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.
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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. Kưster

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



x

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: ,


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

<0
(8)

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

∫ (v (r )− v (r ))(ρ (r )− ρ (r ))dr < 0 ,
1

2

1

2

(9)

where we have denoted the ground-state densities of the corresponding wave functions by and


ρ2 (r ). 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,

F [ρ 0 ] = Ψ 0 Fˆ Ψ 0
= E [ρ 0 ] − ∫ ρ0 (r )v [ρ 0 ; r ]dr

(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


6

Rogelio Cuevas-Saavedra and Paul W. Ayers

Vext [ρ 0 , v1 ] = ∫ ρ 0 (r ) v1 (r )dr

(11)

Therefore the total energy of the system is

Ev [ρ0 ] = F [ρ0 ]+ ∫ ρ 0 (r ) v1 (r )dr
1

(12)

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

Egr ound [ρ1 ] = Ev [ρ1 ] = F [ρ1 ]+ ∫ ρ1 (r ) v1 (r )dr
state

1

≤ Ev [ρ 0 ] = F [ρ 0 ]+ ∫ ρ0 (r )v1 (r )dr
1


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

Eg.s = E v  ρg.s  = min Ev  ρ 
ρ

(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, Ev[], 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 Ev[]). 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


Density Functional Theory

7

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,

F [ρ ] = T [ρ ] + Vee [ρ ]

(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]

T  ρ  = Ts  ρ  + Tc  ρ 
N /2
1

= 2∑ ψ i  ρ ;r  − ∇ 2 ψ i  ρ;r  + Tc  ρ 
2
i=1

(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

Ts  ρ  . is

the correlation-kinetic energy, which represents the correction to the orbital model.
Fortunately,

Tc  ρ  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
N
 1

ˆ
H = ∑  − ∇ i2 + v ( ri ) + w ( ri ) 
2


i =1 

(17)

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


8

Rogelio Cuevas-Saavedra and Paul W. Ayers

 1 2

 − ∇ i + v ( r ) + w ( r ) ψ i ( r ) = ε iψ i ( r )
 2


(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,

{ψ [ρ ; r ]}

N /2

i

i =1


, 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,

E vK S [ρ ] = Ts [ρ ] + ∫ ρ (r )(v (r ) + w (r ))d r
N /2
1
= 2 ∑ ψ i [ρ ; r ] − ∇ 2 ψ i [ρ ; r ] + ∫ ρ (r )(v (r ) + w (r ))dr
2
i =1

(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,

w (r ) =

δ J [ρ]
δ Exc [ ρ ]
+

δρ ( r ) ρ = ρ
δρ ( r ) ρ = ρ
0

0

(20)

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

J [ρ ] =

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

(21)

The second functional on the right-hand side of (20) is referred to as the exchangecorrelation energy functional; it is the only unknown functional in Kohn-Sham DFT. The
exchange-correlation potential is merely the name for the unknown portion of the HohenbergKohn functional,

Exc  ρ  = F  ρ  − Ts  ρ  − J  ρ 
= Vee  ρ  + Tc  ρ  − J  ρ 

(22)


Density Functional Theory


9

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

δ J [ρ ]
δρ ( r )

vJ [ ρ ; r ] =

(23)

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

vxc [ ρ ; r ] =

δ Exc [ ρ ]
δρ ( r )

(24)

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

 1 2

 − ∇ i + v ( r ) + v J [ ρ ; r ] + vxc [ ρ ; r ] ψ i ( r ) = ε iψ i ( r )
 2


N /2

ρ ( r ) = 2∑ ψ i ( r )

(25)

2

i =1

(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 Φ KS (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:

E [v; N ] = Ts  ρ g .s.  + J  ρ g . s.  + ∫ ρ g .s. (r )v (r )dr + E xc  ρ g .s. 
N /2

= 2∑ ε i − J  ρ g .s.  + E xc  ρ g .s.  − ∫ ρ g .s. (r )vxc (r )dr
i =1

= EvKS [ρ ]− J  ρ g .s.  + E xc  ρ g .s.  − ∫ ρ g . s. (r )vxc (r )dr


(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 KohnSham wave function, plus a correction that is written as a density functional.