(Physics research and technology) amlan k roy (ed ) theoretical and computational developments in modern density functional theory nova science publishers, inc (2013)

Chapter 10 Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional Theories in the Presence of a Magnetostatic Field 223 Xiao-Yin Pan and Viraht Sahni Chapter 11 The Construction of

T HEORETICAL AND C OMPUTATIONAL

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or

by any means The publisher has taken reasonable care in the preparation of this digital document, 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 herein This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services

Additional books in this series can be found on Nova’s website

under the Series tab

Additional e-books in this series can be found on Nova’s website

under the e-book tab

T HEORETICAL AND C OMPUTATIONAL

New York

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

1 Density functionals 2 Functional analysis I Roy, Amlan

C ONTENTS

Chapter 1 Density Functional Theory: From Fundamental Precepts

to Nonlocal Exchange- Correlation Functionals

1

Rogelio Cuevas-Saavedra and Paul W Ayers

Chapter 2 Recent Progress towards Improved Exchange-Correlation

Density-Functionals

41

Pietro Cortona

Chapter 3 Constrained Optimized Effective Potential Approach

for Excited States

61

V N Glushkov and X Assfeld

Chapter 4 Time Dependent Density Functional Theory of Core Electron

Excitations: From Implementation to Applications

103

Mauro Stener, Giovanna Fronzoni and Renato De Francesco

Chapter 5 Time Dependent Density Functional Theory Calculations

of Core Excited States

149

Nicholas A Besley

Chapter 6 Density Functional Approach to Many-Electron Systems:

The Local-Scaling-Transformation Formulation

169

Eugene S Kryachko

Chapter 7 Electron Density Scaling - An Extension to Multi-component

Density Functional Theory

189

Á Nagy

Chapter 8 A Symmetry Preserving Kohn-Sham Theory 201

Andreas K Theophilou

Chapter 9 Self-Interaction Correction in the Kohn-Sham Framework 211

T Körzdörfer and S Kümmel

Chapter 10 Hohenberg-Kohn, Kohn-Sham, and Quantal Density Functional

Theories in the Presence of a Magnetostatic Field

223

Xiao-Yin Pan and Viraht Sahni

Chapter 11 The Construction of Kinetic Energy Functionals

and the Linear Response Function

255

David García-Aldea and J E Alvarellos

Chapter 12 Variational Fitting in Auxiliary Density Functional Theory 281

Víctor Daniel Domínguez Soria, Patrizia Calaminici and Andreas M Köster

Chapter 13 Wavelets for Density-Functional Theory and

Post-Density-Functional-Theory Calculationss

313

Bhaarathi Natarajan, Mark E Casida, Luigi Genovese and Thierry Deutsch

Chapter 14 Time-Dependent Density Functional Theoretical Methods

for Treatment of Many-Electron Molecular Systems

in Intense Laser Fields

357

Dmitry A Telnov, John T Heslar and Shih-I Chu

Chapter 15 A Hierarchical Approach for the Dynamics of Na Clusters

in Contact with an Ar Substrate

391

P M Dinh, J Douady, F Fehrer, B Gervais, E Giglio,

A Ipatov, P G Reinhard and E Suraud

Chapter 16 Atoms and Molecules in Strong Magnetic Fields 425

M Sadhukhan and B M Deb

Chapter 17 Chemical Reactivity and Biological Activity Criteria from DFT

Parabolic Dependency E=E(N)

449

Mihai V Putz

Chapter 18 Effect of a Uniform Electric Field on Atomic

and Molecular Systems

485

Santanu Sengupta, Munmun Khatua and Pratim Kumar Chattaraj

Chapter 19 A Quantum Potential Based Density Functional Treatment

of the Quantum Signature aof Classical Nonintegrability

505

Arup Banerjee, Aparna Chakrabarti, C Kamal and Tapan K Ghanty

Chapter 20 Properties of Nanomaterials from First Principles Study 527

Arup Banerjee, Aparna Chakrabarti, C Kamal and Tapan K Ghanty

Chapter 21 The Role of Metastable Anions in the Computation

of the Acceptor Fukui Function

549

Nelly González-Rivas, Mariano Méndez and Andrés Cedillo

Chapter 22 Kinetic-Energy/Fisher-Information Indicators of Chemical Bonds 561

Roman F Nalewajski, Piotr de Silva and Janusz Mrozek

Today, our theoretical understanding of many-electron systems is largely dictated anddominated by Density functional theory (DFT) It plays a unique pivotal role for realisticand faithful treatment of materials in diverse fields such as chemistry, physics and biology.

In many important research areas dealing with atoms, molecules, solids, clusters, terials including organic molecules, biomolecules, organometallic compounds, etc., DFThas become an indispensable and invaluable tool for nearly three and a half decade Range

nanoma-of application is updated almost on a regular basis Numerous exciting developments havebeen made in recent years which render quantum mechanical calculation of larger and largersystems more accurate and computationally approachable, which were otherwise impossi-ble 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 devel-opments that took place lately, which have helped us in extending our knowledge on theelectronic structure of materials Fundamental and conceptual issues, formulation andmethodology development, computational advancements including algorithm, as well asapplications are considered However, a topic as broad as DFT can not be covered in asingle volume such as this The chapters are mostly focused on theoretical, computational,conceptual issues, as the title implies Therefore, purely application-oriented works are notincluded; 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 fromcomplete 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 expressedinterest, 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 correlation (XC) functionals Its exact form remains unknown as yet and must be ap-

exchange-proximated 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 func-tionals Chapter 1 discusses the inadequacy of conventional XC functionals, supposedlyrooted in their inability to recover appropriate behavior for fractional charges as well asfractional spins This arises primarily due to the neglect of dispersion interactions and

strong correlation between non-spatially-separated electrons This leads to the ment of nonlocal 2-point weighted density approximated (2-WDA) functionals which arerigorously self-interaction free, closely mimics the proper fractional charge and spin behav-ior This also produces dispersion interactions with the correct R− 6

develop-form and appears tohold great promise for the future advancements in XC functionals Chapter 2 focuses on alocal (SRC) and two generalized-gradient-type functionals (TCA and RevTCA) The localone offers very similar results as the LDA functional for equilibrium bond lengths while foratomization energies and barrier heights surpasses the LDA results The TCA functionalprovides good results for thermochemistry, geometry and excellent-quality results for hy-drogen bonded systems The last one is found to be quite good for atomization energies andbarrier heights

Although DFT has witnessed remarkable success for ground states, the same for cited states has come much later and somehow rather less conspicuous Chapter 3 presents

ex-a constrex-ained vex-ariex-ationex-al ex-approex-ach bex-ased on the ex-asymptotic projection method ex-along withits applications to the optimized effective potential problem This facilitates the solution

of relevant Kohn-Sham (KS)-type equation to handle appropriate local potential for excitedstates within the framework of both variational and non-variational approaches The useful-ness 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 thecore excitations which are notoriously difficult due to the presence of delicate correlationeffects High accuracy results are obtained in Chapter 4 for fine spectral features of smallmolecules in the gas phase, correctly taking into account the crystal field effect, configura-tion mixing and spin-orbit coupling Chapter 5 sketches the current progress towards theNEXAFS spectra of relatively large systems including biologically significant moleculesthrough TDDFT and development of suitable XC functionals in this regard

Chapter 6 reviews the so-called local-scaling-transformation of DFT for many-electronsystems 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 rele-vance in non-adiabatic processes Here, both the electrons and nuclei can be treated com-pletely quantum mechanically without the use of Born-Oppenheimer approximation Thisgives 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 thescaling 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 rately in terms of the KS orbitals The correlation energy, on the other hand, is not easilyexpressible in terms of the orbitals A simple method to incorporate a major portion ofcorrelation is also given

accu-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 statereproducing 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

Self-interaction remains one of the serious and nagging problems in DFT, and is sumably responsible for many qualitative defects of today’s XC functionals Apparentlythe 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 self-interaction correction that employs orbital-specific potentials Different KS self-interactionapproaches are possible by means of different choices of the unitary transformation Severalsuch schemes are compared and contrasted with the traditional self-interaction approaches

pre-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 amagnetostatic field, B = ∇ × A(r) In presence of an external field, v(r), the basicvariables in all these theories are the ground-state density ρ(r) and physical current den-sity, j(r) This is achieved by proving the relationship between densities {ρ(r), j(r)} andexternal 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 gaugefunction to ensure that the wave function expressed as a functional, is gauge variant Ex-tension of these to other Hamiltonians such as those in spin DFT or in which magnetic fieldinteracts with both orbital and spin angular momentum, etc., is also considered

Chapter 11 reviews some of the most important nonlocal kinetic energy density tionals available today, all of which reproduce the linear response function of a free electrongas General strategies behind the construction of these functionals are discussed, that makethem suitable for use in both extended and localized electron systems It is stressed that thelocal behavior of kinetic energy densities should be used as the guiding factor for designingnew functionals as the latter is closely related to the potential These ideas may also haverelevance for XC functionals as well

func-In Chapter 12, variational fitting of auxiliary densities in DFT is discussed in tail Through an iterative solution of the fitting equations, auxiliary DFT allows accurateand efficient first principles all-electron calculations of complex systems containing 500-

de-1000 atoms A combination of singular value decomposition and preconditioned conjugatemethod offers a viable compromise between numerical stability and accuracy By using dif-fuse 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 dard Fourier-transform techniques are well known for multi-resolution problems with com-plicated boundary conditions However, in the context of quantum chemistry or chemicalphysics, their usefulness and applicability has remained largely unexplored until very re-cently At first, an elementary review of the subject is given Then the authors discuss thetheory behind the wavelet-based BIGDFT code for ground-state DFT and application of thesame in the linear-response TDDFT The possibility of making high-performance comput-ing order-N wavelet-based TDDFT program for practical calculation of larger systems isalso mentioned briefly

stan-Chapter 14 reviews the latest developments in TDDFT front for studying the dynamicalbehavior of many-electron atoms/molecules interacting with a strong laser field Use ofoptimized effective potential plus self-interaction correction facilitates the use of orbital-

independent, one-electron local potential reproducing correct asymptotic behavior ture and dynamics are followed by solving the relevant KS equations quite accurately effi-ciently in a non-uniform, optimal spatial grid by means of the generalized pseudospectralmethod Illustrative results are presented for multi-photon processes in diatomic and tri-atomic molecules through multi-photon ionization, high-order harmonic generation, etc.

Struc-A hierarchical method is presented in Chapter 15 in order to study the dynamics of smallmetal 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 onstructural properties, dynamics in the linear-response regime as well as non-linear dynamicsinduced by strong femto-second laser pulses are given for small Na clusters in contact with

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 chemicalpotential) and chemical hardness, the two first- and second-order parameters respectively.Numerous applications ranging from atomic to molecular systems, as well as chemicallyreactive to biologically active environments are considered Feasibility of DFT parabolicrecipe for describing reactivity-activity principle in open systems (chemical, biological) bymeans of electronegativity and chemical hardness, viewed as “velocity/slope” and “accel-eration/curvature”, in an abstract way, is explored

Effect of a uniform electric field on atomic, molecular systems is investigated in ter 18 within the KS approach Writing the total energy in terms of a Taylor series expan-sion, interesting results are obtained for neutral atoms and ions Explicit expressions aregiven for ionization potential and electron affinity changes in atoms Consequently, it re-veals that electronegativity of an atom exhibits an increment, when immersed in a uniformelectric field And hence the chemical reactivity of a system in such a field will be differentfrom that in absence of the field Molecules in excited states exhibit pronounced geometri-cal changes; excitation energies are decremented under the influence of such a field Effect

Chap-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 thequantum-domain behavior of classically non-integrable systems in Chapter 19 Quantumsignature of classical Kolmogorov-Arnold-Moser-type transitions in different anharmonicoscillators is probed starting from a toroidal to chaotic motion Field-induced barrier cross-ing as well as the chaotic ionization in Rydberg atoms is also analyzed through such aquantum potential-based method In the zero quantum-potential limit, a classical-like sce-nario is restored for a couple of quantum anharmonic oscillators

Theoretical investigation on the structural and optical response properties is madefor various nano-clusters, nano-tubes and nano-cages (C20, C60, C80, C100) through DFTand 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 (Au X, X =

Li, N a, K, Rb, Cs, Cu, Ag) as well as mixture of Ga and P atoms (GanPn) have beenconsidered, 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, connect-ing this to the frequency-dependent dipole polarizability at imaginary frequencies.

Chapter 21 concerns with the computation of vertical electron affinity and acceptorFukui function The role of metastable anions in the latter case is also examined Chemicalreactivity descriptors (local as well as global) for neutral molecules are severely restrictedfor unstable anions The bound state electronic structure of such an unstable anion is sat-isfactorily obtained through an orbital swapping method Applications of the methodologyare 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 applicabilityand 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, takingtheir 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 DFTwhile I was working as a graduate student in his laboratory Fruitful and valuable discus-sion 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 discussionwith the IISER-K colleagues and students has helped me gain a deeper understanding ofthe subject with time This book could not have been possible without the generous sup-port 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

Chapter 1

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: ayers@chemistry.mcmaster.ca, ayers@mcmaster.ca

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)

(2) 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

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

reads

(3)

where we have denoted the external potential by v(r) and grouped the kinetic and

electron-electron repulsion energies in one term, denoted 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,

, which represents the probability than an electron located at has spin , another electron located at has spin , etc

Using this probabilistic interpretation of the wave function, the probability of observing

an electron of either spin at position is given by

(4)

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,

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

different ground-state wave functions and external potentials and Because the ground-state wave functions are different, the external potentials differ by more than a constant shift, Accordingly, to the variational principle we have

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

ˆ

ˆ

N i i

N i i N i i N i i

which simplifies to

(8) Invoking the definition of the electron density, (5), one obtains

where we have denoted the ground-state densities of the corresponding wave functions by and Equation (9) implies that the two systems have different densities because, were the densities the same, the integral would be zero, not negative-valued Since no two external potentials correspond to the same ground state density, the external potential can be written as

a functional of the ground-state density Since the ground-state density determines the number

of electrons and the external potential, it also determines all the properties of an electronic system

A more general discussion of the Hohenberg-Kohn theorem can be found in the works of Levy[12] and Englisch and Englisch.[13]

C The Variational Principle for the Ground-State Electron Density: The Second Hohenberg-Kohn Theorem

The first Hohenberg-Kohn theorem is an existence theorem: it shows that, in principle,

we can obtain the electronic Hamiltonian, the ground-state wave function, and system properties like the ground-state energy from the ground-state electron density But it does not say how to obtain these properties in a practical way Practical calculations in wave function theory are often based on the variational principle There is a similar variational principle for the ground-state electronic density, and this variational principle is also the key to practical DFT calculations

Consider a system of N-electrons with electron density ρ0(r) Consider an external

potential, v1(r), for which ρ0(r) is not a ground-state electron density We have already

observed that the purely electronic contribution to the energy,

(10)

is a universal functional of the electron density and does not depend on the external potential

of interest (F[ρ] is called the Hohenberg-Kohn functional.) It can be easily verified that the

energy due to the electrostatic interaction between the electrons and the external potential is given by

(11) Therefore the total energy of the system is

(12) From the same energy inequality that was used to derive the first Hohenberg-Kohn theorem (cf.(7)),

(13) The equality in equation (13) holds only if is an electron density for the system with

external potential v1(r) This equation is of crucial importance in DFT and is usually referred

to as the second Hohenberg-Kohn theorem[11]

Second Hohenberg-Kohn Theorem A universal functional for the energy in terms of the

electronic density can be defined, valid for any external potential

For any particular external potential, the exact ground-state energy of the system is the global minimum of this functional, and any density that minimizes the functional is a ground-state density

The second Hohenberg-Kohn theorem is the foundation of all practical procedures for finding the ground-state electron density,

(14)

D Orbitals Regained: The Kohn-Sham Equations

The first Hohenberg-Kohn theorem says that the ground-state electron density contains enough information to determine all the properties of an electronic system, including the ground-state energy, but it does not say how that information can be extracted The second Hohenberg-Kohn theorem says that the ground-state energy can be found by minimizing the energy with respect to the electron density, but it does not say how the energy functional can

be determined In wave function theory, the energy was a simple functional of the wave

function, but in DFT the exact energy functional, E v [], is not known in any practical and

explicit form DFT swaps a difficult computational problem (solving the Schrodinger

equation) for a difficult theoretical problem (finding accurate expressions for E v []) This

chapter reviews approaches to this theoretical problem

The first attempts to obtain properties of systems directly from the electron density dates

to the 1920’s and 1930’s, long before the establishment of the Hohenberg-Kohn theorems Some of the most important developments in this direction were due to Thomas and

Fermi[14,15] (kinetic energy), Dirac (exchange energy), Wigner[16] (correlation energy), and Weizsäcker[17] (improved kinetic energy) If we decompose the Hohenberg-Kohn functional into its kinetic and electron-electron repulsion energy contributions,

(15)

it is observed that it is much more difficult to approximate the kinetic energy than it is to approximate the electron-electron repulsion energy For example, if a primitive approximation to the kinetic energy is used, one does not find any chemical bonding whatsoever.[18] However, if the exact kinetic energy functional is combined with a primitive electron-electron repulsion functional (e.g., combining classical Coulomb repulsion, Dirac exchange and Wigner correlation), qualitatively correctly chemical behaviour is observed Accordingly, when pursuing quantitative accuracy from DFT, the idea of expressing the kinetic energy directly in terms of the electron density is usually abandoned Instead, an auxiliary set of orbitals is introduced for the sole purpose of approximating the kinetic energy These orbitals are themselves functionals of the ground-state electronic density and constitute

an accurate approximation to the kinetic energy[19]

(16) The are called the Kohn-Sham orbitals We restrict ourselves to closed shell systems and

so each orbital is doubly occupied The Kohn-Sham kinetic energy is denoted by is the correlation-kinetic energy, which represents the correction to the orbital model Fortunately, is small

To motivate the Kohn-Sham method, consider the form of the electronic Hamiltonian, (3)and notice that if it were not for the electron-electron repulsion term, the Hamiltonian would be expressible as a sum of one-electron operators and the electronic-Schrödinger equation could be solved easily by separation of variables This observation motivates replacing the electron-electron repulsion operator by an average local internal potential

(similar to the idea behind the Hartree-Fock equations) Denoting this potential by w(r), the

Hamiltonian takes the following separable form

(17) Solving the Schrödinger equation associated to this Hamiltonian is equivalent to solving the following one-electron Schrödinger equations

1ˆ

(18) This implies a natural approximation to the ground-state wave function of the system as the Slater determinant of the lowest energy spatial orbitals, , with the appropriate spin factors

How should one choose w(r)? Motivated by the idea that the electron density determines

all properties, including the differences between the properties of the non-interacting model

system (defined by Eq (17)) and the true interacting system, Kohn and Sham defined w(r) so

that the ground-state density from the model Hamiltonian, (17), has the same ground-state electronic density as the interacting system.[19] This implies that the interacting-energy functional,(12), and the non-interacting energy functional,

(19) are minimized by the same electron density (We have included as superscript KS in the functional (19) to specify that it refers to the Kohn-Sham reference system.) The constraint on the optimizing density of the Kohn-Sham system forces the internal potential to be the sum of two functional derivatives,

(20) The first functional on the right-hand side of (20) is the classical electrostatic repulsion energy functional

The functional derivative of the electrostatic repulsion energy is called the Coulomb potential or, somewhat more descriptively, the electronic electrostatic potential

(23) The functional derivative of the exchange-correlation energy is the exchange-correlation potential,

(24) With this notation, we can write the Kohn-Sham equations in their conventional form,[19]

(25)

(26) Since (25) and (26) depend on the Kohn-Sham orbitals, the Kohn-Sham equations have to

be solved self-consistently Solving the Kohn-Sham equations is therefore similar to solving the Hartree-Fock equations

It should be kept in mind that the Kohn-Sham wave function (the Slater determinant formed with the Kohn-Sham orbitals) is not expected to be a good approximation to the exact wave function In fact, the Kohn-Sham wave function has a higher energy than the (variationally optimized) Hartree-Fock wave function, and in this sense is a worse approximation However, because the density obtained from the Kohn-Sham method is exact,

we can correct the errors from the Kohn-Sham functional with a density functional In particular, the energy can be written as:

(27) This equation has the same general form as many equations in Kohn-Sham DFT: the exact value of any property can be written as the value for that property given by the Kohn-Sham wave function, plus a correction that is written as a density functional

E Spin-Density Functional Theory

So far, we have discussed only closed shell systems, that is, systems where the number of up-spin electrons and down-spin electrons are equal and all orbitals are doubly-occupied Especially when the total spin of the system is different from zero, it is more accurate (but it

is not required) to construct dependent Kohn-Sham equations The key elements of density functional theory are:

spin-1 The spin density for the α- and β-spin electrons The spin-densities add up to the total electron density,

2 The exchange-correlation density functional and the corresponding

spin-resolved exchange-correlation potential

3 The unrestricted Kohn-Sham equations

(28)

(29)

4 The energy expression in terms of the spin-density functionals,

(30) For simplicity, we will focus on the original, spin-free, Kohn-Sham method in the remainder of this chapter

F The Exchange-Correlation Energy Functional

The Kohn-Sham method is exact if the exact is used, but unfortunately, the exact exchange-correlation energy functional is not known in any practical explicit form Fortunately, existing approximate exchange-correlation energy functionals usually provide accurate estimates for the energetic properties of importance for chemical kinetics and thermodynamics There is still great interest in developing new approximations for the exchange-correlation energy functional, and this is the focus of the remainder of this chapter Before approximating the exchange-correlation energy functional, it is necessary to understand the form of the exact functional Kohn-Sham DFT relies on a model system of non-interacting electrons, with the energy expression

N i i

(31) that has the same ground-state electron density as the real system of interacting electrons, with the energy expression

(32)

A whole range of other systems, with identical electron densities but different electron repulsion strengths, can be constructed The corresponding energy expression is

electron-(33) Using these systems, one can construct a connection between the ideal non-interacting system (the Kohn-Sham system, λ = 0) and the actual interacting system (λ = 1) Using the fundamental theorem of calculus, one can then derive an expression for the exchange-correlation energy,

(34) Invoking the Hellman-Feynman theorem,[20,21] this simplifies to

0 0

The pair density represents the probability of observing two electrons, one at r and the other one at rʹʹʹ The exchange-correlation energy can then be expressed as

(37) where (r,rʹʹʹ) denotes the value of (r,rʹʹʹ) averaged over the adiabatic connection path By analogy to the way the pair correlation function is modelled in classical liquids, the

(adiabatically averaged) exchange-correlation hole is defined as

(38) The exchange-correlation energy functional can then be written in the deceptively simple form,

(39) Our research focuses on how one may develop new forms for the exchange-correlation energy functional by directly modelling the exchange-correlation hole

The second equality in (39) can be utilized to illustrate an important feature of the exchange-correlation hole It is often assumed that the spherically-averaged exchange-correlation charge,

(40)

is strongly localized near r This is usually a good assumption: it is often true that reaches its

minimum at and rises rapidly and (nearly) monotonically to its asymptotic limit as increases

This suggests that information about the electron density at and near r may be enough to form

an effective approximation to The simplest choice is to express as a function of

the electron density at r This leads to Local Density Approximations (LDAs)[19] If, instead,

is expressed as a function of and at r, this leads to Generalized Gradient

Approximations (GGAs) If is written as a function of the density, its gradient, and the

kinetic-energy density at r, then one has a meta-GGA The exchange component of the

'

' , ' 1

'

xc xc

exchange-correlation hole is known exactly; when some or all of this exact exchange component is included, one has a hybrid-GGA In systems where has a more complicated

structure, with multiple maxima and minima as a function of u, all of these conventional

approximation techniques usually fail

Instead of the exchange-correlation hole or the exchange-correlation charge, sometimes it

is more convenient to model exchange-correlation effects using the pair correlation function,

(41) The pair correlation function is the correction factor that relates to the ideal independent electron model (with electrons that do not experience Coulomb repulsion or Pauli exclusion)

to the exact pair density,

assumes that can be modelled based only on the value of the electron density at r If this is

true, then, based on the idea that E xc is a universal functional of the electron density, any system with the correct electron density can be used to estimate The simplest such system is the uniform electron gas (UEG): an infinite system with the same electron density throughout.[26,27] The exchange-correlation energy functional defined from

The UEG consists of N electrons enclosed in a box of volume V (which is periodically

repeated in space) in the presence of a uniform neutralizing background of positive charge

with density N/V The Hamiltonian of this system is thus

(44) where Λ represents the effect of the background When studying this model one is usually interested in its macroscopic properties, which are obtained in the thermodynamic limit

(where the number of electrons and the volume tend to infinity but their quotient, ρ = N/V,

remains fixed)

In analogy to the natural “atomic units” that are used in molecular quantum mechanics, it

is convenient to define a characteristic scale for length and momentum in the uniform electron

gas The characteristic length scale of the UEG is the Wigner-Seitz radius; this is the radius of

a sphere that encloses, on average, one electron:

Since the UEG system is homogeneous and isotropic, its pair distribution function depends only on the relative position of the electrons and parametrically on its density (and therefore its Wigner-Seitz radius) and its spin polarization This is convenient because it means that the exchange-correlation charge of UEG is already spherically symmetric, which saves us the difficulty of taking its spherical average and makes the LDA an especially simple approximation

B The Exact Exchange Hole of the Uniform Electron Gas

Just as the electron density can be expressed in terms of one-electron wave functions (orbitals), the electron pair density can be expressed in terms of two-electron wave functions (geminals) The mathematical development is simplest for the electron pair density of the non-interacting uniform electron gas.[27,28] For the interacting UEG, the two-electron wave functions are found by solving a scattering problem with a properly chosen effective potential, subject to the constraint that the solutions for the non-interacting UEG are recovered when the effective potential is zero For the sake of simplicity we focus only on the case

Let us select a pair of electrons at random in the spin-unpolarized uniform electron gas There is one chance in four that they will be in the singlet state and three chances in four that they will be in one of the three possible triplet states If there is no electron-electron interaction, the corresponding two-electron wave functions can be expressed

in the center of mass reference frame,

(51)

It is often useful to know the spin-resolved pair distribution functions corresponding to parallel and anti-parallel spin interactions For the unpolarized gas and one-third of the triplet state contributes to the anti-parallel spin correlation function and two thirds of it to the parallel spin correlation function Therefore

(52)

(53) From (49) it is possible to see that the two-electron wave functions in (48) can be expressed in terms of spherical Bessel functions Using (53) and performing the average with

respect to p(k), one finds that

(54) Therefore the exchange hole for the UEG is given by

(55)

C Short-Range Correlation in the UEG: The Overhauser Model

Notice that the approach just presented is entirely equivalent to solving the corresponding radial Schrodinger equation That is, solutions to the radial Schrödinger equation with no interacting potential,

2 1

odd

2 1

sin 2 1

2

1 9

l l

l

F F

kr kr

(56)

and therefore the exchange hole can be obtained following (54) The correlation contribution

to the pair correlation function can be obtained by inserting a suitable electron-electron interaction potential in (56) A simple model for the effective electron-electron interaction was proposed by Overhauser.[27,28]

In the Overhauser model, within a sphere with radius r s (volume ρ-1), the screening

charge density is ρe; outside this sphere the screening charge is zero The resulting electron potential is the electrostatic potential due to a point charge -e at the origin plus a sphere around it of radius r s and of uniform positive charge ρe I.e.,

electron-(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 <r s it is expected that the potential is close to the true potential felt by an electron moving in a uniform electron gas when another electron is fixed

at the origin In the region r >r s 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 r s becomes smaller as the density decreases Finally, at high and intermediate densities the results for the exchange-correlation 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 r s, 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

(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

(61) where corresponds to the pair distribution function when the electron-electron interaction

is In this case, the coupling constant is equivalent to an average over

(62) where we have indicated the usual separation of (and consequently ) into its exchange and correlation contributions

Notice that r s is not part of the arguments for the exchange contribution in (62) This is

because the explicit dependence on r s occurs only when the Coulomb repulsion is taken into account

Both g x and g c 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 short-range 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 non-oscillatory 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,

(66) and the potential energy of correlation,

(67)

in the UEG

The short-range behaviour of the pair correlation distribution function is governed by the

so-called cusp condition[31]

(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

1 1

u= 0

xc u= 0

(69) The corresponding coupling constant average can be obtained by replacing with 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 g xc corresponds to the short-range behaviour

of the static structure factor, which is constrained by the plasmon sum rule[30,33]

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 S xc 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 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

(72) Where

0, ,

3 2

The correlation piece is fit to the form

(74) Where

The constants A x , B x , C x , D x , Ex and F x in equation (73) and a 0, a 1 , a 2 , a 3 , b 2 and 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

2,

d x s

F

n s n

(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

r r

(90)

(91) where

(92)

(93)

(94)

(95) and

0

1 1

1 1

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

(100) where

(101)

(102) The label in (102) refers to the functions and 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