atom or set of atoms); how to avoid long-range effects in the position-space kernel in order to make the functionals applicable to both extended and localized systems in real space, etc.

3.

Nonlocal KEDFs Based in the Von Weizs¨acker Functional

Being the energy of the free electron gas energy exactly given by the TF functional, and being the vW functional exact in the opposite limit, i. e. for systems that can be described by a single spatial orbital, one may wonder if the structure of the vW functional can also be used for building NL terms for KEDFs in the way we have presented before. This question is not only an academic one. We are interested in proposing and studying the quality of the KEDFs beyond the free electron limit we have used so far. The development of nonlocal functionals based on both the homogeneous electron gas — through the Lindhard LR — and a mathematical form able to give correctly the single spatial orbital limit — using it for the structure of the nonlocal term — could provide a forward-looking approach and open the way for KEDFs that can be used in more general applications. So, for these nonlocal KEDFs based in the vW functional we will also impose the three conditions we have used to construct the previous NL functionals: (i) they must give the correct kinetic energy for the homogeneous system and (ii) the exact LR of the free electron gas; (iii) they must be constructed avoiding the long-range effects in the position space in order to make them applicable to both extended and localized systems. As an additional feature, we will use the SNL functional structure, in the same manner as in the STF functionals. Following that nomenclature we will call this family of functionals simple nonlocal vW-based (SvW) functionals. The von Weizs¨acker functional is usually found written in the form given by Eq. (5). This expression is not friendly for constructing nonlocal functionals. We can then rewrite it in other different, but equivalent, mathematical forms in order to propose nonlocal terms for the KEDFs using a a similar procedure to that we did for the TF-based functionals. It is straightforward to obtain one of these new expressions just by using the square root of the electron density, TvW [n] =

1 8

Z

|∇n(r)|2 1 dr = n(r) 2

Z

2 dr ∇n1/2 (r) .

(46)

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On the other side, it is well known that the Laplacian of the electron density integrates to zero in the whole space. Moreover, the Laplacian has the appropriate scaling of a KEDF. So, it is possible to add a term proportional to the Laplacian to the integrand of Eq. (5) to get another expression for the vW functional. After some trivial operations the vW functional becomes Z 1 TvW [n] = − dr n1/2 (r) ∇2n1/2 (r). (47) 2

Now, if we introduce a second integration over r0 together with a Dirac delta function, we have two new mathematical expressions suitable to be used to build nonlocal terms for KEDFs based on the vW functional: Z Z  1 dr dr0 ∇n1/2 (r) δ r − r0 ∇n1/2 (r0 ), (48) TvW [n] = 2

and

1 TvW [n] = − 2

Z

dr

Z

 dr0 n1/2 (r) δ r − r0 ∇2 n1/2 (r0 ).

(49)

Finally, it is possible to rewrite the usual expression for the vW functional with the help of the natural logarithm function, [44] Z Z ∇n(r)∇n(r) 1 1 dr = dr ∇n(r) ∇ ln n(r) TvW −ln [n] = 8 n(r) 8 Z Z  1 dr dr0 ∇n(r) δ r − r0 ∇ ln n(r0 ). (50) = 8 All these three expressions given by Eqs. (46), (47) and (50) just differ locally and are related to the different ways the Fisher information function [45, 46] can be written.

3.1.

Type I (SvW-I)

Using the first previous expression for the von Weis¨acker functional, and substituting the Dirac delta function in Eq. (48) by a kernel, we have the first SvW nonlocal term Z Z  1 nl TSvW −I [n(r)] = dr dr0 ∇n1/2 (r) Ω ζγ (r, r0), r − r0 ∇n1/2 (r0 ), (51) 2

where the two-point scaling factor ζγ (r, r0) is used again [47]. Taking the second functional derivative of this expression, and making it equal to the Lindhard response function, we obtain a direct relationship between FLind (η) and the kernel ω(η) =

FLind (η) . 3η 2

(52)

Examining this equation, we see two important pathologies. Firstly, the kernel has a strong divergence for small values of η. Secondly, it does not go asymptotically to zero as η tends to infinity. We must solve both before proposing a new KEDF. Keeping the nonlocal vW-based term of Eq. (51), if we introduce an explicit TF term in the KEDF the divergence in Eq. (52) when η → 0 will not appear in the equation for the new kernel. On the other

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273

2.0 1.5 ΩHΗL

1.0 0.5 0.0 -0.5 -1.0 0.0

0.5

1.0

1.5 Η

2.0

2.5

3.0

Figure 2. Kernels for the nonlocal SvW functionals as a function of the scaled momentum η. The dashed line corresponds to the kernel obtained without including any explicit semilocal functional in the formulation (i.e., that given by Eq. (52)). The dotted line is the kernel if the TF functional is included while the dashed-dotted line corresponds to the vW functional explicitly included. The solid line is the actual kernel given by Eq. (54), that includes explicitly both TF and vW functionals.

hand, if we also include a vW term in the KEDF the kernel will tend asymptotically to zero. In this way, the SvW functional (type I) becomes [47] nl TSvW −I [n(r)] = TvW [n(r)] + TT F [n(r)] + TSvW −I [n(r)],

(53)

and the corresponding expression of the kernel, ω(η) =

FLind (η) − 3η 2 − 1 , 3η 2

(54)

has no pathologies. Note that this kernel can be directly obtained from a simple (algebraic) expression, contrasting with those of the previous nonlocal functionals, where differential equations were needed. On the other side, as shown in Fig. 2, the kernel is negative for all the values of η, with a value of −8/9 for η = 0. It is also worth to remark that this value for η → 0 makes the response function of TSvW −I [n] to coincide with the GEA2 functional LR (the exact result for low moments).

3.2.

Type II (SvW-II)

On the other hand, if we use the second form of the von Weizs¨acker functional, Eq. (49), and substitute the Dirac delta function by a kernel, we obtain another nonlocal term (type II) for a KEDF [47] Z Z  1 nl dr dr0 n1/2 (r) Ω ζγ (r, r0), r − r0 ∇2 n1/2 (r0 ). (55) TSvW −II [n(r)] = − 2

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Note that this functional has a complete different symmetry than the type I one. Now we evaluate the square root of the electron density at r, whereas the Laplacian of this square root is calculated at r0 . Following the discussion of the previous SvW-I case, it is very convenient to explicitly include both the full vW and the TF functionals in the new proposal. So, we write nl TvW −II [n(r)] = TvW [n(r)] + TT F [n(r)] + TvW −II [n(r)],

(56)

being the type II kernel related to the Lindhard function through a first-order differential equation FLind (η) − 3η 2 − 1 1 . (57) ω(η) − ηω 0 (η) = 3 3η 2 In this case, we again obtain a simpler equation than those previously obtained for the TFbased nonlocal functionals. The kernel ω(η) of the type-II functional has a softer structure than the type-I one — see [47]. In both cases the kernels are negative definite functions and grow monotonically (they have an always positive derivative). Moreover, they verify ωvW −II (η) ≥ ωvW −I (η), being the equality only satisfied for η = 0 and when η −→ ∞.

3.3.

Type ln (SvW-ln)

Following the procedure used for the two previous cases, from Eq. (50) we can propose another nonlocal term that can be used to construct a new SvW functional: Z Z  1 nl dr dr0 ∇n(r) Ω ζγ (r, r0), r − r0 ∇ ln n(r0 ). (58) TSvW −ln [n(r)] = 8

nl This functional form TSvW −ln [n(r)] should have a clear computational advantage, because its evaluation should be easier than for the previous cases, as the gradient of the function ln n(r0 ) is a much smoother function than the gradient or the Laplacian of the electron density (see [44]). Using the same arguments we have presented before for adding to the nonlocal term the complete TF and vW ones, the proposed new KDEF becomes [44] nl TSvW −ln [n(r)] = TvW [n(r)] + TT F [n(r)] + TSvW −ln [n(r)].

(59)

When the second functional derivative is calculated, the relationship between FLind (η) and the kernel is the same algebraic expression previously obtained for the type I functional. It is worth to note here that the kernel of the three SvW nonlocal functionals does not depend on the parameter γ, that determines how the two-point scale factor is calculated. So, we evaluate only one kernel for each of these functionals, instead of calculating the specific kernel for every different value of γ, as the STF nonlocal functionals needed.

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The relative small importance of the parameter γ for these functionals arises when the results obtained with these functionals for closed-shell atoms from He to Ar11 are analyzed: errors in the total kinetic energy are between 0.4% and 0.6% and vary slightly with γ. For the SvW-I functional the best results are found for γ ≈ 0.3, where the mean of relative errors is 0.4%, smaller than any functional of the CAT and STF families [34]. For the SvW-II functional we get also small relative errors, about 0.5%, for γ ≈ −0.5. The results for the SvW-ln functional are very insensitive to the variation of γ, and they are also about 0.5% in this case for γ ≈ 2. Recently, a generalization for the von Weizscker based non local KEDFs has been presented [58]. As already done with the STF, an additional parameter β is introduced to use different powers of the electron density within the SvW functional structure. For some specific values of β (previous cases correspond to β = 1/2), these new families of functionals naturally include logarithms of the electron density. The relative errors obtained with the generalized SvW functionals average to 0.4 − 0.5%, within the best results we have found in our studies.

4.

Evaluation of Functionals as a Single Integral in the Reciprocal Space

There are systems where a constant reference density can be properly defined (e. g. extended systems with a density that shows only relatively small deviations from the mean density). In these cases, this reference density can be used to have a constant scaling within the kernel, through an effective constant Fermi wave vector, instead of using the complicated scaling of the two-point Fermi wave vector ζγ (r, r0). In those systems it is possible to evaluate some of the nonlocal terms of the KDEFs presented before making only one integration in the momentum space (note that this can be done for the STF and SvW functionals but not for the various CAT functional families). The main advantage of a single integral calculation is that the computational cost of the evaluation is almost linear because the most expensive step in the calculations is then the Fourier transform, that scales in a quasilinear form, N ln N , using standard Fast Fourier Transform techniques For the STF nonlocal functionals the calculation in momentum space becomes12 Z nl TST F [n(r)] = CT F dq nβ (q) n5/3−β (q) ω(η), (60) where  nβ (q) is the Fourier transform of the β power of the electron density, i. e. nβ (q) = F nβ (r) . On the other hand, the SvW nonlocal functionals can also be evaluated in momentum space. For the generalized type I and type II SvW functionals [58] (remember that the 11

Densities obtained with both the G AUSSIAN package[39] and approximate orthogonalized Slater orbitals with exponents given by [40, 41] were used in the computations. 12 Expressions in this Section could include an additional prefactor depending of the convention used for the Fourier transform definition. We get the prefactor (2π)2 as a consequence of the definition of the Fourier transform we are using, e2πiq·r , and the presence of differential operators in the position space expression of the energy density functionals.

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previous cases correspond to β = 1/2) we get nl TSvW −(I/II) [n(r)]

(2π)2 = 8β(1 − β)

Z

dq q 2 nβ (q) ω(η) n1−β(q).

(61)

For the the case of the third SvW functional, it is also possible to perform the evaluation in momentum space through Z (2π)2 nl TSvW −ln [n(r)] = dq q 2 n1 (q) nln(q) ω(η), 8 where nln (q) is the Fourier transform of the logarithm of the density, nln (q) = F {ln n(r)}. Note that even the expressions for the evaluation of the nonlocal terms — Eq. (60) for STF functionals and Eq. (61) for SvW ones — coincide for several functionals the corresponding values of T nl [n] actually differ because they have different kernels. Finally, we must comment that a technique developed by Wang, Govind and Carter [32] allows to also get quasilinear computational cost in the evaluation of their nonlocal functionals by making an expansion of the two-point Fermi wave vector scaling factor around the reference density. This or some other techniques — see [48] or the recent procedure given by Huang and Carter [38] — allow to achieve very low computational cost comparing with other methods, making it possible to calculate systems of thousands of atoms in a reasonable computing time [49, 50].

Conclusion In this chapter, we have showed the general ideas behind the construction of nonlocal KEDFs that reproduce the linear response function of the free electron gas. These functionals are currently some of the most sophisticated formulated to date and the most successful ones when the focus is pointed at the quantum properties of many electron systems. We think that means in some way that they are able to mimic the orthogonalization procedure of the individual spin-orbitals. We have found that the KEDs of these KEDFs are the only ones that give better local behavior than the GEA2, yielding at the same time smaller relative errors for the total kinetic energies. As a general conclusion, the evaluation of the total kinetic energies of atoms yields the best results of the relative errors for the SvW functionals, averaging in between 0.4 − 0.6%. Larger errors (1.2 − 2.3%) are found for the CAT and STF functionals, the TF functional gives an average error of 8.7% and the semi-local GEA2 functional yields a mean relative error of 1.1%, whereas the best NL functionals give values for the mean of the relative errors smaller than 0.5% [47, 34]. But we must add a word of caution after studying the local behavior of the KED of all these functionals. We have shown that the nonlocal functionals give in general good results for the total kinetic energies in atoms. In order to test the local performance of the KEDFs, we have also done comparative studies of the KED for a number of GGA semi-local functionals [14], for the CAT and STF functional families [34] and for the SvW families [47], based on the values of a quality factor σ that measures the differences between the KEDs. The optimized values for this quality factor, i. e. the minimized values of σ show

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Table 3. For all atoms from He to Ar, averaged optimized values for the quality factor σ and mean absolute relative error (MARE) of NL functionals (for the best γ in each case), when compared to the exact kinetic energy evaluated with the Slater orbitals He - Ar CAT (β = 1) CAT (β = 2/3) ≡ SNL-TF NLS-vW-I NLS-vW-II NLS-vW-ln TF GEA2

γ −0.5 −0.5 0.25 −0.5 2 − −

MARE (He - Ar) 0.023 0.012 0.004 0.005 0.005 0.083 0.007

σ 0.160 0.141 0.160 0.157 0.167 0.159 0.181

(see Table 3) that the SvW functionals yield smaller values of σ than the GGA semi-local functionals, and close to the values of the STF functionals. So, we have found that fully NL functionals give in general better KEDs than the GEA2 one and clearly better than the GGA semi-local functionals.13 We think all those results can be considered as a positive outcome of the inclusion of fully nonlocal terms in the KEDFs.14 But even we get in general smaller values of σ than with the TF functional, we cannot conclusively say that these NL functionals clearly improves the behavior of the KED of the TF functional. In any case, what is clear is that only the nonlocal functionals are able to give better local behavior than the GEA2 and GGAs functionals, giving at the same time low relative errors for the total KE. On the other hand, we have also reviewed [56, 57]. the first-order gradient correction to the ThomasFermi functional proposed by Haq, Chattaraj and Deb [52, 53, 54, 55], studying two different versions of this approximation [56]. One of these first-order correction functionals gives not only excellent values for the total kinetic energies but also one of the best KED behavior in the literature. We feel this is a quite surprising result, showing the difficult challenge that the study of the KEDFs must face up to: unfortunately this functional is not reliable for general calculations because it is not an universal one, due to its position dependent terms. Even with the relative success of the orbital-free nonlocal KEDFs we have presented in this chapter, the results have been shown to be duly consistent for a number of very different systems (from atoms to jellium surfaces). So, we think this kind of functionals, based on the linear response function of the free electron gas, could be specially useful for future developments of new KEDFs. In our opinion, to reduce the relative errors is not the most important problem to be solved. We think that the improvement of the quality of the local behavior of the KEDs could pave the way for understanding when and why functionals are able or not to describe the characteristic quantum properties of a specific system, as the general performance of 13

We present only the result for the GEA2 functional because we have previously shown in [14] that the local behavior of any GGA KED becomes worse than that corresponding to the Thomas-Fermi functional. 14 We must note that a Laplacian-level meta-GGA for the KED reports better KEDs than those presented here, but giving bigger relative errors for the kinetic energies [51].

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the local kinetic energy density is closely related with the KS effective potential. For that reason, we think that new mathematical forms should be tried for the KEDFs, imposing on them some some other physical constraints that could yield better local behavior for the kinetic energy densities. On the other hand, we feel the ideas we have presented in this chapter should also be applied to the formulation of new exchange and correlation energy density functionals that reproduce the approximate response functions corresponding to these pieces of the total energy.

Acknowledgments We acknowledge the interest in this work of Dr. Pablo Garc´ıa-Gonz´alez. This work has been partially supported by a grant of the Ministerio de Ciencia e Innovaci´on of Spain (reference FIS2010-21282-C02-02). DGA gratefully acknowledges financial support of the Spanish MEC through its grants program.

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[40] P. W. Atkins, Molecular Quantum Mechanics, Oxford University Press, 3rd. ed., Oxford, 1997. [41] E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686 (1963). [42] A. J. Thakkar, Phys. Rev. A 46, 6920 (1992). [43] D. J. Lacks and R. G. Gordon, J. Chem. Phys. 100, 4446 (1994). [44] D. Garc´ıa-Aldea and J. E. Alvarellos, J. Chem. Phys. 129, 074103 (2008). [45] S. Liu, J. Chem. Phys. 126, 191107 (2007). [46] S. B. Sears, R. G. Parr, and V. Dinur, Isr. J. Chem. 19, 165 (1980). [47] D. Garc´ıa-Aldea and J. E. Alvarellos, Phys. Rev. A 77, 022502 (2008). [48] G. S. Ho, V. L. Ligneres, and E. A. Carter, Phys. Rev. B 78, 045105 (2008). [49] L. Hung and E. A. Carter, Chemical Physics Letters 475, 163 (2009). [50] L. Hung and E. A. Carter, The Journal of Physical Chemistry C 115, 6269 (2011). [51] J. P. Perdew and L. A. Constantin, Phys. Rev. B 75, 155109 (2007). [52] S. Haq, P. K. Chattaraj, and B. M. Deb, Chem. Phys. Lett. 111, 79 (1984). [53] B. M. Deb and P. K. Chattaraj, Phys. Rev. A 37, 4030 (1988). [54] P. K. Chattaraj, Phys Rev A. 41, 6505 (1990). [55] B. M. Deb and P. K. Chattaraj, Phys. Rev. A 45, 1412 (1992). [56] T. Mart´ın-Blas, D. Garc´ıa-Aldea, and J. E. Alvarellos, J. Chem. Phys. 130, 034101 (2009). [57] T. Mart´ın-Blas, D. Garc´ıa-Aldea, and J. E. Alvarellos, J. Chem. Phys. 131, 164117 (2009). [58] D. Garc´ıa-Aldea, and J. E. Alvarellos, Phys. Chem. Chem. Phys. 14, 1756 (2012).

In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 Editor: Amlan K. Roy, pp. 281-312 © 2012 Nova Science Publishers, Inc.

Chapter 12

VARIATIONAL F ITTING IN AUXILIARY D ENSITY F UNCTIONAL T HEORY V´ıctor Daniel Dom´ınguez Soria∗, †, Patrizia Calaminici ‡ and Andreas M. K¨oster § Departamento de Qu´ımica, Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional, Col. San Pedro Zacatenco, M´exico D.F.

Abstract This review focuses on the variational fitting of auxiliary densities in Kohn-Sham density functional theory methods. It introduces auxiliary density functional theory (ADFT) as a reliable and efficient alternative to the conventional Kohn-Sham approach. Particular attention is given to the numerical stabilization of the fitting equations and their efficient solutions. Benchmark calculations on systems with more than 1,000 atoms and more than 30,000 auxiliary functions are discussed. As example applications ADFT studies on giant fullerenes and mordenite zeolites are presented. These studies show that first-principle all-electron ADFT geometry optimizations on systems with more than 500 atoms have become routine applications.

PACS 05.45-a, 52.35.Mw, 96.50.Fm Keywords: Coulomb Fitting, Density Functional Theory, Optimization, Fullerene, Mordenite

1.

Introduction

Over the last two decades variational fitting of density distributions has become increasingly popular in quantum chemistry. Density fitting has a long history in Xα [1] and density functional theory (DFT) [2, 3] methods. A least-squares density fitting was introduced first ∗

E-mail address: [email protected] ´ Permanent address: Area de Qu´ımica Aplicada, Universidad Aut´onoma Metropolitana, Azcapotzalco, Av. San Pablo 180, Col. Reynosa-Tamaulipas, 02200 M´exico D.F. ‡ E-mail address: [email protected] § E-mail address: [email protected]

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in the framework of the Xα method by Baerends et al. [4]. It was later transferred to linear combination of Gaussian-type orbital (LCGTO) implementations by Sambe and Felton [5]. Because of its non-variational approximated energy expression least-squares density fitting can compromise the robustness of the underlying electronic structure method. This problem is avoided by the variational fitting. In its original form, it was formulated by Dunlap et al. [6] for the variational fitting of the Coulomb energy in Xα and DFT calculations in order to avoid the calculation of four-center electron repulsion integrals (ERIs). Already in the beginning of the 1990’s this approach was used in popular LCGTO-DFT Kohn-Sham methods like DGAUSS [7, 8] and deMon [9, 10]. The large body of calculations with these programs showed that the accuracy of the variational fitting of the Coulomb energy is within the intrinsic accuracy of the LCGTO-DFT method. Moreover, analytic energy derivatives can be formulated in the framework of the variational fitting of the Coulomb energy. This makes it the method of choice for most modern DFT implementations. Since the middle of the 1990’s the variational fitting has also reached mainstream quantum chemistry under the pseudonym “RI” [11, 12, 13]. Even though, the RI arguments yield expressions identical to the variational fitting, the arguments themselves contain no suggestion of robustness, i.e. variational stability [14]. If the variational fitting of the Coulomb energy is used in Kohn-Sham DFT methods the computational bottleneck shifts from the Coulomb energy calculation to the calculation of the exchange-correlation energy and potential. This has motivated the development of several approximations for the calculation of the exchange-correlation energy and potential [5, 15, 16, 17, 18, 19]. Based on these works, it is now well established that the approximated energy expression remains variational if the auxiliary density from the variational Coulomb fit is used for the calculation of the exchange-correlation energy and potential [17]. This is the foundation of our auxiliary density functional theory (ADFT) approach. The major computational advantage of ADFT is rooted in the substitution of the KohnSham density by the auxiliary density in the numerical integration. This reduces the computational effort at each grid point by one order of magnitude due to the linear scaling of the auxiliary density. As a result, ADFT enables first-principle calculations on systems with thousands of basis functions [20, 21, 22, 23] without jeopardizing the accuracy of the underlying LCGTO-DFT methodology. As for the variational fitting of the Coulomb energy, the accuracy of ADFT is within the intrinsic accuracy of the LCGTO-DFT methodology. More recently, a non-iterative approach of McWeeny’s self-consistent perturbation theory [24, 25, 26] has been formulated within the ADFT framework [27]. So far, this so-called auxiliary density perturbation theory (ADPT) was successfully applied for the calculation of static and dynamic dipole-dipole polarizabilities [28, 29], dipole-quadrupole polarizabilities [30], Fukui functions [31, 32] and nuclear shielding tensors [33]. The variational nature of the corresponding perturbed potential fit was proved, too [34]. ADPT is reviewed in [35]. Parallel ADFT and ADPT implementations are now available in deMon2k [36]. From the large body of ADFT calculations with deMon2k it is now well established that this approximation offers significant speedups without loss of accuracy. The currently more limited number of ADPT molecular property calculations suggest a similar picture for perturbation calculations. Due to the ADPT implementation in deMon2k the number of these calculations will soon increase in the literature. Because the Kohn-Sham matrix construction scales nearly linear in ADFT, linear algebra steps become computational bottlenecks

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already for medium sized systems with a few hundred atoms. At this point it is important to note that the variational fitting introduces new linear algebra steps in the self-consistent field (SCF) energy calculations. In ADFT, the variational fitting introduces two additional inhomogeneous equation systems. Because the dimension of these two equation systems is given by the number of auxiliary functions their solutions become computational bottlenecks for larger systems. So far this problem has been tackled by partitioning approaches [37, 38, 39, 40, 41] that alter the original formulation of the variational fitting. Recently, we have introduced an alternative approach that conserves the variational nature of the fitting [42]. This new algorithm is based on the automatic generation of preconditioners for conjugate gradient (CG) methods [43] proposed by Morales and Nocedal [44]. It can be used for solving a sequence of linear equation systems, A x = bi , i = 1, . . . , t , (1) in which the coefficient matrix A is constant but the right-hand side bi varies. Here, the first problem (i = 1) is solved by a quasi-Newton method [45, 46] that generates the automatic preconditioner for the other linear equation systems in the sequence, (i ≥ 2). After preconditioning, these equation systems are solved with the CG method. This approach is well suited for very large equations systems. Therefore, we have implemented this algorithm in deMon2k for the calculation of the fitting coefficients in the variational fitting of the Coulomb energy. Because in ADFT a second set of fitting coefficients occurs the sequence in equation (1) runs in our implementation over i = 1, 2. In this chapter we review our experiences with the variational fitting in the framework of ADFT. For this purpose we first introduce the LCGTO Kohn-Sham method and the ADFT method in the following two sections. In section 4.1 we discuss the stabilization of the fitting equation systems which is mandatory if large extended auxiliary function sets are used. The iterative solutions of the fitting equation systems according to equation (1) are described in section 4.2 and 4.3. This yields considerable savings in computer time as shown by the performance analysis in section 4.4. As example applications structure optimizations of giant fullerenes and host-guest interactions in mordenite zeolites are reviewed in section 5.2 and 5.3. We have chosen these examples to show that full and constrained ADFT structure optimization for systems with 400 to 800 atoms are feasible with rather moderate computational resources. Final conclusions are drawn in the last section.

2.

The Lcgto Kohn-Sham Method The canonical Kohn-Sham orbital equations are given by [47]:   Z ρ(r0) 1 0 dr + v [ρ] ψi (r) = i ψi (r) − ∇2 + v(r) + xc 2 | r − r0 |

(2)

Here v(r) denotes the external potential of the system and vxc [ρ] the exchange-correlation potential calculated as the functional derivative of the exchange-correlation energy. In the LCGTO ansatz the Kohn-Sham orbitals are expanded in an atomic orbital basis, X ψi (r) = cµi µ(r), (3) µ

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with µ(r) being an (contracted) atomic orbital and cµi the corresponding molecular orbital coefficient. With this expansion the electronic density is given by: ρ(r) =

X

Pµν µ(r) ν(r)

(4)

µ,ν

For clarity of presentation we restrict our discussion to closed-shell systems. The closedshell density matrix elements are defined as: Pµν ≡ 2

occ X

cµi cνi

(5)

i

The corresponding closed-shell LCGTO-DFT Kohn-Sham energy expression is given by: E=

X

Pµν Hµν +

µ,ν

1 XX Pµν Pστ h µν k στ i + Exc[ρ] 2 µ,ν σ,τ

(6)

with Hµν

Atoms X 1 2 ZC = hµ| − ∇ |ν i− hµ| |νi 2 |r−C |

(7)

C

and h µν k στ i ≡

ZZ

µ(r)ν(r)σ(r0)τ (r0 ) dr dr0 |r − r0 |

(8)

The first term in (6) collects the one-electron contributions. It introduces a formal quadratic scaling with respect to the number of basis functions, N , into the LCGTO-DFT energy calculation. The second term represents the Coulomb repulsion between the electrons. To simplify notation the 1/|r1 − r2 | Coulomb operator is denoted by the || symbol. Due to the four-center ERIs this term introduces a formal N 4 scaling into the energy expression. The last term denotes the exchange-correlation energy which, in general, cannot be calculated analytically. Therefore, a three-dimensional numerical integration has to be performed. This introduces a formal N 2 × G scaling into the above energy expression with G being the number of grid points in the numerical integration. The computational bottleneck in LCGTO-DFT energy calculations is, therefore, the evaluation of the four-center ERIs. To avoid the calculation of these integrals the variational fitting of the Coulomb energy was introduced [6]. It is based on the minimization of the following self-interaction term: E2 =

1 2

ZZ

[ ρ(r) − ρ˜(r) ] [ ρ(r0) − ρ˜(r0) ] dr dr0 | r − r0 |

(9)

In deMon2k the approximated density, ρ˜(r), is expanded in primitive Hermite Gaussians [48, 49] that are centered at the atoms: X ¯ ρ(r) ˜ = xk¯ k(r) (10) ¯ k

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In this review we denote the primitive Hermite Gaussian auxiliary functions by a bar. An ¯ (unnormalized) auxiliary function k(r) centered at atom C with exponent ζk has the form: ¯ k(r) =



∂ ∂Cx

k¯x 

k¯y 

∂ ∂Cy

∂ ∂Cz

k¯z

e−ζk (r−C)

2

(11)

Employing the expansions for ρ(r) and ρ˜(r), equations (4) and (10), we can write E2 as: E2 =

XX X 1 XX ¯ x¯ + 1 ¯ ¯li Pµν hµν||ki xk¯ x¯l hk|| Pµν Pστ hµν||στi − k 2 µ,ν σ,τ 2 ¯¯ µ,ν ¯ k

k,l

Here appearing the two- and three-center ERIs are defined by: ZZ ¯ ¯ 0 k(r)l(r ) dr dr0 h¯ k || ¯l i ≡ |r − r0 | h µν || ¯ ki ≡

ZZ

(12)

(13)

¯ 0) µ(r)ν(r)k(r dr dr0 |r − r0 |

(14) (15)

The variational nature of this approach is most obvious by noting that E2 is positive semidefinite. Thus, it holds: XX X 1 XX ¯ x¯ − 1 ¯ ¯li Pµν Pστ hµν||στi ≥ Pµν hµν||ki xk¯ x¯l hk|| k 2 µ,ν σ,τ 2 µ,ν ¯ ¯¯ k

(16)

k,l

Inserting this inequality into the LCGTO-DFT energy expression (6) yields the following approximated SCF energy: E=

X µ,ν

Pµν Hµν +

XX µ,ν

¯ k

X ¯ ¯li + Exc [ρ] ¯ x¯ − 1 x¯ x¯ hk|| Pµν hµν||ki k 2 ¯¯ k l

(17)

k,l

The corresponding Kohn-Sham matrix elements are given by:   X ∂E Kµν = = Hµν + h µν k k¯ i xk¯ + h µ | vxc[ρ] | ν i ∂Pµν x ¯

(18)

k

Here, the derivative of the exchange-correlation energy was obtained as follows [50]: Z ∂Exc[ρ] δExc [ρ] ∂ρ(r) = dr = h µ | vxc[ρ] | ν i (19) ∂Pµν δρ(r) ∂Pµν with vxc [ρ] ≡

δExc[ρ] δρ(r)

(20)

Thus, first- and higher-order analytic derivatives can be obtained from the approximated energy expression with the variational fitting of the Coulomb energy. What remains to be

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done is to determine the fitting coefficients, xk¯ , that are used to expand the approximated density. In Dunlap’s original approach the fitting coefficients are obtained by minimizing the self-interaction E2 (9) with respect to the fitting coefficients. In this minimization the density matrix is kept constant. Thus, we obtain:   X X ∂E2 ¯ mi =− Pµν hµν||mi ¯ + xk¯ hk|| ¯ =0∀m ¯ (21) ∂xm ¯ P ¯ µ,ν k

To cast this equation system into matrix form we now introduce the Coulomb matrix, G, and the Coulomb vector, J, defined as:   ¯i h¯ 1k¯ 1i h ¯1 k ¯2 i . . . h ¯1 k M  h¯ ¯i  ¯ h ¯2 k ¯2 i . . . h ¯2 k M  2k 1i  G= (22) , .. .. .. ..   . . . . ¯ k 1¯ i h M ¯ k 2¯ i . . . h M ¯ kM ¯i hM  X Pµν h µν k ¯1 i  µ,ν  X  Pµν h µν k ¯2 i   µ,ν J=  ..  .  X  ¯i Pµν h µν k M µ,ν

          

(23)

¯ denotes the number of auxiliary functions in the system. With G and J the folHere M lowing inhomogeneous equation system for the determination of the fitting coefficients, collected in x, can be formulated: Gx = J (24) The solution of this equation system is characteristic for the variational fitting of the Coulomb energy and, thus, represents an additional linear algebra step. On the other hand, the four-center ERIs have disappeared from the energy expression (17) and are also not needed for the calculation of the Kohn-Sham matrix elements (18) or fitting coefficients ¯. (21). Thus, the formal scaling of the Coulomb energy calculation is reduced to N 2 × M ¯ Because M is usually in the same range of N , substantial savings are obtained with the variational Coulomb energy fitting. The additional computational time for solving the fitting equation system (24) is for small systems negligible. A particularity of the variational fitting is that the fitting equation system can also be obtained from the approximated energy expression (17):   X X ∂E ¯ mi = Pµν hµν||mi ¯ − xk¯ hk|| ¯ =0∀m ¯ (25) ∂xm ¯ P µ,ν ¯ k

However, the corresponding second derivatives are different. The second derivatives of the self-energy term (9) are positive due to the positive definiteness of the Coulomb matrix:   ∂ 2 E2 = h¯ n||mi ¯ (26) ∂xm ¯ ∂xn ¯ P

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The corresponding derivatives of the approximated energy (17) are for the same reason negative:   ∂ 2E = −h¯ n||mi ¯ (27) ∂xm ¯ ∂xn ¯ P Thus, the minimization of E2 corresponds to the maximization of the approximated SCF energy. As a result, the variational fitting of the Coulomb energy turns the SCF minimization into a MinMax optimization as discussed in [51]. As a result, the converged SCF energy can be approached from below because of the MinMax optimization. Of course, the converged SCF energy remains variational.

3.

Auxiliary Density Functional Theory

As discussed in the previous section the variational fitting of the two-electron Coulomb energy reduces the formal scaling of the Coulomb energy calculations by almost one order of magnitude. In combination with efficient three-center ERI recurrence relations [52] and asymptotic expansions [49] a linear scaling for the Coulomb energy calculation can be achieved [53]. As a result, the computational bottleneck in SCF energy calculations shifts to the numerical integration of the exchange-correlation energy and potential. Thus, a more efficient approach for the calculation of the exchange-correlation energy and potential is needed. Least squares fits with auxiliary functions are often used in this context [4, 5]. However, the non-variational nature of the corresponding energy expression may compromise the numerical accuracy of the underlying methodology. For this reason the direct use of the auxiliary density from the variational fitting of the Coulomb energy for the calculation of the exchange-correlation energy has been investigated over the last decade [14, 16, 17, 18, 19]. The corresponding energy expression takes the form: E=

X µ,ν

Pµν Hµν +

XX µ,ν

¯ k

X ¯ ¯li + Exc [˜ ¯ x¯ − 1 x¯ x¯ hk|| ρ] Pµν hµν||ki k 2 ¯¯ k l

(28)

k,l

It is identical to the energy expression from the variational fitting of the Coulomb potential, except that ρ˜ is used for the calculation of the exchange-correlation energy. The corresponding Kohn-Sham matrix elements are given by:   X ∂E ρ] ¯ ¯ + ∂Exc [˜ Kµν = = Hµν + hµν||kix (29) k ∂Pµν x ∂Pµν ¯ k

For the exchange-correlation energy derivative follows [17]: Z ∂Exc [˜ ρ] δExc [˜ ρ] ∂ ρ˜(r) = dr ∂Pµν δ ρ˜(r) ∂Pµν

(30)

As usual, the functional derivative defines the exchange-correlation potential, now evaluated with the auxiliary density: δExc[˜ ρ] (31) vxc [˜ ρ] ≡ δ ρ˜(r)

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Thus, what remains to be done is to calculate the partial derivatives of the auxiliary density with respect to the density matrix elements: ∂ ρ˜(r) X ∂xk¯ ¯ = k(r) ∂Pµν ∂Pµν ¯

(32)

k

From equation (21) follows for the fitting coefficients: XX ¯ ¯li−1 h¯l||µνiPµν hk|| xk¯ = ¯ l

Thus, we find:

(33)

µ,ν

X ∂xk¯ ¯ ¯li−1 h¯l||µνi = hk|| ∂Pµν ¯

(34)

l

Back substitution of (34) into (32) and then into (30) yields: ∂Exc[˜ ρ] X ¯ −1 hk|v ¯ xc [˜ = hµν||¯lih¯l||ki ρ]i ∂Pµν ¯¯

(35)

k,l

For convenience of notation we now introduce a second set of fitting coefficients, named exchange-correlation fitting coefficients, as: X ¯ ¯li−1 h¯l|vxc [˜ zk¯ = hk|| ρ]i (36) ¯ l

Inserting the exchange-correlation energy derivative (35) into the expression for the KohnSham matrix elements (29) then yields: X ¯ ¯ + z¯ ) Kµν = Hµν + hµν||ki(x (37) k k ¯ k

Remarkable, the ADFT Kohn-Sham matrix depends only on fit coefficients! This is equivalent to the statement that the Kohn-Sham potential in ADFT depends only from the auxiliary density and, therefore, has a similar simple structure as in orbital-free DFT methods. Of course this results in significant computational savings compared with traditional KohnSham implementations. The above described calculation of Kohn-Sham matrix elements as derivatives of the ADFT energy expression (28) is just an example for the formulation of analytic energy derivatives. This can be straightforward extended to the calculation of analytic gradients [17] and other first-order energy derivatives [54]. Also higher-order analytic ADFT energy derivatives can be formulated [27]. Thus, ADFT overcomes the fundamental drawback of the non-variational energy expression of least squares fits to the exchange-correlation contributions. What remains to be proofed is that the ADFT energy is also variationally bounded. This is less straightforward as for the variational fitting of the Coulomb energy. Nevertheless, empirical studies indicate that the ADFT energy is indeed variationally bounded. Table 1 shows typical results of CO ADFT and RI-DFT calculations with the PBE [55]

Variational Fitting in Auxiliary Density Functional Theory

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Table 1. Convergence between ADFT and RI-DFT optimized bond length, dissociation energy, frequency and dipole moment of CO with respect to the auxiliary function set size. The corresponding experimental data are given for reference. See text for details

Auxis GEN-A2 GEN-A2* GEN-A4*

˚ re [A] ADFT RI-DFT 1.137 1.137 1.137 1.137 1.137 1.137

De [kcal/mol] ADFT RI-DFT 266.3 263.1 264.4 264.5 264.6 265.0

ωe [cm−1 ] ADFT RI-DFT 2119 2122 2123 2123 2123 2123

µ[Debye] ADFT RI-DFT 0.197 0.190 0.196 0.194 0.194 0.194

functional, the aug-cc-pVTZ basis [56] set and various auxiliary function sets [57]. The auxiliary function set size increases from GEN-A2 to GEN-A2* and further to GEN-A4*. As already discussed, the exchange-correlation energy and potential is calculated from the fitted density in ADFT, whereas it is calculated from the Kohn-Sham density in the RI-DFT calculations. In both cases, the variational fitting of the Coulomb energy is used. As Table 1 shows, the differences between the ADFT and RI-DFT results diminish systematically with increasing auxiliary function set. Whereas the optimized bond length is already with the GEN-A2 auxiliary function set identical for both approaches, the dissociation energies differ by more than 3 kcal/mol. Enlarging the auxiliary function set to GEN-A2* reduces this difference below 1 kcal/mol. The ADFT and RI-DFT harmonic frequencies are identical with GEN-A2*, too. The difference in the ADFT and RI-DFT CO dipole moment converges smoothly by increasing the auxiliary function set. The convergence described here between ADFT and RI-DFT observables with increasing auxiliary function set size can be observed in all systems. This is a strong empirical indication that ADFT is variationally bounded. At first glance, it seems most obvious that the convergence between ADFT and RI-DFT observables is rooted in the convergence of the auxiliary density, ρ(r), ˜ towards the KohnSham density, ρ(r). However, topological analysis of ρ˜(r) and ρ(r) shows that this is not the case. For this reason we derived an analog of the Hohenberg-Kohn theorem for ADFT. The original Hohenberg-Kohn theorem establishes the mapping [2]: b → E[ρ] ρ(r) → v(r) → H

(38)

Assuming that ρ˜(r) behaves similar to the Kohn-Sham density, i.e. is normalized to the number of electrons, is positive definite and has maxima at the nuclear positions (all these properties can be enforced by the variational fitting procedure), the following mapping exist: b → E[˜ ρ˜(r) → v(r) → H ρ] (39) The proof is identical to that of the original Hohenberg-Kohn theorem [2]. Thus, what remains to be done is to proof the existence of the mapping: ρ(r) → ρ(r) ˜

(40)

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Because the variational mapping defines an inhomogeneous linear equation system with a positive definite coefficient matrix, each Kohn-Sham density maps on one auxiliary density. Thus, we find: b → E[˜ ρ(r) → ρ˜(r) → v(r) → H ρ] (41)

This mapping shows that ADFT differs from conventional DFT only in the definition of the (unknown) universal energy functional. As DFT, ADFT is variationally bounded by its energy functional.

4.

Variational Fitting in ADFT

The two inhomogeneous equation systems that need to be solved in each ADFT SCF cycle are given by equation (24) and (36). In matrix notation these equations take the form: Gx = J

(42)

Gz = L

(43)

¯ xc [˜ Here we have introduced the vector L with elements Lk¯ = hk|v ρ]i that are obtained by numerical integration. Comparison with equation (1) shows that in our case the constant coefficient matrix is the Coulomb matrix G and the varying right-hand side vectors are J and L. In this section we now describe the solution of (42) and (43) as implemented in deMon2k.

4.1.

Stabilization of the Fitting Equation Systems

A very efficient approach for solving (42) and (43) is Cholesky decomposition [58]. The computational time for this decomposition is negligible for systems with less than 5000 auxiliary functions. However, if auxiliary functions sets with more diffuse auxiliary functions are used the positive definiteness of G can be numerically compromised. As a result, Cholesky decomposition will fail. In this case the equation systems must be numerically stabilized. So far, our method of choice is singular value decomposition (SVD). Because our numerically most stable implementation of SVD for the fitting equation systems deviates from the one proposed in the literature [58] we describe it here in more details. The first step of the SVD procedure is the diagonalization of the Coulomb matrix G: G = U D UT

(44)

In (44) D denotes a diagonal matrix containing the eigenvalues, λi, of G. The corresponding orthogonal transformation matrix U collects the eigenvectors, ui , of G in its columns. The next step of the SVD represents the quenching of eigenvalues below a given threshold. By default, this threshold is set to 10−6 in deMon2k. It can be changed with the keyword MATINV [59] (please note that calculations with different SVD thresholds are usually not compatible with each other). Eigenvalues and eigenvectors that are below the

Variational Fitting in Auxiliary Density Functional Theory

291

given threshold are eliminated from D and U. As a result, the quenched diagonal matrix and orthogonal transformation matrix are given by: λ1

0

 0 Dq =   .. . 0

λ2 .. . ···



··· .. . .. . 0

0 .. .

   

(45)

¯ Uq = ( u1 , u2 , · · · , uM−Q )

(46)

0

λM−Q ¯

Here Q denotes the number of quenched eigenvectors from G. Note that the transformation matrix becomes rectangular by this operation. The SVD approximate to G is then given by: ˜ = Uq Dq UTq G (47) ˜ is positive definite. However, we solve equation systems (42) and (43) By construction G ˜ Instead, we transform the equation systems without constructing an explicit inverse of G. into their reduced diagonal representation. For the Coulomb fitting equation system (42) this yields: Uq Dq UTq x = J ⇒ Dq UTq x = UTq J (48) with xq = UTq x

(49)

Jq = UTq J

(50)

Dq xq = Jq ⇒ xq = D−1 q Jq

(51)

and follows further: Thus, the equation system is solved in its diagonal representation. Because Dq and D−1 q are diagonal the numerically delicate couplings are removed. The fit coefficients in the reduced, diagonal representation, xq , are finally back transformed to the original auxiliary function representation by: x = Uq xq (52) The same approach can be applied to the equation system (43). Even though the above described algorithm is formally identical to the direct calculation of the approximate inverse of G by SVD, it is numerically far more stable. Our analysis indicates that this enhanced stability is based on the component representation of the fitting coefficients. It is straightforward to show that the above described SVD algorithm is identical to the following expression of the fitting coefficients, xk¯ , in terms of the eigenvectors of G: x=

¯ M−Q X i=1

(

1 T u J)ui λi i

(53)

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Similar, the exchange-correlation fitting coefficients, zk , can be obtained by substituting J with L in (53): z=

¯ M−Q X

(

i=1

1 T u L)ui λi i

(54)

The described stabilization of the fitting equation systems has proven successful for very large fitting functions sets, such as GEN-A4*, in energy and response calculations.

4.2.

Iterative Solution of the Coulomb Fitting Equation System

The above described stabilization of the fitting equation systems introduces a diagonalization step that scales cubic with the number of auxiliary functions. For larger systems with tens of thousands auxiliary functions, this step becomes a computational bottleneck. Note, that in principle, the SVD has to be performed at each different geometry, e.g. in each step of a geometry optimization or a Born-Oppenheimer molecular dynamics (BOMD) simulation. Clearly, this calls for an alternative approach. As already mentioned, the ADFT fitting equations can be solved with an iterative method proposed by Morales and Nocedal [44] for a sequence of linear equation systems. In our case only two equation systems need to be solved. To introduce this method we first note that the solution of the inhomogeneous equation system (42) is equivalent to the minimization of the convex quadratic function F (x) defined as: X 1X F (x) = Gk¯¯l xk¯ x¯l − Jk¯ xk¯ (55) 2 ¯¯ ¯ k,l

k

The variation of (55) with respect to the fitting coefficients xk¯ yields: ∂F (x) X = Gk¯¯l x¯l − Jk¯ ≡ 0 ∀ k¯ ∂xk¯ ¯

(56)

l

This gradient is just the residual of equation system (42): X rk¯ = Gk¯¯l x¯l − Jk¯

(57)

¯ l

The second derivative of F (x) is given by the Coulomb matrix: ∂ 2 F (x) = Gk¯¯l ∂xk¯ ∂x¯l

(58)

Thus, the positive definiteness of G ensures the minimization of F (x) via a positive definite Hessian. As discussed in the last sections, for auxiliary function sets with diffuse exponents ˜ to ensure a positive definite Hessian. Thus, we need to G in (58) must be substituted by G couple the SVD with the iterative approach from Morales and Nocedal. To avoid the calculation of the Hessian matrix in each optimization step a quasi-Newton method with line-search is employed. In each SCF step the approximated inverse Hessian matrix is initialized by the one computed in the previous SCF cycle. Similar, for the first

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293

SCF cycle of a new geometry, e.g. in structure optimization or BOMD simulations, the inverse Hessian matrix is initialized by the one from the last SCF step of the previous geometry. Thus, only one initial approximation for the inverse Hessian matrix is needed, namely in the very first SCF cycle of a deMon2k run. By default, this initial approximation is calculated via SVD in deMon2k. The initial transformation matrices are then used in all following steps, even if geometries have changed. This has proven to be an excellent compromise between accuracy and performance. If the auxiliary function set is small, e.g. A2 or GEN-A2, the SVD can be avoided. In this case the initial inverse Hessian matrix can be calculated via Cholesky decomposition or simply set to the unit matrix. In any case, matrix diagonalizations are avoided in any following geometries steps. We now describe the individual steps to solve (42) by a quasi-Newton optimization. Different to the description in [42] we include explicitly the transformation matrices of the SVD. In case no SVD was performed these matrices become unit matrices and the description reduces to the one in [42]. The iterative quasi-Newton procedure starts with the calculation of the residual: r = Gx − J (59) This residual is then transformed in the reduced space defined by the SVD in the first SCF cycle: rq = UTq r (60) With an approximate inverse Hessian matrix, B, the search direction in the reduced SVD space is calculated as: ∆xq = −Brq (61) Note that B is given in the reduced space and updated by the inverse BFGS [60, 61, 62, 63] formula:   d gT Bold + Bold g dT gT Bold g d dT old − (62) B = B + 1+ dT g dT d dT g with d = UTq (x − xold) and g = UTq (r − rold )

(63)

The superscript “old” denotes quantities from the previous quasi-Newton cycle. Because the update of B involves only vector-vector and matrix-vector operations it scales quadratically. If SVD is employed, the very first B is set to D−1 q . In the next step the search direction is back transformed, ∆x = Uq ∆xq ,

(64)

and a new set of fitting coefficients are calculated by: x = xold + α∆x

(65)

Here α denotes a line-search parameter that can be calculated analytically due to the quadratic form of F (x) as [64]: α=

rT ∆x (∆x)T G∆x

(66)

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In order to initialize the quasi-Newton optimization procedure, the converged Coulomb fitting coefficients, xk¯ , from the previous SCF cycle are used as initial fitting coefficients. In case of the very first SCF cycle, the Coulomb fitting coefficients, xk¯ , are initialized to zero. In Scheme 1 the pseudocode for the calculation of the Coulomb fitting coefficients, employing the above described quasi-Newton algorithm, is presented. START PSEUDOCODE input: G, J, x, B start loop: calculate residual r (59) test convergence of r exit loop if convergence test was successful transformation of the residual to the reduced SVD space (60) calculate reduced space search direction ∆xq (61) update B (62) back transformation of reduced space search direction ∆xq (64) calculate line-search parameter α (66) calculate new set of Coulomb fitting coefficients x (65) next loop output: B, x END PSEUDOCODE Scheme 1. Quasi-Newton algorithm for the calculation of the auxiliary fitting coefficients xk¯ .

In order to stop the iterative procedure two convergence test are performed. In the first one the normalized length of the residual, r=

|r| , M

(67)

is compared with the convergence threshold εr defined by: εr =

ε0 10 − min(log( εεSCF ), 9) con

(68)

Here εSCF and εcon refer to the current SCF energy error and the requested SCF energy convergence, respectively. ε0 is set to εcon /100. In the first two SCF cycles εr takes the value of the default SCF energy convergence 10−5 . Because εr depends on the SCF error as well as the requested SCF energy convergence, the stopping criteria of the quasi-Newton optimization procedure automatically adapts to the convergence of the SCF solution. The second convergence test is based on the spread of the current normalized length of the residual around the normalized length of the residuals of the last six iterations:

εσ =

6 X

(r − ri )2

(69)

i=1

The convergence threshold εσ is set to 10−8 in the first two SCF cycles, to 10−9 in the following two, and to 10−10 in all others SCF cycles.

Variational Fitting in Auxiliary Density Functional Theory

4.3.

295

Iterative Solution of the Exchange-Correlation Fitting Equation System

As described by Morales and Nocedal [44] the approximated inverse Hessian matrix, B, obtained by the above discussed quasi-Newton method, can be used as preconditioner for the exchange-correlation fitting equation system. Because B exists only in the reduced SVD space we have to transform (43) in this space. This yields: UTq GUq zq = Lq

(70)

zq = UTq z

(71)

Lq = UTq L

(72)

with

and

Note that in general UTq GUq is not diagonal because the transformation matrices are taken from the very first SCF of the run. Preconditioning of (70) yields: BUTq GUq zq = BLq with BUTq GUq ' E

(73)

With this preconditioning the exchange-correlation fitting equation system can be efficiently solved by a conjugated gradient (CG) method [43]. The preconditioned conjugated gradient (PCG) iteration starts with the calculation of the residual of the original exchangecorrelation fitting equation system: r = Gz − L

(74)

This residual is then transformed in the reduced SVD space, rq = UTq r,

(75)

and preconditioned with the approximate inverse Hessian matrix, B, from the quasi-Newton optimization of the Coulomb fitting coefficients: sq = Brq

(76)

In the next step the residual is back transformed to the original auxiliary function space by: s = Uq sq

(77)

The search direction in the CG method is given as [64]: ∆z = −s + β∆zold

(78)

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The parameter β is determined from the conjugate direction criterion: β=

rT s (rold)T sold

(79)

Finally, a new set of exchange-correlation fitting coefficients are obtained by: z = zold + α ∆z

(80)

As in the quasi-Newton method, α denotes a line-search parameter that can be calculated analytically due to the quadratic form of F (z): α=

rT ∆z (∆z)T G∆z

(81)

As in the case of the iterative calculation of the Coulomb fitting coefficients, the converged exchange-correlation fitting coefficients, zk¯ , from the previous SCF cycle are used as initial exchange-correlation fitting coefficients. In case of the very first SCF cycle, the exchangecorrelation fitting coefficients, zk¯ , are initialized to zero. In Scheme 2 the pseudocode for the calculation of the exchange-correlation fitting coefficients, employing the discussed PCG algorithm, is presented. START PSEUDOCODE input: G, L, z, B start loop: calculate residual r (74) test convergence of r exit loop if convergence test was successful transformation of the residual to the reduced SVD space (75) calculate preconditioned residual in the reduced SVD space (76) back transformation of preconditioned residual (77) calculate PCG parameter β (79) calculate PCG direction ∆z (78) calculate line-search parameter α (81) calculate new exchange-correlation fitting coefficients z (80) next loop output: znew END PSEUDOCODE Scheme. 2 PCG algorithm for the calculation of the exchange correlation fitting coefficients zk¯ . It

is important to note that the reduced diagonal representation of the approximated inverse Hessian B is never updated in the PCG procedure. Instead it is used to precondition the reduced diagonal representation of (43).

4.4.

Performance Analysis of the Iterative Fittings

Benchmark calculations using these new implementations were performed with the DFT program deMon2k [36]. Calculations for C60 and three three-dimensional ZSM5 zeolite structures with one, two and three unit cells were employed. The stoichiometry of the zeolite systems is Si96 O200 H80 , Si192 O400 H128 and Si288O600 H176 , respectively. The Coulomb energy was calculated by the variational fitting procedure as described above. The obtained auxiliary density was used for the calculation of the exchange-correlation energy and potential, too. These energies and potentials were numerically integrated on an adaptive

Variational Fitting in Auxiliary Density Functional Theory

C60

Si192 O400 H128

297

Si 96O200 H80

Si 288 O600 H176

Figure 1. Structures of the benchmark systems. grid [65]. The grid accuracy was set to 10−5 a.u. in all calculations. The structure optimizations were performed with the local density approximation (LDA) employing Dirac exchange [66] in combination with the correlation functional of Vosko, Wilk and Nusair (VWN) [67]. DFT optimized all-electron double zeta valence plus polarization (DZVP) basis sets [68] were employed. For the fitting of the density the auxiliary function set A2 was used [68]. A restricted step quasi-Newton method in internal redundant coordinates with analytic energy gradients was used for the structure optimization [69, 70]. The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 10−4 and 10−3 a.u., respectively. The test calculations were performed in parallel [20, 71, 72] with deMon2k. The analytical fit refers to the calculation of G−1 according to [51] whereas the numerical fit refers to the iterative solvers described in section 4.2 and 4.3. We have performed benchmark calculations on zeolite models with up to about 35,000 auxiliary functions. In order to provide a common reference point for the analytical and numerical fitting coefficient calculations we included C60 in the benchmark set. The structures of the benchmark systems along with their stoichiometric formulas are depicted in Figure 1. The larger ones reach well into the nanometric length scale. In Figure 2 the CPU times for the analytical and numerical calculation of the fitting coefficients are plotted against the number of auxiliary functions on a double logarithmic scale. These benchmark calculations were performed in parallel employing 10 2.4 GHz single core Opteron processors. This figure shows that the analytic calculation of the fitting coefficients scales cubic with respect to the num-

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V. D. Dom´ınguez-Soria, P. Calaminici and A. M. K¨oster Numerical Analytical

CPU time [s]

10000

1000

100

10 2000

5000

10000

20000

30000

Number of auxiliary functions

Figure 2. CPU times for the analytical and numerical calculation of the fitting coefficients plotted against the number of auxiliary functions on a double logarithmic scale. ber of auxiliary functions. Due to the linear algebra steps involved and the fact that the Coulomb matrix inversion is performed only one time per energy calculation this behavior is expected. In the case of the numerical calculation of the fitting coefficients the number of SCF cycles influence the timings because the iterative solvers are called in each cycle. As a result a system independent scaling cannot be expected. However, independently from this behavior Figure 2 clearly demonstrates the computational benefits obtained from the numerical calculations of the fitting coefficients. The computational savings increase dramatically with system size. Moreover, for the largest system, Si288 O600 H176, the inversion of the Coulomb matrix consumes about 40 % of the total CPU time for a single-point energy calculation. Thus, the use of the numerical solvers decrease considerably the overall CPU time in this case, too. The benchmark calculations in Figure 2 need on average around 20 SCF cycles for convergence. Thus, for systems with such a SCF convergence behavior a sub-quadratic scaling of around O1.7 for the numerical calculation of the fitting coefficients with respect to the number of auxiliary functions is observed.

5. 5.1.

Example Applications Large Cluster Applications with ADFT

In this section the results of recent all-electron state-of-the-art applications performed with ADFT on selected systems consisting of several hundred atoms and up to more than 30,000 auxiliary functions will be briefly reviewed. The first application will focus on the study of the structure and energy parameters of four large fullerenes, such as C180 , C240 , C320 and C540 . The second application will review the results obtained on structures and intrinsic properties of some mordenite (MOR) type zeolite cationic sites, in particular sodium binding energies and acid strengths. In this investigation it was proposed for the

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299

first time a new methodological approach which could be helpful to consider when the study of these properties for zeolite models is aimed. Our proposed method is based on the construction of very large mordenite type zeolite models, each consisting of more than 400 atoms, as an alternative to the periodic representation of the zeolite when the experimental Si/Al ratio is not very large. In such cases the Al distribution cannot be represented by a periodic model. Finally, we will summarize the results of a very recent study of host-guest interactions of small molecules such as CO and CH3 CN with the large and small cavities of our proposed large cluster MOR models.

5.2.

First-principle Calculations of Large Fullerenes

Fullerenes are synthetic clusters of carbon characterized by highly symmetrical cagelike structures. They are formed when a sheet of graphite is closed with a needed curvature which is achieved intersecting twelve pentagons with a certain number of graphite hexagons [22]. These clusters are known since C60 was discovered, whose structure results as a ball composed by combining hexagons and pentagons, getting as result a structure arrangement such as a geodesic dome or a soccer ball. As more of these systems were synthesized, the term “fullerene” became a generic name for this new class of carbon allotropes which have attracted a lot of attention from both experimental and theoretical point of view. One main reason of the large interest on the study of the fullerenes is certainly to be found in their particularly very appealing geometrical form. Over the last twenty years or so, several experimental works were performed on fullerenes aggregates containing up to around hundred atoms [22]. Nevertheless, experimental results on large fullerenes characterized by several hundred atoms are still very rare or not existing. Those system are also commonly known in our days as “giant fullerenes”. Due to this lack of information, many questions about their structure and properties still are unsolved. Certainly, theoretical investigations can provide important pieces of the missed data which are difficult to extract from experiments. In particular, the understanding about how different properties evolve in large fullerenes is of most relevance for future studies on functionalization and reactivity. A long standing discussion in the literature has questioned whether giant fullerenes exist in a faceted or in spherical arrangement. In order to solve this debate, several theoretical works were performed giving different results [22]. Finally, a few years ago our first-principle ADFT based study has solved this dilemma [22] for a series of four giant fullerenes, namely C180 , C240 , C320 and C540 . The structures of these fullerenes were fully optimized without any symmetry restriction with the deMon2k code employing all-electron DZVP basis sets in combination with the Vosko, Wilk and Nussair (VWN) functional. All calculations were performed on the lowest potential energy surface (PES). Our ADFT first-principle based study clearly shows that the larger systems, C240 , C320 and C540 , possesses a faceted shape and that also for the smallest, C180 , fullerene this topology is preferred over the spheroidal shape. This can be seen in Figure 3 where the resulting optimized structures of these four giant fullerenes are displayed. As Figure 3 shows, the obtained faceted behavior becomes more and more pronounced with increasing fullerene size, as a result of the fact that the curvature (or carbon atom pyramidalization) at the five-membered rings is increasing with cluster size. The optimized geometries of C180 , C240 , C320 and C540 were analyzed in details in order to gain in-

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Figure 3. Optimized structures of C180 , C240 , C320 and C540 . sight into the structure evolution of these clusters as the number of carbon atoms increases. Therefore, the normalized number of bonds of each of these fullerenes were plotted against the bond lengths. We note that C180 , C240 and C540 fullerenes behave very similar since for these carbon clusters a discrete bond length distribution was obtained as can be seen from Figure 4. This result indicates that the symmetry of the electronic structures matches with the expected geometrical symmetry. On the other hand, C320 differs from all others giant fullerenes we have studied. It is characterized by a wide and continuous bond length distribution. This result, which clearly indicates a breaking of the expected high symmetry of the system, is graphically displayed in Figure 5. For C320 calculations with higher multiplicity, such as triplet and quintet, were performed, too. Our results indicate that the C320 fullerene ground state possesses a triplet or quintet multiplicity. However, we have observed that the peculiar behavior of the continuous bond length distribution is still persisting with higher multiplicities. This indicates that the C320 giant fullerene is a good example of a large fullerene to be consider for further studies on reactivity and functionalization. It is important here to underline that such an interesting result of bond length distribution, as the one we observed for the C320 , can be obtained only if a non-symmetry-adapted optimization is employed. ˚ the graphene bond length (dashed line) we obtained from In Figures 4 and 5, at 1.419 A, periodic calculations with deMon2k, employing the Cyclic Cluster Model (CCM) method, is drawn. A detailed analysis of these figures reveals that with increasing cluster size three very important trends can be extracted: 1) The number of different bond lengths increases; 2) There is an accumulation of bond lengths around the graphene bond length; 3) The ˚ for C180 to about longest bond length in the cluster becomes shorter (from about 1.442 A ˚ for C240 and about 1.431 A ˚ for C540 ). This result indicates that the delocalization 1.437 A is increasing with cluster size. We also studied energy properties of these fullerenes with increasing cluster size. The calculated binding energy per C atom for these giant fullerenes increase monotonically with cluster size [22]. This result indicates that these systems become more stable as the number of C atoms increases. In particular, and very interestingly, we notice that the calculated binding energy of the largest fullerene we studied, the C540 , for which the binding energy is 8.75 eV, is comparable to the cohesive energy of diamond (of

normalized number of bonds

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301

C540

normalized number of bonds

1.38

1.39

1.40

1.41 1.42 bond length [Å]

1.43

1.44

1.45

1.43

1.44

1.45

1.43

1.44

1.45

C240

1.38

1.39

1.40

1.41

1.42

normalized number of bonds

bond length [Å]

C180

1.38

1.39

1.40

1.41 1.42 bond length [Å]

normalized number of bonds

˚ for C180 Figure 4. Normalized number of bonds versus the optimized bond length (in A) ˚ (bottom), C240 (middle) and C540 (top). The vertical dashed line at 1.419A is the graphene bond length obtained from periodic calculations. See text for more details.

1.38

C320

1.39

1.40

1.41 1.42 bond length [Å]

1.43

1.44

1.45

˚ for C320 . Figure 5. Normalized number of bonds versus the optimized bond length (in A) ˚ The vertical dashed line at 1.419A is the graphene bond length obtained from periodic calculations. See text for more details.

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V. D. Dom´ınguez-Soria, P. Calaminici and A. M. K¨oster

a)

=

b)

c)

Figure 6. Schematic representation of the building pattern for zeolites: a) tetrahedral SiO4 unit; b) two tetrahedral units connected by their corners; c) example of a resulting zeolite framework. 8.78 eV) indicating that there is hope that indeed giant fullerenes could be experimentally prepared in the future. However, the C540 binding energy is considerably smaller than the one calculated for graphene of 8.91 eV. This indicates that even the largest giant fullerene we have studied here is still a metastable system.

5.3.

Structure, Properties and Confinement Effects in Mordenite Models Type Zeolite

Zeolites are materials of central interest to a large number of very important industrial processes, such as catalysis, gas separation and purification, in between others [73, 74]. These materials can be described as microporous crystalline solids with a well-defined structure. They contain several elements such as silicon, aluminium and oxygen in their framework, whereas cations, water, and/or small molecules could be present in their pores. Zeolites occur naturally as minerals and are extensively mined in many parts of the world. Many others are synthetic and are commercialized for different and specific uses or produced by research scientists who have tried to understand more about their chemical and physical properties. Because of their characteristic porous properties, zeolites are used in a large variety of applications as in petrochemical cracking, ion-exchange and in the separation and remotion of gases and solvents. A defining feature of zeolites is that their framework is built up by connected SiO4 units. One way to think about this is in terms of tetrahedra with a silicon atom in the middle and oxygen atoms at the corners. One of this tetrahedric unit is illustrated in Figure 6a. These

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303

B T1

T3

Atom Legend: Grey atoms = Si

C

D

Red atoms = O White atoms = H

A T2

T4

Figure 7. Structure of a pure mordenite cluster model as the ones proposed in this study. The four independent tetrahedral sites (T1, T2, T3, and T4) are indicated by arrows and the four types of rings present in this zeolite by bold letters: A (12-membered ring), B (8-membered ring), C (5-membered ring), and D (4-membered ring), respectively.

kinds of tetrahedral units can be linked together by connecting their corners (see Figure 6b) in order to form a rich variety of beautiful structures, as the one depicted in Figure 6c. The framework structure may contain linked cages, cavities or channels which are of the right size to allow small molecules to enter, i.e. to be confined in the zeolite framework. Up to now, over 130 different zeolite framework structures are known. In between all those, mordenite type zeolite are particularly attractive from a structural point of view due to their very peculiar pore system. Mordenites are natural and synthetic zeolites with Si/Al ratios of 4.3-6.0 in the first case and 5.0-12.0 in the second one [75, 76]. The mordenite structure can be described as composed of edge-showing five-membered (5m) rings of tetrahedra forming chains along the c crystallographic axis [77]. The mordenite zeolite framework is composed by large mono-directional 12-membered ring channels formed by TO4 tetrahedra, where T stands either for Si or Al atoms, and small 8-membered ring channels, which are interconnected through 8-membered ring tubes. The large 12˚ run along the membered rings of elliptical shape whose dimensions are of about 6.7×7.0 A

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c crystallographic axis and form the commonly known “main channel”. The 8-membered ˚ run along the b crystallographic rings characterized with an aperture of about 3.4×4.8 A axis, and therefore perpendicular to the main channels. They are known as “side pockets” [78]. Apart 12- and 8- membered rings, 5-membered and 4-membered rings are also present in mordenite zeolites. The topological symmetry of mordenites is orthorhombic with space group Cmcm. The mordenite unit cell is characterized by four symmetrical independent tetrahedral sites commonly named as T1, T2, T3, and T4. T1 and T2 sites connect four different rings, whereas the T3 and T4 sites constitute the 4-membered rings of the zeolite framework [77]. Figure 7 illustrates one of the cluster models we have proposed for a pure mordenite type zeolite (i.e. when there is no exchange of Si with Al). In this figure the four independent tetrahedral sites described above are indicated by arrows. In this model the four different types of rings present in this zeolite are marked with bold letters, A (12membered ring), B (8-membered ring), C (5-membered ring), and D (4-membered ring), respectively. As aluminium substitution of the tetrahedral sites might occur the zeolite will be active. In this situation, in order to balance the charge of the full system, small cations such as Li+ , Na+ , etc. are inserted in the zeolite pores. Moreover exchange of these alkali cations with protons H+ can also occur. This confers the acidic property to the zeolite. Finally, by the confinement of small guest molecules within the pores at the proton site a catalytic process can be activated. Theoretical studies based on ab-initio methods are very useful as they can furnish detailed information about the elementary steps of catalytic processes. The main limitation of such investigation is related to the size of the used model which is generally very small and consequently is a rather poor description of the zeolite framework. Fortunately, modern ab-initio codes, such as the deMon2k program, allow now the geometry optimization of very large zeolite models. These calculations can be performed in relatively short times using common parallel compute clusters and are characterized by very moderate disk and RAM requirements [71, 72]. Due to this impressive development over the last years we have been able to focus part of our investigation on the study of structural parameters and energy properties of different Na- and H-mordenites using cluster models of the type displayed in Figure 8, each of which consist of more than 400 atoms [21, 23]. These state-of-the-art calculations were performed within the framework of auxiliary density functional theory, using both, the local density approximation as well as the generalized gradient approximation in combination with all-electron basis sets [68]. Initially, the most populated T3, T4, and T1 Al sites have been investigate using two different mordenite models, each containing two isolated Al sites, T4 and T3 in the first case and T4 and T1 in the second case. In the literature the T4 site is proposed as the most probable candidate for Al substitution in mordenite zeolites [79]. Our strategy to build the resulting models to be able to determine the structure and the intrinsic properties of the catalyst, has been to use models containing 120 tetrahedra, which are therefore large enough to enclose the main 12-membered rings and side pocket 8-membered rings. The original cluster models were built by cuts from a solid with the adequate Al distribution. Those models were generated using the CERIUS2 package [80] and were terminated with hydrogens. In order to maintain the structure of the cluster models to that of the solid during the geometry optimization procedure, the coordinates of the terminal hydrogen atoms, positioned along the Si-O bonds, were fixed. All other coordinates were relaxed during the subsequent opti-

Variational Fitting in Auxiliary Density Functional Theory

Na−MOR1

a)

305

Na−MOR2

b)

Figure 8. Structures, T and O legend, and crystallographic axes a, b, c of the studied mordenite models. mization steps. In these two models Na cations and protons were used as counterions to the Al substitution. The models are named Na-MOR1, Na-MOR2 and H-MOR1, H-MOR2, respectively. The structures of the resulting Na-MOR1 and Na-MOR2 models are shown in Figure 8a and 8b, respectively. The orientation of the three crystallographic axes a, b and c, is also shown in Figure 8. The first purpose of our study was to investigate if from the geometry optimization of our proposed models averaged bond and angle data could be simulated and compared with those provided by X-ray experiments performed on natural and synthetic mordenites with a similar Al/Si ratio [81, 82]. Therefore, the optimized structural parameters (T-O bonds and T-O-T bond angles) have been averaged over the Na-MOR1 and Na-MOR2 systems, namely, considering different Al distributions. This procedure mimics the experimental averaging over the non distinguishable Si and Al positions. As it was shown in Table II of ref. [21] the calculated averaged bond lengths are very comparable with the experimental ˚ Moreover, also the averaged values of the T-O-T results. The accuracy is within 0.02 A. bond angles result in good agreement with experiment, within 4◦ . Therefore, the models we proposed containing each two Al sites, are well comparable with experimental mordenites which contain a low population of Al (4-6 Al) per unit cell. The substitution of Si by Al in the zeolite framework generates one negative charge which has to be compensate by the presence of one counter-ion. The negative charge results distributed among the framework

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of oxygens of the tetrahedral site where the Si-Al substitution has occurred, more particularly to the four oxygens adjacent to Al. In order to locate the most probable positions of the counterions, the well known strategy to approximate the cation position by calculating the largest electrostatic interaction (MEP) of the system density with a positive point charge, was used. The MEP values were therefore calculated for the optimized structures of the corresponding anion at T4 and T3 sites of the Na-MOR1 model, and at T4 and T1 sites in the Na-MOR2 model, respectively. For Na-MOR1, the lowest MEP minimum value calculated at T4 is located between the oxygens O2 and O10 (Figure 8a) with a value of -4.13 eV, whereas the lowest minimum at T3 is located close to the oxygen O9 and has a value of -4.22 eV, respectively. In the case of Na-MOR2, the minimum value of the MEP at T4 is the same as the one calculated for Na-MOR1, whereas the MEP well at T1 results essentially located within the side pocket, with the lowest minimum value of -3.94 eV close to the oxygen O6 (Figure 8b). Once the spatial Cartesian coordinates of these minimum MEP values were determined, we positioned Na cations at these coordinates and the resulting structures were optimized. In this way the Na binding energies with respect to the framework have been calculated. The trend of the calculated Na binding energies follows the ordering of the MEP minima, i.e. T3> T4 ≈ T1. This result indicates that electrostatic effects govern the cation framework binding. Moreover, the calculated binding energies of the Na cation follows also the same ordering as the one of the population of Al T sites, as derived from X-ray experiments on synthetic Na-mordenites. This result shows the existence of a synergic effect between cations and Al, Si ordering during the growth of the solid, as it is suggested experimentally [82]. As for the Na-MOR models, the protonated structures have been optimized as well, with the original proton positions being those of the respective lowest MEP minima, and the proton affinity values at each T site have been calculated. The obtained results allow us to conclude that the sites belonging to the main channel and to the side pocket possess similar proton affinities. This shows that the strength of the O-H bond is not depending on the local structure of the studied sites and indicates that these Br¨onsted sites have similar acid strengths [21]. This conclusion suggests that the MOR acidic properties are govern by the interaction of the zeolite framework with the base instead of the local acidity of the OH sites. Therefore, in order to be able to evaluate any difference in local acidity the presence of the associate base, such as CO, CH3 CN, NH3 , etc. has to be taken into account, as well. For this reason we have subsequently also used small probe molecules like CO and CH3 CN to estimate the acidic strength of Br¨onsted sites in mordenite (H-MOR) models, depending on their framework location, main channel or side pocket [23]. Experimental data concerning the interaction of these small probe molecules with mordenite type zeolite frameworks can be summarized as follows. Interaction of CO with Br¨onsted sites of HMOR systems has been commonly used to analyze their acidic strength in particular, with the aim to distinguish between the 8-ring OH sites in the main channel and those located in the side pocket. Upon adsorption of CO in H-MOR, infrared spectra show the appearance of two bends one in the γ(O-H) region and the other in the γ(C-O) region. The higher bends are assigned to the OH· · · CO complexes in the main channels. The lower frequencies that are about 10 cm−1 below are assigned to the side pocket hydroxils [83]. The experimental studies involving the adsorption of nitriles in zeolite have been more detailed and extensive

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than those using CO as a probe for testing acid properties. The most intriguing result is the fact that the nitrile shows larger affinity towards the mordenite side pocket than for the main channel [84, 85, 86, 87, 88, 89]. The studied mordenite models interacting with either a single CO or CH3 CN molecule at T1,T2, T3 and T4 site were optimized and the adsorption energy at these different sites were calculated. In this study, in order to take into account neutron powder diffraction results on mordenite Br¨onsted sites [90], the adsorption study of CO and CH3 CN at the T2 tetrahedral site has been also included. In this way, the interaction of the two single guest molecules, CO and CH3 CN, with hydroxils located in main channels (T2 and T4 tetrahedral sites) and in side pockets (T1 and T3 tetrahedral sites) of a mordenite type zeolite was fully explored in our work for the first time [23]. Our results show that the O-H bonds at the four sites are elongated when the CO guest molecule adsorbs. However, this elongation is moderate with respect the O-H bonds of the H-MOR system, being of only ˚ at T1, 0.008 A ˚ at T2, 0.011 A ˚ at T3 and 0.014 A ˚ at T4, respectively. On the 0.008 A contrary, the comparison of the results obtained when a single CO or CH3 CN molecule is interacting with the four hydroxils sites shows that this elongation is much more pronounced ˚ For the calculated when the CH3 CN molecule is considered, being of about 0.04-0.05 A. energy properties the situation is different. In fact, the adsorption energies of CH3 CN show, as for CO adsorption, very comparable values at T3 and T4 and a weaker value at T1 but a much larger stability at T2. Also we investigated the charge perturbation by the surrounding zeolite by applying a Mulliken charge analysis. Our results indicate that the charge perturbation is manifested by the increase of positive and negative charges for CO by about 0.02e. The perturbation results much larger for CH3 CN but not specific with respect to the adsorption site. The most pronounced changes we observed correspond to a strong increase of the negative N charge (-0.8e) at the T1 and T3 sites. Therefore, the proposal that the acetonitrile is more polarized in small rather than in large cavities is indeed supported by our study. However, our results show that the host-guest interaction is not favored by this polarization. Putting together these results we conclude that the local electronic host-guest interactions lead to similar total adsorption energies in the main channel and in the side pocket, despite the fact that a stronger molecule-acid site complex occurs in small cavities. This unexpected result is in agreement with recent experimental studies [89] which show that confinement is more related with higher concentration of host molecules in small rather than in large cavities at high pressure. Theoretical studies aiming to simulate this phenomena under these environmental conditions should be considered in the future.

Conclusion With the above discussed development of iterative approaches for the solution of the fitting equation systems in auxiliary density functional theory first-principle all-electron calculations of systems with 500 to 1,000 atoms become feasible. The described combination of singular value decomposition and preconditioned conjugate gradient method represents a successful compromise between numerical stability and accuracy. It permits the use of diffuse auxiliary functions sets for large systems. This is particularly important for response property calculations of large systems. Benchmark calculations using these

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new methodology shows the computational benefits that can be obtained if such implementations are applied. The example applications demonstrate the current state-of-the-art for auxiliary density functional theory structure optimizations. By using moderate (up to 64 cores) parallel architectures structure optimizations of systems with more than 500 atoms have become routine applications. The example applications show that such calculations can provide an unprecedented insight into the structure and interaction of nanomaterials, such as the discussed giant fullerenes and zeolites. Thus, it is fair to say that first-principle all-electron LCGTO-DFT methods have reached nanoscience.

Acknowledgments The authors gratefully acknowledge support from CONACyT (60117-4, 130726) and from ICyTDF (PIFUTP08-87, PICCO-10-47), V.D. Dom´ınguez-Soria specially acknowledge CONACyT retention program RUA.513.2011.

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In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 313-356

Chapter 13

WAVELETS FOR D ENSITY-F UNCTIONAL T HEORY AND P OST-D ENSITY-F UNCTIONAL -T HEORY C ALCULATIONS Bhaarathi Natarajana,b∗, Mark E. Casidaa†, Luigi Genoveseb‡ and Thierry Deutschb§ a Laboratoire de Chimie Th´eorique, D´epartement de Chimie Mol´ecularie (DCM, UMR CNRS/UJF 5250), Institut de Chimie Mol´eculaire de Grenoble (ICMG, FR2607), Universit´e Joseph Fourier (Grenoble I), Grenoble Cedex 9, France b UMR-E CEA/UJF-Grenoble 1, INAC, Grenoble, France

Abstract We give a fairly comprehensive review of wavelets and of their application to densityfunctional theory (DFT) and to our recent application of a wavelet-based version of linear-response time-dependent DFT (LR-TD-DFT). Our intended audience is quantum chemists and theoretical solid-state and chemical physicists. Wavelets are a Fourier-transform-like approach which developed primarily in the latter half of the last century and which was rapidly adapted by engineers in the 1990s because of its advantages compared to standard Fourier transform techniques for multiresolution problems with complicated boundary conditions. High performance computing wavelet codes now also exist for DFT applications in quantum chemistry and solid-state physics, notably the B IG DFT code described in this chapter. After briefly describing the basic equations of DFT and LR-TD-DFT, we discuss how they are solved in B IG DFT and present new results on the small test molecule carbon monoxide to show how B IG DFT results compare against those obtained with the quantum chemistry gaussian-type orbital (GTO) based code DE M ON 2 K. In general, the two programs give essentially the ∗

E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected]

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Bhaarathi Natarajan, Mark E. Casida, Luigi Genovese et al. same orbital energies, but the wavelet basis of B IG DFT converges to the basis set limit much more rapidly than does the GTO basis set of DE M ON 2 K. Wavelet-based LR-TDDFT is still in its infancy, but our calculations confirm the feasibility of implementing LR-TD-DFT in a wavelet-based code.

PACS: 31.15.A-, 33.20.-t, 31.15.E-, 02.70.Hm Keywords: Wavelets, Density-functional theory, Time-dependent density-functional theory, Linear-response time-dependent density-functional theory, Orbital energies, Electronic excitation energies

1.

Introduction

The broad meaning of “adaptivity” is the capacity to make something work better by alternation, modification, or remodeling. Concepts of adaptivity have found widespread use in quantum chemistry, ranging from the construction of Gaussian-type orbital (GTO) basis sets, see e.g., the development of correlation consistent bases [1, 2, 3], to linear scaling methods in density functional theory (DFT) [4, 5, 6, 7, 8, 9], selective configuration interaction (CI) methods [10, 11] and local correlation methods based on many-body perturbation theory or coupled cluster (CC) theory [12, 13]. This chapter is about a specific adaptive tool, namely wavelets as an adaptive basis set for DFT calculations which can be automatically placed when and where needed to handle multiresolution problems with difficult boundary conditions. Let us take a moment to contrast the wavelet concept of adaptivity with other types of adaptivity. In other contexts, the adaptive procedure is typically based on a combination of physical insights together with empirical evidence from numerical simulations. A rigorous mathematical justification is usually missing. This may not be surprising: Familiar concepts lose a lot of their original power if one tries to put them in a rigorous mathematical framework. Therefore, we will not shoulder the monumental and perhaps questionable task of providing a rigorous mathematical analysis of all the adaptive approaches used nowadays throughout quantum chemistry. Instead we will concentrate on the mathematical analysis of a particular electronic structure method which lends itself to a rigorous mathematical analysis and application of adaptivity. In contrast with other adaptive methods, multiresolution analysis (MRA) with wavelets can be regarded as an additive subspace correction and their wavelet representations have a naturally built-in adaptivity which comes through their ability to express directly and separate components of the desirable functions living on different scales. This combined with the fact that many operators and their inverses have nearly sparse representations in wavelet coordinates may eventually lead to very efficient schemes that rely on the following principle: Keep the computational work proportional to the number of significant coefficients in the wavelet expansions of the searched solution. As there are a lot of different wavelet bases with different properties (length of support, number of vanishing moments, symmetry, etc.) in each concrete case we can choose the basis that is most appropriate for the intrinsic complexity of the sought-after solution. This fact makes the wavelet-based schemes a very sophisticated and powerful tool for compact representations of rather complicated functions. The expected success of wavelet transforms for solving

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electronic structure problems in quantum mechanics are due to three important properties: (a) the ability to choose a basis set providing good resolution where it is needed, in those cases where the potential energy varies rapidly in some regions of space, and less in others; (b) economical matrix calculations due to their sparse and banded nature; and (c) the ability to use orthonormal wavelets, thus simplifying the eigenvalue problem. Of course, this might lead to adaptive methods which are fully competitive from a practical point of view, for example, working with a systematic basis instead of GTO bases requires from the onset larger basis sets and the benefit of systematic improvement might be a distant prospect. However, we have the more realistic prospect that our rigorous analysis provides new and hopefully enlightening perspectives on standard adaptive methods, which we reckon cannot be obtained in another way. On the other hand advances in computational technology opened up new opportunities in quantum mechanical calculation of various electronic structures, like molecules, crystals, surfaces, mesoscopic systems, etc. The calculations can only be carried out either for very limited systems or with restricted models, because of their great demand of computational and data storage resources. Independent particle approximations, like the Hartree-Fock based [14, 15, 16, 17] algorithms with single determinant wave functions, leave out the electron correlation and need operation and storage capacity of order N 4 , if N is the total number of electrons in the system. If inclusion of the electron correlation is necessary, CI or CC methods can be applied, with very high demand of computational resources (O(N 6 ) to O(N !)). An alternative way is to use MBPT. The second order perturbation calculations can be carried out within quite reasonable time and resource limits, but the results are usually unsatisfactory, they just show the tendencies, while the 4th order MBPT needs O(N 7 ) to O(N 8 ) operations. All these algorithms use the N -electron wave function as a basic quantity. Another branch of methods use electron density as the primary entity. Pioneers of this trend, like Thomas [18], Fermi [19, 20], Frenkel [21] and Sommerfeld [22] developed the statistical theory of atoms and the local density approximation (LDA). The space around the nuclei is separated into small regions, where the atomic potential is approximated as a constant, and the electrons are modeled as a free electron gas of Fermi-Dirac statistics [23, 24, 21]. Dirac included electron correlation [25], which improved the results. After the Hohenberg–Kohn theorems had appeared [26], and Kohn and Sham had offered a practically applicable method [27] based on their work, many scientists were motivated to work on the theory, and DFT developed into one of the most powerful electronic structure methods. Despite the success of density functional theory, it has some drawbacks. The exact formula of the exchange-correlation potential is not known, thus chemical intuition and measured data are necessary in order to approximate it, and the kinetic energy functional is hard to calculate. Powerful approximating formulas are available (see, e.g. [28]), like the Thomas–Fermi functional based gradient and generalized gradient expansions, where the energy functionals are expressed as a power series of the gradient of the density (the first such suggestion was [29].) Considering the historical development of sophisticated N -electron methods, a typical trend can be observed. Starting with a very simple model, new details are introduced in order to improve the results. This scheme is followed in the linear muffin tin orbital

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method (LMTO) [30] where the inter-atomic regions is replaced by the spherical orbital of an atomic potential around the nuclei. Similarly, the linearized augmented plane wave method (APW) [31] and the plane wave pseudopotential approach [32] describe the details of the crystal potential differently in different spatial regions. Although they are rather successful, for applying any of these models, chemical intuition is needed, free parameters, like the radius of the bordering sphere between the two types of potentials, and the boundary conditions have to be set. A systematic method, which can handle the different behaviors of the electron structures at different spatial domains, or either at different length scale [33], is the long-term requirement of any physical chemists. Multiresolution or wavelet analysis, this rapidly developing branch of the applied mathematics, is exactly the tool for satisfying all the need of any chemical physicists/physical chemists. From mathematical point of view wavelet analysis is a theory of a special kind of Hilbert space basis sets. Basis sets are commonly used in all electron structure calculations, as the wave function is usually expanded as linear combination of some kind of basis functions. Thus the operator eigenvalue problem is reduced to an algebraic matrix eigenvector problem. The resulting algebraic equations are easier to solve, well known algorithms and subroutine libraries are available, however, the difficulty of choosing the proper basis set arises. If linear combination of atomic orbitals (LCAO) is used, the atomic basis functions are Slater or Gaussian-type of functions [34, 35], the selection of atomic orbitals needs chemical intuition, which is a result of long time’s experience, and can not be algorithmized. Both basis sets are non-orthogonal, and lack the explicit convergence properties [36]. Moreover, calculation of operator matrix elements with Slater-type orbitals is complicated, their integrals have to be treated numerically. Although integrals of Gaussian functions are analytically known, the Gaussian-type basis does not reflect the nuclear cusp condition of Kato [37], which reflects on singularities of the N -electron wave function in the presence of Coulomb-like potentials. Since then it turned out that for high precision numerical calculations it is essential to satisfy these requirements. However, while the nuclear cusp condition is relatively easy to fulfill by Slater-type orbitals (STO), the electron-electron cusp is extremely hard to represent. In general, GTO-based/STO-based DFT codes gives reliable results with a relatively small number of basis functions, making them optimal for large scale computations where high accuracy is less crucial. On the other hand there is no consistent way to extend these basis sets and thereby converge the results with respect to the size of the basis. The second type of basis set covers the system-independent functions such as plane waves [32] or wavelets [38]. The main advantage of these basis sets, is that their size can be systematically increased until the result of the calculation has converged, and are generally considered to be more accurate than the former type. The number of basis function required to obtain convergence is normally so large that direct solution of the matrix eigenvalue problem within the entire basis space is not possible. Instead one has to use iterative methods to determine the lowest (occupied) part of the spectrum [32]. In solid state physics, where more or less periodic systems are studied, choosing plane wave basis sets is rather usual. These basis functions are system independent and easily computable, but the results are not always convincing and the number of necessary basis functions is almost untreatable. (Theoretically, plane waves could also be used for describing molecules, since the two-electron integrals and the expectation values are connected to the Fourier transform, thus they are

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easily computable, and this could balance the large number of necessary basis functions.) The reason, why so many plane waves are needed is that the wave functions around the nuclei need very high frequency terms, i.e. high resolution level, for reproducing the nuclear cusps. In the framework of Fourier analysis, the whole space has to be expanded at the same resolution, despite that in most of the space low frequency terms would be sufficient. Fully-numerical “basis-set free” Hartree-Fock (HF) calculations of atoms have been known since the 1960s (Vol. 1, pp. 322-326 and Vol. 2, pp. 15-30 of Ref. [39] and Refs. [40, 41, 42, 43]) and have proven helpful in constructing efficient finite basis sets for molecular calculations. In the late 1980s, Axel Becke used a fully-numerical densityfunctional theory (DFT) program for diatomics to show that many of the problems of DFT calculations at that time were due not to the functionals used, but rather numerical artifacts of the DFT programs of the 1970s [44].) Since that time, fully-numerical DFT codes have been implemented for polyatomic molecules using the finite element method (FEM), with PARSEC from the chemists point of view or OCTOPUS from the view of physicists being a notable example. B IG DFT the pseudo potential code for bigger systems based as it is on traditional Hohenberg-Kohn-Sham DFT [26, 27], could only calculate ground-state properties with an eye to order-N DFT. As a step to increase the feasibility of the code we formulated the wavelet-based linear-response time-dependent density-functional theory (TD-DFT) and here we support our first implementation for calculating electronic excitation spectra [45]. Electronic excitation spectra can be calculated from TD-DFT [46] using time-dependent linear response (LR) theory [47, 48]. Casida formulated LR-TD-DFT (often just referred to as TD-DFT) so as to resemble the linear-response time-dependent HF equations already familiar to quantum chemists [48]. That method was then rapidly implemented in a large number of electronic structure codes in quantum chemistry, beginning with the DE M ON family of programs [49] and the T URBO M OL program [50]. Among the programs that implemented “Casida’s equations” early on was the FEM DFT program PARSEC [51] and also be found in the FEM DFT program O CTOPUS [52]. See Ref. [53] for a recent FEM implementation of TD-DFT. Since a wavelet-based program offers certain advantages over these other FEM DFT programs, it was deemed important to also implement LR-TD-DFT in B IG DFT. In the next section we give a detailed description of the idea behind the multiresolution analysis and wavelets, with a historical note. Sec. 3 and Sec. 4, briefly presenting the theoretical introduction to DFT and TD-DFT, and Sec. 5, talks about the well-known Krylov space methods for solving eigenvalue equations involved in our implementation. Sec. 6 and Sec. 7, gives the numerical implementation of DFT and how we have implemented TD-DFT from the aspects of theoretical and algorithmic point of view on wavelets based pseudopotential electronic structure code B IG DFT, and in Section 8 we give the results of detailed comparisons between TD-DFT excitation spectra calculated with B IG DFT and with the implementation of Casida’s equations in the GTO-based program DE M ON 2 K. The conclusion were drawn for future applications in the field of chemistry and some of the other problems are reviewed to draw chemists’ greater attention to wavelets and to gain more benefits from using wavelet technique.

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Wavelet Theory

The mathematics of wavelets is a fairly new technique, it can generally be used where one traditionally uses Fourier techniques. They incorporate the feature of having multiple scales, so very different resolutions can be used in different parts of space in a mathematically rigorous manner. This matches many systems in nature well, for example molecule where the atomic orbitals are very detailed close to the cores, while they only vary slowly between them. Wavelet analysis can quite generally be viewed as a local Fourier analysis. From the wavelet expansion, or wavelet spectrum, of a function, f , it can be inferred not only how fast f varies, i.e. which frequencies it contains, but also where in space a given frequency is located. This property has important applications in both data compression, signal/image processing and noise reduction [54]. Wavelet methods are also employed for solving partial differential equations [55, 56], and in relation to electronic structure methods a complete DFT program based on interpolating wavelets has been developed [57].

2.1.

The Story of Wavelets

Most historical versions of wavelet theory however, despite their source’s perspective, begin with Joseph Fourier. In 1807, a French mathematician, Joseph Fourier, discovered that all periodic functions could be expressed as a weighted sum of basic trigonometric functions. His ideas faced much criticism from Lagrange, Legendre and Laplace for lack of mathematical rigor and generality, and his papers were denied publication. It took Fourier over 15 years to convince them and publish his results. Over the next 150 years his ideas were expanded and generalized for non-periodic functions and discrete time sequences. The fast Fourier transform algorithm, devised by Cooley and Tukey in 1965 placed the crown on Fourier transform, making it the king of all transforms. Since then Fourier transforms have been the most widely used, and often misused, mathematical tool in not only electrical engineering, but in many disciplines requiring function analysis. This crown however, was about to change hands. Following a remarkably similar history of development, the wavelet transform is rapidly gaining popularity and recognition. The first mention of wavelets was in a 1909 dissertation by Hungarian mathematician Alfred Haar. Haar’s work was not necessarily about wavelets, as “wavelets” would not appear in their current form until the late 1980s. Specifically, Haar focused on orthogonal function systems, and proposed an orthogonal basis, now known as the Haar wavelet basis, in which functions were to be transformed by two basis functions. One basis function is constant on a fixed interval, and is known as the scaling function. The other basis function is a step function that contains exactly one zero–crossing (vanishing moment) over a fixed interval (more on this later). The next major contribution to wavelet theory was from a 1930s French scientist Paul Pierre L´evy. More correctly, L´evy’s contribution was less of a contribution and more of a validation. While studying the ins and outs of Brownian motion in the years following Haar’s publication, L´evy discovered that a scale–varied Haar basis produced a more accurate representation of Brownian motion than did the Fourier basis. L´evy, being more of a physicist than mathematician, moved on to make large contributions to our understanding of stochastic processes.

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Contributions to wavelet theory between the 1930s and 1970s were slight. Most importantly, the windowed Fourier transform was developed, with the largest contribution being made by another Hungarian named Dennis Gabor. The next major advancement in wavelet theory is considered to be that of Jean Morlet in the late 1970s. Morlet, a French geophysicist working with windowed Fourier transforms, discovered that fixing frequency and stretching or compressing (scaling) the time window was a more useful approach than varying frequency and fixing scale. Furthermore, these windows were all generated by dilation or compression of a prototype Gaussian. These window functions had compact support both in time and in frequency (since the Fourier transform of a Gaussian is also a Gaussian.) Due to the small and oscillatory nature of these window functions, Morlet named his functions as “wavelets of constant shape”. In 1981, Morlet worked with Croatian–French physicist Alex Grossman on the idea that a function could be transformed by a wavelet basis and transformed back without loss of information, thereby outlining the wavelet transformation. It is of note that Morlet initially developed his ideas with nothing more than a handheld calculator. In 1986, St´ephane Mallat noticed a publication by Yves Meyer that built on the concepts of Morlet and Grossman. Mallat sought Meyer’s consult, and the result of said consult was Mallat’s publication of multiresolution analysis. Mallat’s MRA connected wavelet transformations with the field of digital signal processing. Specifically, Mallat developed the wavelet transformation as a multiresolution approximation produced by a pair of digital filters. The scaling and wavelet functions that constitute a wavelet basis are represented by a pair of finite impulse response filters, and the wavelet transformation is computed as the convolution of these filters with the input function. The importance of Mallat’s contribution cannot be overstated. Without the fast computational means of wavelet transformation provided by the MRA, wavelets, undoubtedly, would not be the effective and widely used signal processing tools that they are today. In 1988, a student of Alex Grossman, named Ingrid Daubechies, combined the ideas of Morlet, Grossman, Mallat, and Meyer by developing the first family of wavelets as they are known today. Named the Daubechies wavelets, the family consists of 8 separate wavelet and scaling functions (more on this later). With the development of pair Daubechies wavelet and scaling functions is orthogonal, continuous, regular, and compactly supported, the foundations of the modern wavelet theory were laid. The last ten years mostly witnessed a search for other wavelets with different properties and modifications of the MRA algorithms. In 1992, Albert Cohen, Jean Feauveau and Daubechies constructed the compactly supported bi-orthogonal wavelets, which are preferred by many researchers over the orthonormal basis functions, whereas R. Coifman, Meyer and Victor Wickerhauser developed wavelet packers, a natural extension of MRA.

2.2.

Multiresolution Analysis

A suitable gateway to the theory of wavelets is through the idea of MRA. A detailed description of MRAs can be found in Keinert [58], from which a brief summary of the key issues are given in the following. A multiresolution analysis is an infinite nested sequence of subspaces L2 (R) Vj0 ⊂ Vj1 ⊂ ... ⊂ Vjn ⊂ ...

(1)

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with the following properties • Vj∞ is dense in L2 • f (x) ∈ Vjn ⇐⇒ f (2x) ∈ Vjn+1 0 ≤ n ≤ ∞ • f (x) ∈ Vjn ⇐⇒ f (x − 2−n l) ∈ Vjn 0 ≤ l ≤ (2n − 1) • There exists a function vector ϕ of length j + 1 in L2 such that forms a basis for Vj0 .

{ϕj (x) : 0 ≤ k ≤ j}

This means that if we can construct a basis of Vj0 , which consists of only j +1 functions, we can construct a basis of any space Vjn , by simple compression (by a factor of 2n ), and translations (to all grid points at scale n), of the original j + 1 functions, and by increasing the scale n, we are approaching a complete basis of L2 . Since Vjn ⊂ Vjn+1 the basis functions of Vjn can be expanded in the basis of Vjn+1 ϕnl (x)

X def n/2 2 ϕ(2n x − l) = h(l) ϕn+1 (x) . l =

(2)

l

(l)

where h s are the so-called filter matrix that describes the transformation between different spaces Vjn . The MRA is called orthogonal if hϕn0 (x), ϕnl(x)i = δ0l Ij+1 ,

(3)

where Ij+1 is the (j + 1) × (j + 1) unit matrix, and j + 1 is the length of the function vector. The orthogonality condition means that the functions are orthogonal both within one function vector and through all possible translations on one scale, but not through the different scales. Complementary to the nested sequence of subspaces Vjn , we can define another series of spaces Wjn that complements Vjn in Vjn+1 Vjn+1 = Vjn ⊕ Wjn

(4)

where there exists another function vector φ of length j + 1 that, with all its translations on scale n forms a basis for Wjn . Analogously to Eq. (2) the function vector can be expanded in the basis of Vjn+1 φnl (x)

X def n/2 2 φ(2n x − l) = g (l)φn+1 (x) . l =

(5)

l

with filter matrices g (l). The functions φ also fulfill the same orthogonality condition as Eq. (3), and if we combine Eq. (1) and Eq. (4) we see that they must be orthogonal with respect to different scales. Using Eq. (4) recursively we obtain Vjn = Vj0 ⊕ Wj0 ⊕ Wj1 ⊕ ... ⊕ Wjn−1 . which will prove to be an important relation.

(6)

Wavelets for Density Functional Theory ...

2.3.

321

Wavelets

There are many ways to choose the basis functions ϕ and φ (which define the spanned spaces Vjn and Wjn ), and there have been constructed functions with a variety of properties, and we should choose the wavelet family that best suits the needs of the problem we are trying to solve. (Wavelets are often denoted by ψ in the literature but the choice has been made here to denote them by φ so as to avoid confusion with the Kohn-Sham orbitals.) Otherwise, we could start from scratch and construct the new family, one that is custommade for the problem at hand. Of course, this is not a trivial task, and it might prove more efficient to use an existing family, even though its properties are not right on cue. There is a one-to-one correspondence between the basis functions ϕ and φ, and the filter matrices h(l) and g (l) used in the two-scale relation equations Eq. (2) and Eq. (5), and most well-known wavelet families are defined only through their filter coefficients. In the following we are taking a different, more intuitive approach, for defining the scaling space Vjn as the space of piecewise polynomial functions Vjn

def =

{f : all polynomials of degree ≤ j on the interval(2−n l, 2−n(l + 1)) for0 ≤ l < 2n , f vanishes elsewhere} .

(7)

Figure 1. Wavelets (bottom) and scaling function (top). It is quite obvious that one polynomial of degree j on the interval [0, 1] can be exactly reproduced by two polynomials of degree j, one on the interval [0, 12 ] and the other on the interval [ 12 , 1]. The spaces Vjn hence fulfills the MRA condition Eq. (1), and if the polynomial basis is chosen to be orthogonal, the Vjn constitutes an orthogonal basis.

2.4.

An Example: Simple Haar Wavelets

The basic wavelet ideas that we need can be easily explained using Haar wavelets [59]. These are simply the box functions shown in Fig. 2. We begin with a compact “mother

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Figure 2. Haar scaling functions. scaling function,” in this case the Haar function,  x>1  0 ; ϕ(x) = 1 ; 00 and LUMO I(0),

(28)

A(F) > A(0).

(29)

and

In order to corroborate the behavior predicted by eqs. (28) and (29), we have performed some calculations on atomic systems. The SVWN [20], PBE [21] and B3LYP [22] exchange-correlation functionals are used in this section, as examples of the LDA, GGA and hybrid approximations, respectively. Two basis sets functions were coupled with these exchange-correlation functionals, the 6311++G**[23] and the aug-cc-pVTZ[24]. Additionally, we imposed a UEF strength of 0.005 a. u. All calculations were done with a development version of the NWChem suite code v5.1 where we implemented the contribution of the UEF in the KS equations [25]. We are reporting in Table 1 the experimental data of I, and the estimations obtained with the proposed methodologies for UEF=0. Additionally, we are including the ERROR, defined as the average of the absolute difference obtained between the experimental information and the predicted value. From Table 1, it is clear that when UEF=0 the two basis sets show the same performance and, with regard to the exchange-correlation functional the LDA gives the worst estimation whereas the best one is reached by the PBE functional. For A, we are reporting in Table 2 the available experimental information, the corresponding DFT estimations and the ERROR. For this property the hybrid exchange-correlation functional, B3LYP, gives the best prediction and LDA again gives the worst one. Curiously, the smallest basis set gives slightly the best prediction with regard to the aug-cc-pVQZ. When we apply a UEF (strength of 0.005 a. u.) we found a small response for I. For all atoms I increases, as it was predicted by eq. (28), and for the Li, Na and K we found the biggest response (from 1 to 2 %), for the rest of the atoms the increment was less than 1 %. The response exhibited by A is quite different to that observed for I. In Table 3 we are reporting the relative error, in percent, of A(|F|=0.005 a. u.) with regard to A(|F|=0.000 a.u.). The atoms of the groups 1 and 13 present the biggest response when the UEF is applied. This result is very important since some atoms that show small A, under the action of a UEF it can be drastically increased. The electronegativity and the hardness, obtained from eqs. (19) and (20), can be expressed also in a Taylor series as

1   N 1   ijN 1 FiF j , 4 i, j x,y,z ij

(30)

1  2 N   ijN 1   ijN 1 FiF j . 4 i, j x,y,z ij

(31)

F  0 





and

F  0 





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Guillermo Nieto-Malagón, Julio M. Hernández-Pérez, Rubicelia Vargas et al.

From these equations arises an important result: the electronegativity for an atom will be increased when it is submitted under a UEF. We can give this important conclusion because 1 1 always the polarizability for an anion,  ijN , is bigger than that exhibited by a cation,  ijN . Table 1. Ionization potential, in eV, for zero uniform electric field Atomic Number

b

aug-cc-pVQZ

SVWN

PBE

B3LYP

SVWN

PBE

B3LYP

Expa

1

13.02

13.60

13.67

13.02

13.60

13.67

13.60

2

24.30

24.47

24.94

24.30

24.47

24.94

24.59

3

5.46

5.57

5.62

5.47

5.58

5.63

5.39

4

9.02

9.00

9.12

9.02

9.00

9.11

9.32

5

8.64

8.66

8.73

8.64

8.67

8.74

8.30

6

11.70

11.54

11.55

11.69

11.54

11.54

11.26

7

15.03

14.74

14.68

15.01

14.73

14.66

14.53

8

14.00

14.04

14.13

14.00

14.06

14.14

13.62

9

18.01

17.68

17.77

17.96

17.66

17.73

17.42

10

22.30

21.74

21.80

22.19

21.66

21.71

21.56

11

5.40

5.35

5.42

5.37

5.36

5.43

5.14

12

7.73

7.62

7.73

7.72

7.61

7.73

7.65

13

6.03

6.06

6.01

6.03

6.08

6.02

5.99

14

8.23

8.20

8.11

8.23

8.20

8.11

8.15

15

10.58

10.50

10.39

10.58

10.49

10.39

10.49

16

10.52

10.38

10.51

10.57

10.43

10.56

10.36

17

12.21

12.98

13.08

13.20

12.97

13.07

12.97

18

16.04

15.77

15.85

15.99

15.71

15.79

15.76

4.50

b

b

b

4.34

b

b

6.11

19

a

6-311++G**

4.52

4.44

20

6.23

6.08

6.15

b

31

6.08

6.01

6.03

6.07

6.00

6.02

6.00

32

8.04

7.93

7.90

8.03

7.91

7.89

7.90

33

10.06

9.92

9.87

10.04

9.90

9.85

9.82

34

9.90

9.65

9.82

9.92

9.67

9.84

9.75

35

12.12

11.84

11.96

12.10

11.82

11.94

11.81

36

14.40

14.12

14.22

14.35

14.05

14.16

14.00

ERROR

0.29

0.13

0.17

0.27

0.13

0.17

Ref. [26] The aug-cc-pVTZ basis set is not reported for these atoms.

Effect of a Uniform Electric Field on Atomic and Molecular Systems

493

Table 2. Electron affinity, in eV, for zero uniform electric field Atomic Number

6-311++G**

aug-cc-pVQZ Expa

SVWN

PBE

B3LYP

SVWN

PBE

B3LYP

1

0.87

0.65

0.87

0.93

0.71

0.91

0.75

3

0.59

0.52

0.56

0.59

0.51

0.56

0.62

5

0.67

0.54

0.41

0.74

0.62

0.46

0.28

6

1.78

1.56

1.36

1.81

1.59

1.37

1.26

8

1.96

1.67

1.61

2.06

1.78

1.68

1.46

9

4.06

3.58

3.49

4.13

3.65

3.53

3.40

11

0.61

0.55

0.58

0.62

0.55

0.59

0.55

13

0.60

0.50

0.39

0.64

0.59

0.46

0.43

14

1.57

1.47

1.33

1.50

1.49

1.34

1.39

15

0.99

0.82

0.91

1.03

0.88

0.96

0.75

16

2.38

2.15

2.19

2.39

2.17

2.20

2.08

17

3.98

3.69

3.72

3.93

3.65

3.67

3.61

0.52

b

b

b

0.50

b

b

0.02

19

-1.40

0.52

20

0.08

0.09

0.02

b

31

0.64

0.55

0.45

0.60

0.51

0.43

0.43

32

1.60

1.43

1.33

1.56

1.44

1.33

1.23

33

1.08

0.86

0.98

1.09

0.88

1.00

0.81

34

2.40

2.14

2.21

2.38

2.13

2.20

2.02

35

3.81

3.54

3.59

3.77

3.49

3.54

3.36

ERROR

0.38

0.12

0.10

0.32

0.15

0.12

a

Ref. [26] b The aug-cc-pVTZ basis set is not reported for these atoms.

We cannot say the same for the hardness, since not always

 ijN 

1 N 1    ijN 1 . We have 2 ij





corroborated the equation (30) for several values of |F|, however we do not want to extend the length of this section; the details corresponding to these results will be published elsewhere.

IV. Uniform Electric Field Effect on Excitation Energies in Molecular Systems IVa. The Time-Dependent DFT One extension of the DFT is that related with the estimation of the excitation energies. This approach is based on the time dependence of the electron density, which is written as

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 r, t    f i  i r, t  , 2

(32)

i

where f i represents the orbital occupancy and the time-dependent orbitals,  i r,t  are the solutions of the time-dependent Kohn-Sham equations,

  1 2     r,t   2  







Axc  ,       i r,t   i i r,t , dr r  r  r,t   t

 r,t 

(33)



Table 3. Relative error for the electron affinity defined as 100(A(|F|=0.005 a. u.)A(|F|=0.000 a. u.))/ A(|F|=0.000 a. u.)

a

6-311++G**

aug-cc-pVQZ

Atomic Number

SVWN

PBE

B3LYP

SVWN

PBE

B3LYP

1

0

0

0

1

2

1

3

55

62

54

64

75

62

5

4

6

7

9

12

5

6

0

1

1

1

1

1

8

0

0

0

0

0

0

9

0

0

0

0

0

0

11

55

62

54

77

91

76

13

7

8

10

26

29

36

14

1

1

1

2

3

3

15

0

1

0

2

4

2

16

0

0

0

0

0

1

17

0

0

0

0

0

0

19

3

4

3

a

a

a

20

98

111

104

a

a

a

31

752

715

102

30

36

39

32

71

87

116

3

3

3

33

1

840

2

2

2

2

34

1

1

1

0

1

1

35

1

1

0

0

0

0

The aug-cc-pVTZ basis set is not reported for these atoms.

Effect of a Uniform Electric Field on Atomic and Molecular Systems where the external potential,

r,t , has now a time dependence [27-29].

495

As Exc, there is no

exact form for Axc , however it is usual to assume the adiabatic approximation,



Axc  ,   r,t 

  E  t ,  t. 

xc



(34)

 r,t 

Thus, in the case of an external potential, which varies slowly in time, Eq. (33) is just the time-dependent analogue of the time-independent Kohn-Sham equations. In the adiabatic approximation, there are not retardation effects because the self-consistent field responds instantaneously to any change on the charge density. Consequently the adiabatic approximation is formally a low-frequency approximation, like LDA for E xc , which was originally derived for charge densities varying slowly in space, the useful range of validity of the adiabatic approximation for real systems can only be determined on the basis of the formal conditions. Now let us consider a system initially in its ground state. The linear response to a perturbation, wt  , turned on slowly at some time in the distant past is just given by

 r,     i r Pij   j r ,

(35)

ij

where

Pij   

f j  f i

  wij     K ij ,kl Pkl      j   kl ,

   i

(36)

is written from the unperturbed molecular orbitals (j and l refer to unoccupied orbitals and i, k to the occupied ones, while  and  are spin indices). The coupling matrix K describes the linear response of the self-consistent field  , which corresponds to the last two terms of equation (33), to the changes in the charge density, SCF

K ij ,kl

v ijSCF    Pkl

1

   r   r  r  r  r  rdrdr  i



j

k



 2 E xc  , 

 l



(37)

   r   r   r  r  r  rdrdr    i

j

k

 l

It is important to remark that the functional derivative is evaluated on the unperturbed charge densities, and K is time-independent as a consequence of the adiabatic approximation, eq. (34). In order to obtain explicit expressions for xz-component of the polarizability, let us consider an electric perturbation

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Guillermo Nieto-Malagón, Julio M. Hernández-Pérez, Rubicelia Vargas et al.

wr, t   z z t ,

(38)

where  z t  is the time-dependent EF strength, the xz-component of the dynamic polarizability can be computed from

 xz    2

f i  f j 0

 x  Re P    /   , ji

ij

(39)

z

ij

where x ji represents the matrix elements of xˆ , in the basis of the unperturbed molecular orbitals. Note that only the real part of the response of the density matrix is needed for the present purposes. Solving Eq. (36) for P , and separating the real and imaginary parts, the polarizabilty can be expressed as

 xz    2x † S 1 / 2 Ω   2 1 S 1 / 2 z , 1

(40)

where

  ,  i ,k  j ,l 0  f k  f l  l   k 

S ij ,kl  and

 ij ,kl    ,  i ,k  j ,l  l   k   2 2

 f k

 K ij , kl

f

i

(41)

 f j  j   i 

 f l  l   k .

(42)

Equation (40) may be compared with the sum-over-states formula for the mean dynamic polarizability,

    trα     1 3

I

fI ,  2

(43)

2 I

where the difference

 I  E I  E0

(44)

represents a vertical excitation energy and

fI 

2 E I  E 0  0 xˆ I  3

2

 0 yˆ I

2

 0 zˆ I

2

 

(45)

is the corresponding oscillator strength. A comparison between the equations (40) and (43) shows that excitation energies and oscillator strengths can be obtained by solving

ΩFI   I2 FI

(46)

Effect of a Uniform Electric Field on Atomic and Molecular Systems

497

and

fI 

2  † 1 / 2  x S FI 3

2

 y † S 1 / 2 FI

2

2 2  z † S 1 / 2 FI  / FI . 

(47)

Naturally, the matrix Ω could become rather large, however, since one is usually interested in the lowest few excitations, efficient algorithms such as the Davison algorithm can be used [30]. Two important issues must be mentioned when one is interested on the excitation energies for atomic and molecular systems. 1) The exchange-correlation potential: it is wellknown that exchange-correlation potentials with the proper asymptotic behavior can give good estimations of the lowest lying excitation energies [31-33]. Unfortunately, many of the current approximations do not show the correct asymptotic behavior [31]. Thus, it is convenient to consider the approaches reported for the correction to the exchange-correlation tail. 2) For molecular systems the representation of the KS orbitals on a basis set functions is crucial, as it was discussed in section IIa, mainly to obtain a good description of the unoccupied orbitals and consequently give better estimations of the excitation energies.

IVb. UEF Effects on Excited States of Intramolecular Charge-Transfer Systems Without doubt the dual florescence exhibited by several systems is one of the most interesting optical phenomena. All arguments on the dual florescence explanations agree in the existence of an intramolecular charge-transfer (ICT) excited state exhibiting a large dipole moment [34]. Various mechanisms have been suggested to explain the formation of the ICT. Lippert's original hypothesis stems on the existence of a state reversal induced by a solvent [35]. Grabowski and coworkers proposed a more general model called twisted intramolecular charge-transfer (TICT) [36]. Also, there exist the RICT model, which involves a rehybridized of the cyano group. Zachariasse and coworkers suggest a pseudo-Jahn-Teller interaction [37,38]. Nowadays, experimental and computational studies are in favor of the TICT model. The TICT model assumes the existence of two conformers for the intramolecular chargetransfer excited state. After absorption, the excited state starts in a non-twisted conformation. Then, in a radiationless process, it goes on the hyper surface and transforms into a twisted structure, which is a local minimum on the potential energy surface (PES). Naturally, the role of the solvent on the ICT excited states is important and consequently in many experimental and theoretical studies this issue has been considered explicitly. In this chapter we adopt the following position about the role of the solvent on the dual fluorescence: It is well-known that a solvent or a molecular network induces an electrostatic potential on one molecule immersed within this environment. Thus we study the ICT excited states by immersing a molecular system within a UEF, in this way, one can tune the electrostatic potential on the excited states and on the whole PES exhibited by the excited states. We decide to use this approach in order to avoid the parameters involved with many solvent models. Additionally, it is clear that the molecular response to the external EF provides wide information about the properties of the system under study, molecular structure, optical properties, etc. Besides, it is known that an EF has a large influence on transition dipole moments, which determine absorption and emission spectral lines. In this section, we study

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the ICT under a UEF on three molecular systems; the 3-methyl-4-amino-benzonitrile, 3M4AB-CN; the 3,5-dimethyl-4-(N,N-dimethylamino) benzonitrile, TMAB-CN and the ethyl 4-(N,N-dimethylamino) benzoate, 4DMAB-COOEt (Figure 1). Experimental dual fluorescence for 4DMAB-COOEt has been reported, while TMAB-CN shows single activity. No data have been reported in literature for 3M4AB-CN, but theoretical studies predict single fluorescence. We must remark the work developed by Jamorski and coworkers who have studied, from a theoretical point of view [34,39,40], many systems in order to give criteria to establish the systems that could present dual florescence or does not. Thus, we have decided to show the role of a UEF on the optical properties of just three systems and we connect our results with those reported by Jamorski. Before the discussion around the EF effect on the excitation energies, we want to stress the impact of the exchange-correlation potential and the basis set functions on the excitation energies with an EF absence, as it was pointed out before. For this chapter we use the TimeDependent Density Functional Theory (TDDFT), summarized above, for the estimation of the excitation energies by using three exchange-correlation functionals, the PBE [21] and the B3LYP.[22] Furthermore, the asymptotic correction proposed by van Leeuwen and Baerends (LB94) [41] has been applied to the PBE. Two basis sets functions were coupled with these exchange-correlation potentials, the TZVP [42] and the same TZVP plus one diffuse function which we have included for each angular moment (TZVP+diff).

3M4AB-CN

4DMAB-COOEt

TMAB-CN

1234

Figure 1. Molecular systems analyzed in this chapter and the dihedral angle, 1234, considered for a scanning of the PES for the ICT.

In Table 4 we are reporting the lowest excitation energies, S1 and S2, of the three systems considered. Additionally to the excitation energies, we are including their corresponding oscillator strength (OS), all numbers reported were obtained after a full geometry optimization of each system. We see from Table 4 that the diffuse function gives narrower excitation energies than those obtained without this additional function. We found that the most important effect of the diffuse function was on the virtual orbital energies and consequently the excitation energies are reduced. By other side, the B3LYP exchange correlation potential predicts the biggest excitation energy values. This result is related with the spurious contribution of the self-interaction energy non-cancelled for the virtual orbitals in this potential [32], thus this potential will always give larger excitation energies than those obtained with local-multiplicative potentials.

Effect of a Uniform Electric Field on Atomic and Molecular Systems

499

Table 4. Lowest excitation energies (in eV), S1 and S2, for the 4DMAB-COOEt, TMABCN and 3M4AB-CN obtained with several exchange-correlation potentials and two basis set functions PBE TZVP+diff S1 O.S. S2 O.S.

3.92 0.46 4.11 0.02

S1 O.S. S2 O.S.

3.42 0.20 4.20 0.01

S1 O.S. S2 O.S.

4.27 0.04 4.55 0.32

PBE+LB94 TZVP TZVP+diff TZVP 4DMAB-COOEt 3.96 3.78 3.79 0.46 0.45 0.44 4.12 4.12 4.13 0.01 0.01 0.01 TMAB-CN 3.45 3.38 3.40 0.16 0.15 0.15 4.35 3.70 4.57 0.01 0.01 0.02 3M4AB-CN 4.30 4.26 4.27 0.03 0.02 0.02 4.62 4.55 4.57 0.33 0.32 0.31

B3LYP TZVP+diff TZVP 4.28 0.53 4.53 0.011

4.40 0.53 4.53 0.01

3.97 0.19 4.36 0.01

3.99 0.18 4.38 0.01

4.63 0.03 4.89 0.33

4.64 0.03 4.92 0.35

Figure 2. S0 (solid line), S1(dashed line) and S2(dotted line) as a function of the dihedral angle defined in Figure 1.

In Figure 2, the PES as a function of the dihedral angle, 1234, is presented for the 4DMAB-COOEt system with the PBE+LB94/TZVP+diff method. In order to generate this plot, we fixed the dihedral angle 1234 and relaxed the remained geometrical parameters, then on the resultant geometry we applied the TDDFT. From this figure it is clear that the 4DMAB-COOEt molecule exhibits an almost flat geometry (1234=10o) when it is on the ground state. However, if the system is excited to the S1 or S2 states the system exhibits a

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Guillermo Nieto-Malagón, Julio M. Hernández-Pérez, Rubicelia Vargas et al.

conformation where the dihedral angle changes in 90 degrees, showing in this way the twisting exhibited by this system. According to this figure, the emission spectra of the 4DMAB-COOEt system exhibits two bands, one around of 328 nm (E= 3.78 eV) and other around of 442 nm (E= 2.81 eV), which is in reasonable agreement with the experiment since for this system the experimental data are E=3.60 eV and E=2.48 eV (data reported in nhexane) [43]. Now, we want to discuss the effect of the UEF on the results presented in the last paragraph. In Table 5 the same quantities than those reported in Table 4 are depicted, the difference between these two tables is the UEF (0.005 a. u.) considered in Table 5. The data in this table were obtained by aligning the UEF on the dipolar moment of the molecule, it means that for each value of 1234 the dipole moment was recorded in order to align the UEF on this vector. By comparing Table 4 and Table 5, it is evident that the UEF induces a reduction on the excitation energies for the three systems and with any method considered in this chapter. Thus, the absortion spectra predicted by the TDDFT without the presence of a UEF will be moved to highger wavelengths when a UEF is considered. In the Figure 3 we are contrasting the PES with that presented in Figure 2. The conclusion obtained in the last paragraph can be applied also on the PES since we see a reduction on the excitation energies along whole PES and consequently for systems exhibiting the TIC, a translation is predicted on the emission spectra to higher wavelengths. Thus, according to the Figure 3 we predict a dual fluorescence, for the 4DMAB-COOEt, with the first pick around 353 nm (E= 3.52 eV) and the second one around 547 nm (E= 2.27 eV), if it is immersed in a UEF with strength of 0.005 a.u.

Figure 3. S0 (solid line), S1(dashed line) and S2(dotted line) as a function of the dihedral angle defined in Figure 1. The lines with markets correspond to the presence of a UEF of magnitud equal to 0.005 a. u.

Effect of a Uniform Electric Field on Atomic and Molecular Systems

501

Table 5. Lowest excitation energies (in eV), S1 and S2, for the 4DMAB-COOEt, TMABCN and 3M4AB-CN obtained with several exchange-correlation potentials and two basis set functions in presence of a UEF with strength of 0.005 a.u PBE TZVP+diff TZVP S1 O.S. S2 O.S.

3.66 0.46 3.96 0.01

3.69 0.46 3.98 0.01

S1 O.S. S2 O.S.

3.17 0.20 3.43 0.01

3.11 0.16 4.59 0.02

S1 O.S. S2 O.S.

4.20 0.05 4.41 0.35

4.27 0.04 4.47 0.33

PBE+LB94 TZVP+diff TZVP 4DMAB-COOEt 3.52 3.53 0.45 0.43 3.99 3.99 0.01 0.01 TMAB-CN 3.13 3.06 0.18 0.15 3.43 4.47 0.01 0.02 3M4AB-CN 4.20 4.24 0.04 0.03 4.37 4.41 0.31 0.30

B3LYP TZVP+diff TZVP 4.01 0.55 4.41 0.01

4.15 0.56 4.40 0.02

3.63 0.20 4.07 0.01

3.64 0.21 4.91 0.04

4.55 0.05 4.77 0.35

4.53 0.07 4.72 0.36

Conclusion The effect of a uniform electric field on atoms and molecules has been discussed within the Kohn-Sham approach. By the side of atoms we found interesting results when the total energy is written as a Taylor series for neutral and ion atoms. As a consequence of this analysis the electronegativity for atoms, will exhibit an increment when an atom is immersed in a uniform electric field and consequently its chemical reactivity will be different with regard to the absence of such a field. The molecular systems analyzed in this chapter are related with optical properties. The decrement on the excitation energies, when the molecules are under the action of a uniform electric field, could be mentioned as the principal result, which must be considered for future studies, in particular its link with the solvent effect.

Acknowledgments All calculations were done with the supercomputer installed in the Laboratorio de Supercómputo y Visualización en Paralelo at the Universidad Autónoma MetropolitanaIztapalapa. J. G. thank CONACYT for the partial funding to this work trough the project 60614.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24]

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[27]

Buckingham, A. D.; Fowler, P. W.; Hutson, J. M. Chem. Rev. 88, 1988, 963. Schiff, L. I. Quantum Mechanics. McGraw-Hill: Singapore, 1968. Rao, J. G.; Liu, W. Y.; Li, B. W. Phys. Rev. A 50, 1994, 1916. Kar, R.; Pal, S. Chemical Reactivity Theory: A Density Functional View, Chapter 25. CRC Press: Florida, 2009. Mahan, G. D.; Subbaswamy, K. R. Local Density Theory of Polarizability. Plenum Press: New York, 1990. Sttot, M.; Zaremba, E. Phys. Rev. A 21, 1980, 12. Garza, J.; Robles, J. Phys. Rev. A 47,1993, 2680. Kar, R.; Chandrakumar, K. R. S.; Pal, V. J. Phys. Chem. A 111, 2007, 375. Kar, R.; Pal, S. Theor. Chem. Acc. 120, 2008, 375. Parthasarathi, R.; Subramanian, V.; Chattaraj, P. K. Chem. Phys. Lett. 2003, 382, 48. Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules. Oxford University Press: New York, 1994. Kohn, W.; Sham, L. J . Phys. Rev. 140, 1965, A1133. Hohenberg, P.; Kohn, W. Phys. Rev. 136, 1964, B864. Roothaan, C. C. J. Rev. Mod. Phys. 23, 1951, 69. Perdew, J. P.; Schmidt, K. AIP. Conf. Proc. 577, 2001, 1. Görling, A.; Levy, M. J. Chem. Phys. 106, 1997, 2675. Dykstra, C. E. Ab-initio Calculations of the Structures and Properties of Molecules. Elsevier Science: Amsterdam, 1988. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 68, 1978, 3801. Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 105, 1983, 7512. Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 58, 1980, 1200. Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 77, 1996, 3865. Becke, A. D. J. Chem. Phys. 98, 1993, 1372; 98, 1993, 5648. a) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 72, 1980, 650. b) Curtiss, L. A.; McGrath, M. P.; Blandeau, J.-P.; Davis, N. E.; Binning, R. C.; Radom, L. J. J. Chem. Phys. 103, 1995, 6104. c) Glukhovtsev, M. N.; Pross, A.; McGrath, M. P.; Radom, L. J. J. Chem. Phys. 103, 1995, 1878. d) Blaudeau, J.-P.; McGrath, M. P.; Curtiss, L. A.; Radom, L. J. J. Chem. Phys. 107, 1997, 5016. a) Dunning Jr., T. H.; J. Chem. Phys. 90, 1989, 1007. b) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 100, 1994, 2975. c) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 98, 1993, 1358. d) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Dunning Jr., T. H. J. Chem. Phys. 110, 1999, 7667. e) Koput, J.; Peterson, K. A. J. Phys. Chem. A 106, 2002, 9595. Bylaska, E. J.; et. al., NWChem, A Computational Chemistry Package for Parallel Computers (Pacific Northwest National Laboratory, Richland, Washinton, 2007). Linstrom, P.J.; Mallard, W.G. Eds. NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standards and Technology: Gaithersburg MD, 20899, http://webbook.nist.gov, 2011. Casida, M.E. in Recent Advances in Density Functional Methods, edited by Chong, D. P. (World Scientific) 1993.

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[28] van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J. Comput. Phys. Commun. 118, 1999, 119. [29] Marques, M. A. L.; Ullrich, C. A.; Nogueira, F.; Rubio, A.; Burke, K.; Gross, E. K. U. Eds. Time-Dependent Density Functional Theory (Lecture Notes in Physics). Springer: Berlin, 2006. [30] Davidson E.R., J. Comput. Phys. 7, 1993, 519. [31] Garza, J.; Nichols, J. A.; Dixon, D. A. J. Chem. Phys. 112, 2000, 7880. [32] Garza, J.; Nichols, J. A.; Dixon, D. A. J. Chem. Phys. 113, 2000, 6029. [33] Garza, J.; Vargas, R.; Nichols, J. A.; Dixon, D. A. J. Chem. Phys. 114, 2001, 639. [34] Jamorski, C.; Lüthi, H.-P. J. Chem. Phys. 117, 2002, 4146; 117, 2002, 4157. [35] Lippert, E.; Lüppert, W.; Moll, F.; Nägele, W.; Boos, H.; Prigge, H.; SeiboldBlankenstein, I. Angew. Chem. 73, 1961, 695. [36] Rotkiewicz, K.; Grellmann, K. H.; Grabowski, Z. R. Chem. Phys. Lett. 19, 1973, 315; 21, 1973, 212. [37] Schuddeboom, W.; Jonker, S. A.; Warman, J. M.; Leinhos, U.; Kühnle, W.; Zachariasse, K. A. J. Phys. Chem. 96, 1992, 10809. [38] Zachariasse, K.; van der Haar, T.; Hebecker, A.; Leinhos, U.; Kühnle, W. Pure Appl. Chem. 65, 1993, 1745. [39] Jamorski, C.; Lüthi, H.-P. J. Chem. Phys. 119, 2003, 12852. [40] Jamorski, C.; Casida, E. M. J. Phys. Chem. B, 108, 2004, 7132. [41] van Leeuwen, R.; Baerends, E. J. Phys. Rev. A 49, 1994, 2421. [42] Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Can. J. Chem. 70, 1992, 560. [43] Retting, W.; Lippert, E. Ber. Bunsenges. Phys. Chem. 83, 1979, 692.

In: Theoretical and Computational Developments ... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 505-526

Chapter 19

A Q UANTUM P OTENTIAL B ASED D ENSITY F UNCTIONAL T REATMENT OF THE Q UANTUM S IGNATURE OF C LASSICAL N ON -I NTEGRABILITY Santanu Sengupta1, Munmun Khatua2 and Pratim Kumar Chattaraj2∗ 1 Department of Physics, C. V. Raman College of Engineering, Bidyanagar, Mahura, Janla, Bhubaneshwar 2 Department of Chemistry and Center for Theoretical Studies Indian Institute of Technology, Kharagpur, India

Abstract This chapter reviews the possible uses of quantum potential based approaches like quantum fluid dynamics and quantum theory of motion in understanding the quantum domain behavior of classical non-integrable systems. Quantum signatures of classical Kolmogorov-Arnold-Moser type transitions from toroidal to chaotic motion in different anharmonic oscillators, field induced barrier crossing as well as the chaotic ionization in Rydberg atoms are analyzed with the help of those theories. A quantum fluid density functional theory may be used to analyze the behavior of many-electron systems. The zero quantum potential limits of a couple of quantum anharmonic oscillators are also explored.

PACS: 05.10.-a Keywords: Quantum potential, Bohmian trajectories, H´ enon-Heiles oscillator, KAM transition

1.

Introduction

Chaotic behavior in classical mechanics has been identified by Henri Poincare [1] who while studying the three body problem came across nonperiodic orbits which were bounded but not approaching a fixed point. Seminal studies were carried out by Birkhoff in the three ∗

E-mail address: [email protected]

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Santanu Sengupta, Munmun Khatua and Pratim Kumar Chattaraj

body problem leading to famous Birkhoff series [2], Kolmogorov in relation to studies on the structure of turbulence and the famous Kolmogorov-Arnold-Moser (KAM) theorem [3][5] while studying transition to chaos, Mitchell Feigenbaum through his study of logistic maps and his discovery of universality in chaos and a host of other investigators. It was Einstein back in 1917 who raised the question of chaos in quantum systems as unlike the classical equations of motion the Schr¨ odinger wave equation is essentially linear the role played by non-linearity in quantum domain is not easy to discern. However, Bohr’s correspondence principle which basically considers classical domain behavior to be obtainable from the quantum description of a system for large quantum numbers or in the limit ~ → 0 offers the semi-classical propagator route to the study of quantum chaos. Gutzwiller [6] derived the celebrated Gutzwiller trace formula from the Van-Vleck semiclassical propagator and applied it to the anisotropic Kepler problem to successfully obtain [7] the quantum spectra from the classical periodic orbits of the system. Following his positive results the trace formula route to the study of quantum chaos has been successfully pursued by many investigators. A different route to the study of quantum chaos is through the random matrix theory (RMT) originally developed to characterize the spectra of complex nuclei. The energy levels in the spectra of quantum systems are theoretically predicted to be characterized by the statistics of the random matrix eigenvalues. Berry and Tabor [8] have shown that for systems with few degrees of freedom if the motion is regular non-chaotic the nearest neighbor spacing between energy levels obey a Poisson distribution in the quantum domain. It has been shown [9] that for quantum systems with time reversal symmetry exhibiting chaos in the classical domain the spectral energy levels, the normalized nearest neighbor spacing distribution is Gaussian in nature. The wave packet dynamics studies initiated by Heller and his group are at present one of the pre-eminent methods for investigating quantum dynamics for various systems. The cornerstone of wave packet dynamics is the Ehrenfest’s theorem which supplements the particle trajectory with the wave packet path. Heller and Tomsovic [10] have studied the dynamics and the eigenstates [11] of a stadium billiard which exhibits strong chaos in the classical domain. The wave packet dynamics method has been extensively used to look for signatures of chaos in quantum systems which are chaotic in the classical domain [12]. By far the most intuitive signature of dynamical chaos is the sensitive dependence of the state of the system upon its initial conditions. Implicit within this definition is the concept of trajectories which, however, does not exist in the true sense in the various approaches discussed above. There is, however, another interpretation of quantum mechanics which allows for the concept of trajectory in the quantum domain in a natural way. This is the Madelung’s quantum hydrodynamical equations which with some further modifications lead us to the Bohm’s trajectory approach to quantum mechanics. This review essentially focuses on some representative examples through the study of chaos and nonlinear systems carried out by our group using the quantum fluid dynamics and the Bohm trajectory concepts with a special emphasis on the zero quantum potential limit of these systems.

A Quantum Potential Based Density Functional Treatment ...

2.

507

Madelung-Bohm Interpretation of Quantum Mechanics

Soon after the introduction of the time dependent Schr¨ odinger equation Madelung obtained the hydrodynamic version [13] of it by using the ansatz ψ(x, y, z, t) = R(x, y, z, t)eıS(x,y,z,t)/~ in the Schr¨ odinger wave equation where R(x, y, z, t) and S(x, y, z, t) are both real functions with the added assumption R(x, y, z, t) ≥ 0. Equating the real and imaginary parts the following equations are obtained  2  ∂R2 R ∇S = −∇. (1) ∂t m ~2 1 2 1 ∂S = ∇ R− (∇S)2 − Vcl . (2) ∂t 2m R 2m The probability density function is then obtained as ρ = R(r, t)2, while the flux velocity of the probability current is v(x, y, z, t) = ∇S m and the probability current density is given as j = ρv. The first of the above two equations can be then rewritten as the continuity equation of the probability fluid ∂ρ(x, y, z, t) + ∇.j(x, y, z, t) = 0 ∂t

(3)

while the second of the two assumes the following Euler-Lagrange form 1 ~2 1 2 ∂S(x, y, z, t) =− (∇S)2 − Vcl + ∇ R. ∂t 2m 2m R

(4)

In this form the equation is the same as the classical Hamilton-Jacobi equation [14] save ~2 ∇2 R for the last term called the quantum potential defined as Vqu = − 2m R . The quantum potential essentially couples the two equations and is responsible for all quantum effects, accordingly the last equation is known as quantum Hamilton-Jacobi equation of motion [14]. The most significant aspect of the quantum potential was identified by David Bohm as being the reason behind the quantum non-locality, for along with the classical force obtained from the classical potential function we now have a quantum force which arises as a function of the probability amplitude R(x, y, z, t) which is nonlocal. So the motion of the probability fluid elements are correlated. The preceding equations are in the Eulerian frame of reference where the observer is on a point fixed in space, to move over to the Lagrangian frame of reference attached to the fluid element trajectory one has to use the well known relationship between the total time derivative of a function f (x, y, z, t) to its partial time derivative introducing the advective term in the process as ∂f (x, y, z, t) df (x, y, z, t) = + v · ∇f (x, y, z, t). dt ∂t So that the quantum Hamilton-Jacobi equation in the Lagrangian frame has the form  2 dS 1 ∂S = − (Vcl + Vqu ). dt 2m ∂x

(5)

(6)

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Again the quantum Hamliton-Jacobi equation in the Eulerian frame can be partially differentiated with respect to position to yield m

dv = −∇(Vcl + Vqu ). dt

(7)

The above equation in the Lagrangian frame of reference is the quantum equation of motion where the total force acting on a fluid element of mass m has the usual classical force −∇Vcl augmented by a quantum force −∇Vqu . The quantum equations of motion can be theoretically solved by computing the Vcl and Vqu at successive positions using the probability density ρ and the action function S obtained from the equation of continuity and the quantum Hamilton-Jacobi equation. We have some flexibility though in choosing the set of equations we prefer to solve for instead of the quantum equation of motion we can as well use the equation for the trajectory velocity in terms of the action function S v(r, t) =

3.

dr 1 = ∇S. dt m

(8)

Solution of the Quantum Hydrodynamic Equations

Takabayashi [15] in a series of papers in 1952 highlighted the hydrodynamic aspects of quantum mechanics. The quantum hydrodynamic equations offered novel prospects in the study of quantum mechanics of atom. Molecular interactions were identified by various workers around the late sixties and the seventies. In a series of papers Hirschfelder has introduced the idea of quantum streamlines using the quantum hydrodynamic interpretation [16] and has rederived and elaborated [17] the concept of quantum vortices first introduced by Dirac. The quantum vortices are shown to occur when in course of collision processes the incident and reflected wave functions interfere and the streamlines give rise to vortices around a node where ψ = 0. Vortices have been previously observed by them [18] when they integrated the quantum mechanical equations of motion for the collinear collision of a hydrogen atom and a hydrogen molecule. This was borne out subsequently by Kuppermann and his associates [19]. Wyatt had referred to them as quantum whirlpools. Consistent with the Madelung-Bohm theory defining v =H ∇S m as the fluid element velocity it was shown that the circulation of the velocity v viz. v · dr along any closed loop around a node is quantized if the loop does not pass through the node itself. A nonzero circulation implies a certain quantum of angular momentum associated with a quantum vortex, which has been shown to have experimentally observable effects. Weiner and Partom proposed the particle-in-cell (PIC) method [20] to solve the time dependent Schr¨ odinger equation in one spatial dimension using the hydrodynamic interpretation of Madelung for one dimensional potentials like a parabolic potential and a piecewise quadratic potential. In this method an ensemble of N particles are propagated whose linear density is taken as the wave function modulus squared |ψ(x, t)|2. This approach was further refined [21] and applied successfully to two dimensional potential systems. This method uses as its basis the fact shown by Levi-Civita [22] that if the flow and deformation of a continuous body is homogeneous then a material particle at its center of mass retains its position at all times. If the space under consideration is partitioned off into volume elements

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(cells) small enough so that their deformation can be considered to be approximately homogeneous then particle initially placed at the center of mass of these cells will remain at the center of masses of the respective cells even as the cells deform taking the particles along with it. The individual particles are propagated using the quantum equations of motion where the quantum potential is computed using the Wilkins’ method [23] of numerically differentiating functions whose values are provided at irregularly spaced points.

4.

Recent Developments in Quantum Hydrodynamics

The work carried out by Weiner and associates [20] is the first instance of what is now called the “synthetic” approach in the field of Bohmian quantum mechanics. In this approach the quantum hydrodynamic equations are solved on the fly generating the quantum trajectories [24] and the wave function at every time step is “synthesized” from the probability amplitude R(r, t) and S(r, t). This is in marked contrast to the work carried out by Dewdney and others [25] where the time dependent wave function is calculated using conventional numerical techniques to solve the TDSE and the Bohm trajectories are obtained from it. After the work of Weiner and his associates there was however a marked lull in investigations along those lines mainly because the algorithms used were not sophisticated and robust enough to treat more realistic problems with anharmonic potentials. The main impediments to a straightforward solution of the quantum hydrodynamical equations are the evaluation of the derivatives at irregular grid points required to compute the quantum force or the quantum potential and the “node problem” encountered while dealing with regions where ψ(r, t) = 0. The synthetic method was resurrected at the end of nineties with two different groups working independently in studies where they evolved ensemble of quantum trajectories on anharmonic potentials. The group of Lopreore and Wyatt [26] used function fitting techniques to find the spatial derivatives of the probability density and the fluid velocity using irregular grid data obtained in the Lagrangian picture, while the group of Rabitz [27] used finite element techniques to compute the same. Since then the fundamental problem of evaluating function derivatives for unstructured grids has seen some rapid progress with use of sophisticated techniques for curve fitting which have been applied in quantum trajectory computations. These include the moving weight least squares (MWLS) method applied to a one dimensional metastable mode interacting with a multi-mode harmonic reservoir [28]. Dynamic least squares (DLS) methods were used [29] to study the scattering of an initial Gaussian wave packet from an Eckert barrier. Distributed approximating functionals (DAFs) were applied [30] to integrate the equations of motion for quantum trajectories undergoing barrier scattering, while Nerukh and Fredrick have introduced [31] tessellation methods of curve fitting to solve the quantum hydrodynamical equations to solve the problem of scattering off an Eckert barrier augmented by harmonic potentials. A very detailed treatment of the various computational and theoretical aspects of the quantum trajectory approach to quantum mechanics is provided in the excellent book by Robert Wyatt [32].

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Quantum Hydrodynamics for Many Particle Systems

It must be noted that the extension of the hydrodynamic formalism for many particle systems is not straightforward but within the ambit of Hartree theory [15], Hartree-Fock theory [33],[34] and natural orbitals theory [35] this has been accomplished. Along with this the time dependent density functional theory (TDDFT) has been developed which formally allows any time-dependent quantity to be expressed as a functional of the charge density or the probability density ρ(r, t) and the current density j(r, t). It has been shown that the quantum hydrodynamics or quantum fluid dynamics and the time dependent density functional theory are formally equivalent [36],[37]. Quantum fluid density functional theory (QFDFT) [38]-[40] arises from the combination of the TDDFT and the QFD. Here an N-particle system is mapped onto a system of N noninteracting particles moving under the influence of a nonlinear one-body effective potential Vef f (r, t). In this case the governing equation is the generalized nonlinear Schr¨ odinger equation (GNLSE) [38]-[40]   ∂φ(r, t) 1 (9) − ∇2 + Vef f (r, t) φ(r, t) = ı 2 ∂t arrived at by combining Eq. (3) and Eq. (4) for many-particle systems and various time dependent density functionals. Here the explicit form of the Vef f (r, t) depends on the Hohenberg-Kohn functional [41]. The charge and current densities are expressed as ρ(r, t) = |φ(r, t)|2

(10)

j(r, t) = [φre ∇φim − φim ∇φre ]

(11)

and

The function φ(r, t) should not be considered as the wave function in general except for single particle systems. The dynamics of any physical entity can now be monitored using the functions ρ(r, t) and j(r, t). QFDFT has been successfully applied in studies of ion-atom collisions [38], atom field interactions [39] and dynamics of chemical reactivity parameters [40] such as electronegativity, chemical hardness, entropy and polarizability in chemical reactions.

6.

Quantum Domain Behavior of Some Chaotic Anharmonic Oscillators

enonIn the fields of stellar dynamics [42] and molecular dynamics [43] use of the H´ Heiles oscillator has been well known. The H´ enon-Heiles potential function has been used in the study of chaos in a molecular system using the method of avoided crossings [44]. The generalized Henon-H´ eiles system potential has been expressed as [45]   1 1 VHH (x, y) = (Ax2 + By 2 ) + λ Cx2 y + Dy 3 (12) 2 3

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where λ is a parameter that determines the non-linearity and non-integrability of the Hamiltonian system and can be constructed as a time-like parameter [46]. In the commonly used Henon-H´ eiles potential the value of λ = A = B = C = 1 and D = −1. However as pointed out by Gutzwiller [5] the value of λ can be varied keeping ~ = 1 ( in atomic units) or fix λ = 1 and change the value of ~. So the problem involves two parameters, λ and ~. While the former is a purely classical parameter the latter one takes care of the quantum nature of the problem. So that in the limit ~ → 0 the classical dynamics is recovered. It must be noted however that both these parameters cannot be varied at the same time. The generalized H´ enon-Heiles oscillator requires three physical constant parameters, viz., mass, frequency and time in order that the H´ enon-Heiles Hamiltonian represents a complete model of a nonlinear system. So λ no longer represents the degree of non-linearity but fixes the length scale [5] as it is inversely proportional to length. However in this case λ can assume any value that cannot be scaled to unity. The quantum fluid dynamics equations introduced before can be obtained [47]-[50] as Hamilton’s equations of motion [47]-[51] when the classical Hamiltonian is replaced by an appropriate energy functional from density functional theory (DFT) and ρ and −S are considered as canonically conjugate variables. The following Hamilton’s equations can then be used in deriving the QFD equations for the H´ enon-Heiles system. δρ δH[ρ, S] =− δS1 δt δH[ρ, S] δS1 = δρ δt where S1 is −S and the Hamiltonian functional H[ρ, S] is expressed as [47]-[49] Z Z Z 1 ∇ρ · ∇ρ 1 ρ(∇S)2dτ + dτ + VHH ρdτ H[ρ, S] = 2 8 ρ

(13)

(14)

Here the first term is the macroscopic kinetic energy, second term refers to the intrinsic kinetic energy whose functional form was suggested first by von Weizs¨ acker [52] and the last term is the potential energy. The corresponding variational principle for the Lagrangian in this case can be expressed as Z t2 δ L[ρ, S]dt = 0 (15) t1

where the Lagrangian functional is given in terms of the Hamiltonian functional introduced above as [47]-[49]  Z  ∂S1 L[ρ, S] = ρ dτ − H[ρ, S] (16) ∂t It may be noted that the classical nonlinear systems have been extensively studied [4],[5],[54] using phase space plots in terms of the two classical canonically conjugate dynamical variables. Numerical solution of the QFD equations for the H´ enon-Heiles system likewise allows one to study the quantum domain behavior in terms of the ρ versus S1

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plots. Numerical solutions carried out [47] revealed interesting features in phase plots of the canonically conjugate variables ρ and S1 with the system chaotic in the classical domain exhibiting certain self-similar fractal like structure a feature that is absent in the regular integrable domain thus geometrically characterizing chaos. Signatures of non-linearity are also observed in such entities as the Shannon entropy, density correlation and the macroscopic kinetic energy showing the effectiveness of the QFD route to quantum chaos [47]-[49]. As discussed earlier the fundamental feature of Bohm’s interpretation of quantum mechanics is the introduction of quantum trajectories of an ensemble of the system being studied such that the trajectory of each and every ensemble is governed by the nonlocal quantum (Bohm) potential augmenting the classical potential. It is this concept of trajectories that allows one to check if there is any “sensitive dependence on initial conditions”, a signature of chaos. Accordingly a phase space distance function has been defined [49, 50, 55, 56] which in a 4- dimensional phase space is expressed as D(t) = (x1 (t) − x2 (t))2 + (px1 (t) − px2 (t))2 + (y1 (t) − y2 (t))2 + (py1 (t) − py2 (t))2

1/2

(17)

where (x, px, y, py ) are the phase space coordinates of two trajectories numbered 1 and 2 obtained by solving the equation of motion viz. Eqn. (8), with slightly different initial values. Following the diagnostic tools used in the study of classical chaos a generalized Lyapunov exponent has been defined [55] while studying the Bohmian trajectories in a kicked rotor to quantify the exponential separation of two initially close quantum trajectories with time as   1 D(t) Λ = lim ln (18) D(0) D(0)→0 t t→∞

with a variant of this provided elsewhere [57]. The associated Kolmogorov-Sinai (KS) entropy function has also been introduced [55] to quantify chaos exhibited by the quantum trajectories as X H= Λ+ (19) Λ≥0

Using the preceding diagnostics it is now possible to define the quantum dynamics of the trajectories as “chaotic” in a given region of phase space if the associated KS entropy is large positive. These tools have been applied in our extensive studies [50] on anharmonic oscillators that exhibit chaotic as well as regular integrable dynamics depending on the parameter values and the initial conditions. A Gaussian wave packet representing the particle was propagated in the relevant potential by solving the time dependent Schr¨odinger equation. The wave packet propagation was carried out using the well known Cayley-Hamilton scheme for approximating the quantum propagator. Once the wave function was computed at different times the Bohm trajectories were obtained using Eqn. (8). To illustrate the results obtained using this approach phase space distance function defined in Eqn. (17) has been shown in Figures 1(a) and 1(b) for H´ enon-Heiles and Chang-Tabor-Weiss [58] (CTW) oscillators respectively. It is seen

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Figure 1(a). Plot of distance function (D(t)) as a function of time (t) in H´ enon-Heiles oscillator (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 74, 758, 1998. Copyright 1998 Indian Academy of Sciences.)

Figure 1(b). Plot of distance function (D(t)) as a function of time (t) in Chang-TaborWeiss (CTW) oscillator (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 74, 758, 1998. Copyright 1998 Indian Academy of Sciences.) that for the CTW oscillator, a classically nonlinear but integrable system, the phase space distance remains small all along while the H´ enon-Heiles oscillator in the non-integrable domain exhibit very large phase space distance values indicating that the initially close trajectories are no longer correlated. The KS entropies for the two cases (Figures 2(a) and 2(b)) as expected reflect this fact and more clearly brings out the chaoticity of the H´ enon-Heiles system where H abruptly increases and continues to increase as compared to the CTW oscillator for which the KS entropy remains more or less constant and close to zero. Chaos in Bohmian trajectories has been studied in various other physical systems like a charged particle in a periodically driven potential confined to a square which is the quantum cat map [55], the parabolic barrier [59], hydrogen atom subjected to an oscillating electromagnetic field [60] as well the quantum pinball monitored by measuring devices [25]. However, it has been observed that even when the underlying classical dynamics is not integrable with suitably chosen wave function quantum trajectories generated can be chaotic. In case of a particle in a two dimensional box the so called “square billiard” it

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Figure 2(a). Plot of KS entropy (H) as a function of time (t) in H´ enon-Heiles oscillator (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 74, 758, 1998. Copyright 1998 Indian Academy of Sciences.)

Figure 2(b). Plot of KS entropy (H) as a function of time (t) in Chang-Tabor-Weiss (CTW) oscillator (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 74, 758, 1998. Copyright 1998 Indian Academy of Sciences.)

was shown [61] that for the two different initial wave packets prepared from two different sets of wave functions the dynamics of the quantum trajectories generated were entirely different in character. Depending on the initial wave packet the dynamics was either integrable or chaotic. Makowski et al. [62] showed that a combination of two suitable chosen stationary state wave functions were sufficient to generate chaos in the quantum trajectories, while Frisk [63] has studied autonomous and non autonomous flows and how quantum trajectories can exhibit chaos even when the classical potential is zero. Billards in a right angled triangle exhibiting regular integrable behavior also reveal chaos [64] in the Bohmian trajectories. In such systems it is the quantum potential which is responsible for chaotic quantum trajectories. Studies on the rectangular billiards reveal [65] that the formation of wave function nodes plays an important role in the formation of quantum trajectories. The formation of quantized vortices that form around the wave function nodes and the nature of the quantum trajectories in their vicinity have been investigated [66] for hydrogen atom wave packets where it has been shown that quantum trajectories exhibit that spiral regular behavior trapped around a nodal line but are chaotic during the intervening period when such a trajectory leaves one nodal line and is trapped by another.

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515

External Field Induced Quantum Chaos in Rydberg Atoms

Rydberg atoms in an oscillating field has been a rich source of experimental and theoretical results related to quantum chaos. It has been aptly remarked [67] that they are “veritable gold mines for exploring the quantum aspect of chaos”. It is because that the hydrogen atom a simple example of solvable quantum mechanical system exhibits chaotic dynamics [68] like its classical mechanical counterpart the planetary Keplerian system when a hydrogen atom is subjected to a suitably strong external field. Investigations were carried out [69] to look for the signatures of chaos in a hydrogen atom excited with an externally applied oscillating electric field using the quantum trajectory concepts. Extensive theoretical [70] and experimental [71] efforts have been directed at the problem of quadratic Zeeman effect for this system, a beautiful review of which can be found in the reference [67]. In the quantum trajectory study of Chattaraj and Sengupta [69] the time evolution of the ground state (n = 1) and an excited electronic state (n = 20) of a hydrogen atom subjected to an oscillating electric field has been analyzed. While the temporal evolution of Shannon entropy and correlation function quite easily demarcate the chaotic nature of the dynamics for the n = 20 state as compared to the integrable n = 1 state it is through the Bohm trajectories that full flavor of the chaotic highly excited state can be discerned. Sensitive dependence on initial conditions is understood in terms of the phase space distance of two initially close trajectories and the related Kolmogorov-Sinai entropy. For the n = 1 state two initially close trajectories remain close for the long time the system is studied, however for n = 20 the distance between two initially close trajectories shows erratic oscillations becoming very large at times. The KS entropy remains practically constant at zero value for n = 1 while for the n = 20 state the KS entropy rapidly assumes a very large positive value identifying chaos. Iacomelli and Pettini [60] have studied the same model subject to an oscillating electromagnetic field through a classical model and a truncated N-level quantum model using quantum trajectory concepts respectively. Using Bohmian trajectories a chaotic transition in the quantum dynamics and a possible suppression of classical chaos has been observed. As pointed out by Holland [14] for many particle systems the quantum theory of motion derived from the conventional quantum mechanics does not treat the wave function and the particle at the same footing. In order to completely describe an N-particle system a wave function defined in the 3N-dimensional configuration space is required along with N point particles moving in a 3-dimensional configuration space to represent each of the ensemble elements. Possibility of a quantum trajectory approach for a many-particle system has been explored [72] within a quantum fluid density functional theory framework. As in the QFDFT formalism the “wave function” φ(r, t) is 3-dimensional even for an N-particle system both the particle and the wave aspects of the quantum theory of motion can be described in the 3-dimensional Euclidean space. Some progress in these directions have been made using the generalized nonlinear Schr¨ odinger equation (Eqn. 9) for the interaction of a helium atom with an external intense laser field, a problem of recent interest [73, 74] on the chaotic ionization of highly excited helium atoms.

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Figure 3. Classical stroboscopic plot of the momentum and position variables for the double-well oscillator ( at t = nT , n = 0, 1, 2, . . ., T = 2π/ω0 ) for two different initial conditions, (x, px) viz. (a) (−2.0, 0.0) and (b) (2.0, 0.0) with the set of parameters, (a = −0.5, b = 10.0, c = 10.0 and ω0 = 6.07) (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 76, 1371, 1999. Copyright 1999 Indian Academy of Sciences.)

Figure 4. Quantal phase space trajectories for the double-well oscillator for two different initial conditions, (x, px) viz. (a) (−2.0, 0.0) and (b) (2.0, 0.0) with the set of parameters, (a = −0.5, b = 10.0, c = 10.0 and ω0 = 6.07) (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 76, 1371, 1999. Copyright 1999 Indian Academy of Sciences.)

8.

Quantum Chaos Associated with the Barrier Crossing in a Double-Well Potential in Presence of an External Field

Barrier penetration in a double well potential has far reaching ramifications in several areas of chemical physics [75] and has been intensively studied [53],[54]. It has been ob-

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served that classical chaos as well as quantum tunneling are present simultaneously leading to the coherent oscillatory nature of the diffusion between two stable KAM tori. Such a system has been studied [76] using quantum trajectory methods to analyze the quantum domain analogue of the classical domain chaotic dynamics associated with the barrier penetration by a particle subject to an oscillating monochromatic external field with increasing amplitude. The classical Hamiltonian of a particle in a double-well oscillator excited with an oscillating electric field is expressed as H=

p2x + ax4 − bx2 + cx cos(ω0 t) 2m

(20)

For given sets of parameter values and chosen initial values of position and momentum the classical equations of motion can be solved easily to generate various trajectories. Phase space may comprise a stable integrable region bounded by KAM surfaces or a chaotic region, depending on the choice of initial position and momentum coordinates. The associated time dependent Schr¨ odinger equation has been solved [76] to obtain the time dependent wave function from which one computes the Bohmian trajectories using Eqn. (8). Figure. 3 exhibits the classical stroboscopic plots of the momentum and position variables ( obtained at times t = nT , where n = 0, 1, 2, . . . and T is the driving period given by T = 2π/ω0 ) for two different initial conditions, (x, px) viz. (a) (−2.0, 0.0) and (b) (2.0, 0.0) with the set of parameter values chosen following the work of Lin and Ballentine [53], (a = −0.5, b = 10.0, c = 10.0 and ω0 = 6.07). Under these conditions the case (a) represents an integrable system while case (b) gives rise to chaotic motion as depicted in Figure. 3. The corresponding plots shown in Figure. 4 obtained from the quantum trajectory generated seem to reflect the classical dynamics with a regular cantorus-like structure [77] the quantum equivalent of the classical KAM torus while the rest of the points are scattered over the phase space echoing the classical chaotic situation.

Figure 5(a). Plot of KS entropy (H) as a function of time (t) associated with the classical motion for the double-well oscillator in presence of external field with c = 10, c = 20, and c = 40 with initial condition (x0 = −2.0, px0 = 0.0) (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 76, 1371, 1999. Copyright 1999 Indian Academy of Sciences.)

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Figure 5(b). Plot of KS entropy (H) as a function of time (t) associated with the quantal motion for the double-well oscillator in presence of external field with c = 10, c = 20, and c = 40 with initial condition (x0 = −2.0, px0 = 0.0) (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 76, 1371, 1999. Copyright 1999 Indian Academy of Sciences.)

To study the breakdown of KAM torus [78] and the possible suppression of ensuing chaotic dynamics the double well system was studied with the same initial conditions as in case (a) which originally corresponded to regular integrable motion but now with different higher amplitudes, viz. c = 20 and 40 of the driving field. Time evolution of the KS entropies associated with the classical motion and the same obtained from the quantal trajectories is presented in Figures. 5(a) and 5(b) respectively while the corresponding phase volumes, defined as [78],[79] Vps (t) =

q h(x − hxi)2 ih(px − hpx i)2 ih(y − hyi)2 ih(py − hpy i)2 i

(21)

are shown in Figure. 6. It is observed that while the classical system exhibits chaotic nature for c = 20 the quantum analogue behaves regularly similar to the situation for c = 10. This confirms clearly a suppression of chaos in quantum domain in this system. Such effect has also been noted [80] in an N-component φ4 -oscillator in the presence of an external field. It has been explained that the effect is due to the removal of a hyperbolic fixed point from the effective quantum dynamics which is a major source of chaotic behavior in the classical case. Both classical and quantum domain behaviors are, however, chaotic for c = 40. Polavieja [81] has also observed exponential divergence of initially close quantum trajectories while studying a driven quartic oscillator in the coherent state representation. Interestingly these observations are somewhat opposite to those of Makowski [62], Frisk [63] and Sales [64] who have shown that the quantum trajectories exhibit chaos even when the classical dynamics do not exhibit any chaos, which can be perhaps termed as “classical suppression of Bohmian chaos”. Clearly this aspect of chaos arising out of the classical potential itself or that arising out of the convoluted nature of the quantum potential needs further study.

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Figure 6. Plot of phase volume (Vps) as a function of time (t) associated with the quantal motion for the double-well oscillator in presence of external field with c = 10, c = 20, and c = 40 with initial condition (x0 = −2.0, px0 = 0.0) (Reprinted with permission from P. K. Chattaraj, S. Sengupta and A. Poddar, Curr. Sci. 76, 1371, 1999. Copyright 1999 Indian Academy of Sciences.)

9.

The Zero Quantum Potential Limit of a Quantum Anharmonic Oscillator The polar form of time dependent wave function in two dimension can be written as ψ(x, y, t) = R(x, y, t)eiS(x,y,t)

(22)

where R(x, y, t) and S(x, y, t) are real valued functions. The probability density has the form ρ(x, y, t) = R(x, y, t)2 where R(x, y, t) > 0. Substitution of the polar form of the wave function in the time dependent Schr¨odinger equation ∂ψ(x, y, t) ˆ Hψ(x, y, t) = i ∂t

(23)

yields the continuity equation ∂ρ(x, y, t) =− ∂t



∂ ∂ + ∂x ∂y



→ {ρ(x, y, t)− v (x, y, t)}

(24)

→ where − v (x, y, t), the velocity of the probability fluid given by − → v (x, y, t) =



∂ ∂ + ∂x ∂y



S(x, y, t) = Re



−i 5 ψ(x, y, t) ψ(x, y, t)



(25)

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Figure 7. Phase portraits for classical, quantum and adjusted quantum H´ enon-Heiles oscillators subjected to external electrical field of strength g = 0.0, 0.5, 1.0 and frequency ω = 0.3. and the quantum Hamilton-Jacobi equation turns out to be −

∂S(x, y, t) = (5S(x, y, t))2 + V (x, y, t) + Q(x, y, t) ∂t

(26)

where V (x, y, t) is the classical potential and Q(x, y, t) is the Bohm quantum potential having the form Q(x, y, t) = −

52 ρ1/2 (x, y, t) 2ρ1/2(x, y, t)

(27)

In the present study we have considered the H´ enon-Heiles oscillator in presence of an axial external field in the y-direction with the classical Hamiltonian of the form   1 2 1 2 x2 2 2 2 H(x, y) = (px + py ) + (x + y ) + λx y − − gycos(ωt) (28) 2 2 3 where the first term is the kinetic energy, the second and third terms are the harmonic and anharmonic parts of the potential energy respectively, and the last term represents the interaction of the oscillator with the axial field and λ is the non-linearity and non-integrability parameter having the value 0.1118034 [42],[75],[77] and the system is quantized with ~ = 1, m = 1. Here the effective potential is given as Vef f (x, y, t) = Vclassical − Q(x, y, t)

(29)

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Figure 8. Phase portraits for classical, quantum and adjusted quantum H´ enon-Heiles oscillators subjected to external electrical field of strength g = 0.0, 0.5, 1.0 and frequency ω = 0.7. so that the zero quantum potential limit is retrieved and we call it the adjusted quantum case. The mesh sizes adopted are 4x = 4y = 0.08 and 4t = 0.128 and 4t = 0.0128 for classical and quantum cases respectively. The calculations have been carried out over a spatial grid of −10 ≤ x, y ≤ 10 for about 105 time steps. The trajectory is determined by solving Eqn.(25) for the initial condition with center (x0 , y0 , px0 , py0 ) at (0.390625, 0.390625, 0.0, 0.0). Here the field strength values used are g = 0.0, 0.5, 1.0 with field frequency ω = 0.3, 0.7. The relevant classical and quantum phase plots are provided in Figures 7 and 8 respectively. Figure 7 represents the classical, quantum and adjusted quantum phase portraits (y vs py ) for different field intensities with ω = 0.3 and Figure 8 represents the same with ω = 0.7. In both the cases for g = 0.0 the classical phase portraits show smooth tori reflecting the integrability of the system. As the field strength increases the tori gradually become diffused indicating loss of integrability. The quantum phase portraits show similar transition from integrability to chaos with increase in field strength for both ω = 0.3 and 0.7. For the adjusted quantum case at g = 0.0 the phase portraits are similar to the classical and quantum cases for both ω = 0.3 and 0.7. The adjusted quantum phase portraits lie somewhere in between the classical and quantum cases in most of the situations. This highlights the fact that the zero quantum potential limit of a quantum anharmonic oscillator is more classical like.

Conclusion It is seen that the quantum trajectory methods based on quantum potential based approaches to quantum mechanics offer interesting insights into the quantum domain behavior

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of classically chaotic systems. At the same time it throws up new challenges in interpreting the results obtained. Chaos as observed in quantum trajectories may have a classical root as determined by the classical potential or a quantum potential root or even arises due to a combination of the two. Further so far as the trajectories are “guided” by the probability density the chaos arising out of the complicated nature of the quantum potential might well be an artifact of the particular wave packet chosen. How to resolve these issues or even to ensure that it is indeed an issue that can be meaningfully raised is something that can be revealed only after further work in these directions. It is gratifying to note that a classical like situation is retrieved in the zero quantum potential limit.

Acknowledgments We thank CSIR, New Delhi for financial assistance and Drs. A. Poddar, B. Maiti and U. Sarkar for their help in various ways. PKC thanks the DST, New Delhi for the Sir J. C. Bose Fellowship.

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[77] J. D. Hanson, J. R. Cary and J. D. Meiss, J. Stat. Phys. 39, 327, 1985; J. D. Meiss, Journal of Particle Accelerators, 19, 9, 1986; R. S. McKay and J. D. Meiss, Phys. Rev. A, 37, 4702, 1988. [78] R. Graham and H. Hohnerbach, Phys. Rev. A., 433, 3966, 1991; Phys. Rev. Lett., 64, 637, 1990; S. Chaudhuri, G. Gangopadhyay and D. S. Ray, Indian J. Phys., 69B, 507, 1995. [79] M. D. Feit and Jr. J. A. Fleck, J. Chem. Phys., 80, 2578, 1984; J. D. Hanson, J. R. Cary and J. D. Meiss, J. Stat. Phys. 39, 327, 1985; J. D. Meiss, Journal of Particle Accelerators, 19, 9, 1986; R. S. McKay and J. D. Meiss, Phys. Rev. A, 37, 4702, 1988; W. A. Lin and L. E. Ballentine, Phys. Rev. Lett., 65, 2927, 1990. [80] L. Casetti, R. Gatto and M. Modugno, Phys. Rev. E, 57, R1223, 1998. [81] G. C. Polavieja, Phys. Rev. E, 55, 1451, 1997.

In: Theoretical and Computational Developments... ISBN: 978-1-61942-779-2 c 2012 Nova Science Publishers, Inc. Editor: Amlan K. Roy, pp. 527-548

Chapter 20

P ROPERTIES OF N ANOMATERIALS FROM F IRST P RINCIPLES S TUDY Arup Banerjee1∗, Aparna Chakrabarti2 , C. Kamal2 and Tapan K. Ghanty3 1 BARC Training School at RRCAT, Raja Ramanna Centre for Advanced Technology, Indore, India 2 Indus Synchrotrons Utilization Division, Raja Ramanna Centre for Advanced Technology, Indore, India 3 Theoretical Chemistry Section, Chemistry Group, Bhabha Atomic Research Centre, Mumbai, India

Abstract In this chapter we review the theoretical results for the various ground-state and optical response properties of variety of nano-clusters, nano-tubes and nano-cages. In particular we consider nano-clusters made of alkali metal atoms (Nan and Kn ), noble metal gold atom doped with alkali and other coinage atoms (Au19 X, X = Li, Na, K, Rb, Cs Cu, Ag), and mixture of gallium and phosphorus atoms (Gan Pn ). We study the properties of nano-cages, namely C20 , C60 , C80 , and C100 and carbon nano-tubes of various lengths and diameters. The optimized structures and the ground-state properties have been obtained by employing density functional theory and its time dependent counterpart time-dependent density functional theory has been used to obtain optical response properties. Using the optimized structures we calculate ground-state properties like binding energy, HOMO-LUMO gap, ionization potential, and electron affinity to characterize the stabilities of these nano-structures. We have calculated linear polarizability of the above mentioned nano-structures and studied the size-to-property relationship for these systems. The van der Waals coefficient related to the dipole-dipole interaction between two polarizable systems, which plays an important role in the description of cluster-cluster collision and in characterizing the orientation of clusters in bulk matter. We calculate van der Waals coefficient by using Casimir-Polder expression which relates this coefficient to the frequency dependent dipole polarizabilities at imaginary frequencies. Here we discuss in detail the effect of exchange-correlation ∗

E-mail address: [email protected]

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PACS 36.40.-c, 36.40.Cg, 36.40.Mr, 36.40.Vz, 61.46.Fg, 32.10.Dk, 31.15.EKeywords: Atomic Clusters, Carbon nano-structures, Polarizability, van der Waals coefficients

1.

Introduction

There is a recent surge in both theoretical and experimental research activities devoted to understand the nano-structures - materials with ultra-small dimensions of the order 1 100 nm ushering in birth of a new field, now commonly known as nanoscience. This field has been attempting to address various important issues ranging from fundamental ones associated with physical and chemical properties of various nano-structures to exploring their various technological applications for which the word nanotechnology is frequently used. The purpose of nanoscience and nanotechnology is to comprehend, control, and manipulate objects of a few nanometers in size. Among nano-objects, nano-clusters and nano-tubes occupy a very vital place as they are the basic building blocks of many nanodevices. Nano-clusters are complex many-electron systems in between atoms or molecules and the bulk. They are aggregate of atoms or molecules that are too large to be referred to as molecules and too small to be treated like bulk. Being small objects, nano-clusters, nano-cages, and nano-tubes have very high surface to volume ratio. For example, even for a cluster containing 2000 atoms, around 20% of the atoms lie on the surface. The atoms located at the surface see a different environment from those in bulk. It is due to this reason that the properties of these nano-clusters are markedly different from those of bulk and also from their constituent atoms. Therefore, they provide an excellent way to study and understand how physical and chemical properties evolve in the transition from an atom to a molecule to a cluster to small particles and finally to a bulk solid. Clusters are different from molecules as they do not have fixed geometry or composition. For example, the water molecule contains one oxygen and two hydrogen atoms, which are placed at a well defined angle to each other. On the other hand, a cluster of alkali metal atom or even water clusters may contain any number of constituent particles and for a given size, exhibit a variety of morphologies (for example, see Figure 1). The properties like geometric structure, binding energy (BE), ionization potential (IP), and melting temperature of small clusters containing few hundreds of particles show strong size-dependent behavior. On the other hand, larger clusters, with many thousands of atoms have smoothly varying behavior, which tends to the bulk limit as the size increases. Similarly, the nano-tubes and nano-cages, (see Figure 2) mainly due to their large surface to volume ratio, show variety of properties which are largely different from their bulk counterpart. The rapid progress in the research work on nanoscience in recent times has been due to the availability of experimental methods for producing nano-objects in very controlled fashion and also characterizing them unambiguously along with the development of sophisticated theoretical tools to handle such finite fermion systems at the ab initio level. We note here that nanoscience is a field which requires a synergy between the experiment and theory for characterization as well as for understanding the properties of clusters. Our research

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20_0

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20_3

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20_2

20_4

Figure 1. Various isomers of 20-atom potassium cluster K20 with different symmetries. The structure K20 4 is the most stable isomer and K20 0 represents the least stable one. work in the area of nanoscience is primarily aimed at theoretical understanding of various ground-state electronic and optical response properties of variety of nano-objects. In this connection we emphasize here that nano-objects, even a cluster of few atoms is a complex many-electron system especially from computational point of view. This complexity arises from the electron-electron interaction which plays a dominant role in the physics of these nano-objects. Therefore, for correct description of these systems it is necessary to employ methodologies which take into account effect of electron-electron interaction accurately and this is also essential for correct determination of optical response properties. Density functional theory (DFT) [1] and its time dependent counterpart time-dependent DFT (TDDFT) [2] provide efficient ways to perform both model-based and ab initio calculations taking into account electron-electron interaction and that is why these two methods have found wide applications in determining electronic structures and response properties of atoms, molecules, clusters and solids. We wish to mention here that there also exist a number of wave-function-based quantum chemical methods for calculating electronic structure and optical response properties at varying level of sophistication taking electron correlation into account. However, these methods turn out to be computationally very expensive for systems containing large number of atoms. On the other hand, DFT/TDDFT with suitable choice of exchange-correlation (XC) potential can be of very high accuracy and significantly less cumbersome from the computational point of view, making it possible to handle much larger clusters. For example, calculations on metal clusters including transition metals and noble metals with more than hundred atoms have been reported in the literature [3].

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Figure 2. Some of the nano-tubes and nano-cages used in the present work. Carbon nanotubes (CNT) are terminated by hydrogen atoms

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Although DFT/TDDFT methods are exact in principle but calculations by these methods are performed with approximate forms for the XC potentials. The choice of XC potential has to be tested for each system. In next section we discuss the effect of different exchangecorrelation functionals in determining some of the properties. We carry out studies on nano-clusters and nano-tubes by performing calculations of various properties for metal (Nan , Kn , and Au19 X, X = Li, Na, K, Rb, Cs, Cu, Ag), semiconductor (Gan Pn ), carbon (C20 , C60 , C80 and C100 ) clusters and carbon nano-tubes (CNT). For each system we first carry out DFT-based geometry optimization calculation to obtain energetically stable structures. Using these optimized structures we calculate ground-state properties like BE, HOMO-LUMO gap (∆EHL ), IP, and electron affinity (EA) to characterize their stabilities. Optical response properties like dipole polarizability and VIS-UV optical absorption spectrum are routinely measured for characterization of nano-systems. Motivated by this we devote our significant research endeavor on calculating these response properties by employing TDDFT. The van der Waals coefficient C6 is associated with the interaction potential which decays as R−6 (where R is the intermolecular distance) and it describes the dipole-dipole interaction between the two polarizable systems. We calculate van der Waals coefficient C6 by using Casimir-Polder [4] expression which relates this coefficient to the frequency dependent dipole polarizabilities at imaginary frequencies which can be computed by employing TDDFT. According to this expression, the orientation averaged dispersion coefficient C6 between two molecules A and B is given by[4, 5]. 3 C6 (A, B) = π

Z



dω α ¯ A (iω)¯ αB (iω)

(1)

0

where α ¯ j (iω) is the isotropic average dipole polarizability of the j-th molecule and j j j αxx (ω) + αyy (ω) + αzz (ω) . α ¯ (ω) = 3 j

(2)

In the above expression αxx (ω), αyy (ω) and αzz (ω) are diagonal elements of the dipole polarizability tensor. The Casimir-Polder integral Eq. (1) has been evaluated by employing thirty point Gauss-Chebyshev quadrature scheme. The convergence of the results has been checked by increasing number of frequency points. We employ TDDFT to calculate the frequency dependent dipole polarizability for a range of frequencies and then use Casimir-Polder expression to calculate C6 for interaction between various pairs of clusters of different sizes and chemical compositions. The DFT and TDDFT based ab initio calculations of various ground-state and response properties of the above-mentioned systems have been performed by employing Amsterdam Density Functional (ADF) program package [6] and GAMESS electronic structure code [7] has been employed for carrying out post-Hartree-Fock MP2 and CCSD(T) calculations. For each type of nano-cluster we optimize geometries of several possible isomers (wherever applicable) of the clusters. The geometry optimizations of all the systems have been performed through DFT based calculations by employing appropriate basis set. For most of the systems we use triple-ξ Slater-type orbital (STO) basis set with two added polarization functions (TZ2P basis set of ADF basis set library).

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Results Effect of Exchange-Correlation Potentials on Static Polarizability

We note here that a TDDFT based response property calculation requires approximating the XC functional at two different levels. The first one is the static XC potential needed to calculate the ground-state Kohn-Sham (KS) orbitals and their energies. The second approximation is needed to represent the XC kernel fXC (r, r′ , ω) which determines the XC contribution to the screening of an applied field. For the XC kernel, we use reasonably accurate adiabatic local density approximation (ALDA)[2]. Therefore, the accuracy of the results for response properties crucially depend on the nature of XC potential/functional employed for the ground-state calculations especially its behavior in the asymptotic region [8, 9]. In order to study the effect of XC potential we have calculated the static dipole polarizability of various clusters made of Gallium Phosphide (Gan Pn ) as well as alkali metals sodium (Nan ) and potassium (Kn ) atoms, by employing various ab initio wave function based and DFT/TDDFT methods. A systematic investigation is carried out to analyze the performance of different XC functionals used in DFT/TDDFT method in determining static dipole polarizability of these clusters. The results for most stable isomers of Gan Pn show that, the DFT/TDDFT method with different XC functionals underestimate the values of polarizability in comparison to the results of MP2 method, which is a wave function based method and is known to yield accurate results [10]. Among the several XC functionals, the performance of XC functional of Perdew-Burke-Ernzerhof [12](PBE) within generalized gradient approximation (GGA) has been the best when compared to the results from MP2 method. In comparison to the results produced by the MP2 method, the values of polarizability obtained by the DFT/TDDFT calculation with a model potential - statistical average of orbital potential (SAOP) - possessing correct behaviors both in the asymptotic and inner regions of the molecule [13, 14], are found to be worse than those obtained with the LDA and GGA XC functionals. In this connection we mention that the correct asymptotic behavior of SAOP is capable of yielding quite accurate results for the low lying excited states[11]. However, for high lying excited states it is not very accurate. This may be one reason for the discrepancy between the results for polarizability obtained via TDDFT (XC SAOP) and MP2 methods. On the other hand the closeness of LDA and GGA results to MP2 data may be attributed to the following. The LDA and GGA XC potentials underestimate the contributions to the polarizability arising from transition to bound Rydberg type states and overestimate those from continuum. The cancellation of errors in these two contributions to the polarizability sometimes yields good results for it accidentally [11, 10]. Therefore, the closeness of LDA and GGA results with MP2 values in some systems, as for example here, may be fortuitous. However, no such cancellations of errors occur when excited states are obtained with SAOP XC potential. Hence it may be due to this lack of cancellation together with inaccuracy in predicting high lying excites states by SAOP that leads to results for the polarizability which deviate more from MP2 data than the corresponding LDA and GGA values in case of Gan Pn clusters. Now it is important to note here that unlike III-V semiconductor clusters, for alkali metal clusters of Na and K, the values of polarizability obtained by employing SAOP are

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very close to data from both accurate wave function based methods (MP2 and CCSD(T)) [15] and experiments[16, 17]. Moreover, for these systems the SAOP results are also significantly higher than both LDA and GGA values. For these systems, only few low lying excited states contribute to the polarizability as these few states are sufficient to saturate the oscillator strength of excitations. It is exactly these low lying states which are well reproduced by SAOP yield very accurate values of polarizability of alkali metal clusters. Hence, our studies on clusters made of different classes of elements clearly show that the choice of XC functional for the ground-state calculation should be made judiciously depending on the system under consideration for obtaining accurate results for the response properties.

2.2.

Interesting Size-to-Property Relationship

2.2.1. Polarizability and van der Waals Coefficients for Alkali Metal Atom Clusters Having discussed the role of XC functional on the accuracy of the results for polarizability we now present and discuss in detail various aspects of the static polarizability (α ¯ (0)) of different clusters using DFT/TDDFT. In this subsection, we present the data for the alkali metal atom clusters, consisting of Na and K atoms [16, 17]. The results are displayed in Figures 3 and 4 for Nan and Kn clusters respectively. The upper panel in Figure 3 shows the plot of average static polarizability of Na atoms clusters, Nan , n being an even number and varying from 2 to 20. Along with the calculational results we also plot the corresponding experimental results [18]. The data in Figure 3 clearly shows that though the results obtained by ab initio method are slightly lower than the corresponding experimental data, the two data, specifically the increasing trend of the α with the number n match very well with each other. The clusters under present study are essentially non-spherical in nature, consequently the polarizability tensors are anisotropic. It is then natural to investigate how the anisotropy in polarizablity evolves with the size of the cluster. For this purpose, we carry out calculations of anisotropy in polarizability given by "

3T rα2 − (T rα)2 |∆α| = 2

#1/2

(general

axes)

(3)

where α is the second-rank polarizability tensor. The middle panel of Figure 3 shows the plot of anisotropy ∆α of Na atom clusters as a function of n. It is interesting to see that there is a clear size (or number)-to-property relationship. The clusters with magic number of atoms (n = 2, 8, 20) show a very small anisotropy in α, which corroborates with the fact that the clusters with magic number of atoms are more symmetric in structure, naturally leading to a smaller value of anisotropy compared to the other clusters with non-magic number of atoms. The third panel shows the plot of α ¯ (0) as function of total cluster volume. Note that the volume of the clusters has been obtained by using the method of Tomasi and Persico [19]. It is interesting to see that a good linear fitting with the volume exists with a correlation coefficient close to one. This indicates that the α ¯ (0) value exhibits a linear dependence on the volume of the cluster. In Figure 4, we show the same properties for Kn , n being from 2 to 20. Top, middle and bottom panels show α ¯ (0), ∆α and the volume dependence of α ¯ (0), respectively. It is observed that the results for potassium clusters are very similar to those of sodium clusters.

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Figure 3. Results for static polarizability and anisotropy in polarizability for sodium atom cluster Nan . Upper panel shows comparison of our theoretical results with MP2 level calculation [15] and experimental data from Ref. [18] for clusters with up to 20 atoms. Middle panel shows the anisotropy. The X-axis for upper and middle panels is the number of atoms in the cluster. Bottom panel shows polarizability as a function of volume in (a.u)3 . Lines are guides to the eyes. Linear fit to volume is represented by black line. After finding the trend for α and ∆α with cluster size and volume we now move on to discuss about the van der Waals interaction between these clusters. To this end we calculate lowest order van der Waals coefficient C6 and also study its scaling behavior with the volume of the clusters (size-to-property relationship). We demonstrate that both for Na and K clusters C6 varies quadratically with the volume of the clusters [16, 20]. It is important to note that this scaling behavior is applicable for both spherical as well as non-spherical geometries. The scaling behavior of C6 can be understood by considering the expression for it within London approximation. Within this approximation the dispersion coefficient C6 between two molecules is represented in terms of an effective or a characteristic frequency ω1 and the static polarizability α ¯ (0) as C6 =

3ω1 α ¯ (0)2 4

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Figure 4. Results of static polarizability and anisotropy of polarizability for K atom clusters. Upper panel shows comparison of our theory (both DFT/TDDFT and MP2) with the available experimental data (for n=2, 8 and 20) from Ref. [18] for clusters with up to 20 atoms. Middle panel shows the anisotropy. The X-axis for upper and middle panels is the number of atoms in the cluster. Bottom panel shows polarizability as a function of volume in (a.u)3 . Lines are guides to the eyes. Linear fit to volume is represented by black line.

it is expected C6 scales quadratically with the volume. To check the validity of the abovementioned assumption, we calculate the characteristic frequencies ω1 of potassium clusters by employing Eq. (4) using the values of C6 and α ¯ (0) obtained with SAOP [20]. We find that these frequencies lie within a range from 0.061 to 0.078 a.u. around the mean of 0.070 a.u.. This narrow distribution of ω1 values around its mean value warrants the above mentioned assumption and consequently a very good linear fit between C6 and square of the volume of the cluster has been obtained. This is elucidated in Figure 5 where we plot C6 for both Nan (upper panel) and Kn (lower panel) clusters obtained by using SAOP as a function of (volume)2 along with the least square fitted line. It can be clearly seen from Figure 5 that a linear relation exists and a good fitting is obtained with the correlation coefficient value close to one (above 0.99). This suggests that both for sodium [16] and potassium clusters [20], a good linear correlation exists between the van der Waals coefficient and the square of the cluster volume. Hence, the results above on α ¯ (0) and C6 for different clusters establish

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Figure 5. Plot of van der Waals coefficients C6 as a function of square of volume in atomic units. Upper and bottom panels show the same for Na and K atom clusters, respectively. The data points are shown by dots (joined with lines for guide to the eye), linear fit to square of the volume is represented by black lines. important examples of the interesting size-to-property relationship in case of nano-clusters. 2.2.2. Polarizability and van der Waals Coefficients of Carbon Nano-Structures In this section we discuss the results for the van der Waals coefficient, C6 , between different finite carbon cages and finite-length single-walled H-terminated carbon nano-tubes (CNT) with different size and chirality, containing maximum of about 100 atoms [21]. Our aim is to accurately estimate the comparative strength of the long-range van der Waals interaction between these carbon nano-systems (CNS). Keeping in mind the role played by the XC potential, specifically in the asymptotic regime, we employ in our calculations the asymptotically correct XC potential SAOP. From our calculations, we find that for the CNTs considered by us, the average static polarizability increases with the length of the tube in a non linear fashion. Figure 6 shows the parallel or longitudinal and the perpendicular or transverse components of the polarizability for arm-chair (CNT(3,3)) and zig-zag (CNT(6,0)) as a function of length of the tubes. The upper panel which shows the parallel component of the polarizability clearly indicates that it varies non linearly with the length which is similar to the length-dependence of the total polarizability. On the other hand, in the lower panel the perpendicular component is seen to have a linear dependence with length of the tube. These quasi one-dimensional structures also show large anisotropy in polarizability. For calculating C6 , we need the frequency dependent polarizabilities. The C6

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Figure 6. Plot of (a) longitudinal and (b) transverse components of static polarizability as a function of length of the tube for two typical nano-tubes, chirality index for the zig-zag tube is (6,0) and the arm-chair tube is (3,3). Symbols correspond to the data and lines are guide to the eyes.

values increase linearly with the diameter of the CNTs, while these values show a quadratic dependence on the length of the tubes as is shown in Figure 6. It has also been observed that there is a quadratic dependence of the C6 with respect to the polarizability for both the cages and the tubes. It is observed that the zigzag tubes have larger polarizability and C6 compared to the armchair tubes[21]. This is due to the fact that zigzag tubes have more number of bonds along the tube axis, making them more polarizable which in turn leads to stronger dispersion interaction between the zigzag tubes as is observed from our results. The loosely bound π-electrons play a major role in the polarizability of carbon nano-structures. The bigger the system in size the larger is the number of the π-electrons and the volume over which these electrons are spread. Hence, the long-range interaction in these highly polarizable systems with a large number of carbon atoms is expected to be strong. The carbon cages are less polarizable than the CNTs due to the compact shape of the former, hence we observe that the cages with similar number of atoms are having C6 and polarizability values which are much lower as compared to the corresponding mean values for CNTs. For nano-structures with 60 and 80 atoms, it is found that the C6 and polarizability are about 40 - 50 % lower in the cages compared to the tubes[21]. Before proceeding further, we wish to point out that many authors have observed that conventional XC functionals such as LDA and GGA overestimate the values of (Hy-

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per)polarizabilities for pi-conjugated long-chain systems[22, 23, 24]. The reason for the overestimation is due to the fact that these XC functionals possess incorrect description of electric field dependence of the response part in them[24]. We have carried out calculations of polarizability for two different tubes, namely, the zigzag CNT(4,0) and arm-chaired CNT(3,3), as a function of length, by employing the Hartree-Fock (HF) method (using the package GAMESS [7]) to assess the role of the non-local field-counteracting potential in the presence of an external electric field in the calculations of polarizability. Since HF method takes the effect of exchange exactly, it does not have the above-mentioned shortcoming. A comparison of polarizability obtained by DFT and HF methods is displayed in Figure 7. It is to be noted here that the lengths of all finite CNTs considered in the present ˚ In this length regime (≤ 20 A), ˚ we find from the HF work are less than or about 20 A. results, that the trend in variation of parallel component of polarizability with the length of CNT is the same as that obtained by the DFT (SAOP) method. We also find that that the parallel component of polarizability (polarizability per length) increases quadratically ˚ Thus in this length regime (linearly) with the length of the tube up to a length of about 20 A.

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Table 1. Results for C 6 × 10−3 between sodium clusters and C60 molecule in atomic units. The theoretical results obtained via B3PW91 are taken from Ref. [25] and the experimental results are taken from Ref. [26] Nan 2 4 6 8 10 12 14 16 18 20

SAOP 15.27 30.55 43.42 53.36 68.39 82.00 94.38 103.7 113.7 134.7

B3PW91 15.36 29.94 43.33 54.33 65.67 81.81 98.25 111.7 124.0

EXPERIMENTAL 17.62± 5.11 25.05± 7.44 38.91 ± 12.06 55.01± 15.95 63.71 ± 19.75 92.52 ± 28.68 108.3 ± 33.56 117.8 ± 37.72 139.0 ± 43.10 169.2 ± 52.45

the effect of non-local field-counteracting potential in the presence of external electric field ˚ is presumably negligible. We observe that when we go beyond the length regime of 20 A, the results for polarizability obtained from both the methods start showing differences. The HF data exhibit an onset of saturation with increasing length, specifically for the CNT(3,3) case. We conclude this subsection by noting that the nonlinear increase of the C6 values of the carbon nano-structures with length implies a much stronger vdW interaction between the longer carbon nano-structures compared to the shorter ones. These results will aid us in understanding the cluster-cluster and cluster-surface interactions in carbon-based systems at the nanometer level as well as in the formation of superstructures from these nano-objects. The existence of large anisotropy in polarizability in carbon nano-tubes can play an important role in electric field aligned growth of these systems.

2.3.

Van der Waals Coefficients between CNS and Alkali Atom Clusters as Well as Small Molecules

In this section, we first report the van der Waals interaction, in terms of the C6 coefficients, between the carbon cage C60 and alkali metal atom clusters, namely, Nan as this force plays an important role in describing many physical and chemical phenomena. In Table 1, we present results of our calculations along with other theoretical results [25] and also the experimental data fitted with London’s formula [26]. Model XC potential SAOP has been used for reasons discussed above. The values of C6 between the C60 and Nan are seen to agree well with the earlier theoretical as well as the experimental results; these data are seen to lie well within the experimental error bars [27]. All the results indicate an increase in C6 value as the size of the cluster increases. Furthermore, the knowledge of the van der Waals coefficient C6 between the carbonbased nanomaterials with large surface-to-volume ratio and different molecules of envi-

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ronmentally important gases can be highly useful for possible practical applications of the carbon nano-structures as absorbers of these molecular gases. These gases include water vapor (H2 O), carbon dioxide (CO2 ), carbon monoxide (CO), sulfur dioxide (SO2 ), nitrous oxide (N2 O), methane (CH4 ), ozone (O3 ), carbon tetra fluoride (CF4 ), carbon tetra chloride (CCl4 ). Keeping this in mind we carry out calculations to estimate C6 co-efficient for interactions between CNTs and above mentioned molecules. Along with the molecules listed above, we also present the results for van der Waals coefficient between carbon nanostructures and inert gas atoms (He, Ne, Ar, and Kr), N2 , F2 , Cl2 as well as H2 molecules. For the carbon cages, we consider two fullerene molecules with 60 and 100 atoms and for the carbon nano-tubes, we examine the zig-zag and arm-chair nano-tubes with the chirality indices, (2,2) to (5,5) as well as (4,0) to (10,0) in two sets, one set with about 60 and the other set with about 100 atoms [21]. In Figures 8 and 9, we present the results of our calculations for carbon nano-structures with 60 and 100 atoms, respectively. Since for all the nano-tubes the trend is similar, we only show the results for some of the typical nano-tubes. In Figure 8 we show the data for tubes with chiralities (2,2), (3,3), (6,0), (4,4) and (5,5) as well as C60 . On the other hand, in Figure 9, we display the results for tubes with chiralities (3,3), (4,4) and (10,0) as well as C100 .

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Figure 9. The vdW coefficient C6 between the small molecules and the C100 as well as the CNTs as a function of molecular weight of the small molecules. The chirality index is given in brackets. The results for C100 are denoted by dark circles. The different CNTs are shown by different symbols: CNT(3,3) by diamond, CNT(4,4) by star, and CNT(10,10) by squares. From Figures 8 and 9 it can be clearly seen that for inert gas atoms interacting with cages and nano-tubes, yield minimum values for the van der Waals coefficient. It is well known that due to the shell filling the rare gases are non-polar, chemically very inert and have low polarizability. For systems with larger sizes the valence electrons are farther from the nucleus and thus it becomes easier to polarize the electron cloud. So the magnitude of the dispersion forces increases with increasing system size due to the fact that the dispersion coefficient is dependent on the polarizability of the system. Hence, the C6 coefficients between these atoms and carbon nano-structures are expected to increase with the size of the atom as the value of polarizability increases when one moves down the group from He to Kr. Indeed in Figures 8 and 9, the C6 coefficients between the nano-structures and the rare or noble gas atoms are seen to be increasing with the size of the atom, as one moves from He to Kr, for each carbon nano-structure. We also observe from both Figures 8 and 9, that the values of C6 between the rare gases and the carbon cages are lower than the ones between these atoms and the elongated carbon nano-structures, namely, the nano-tubes. This can be attributed to enhancement of loosely bound π electron density along the tube axes which leads to higher polarizability of the CNTs over the cages with more compact shapes, as discussed above. Next we compare C6 coefficients between the nano-structures and all the other small molecules which include H2 and primarily the environmentally important gases, as listed

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above. First of all, it is observed that the C6 coefficients of the carbon cages with all the molecules are consistently smaller than those of the nano-tubes. This is due to fact that the nano-tubes are clearly more polarizable compared to the finite cages. Furthermore, it is interesting to see from Figures 8 and 9, that for H2 , the values of C6 are much larger compared to those of He even though He and H2 have the same number of electrons. These reasonably high values of C6 between the H2 and the carbon nano-structures, including cages, expectedly, lead to the easy adsorption of H2 on these types of structures with high surface-to-volume ratio. This observation is in conformity with the numerous studies on CNT-derived and fullerene-derived systems as possible hydrogen storage materials. Furthermore, Figures 8 and 9 show that the hazardous gases such as methane, nitrous oxide, carbon dioxide, carbon tetrachloride, carbon tetrafluoride, and sulphur dioxide interact strongly with the carbon nano-structures compared to the rare gases or other molecules. We observe that the interaction of carbon tetrachloride, which is an ozone-depleting and a greenhouse gas, is much higher than the other hazardous gas molecules considered here. This molecule is also having higher values of static dipole polarizability and interacts strongly with itself. Carbon tetrachloride and chlorine molecule possess very high values of C6 with themselves as well as with the carbon nano-structures when compared to their counterparts comprising of fluorine atom (namely, carbon tetrafluoride and fluorine molecule). This is due to the presence of the 7 valence electrons in chlorine atom, however, fluorine molecule and CF4 show exception due to the smaller atomic number of fluorine which leads to more tightly bound valence electrons to the core. Here we would like to mention in support of our result of reasonably high value of C6 for various gases that Cinke et. al. [28] observed that SWCNTs adsorb nearly twice the volume of CO2 compared to activated carbon; Kowalczyk et. al.[29] showed that the carbon nano-tubes are better adsorbents of CF4 than currently used activated carbons and zeolites. We conclude this section by noting that, there are two competing forces those come into play if a molecule interacts with a carbon nano-structure - first is the long-range interaction that is characterized by the C6 coefficient, second is the repulsive interaction arising due to the overlapping charge distributions of the two systems. Hence it is not possible to infer from the results of polarizability and C6 alone as to whether the molecules will adhere to the carbon nano-structures exohedrally or not. Furthermore, the kinetic aspects which are also closely involved in a physisorption process cannot be addressed and accounted for from the present study.

2.4.

Effect of Alkali Metal Atom Doping in Gold Nano-Clusters

Gold is one of the most unique elements in the periodic table with a wide range of applications in chemistry, physics, and material science including clusters and nanomaterials. In recent years gold clusters as well as gold clusters doped with impurity atoms of alkali metal or transition metal have attracted attentions of both theoreticians and experimentalists working in the field of cluster science [30, 31]. It is the discovery of catalytic effect [31] in gold cluster toward oxidation of CO which has prompted large number of studies to understand and characterize the structure and properties of these clusters. Besides being used as catalysts, gold clusters are also finding applications in many other areas like material science [32], molecular electronics devices, medical and biological diagnostics [33, 34].

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Figure 10. Ground-state geometry of tetrahedral Au20 clusters with Td symmetry.

Due to the relatively free-electron nature of the valence electron in alkali metal atoms, the gold nano-clusters doped with these atoms are expected to exhibit interesting physical and chemical properties. In this section we will report the results of our study on the alkalimetal atom doped nano-clusters of gold atoms. We mainly investigate how the stability and chemical inertness of highly stable Au20 cluster are affected when a single gold atom is replaced by an alkali atom. To this end we have carried out a systematic study of electronic properties of doped neutral gold clusters Au19 X ( X = Li, Na, K, Rb, and Cs) [35]. In the tetrahedral structure of the Au20 cluster, the sites can be grouped into three categories: four equivalent vertex sites (Au1−4 as shown in Figure 10), four equivalent face-centered or surface sites (Au17−20 as shown in Figure 10), and twelve equivalent sites located on the edges (Au5−16 as shown Figure 10). To study the doped clusters we replace a single gold atom from one of these three unique sites by an alkali atom (Li, Na, K, Rb, Cs), or a coinage metal atom (Ag, Cu). The ground state structures and the electronic properties of doped and undoped gold cluster with 20 atoms, have been calculated by employing ab-initio method within the realm of scalar relativistic DFT as relativistic effects play important role in the chemistry of gold. The relativistic effects are taken into account by employing zero-order regular approximation (ZORA). The starting geometries of Au19 X for structure optimization are generated from tetrahedral geometry of Au20 cluster by replacing the gold atom from one of the three distinct groups mentioned above by an alkali atom. These structures are then optimized by employing DFT based geometry optimization scheme with PW91 XC potential within GGA and TZ2P basis set. The optimized structures of Au19 X clusters based on tetrahedral geometry as obtained from our calculations are displayed in Figure 11.

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Figure 11. Some of the gold clusters, with a replacement doping of one alkali metal atom. V, E and S represent vertex, edge and surface sites of the gold cluster.

The optimized structures of clusters with dopant atoms located at surface and the vertex locations possess C3V symmetry whereas for dopant atoms located at edges have CS symmetry. In order to check the stability of these geometries we carry out vibrational analysis. None of the clusters considered in this work possesses any imaginary harmonic vibrational frequencies, indicating stable structures for all. We observe that the Cs atom substituted clusters show maximum deviation from the tetrahedral structure, hence these are expected to be least stable structures; Li atom substituted clusters are among the least distorted ones compared to the pure Au20 structure and hence are more stable. In order to make our search for the stable isomers of doped Au19 X clusters more exhaustive we also consider cage-like structures in which dopant atom X is located at an endohedral position. To analyze and compare the stability and the chemical inertness of the doped clusters with the pure Au20 cluster, we calculate BE energy per atom, the interaction energy (IE) of the substituted atom with the Au19 cluster, vertical IP (VIP), vertical EA (VEA), and HOMO-LUMO gaps of the doped clusters. All these results are compiled in Table 2. We observe that for gold clusters doped with Li (located at the edge and surface positions), the BE per atom is higher than the corresponding value for Au20 . For all other alkali atoms the BE is lower than that for

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Table 2. The binding energy per atom (BE), vertical ionization potential (VIP), vertical electron affinity (VEA), and the energy difference between HOMO and LUMO (∆EHL ) of exohedrally doped Au19 X clusters. IE denotes the interaction energy of the doped atom in the cluster System Au20 Au19 Li(V) Au19 Li(E) Au19 Li(S) Au19 Na(V) Au19 Na(E) Au19 Na(S) Au19 K(V) Au19 K(E) Au19 K(S) Au19 Rb(V) Au19 Rb(E) Au19 Rb(S) Au19 Cs(V) Au19 Cs(E) Au19 Cs(S) Au19 Cu(V) Au19 Cu(E) Au19 Cu(S) Au19 Ag(V) Au19 Ag(E) Au19 Ag(S)

BE (eV) 2.369 2.352 2.375 2.398 2.333 2.346 2.362 2.339 2.350 2.364 2.340 2.348 2.361 2.344 2.352 2.366 2.359 2.375 2.386 2.336 2.339 2.348

IE(eV) 3.23 2.89 3.35 3.81 2.52 2.77 3.09 2.64 2.86 3.13 2.64 2.81 3.08 2.73 2.89 3.17 3.03 3.30 3.57 2.57 2.63 2.82

VIP (eV) 7.188 6.66 6.91 7.03 6.55 6.84 6.98 6.30 6.68 6.83 6.22 6.61 6.77 6.12 6.54 6.71 7.08 7.09 7.14 7.03 7.08 7.12

VEA(eV) 2.587 2.451 2.548 2.570 2.322 2.464 2.517 2.197 2.380 2.449 2.139 2.337 2.410 2.100 2.312 2.370 2.565 2.567 2.574 2.521 2.526 2.552

∆EHL (eV) 1.786 1.402 1.512 1.671 1.446 1.531 1.670 1.333 1.470 1.615 1.324 1.452 1.596 1.279 1.419 1.580 1.699 1.662 1.748 1.707 1.693 1.756

the Au20 cluster. On the basis of the BE and geometrical analysis, it is established that the clusters with dopant atom located at the surface are the most stable structures as compared to the edge and vertex sites. We note that for a particular dopant atom X, VIP follows the trend VIP(V) < VIP(E) < VIP(S). For alkali metal atom doping we find that both VIP and HOMO-LUMO gap decrease as we move down the group from Li atom to Cs atom. Like VIP and ∆EHL , VEA also follows a similar trend. The BE analysis show that for small dopant atoms like Li, Na, few cage-like endohedral structures of Au19 X clusters have stability comparable to the corresponding exhohedrally doped Au19 X clusters. However, among all the structures considered in this paper exhohedrally doped Au19 X clusters with dopant atom (X = Li, Na) sitting at one of surfaces of tetrahedral (Au20 ) structure correspond to the most stable isomers of Au19 X. For Li an equally stable (BE same as that of corresponding exohedral structure) cage-like isomer with C(3V) symmetry and endohedral doping is also found. For larger alkali atoms (K, Rb, and Cs) all the cage-like structures

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with endohedral doping are found to be significantly less stable than the corresponding exohedral structures. At this point we note that endohedral complexes formed between highly symmetric Au32 cluster and alkali atom cations (Li+ - Cs+ ) have also been investigated by Jaysekharan and Ghanty [36]. This study has revealed that the larger cations (K+ - Cs+ ) prefer to occupy the centre of Au32 icosahedron cage without disturbing the Ih symmetry, however, smaller cations (Li+ and Na+ ) modify the symmetry from Ih to Cs . We have also carried out Mulliken charge analysis to study the charge distribution and charge transfer in the cluster. It is observed that the dopant atoms change the atomic charge of a particular site considerably. For the surface doping, the vertex atoms undergo the maximum change in the charge distribution. A substitution at the vertex site causes the maximum change in the charge for the adjacent edge atoms. Doping at any edge site leads to the maximum changes in the overall charge distribution, affecting all the atoms. In our recent study we have also found that the maximum change in the charge distribution of the neighboring atoms due to doping by alkali atoms at vertex site also yields large first-order nonlinear optical coefficient for these clusters [37] We conclude that the replacement of a Au atom at the surface site of the tetrahedral structure of the pure Au20 cluster by a Li atom among other alkali metal atoms results in a highly stable cluster. Therefore it is expected that an anion of this cluster (Au19 Li− ) might be highly reactive and a potential candidate for catalysis.

Acknowledgments A.B., A.C., and C. K. wish to thank Dr. S. C. Mehendale, Dr. S. M. Oak, Dr. S. K. Deb for their encouragement and support. We all wish to thank Mr. Pronobesh Thander, RRCAT Computer Center, and Computer Division, BARC for providing computational facilities. T.K.G. would like to thank Dr. S. K. Ghosh and Dr. T. Mukherjee for their constant encouragement and support.

References [1] Parr, R.G.; Yang, W. Density functional theory of atoms and molecules; Oxford; New York 1989.; Dreizler, R. M.; Gross, E. K. U. Density functional theory; Springer; Berlin 1990. [2] Gross, E. K. U.; Dobson, J. F.; Petersilka, M. In Density Functional Theory, Topics in Current Chemistry; Editor, R. F. Nalewajski; Springer, Berlin, 1996; Vol. 181, pp. [3] Stener, M.; Nardelli, A.; Francesco, R.; Fronzoni, G. J. Phys. Chem. C 2007 111, 11862. [4] Casimir, H. B. G.; Polder, D. Phys. Rev. 1948, 73, 360. [5] Stone, A. J. The Theory of Intermolecular Forces, Clarendon: Oxford, 1996. [6] Baerends, E. J.; Autscbach, J.; Berces, A.; et al. Amsterdam Density Functional, Theoretical Chemistry, Vrije Universiteit, Amsterdam, URL http://www.scm.com.

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[7] Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993 14, 1347. [8] van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys., 1998 109, 10657. [9] Banerjee, A.; Harbola, M. K. Phys. Rev. A 1999, 60, 3599. [10] Kamal, C.; Ghanty, T. K.; Banerjee A.; Chakrabarti, A. J. Chem. Phys. 2009, 130, 024308. [11] Gruning, M.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J. J. Chem. Phys. 2002 116, 9591. [12] Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996 77, 3865. [13] Gritsenko, O. V.; Schipper, P. R. T.; Baerends, E. J. Chem. Phys. Lett. 1999 302, 199. [14] Schipper, P. R. T.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J. J. Chem. Phys., 2000 112, 1344. [15] Chandrakumar, K. R. S.; Ghanty, T. K.; Ghosh, S. K. J. Chem. Phys. 2004, 120, 6487. [16] Banerjee, A.; Chakrabarti, A.; Ghanty, T. K. J. Chem. Phys., 2007 127, 134103. [17] Banerjee A.; Ghanty, T. K.; Chakrabarti, A. J. Phys. Chem. A 2008, 112, 12303. [18] Knight, W.D.; Clemenger, K.; de Heer, W. A.; Saunders, W. A. Phys. Rev. B, 1985, 31, 2539. [19] Tomasi J.; Persico, M. Chem. Rev., 1994, 94, 2027. [20] Banerjee A.; Chakrabarti, A.; Ghanty, T. K. Int. J. Quant. Chem. 2009, 109, 1376. [21] Kamal, C.; Ghanty, T. K.; Banerjee A.; Chakrabarti, A. J. Chem. Phys. 2009, 131, 164708. [22] Champagne, B. E.; Perpete, A.; Jacquemin, D.; van Gisbergen, S. J. A.; Baerends, E. J.; Ghaoui, C. S.; Robins, K. A.; Kirman, B. J. Phys. Chem. A, 2000 104, 4755. [23] Karolewski, A.; Armiento, R.; Kummel, S. J. Chem. Theory Comput. 2009 5, 712. [24] van Gisbergen, S. J. A.; Schipper, P. R. T.; Gritsenko, O. V.; Baerends, E. J.; Snijders, J. G.; Champagne, B.; Kirman, B. Phys. Rev. Lett., 1999 83, 694. [25] Jiemchooroj, A.; Norman, P.; Sernelius, B. E. J. Chem. Phys, 2006 125, 124306. [26] Kresin, V.; Tikhonov, G.; Kasperovich, V.; Wong, K.; Brockhaus, P. J. Chem. Phys., 1998 108, 6660. [27] Banerjee, A.; Autscbach, J. A.; Chakrabarti, A. Phys. Rev. A 2008 78, 032704.

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[28] Cinke, M.; Li, J.; Bauschlicher Jr., C. W.; Ricca, A.; Meyyappan, M. Chem. Phys. Lett., 2003, 376, 761. [29] Kowalczyk, P.; Holyst, R. Environ. Sci. Technol., 2008 42, 2931. [30] Pyykk¨ o, P.; Angew. Chem. Int. Ed., 2004 43, 4412. [31] Nanocatalysis, Heiz, U.; Landman, U. Eds. Springer-Verlag, Berlin, 2007. [32] Dyson, P. J.; Ningos, D. M. P. Gold: Progress in Chemistry, Biochemistry and Technology; Schmidbaur, H. Ed.; Wiley, New York, 1999, 511. [33] Pyykk¨ o, P.; Runeberg, N. Angew. Chem. Int. Ed., 2002, 41, 2174. [34] Shaw III, C. F. Chem. Rev., 1999, 99, 2589. [35] Ghanty, T.; Banerjee, A.; Chakrabarti, A. J. Phys. Chem. C, 2010, 114, 20. [36] Jayasekharan, T; Ghanty, T. J. Phys. Chem. C, 2010, 114, 8787. [37] Banerjee, A.; Ghanty, T.; Chakrabarti, A.; Kamal, C. in preparation.

In: Theoretical and Computational Developments … Editor: Amlan K. Roy, pp. 549-559

ISBN: 978-1-61942-779-2 © 2012 Nova Science Publishers, Inc.

Chapter 21

THE ROLE OF METASTABLE ANIONS IN THE COMPUTATION OF THE ACCEPTOR FUKUI FUNCTION Nelly González-Rivas,a Mariano Méndezb and Andrés Cedillob,1 a

Centro Conjunto de Investigación en Química Sustentable UAEM-UNAM Carretera Toluca-Atlacomulco Km. 14.5, Unidad San Cayetano, Toluca, Estado de México, MÉXICO b Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa San Rafael Atlixco 186, México DF, MÉXICO

Abstract Estimation of chemical reactivity descriptors for neutral molecules is limited if the corresponding anion is unstable. When the electronic structure of an unstable anion corresponds to a bound state, the anion is a metastable or a temporary species. An orbital swapping methodology is proposed to obtain a bound state electronic structure for unstable anions. The electronic structure of the metastable anion allows the estimation of global and local reactivity descriptors for neutral chemical species with unstable anion. This chapter focuses in the computation of the vertical electron affinity and the acceptor Fukui function. We apply our methodology to some small molecules, carbonyl organic compounds and inorganic Lewis acids.

PACS: 31.15.es, 33.15.Ry Keywords: Metastable anions, Chemical reactivity, Fukui function

Introduction Prediction of the chemical reactivity of molecules is an important issue in chemistry. Over the years, chemists have been able to classify species into families, taking into account 1

E-mail address: [email protected]

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the characteristic reactivity pattern among each group of molecules. Following the same goal, theoretical chemistry defines reactivity descriptors to predict the behavior of the chemical species when they interact with other compounds. In this context, density functional theory (DFT) provided a very successful background to construct chemical reactivity parameters [16]. Many chemical reactivity descriptors have been identified as derivatives of some molecular property, that is, they can be identified as response functions. For example, the electronegativity [8] the chemical hardness [9] and the Fukui function [10] represent the derivative, with respect to the number of electrons, of the negative of the energy, the chemical potential and the electron density, respectively. Reactivity parameters also come from the derivatives with respect to the external potential [7], for example, the molecular electrostatic potential corresponds to the derivative of molecular energy. Derivatives with respect to the number of electrons are usually evaluated by a finite differences approach. In this case, the electronic structure of species with an additional electron and without one electron is also required, at the same geometry of the reference species. In general, theoretical chemistry can accurately predict the electronic properties of neutral molecules and its corresponding positive ions, however the computation of the electronic structure of the anions presents additional challenges [11-12]. Anions can be classified from their stability with respect to the neutral species. The relative stability is usually estimated from the electron affinity, which is defined as a difference of energies, namely A  E neutral  E anion . There are two kinds of anions, the stable and unstable ones. A stable anion presents positive electron affinity, that is, this kind of anion is more stable than the corresponding neutral species. On the other hand, the electron affinity of an unstable anion is negative, since the anion is less stable than the corresponding  unstable anions can be metastable or temporary species, since neutral species. In addition, they decay to the neutral species by electron detachment. The lifetime of temporary anions is very small, usually less than 10-12 s [11]. In general, the theoretical estimation of the electron affinity is a complex task, even for molecules with a stable anion. The use of post Hartree-Fock methods, such as, CI, MP, CC, for the computation of the electron affinities of atoms and molecules is appropriate and, in many cases, the CC methods give the most accurate results [12]. On the other hand, the estimation of the electron affinity by using DFT methodologies provides satisfactory predictions. Specifically Rienstra et al. report differences around 0.1 and 0.2 eV (2-5 kcal/mol) compared to experimental values when the B3LYP hybrid functional is used and similar estimations from B3LYP and MP2 methods [12]. They also emphasized the importance of the use of diffuse basis functions to obtain representative results. The most challenging issue is the estimation of negative electron affinities. The anion of a closed shell neutral molecule is a radical with one unpaired electron in an outer shell and, in most of the cases, the anion is less stable that the neutral molecule. When the unstable anion possesses a bound state, a metastable or temporary state exists, however it will eventually decay to the neutral state and a free electron. In general, theoretical models are not able to make an appropriate description of the electronic structure of these anions. For a temporary anion it is usually observed that additional electron is found far from the molecular region and the corresponding energy turns out to be close to that of the neutral species. In fact, this effect increases when the basis set size grows [13]. On the other hand, basis sets without diffuse functions provide reasonable estimations of the electron affinity [14]. This unusual behavior

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derives from the minimization procedure within the SCF algorithm, since the anion is less stable that the neutral species. For an unstable anion and a large basis set, the electronic structure evolves along the SCF process to resemble the neutral species and a non bound electron, which is the minimal solution for this number of electrons. Limited basis sets cannot represent a non bound electron and, consequently, the electron structure approximates the temporary state. On the opposite, as the basis set increases with more diffuse functions, the anion description becomes closer to the neutral species and the non bound electron locates farther from the molecule. Some alternatives have been recently proposed to estimate the electronic properties of the metastable anions within DFT. Tozer et al. used the discontinuity in the exchange and correlation potential [14]. Vargas et al. presented a Koopmans-like approach in the context of the Kohn-Sham method [15]. Szarka et al. proposed the use of an alternative electronic configuration for the temporary anion [13]. In Szarka’s method they form a new KS determinant for the anion by exchanging two orbitals, the HOMO and an empty localized orbital. Their suggestion for the new orbital is a * orbital for saturated molecules, while a * orbital for unsaturated ones. They restart the SCF convergence process with the new electronic configuration. This orbital swap procedure provides a better estimation of the electronic structure of the metastable anion and the corresponding vertical electron affinity. The inaccurate description of the temporary anions restricts the use of those DFT reactivity descriptors related to the electronic structure of the negatively charged species. For global reactivity parameters one can find some options in the literature, from the use of the orbital energies (using Hartree-Fock expressions in Kohn-Sham context) up to models to estimate the electron affinity within the Kohn-Sham scheme [14-15]. Fewer choices are available for the local properties. The frontier orbital approximation is widely used, but it completely neglects the electronic relaxation coming from the change in the number of electrons. The use of a potential barrier to confine the electron density has been also explored [16]. Fukui function describes the changes in the electron density when the number of electrons is modified [10] and it is used to describe the changes in the electronic distribution when electron-acceptor/electron-donor interaction processes take place. This kind of processes usually appear in chemistry and they are recognized with different names, for example Lewis acid-base reactions, electrophile-nucleophile reactions, donor-acceptor complexes, etc. From the discontinuity of the chemical potential at an integer number of electrons [17], there are two Fukui functions: the acceptor Fukui function (  f  N0 1  N0 ), when the number of electrons of the chemical species increases, and the donor Fukui function ( f   N0  N0 1 ), associated with a decrease in the number of



electrons, here N0 is the number of electrons of the species of interest and N represents the density of the species with N electrons [10]. Then, the evaluation of the Fukui function requires the electron density of the species of interest and the species with N0+1 and N0-1 same geometry. This procedure takes into account the electronic relaxation electrons, at the from the variation in the number of electrons. A simpler method only involves the frontier orbitals, but it implies the neglect of the electronic relaxation, which can be relevant in specific cases [18]. The evaluation of the chemical reactivity descriptors of neutral molecules also requires the electronic structure of the positively and negatively charged species. When the anion is a

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temporary species, the electronic structure of a bound state of this kind of anions is needed. In this chapter we present a procedure to handle this issue.

Model For unstable anions, the electronic structure computation usually leads to a non bound state. In this case the electron affinity is very small, compared with the experimental value. Szarka et al. reported a solution to this problem, they proposed to construct a different configuration by swapping the HOMO and an empty orbital with the appropriate symmetry [13]. However, this procedure becomes cumbersome when the molecule possesses low symmetry. One way to identify that the anion is in a non bound state comes from the analysis of the acceptor Fukui function. Since this Fukui function represents the change in electron distribution when one electron is added, in a non bound anion, usually this additional electron is not localized. That is, the acceptor Fukui function presents large values beyond the molecular region. Figure 1 shows the acceptor Fukui function computed with the electron density of the anion in a non bound state.

Figure 1. Isosurfaces of the donor (f -) and acceptor Fukui function (f +) for ethylene (contour level 0.01).

In order to find a bound state of the temporary anion, one needs a localized empty orbital. The variance of the orbital density,

2

2

2i  i r2 i  i r i  r2  r , measures of

the degree of localization of the orbital. In our entire test we found that the expected value

r is very close to zero, then we approximate the variance by  2i  r 2 . For large or



branched molecules this approximation may fail. Table 1 shows the variance values of some  orbitals corresponding to the ethylene anion. One can note that the variance value of the HOMO is much larger than the other occupied orbitals. The same behavior is observed on the  empty orbital set, except for orbital 13, which shows a variance value comparable with those on the occupied set. For the ethylene anion, a new electronic configuration is built by swapping orbitals 9 and 13. The orbital ordering is locked and the SCF procedure is restarted up to convergence. For the new configuration, the acceptor Fukui function is analyzed and we found that it is localized around the molecular region, see Figure 1. Following Szarka’s method, we found that the vertical electron affinity is -1.74 eV, which is close to the experimental report (-1.78

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eV) and very different from the value from the non bound state (-0.87 eV). In addition, Figure 1 also shows that the acceptor Fukui function presents  symmetry, which is consistent with Szarka’s proposal. This procedure is applied for a set of molecules with unstable anion and reported vertical electron affinity. The set is selected from the molecules considered in Szarka et al. [13] and Tozer et al. [14] previous reports. The electronic structure is computed with the model B3LYP/6-311++g(3df, 3pd), using NWChem 5.1.1 program. [19] This computational model was previously test for a set of molecules with stable anion and provided excellent adiabatic electron affinity estimations [20]. Table 1. Orbital properties for the ethylene anion computed with the method B3LYP/6-311++g(3df,3pd) orbital

Occupation

4 5 6 7 8 9 10 11 12 13

1 1 1 1 1 1 0 0 0 0

Energy

r2

-0.4424 -0.3174 -0.2859 -0.2169  -0.1403 0.0673 0.0993 0.1000 0.1252 0.1253

2.1 1.9 1.7 2.4 1.6 16 19 18 24 4.5

Table 2. Keto and enol tautomers

1a 1b 1c 1d

2a 2b 2c 2d

R H CH3 F OH

In addition to the test set, our procedure is applied to the anions of some aliphatic organic compounds present in the keto-enol tautomeric equilibrium. (Table 2) and the anions of some substituted boranes (BX3), which act as Lewis acids. In the former case, all the anions are unstable, while, in the latter, we select those with unstable anion (boranes with electronreleasing groups).

Results The electronic structure of the anion is explored through the analysis of the acceptor Fukui function. We identify that an unstable anion is poorly described when this Fukui

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function is mainly located beyond the molecular region, that is, when the additional electron is not bound to the molecule.

Figure 2. Isosurfaces of the donor (f -) and acceptor Fukui function (f +) for dichloromethane (contour level 0.002).

Dichloromethane Fukui functions are shown in Figure 2. One can note large contributions of the Fukui function f + beyond the chlorine atoms. After swapping orbitals 22 and 37, Fukui function becomes localized in the molecular region. The orbital swapping procedure radically changes the vertical electron affinity estimation, from -0.51 eV to -1.18 eV. This result is in good agreement with the reported experimental value (-1.23 eV). The vertical electron affinity of the molecules from the test set can be found in Table 3. Note that chloroform, formaldehyde and nitrogen anions are unstable, but their electronic structure corresponds to a bound state (see Figure 3). For anions of these molecules the orbital swapping is not preformed. Table 3. Vertical electron affinities, in eV Molecule

Ref. 13a

non bound stateb

bound stateb

experimental valuec

CH3Cl CH2Cl2 CHCl3 H2CO N2 Ethylene

-1.95 -1.36 -0.55 -0.96 -2.14 -1.88

-0.58 -0.51 -0.87

-2.13 -1.18 -0.34 -0.80 -2.04 -1.74

-3.45 -1.23 -0.35 -0.86 -2.2 -1.78

a) B3LYP/D95+(d). b) B3LYP/6-311++g(3df,3pd). c) Taken from reference 13.

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Figure 3. Isosurfaces of the donor (f -) and acceptor Fukui function (f +) for nitrogen and formaldehyde (contour level 0.03).

Chloroform is an intricate case. The analysis of the Fukui function shows large contributions in both the molecular zone and beyond (see Figure 4). In addition, the computed vertical electron affinity is very close to the experimental value. The swapping orbital procedure always converges to the original Kohn-Sham results, suggesting that no other minimum of the electronic structure is found. As an additional test we compare the LUMO of the neutral species with the acceptor Fukui function and we find that both are very similar. From this analysis we conclude that the electronic structure of the unstable anion is correctly described and it corresponds to a bound state. Acceptor Fukui function contribution outside of the molecular region seems to come from the lone electron pairs of the chlorine atoms. It is very important to remember that the acceptor Fukui function is not equal to the LUMO density, since the Fukui function includes the electronic relaxation effects from the addition of one electron.

Figure 4. Isosurfaces of the acceptor Fukui function, f +, (contour level 0.002) and LUMO density (contour level 0.01) for chloroform.

The estimation of the electron affinity and the acceptor Fukui function offers good results for the unstable anions. The same methodology will be applied to a set of molecules whose electron affinity is not reported.

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Nelly González-Rivas, Mariano Méndez and Andrés Cedillo Table 4. Vertical electron affinities of keto and enol tautomers, in eV molecule Anon bound -0.60 1a -0.47 1b -0.50 1c -0.52 1d

Abound -1.07 -0.90 -1.01 -1.20

molecule Anon bound -0.69 2a -0.54 2b -0.48 2c -0.46 2d

Abound -2.07 -1.35 -2.07 -2.31

The first set corresponds to the molecules involved in the keto-enol tautomeric equilibrium. Structures of these compounds are listed in Table 2. Substituted aliphatic carbonyl and vinyl compounds interconvert by a proton migration and their corresponding anions are unstable. The estimated vertical electron affinities are shown in Table 4 and all are negative. The analysis of the acceptor Fukui function points out that the electronic structure corresponds to a non bound state, since it is located beyond the molecular region. We apply the orbital swap procedure to the each molecule of this set and it leads to a bound state electronic structure, with a more negative electron affinity (see Table 4) and a localized acceptor Fukui function. For the carbonyl molecules, the acceptor Fukui function is mainly located on the oxygen and carbon atoms of the carbonyl group. Figure 5 displays the acceptor Fukui function for the acetaldehyde molecule, which is representative of the carbonyl molecules. The carbonyl group is the reactive part of this kind of molecules and this group presents the largest values of the Fukui function. For the enol tautomers, the acceptor Fukui function localizes on the carbon atom of the carbonyl group (see Figure 5).

Figure 5. Acceptor Fukui function isosurface for acetaldehyde and vinyl alcohol (contour level 0.02).

By using this methodology we have a greater confidence to describe a bound state of the system. The results shown in Table 4 allow one to study the substituent effect in the stability of the anions. The methyl group stabilizes the anion (1b), compared to the case where the substituent is H group (1a). In contrast, the hydroxyl group destabilizes the anion (1d). Electron deficient species, like boranes, act as electron pair acceptors, that is, Lewis acids. Lewis acidity scales often involve the reaction of the acid with a reference Lewis base and the hydride ion is the smallest one. The hydride affinity (HA) of a Lewis acid is the negative of the enthalpy of the reaction between the acid and the hydride ion, in the gas phase. This Lewis acid-base reaction supports the HA acidity scale. A thermodynamic cycle relates the adiabatic electron affinity (EA) of a Lewis acid (M) with its HA, HA(M) = D(MH-) − EA(H) + EA(M), where D(M-H-) represents the homolytic bond dissociation energy of the B-H bond in the species MH- [21]. Since the bond dissociation energy variation is small

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among a set of similar compounds, EA is also a measure of the gas phase Lewis acidity [2122].

Figure 6. Comparison between the electron and the hydride affinities for the substituted boranes. The open squares correspond to the vertical electron affinity with no orbital swapping for the unstable anions, while the filled circles represent the modified estimation by the swapping procedure. The orbital swap is only applied for unstable anions. These results are computed with the BLYP/6311++g(3df,3pd) method.

The electronic nature of the substituent groups strongly affects Lewis acidity and basicity of the chemical species. In the substituted boranes, electron-attracting groups remove electron density from the borane atom and they usually increase the acidity, while the electronreleasing groups lower it. In agreement with these general trends, boranes with electronreleasing groups, like methyl, hydroxyl and phenyl groups, have a lower HA that BH3, while strong electron-attracting groups, like CF3 and C6F5, increase the HA [22]. Fluoride group presents an intriguing character, it moderately lowers the HA. This behavior comes from a balance between its large electronegativity and the electronic conjugation of its lone electron pair with the empty p-orbital from the boron atom. From this set of boranes, only fluoro-, hydroxy- and methylboranes present negative values of the vertical electron affinity. The analysis of the electronic structure and the orbital swapping procedure is applied to the unstable borane anions. For the methyl-, difluoro- and all the hydroxyboranes, a more localized electronic structure is found. Fluoroborane anion already corresponds to a bound state. For dimethyl-, trimethy- and trifluorolborane, the orbital swapping does not produce a more localized state and the electronic structure of the anion was not replaced for these compounds. Figure 6 shows the comparison between EA and HA, before and after the swapping procedure. A better trend for the unstable anions comes after the orbital swapping. Note that boranes with a stable anion are also plotted to show the general trend, but no swap is made for them.

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Figure 7 presents the acceptor Fukui function of the fluorboranes (H3-nBFn, for n = 1, 2, 3). As we mention above, the Kohn-Sham electronic structure of the anion of fluoroborane (H2BF) already represents a bound state. The orbital swapping is applied to difluoroborane (HBF2) and trifluoroborane (BF3) anions. For the former we find a more localized electronic structure, while for the latter the procedure does not produce a more localized structure and we keep the original results. In the three molecules one can note that the Fukui function at boron atom presents significant values. Substituted boranes act as Lewis acids and the boron atom represents the acidic site.

Figure 7. Acceptor Fukui function for fluoro, difluoro- and trifluoroborane (contour value 0.002)

This study leaves an open question. If the estimated electron affinity values are close to those reported experimentally, does it imply that the anion is correctly described? The acceptor Fukui function provides us a guide on this question, but one can find a case where there is not an absolute answer. It is important to remark that there are only a few measurements of vertical electron affinities for unstable anions. Therefore, our proposal tries to identify the non bound character on the electronic structure of anion. In this case, it is necessary to look for a localized empty orbital and use it to generate an alternative electronic configuration for the unstable anion.

Conclusion Szarka’s methodology is modified to apply it to molecules with low symmetry. We use the variance of the orbital density as a measure of the degree of localization among the empty orbital set and it is useful to select a localized one. An orbital swapping allows one to obtain a bound state for the temporary anions, which enhances the quality of the estimated vertical electron affinity and the acceptor Fukui function. Even when this procedure properly predicts the electron affinity, it is always advisable to check that the orbitals and the Fukui function of the anion correspond to a bound state. Our orbital swap methodology was successfully applied to several molecules with unstable anion. In the test set, the vertical electron affinity turns out to be close the experimental results. Chemical reactivity of the carbonyl compounds correctly reflects on the

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Fukui functions, in particular on the acceptor one. For the substituted boranes, the gas phase Lewis acidity trend is recovered for the compounds with unstable anions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

[20] [21] [22]

Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford: New York, NY, 1989. Chattaraj, P. Chemical Reactivity Theory: A Density Functional View; CRC: Boca Raton, FL 2009. Toro-Labbé, A. Theoretical Aspects of Chemical Reactivity; Elsevier: Amsterdam, 2006. Chermette, H. J. Comput. Chem. 1999, 20, 129-154. Geerlings, P.; De Proft, F.; Langenaeker, W. Chem. Rev. 2003, 103, 1793-1873. Chattaraj, P.; Sarkar, U.; Roy, D.; R. Chem. Rev. 2006, 106, 2065-2091. See for example Cedillo, A. In Theoretical Aspects of Chemical Reactivity; ToroLabbe, A.; Ed.; Elsevier: Amsterdam 2006; pp 27-35. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E.; J. Chem. Phys. 1978, 68 38013807. Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512-7516. Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1984, 106, 4049-4050. Jordan, K. D.; Burrow, P. D. Chem. Rev. 1987, 87, 557-588. Rienstra-Kiracofe, J. C.; Tschumper, G. S.; Schaefer III, H., F. Chem. Rev. 2002, 102, 231-282. Szarka, A. Z.; Curtiss, L. A.; Miller, J. R. Chem. Phys. 1999, 246, 147-155. Tozer, D. J.; De Proft, F. J. Chem. Phys. 2007, 127, 034108. Vargas, R.; Garza, J.; Cedillo, A. J. Phys. Chem. A 2005, 109, 8880- 8892. Tozer, D. J.; De Proft, F. J. Phys. Chem. A 2005, 109, 8923-8929. Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Phys. Rev. Let. 1982, 49, 1691-1694. Bartolotti L. J.; Ayers, P. W. J. Phys. Chem. A 2005, 109, 1146-1151. Valiev, M.; Wang, D.; Apra, E.; Windus, T. L.; Hammond, J.; Nichols, P.; Hirata, S.; Hackler, M. T.; Zhao, Y.; Fan, P.-D.; Harrison, R. J.; Dupuis, M.; Smith, D. M. A.; Nieplocha, J.; Tipparaju, V.; Krishnan, M.; Vazquez-Mayagoitia, A.; Wu, Q.; Van Voorhis, T.; Auer, A. A.; Nooijen, M.; Crosby, L. D.; Brown, E.; Cisneros, G.; Fann, G. I.; Fruchtl, H.; Garza, J.; Hirao, K.; Kendall, R.; Nichols, J. A.; Tsemekhman K.; Wolinski, K.; Anchell, J.; Bernholdt, D.; Borowski, P.; Clark, T.; Clerc, D.; Dachsel, H.; Deegan, M.; Dyall, K.; Elwood, D.; Glendening E.; Gutowski, M.; Hess, A.; Jaffe, J.; Johnson, B.; Ju, J.; Kobayashi, R.; Kutteh, R.; Lin, Z.; Littlefield, R.; Long, X.; Meng, B.; Nakajima, T.; Niu, S.; Pollack, L.; Rosing, M.; Sandrone, G.; Stave, M.; Taylor, H.; Thomas, G.; van Lenthe, J.; Wong, A.; Zhang, Z. NWChem, A Computational Chemistry Package for Parallel Computers, Version 5.1.1 (2009), Pacific Northwest National Laboratory, Richland, WA 99352-0999, USA. González-Rivas, N.; Cedillo, A. Int. J. Quantum Chem. 2009, 109, 1031-1035. Vianello, R.; Maksic, Z. B. Inorg. Chem. 2005, 44, 1095-1102. Méndez-Chávez, M.; Cedillo, A. Inorg. Chem. (submitted).

In: Theoretical and Computational Developments… Editor: Amlan K. Roy, pp. 561-587

ISBN: 978-1-61942-779-2 © 2012 Nova Science Publishers, Inc.

Chapter 22

KINETIC-ENERGY/FISHER-INFORMATION INDICATORS OF CHEMICAL BONDS Roman F. Nalewajski,a,1 Piotr de Silvab and Janusz Mrozekb a

Department of Theoretical Chemistry and Department of Computational Methods in Chemistry, Faculty of Chemistry, Jagiellonian University, Cracow, Poland b

Abstract The kinetic energy (contragradience) criterion, related to the non-additive Fisher information in the resolution determined by the basis-functions χ ={i}, e.g., the Atomic-Orbitals (AO), is used to localize chemical bonds in molecules. The interference, non-additive (nadd.) contribution to the molecular Fisher-information density, fnadd.[χ; r] = ftotal[χ; r]  f add.[χ; r], where f total[χ; r] = f [; r] is the overall distribution for the molecular electron density  and fadd.[χ; r] = i f[i; r] denotes its AO-additive part, is used to determine the bonding regions in molecules. These closed basins of the physical space, for which fnadd. < 0 identify regions of a diminished Fisher information (increased delocalization) content, compared to the reference AO-additive value. Indeed, such volumes represent a locally-decreased gradient content of the system wave-function thus reflecting less “order” (more “uncertainty”) in the molecular distribution of electrons, and hence their increased delocalization via the system chemical bonds. This suggests the use of the zerocontra-gradience surface f nadd.[χ; r] = 0, as the sensitive detector of the local presence of chemical bonds. The representative results from the minimum-basis-set (STO-3G) SCF MO calculations are reported for representative diatomics (H2, N2, HF, HCl, NaCl, CO) and selected polyatomic systems (ethylene, acetylene, ethane, butadiene, benzene, diborane and small propellanes). These illustrative examples convincingly validate the applicability of this contragradience probe in exploring the bonding patterns in molecules from the novel perspective of the Fisher-information/kinetic-energy redistribution.

Keywords: Bonding regions in molecules, Information origins of chemical bonds, Kinetic energy as bond criterion, Localization of bonds in molecules, Non-additive Fisher information

1

E-mail address: [email protected]

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1. Introduction In chemistry an adequate identification of chemical bonds in a complicated molecular system is of paramount importance for both the structural and reactivity purposes. In the past various local probes have been suggested for exploring the molecular electron distributions in chemical terms, such as the bonding and lone-electron pairs, etc. For example, the densitydifference diagrams, relative to the promolecular reference, and the Electron Localization Function (ELF) [1,2] have been shown to provide attractive tools for delineating and visualizing such structural building-blocks of the system electronic structure. Alternatively, the bond-multiplicity indices formulated in the MO theory can be used to quantify the bondconnectivities between constituent atoms [3-11]. It has recently been shown that various concepts and techniques of the Information Theory (IT) [12-18] can be successfully used to extract chemical interpretations of the known electron probabilities in molecular systems [1929]. In particular, the Orbital Communication Theory (OCT) of the chemical bond [2024,31,32] has uncovered the entropy/information roots of the bond covalency/ionicity and the new bond-localization probe based upon the contragradience (CG) criterion related to the non-additive Fisher information [12,13] (kinetic energy) has been proposed [20,21,24-27]. In OCT the additional indirect (through-bridge) bonding mechanism has been identified [33-36], which complements the familiar direct (through-space) component of the chemical bonds. The Fisher information for locality [12], called the intrinsic accuracy, contained in the normalized, continuous probability distribution p(r), I[p] =  [p(r)]2/p(r) dr,

(1)

historically predates the familiar Shannon entropy, S[p] =   p(r) log p(r) dr,

(2)

by about 25 years, being proposed in about the same time when the final form of the quantum mechanics was formulated. The Fisher functional, reminiscent of von Weizsäcker’s [37] inhomogeneity correction to the electronic kinetic energy in the Thomas-Fermi theory, characterizes the narrowness (order) of the probability density while the Shannon functional reflects the distribution spread (disorder). For example, the Fisher information of the normal distribution measures the inverse of its variance, called the invariance, while the complementary Shannon entropy is proportional to the logarithm of variance, thus monotonically increasing with the spread of the Gaussian distribution. The Shannon entropy and the Fisher information for locality thus describe the complementary aspects of the probability density: the former reflects distribution’s “spread”, while the latter measures its compactness. The expression for the intrinsic-accuracy can be simplified by expressing it as functional of the associated probability amplitude A(r) =

p(r ) of the probability distribution p(r):

I[p] = 4 [A]2 dr = I[A] =  f(r) dr.

(3)

Therefore, I[A] in fact measures the gradient content of the probability-density amplitude. For the complex probability amplitude of a single electron in quantum mechanics, the wavefunction (r), the integrand in this equation is replaced by 2 [13,38]:

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I[] = 4  (r)2 dr =  4     ˆi .

(4)

In the preceding expression we have introduced the one-electron Fisher-information operator ˆi( r ) , which defines the associated N-electron operator N

ˆI( N )   ˆi( rj ) .

(5)

j 1

The IT was also proven useful in tackling the definition of Atoms-in-Molecules (AIM) [19,39-45]. It has justified Hirshfeld’s [46] “stockholder” division of the molecular electron density into atomic contributions. The Communication Theory of the Chemical Bond (CTCB) [19,47-54] and its more recent AO extension called Orbital Communication Theory (OCT) [20-24,31] provide the entropy/information measures of the chemical bond covalency/ionicity, which characterize the molecular information (communication) channels. These indices have also been used to characterize the AIM promotion due to the orbital hybridization [53]. The thermodynamic-like description of the electronic “gas” in molecular systems has also been explored [19,39,55]. In physics IT plays the unifying role by facilitating a derivation of its basic laws from the common Extreme Physical Information (EPI) principle, which uses the Fisher information measure [13,19,38,56,57]. It has been applied to problems in chemical kinetics [58], to issues of the electron localizability and transferability in molecules [30,59-61], and in the “surprisal” analysis and synthesis of the molecular electron density [62-64]. The IT concepts have been used in the field of the Compton profiles and momentum densities [65-67], in the Density Functional Theory (DFT) [68-71], and to describe the electron correlation [72-74]. In the molecular structure theory they have been successfully used in devising topological descriptors of the chemical structure formulated in the molecular graph theory [75-77]. Of great interest also are the recent information probes of the elementary reaction mechanisms [78,79]. The ELF has been shown to explore the non-additive part of the Fisher information in the Molecular Orbital (MO) resolution [30], while a similar approach in the Atomic Orbital (AO) representation [38] proposes the use of the so called “contragradience” criterion to explore the chemical bonds in the molecule. It is related to the AO matrix representation of the electronic kinetic-energy operator. It has been qualitatively demonstrated that this AO-phase sensitive index is capable of distinguishing the bonding, non-bonding, and anti-bonding electronic states in H2, as do the off-diagonal Charge and Bond-Order (CBO) matrix elements of quantum chemistry. It could also be used to delineate the bonding regions of the molecule [20,21,24-27]. It is the main purpose of this review to introduce and demonstrate the applicability of this novel, local probe of the direct chemical bonds.

2. Local Contra-Gradience Criterion for the Presence of the Direct Chemical Bond Let us consider the elementary chemical interaction between two Löwdin-orthogonalized AO (OAO), χ(r) = {A(r), B(r)}, centered on atoms A and B, respectively, which contribute a single electron each to form the chemical bond AB, e.g., the covalent  bond in H2 or the 

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bond in ethylene. Their interaction gives rise to the bonding (b) and anti-bonding (a) MO combinations,

b = P A  Q B ,

a =  Q A  P B ,

P + Q = 1,

(6)

where one of the complementary probability parameters P and Q = 1  P uniquely control the shape (polarization) of both MO. The three orbital configurations of this two-electron model, N = 2, the ground-state 0(2) = [b2], the singly-excited state 1(2) = [b1a1], and the doubly-excited state 2(2) = [a2], mark the reference bonding (attractive), non-bonding (neutral), and the anti-bonding (repulsive) interactions between the two atoms. They are respectively distinguished by the positive, zero, and negative values of the off-diagonal element A,B of the corresponding CBO matrix, MO

γ = χ  ns Pˆ s χ,

Pˆ s  s s ,

(7)

s

the OAO representation of the projection operator onto the subspace of all occupied MO, which exhibit the non-vanishing occupation numbers in the configuration under consideration: ns > 0, s ns = N. The CBO matrices {γ} for these three orbital configurations {(2),  = 0, 1, 2} read:

 P  PQ

γ0 = 2 

PQ  , Q 



1 0  , 0 1 

Q  PQ

γ2 = 2 

γ1 = 

 PQ  . P 

(8)

It should be recalled at this point, that the recognition of the bonding/non-bonding/antibonding character of these prototype chemical interactions is lost in the Wiberg-type bondmultiplicity index [3-11] which in the 2-OAO model measures the square of A,B. The same shortcoming characterizes the non-projected bond indices from CTCB [19]. In a search for alternative IT indices, in which this distinction is preserved, the nonadditive Fisher information in AO resolution has been proposed as a good candidate for a sensitive detector of the MO bonding character [38]. In the 2-OAO model the relevant partition of the Fisher information densities (per electron) of the bonding and anti-bonding MO gives: fb = 4(b)2  f χ

total

[b] = 4[P(A)2 + Q(B)2] + 8 PQ A  B 

fa = 4(a)  f 2

total [a] χ

f χadd.

[b] +

f χnadd.

[b],

= 4[Q(A) + P(B)2]  8 PQ A  B 2

 fχ

add.

[a] + f χ

nadd.

[a].

(9)

In the strong (positive) overlap region between the two atoms, where A  B  0 (see Fig. 1), the corresponding non-additive contributions to the Fisher information density thus exhibit the following signs identifying the specific nature of the AO coupling into MO:

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f χnadd. [b] = 8 PQ A  B < 0, bonding; f χnadd. [a] =  8 PQ A  B > 0, anti-bonding.

(10)

Therefore, the covalent chemical bond, which exhibits the electron delocalization onto the bonding partner and hence decreases the “structure” (inhomegeneity) content in the molecular electron distribution compared to the free-atom distributions of the system “promolecule”, effectively lowers the Fisher information density of b in the bonding region, while the anti-bonding interaction between the two AO in a increases this Fisher information density between the two atoms. Since the Shannon entropy (direct measure of the electron delocalization, i.e., the “spread” in the electron probability distribution) represents the complementary descriptor to the Fisher information (measure of the electron localization, i.e., the “compactness” of the electron probability density), the covalent chemical bonding of b increases the density of the Shannon entropy S[p] in the bonding region, while the antibonding interaction of a decreases the density of S[p] in this spatial location [19,29].

Figure 1. The circular contour in (x,z)-plane of the vanishing CG integrand in the 2-OAO model of H2,

A,B(r) = 0, which separates the bonding region inside the contour circle passing through both nuclei, where A,B(r) < 0, from the region of positive contributions A,B(r) > 0, outside the circle. The two

vectors rA and rB, from the corresponding nuclei to the current electron location in space, are perpendicular on this dividing CG surface. For the specified location inside this spherical bonding region, the more anti-parallel are the gradients of two AO interacting in the bonding MO b, the stronger is the local bonding effect. Therefore, their scalar product gives rise to the negative nonadditive Fisher information density inside the limiting zerocontra-gradience surface. The opposite trends are predicted for the anti-bonding interaction of two AO in a [Eq. (10)]. In the bonding region A exhibits the negative projection on B, which justifies the name of the CG criterion itself.

The minimum basis set in H2 involves two 1s functions {A(r), B(r)} centered on nuclei A and B, respectively, and the scalar product of their gradients, A(r )  B(r ) , represents the density A,B(r) = A,B(r; R) of the CG integral, a function of the assumed bond length R [38], 2 I Ac , Bg ( R)   A(r ; R)  B(r ; R) dr    A, B (r ; R) dr = (1 + R  R /3) exp(R),

(11)

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Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

which determines the displacement in the system average kinetic energy of electrons T(R) with respect to the separated-atom value T() = 1(a.u.): T(R)  T() = I Ac,Bg ( R ) . The volume of the negative values of the non-additive Fisher-information density in H2, shown in Fig. 1, determines the bonding region of H2. It is enclosed by the spherical A,B(r) = 0 surface, which separates the bonding region of locations exhibiting A,B(r) < 0, inside the critical circular contour of the figure, from its anti-bonding surroundings exhibiting A,B(r) > 0. Indeed, for H2 the sign of A,B(r) is determined by the scalar product of two vectors rA and rB originating from the nuclei and pointing to the current location of an electron in space, so that A,B(r) = 0 surface constitutes the locus of points for which these vectors are mutually perpendicular. This condition thus gives rise to the circular-contour in the plane of section of Fig. 1, with the center at the bond mid-point, passing through both nuclei, and hence - the spherical bonding region between the two nuclei (see also Fig. 2). In the familiar orbital approximation of the SCF MO theory the ground-state wavefunction of the closed-shell system consisting of N = 2n electrons is approximated by the Slater determinant (N) = 11 ... n n of n lowest, doubly occupied MO, φ = (1, …, n) = χU, where χ = (1, …, m) groups the OAO (basis functions). The expectation value of the N-electron operator for the Fisher information ˆI( N ) [Eq. (5)], is then given by the trace of the product of the CBO matrix γ = 2UU† and the AO matrix representation of the one-electron Fisher-information operator ˆi( r ) = 4 [Eq. (4)]: n

n

s 1

s 1

I   f (r ) dr   ˆI   2  s ˆi  s  8  s



2

 tr ( γI )  8 T ,

I = χ ˆi χ  I i , j  4  i   j  4  i*   j

.

(12)

The Fisher information density per electronic pair,

f (r ) 

n 1 2 f ( r )  4   s ( r ) 2 s 1 m m

 2

i 1 j 1

i* (r )   j (r )

(13)

 n  *  2U j ,sU i ,s   2tr[ω( r ) γ ] ,  s 1 

where ω(r ) = {i,j(r) =  i* ( r )   j ( r ) }, can be then decomposed into the additive and non-additive components in this AO resolution [see Eq. (9)]: m m

m

m 1 m

i 1 j 1

i 1

i 1 j i 1

f ( r )  2 i , j ( r )  j ,i  2 i ,i ( r )  i ,i  4   f add . ( r )  f nadd . ( r ) .

 i , j (r )

 j ,i

(14)

Kinetic-Energy/Fisher-Information Indicators of Chemical Bonds In what follows, we shall explore the bonding basins {} of f

nadd.

567

( r ) , represented by

the largest (dominating) valence-shell volumes satisfying the inequality f nadd. ( r ) < 0, for a series of illustrative molecular systems. They are enclosed by the associated (closed)

f nadd. ( r ) = 0 surfaces and represent the crucial bonding regions of the molecule, which can be used to detect and qualitatively visualize the presence or absence of the direct bonds in the spatial region of interest. However, relatively small inner-shell basins, which reflect the information polarization of atomic cores and the nodal structure of the outer AO, should also be expected. Clearly, by integrating f

nadd.

( r ) over the specified bonding region  one could

generate the associated (condensed) index (“occupation”) of the  bond [24],

I 

f

nadd.

( r ) dr ,

(15)



which quantitatively reflects the lowering of the Fisher information in the chemical bond in question. Such descriptors, which will not be reported in the present work, should facilitate a comparison of a relative magnitude of the information changes involved in different bonds [26].

3. Numerical Results and Discussion Standard SCF MO calculations using the minimum (STO-3G) Gaussian basis set and the GAMESS software have been performed to explore the bonding regions in representative molecules. The relevant contour analysis has been carried out for the optimized geometries of all molecules. All contour maps are reported in atomic units. Some negative CG basins have also been shown in the perspective plots. For the visualization purposes we have used the Matpack and DISLIN graphic libraries.

3.1. Diatomics The representative homo- and hetero-nuclear diatomic molecules selected for the present analysis exhibit both the single (H2, HF, HCl, NaCl) and multiple (N2, CO) bonds. In Fig. 2 the contour map of the axial cut of the non-additive density f nadd. ( r ) of the Fisher information is shown for H2. This CG density is seen to be lowered in the spherical bonding region between the two nuclei, due to electron delocalization towards the other atom, which accompanies the covalent HH bond, as indeed qualitatively predicted in Fig. 1. At the same time the accompanying increases in these information/kinetic energy densities are observed in the non-bonding regions of each hydrogen atom, signifying the increased localization/structure in this homonuclear diatomic due to the axial polarization of the initially spherical atomic densities. It should be recalled that for the equilibrium internuclear distance the integral of the CG density over the whole physical space must be positive thus giving rise

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Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

to the overall “production” of the non-additive Fisher information in molecular hydrogen. This familiar result from the virial-theorem reflects the net increase in the average kinetic energy relative to the separated-atom value at the equilibrium bond length [38]. A similar analysis for HF is presented in Fig. 3. Both the perspective view of the nadd. f ( r ) < 0 volumes and the contour map of the axial cut of f nadd. ( r ) are shown in the figure. As shown in its two panels, there are three basins of decreased non-additive Fisher information: a large dominating valence bonding region 1 between the two atoms, and the remaining two small volumes in the inner-shell of F. The shape of the bonding volume exhibits its polarization towards the more electronegative fluorine atom, but the core electrons of the latter shift the valence bonding region towards the proton. One again detects the axial polarization of both atoms in their non-bonding regions, with the dominating information redistribution being observed on the heavy atom, a clear sign of its sp-hybridization promotion in the valence shell.

Figure 2. The contour map of the CG density

f (r )

for H2 (see also Fig. 1). The negative contour values

correspond to broken lines, while solid lines represent positive values of the non-additive density of the Fisher information.

In another heteroatomic HCl system, which is the subject of Fig. 4, again three regions of the decreased contra-gradience are observed. The softer heavy atom is now seen to undergo a more substantial inner-shell reconstruction of the non-additive Fisher information, with a

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569

relatively large (n = 2) “bonding” basin 2 being observed in addition to the largest, valence (n = 3) truly bonding volume 1 between the two nuclei; here n stands for the atomic principal quantum number. Again, there is the axial build-up of f nadd. ( r ) being observed in the non-bonding regions, particularly on hydrogen atom. It should be realized that the soft chlorine atom when combiner with the hard hydrogen generates a relatively stronger ionic (electron-transfer) component HCl, ultimately giving rise to the ionic pair H+Cl, and hence the smaller covalent (electron-sharring) component HCl in the resultant chemical bond, compared to that in HF where both atoms exhibit the “hard” (difficult to polarize) electron distributions in their valence shells.

Figure 3. The perspective view and the contour-map of the CG density distribution for HF.

In NaCl (Fig. 5) both atoms can be classified as “soft” (easy to polarize) in their n = 3 valence shells. Although this combination is conducive for a formation of a strong covalent component, the electronegativity difference between the two atoms should again generate a

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Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

relatively strong bond ionicity: NaCl. This is indeed observed in Fig. 5, where a smaller, valence bonding volume is detected in addition to two inner shell regions of a diminished contra-gradience. Notice, that in the donor, sodium atom this core basin is placed in the direction of the bond partner, while that on the acceptor, chlorine atom is located in the nonbonding axial direction of the latter. The opposite trend is observed in the inner shell regions of the increased contragradience: the one on the donor atom is located in its non-bonding region, while that on the acceptor atom is facing the other atom. This sign alternation reflects both the direction of the electron transfer, the sp-hybridization promotion, and the accompanying polarization of the Fisher information carried by the inner-shell electrons.

Figure 4. The same as in Fig. 3 for HCl.

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Consider next the triple chemical bond in CO (Fig. 6), where a large valence bonding basin is now distinctly extended away from the bond axis, due to the presence of two  bonds accompanying the central  bond. The accompanying small core-polarization basins, now almost symmetrically distributed near each constituent atom along the bond axis, are again observed in the perspective view, while the sp-hybridization reconstruction of the nonbonding regions on both atoms is again much in evidence in the accompanying contour map.

Figure 5. The same as in Fig. 3 for NaCl.

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Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

Figure 6. The same as in Fig. 3 for CO.

A similar bonding pattern of f nadd. (r ) is seen in N2 (Fig. 7), where again the dominating (bonding) region around the bond middle-point is now “squeezed” between the two cores of nitrogen atoms. Of interest also are the four small core volumes of the negative contragradience density around each nucleus. The marked extension of the valence bonding volume perpendicular to the bond-axis again reflects the presence of both  and  bonds and the large increases of the contra-gradience density in the non-bonding regions of two atoms reflect the sp hybridization involved in their axial promotion accompanying the bond-formation process. The small, axially placed core regions of the depleted contra-gradience are seen to be

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573

surrounded by the volumes of the positive values of this information density in transverse directions. They reflect the charge displacements accompanying the  bonds, which are also seen in the familiar density difference diagrams. The same feature transpires from the contour map for CO.

Figure 7. The same as in Fig. 3 for N2.

One thus concludes that the CG criterion for detecting the valence basins of a diminished non-additive Fisher information in AO-resolution indeed provides an efficient tool for locating the bonding regions in typical diatomics. It is also capable of revealing the atomic promotion/polarization processes accompanying the formation of chemical bonds, and the associated contour maps can be used to separate the effects due to the  and  bonds.

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Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

3.2. Polyatomics The illustrative selection of polyatomic systems includes water molecule (Fig. 8), representative hydrocarbons, ethane (Fig. 9), ethylene (Fig. 10), acetylene (Fig. 11), butadiene (Fig. 12) and benzene (Fig. 13), diborane (Fig. 14), and a series of small propellanes exhibiting steadily increasing length of the carbon bridges, [1.1.1], [2.1.1], [2.2.1], and [2.2.2] (Fig. 15), for which the CG analysis is summarized in the contour maps of Fig. 16. For butadiene, benzene, diborane and the four propellanes only the contour maps in selected cuts are reported. The position of the bonding basins in these maps is clearly seen as regions of the negative density of the non-additive Fisher information, which are represented by the brokenline contours.

Figure 8. The same as in Fig. 3 for H2O.

In water molecule of Fig. 8 one detects two (overlapping) basins of the negative nonadditive Fisher information in the OH bonding regions, and two small inner shell-regions of the negative contra-gradience. The bonding basins are located between the corresponding pairs of nuclei, which define these localized single bonds, and the lowering of the contragradience in each bond is seen to be the strongest in the direction linking the two nuclear attractors. The overlapping character of these two regions of the negative kinetic-energy displacement relative to separated atoms, reflected by the present non-additive Fisherinformation probe, indicates delocalization of the bonding electrons of one OH bond into

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575

the bonding region of the other chemical bond. The contour map in the molecular plane cut also reveals a strong buildup of this information/kinetic-energy quantity in the lone-pair region of oxygen, and – to a lesser degree – in the non-bonding regions of two hydrogens. This reflects a near tetrahedral sp3-hybridization promotion on the heavy atom.

Figure 9. The same as in Fig. 3 for ethane.

Next, let us examine a series of typical hydrocarbons exhibiting a monotonically increasing bond multiplicity between the two carbon atoms: ethane, ethylene, acetylene (Figs. 9-11). As witnessed by the perspective view of the bonding regions in Fig. 9, determined by the associated f nadd. (r )  0 surfaces, the CG criterion efficiently locates the valence basins of all localized CH and CC bonds in ethane, with the former being shifted more towards the proton, due to the presence of the inner shell in the carbon atom, which also reveals a small core region of the negative non-additive Fisher information. The contour map in the section passing through the central CC bond and the two adjacent (trans) CH bonds, also shows an increased CG density on each carbon atom in the non-bonding region between two

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Roman F. Nalewajski, Piotr de Silva and Janusz Mrozek

CH bonds located below and above the plane of section, respectively. This feature again reflects the tetrahedral (sp3-hybridization) promotion of the carbon atoms in this molecular framework.

Figure 10. The same as in Fig. 3 for ethylene.

Similar conclusions follow from examining Fig. 10 reporting results of the related analysis for ethylene. The first panel shows the perspective view of the negative contragradience, bonding regions in this molecule, while the contour map corresponds to cut in the molecular plane. One again detects small negative contra-gradience volumes in the inner shells of carbon atoms. The buildup of the positive contra-gradience near the carbon nuclei reflects the trigonal (sp2-hybridization) promotion of these two heavy atoms in the molecule. In acetylene (Fig. 11) the two cylindrical bonding regions of the CH bonds, axially extended due to the linear (sp-hybridization) promotion of both carbon atoms, and the central bonding basin due to the triple CC bond, transversely extended in the directions perpendicular to the bond axis, can be seen in both the perspective view of the negative contra-gradience volumes and in the accompanying contour map.

Kinetic-Energy/Fisher-Information Indicators of Chemical Bonds

577

Figure 11. The same as in Fig. 4 for acetylene.

Small inner shell volumes can again be detected. The f nadd. ( r ) > 0 regions on each carbon atom, very much resembling the atomic 2p distributions, reflect the presence of the bond  component. Indeed, the transfer of the valence 2p electrons from these spatial locations to the cylindrical bonding region between the two nuclei must result in the positive shift of the kinetic energy density, i.e., the positive contra-gradience reflecting the increased non-additive Fisher information density. In other words, the depletion of the 2p electron density near the carbon nuclei generates more structure in the electron -donating (non-bonding) regions of both carbon atoms, and hence less structure (more delocalization) in the -accepting (bonding) volume between the two nuclei. Among the three contour diagrams for butadiene (Fig. 12) the first panel corresponds to the molecular plane of section, which passes through all the nuclei and hence misses the effects due to the system of the delocalized  bonds between carbon atoms. In order to compare the peripheral (second panel) and central (third panel) -bonds in this molecule two

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additional contour maps have been generated in cuts perpendicular to the molecular plane, passing through the relevant pair of the carbon nuclei. The same map convention has been adopted in Fig. 13, reporting the associated results for benzene, with the first diagram presenting the cut in the molecular plane and the second, perpendicular cut exploring the effects of the -bond between the neighboring carbon atoms. The first diagrams of these figures again testify to the efficiency of the present CG probe in locating all localized CH and CC  bonds in these two -electron systems: nine bonds in butadiene and 12 in benzene.

Figure 12. The same as in Fig. 2 for butadiene. In the first diagram the map in the molecular plane is shown, while the middle and bottom diagrams report perpendicular cross sections along the peripheral and central CC bonds, respectively.

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Figure 13. CG analysis of the chemical bonds in benzene: the molecular plane section (first panel), and the perpendicular cut through the CC bond between the neighboring carbon atoms (second panel).

The non-bonding, positive values of f nadd. ( r ) near each carbon atom displayed in these molecular-plane cuts reveal the trigonal hybridization promotion of each heavy atom, and the axial polarization of each bonded hydrogen is detected.

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Figure 14. Two mutually perpendicular cuts of the non-additive Fisher information densities in diborane B2H6 passing through the two boron atoms and including the hydrogen bridges (upper panel) and involving atoms of both peripheral BH2 groups (lower panel).

The diborane (B2H6) results reported in Fig. 14 include the two mutually perpendicular cuts, in the planes of section passing through the two boron atoms: the first diagram corresponds to the plane including the hydrogen bridges while the second panel uses the cut involving all atoms of both peripheral BH2 groups. The three-center character of the bridging bonding region BHB is clearly seen in the first plot of the figure. This panels also reveals some hybridization promotion of the boron atoms, in their non-bonding locations, between the localized BH bonds of the two terminal BH2 groups. Their bonding patterns are exposed in a more detail in the second contour diagram of the figure. The associated contours

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are seen to be distributed very much like those in the localized CH bonds in hydrocarbons. These two plots also confirm that the direct bond between the boron bridgeheads is missing in diborane.

Figure 15. The propellane structures and the planes of sections containing the bridge and bridgehead (C’) carbon atoms, identified by black circles.

Finally, the bonding patterns in a series of four small propellanes shown in Fig. 15, which were recently the subject of the information-theoretic [19,29] and ELF [30] studies, are examined in Fig.16. Each row of this figure is devoted to a different propellane, arranged from the smallest [1.1.1]-molecule, exhibiting three single-carbon bridges, to the largest [2.2.2]-system, consisting of three two-carbon bridges. The main result of the previous studies, namely, the lack of the direct bond between the carbon bridgeheads in the [1.1.1]and [2.1.1]-systems, and the presence of practically the single bond in the [2.2.1]- and [2.2.2]propellanes [12,19,21], remain generally confirmed by the present CG probe, but this transition is now less sharp, with much smaller bonding basins between bridgeheads being observed also in the two smallest molecules. Indeed, this transition from the missing central bonding in the [1.1.1]-system to the full central bond in [2.2.2]-system is now seen to be less abrupt: a very small bonding basin identified in the former case is steadily evolving into that associated with the full bond in both [2.2.1]- and [2.2.2]-propellanes. The localized chemical bonds of the carbon bridges and the CH are again seen to be perfectly localized by the vanishing CG surfaces. This analysis generally confirms the previous IT probes into these systems summarized in Fig. 17. Indeed, it follows from this comparison that no accumulation of the electronic density or the related information densities is observed between the two bridgehead carbons in the two smallest propellanes, a clear sign of the absence of any direct bending between these two atoms. It is observed in the two largest systems, thus confirming the presence of the direct bonding when the number of the two-carbon bridges is sufficient. It should be emphasized, however, that all systems, including the smallest propellanes, exhibit the indirect central bond, realized via the carbon bridges [33-36].

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[1.1.1]: C5H6

[2.1.1]: C6H8

Figure 16. (beginning).

Conclusion In this review we have demonstrated that the density of the non-additive Fisher information in AO resolution, which we call the “contra-gradience” distribution [38], does indeed provide a sensitive and efficient probe for locating the chemical bonds. More specifically, the valence regions of its negative values mark the location of bonds in the molecule. This novel visualization tool should prove useful in exploring bonding patterns of controversial molecular systems, the bonding structure of which still remains a matter for scientific debate, with alternative bond criteria giving conflicting answers to the very

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qualitative question of the existence or non-existence of the disputed chemical bonds between the specified atoms in the molecular environment under consideration.

[2.2.1]: C7H10

[2.2.2]: C8H12

Figure 16. (conclusion). The CG analysis of chemical bonds in the four propellanes of Fig. 15 focusing on the central and bridge CC bonds. The planes of sections in the left panels correspond to the plane perpendicular to the central bond between the bridgehead carbons, passing through its midpoint, while the right panels contain the central and a bridge CC bonds.

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Figure 17. A comparison between the (equidistant) contour maps of the density-difference function, (r) = (r)  0(r) (first column), the information-distance density of Kullback and Leibler [16,17], s(r) = p(r)log[p(r)/p0(r)] (second column), and the density of the Shannon-entropy displacement, ℎ(r) = 0(r) log0(r)  (r) log(r) (third column), for the four propellanes of Fig. 15; here, (r) = N p(r) and 0(r) = Np0(r) stand for the molecular and promolecular electron densities, while their “shape” factors p(r) and p0(r) denote the associated probability distributions.

Most of existing theoretical interpretations of the origins of the covalent chemical bond emphasizes exclusively the potential (interaction) aspect of this phenomenon, focusing on the mutual attraction between the accumulation of electrons between the two atoms (a negative “bond-charge”) and the partially screened (positively charged) nuclei. The CG criterion, reflecting the non-additive Fisher information in the basis-function/AO, resolution, adopts the complementary view by stressing the importance of the kinetic-energy bond component. This is similar to ELF [1,2], which has also been shown to reflect the inverse of the non-additive Fisher information in the MO resolution [30]. The pattern of the non-additive Fisher information density is always very much polarizational in character, with the closed bonding-regions of the negative CG being separated by the molecular environment of the positive values of this quantity, marking the system non-bonding regions. Thus, the depletion of the CG density in the bonding region is accompanied by its increases in the nearby regions, so there is but little net increase in the overall Fisher information of the molecular system compared to the atomic promolecule. Indeed, the overall Fisher information content of the molecule is not conserved since it is related to the average kinetic energy of electrons [38]. Since the molecule exhibits a higher

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degree of “structure”, due to the net overall contraction of the atomic electron distributions, it exhibits the net increase in the Fisher information content, and hence also in the average kinetic energy of electrons, as indeed implied by the molecular virial theorem. More specifically, for the equilibrium geometry of the molecule, i.e., the vanishing forces acting on the system nuclei, T = E. Therefore, the negative displacement in the average electronic energy of the given molecule M relative to its atomic promolecule M0, due to formation of chemical bonds, E = E  E0 < 0, indeed implies the associated positive displacement in the system average kinetic energy, T = T  T0 > 0, which is proportional to the overall Fisher information: T  I = I  I0. However, that the overall kinetic/Fisher-information component only blurs the picture of the true information origins of chemical bonds, stressing its increase as a result of the chemical bond formation, in accordance with the virial theorem. Only by focusing on the non-additive part of this component can one uncover the real information origins of chemical bonds, and define the useful local probes for their localization and - eventually - for their quantitative information descriptors.

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