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Basic Density-Functional Theory—an Overview

U. von Barth!

Departamento de Fisica Teorica, Facultad de Ciencias, Universidad de Valladolid, 47011 Valladolid, Spain

Received June 13, 2003; accepted February 6, 2004

PACS Ref: 71.15.Mb, 71.15.Nc, 71.45.Gm

Abstract

In these notes I have given a personally flavored expose of static density-

functional theory (DFT). I have started from standard many-body physics at a

very elementary level and then gradually introduced the basic concepts of

DFT. Successively more advanced topics are added and at the end I even

discuss a few not yet published theories.

The discussion represents many of the personal views of the author and there

is no attempt at being comprehensive. I fully realize that I am often ‘unfair’ in

treating the achievements of other researchers.

Many topics of standard DFT are deliberately left out like, e.g., time-

dependence, excitations, and magnetic or relativistic effects.

These notes represent a compilation of a series of lectures given at at the

EXC!TING Summer School DFT beyond the ground state at Riksgransen,

Sweden in June of 2003.

1. Background

Many reviews and articles on the topic of density-functional theory (DFT) start by the proclamation that,over the past so and so years, DFT has become, by far, themost prominent tool for the calculation of the ground-stateproperties of electronic systems. Although somewhatvacuous the statement is definitely true. In fact, DFTcalculations of the electronic properties of real materialshave nowadays turned into an extensive industrial endea-vor.The idea of using the density as the basic variable for the

description of the energies of electronic systems goes backalmost to the advent of quantum mechanics and therealization that the solution of the full equation ofSchrodinger was beyond reach in most cases. The statisticalatom of Gombas [1] and the approximations by Thomas [2]and Fermi [3] were early attempts in this direction. Then, ofcourse, came the Hohenberg and Kohn [4] theorems in themid sixties followed by the work by Kohn and Sham [5].They demonstrated that the electron density of a fullyinteracting system could actually, in a rigorous way, beobtained from simple one-electron theory. At that time,most researchers involved with the calculation of theelectronic properties of atoms, molecules, and solids wherestrongly influenced by the school of J. C. Slater. Propertieswere calculated from one-electron theory using a statisticalapproximation but only for the effect of exchange andcorrelation. The latter was obtained in 1951 by Slater [6] asan average over the Fermi sea of the self-energy of thehomogeneous electron gas treated within the Hartree–Fockapproximation. As a matter of fact, from a numerical pointof view, that theory was not very different from modernDFT within the local-density approximation (LDA).

In the beginning of the seventies, Slater’s so called X!method still dominated the scene and few people realizedthe immense conceptual importance of the new DFT. Ayounger scientist at IBM Research, A. Williams, was,however, one of those propagating the idea that if there isan, in principle, exact theory, why not try to use it.Eventually, Williams and collaborators produced a compi-lation of LDA results for almost the entire periodic table ofelements—a very useful piece of work which is often usedas a reference even today [7]. DFT, and then within theLDA, slowly gathered momentum until it, in the earlyeighties, by many people, was thought to be the answer toalmost all questions. The journals were full of LDAcalculations where the agreement with experiment wasalmost perfect and where remaining discrepancies wereblamed on numerical flaws. Gradually came a realizationthat there were severe problems with the LDA. Perhapsmainly from solid-state physicist dealing also with morelocalized systems like molecules. Not really by thetheoretical chemists who, at the time, were quite immuneto the ideas of DFT. It was realized that the LDA alwaysresulted in a certain overbinding depending on the degreeof localization of the constituent electrons.

Because the LDA is based on the homogeneous butinteracting electron gas, the inclusion of gradient correc-tions was the natural way to proceed in order to go beyondthe LDA. These kind of corrections were actually workedout already in the original paper by Hohenberg and Kohn[4]. The next important steps were taken by Langreth,Perdew, and Mehl [8,9] in the late seventies and earlyeighties and John Perdew and collaborators have ever sincebeen perusing the painstaking work which has carriedthose corrections to the level of sophistication where we aretoday [10].

But it was not John Perdew who turned the quantumchemists around. That achievement was signed Axel Becke[11]. One of his ideas is that molecules show noresemblance to the electron gas and gradient correctionsshould be designed to reproduce accurate results forprototype localized systems. With Becke’s gradient correc-tions, and those of others, the atomization energies of alarge number of well known molecules improved to thepoint where most established and commercially availablecodes for molecular calculations today can be run also inDFT mode. In fact, after the beginning of the nineties thepopularity of DFT within the community of quantumchemists grew to the point where Walter Kohn wasawarded the Nobel Prize in Chemistry, in 1998. ProfessorKohn shared the prize with the chemist John Pople [12]!Permanent address: Lund University, SE-22362 Lund, Sweden

Physica Scripta. Vol. T109, 9–39, 2004

# Physica Scripta 2004 Physica Scripta T109

who is the major player in the development of computercodes for electronic structure calculations in molecules.I am inclined to hand out a mark of ‘‘not passed’’ to the

Nobel Committee for physics and perhaps to the entirephysics community for allowing themselves to be beaten bychemists in realizing the importance of DFT. After all,DFT is more accurate and perhaps more indispensable insolids as compared to the case of molecules.These notes could rather easily be mistaken for a review

article on DFT—and then a comparatively inferior andincomplete one. But they are definitely not intended as one.Instead, my aim is to guide a beginner through the basicprinciples of DFT up to a level where he or she will haveacquired a basic understanding of what is involved, whichare the major methods, the main obstacles, the predomi-nant fields of applications, and what kind of accuracy onecan expect. It would be great if a follower of these noteswould also end up having some idea about in whatdirection the field will develop in the near future. There aremany topics which will not be discussed here at all becausethey are covered by other contributors to the presentmeeting. Typical such topics are the the spin-dependentgeneralization [13,14], the pseudizing of the theory [15–17],excited states and the time-dependent version of the theory(TDDFT) [18,19], current-density functional theory [20],relativistic DFT [21,22], DFT at higher temperatures [23],and many-component DFT [24].Looking through the list of references you will find that

the majority of them are relatively old. This is connected, inpart, to my opinion that there has not been so much noveltheoretical development over the past ten years. Instead wehave seen an explosion in the production of results for avery large number of different physical systems of interestto material science, magnetic properties, bio-active mole-cules and drugs, nano-systems and devices, etc. etc. Sincemy interest has been in the development of new theory,these advances are only mentioned here.The list of references given here is in no way complete or

‘fair’ to many workers in the field. The list reflects mypersonal preferences and to some extent also my lack ofknowledge caused by my absence from the field in lateryears. I have basically referred only to work which has hadsome impact on my own work within the field. For a morecomprehensive coverage of DFT and a better descriptionof other peoples work, I refer the interested reader to anumber of excellent review articles in the field [25–39].

2. Many-electron theory

2.1. The total energy

Most presentations of DFT follow the historical path ofthe founding fathers. Unfortunately, many beginners findthis formulation rather abstract and difficult to grasp. In abeautiful but very short paper from 1979 [40], Mel Levygave a much less abstract presentation of the basicfunctionals of the theory. For pedagogical reasons, weprefer the Levy approach and will later make extensive useof this approach. But we will begin by investigating how farwe can proceed without any use of DFT. We believe thatour approach has certain pedagogical advantages in so far

as it builds up expectations to the point where DFTbecomes the most natural next step.

The problem we like to address is that of finding theground-state energy of a many-electron system. As usual inquantum mechanics, we then solve the eigenvalue problem,

H! " E!

for the Hamiltonian H and look for the lowest eigenvalue.Unfortunately, in the case of a solid or a large molecule theHamiltonian has a dreadful appearance,

H "XN

i"1

# 1

2r2i $

XN

i"1

w%ri& $1

2

XN

i 6"j

v%ri # rj&:

Here, the first term represents the kinetic energy of all theN electrons in the system. The second term represents theinteraction with the external (to the electrons) charges andfor a molecule or a solid it would, e.g., look like

w%r& " #XM

""1

Z"v%r# R"&

where Z" are the different number of protons on thedifferent constituent M atoms and v%r& " 1=r is theordinary Coulomb interaction. Notice that we have heretransformed ourselves to the world of atoms by puttingm " h" " e2=%4#"o& " 1: In this world the unit of length isone Bohr radius equal to 0.529 A and the unit of energy isone Hartree or 27.21 eV. Finally, the last term is the verystrong and important repulsion energy between all theelectrons (the factor of 1/2 makes sure that everyinteraction is included only once).

For any system of practical interest the number ofelectrons would range from ten to 1023 and the number ofdegrees of freedom in the Hamiltonian is prohibitivelylarge. Still, in the absence of the last term in theHamiltonian, it would actually have been feasible to findthe total energy of the system from simple one-electrontheory. This follows from the fact that, without theelectron-electron interaction terms, the Hamiltonianwould have been a sum of identical one-electron Hamilto-nians—one for each electron. And they must all beidentical because all electrons in nature are indistinguish-able. The eigenstates of such a Hamiltonian can be shownto be Slater determinants consisting of one-electronorbitals which are solutions to the mentioned one-electronHamiltonian and the energies of the many-body states arejust the sum of the eigenvalues of the constituent orbitals.We are, however, not that fortunate but we might considerthe possibility of approximating the inter-electron part ofthe Hamiltonian by a sum of terms corresponding to eachelectron moving in some average field of all the otherelectrons. This was exactly what was done in the so calledX! approach mentioned in the introductory section andalso Hartree–Fock theory has this form. And, as we shallsee later, also DFT can be cast in this mean-field formalthough the resulting equations are no approximationsbut give exact answers to certain well defined questions.

For the time being, we notice that the total ground-stateenergy can be obtained as the expectation value of the

10 U. von Barth

Physica Scripta T109 # Physica Scripta 2004

Hamiltonian with respect to some exact or approximatewave-function ! according to

E " h!jHj!i:

But in the process of evaluating this expression we will findthat most degrees of freedom integrate out and only somepartial knowledge of the full wave function is needed inorder to obtain the energy. Let us first consider the simplestterm in the Hamiltonian, i.e., the energy arising from theinteraction with the external potential. We have, in obviousnotation,

W " h!jWWj!i "XN

i"1

h!jw%ri&j!i

"!d3r w%r&

XN

i"1

h!j$%r# ri&j!i "!d3r w%r&h!jnn%r&j!i

where we have defined the density operator nn%r& throughthe relation

nn%r& "XN

i"1

$%r# ri&:

Thus, defining the ground-state density n%r& of the systemby

n%r& " h!jnn%r&j!i

we have the very intuitive and very classical relation

W "!d3r w%r& n%r&:

The important message displayed by this relation is that,for this part W of the energy, only a very limited piece ofinformation from the full wave function is needed, in thiscase, the density given by

n%r& " N

!d3r2 . . . d

3rNd%2 . . . d%N

' j!%r; %; r2; %2; . . . ; rN; %N &j2:

Here, the factor N; i.e. the total number of electrons, entersbecause the symmetry of the wave function makes all termscontribute an equal amount in the sum over electronsabove. (We have written the discrete sum of the two valuesof the spin variables %i as a generalized integral.)Turning next to second most complicated term in the

total energy, i.e. the kinetic energy T; we obviously obtain

T" h!jTTj!i "XN

i"1

h!j# 1

2r2i j!i

"#N

2

!!!%x1;x2;. . .; xN &r2

1!%x1;x2;. . .; xN &dx1 dx2. . .dxN:

In the latter integral we have used the short-hand notationxi for the set of variables %ri; %i&: Thus, defining what in the

trade is known as the one-particle density matrix accordingto

#%1&%;%0 %r; r0&

"!!!%r; %; x2; . . . ; xN &!%r 0; %0; x2; . . . ; xN &dx2 . . . dxN

we have

T " # 1

2

X

%;%0

!r2#%1&

%;%0%r; r0&h i

r 0"rd3r:

We see again that a rather limited information from the fullwave function is needed in order to calculate the totalkinetic energy of the system. For future reference and fordealing also with magnetic systems, we notice that thediagonal components of the one-particle density matrixrepresent the spin densities n%%r&;

n%%r& " #%1&%;%%r; r&

in terms of which the density is just

n%r& "X

%

n%%r&:

Finally, we turn to the most complicated part of the energy,i.e., the interaction energy U which, in terms of the fullwave function, is given by

U " 1

2

XN

i 6"j

h!jv%ri # rj&j!i

" 1

2N%N# 1&

'!j!%r1; %1; r2; %2; . . . ; rN; %N &j2v%r1 # r2&dx1 dx2 . . . dxN:

Also in this case, the particular contraction of the totalwave function giving the interaction energy, has a specialname. It is called the diagonal of the two-particle densitymatrix and it is given by

#%2&%;%0 %r; r 0& "N%N# 1&

'!j!%r; %; r 0; % 0; x3; . . . ; xN &j2 dx3 . . . dxN:

This quantity has a very physical interpretation. Theoperator representing the density of electron i at the point rin space with spin % is obviously $%r# ri&$%;%i : And there is asimilar expression giving the density of electron j at anotherpoint r0 with spin %0: We can then construct the densitycorresponding to any electron at the point r with spin %given the fact that there is another one at the point r0 withspin %0: This quantity must obviously have contributionsfrom all pairs of electrons and we write it

nn%;%0%r; r 0& "XN

i6"j

$%r# ri&$%;%i$%r 0 # rj&$% 0;%j :

Basic Density-Functional Theory—an Overview 11


We have here excluded the term with i 6" j because noelectron can be at two different points in space at the sametime. Calculating the expectation value of this two-electrondensity in the ground-state state gives,

#%2&%;%0 %r; r 0& " h!jnn%;%0%r; r 0&j!i:

The quantity #%2& has a number of intuitively under-standable properties. According to the Pauli exclusionprinciple, two electrons can never be at the same place ifthey have the same spin. Thus,

#%2&%;%%r; r& " 0

which follows directly from the antisymmetry of the wavefunction. This symmetry also shows that #%2& is symmetricin its arguments

#%2&%;%0 %r; r 0& " #%2&

%0;%%r 0; r&:

The operator nn%r; r 0& is easily seen to be a positive definiteoperator so that

#%2&%;%0 %r; r 0& ( 0:

Let us then integrate one of the spatial variables over allspace to obtain

!nn%;%0%r; r 0&d3r 0"

XN

i6"j

$%r# ri&$%;%i$%0;%j

"XN

i; j

$%r# ri&$%;%i$%0;%j #XN

i"1

$%r# ri&$%;%i$%0;%i

" nn%%r&NN%0 # nn%%r&$%;%0

where the operator nn%%r&; of course, corresponds to the spindensity at r;

nn%%r& "XN

i"1

$%r# ri&$%;%i

The operator NN% counts the total number of electrons withspin %;

NN% "XN

i"1

$%;%i :

The latter operator is just a constant %2=h" & times thez-component of the total spin. Since the Hamiltonian doesnot contain any spin variables it commutes with the totalspin and, in a finite system, one can then always considerthe ground state as an eigenfunction of the total spin.Writing %h" =2&N% for the corresponding eigenvalue andtaking an expectation value with respect to the groundstate of the above operator relation we obtain,

!#%2&%;%0%r; r0& d3r0 " n%%r&fN% # $%;%0 g:

Due to the symmetry of #%2& we would certainly haveobtained a similar result by instead integrating over r andwe realize that the spin densities n%%r& can be considered to

be factors in #%2&: We thus define a new quantity g; calledthe pair-correlation function, according to the relation

#%2&%;%0 %r; r 0& " n%%r&n%0%r 0&g%;%0 %r; r 0&:

The function #%2& is an expectation value of a two-particledensity and thus more related to densities than toprobabilities. Dividing by the spin-densities the pair-correlation function g becomes more related to probabil-ities. In fact, if the electrons were completely independentthe two-particle density would be just the product of theone-electron densities and the pair-correlation functionwould equal unity. Thus, the degree of correlation betweenthe electrons is measured by how much smaller than unity gis. And, from the definition, it is certainly always positiveas one would expect from a probability.

Using the new definition of g and the obvious relation

!n%%r& d3r " N%;

our previous integral formula for the pair density #%2&

becomes

!n%0%r 0&fg%;%0%r; r 0& # 1g d3r 0 " #$%;%0

after division by n%%r&: Consequently, the quantity n)g# 1*represents a negative particle density, i.e., a lack ofelectrons which integrates up to negative unity for electronsof the the same spin and to zero for those of opposite spin.This is the famous exchange-correlation hole surroundingevery electron in the system. Using g; the formula for theinteraction energy U becomes

U " 1

2

X

%;%0

!n%%r&n%0 %r 0&g%;%0%r; r 0&v%r# r 0& d3rd3r 0:

We see that this energy involves a sum over spin indicesand we do not need the detailed spin structure of the pair-correlation function g in order to calculate U: It is,therefore, convenient to define the spin averaged pair-correlation function g by means of the relation

n%r&n%r 0&g%r; r 0& "X

%;%0

n%%r&n%0%r 0&g%;%0 %r; r 0&:

It should be remembered, however, that the full spinstructure is of interest in magnetic system and is alsoimportant to attempts to approximate the pair-correlationfunction. This is due to the fact that the correlationsbetween electrons of different spin is much stronger anddifferent in character as compared to the case of the likespin electrons. The latter are kept apart by the Pauliexclusion principle thus reducing their mutual Coulombrepulsion.

The largest part of the interaction energy is the classicalelectrostatic Coulomb interaction U0 between the elec-trons. In most electronic systems, this is a huge part of thetotal energy, much larger than the energies associated withexchange and correlation, and it is not very difficult to

12 U. von Barth


calculate. It is given by

U0 "1

2

!n%r&n%r 0&v%r# r 0& d3r d3r 0:

It is thus customary to separate out this part of theinteraction energy. In terms of our spin averaged pair-correlation function g the interaction energy U becomes

U " U0 $1

2

!n%r&n%r 0&fg%r; r 0& # 1gv%r# r 0& d3r d3r 0:

The last term in this expression is referred to as theexchange-correlation part if the interaction energy oftenwritten Uxc: In terms of the spin averaged pair-correlationfunction g the famous sum rule for the exchange-correlation hole becomes

!n%r 0&fg%r; r 0& # 1gd3r 0 " #1:

We see from this that whereas the classical Coulombenergy, also sometimes called the Hartree energy, is aninteraction between N electrons and N electrons, theexchange correlation part Uxc of the interaction energy isthe classical Coulomb interaction between each of the Nelectrons with one negative unit of charge, i.e., with itsexchange-correlation hole.Collecting all pieces of the total energy we have

E " # 12TR )r

2#%1&* $!wn$ 1

2

!nvn0 $ 1

2

!nn0fg# 1gv;

where we have suppressed all integration variables andused the symbol TR to signify the operations of making thearguments equal and integrating over all space, in analogywith the concept of the trace of an ordinary matrix.

2.2. The Hellman-Feynman theorem

As we shall see shortly, it is not difficult to find ratheraccurate approximations to the spin averaged pair-correlation function of many electronic systems. Theseapproximations will give errors in the already rathersmall exchange-correlation energies of the order of 5% orless. It has, however, proved to be difficult to findapproximations to the one-particle density matrix #%1&

which are accurate enough for calculating the very largekinetic energy of the system. Fortunately, one can use thewell known Hellman–Feynman theorem of textbookquantum mechanics to transform the calculation of thetotal kinetic energy into a calculation of the kineticenergy of non-interacting electrons plus a small correctionto the correlation part of the interaction energy. It isactually rather simple to demonstrate how this is done.We imagine that we scale down the strength—now calledl—of the Coulomb interaction between the electrons. Forinstance, in an atom this would have a drastic effect onthe density of the electrons which would then becomemuch more compact and pile up close to the nucleus.This effect would result from the reduction in thescreening by the other electrons of the attraction of oneelectron to the nucleus. We will therefore add an externall-dependent one-body potential to the system in such a

way that the density is not affected by the reduction inthe inter-electron Coulomb interaction. It is perfectlylegitimate to ask if this is actually possible to achieve.This is a very difficult question and is referred to as thew-representability problem of DFT. It is normallyassumed to be possible but it is very difficult to provein an exact way. But most physicists are content with thefact that, in practice, it has always been possible toachieve w-representability within the desired degree ofaccuracy.

The energy El of the system will, of course, change as aresult of the reduction in the Coulomb interaction. Forobtaining the derivative of El with respect to the parameterl, the Hellman–Feynman theorem states that it is sufficientto differentiate the explicit l-dependence in the Hamilto-nian. Due to the stationary property of the ground-stateenergy El; the l-dependence of the ground-state wavefunction !l does not enter. Thus, in obvious notation,

@El

@l" !l

@Hl

@l

""""

""""!l

# $" !l

@WWl

@l

"""""

"""""!l

* +

$ !ljUUj!l

D E

"!@wl%r&@l

nl%r& d3r$12

!nl%r&nl%r 0&gl%r; r 0&v%r#r 0& d3r d3r 0:

But the l-dependence of wl%r& was defined to be such thatthe density remained constant and equal to the density n%r&of the fully interacting system at all l: Therefore,integrating again with respect to l; we obtain

E# Eo "!)w%r& # w0%r&*n%r& d3r

$ 12

!n%r&n%r 0& ~gg%r; r 0&v%r# r 0& d3r d3r 0:

Here, Eo is the total energy of a system of N non-interacting electrons moving in the external potential wo%r&and having the same density n%r& as the fully interacting%l " 1& system. This means that we find Eo by solving theone-electron Schrodinger equation

#12r

2 $ wo%r&% &

&k%r& " "k&k%r&

to obtain

Eo "XN

k"1

"k "XN

k"1

&k #12r

2"" ""&k

' ($!wo%r&n%r& d3r

where

n%r& "XN

k"1

j&k%r&j2:

The quantity ~gg is just the normal pair-correlation functiong but averaged over all values of the interaction strength l:Thus,

~gg%r; r 0& "!1

0gl%r; r 0& dl:



Again adding up the pieces of the total energy we find

E "XN

k"1

&k #12r

2"" ""&k

' ($!wn$ 1

2

!nvn 0 $ 1

2

!nn 0f ~gg# 1gv;

where we suppressed integration variables as we didpreviously.This is a truly remarkable results. It means that we can

always find the total energy of an interacting many-electronsystem by doing one-electron theory. And we have not evenmentioned density-functional theory (DFT) at this stage.What we have to do is to find a one-electron potential thatgives the corresponding non-interacting system of Nelectrons a density which is identical to the true densityof the interacting system. And we must find the exactaverage pair-correlation function ~gg giving rise to thecomparatively small exchange-correlation energy.As we shall see in the next section, is not very difficult

to find rather accurate approximations to the latterquantity. The first task is, however, considerably moredifficult in the sense that it requires the formal machineryof DFT. Fortunately, an accurate total energy actuallydoes not require a very accurate one-electron potential.This is due to the variational property of the total energyas a functional of the density as will be explained inSection 3.We will end this section by making a comment which we

find both interesting and amusing. By means of the local-density approximation (LDA) of the following section wehave an explicit expression for the last term, i.e., theexchange-correlation part of the total energy from the lastequation. We could, of course, then do exactly as inThomas–Fermi theory and minimize our, by now, explicitexpression for the total energy by varying the density. Thiswould certainly entail varying the one-electron potential wo

giving rise to that density and we would find the lowestenergy when wo is chosen equal to what is nowadaysreferred to as the total Kohn–Sham potential within theLDA. Consequently, we could have started to do local-density calculations without ever having heard about DFT.Of course, in order to proceed toward an exact theoryallowing us, e.g., to go beyond the LDA we need DFT.And one should certainly not underestimate the tremen-dous conceptual impact of the theorems by Hohenberg,Kohn, and Sham.

2.3. The local-density approximation

We end this section by a short discussion on the well knowlocal-density approximation (LDA). Of course, the LDA isalmost always associated with DFT—and rightfully so. Wewill, however, introduce it already here in an attempt todemonstrate how far toward accurate total energies onecan actually get without invoking DFT. The purpose ofthis approach is certainly not to rob the revered fathers ofDFT of any glory. It is rather to build up understandingand expectations to the point where DFT becomes the oneand only way to proceed.Our task is here to approximate the average pair-

correlation function ~gg of the inhomogeneous and interact-ing system in order to obtain a good approximation to

what we from now and on will refer to as the exchange-correlation part of the total energy, Exc;

Exc "1

2

!n%r&n%r 0&f ~gg%r; r 0& # 1gv%r# r 0& d3r d3r 0:

We begin by approximating the exact ~gg by that of thehomogeneous but interacting electron gas which we call ~ggh:But this function is also a function of the density of thehomogeneous gas and one is immediately faced with theproblem of choosing some effective density "nn for theapproximation. This is actually a standard problem whenone wants to, as is often done, approximate two-pointfunctions in inhomogeneous systems with the correspond-ing quantities from the homogeneous gas. The choicewhich leads to the simplest algebra is "nn " n%r&; which ishere the preferred choice. We notice that these approxima-tions for ~gg will become exact when the density approaches aconstant. We finally complete our approximation by alsoreplacing the argument r 0 in the second density by r: Againthis approximation becomes exact when the density tendsto a constant and we can also argue that the largestcontribution to the integral is coming from regions wherethe density is large. It can be shown that the function ~gg# 1differs little from zero when the two arguments r and r 0 arefurther apart than the average distance between electrons.Thus, the integral has large contributions only fromregions when r and r 0 are close which means that it isquite reasonable to replace the argument r 0 by r in thesecond density if the density does not vary appreciably overan inter electronic distance. We have completed the LDAand obtain (notice that, due to the complete translationalsymmetry of the homogeneous gas, the function ~ggh canonly be a function of r# r 0)

ELDAxc " 1

2

!n%r&n%r&f ~ggh%r# r 0; n%r&& # 1gv%r# r 0& d3r d3r 0

" 1

2

!d3r n%r&n%r&f ~ggh%r 0; n%r&& # 1gv%r 0& d3r 0

"!d3r n%r&" xc%n%r&&:

Here, the function "xc%n& is just the exchange-correlationenergy per electron of the homogeneous and interactingelectron gas obtained by evaluating the exact expression forExc above at a constant density n and then dividing by thetotal number N of electrons. (Notice that N " $n where $is the total volume of the gas).

We could actually have arrived at this expression in amore direct way as follows. If the density of aninhomogeneous system does not vary that rapidly we cansplit the whole system into small boxes of volume d3r andconsider each box as containing a homogeneous electrongas with a density equal to the local density n%r& at thecoordinate r of the box. The contribution to the energy ofeach box is, of course, the exchange-correlation energy perparticle "xc%n%r&& of the gas in that box times the totalnumber, n%r& d3r; of electrons in the box. Summing over allboxes gives us back the LDA above.

In order to use the LDA in practical calculations, wemust certainly supply a table of accurate exchange-correlation energies of the homogeneous gas as a function

14 U. von Barth


of the density n: Such results are obtained from large-scaleMonte Carlo simulations [41] and parameterized forconvenient access in computer codes. Just one word ofadvice. There are many parameterizations in the literatureand most of them are, for all practical purposes, next toequivalent [26,42,43]. But avoid those which are based onvery complicated analytical expressions which use up anundue amount of computer time.We will give a detailed account of the successes and

failures of the LDA later on.

3. Density-functional theory

3.1. The kinetic-energy functional

As the name density-functional theory (DFT) suggests,DFT is a many-body theory based on the idea of using onlythe density as the basic variable for describing many-electron systems. And it is a formulation in terms offunctionals of the density. As mentioned in the introduc-tion, the rather abstract functionals of DFT wereformulated in a rather hands-on way by Mel Levy in1979 [40]. And we will here prefer this so called constrainedsearch approach by Levy to the original formulation byHohenberg, Kohn, and Sham. We will begin by definingthe functional for the kinetic energy, To:Given an arbitrary particle density n%r&; we think of all

anti-symmetric many-body wave functions which yield thatdensity n: In mathematical language this set M of states iswritten

M%n& " fj!i j h!jnn%r&j!i " n%r&g:

Here, nn%r& is the density operator for the N electrons asdefined in Section 2.1. Although the density n%r& could bechosen freely, it should obey certain conditions like beingeverywhere non-negative and integrating up to N electrons.It is appropriate to ask whether there is always an anti-symmetric state yielding the density n%r& or, in other words,if the set M%n& is not empty. The answer to this questionis actually yes. This problem is referred to as theN-representability problem of N-electron densities andwe will not further dwell on this rather complicated issue.The operator TT corresponds to the kinetic energy of the

N electrons and we will now study its expectation valueswith respect to states in the set M%n&: And we search for thesmallest possible such expectation value. These expectationvalues are bounded from below because the kinetic energyis a positive definite operator. This means that the set ofsuch expectation values has an infimum meaning that thereis a number, say To; such that all the expectation values arelarger than or equal to To and that there is always at leastone expectation value smaller than To $ "; no matter howsmall " > 0: We write

To)n* " infimumj!>2M%n&

h!jTTj!i:

From the point of view of a physicist it is enough to knowthat To)n* is the smallest possible kinetic energy which canbe achieved among many-body states all having the densityn%r&: In other words, the value To)n* is the minimum of allexpectation values h!jTTj!i provided also h!jnn%r&j!i

" n%r&: Consequently, in order to find T0)n*; we are facedwith a minimization under constraints, and there is oneconstraint for each point in space since the expectationvalue of the density operator at each point r in space has toequal the prescribed density n%r&: Such a constrainedminimization is most easily achieved through the use ofLagrangian multipliers. And we must have one Lagrangianparameter V%r& for each point r in space and then minimize

h!jTTj!i$!V%r&n%r& d3r " h!jTTj!i$

!V%r&h!jnn%r&j!id3r

" h!jTT$ VVj!i

with respect to arbitrary variations of the wave function orstate j!i (notice, however, that it still has to obey the anti-symmetrization postulate). Here, the ‘one-body’ quantummechanical operator VV is defined through the relation

VV "XN

i"1

V%ri&

But this is a common problem in quantum mechanics thatwe know how to solve. It is the problem of finding theground-state energy of the Hamiltonian

Ho " TT$ VV

which is a sum of one-electron Hamiltonians. Thus, we justhave to construct all eigensolutions to the one-dimensionalSchrodinger equation

#12r

2 $ V%r&% &

&k%r& " "k&k%r&

and then we know that the ground-state wave function isthe normalized Slater determinant consisting of the Nsolutions with the lowest possible one-electron eigenvalues(N is the number of electrons in the system). The ground-state energy is just the sum of those lowest eigenvalues and

To)n* " h!jTTj!i "XN

k"1

&k #12r

2"" ""&k

' (

Then, of course, the infinite set of Lagrangian parametersV%r&; which, in this case, obviously acts as the external one-electron potential has to be adjusted until the density givenby

n%r& "XN

k"1

j&k%r&j2

becomes equal to the chosen one.The whole procedure can now be summarized in words.

In order to find the value of the functional To)n* of thedensity n at a particular density n%r&; we just solve the one-electron Schrodinger problem in some potential V%r& whichis then adjusted until the sum of the squares of thecorresponding one-electron orbitals agrees with the chosendensity. The value of the functional for the kinetic energy isthen the total kinetic energy of this non-interactingproblem. Notice that the procedure relies on the possibilityof always being able to find a potential (Lagrangian



parameters) which reproduces the chosen density. This isvery difficult to show in an exact way and requires verystrict conditions on the chosen densities. We will not go into this rather complicated issue here. This problem isreferred to as the non-interacting w-representabilityproblem in the density-functional literature. We will becontent by the fact that, in practice, one has always beenable to find to such a potential within some rather highdegree of accuracy. And many such calculations have beencarried out since I started the activity in 1982 (see e.g. Ref.[32]).The next step in our formal development of DFT is to

study how the functional To changes when the density nundergoes a small change to n$ $n: The Lagrangianmultipliers must obviously change to, say V$ $V; inorder to produce the new density n$ $n: The first-orderchange in To becomes

$To " $XN

k"1

"k #!V%r&n%r& d3r

( )

"XN

k"1

$"k #!$V%r&n%r& d3r#

!V%r&$n%r& d3r:

But, when the one-electron potential V%r& changes, the one-electron eigenvalues also change and from first-orderperturbation theory we know that

$"k " h&kj$Vj&ki

Hence,

XN

k"1

$"k "!$V%r&n%r&d3r

and we obtain

$To " #!V%r&$n%r& d3r;

an equation which is close to the defining equation to whatis know as the functional derivative of the functional To

with respect to the density. There is only one slight catch.When defining a functional derivative, one must allow forcompletely arbitrary variations in the density n%r& but, here,we are only allowed to consider density changes whichconserve the number of electrons. For such densitychanges, we see that adding a constant to the potentialV%r& will produce no change in To because

!$n%r& d3r " 0:

We are thus only allowed to infer

$To

$n%r& " #V%r& $ "0

with some unknown constant "0:We can carry our formal analysis one step further by

also studying second-order changes in the kinetic energy aswe change the density. As we shall see later, this will be ofimportance when we do linear response theory and

gradient expansions. If we change the potential V%r& bythe amount $V%r& the resulting density will change ton%r& $ $n%r&: From the above so called Euler equation weobtain

!$2To

$n%r&$n%r0& $n%r0& d3r0 " #$V%r& $ $"0:

On the other hand, as a matter of definition, the first-orderchange $n%r& in the particle density resulting from a change$V%r& in the total one-electron potential is proportional tothis $V with a ‘constant’ of proportionality called the non-interacting density response function 'o%r; r0& of the system,i.e.,

$n%r& "!'o%r; r0&$V%r0& d3r0:

The non-interacting response function '0 is often alsocalled the Lindhard [44] function and is easily obtainedfrom standard second-order perturbation theory asdescribed in most text books on quantum mechanics.Combining the last two equations leads to the conclusion

$2To

$n%r&$n%r0&" #'#1

0 %r; r0&;

meaning that the second functional derivative of the kineticenergy with respect to the density is the negative of theinverse of the non-interacting static density responsefunction of the system. Notice that, in order to draw thisconclusion, we have assumed that

!'0%r; r0& d3r0 " 0

which allows us to ignore the constant $"0: This relationis, however, valid in all finite systems on account ofparticle conservation. (In a metallic system, '0 integratesto a finite value actually proportional to the density ofstates at the Fermi level and one has to be a little morecareful in handling the constant $"0: The conclusion withregard to the second derivative of To remains, however,unaltered.)

We end the present subsection with the remark that To isobviously the kinetic energy of non-interacting electrons—despite the fact we started out searching for the minimumkinetic energy among arbitrary many-body wave-functionsof density n! The underlying reason for this somewhatlimited result is the one-body character of the operator forthe kinetic energy being a sum of terms each containing thevariables of only one of the electrons. As we shall see in thenext subsection, this will not be true about the many-bodyfunctional F)n* defined there.

3.2. The functional F)n*An important ingredient in defining the functional To forthe kinetic energy was the fact that the correspondingoperator TT is positive definite and thus has a spectrum

16 U. von Barth


bounded from below. The operator TT$ UU; where UU is theoperator

UU " 12

XN

i 6"j

v%ri # rj&

corresponding to the total interaction energy between theelectrons, has the same property, i.e., it is positive definite.(In the definition of UU; v is the Coulomb interactionv%r& " 1=r; as discussed in subsection 2.1). The operatorTT$ UU defines the functional F)n* of the density n in a waycompletely analogous to the definition of To above. Wedefine

F )n* " infimumj!>2M%n&

h!jTT$ UUj!i

We point out that the operator TT$ UU contains termsinvolving the coordinates of two electrons, i.e., it is a ‘two-electron’ operator. We find F)n* by a minimization of

h!jTT$ UU$ VVj!i

with respect to arbitrary variations of the wave function orstate j!i: Here, as above, the operator VV is the one-bodyoperator acting as external potential but corresponding tothe Lagrangian multipliers needed to keep the densityequal to the chosen one during the minimization. As above,this problem is identical to the quantum mechanicalproblem of finding the ground-state energy of theHamiltonian H " TT$ UU$ VV: And, then, just as above,the potential V%r& (" the Lagrangian multipliers) has to beadjusted such that the ground-state density of H becomesequal to the chosen density n%r&: And we will here assumewithout proof that this is always possible to achieve. Wehave thus established that the value of the functional F)n* isjust the sum of the kinetic and the interaction energies of asystem of interacting electrons in an external one-bodypotential which gives the system a ground-state densityequal to the chosen density n: If the ground-state energy ofthe Hamiltonian H is E; this minimal value obviously hasto be

F)n* " E#!V%r&n%r& d3r:

As we did in the case of the functional To above, we canalso discuss how the functional F)n* changes due to smallchanges $n%r& in the density n: In order to have a slightlydifferent density, the Lagrangian parameters have tochange from V to V$ $V; and first-order perturbationtheory then says that the ground-state energy E of Hchanges by $E where

$E "!n%r&$V%r& d3r:

Consequently, the corresponding change $F in F is given by

$F)n* " #!V%r&$n%r& d3r

from which we, in analogy with above, conclude that

$F

$n%r& " #V%r& $ "

with some unknown constant ":For the purpose of later doing response theory and

gradient expansions we can, in analogy with the case of thekinetic energy, also conclude that

$2F

$n%r&$n%r 0&" #'#1%r; r 0&;

where '#1 is the inverse of the full static density responsefunction of the interacting system. (Static because there isno time dependence in the perturbation).

3.3. The minimum of the energy functional

Having defined the functional F)n* of the density n; itappears very natural to define the functional E)n;w*;

E)n;w* " F)n* $!n%r&w%r& d3r;

of the two independent variables n and w: At a fixed‘potential’ w; this functional has a minimum at the ground-state density n of that many-electron system which has w asexternal potential. And the value at the minimum is theground-state energy of that system. In order to demon-strate this minimal property we let j! > be the ground stateof the Hamiltonian H " TT$ UU$ WW with the ground-statedensity n " h!jnnj!i: We also introduce the ground statej!1i of the Hamiltonian H " TT$ UU$ WW1 having adifferent ground-state density n1: The two potentials wand w1 must, obviously, differ by more than a constant inorder for the two densities n and n1 to be different.According to our definitions in the previous subsectionsand to the variational property of the ground state j!i; wehave

E)n;w* " F)n* $!nw " h!jTT$ UUj!i$

!nw

" h!jTT$ UU$ WWj!i +h !1jTT$ UU$ WWj!1i

" F)n1* $!n1w " E)n1;w*

where we, for simplicity, have suppressed the integrationvariables.

For the first functional derivative of the functionalE)n;w* with respect to the density n at the ground-statedensity corresponding to the external potential w we obtainfrom above,

$E

$n%r&

) *

w

" $F

$n%r&$ w%r& " ":

This is the ‘quasi’ stationary property of the total energy asa function of the density at a fixed external potential w:(The quantity which is strictly stationary is, obviously,E# "N; where N is the total number of electrons—but thisamounts to the same thing when N is kept fixed). Noticethat we are free to study arbitrary variations of the energy



functional E)n;w* with respect to both variables n and w:By choosing that density which renders the functionalE)n;w* stationary with respect to the density at a fixedexternal potential w; the variables n and w become coupledsuch that the density n becomes the ground-state densitycorresponding to the external potential w: We can thenwrite E)n* " E)n;w)n** and we can consider the ground-state energy of any many-electron system as a functional ofonly the density n: This latter functional has, of course, ingeneral, no stationary point at some density n:

3.4. Kohn–Sham theory

We are now in the position to appreciate the impact ofthese functionals of DFT on our previous discussion of therelevance of one-electron equations to the total energies ofmany-electron systems. Assume that n%r& is the ground-state density of a many-electron system subject to theexternal potential w%r&: And let again wo%r& be thatpotential which produces the same density n%r& in a non-interacting system. According to the discussion above, wehave the following two expressions for the total energy E ofthe many-body system,

E"To)n*$!n%r&w%r& d3r$1

2

!n%r&n%r0&v%r# r0& d3rd3r0$Exc)n*

where the exchange-correlation energy Exc is defined inSubsection 2.3, and

E " F)n* $!n%r&w%r& d3r:

Consequently,

F)n* " T0)n* $ 12

!n%r&n%r0&v%r# r0& d3r d3r0 $ Exc)n*:

Notice that this equation clearly demonstrates Exc to be afunctional of the density n because all other terms are.Furthermore, for the functional derivatives of To and Fwith respect to the density taken at the ground-statedensity n%r& we have,

$To

$n%r& " #wo%r& $ "o

and

$F

$n%r& " #w%r& $ "

Inserting these two equations into that obtained byfunctionally differentiating the functional F)n* above, wehave

wo%r& " w%r& $ VH%r& $ vxc%r& $ $":

Here, the so called Hartree potential VH is just thefunctional derivative of the classical Coulomb interactionenergy,

VH%r& "!v%r# r 0&n%r 0& d3r 0

and the so called exchange-correlation potential vxc is thefunctional derivative of the exchange-correlation energyExc with respect to the density,

vxc%r& "$Exc

$n%r&:

The quantity $" is some yet undetermined constant of littlerelevance since the potential floor is of no consequence innon-relativistic physics. We have thus arrived at ourdesired recipe for constructing the potential wo%r& whichproduces the correct ground-state density of the fullyinteracting system in a completely non-interacting fashion.

From our exact expression for the exchange-correlationenergy Exc we obtain the following exact expression for theexchange-correlation potential vxc%r&;

vxc%r& "!n%r 0&f ~gg%r; r 0& # 1gv%r# r 0& d3r 0

$ 12

!n%r&n%r 0& $

~gg%r; r 0&$n%r&

v%r# r 0& d3r d3r 0:

The predominant way of implementing DFT is to constructsome reasonable approximation to the exchange-correla-tion energy Exc like, e.g., the LDA and then to constructvxc%r& by functionally differentiating the approximate Exc

with respect to the density. In the case of the LDA thisgives

vxc%r& " "xc%n%r&&

where "xc%n& is the exchange-correlation part of thechemical potential of the homogeneous electron gas atdensity n: It actually has certain computational advantagesto have the exchange-correlation energy and potentialbeing consistent in this way. But there are also manydrawbacks as will be discussed later. For instance, in thecase of several accurate gradient approximations, thepotential produced by functional differentiation is oftensingular and can actually drive the density away from thecorrect result. In such cases we have, for many years,advocated the practice of doing separate approximationsfor the potential vxc directly from the exact expressionabove. This practice is strongly supported by the varia-tional principle discussed in Section 3.3.

In modern work, however, a different exact equation,referred to as the Sham–Schluter equation [45], forms thestarting point for the construction of accurate exchange-correlation potentials vxc:

3.5. The gradient expansion

Our previous discussion of the total energy of the many-electron systems has led us to the point where all remainingdifficulties are deferred to finding reasonable approxima-tions to the exchange-correlation energy Exc of the system.And we have already introduced the LDA which goes along way toward this end. By construction, the LDA iscorrect for the homogeneous electron gas and accurate fordensities that vary little over distances smaller than aninverse Fermi wave vector. A natural step beyond the LDAwould, therefore, be to allow the energy functional todepend not only on the local value of the density but also

18 U. von Barth


on its different local gradients. By adding successivelyhigher-order gradients, the gradient expansion is supposedto correct the results of the LDA when the density variesmore and more rapidly. One should, however, not have toassume that the deviations from a constant density are alsosmall. But the gradient expansion should, of course, bevalid also in that case and it is actually easier to determinethe different gradient corrections in this regime. Thus, wewill here apply response theory to the homogeneouselectron gas in order to see how such gradient correctionscan be obtained.We expose the gas to a weak and slowly varying external

perturbation and expand the exchange-correlation energyExc in a Taylor series in the deviation $n%r& from thehomogeneous density no;

Exc)n* " Exc)no* $!vxc%no&$n$ 1

2

!Kxc $n $n0

$ 16

!Lxc$n $n0$n00 $ . . . :

Here, we have suppressed integration variables, and

Kxc%r; r 0& "$2Exc

$n%r& $n%r 0& ;

and

Lxc%r; r 0; r 00& "$3Exc

$n%r& $n%r 0& $n%r 00&:

Due to the full translational symmetry of the electron gasthe potential vxc is a constant, Kxc can only depend on thedistance between r and r 0; and Lxc must be invariant underthe operation of displacing all arguments by the sameamount. Since we are considering particle conservingperturbations the linear term in the expansion vanishes.Furthermore, we are interested in the corrections to theLDA and thus expand the LDA result in a similar manner.We obtain

ELDxc )n* "

!no"xc%no& $

!@%n"xc&@no

$n$ 12

!@2%n"xc&@n2o

%$n&2

$ 16

!@3%n"xc&@n3o

%$n&3 $ , , ,

Our purpose here, is to illustrate the origin of gradientterms rather than to develop a full fledged gradientexpansion. Consequently, in what follows, we will, forsimplicity, restrict the analysis to linear response, i.e., toterms of second order in the deviations from the homo-geneous limit. Subtracting the LDA result from the exactone and noting that "xc " @%n"xc&=@n gives the followingsecond-order correction to the LDA,

Exc)n* " ELDxc )n* $ 1

2

!$nfKxc # "0

xc, $%r# r 0&g$n0:

In Fourier space this becomes

Exc)n* " ELDxc )n* $ 1

2

!d3q

%2#&3fKxc%q& # Kxc%0&gj$nqj2

where we have used the exact relation Kxc%0& " @"xc=@nwhich follows from the compressibility sum rule of theelectron gas [46]. If we now make the crucial assumptionthat the perturbed density has appreciable Fouriercomponents only at small q; we can approximate Kxc byits small-q Taylor series

Kxc%q& " Kxc%0& $ 2Bxc%no&q2 $ , , ,

and going back to real space we obtain

Exc)n* " ELDxc )n* $

!Bxc%no&jrnj2 d3r:

This gives the lowest-order gradient correction to the LDA.Of course, in a strongly inhomogeneous system like anatom or a molecule, we might ask ourselves what number%no& to use for the homogeneous density. It can be shownthat it is quite appropriate to again use the local densityn%r& rather than some average density for no: In fact, thisprocedure eliminates the necessity to include many of thehigher-order gradients from higher-order response func-tions. But the validity of this procedure as well as of theentire gradient expansion depends in a crucial way onsatisfying certain consistency relations between responsefunctions of different order. For instance, one suchrequirement is the exact relation [47]

@Kxc%q&@n

" Lxc%q; q&:

We have so far said little about how to calculate thedifferent so called exchange-correlation kernels like Kxc

and Lxc on which the gradient expansion rests. But inprevious sections we have laid down the machinery fordetermining these kernels. For instance, taking the secondfunctional derivative with respect to the density of therelation connecting the functional F)n* to the functionalExc)n* for the exchange-correlation energy and insertingour results for those derivatives gives

#'#1 " #'#1o $ v$ Kxc:

This very important relation can be used in either of twoways.

(i) From, e.g., many-body perturbation theory we mightbe able to calculate an accurate static density responsefunction ' of the electron gas. Then, the above relationdetermines Kxc and its small-q Taylor expansion inreciprocal space and thus a whole set of gradientcoefficients according to the above analysis. Suchcalculations were done as early as 1970 by Geldartand Taylor [48] and repeated more accurately by Engeland Vosko in 1990 [49] but including only effects of theinter-electronic Coulomb interaction to lowest order.Including also correlation effects to the level of the socalled random-phase approximation (RPA) the gradi-ent coefficient Bxc%n& has been calculated by Ma andBrueckner [50] in the high-density limit and later byGeldart and Rasolt [51] at metallic densities. Rasoltcalculated the same coefficient for spin-polarizedsystems in 1977 [52].



(ii) Having some accurate expression for the exchange-correlation energy Exc)n* of the inhomogeneous system,we can form the second derivative with respect to thedensity to obtain the kernel Kxc and thus the fulldensity response function ' of the system. This way ofusing the equation for the response function demon-strates the important fact that the exact static densityresponse function of any electronic system can, inprinciple, always be obtained from DFT.

With regard to the second-order response function Lxc

which gives rise to a gradient correction of the form

%Exc)n* "!Cxc%n%r&&

jrn%r&j2r2n

jn%r&j3d3r;

much less work has been done. A student of minecalculated the coefficient Cx including only exchangeeffects [47] and, to my knowledge, nothing is knownabout the effects of correlations on this coefficient. Ourcoefficient was later incorporated in a Meta-GGA byPerdew and collaborators [53]. This particular GGA wasmore successful than others and it appeared to have abetter ability to distinguish between single, double, andtriple bonds in molecules (see Section 5.6).In cases when the density varies more rapidly and does

have appreciable amplitude at larger q-vectors, one cannotuse the Taylor expansions of the response functions aroundq " 0: The derivation above suggests, however, that it isnot at all necessary to use a gradient approximation. Usingthe fact that

!fKxc%r# r 0; no& # "0

xc, $%r# r 0&g d3r 0 " 0;

the second-order correction to ELDxc can readily be rear-

ranged to read ([4])

Exc)n* " ELDxc )n* # 1

4

!Kxc%r# r 0; no&fn%r& # n%r 0&g2 d3r d3r 0:

This correction to the LDA is obviously valid forarbitrarily rapid but small density variations. It representsan infinite summation of the gradient terms. When,however, we want to apply this correction to a stronglyinhomogeneous system like, e.g., an atom, we are forced todecide what to use as an average density no in the kernelKxc: In a solid we could, e.g., use the average density of thevalence electrons. In an atom it becomes more natural tothink of a local density but, since the correction involvestwo points in space (r and r 0), there are a number ofpossible choices, e.g., n%%r$ r 0&=2& or %n%r& $ n%r 0&&=2:Gunnarsson et al. [54] have shown that the first choicegives an infinite correction for an atom whereas the secondchoice gives a reasonable result. These conflicting results oftwo similar and seemingly sound approximations could beviewed as a complete breakdown of gradient expansions.The origin of the sensitivity to the choice of average densityis the use of linear response theory underlying the entirediscussion above. Thus, our conflicting results is merely areflection of the fact that an atom or, for that matter, asolid is not a linear perturbation of the homogeneouselectron gas. As we shall see shortly, things are not as bad

as they seem. Gradient expansions are not convergent inthe sense that successively more accurate results can beobtained by adding more terms. They represent asymptoticexpansions, meaning that an approximation with a fewgradient terms added will be more accurate than the LDAin cases with relatively slow density variations. When thedensity varies more rapidly, we might be better off withoutgradient terms. As we shall see, the criterion for the validityof the LDA is not as strict as the corresponding criterionfor the gradient expansion. Over the past fifteen years,people have tried to circumvent this problem by applyingdifferent cut-off procedures designed to retain the advan-tages of gradient corrections for moderate density varia-tions without displaying the breakdown characteristic ofthe straightforward gradient expansion in cases with rapiddensity variations.

4. The performance of the LDA

4.1. The realm of the LDA

The extreme usefulness of the LDA has been commonknowledge for many years. We will here give a shortresume of results obtained from the LDA for differentphysical properties and state some general conclusionsconcerning the quality of these results.

We will, however, first remind the reader that static DFtheory, with a few exceptions, is strictly applicable only toground-state properties and, secondly, we will say a fewwords about to what systems the theory should be applied.

As a test of the quality of new approximations to Exc;DF theory is often applied to atoms and small molecules.With regard to this practice we offer the following remarks:

(i) These systems are, for their physical properties, besttreated by other many-body techniques such as Many-Body-Perturbation Theory (MBPT) or Configuration-Interaction expansions (Cl) which offer a well definedroute to successively more accurate answers.

(ii) The interior of atoms is extremely difficult to handleby DF techniques due to the rapid density variationsin these regions. Fortunately, the physics of theseregions have little to do with the physics of everydaylife (chemical bonding etc.).

(iii) The almost non-vanishing densities in the exterior ofatoms cause gradient corrections to diverge andapproximate xc-holes to be misplaced and deformedbut contain little energy. More importantly—suchregions do exist only in very sparse solids. Conse-quently, atoms and small molecules represent verysevere tests on the quality of different approximationsand care must be exercised in order not to have theconclusions obscured by irrelevant difficulties.

The application of Cl expansions or MBPT to complicatedsystems of considerable interest such as solids with manyatoms per unit cell, surfaces, atoms or molecules adsorbedon surfaces, and large molecules, is computationallyprohibitive. This is the realm of DF theory provided itcan be made accurate enough.

20 U. von Barth


4.2. Results

The following pages with tables I–X demonstrate thesuccesses of the Local-Density Approximation (LDA) asapplied to a variety of physical properties of solids. Thesepages also indicate some of the deficiencies of the LDA.Some of the results have been taken from the literature andin most cases I have given the source. Other results havebeen produced in the course of my own research.

4.3. Successes

The most striking feature discernible in the data shownbelow, is the remarkable accuracy of the LDA as applied toa variety of physical properties in many different systems.We will here summarize the existing experience fromnumerous applications of the LDA as follows:

. Binding energies are often better than 1 eV but in somes–d bonded systems the error can be twice or even threetimes as large. There is a systematic over binding.

. Equilibrium distances are generally accurate to within0.1 A They are systematically too short.

. Vibrational frequencies are accurate to within 10–20%.There exist occasional cases with larger errors.

. Charge densities are better than 2%.

. Geometries are accurate.

. LDA results are nearly always much better than those ofthe Hartree–Fock (HF) approximation.

. Most importantly, physical trends are generally correct.

We end this subsection by stressing that whereas the errorsin the binding energies of the LDA can be large, thebinding distances are usually very accurate. Thus, theenergy surfaces of the LDA are rather parallel to the trueones.

4.4. Deficiencies

The results shown above, however, also indicate somecumbersome deficiencies. Most notable is the systematicoverbinding predicted by the LDA, particularly for the s-dbonded systems. The overbinding is also, although to amuch lesser extent, reflected in a small but relativelysystematic underestimate of the bonding distances. We endthis section with a small list of systems for which the LDApredicts the wrong ground states.

. The transition-metal oxides FeO and CoO are erro-neously predicted to be metallic. But, MnO and NiOcome out as anti-ferromagnetic insulators in accordancewith experiment [71].

. Solid Fe is predicted to be an fcc paramagnet [72] but is abcc ferromagnet at low temperatures.

. In many semiconductors, the LDA gives the metal-insulator transition at much too large volumes. [73]

. The LDA predicts the wrong dissociation limits for alarge number of molecules. [29,35,37]

. The LDA predicts incorrect ground states for manyatoms. [29,32,37]

. The LDA gives unstable negative atomic ions in manycases when these are stable. [32]

. Et cetera, et cetera..

4.5. s-p Transfer energies

Gunnarsson and Jones [74] suggest that the root of theproblem is the inability of the LDA to properly account forso called s-to-p and s-to-d transfer energies. When a solidor molecule is formed even from rather ‘‘round’’ atoms theelectronic density stretches out toward neighboring atomsto form bonds. In one-electron calculations the stretchingis accomplished by transferring electrons from states oflower angular momentum to states with higher angularmomentum (e.g. s-d) in the bonding process. The physicalpicture is illustrated, e.g., by the case of Si. The Si atom hasfour valence electrons 3s2; 3p2: In forming the solid wewould, to a first approximation, promote one of the selectrons to a p orbital and then form the tetrahedrallyarranged hybrids. Thus, the atom in the solid has theapproximate configuration 3s; 3p3 and bonding hasresulted in a transfer of one electron from an s- to a p-

Table II. Ground-state properties of the molecules H2O, NH3, and CO2, as obtained from the LDA and from experiment.The results are taken from the work by Muller, Jones, and Harris [55]. We assume that the numerical errors involved inobtaining the LDA results are negligible in comparison to the deviation between theory and experiment. The equilibriumdistances in Table II are probably exceptions to our assumption because they do not conform to the general expectations ofbond distances being too short within the LDA.

H2O NH3 CO2

LDA exp LDA exp LDA exp

d 1.84 1.81 1.94 1.91 2.21 2.20( 106 105 108 107 180 180!s 3680 3657 3335 3337 1420 1388!b 1590 1595 820 950 730 667" 0.732 0.730 0.564 0.583 0 0

d is the equilibrium distance in atomic units.( is the equilibrium bond angle in degrees.!s is the stretching frequency in cm#1.!b is the bending frequency in cm#1." is the dipole moment in atomic units.

Table I. Total energies (in eV) of a few atoms.

Atom LDA exp

H 13.3 13.6H# 14.4 14.4Al 6567 6592



orbital. This costs energy but, in the LDA, we pay toosmall a price for the transfer resulting, on an absolute scale,in too low an energy for the solid. Consequently, the LDAoverestimates the cohesive energy of Si and in this case theactual error is 0.5 eV or 11% (See Table IV). A similarsituation exists in the transition metals although thetransfer in that case is from s, p to d [7]. According to

Gunnarsson and Jones [74] the errors in the s-p or s-dtransfer energies are due to the insensitivity of the LDA tothe nodal structure of the one-electron wave functions.Consider e.g. the fluorine atom which has a 2P ground stateconstructed from the configuration 2s"j2s#2p

3"j2p

2#: An s-p

transfer is achieved by promoting one of the 2s electrons tothe empty hole in the 2p-shell resulting in a 2S excited state.From HF theory, the accompanying change in theexchange energy of the L-shell is easily found to be

23G1%s; p& # 9

25F2% p; p& $ 12)Fo%s; s& # Fo% p; p&*

in terms of the usual Slater [75] integrals. Inserting theactual orbitals of F gives a 6.7 eV increase in the exchangeenergy. This rather large value is associated with the extraangular node of the 2p- relative to the 2s-orbital. In theLDA the exchange energy is given by

ELDx )n*; n#* " Ax

X

%

!)n%%r&*4=3 d3r

where n" and n# are the spin-up and spin-down densitiesand where Ax is equal to #%3=4&%6=#&1=3 in Hartrees. Thus,a substantial change in the exchange energy associated withthe s-p transfer requires the square of the s- and p-orbitalsto be rather different. This is not the case in the F atombecause both orbitals belong to the same principal-quantum number shell (the L-shell). Consequently, thecost of the transfer is much too small in the LDA%-0:6 eV&: Although the total error is considerably reduced[32] (to -2:6 eV) by correlation effects and self-consistency,it is still large enough to explain a large part of the error(1.7 eV from Table VI) in the binding energy of the F2

molecule.Clearly, the LDA in which the exchange-correlation

energy only depends on the local density cannot accountfor changes in the nodal structure of the wave functions.One could, however, hope that xc-functionals based ongradients would be capable of picking up such nodaldependencies. Unfortunately, this does not seem to be thecase as can be seen in Table XI showing results for theaforementioned s-p transfer treated by different methods.The Generalized Gradient Approximation (GGA) byLangreth, Perdew, Mehl and Hu (LPM) [8,9,56] inconjunction with spherically averaged densities performalmost as badly as the LDA.

One could guess that this disappointing result might be aconsequence of the spherical averaging, a procedure whichcertainly softens the nodal structure closely connected tothe angular dependence. Therefore, it is even moredisappointing to see the results marked LPM-NS inTable XI. They are obtained from the LPM scheme bytaking full account of the asphericities of the spin-densities.The results are marginally better than those of the sphericalapproximation and we conclude that a functional con-structed from the density and its first spatial derivatives isnot able to respond to a change in the nodal structure ofthe wave functions. Indeed, s-d transfer energies have beencalculated in transition-metal atoms by Kutzler and Painter[76] with the same disappointing outcome. They also testedan improved generalized-gradient approximation by Per-dew and Wang [77,78] to be discussed in Section 5.4 and

Table V. Lattice parameters in atomic units for a few solidsas obtained from the LDA and from experiment. With oneexception, Si, the data are taken from the book by Moruzziet al. [7]. The Si result is from Ref. [57].

Na Mg Si Ti Zr Ni

exp 8.0 8.5 10.3 7.8 8.2 6.7LDA 7.7 8.4 10.2 7.6 8.2 6.6

Table VI. The binding energies (in eV) of the first-rowdimers as obtained from the LDA, from the LPM scheme[8,9], from the GGA according to PBE [39,61,62], andfrom experiment. All results (except PBE) are taken fromthe work by Becke [63,64].

H2 Li2 B2 C2 N2 O2 F2

exp 4.8 1.1 3.0 6.3 9.9 5.2 1.7LDA 4.9 1.0 3.9 7.3 11.6 7.6 3.4LPM 5.0 0.6 3.3 6.1 10.2 6.4 2.4PBE 4.5 0.9 — — 10.5 6.2 2.3

Table VII. Equilibrium distances in Bohr of the first-rowdimers. The LDA results are from the work by Becke [63].

H2 Li2 B2 C2 N2 O2 F2

exp 1.40 5.05 3.00 2.35 2.07 2.28 2.68LDA 1.45 5.12 3.03 2.35 2.07 2.27 2.61

Table IV. Cohesive energies (in eV) of a few solids asobtained from the LDA and from experiment. The Si result isfrom Ref. [57]. The other results are taken from the book byMoruzzi et al. [7] and from Refs. [58–60].

Na Mg Si Ti Zr Ni

exp 1.1 1.5 4.6 4.9 6.3 4.4LDA 1.1 1.6 5.1 6.1 6.8 5.7

Table III. Ionization potentials (in eV) of a few atoms.LPM designates results from the gradient corrected schemeby Langreth, Perdew and Mehl [8,9] and Hu and Langreth[56]. In the cases of Mn and Fe we refer to the removalenergy of a 4s electron.

Atom exp LDA LPM

He 24.6 24.2 24.8Li 5.4 5.6 5.6Be 9.3 8.4 9.0Na 5.1 5.3 5.1Ca 6.1 6.3 6.0Mn 7.4 7.9 7.1Fe 7.9 8.4 7.8

22 U. von Barth


there referred to as the PW86 scheme. Although thisscheme is somewhat better than both the LDA and theLPM scheme, the improvement is marginal. (In this casethe LDA is actually better than the LPM scheme.)

The s-d transfer errors in atoms is one possiblemechanism responsible for the cohesive energy errors ofextended systems. Correlations, however, strongly reducethe effect already in atoms and screening as well assmoother densities in the solids will reduce the effect evenfurther. As we saw above, two proposed generalized-gradient approximations which are known to be superiorto the LDA with regard to binding energies of moleculesand solids (see Sections 5.3 to 5.6) fail to improve the s-dtransfer energies. From this we conclude that the s-p or s-dtransfer mechanism is probably not the most importantmechanism behind the failure of the LDA with regard tocohesive energies.

4.6. The near-degeneracy problem

Cl and MBPT dominate the world of Quantum Chemistry(QC), i.e. the physics of small to large molecules. Thesemethods are reliable and can often provide the 1 kcal/mol%1 eV " 23 kcal=mol& accuracy of interest to the physics ofchemical reactions [12]. They are, however, enormouslytime-consuming and there would be a breakthrough in QCif DF methods could be brought to chemical accuracy. It istherefore rather disappointing to see the LD results for thebinding energies of the first row dimers in Table VI. Themuch larger binding-energy errors in these finite systemscan be understood in terms of their more rapidly varyingdensities. Clearly, the explanations based on the s-ptransfer mechanism becomes more relevant but also othereffects come into play. One such effect can be referred to asthe near-degeneracy problem in the DF theory. Considerthe H$

2 -molecule at large internuclear separation R: Thecorrect wave function is close to

%r& "++12

qf&%r& $ &%r# R&g "

++12

qf&a $ &bg

where &%r& is the wave function of hydrogen: & " e#r=+++#

p:

The expectation value of the full Hamiltonian

H " # 1

2r2 # 1

ra# 1

rb$ 1

R

with respect to this wave function is, to exponentialaccuracy, equal to h&jHj&i "# 1Ry: Thus, in real life,the system cares very little about whether we break thesymmetry of the molecule and put the electron on one ofthe sites or we keep the symmetry and put half an electronon each site—the corresponding energies are very much thesame. In all approximations to DF theory which areconstructed by using only the local density and itsgradients, however, the situation is quite different. Inaddition to the energy terms considered above, the energyof the effective one-electron Hamiltonian of DF theorycontains the terms

12

! 2 v 2 $ Exc) 2*:

Table VIII. Vibrational frequencies in cm#1 of the first- andsecond-row dimers as obtained from the LDA and fromexperiment. The results are taken from the work by Becke[63].

H2 Li2 B2 C2 N2 O2 F2

exp 4400 350 1050 1860 2360 1580 890LDA 4190 330 1030 1880 2380 1620 1060

Table IX. Heat of formation (in eV) of a few compounds asobtained from the LDA and from experiment. The data aretaken from the work by Williams, Kubler, and Gelatt[65,66].

MgAg AlZr SiZr NiAl CuZn

exp 0.19 0.44 0.81 0.61 0.12LDA 0.24 0.44 0.73 0.74 0.14

Table X. a The saturated magnetic moment (Bohr magne-tons), b the hyperfine field (kiloGauss), and c the spin-susceptibility enhancement factor for a few metals. The dataare taken from the work by Janak et al. [67–70].

X.a Magnetic momentFe Co Ni

exp 2.22 1.56 0.61

LDA 2.15 1.560.59

X.b Hyperfine fieldFe Co Ni

exp 339 217 75

LDA 260 22080

X.c Susceptibility enhancementLi Na K

exp 2.50 1.65 1.70LDA 2.25 1.71 1.95

Table XI. Exchange energies (#Ex, in eV) in two differentconfigurations of the fluorine atom as obtained fromHartree–Fock (HF), from the LDA, and from the LPMscheme (Ref. [9]) evaluated at spherically averageddensities. LPM-NS designates LPM exchange energiesevaluated at the correct non-spherical densities. The numbersillustrate the failure of all DF approximations in describingthe correct (HF) s-to-p transfer energy.

2s*2s#2p3*2p2# 2s*2p3*2p

3# %Ex

HF 272.5 264.4 8.1LDA 246.4 244.1 2.2LPM 267.4 264.6 2.8LPM-NS 267.6 264.6 3.0



If we have found a functional which is very accurate for theatom and which achieves a high degree of cancellation ofthe so called self-interaction, this becomes

12

!&2 v &2 $ Exc)&2* " U# J . 0

in the hydrogen-plus-proton case. Thus, for this case, theLDA provides a good answer. In the symmetric case, weinstead obtain

21214

!&2v&2 $ 212

14

!&2av&

2b $ Exc

12 &2

a $ &2b $ 2&a&b

, -. /

. 12U$ 1

4R$ 2Exc

12&

2. /

i.e., a quantity with a slow variation with R: This result isfundamentally incorrect, thus demonstrating that thecorrect functional must depend on the density in a morecomplicated way than just through its local value and itsgradients. If we, e.g., use the LDX approximation (Local-Density-eXchange-only) given in Section 4.5, we obtain anLD error of

%ELD " 12U$ 1

4R# 2#1=3J . 0:25=R# 0:3U;

and, since U is - 8 eV for hydrogen, we obtain -2:4 eVoverbinding in the LDA at large separation. Even thoughthe other spurious term %1=4R & partly reduces the error atsmaller R; an effect might still remain at the equilibriumdistance. (As a matter of fact, the LDA error in the bindingenergy of H$

2 is only 0.15 eV at equilibrium [79].) Notice,however, that an essential ingredient of the mechanismdiscussed above is the existence of an unpaired electron.Consequently, the effect does not affect the binding energyof the dimers but it does affect their ionization potentialsand also the binding energies of the dimer cations asdemonstrated by Merkle et al. [80].The accurate results achieved by the LDA for the

‘billiard-ball’ systems Be2 (error -0:46 eV) [64] and Mg2(error -0:12 eV) [81], and the difficulty for the LDA toprovide good answers in cases with near degeneracies isopposite to the situation encountered with Cl methods. Theso called dynamical correlations responsible for, e.g.,binding in Be2 require a very large number of configura-tions for their accurate description whereas the near-degeneracy cases are often taken care of by a fewconfigurations. This immediately suggests a hybrid method[82] in which a local-density correction is added to a smallCl calculation.The discussion above illuminates another peculiar

feature of the LDA and approximations based on densitygradients. Even though the LDA does not work well for asystem with some particular symmetry, one can often find a‘‘neighboring’’ system with nearly the same energy (e.g. asymmetry broken solution) for which the LDA provides anaccurate answer. For example, the LDA gives a reasonabledescription of the H$

2 molecule at large nuclear separationif we break the symmetry and put one electron on one ofthe protons. A more interesting example is furnished by theCr2 molecule whose bond length is accurately given by theLDA provided the molecule is assumed to be anti-ferromagnetic, i.e., symmetry broken [83]. The bindingenergy is reasonable but too large as usual. In this case, one

of the largest and most sophisticated Cl calculations isinferior to the LDA [84].

5. Gradient approximations

As we saw in the previous paragraph, the LDA is oftenquite adequate while, in other cases, a higher accuracy isdesired. An error in a binding energy of the order of 20%or 1 eV is, e.g., not acceptable in the study of chemicalreactions. In this field we would like binding-energy errorsto be of the order 0.05 eV or less. The simplicity of DFmethods as compared to traditional many-body techniqueshas, however, spawned considerable efforts to improve onthe LDA. It would, e.g., mean a breakthrough in quantumchemistry if DF methods could be brought to the samelevel of accuracy as large-scale configuration-interaction(Cl) calculations for larger molecules. One could thenconfidently go on to treat very large molecules as well asextended systems which presently are beyond our compu-tational capabilities.

Attempts to go beyond the LDA are based either on animproved description of the exchange-correlation hole (seeSections 2.1 and 2.3) in real space or on a description ofexchange-correlation energies in reciprocal space usuallyleading to so-called generalized gradient approximations(GGA). Some times a mixture of the two approaches havebeen considered [77,78]. In recent years, the largest efforthas gone into the reciprocal-space approach which so farhas been the most successful ab initio DF method. As weshall see later, ab initio density functionals are still far fromthe 0.05-eV goal mentioned above but the results of newsemi-phenomenological functionals are not too far away.

5.1. Straightforward gradients

As discussed in Section 3.5, the natural extension of theLDA is a straightforward gradient expansion. The startingpoint is then the homogeneous electron gas and thegradient terms and their coefficients are determined bythe small wave-vector expansions of the density responsefunctions of different orders pertinent to the gas. Due tothe translational as well as the rotational invariance of thegas, one can show [47] that only rather few gradient termsexist in each order. For instance, all terms up to andincluding fourth-order gradients can be summarized as

Exc)n* " ELDAxc )n* $

!B%2&xc %n%r&&

jrn%r&j2

jn%r&j4=3d3r

$!B%4&xc %n%r&&

jr2n%r&j2

jn%r&j2d3r

$!Cxc%n%r&&

jrn%r&j2r2n%r&jn%r&j3

d3r

$!Dxc%n%r&&

jrn%r&j4

jn%r&j4d3r:

It is not difficult to realize that the terms containing thecoefficients labeled B originate in linear response, theC-term comes from second-order response, and thatinvolving Dxc from third-order response. The significantfeature determining the origin is the total power of thosedensities subject to differentiation. The coefficient B%2&

xc has

24 U. von Barth


been calculated in an approximation involving the effectsof both exchange and correlation whereas the coefficientsB%4&xc and Cxc are known in the exchange-only approxima-

tion. The coefficient Dxc is, however, yet unknown.It is not hard to see that this kind of gradient expansion

can never be applied to a finite system. Only the lowest-order gradient term will give a finite result for theexchange-correlation energy. The integrands of all thefourth-order terms will tend to some constant in theregions of exponentially decaying densities in the outskirtsof any finite system, thus producing an infinite result in thesubsequent integration. There will also be problems at thenucleus particularly from the term containing r2n: Becauseof the Coulomb potential from the nucleus, the density willhave a so called cusp at the nucleus. This means that thedensity has a term linear in the distance to the nucleus andthat will cause a singularity of the form const/distance inthe gradient part of the energy density. Consequently, thegradient correction will act as an additional contribution tothe nuclear charge, a result which is definitely wrong from aphysical point of view.In a solid treated with pseudo potentials both the

‘surface’ problem and the ‘core’ problem go away and thereis hope for obtaining accurate total energies from thegradients. Such tests have been carried out for differentmodel solids treated with pseudo potentials and includingonly exchange energies. Of course, in the case of exchangeonly, comparisons with exact results can be made whichallows for an precise assessment of the quality of astraightforward gradient expansion. And the gradientexpansion of the correlation energy is not expected tobehave in vary a different way in this regard. These testdemonstrate the extraordinary quality of the total energiesof ‘pseudo’ solids obtainable from such expansions. Thiscan be seen in Table XII which displays only the smallerexchange part of the energies of a number of metals. Theerrors are of the order of 1 Milli-Rydberg or less and muchsmaller than those resulting from the use of commonGGA:s to be discussed later. In Table XII, the quantitieslabeled "LDX

x ; "GE1x ; and "GE2x refer to the contribution fromthe LDA in the exchange only version and the contribu-tions including linear and second order responses—respectively.The significant features of all the systems presented in

Table XII is that they are metals and there are no bandgaps. Unfortunately, the very positive impression conveyed

by Table XII is drastically altered when we turn to systemswith band gaps. This is demonstrated in Table XIIIdisplaying similar data as in Table XII but now for systemsall having band gaps. In the first case labeled Si, we have tothe homogeneous gas added a pseudo potential whichresults in a band structure very close to that of real bulksilicon [85]. The case labeled SiX is similar to Silicon butthe pseudo potential has been scaled up resulting in a largerband gap. In these two examples the density has non-negligible Fourier components also on multiples of thefundamental harmonics of the lattice potential%V111;V220;V311& and it could be argued that gradientsalso from third- and higher order response theory shouldhave been included in the gradient expansion. Thiscriticism can, however, not be raised against the tworemaining cases labeled M1 and M2 in which care has beentaken to avoid larger amplitudes on any of the morerapidly varying density components. Thus, these cases haveslowly varying densities but still possess band gaps. And inall four cases the energy errors are an order of magnitudelarger as compared to the metallic cases and actually worsethan those of the GGA:s as exemplified by that of Becke[11]. The conclusion is now inescapable. The straightfor-ward gradient expansion breaks down as soon as a bandgap appears. This should, however, not come as a surprise.The entire gradient expansion originates in responsetheory, i.e., in perturbation theory which is known tobreak down in the presence of band gaps [86]. Theappearance of a band gap is actually an extremely non-local property of the system and can never be predictedfrom a knowledge of the density or its gradients in onepoint in space. In my opinion, even a GGA can never bemade very accurate and independent on the system towhich it is applied, unless the information about thepresence of a band gap is, somehow, fed into theconstruction of the GGA. It is thus more surprising thatthe GGA works so well for the systems with band gaps inTable XIII.

5.2. The kinetic energy

In this subsection we will discuss a topic which is somewhatoutside our stated task of finding successively betterapproximations to the exchange-correlation energies ofinhomogeneous electronic systems. We will, for a moment,digress and discuss gradient approximations also to theirnon-interacting kinetic energies. We have chosen to do sobecause this topic is closely connected to the straightfor-ward gradient expansion discussed in the previous subsec-tion. Moreover, interest in this topic has increased over thepast decade due to the desire to treat successively largersystems where it becomes difficult and expensive to solveeven the simple one-electron Schrodinger problem ofstandard DFT. If we can construct accurate approxima-tions to the kinetic energy from just the density alone wewould have a theory without the need for wave functionsand a large part of the computational effort of standardDFT would disappear.

Such a scheme has been a goal for many researcher sincethe time of Thomas and Fermi [2,3] and the gradientexpansion of the kinetic energy including gradients up toorder six has been known for years (see e.g. Ref. [36]). Testsmainly on atoms have, however, shown that the kinetic

Table XII. Calculated exchange energies in Rydberg perelectron for different values of the effective potential V. V111

(" V200) in units of the Fermi energy, "F.

V111 Exact "LDXx "LDX

x $ "GE1x "LDXx $ "GE1x $ "GE2x

GGA(Becke)

0.023313 #0.3303 #0.3302 #0.3304 #0.3304 #0.33060.069938 #0.3376 #0.3354 #0.3376 #0.3377 #0.33900.116563 #0.3515 #0.3460 #0.3518 #0.3517 #0.35500.163188 #0.3704 #0.3613 #0.3720 #0.3711 #0.37700.186500 #0.3809 #0.3702 #0.3836 #0.3820 #0.38920.209813 #0.3919 #0.3796 #0.3959 #0.3931 #0.40190.233126 #0.4030 #0.3893 #0.4086 #0.4043 #0.41470.256438 #0.4140 #0.3991 #0.4215 #0.4155 #0.42750.279751 #0.4248 #0.4089 #0.4346 #0.4264 #0.4402



energies obtainable from gradient expansions do not havethe accuracy to which we have been accustomed previouslyin these notes (e.g. of the LDA). Learning about theaccurate exchange energies I had obtained from straight-forward gradient expansions in metallic system, MalcolmStott [87] suggested that I should try the same techniquealso to the full kinetic energies. The main new idea whichwould allow us to disregard the negative results previouslyobtained for atoms was that we proposed to apply thegradient expansion only to pseudo atoms and not to realatoms. After all, most s-p bonded solids and eventransition metal compounds are nowadays treated withpseudo potentials and plane-wave codes. Thus, we exposedthe electron gas to a periodic array of local pseudopotentials and calculated the kinetic energy both from fullone-electron band theory and from the straightforwardknown gradient expansion. In Table XIV we display theastonishingly accurate result for our first test casesimulating an Aluminum metal as far as concerns theaverage electron density and the valence band structure. Inthese calculations we included up to fourth-order gradientsalthough the expansion is known at least up to sixth order.Table XIV also shows the results for our pseudo Siliconsystem discussed in the previous subsection. The case ofSilicon again demonstrates the breakdown of perturbationtheory as soon as band gaps occur although the error isonly 0.007Ry/electron.In Table XIV, the column marked tTF is the Thomas–

Fermi approximation which is equivalent to the LDA forthe kinetic energy. The column marked tTF $ tGE1 is theresult obtained including gradients to fourth order fromlinear response theory and the column markedtTF $ tGE1 $ tGE2 contains results including all fourth-order gradients. Thus, the data demonstrate the relativelylesser importance of higher-order responses for theseparticular systems.In order to obtain the results in Table XIV, we have

calculated the gradient corrections from the accuratedensity obtained from the full band structure calculation.Thus, from a computational point of view, nothing isgained by this procedure which still requires the calculationof all the occupied wave functions. Therefore, we went onestep further and minimized the total-energy expressioncontaining only the density and up to fourth-order

gradients. The minimization was carried out using a Car–Parrinello code [88] for minimization through simulatedannealing. From older test on atoms it is well known (see,e.g., Ref. [36]) that a minimization of the gradient-expanded total energy is relatively unstable and givessingular or bad densities which, in turn, produce inaccuratetotal energies. And the problem is aggravated by includinggradients of high order. This was actually one of thereasons why we decided not to go to sixth-order gradients.The rather small energy contributions from higher-ordergradient terms (see Table XIV) in our test systems suggestthat the minimization procedure could include onlysecond-order gradients in order to make it more stable.Then, as a last step, one could apply also the fourth-ordergradients to the minimizing density. Possibilities of thiskind could be investigated as the method becomes morerefined but, at the moment, we were quite pleased to findthat the minimization procedure, at odds with the atomicexperience, led to rather accurate energies of our pseudosystems. The error in Aluminum, some 20 milli Ry, is still alittle too high but might be remedied by introducingGGA:s also for the kinetic energy. In Silicon, the originalerror from the gradient expansion was actually almostcanceled by the minimization procedure which we, how-ever, consider to be a lucky fluke. Nevertheless, we thinkthat the results presented here offer a lot of hope for futuretotal-energy calculations in solids without the use of wavefunctions—at least in systems which can be described bymeans of pseudo potentials.

5.3. The LPM scheme

In Section 3.5 and in the previous section we haveobviously made a strong case against the use of gradientsas a tool for going beyond the LDA—except in the varyfavorable case of metals. As we shall see here, the situationis not that bad. One of the first attempts to improve on theLDA by means of gradients but without much theoreticalsupport was due to Herman et al. [89]. They added thesecond-order gradient correction with a variable coefficientB%2&x chosen to reproduce atomic exchange energies. In this

way, they were able to accurately reproduce Hartree–Fockenergies of atoms but they had to use a coefficient whichwas rather far from the value predicted on the basis ofgradient expansions for the electron gas. This representsthe first so called Generalized Gradient Approximation(GGA) and this kind of approach is used extensively eventoday. The exchange-correlation (xc) energy is written assome function of the density and its first gradient includinga large number of adjustable parameters. These aresubsequently determined by requiring the expression toreproduce nearly exact results in a number of known cases.Embarrassingly enough, this rather ad hoc method is stillprobably the most accurate approach within DFTalthough results are never consistently good. One oftenencounters surprises when treating systems different incharacter as compared to those of the reference group.

But there are certainly more fundamental approachestoward improvements via GGA:s. As we discussed inSection 2.3, xc energies can be obtained from a modeling ofthe exchange-correlation (xc) hole in real space. Anobvious complementary idea is to go to reciprocal spaceby means of Fourier transforms and to study xc energies

Table XIII. Exchange energies in Rydberg per electron forSi and Si-related models

Model Exact "LDXx "LDA

x $ "GE1x "LDXx $ "GE1x $ "GE2x

GGA(Becke)

Si #0.5321 #0.5041 #0.5249 #0.5173 #0.5263SiX #0.5539 #0.5186 #0.5545 #0.5320 #0.5468M1 #0.5340 #0.5107 #0.5250 #0.5215 #0.5291M2 #0.5550 #0.5267 #0.5471 #0.5393 #0.5495

Table XIV. Kinetic energies in Rydberg per electron forpseudo Si and pseudo Al.

Model Exact tTF tTF$ tGE1 tTF$ tGE1$ tGE2 t(minimized)

Al 0.466 0.436 0.465 0.464 0.486Si 0.716 0.692 0.725 0.723 0.717

26 U. von Barth


coming from different regions of momentum space. Fromthe discussion in Section 3.5 we also know that there is aclose connection between momenta and gradients of thedensity—at least in the linear regime. We are thus led tobelieve that a decomposition of xc energies with regard tomomentum transfers could, somehow, be converted intoexpressions involving gradients. This is actually a viableidea as we will now argue.The first attempts in this direction is due to Langreth,

Perdew, Mehl, and Hu [8,9,56] (LPM). I consider theirwork as the start of the activity to construct GGA:s fromfirst principles. It would carry too far to present the rathercomplicated theory underlying the LPM correction to theLDA. We will here be content with presenting some oftheir key ideas and some results for different physicalsystems. Langreth and Perdew [8] start from the basicequation for the exchange-correlation energy of theinhomogeneous system (see Section 2.3). In order to findthe interaction-averaged pair-correlation function ~gg of theinhomogeneous system they first use the RPA and computethe frequency dependent density-density response functionto second order in a perturbing external potential w:Within the RPA, the required integration over the strengthof the Coulomb interaction can be carried out analyticallyand from the zero-temperature version of the fluctuation-dissipation theorem [46] they then obtain Exc for theslightly inhomogeneous system in the form of an integral inreciprocal %k#& space. In this way, they can study thecontributions from different wave vectors both to the LDA%ELD

xc & and to the xc-energy obtained by adding the lowestorder gradient correction. They find that the gradientcorrection works well for large wave vectors but stronglyoverestimates the contributions to the xc-energy at small k:The crossover occurs around a q-vector approximatelyequal to q " jrn%r&j=%6n%r&&: In principle, their full resultfor Exc could be used in actual calculations although thecomputational effort would be considerable. Therefore, inorder to obtain an easily applicable scheme based ondensity gradients, Langreth, and Mehl [9] suggested anapproximation to the full results by Langreth and Perdew.They proposed to keep the full gradient correction down toa cut-off - q and to neglect the gradient correctionaltogether below this cut-off. In real space this results inthe following approximation for Exc :

Exc)n* " ELDxc )n* $ a

!)n%r&*#4=3)rn%r&*2 e#F # 7

18

% &d3r

where

F " b , jrn%r&j , )n%r&*#7=6; a " #

8%3#2&#4=3; b " %9#&1=6f

in atomic units, i.e., Hartree.The quantity f is a ‘‘fudge’’ factor of order 0.15–0.17 and

results are relatively insensitive to the choice of f in thisrange. Notice that surface energies indeed are sensitive tothe choice of f, indicating, not a breakdown of the abovetheory, but rather of the simple cut-off procedure leadingto the practical formula above.It is important to stress one of the basic assumptions

underlying the parameterized version of the wave vectordecomposition due to LPM. It is assumed that the physical

system under study is dominated by a single length scaleapproximately given by 1=q: This is probably correct in anatom where that length would be, e.g., the radius of anatomic orbital. As a result, the LPM scheme yields quiteaccurate total energies for atoms, the error being of theorder of a few parts in a thousand. As it turns out, themajor source of error comes from the treatment ofexchange. For this part of the energy the LPM approxima-tion gives an error of -3% which should be compared toan error of the order of -10% in the LDA. In the case ofatomic correlation energies only, the LPM errors are-10% compared to the infamous > 100% overestimateresulting from the LDA.

In molecules, on the other hand, there is clearly at leastone additional important length scale namely that of themolecular bond. As a result, the performance of the LPMscheme is not nearly as good in the case of binding energiesof molecules as illustrated in Table VI. Also, in this case,there does not seem to be any advantage in treatingexchange exactly and only leave the correlation part of thebinding energies to the LPM scheme (see Table XVI).

As pointed out in Section 3.5, approximations based ona theory for systems with relatively small and slow densityvariations like, e.g., the LDA and different gradientcorrections to it, can perhaps not be expected to workwell for strongly inhomogeneous systems like atoms andsmall molecules. Thus, tests on the ground-state propertiesof solids are highly desirable. Unfortunately, such tests arerare in the literature. The first such test was carried out byvon Barth and Car [57] in bulk silicon and they obtained acohesive energy of 4.89 eV compared to 5.19 eV in the LDAand 4.63 eV from experiment. The lattice parameter of theLPM calculation agrees with experiment and is 0.5% largerthan that of the LDA. Within the numerical accuracy, thebulk moduli of the LDA and the LPM calculations bothagree with experiment. Later von Barth and Pedroza [90]tried the LPM approximation in bulk beryllium andobtained a cohesive energy of 3.20 eV compared to3.65 eV in the LDA and 3.32 eV from experiment. Morerecently Bagno et al. [91] have applied the LPM scheme tothe structural properties of several elemental solidsincluding transition metals in which the general tendencyof the LDA to over-bind is particularly pronounced. Theirresults are similar to those reported above. The over-binding is reduced and the lattice parameters are slightlyimproved by the LPM scheme. In particular, this schemegives the correct ground state of iron, i.e., a ferromagneticbcc structure whereas the LDA erroneously predicts aparamagnetic fcc structure.

We can summarize our experience of the performance ofthe LPM scheme in solids and molecules by saying that thisscheme is superior to the LDA and that errors of the latterare typically reduced by a factor of two by the LPMapproximation. Thus, the new scheme is an improvementbut does certainly not represent a breakthrough within DFtheory.

Before ending this section on the LPM scheme, we mustmention a few technical details of relevance to the practicalapplication of the scheme. In most cases, especially in finitesystems, it is necessary to have a spin-polarized version ofthe theory. Such a generalization has been worked out byHu and Langreth [56]. For self-consistent calculations the



potential corresponding to the parameterized LPMscheme, i.e., the functional derivative of Exc with respectto the density, must be derived and an explicit formulainvolving second spatial derivatives of the density can befound in Ref. [9].The LPM theory for the slightly inhomogeneous electron

gas is based on the RPA. According to Langreth et al.,[9,92] there is a large cancellation between local andnonlocal contributions to the xc-energy also beyond theRPA and, in order not to lose this advantage by using atheory beyond the RPA for just the LDA, the RPA versionof the latter should be used in conjunction with the LPMgradient correction.As a byproduct of their wave vector analysis Langreth et

al. found two very useful criteria determining the validityof the LDA and of the straightforward gradient correction.They found that the LDA is valid when

jrn%r&jkF%r&n%r&

/ 6

whereas the gradient expansion is valid when

jrn%r&jkTF%r&n%r&

/ 1:

Here, kF%r& is the local Fermi momentum %k3F " 3#2n& andkTF%r& is the inverse of the local screening length, i.e., theThomas–Fermi wave vector given by kTF " 2

++++++++++kF=#

p: In

real systems the last criterion is usually an order ofmagnitude more severe than the first which explains whythe LDA can provide reasonable answers while theinclusion of gradient terms will make things worse.

5.4. The Perdew–Wang scheme

In the discussion of the LPM scheme we noted that thelargest errors in that scheme originates in their treatment ofexchange. For this part of the energy Langreth et al.essentially use a straightforward gradient correction. Beingconcerned about the shortcomings of the previous treat-ment Perdew [93,94] introduced a new idea comprising acombination of real- and reciprocal space arguments. InRef. [54] as well as in the work of many investigators, thesuccess of the LDA is attributed to various exact propertiesof its exchange-correlation (xc) hole like the sum rule or thenegativity of the exchange hole. In the wave vector analysisof Langreth et al. we can say that we study andapproximate the xc-hole in reciprocal space and it becomesless evident how to implement, e.g., the sum rule. Aparticular approximation for the xc-energy in reciprocalspace can, however, often be transformed into anapproximation for the xc-hole in real space based on thedensity and its gradients. Perdew showed, e.g., that thesecond order gradient expansion for the xc-energy does notcorrespond to an xc-hole that obeys the sum rule. This isyet another way in which we can understand why the LDAin some cases can yield an accurate result when the

Table XVI. The errors (in eV) in the correlation-energy contributions to the binding energies of the first-row dimers asobtained from different correlation functionals defined below. % is the average absolute error for each functional.

Li2 Be2 B2 C2 N2 O2 F2 %

LDC # 0.1 # 0.3 # 1.9 # 4.1 # 2.4 # 2.9 # 2.5 2.0LPMC 0.1 0.1 # 0.8 # 2.7 # 1.0 # 1.2 # 1.4 1.0PW91C # 0.3 # 0.2 # 1.5 # 3.9 # 2.4 # 2.1 # 2.4 1.8

LDC the local-density approximation for correlation energies. Data from Savin et al. in Ref. [33].LPMC the correlation part of the functional by Langreth et al. [8,9,56]. Data from Savin et al. in Ref. [33].PW91C the predecessor [97] to the PBE [61,62] by Perdew et al. Data from Ref. [119].

Table XV. The errors (in eV) in the binding energies of the first-row dimers as obtained from different density functionalsdefined below. % is the average absolute error for each functional.

Li2 Be2 B2 C2 N2 O2 F2 %

LDA # 0.1 0.5 0.8 1.0 1.7 2.4 1.7 1.2LPM # 0.5 0.3 0.2 # 0.2 0.3 1.2 0.7 0.5PW86 – – 0.2 # 0.1 – 0.7 0.5 0.4PW91 # 0.1 0.3 0.3 # 0.1 0.7 1.0 0.7 0.5B86 # 0.1 0.1 # 0.5 # 0.8 0.2 0.2 0.0 0.3B I # 0.1 0.1 # 0.3 # 0.7 0.3 0.3 0.2 0.3B II # 0.3 – – – 0.4 0.8 0.5 0.5B III # 0.3 – – – # 0.1 0.2 # 0.1 0.2

LDA the local-density approximation. Data from Ref. [63].LPM the functional by Langreth, Perdew, Mehl and Hu [8,9,56]. Data from Ref. [64].PW86 the older functional by Perdew and Wang [77,78]. Data from Ref. [103].PW91 the functional by Perdew et al. [97] preceding the PBE [61]. Data from Ref. [80].B86 an older exchange approximation by Becke [64] plus correlation from Stoll et al. [107]. Data from Ref. [64].B I the functional of ‘‘Becke: Thermo-chemistry I’’ [106]. Data from Ref. [80].B II the functional of ‘‘Becke: Thermo-chemistry II’’ [109]. Data from Ref. [109].B III the functional of ‘‘Becke: Thermo-chemistry III’’ [111]. Data from Ref. [111].

28 U. von Barth


addition of the lowest order gradient term can destroy thataccuracy.From the work by Gross and Dreizler [95] on gradient

expansions it is not difficult to obtain one for the exchangehole. A second order expansion for the hole does not obeyeither the sum rule or the negativity condition (See Section2.1). Therefore, Perdew [93] multiplied the result by Grossand Dreizler by an appropriate cut-off function to ensurethe satisfaction of both the sum rule and the negativitycondition. The resulting approximation for the exchangehole looked somewhat complicated and difficult to use butin a later paper by Perdew and Wang [77], the result wassimplified by partial integration and parametrized for easyuse in DF calculations. Their functional for the exchangeenergy became

Ex)n* " Ax

!)n%r&*4=3Fx%s&d3r;

where Ax is a constant determined by the electron-gas limitand given by

Ax " # 3

4#%3#2&1=3:

The quantity Fx%s& is a function of the dimensionlessdensity gradient

s " jrn%r&j2kF%r&n%r&

:

The function Fx is originally obtained numerically in orderfor the x-hole to obey the sum rule and the negativitycondition and later fitted to an analytical formula:

Fx%s& " %1$ as2=m$ bs4 $ cs6&m

with the constants

a " 781 ; b " 14; c " 1

5 ; m " 115 :

Energies are given in Hartrees. Notice that Fx%s& " 1corresponds to the LDA for exchange only. Notice alsothat many systems are spin-polarized, particularly those offinite extent and that, therefore, the spin-polarized versionsof the functionals must be used. In the case of exchangeonly this is easily obtained as

Ex)n"j; n#* " 12fEx)2n"j* $ Ex)2n#*g;

As a side issue, we finally notice that the generalizedgradient approximation by LPM of the previous subsec-tion, and that of Becke [11,64,96] (see next subsection) aswell as, really, all GGA:s can be cast in the same form. Andthe different GGA:s (excluding, of course, the Meta-GGA:s involving also the Laplacian of the density) differonly in their particular form of the function Fxc%s&:The particular parametrization of the Perdew–Wang

scheme presented above has the property that an incorrectsecond-order gradient term is obtained in the slowlyvarying limit (small s). In this context, ‘‘incorrect’’ refersto a deviation from the gradient coefficient given exactly byEngel and Vosko [49]. Since the small-q limit of theexchange-only kernel Kx%q& (see Section 3.5) is better

described over a larger range of q-values by a correctsecond-order gradient coefficient for exchange, the func-tion Fx%s& in the Perdew–Wang scheme above should bemodified to give this slowly varying limit. Settinga " 10=81 in the parametrization above will achieve thedesired result. In later work [97], Perdew et al. madeadditional modifications of the function Fx%s& and essen-tially made it much more similar to the Becke exchangeapproximation [11] (see also below). The slightly modifiedfunctional of Ref. [97] was tried in solid Li and Na and inatoms and small molecules [80,97]. This new GGAperformed better but not much better.

Having improved on the LPM description of non-localcontributions to exchange energies, Perdew [78] proceededto introduce some beyond-RPA effects into the LPMscheme for the correlation energies. Apart from takingaway a spurious contribution of exchange origin added byLangreth et al. in order to separate exchange andcorrelation in finite systems, Perdew’s modification essen-tially amounts to a rescaling of the LPM result in order toretrieve the correct beyond RPA second-order gradientcoefficient calculated by Geldart and Rasolt [51]. Theexplicit formula is very similar to the LPM formula of theprevious subsection, but without the 7/18:th. We refer thereader to Ref. [78] for details. Having a beyond RPA resultfor the non-local correlation energy allowed Perdew to addthe state-of-the-art version of the LDA for the correlationenergy, e.g., a parametrized version of the Monte–Carloresults for electron-gas correlation energies by Ceperleyand Alder [41].

The Perdew–Wang (PW86) correction to the LDAdescribed above has been employed in several electronicstructure calculations of solids. Kong et al. [98] calculatedthe ground-state properties of solid Al, Si, and C andconcluded that the PW86 functional gives a substantialreduction of the overbinding characteristic of the LDA.The bulk moduli are almost the same as those of the LDAbut there was a small increase in the lattice parametersresulting in closer agreement with experiment. Notice,however, that these quantities are very close to experimentalso in the LDA. Bagno et al. [91] tried both the PW86functional and the LPM functional on solid K, Ca, V, Fe,and Cu. Their conclusions were similar to those of Kong etal. The LDA errors in the binding energies are reduced bytypically a factor of two and the lattice parameters and thebulk moduli are also improved by the two gradientcorrected schemes. In particular these schemes predict thecorrect ground state of iron, i.e., ferromagnetic bcc. On theother hand one finds that one of the well known failures ofthe LDA, i.e., the prediction of metallic ground states forthe transition metal oxides FeO and CoO, are not correctedby the PW86 functional [99]. Comparing the PW86 and theLPM functionals, Bagno et al. found that the first had aslight edge on the latter although they noted a tendency ofthe PW86 functional to over-correct the LDA errorsespecially for the more weakly bound systems. The sametendency has been reported by Garcia et al. [100] in a studyof the cohesive properties of solid Al, Si, Ge, GaAs, Nb,and Pd. In particular these researchers report that thePW86 functional in many cases gives lattices which are toosoft. Similar conclusions have been reached by others[101,102].



As far as molecules are concerned, there is someexperience with the PW86 functional [103]. It seems clear,however, that this functional performs slightly better thanthe LPM functional also in the molecular case and that itthus represents a substantial improvement on the LDA. Aswill be discussed in the next section, this is a somewhatspurious result most likely of unphysical origin.As mentioned above, further improvements on the

Perdew–Wang scheme was made in Ref. [94]. There,Perdew and collaborators incorporated the sum rule forthe correlation hole, i.e., the fact that this hole mustcontain zero charge. This follows from the sum rule for thetotal hole and the unit charge contained in the Fermi (orexchange) hole. They have also modified their approxima-tion to the exchange energy through a different choice ofthe function Fx%s& as discussed above. The latest Fx%s&behaves as 1$ 10=81 s2 at small s corresponding to thecoefficient of Engel and Vosko [49] for the straightforwardsecond-order gradient correction for the exchange energyof the homogeneous electron gas. Probably inspired by thesuccess of one of Becke’s [11] gradient corrections forexchange (see next section), the function Fx%s& closelyfollows the Becke approximation at intermediate s: Incontrast to the latter approximation, Fx%s& is made tovanish as const=s2 at large s in order to obey certaininequalities for exact exchange. The sum rule on the x-holeis not enforced but is obeyed rather closely by the resultingfunctional referred to as PW91 in what follows. The PW91functional was tested by Perdew and his collaborators onsolid Li and Na. In both cases the PW91 functionalcorrects the unusually large LDA errors in the latticeparameters of these systems (0.11 A in Li and 0.17 A inNa). The PW91 functional has, by now, been tried in manysystems. It is rather similar to the PW86 functional and, insolids, it gives results of comparable accuracy or slightlybetter. Also in molecules the PW91 functional is compar-able to the PW86 functional as reported by Merkle, Savin,and Preuss [80].The experiences from the functionals of GGA type

which we have referred to as PW86 and PW91 have beencollected by Perdew and his collaborators in the GGA-functional called PBE [61,62]. This functional is today astandard ingredient in most readily available density-functionals codes for solids and molecules. It performswell in both finite and infinite systems without beingspecialized for either application. This is in contrast tomany new functionals of Quantum Chemistry which mostoften are designed for molecular applications. It differsrather little from the PW91 but it is much easier to programand its derivation is much simpler. A particular advantageis that it gives a rather good representation over a range ofmomenta %q& of the static linear density response functionof the electron gas as indicated by quantum Monte Carlocalculations [104,105].We will now attempt to summarize our experience from

the gradient corrected density functional methods. From atheoretical point of view, many of these schemes are aresult of a careful analysis of exchange and correlationeffects in a weakly perturbed electron gas (basically linearresponse theory). Through the use of sum rules and otherconsistency requirements, effects of higher order responsesare brought into the approximations. The results are

encouraging and usually much superior to the LDA. Errorsin binding energies are reduced by a factor of two to threeor even more. In the case of properties which are very welldescribed already by the LDA the picture is less consistentand gradient corrections can lead to worse results althoughthe difference is marginal. In going from the LPM gradientcorrected functional to the PW86, the PW91, and later tothe PBE and, especially, to the Meta-GGA functionals (tobe discussed in Section 5.6) there is an increase intheoretical sophistication. Tests on different systems andfor different physical properties indicate a minor andinconsistent improvement going from the LPM to thePW91 scheme. Some additional improvement, a simplerderivation and a smoother expression and thus a lesssingular potential is obtained by going on to the PBE. Afurther improvement results from the introduction of theMeta-GGA. The gradient corrected schemes are margin-ally more difficult to apply as compared to the LDA and inview of aforementioned properties we recommend that thelater functionals, e.g. the PBE or the Meta-GGA, to beused in all DF calculations.

In Section 4.5, several of the failures of the LDA wasblamed on the the inability of the LDA to account for thes-p or the s-d transfer energies. Tests on the gradientcorrected functionals [32,76] show that they suffer from thesame deficiency but still yield much better binding energies.This somewhat unexpected result is the basis for ourcomment in Section 4.5 that the s-d transfer problem maynot even be the major source of error in the LDA.

In small molecules it is relatively easy to compute theexact exchange energy thus leaving only the much smallercorrelation energy to be treated by DF methods. Such aseparation leads to disastrous results as can be seen inTable XVI. The usual argument for treating exchange andcorrelation together is based on the large cancellationoccurring between exchange and correlation energies dueto the much smaller extent of the xc-hole as compared tothe exchange and correlation holes taken separately. Thisargument is certainly valid in extended systems but itsvalidity is doubtful for small systems where already the sizeof the system limits the extent of both holes. The badresults shown in Table XVI and resulting from a separatetreatment of exchange and correlation, rather indicate apoor description of the correlation hole at intermediatedistances—a region not amenable to gradient correctionsbased on electron-gas theory. At this stage, the relativelygood results obtained for the combined DF treatment ofexchange plus correlation must be considered as fortuitous.Indeed, as we shall see in the next section, Becke [106] hasmanaged to produce accurate correlation energies for alarge number of molecules using an approximation basedon arguments relevant only to exchange energies.

5.5. The Becke approach

In the previous subsections we have described attempts todesign approximations beyond the LDA which are basedon a careful analysis of the effects of exchange andcorrelation in inhomogeneous systems. From early on,Becke has taken a different approach to DF theory. Inprinciple we seek to map out the functional Exc)n* insystems of interest which, in Becke’s case, have beenatomic and molecular systems. Suppose that we have

30 U. von Barth


designed a functional based on, e.g., the density, itsdifferent gradients, and different integrals thereof, con-taining a large number of parameters. Suppose further,that we were able to determine those parameters in sucha way that the chosen functional, within a certainaccuracy, reproduces the exact Exc for a large numberof physical systems with rather different properties withregard to exchange and correlation. Then, we wouldcertainly believe that we had obtained an accuratedescription of the true Exc)n*; at least for densities notdeviating in a qualitative way from those already studied.We would feel comfortable in applying the constructedfunctional to unknown systems and we would have someconfidence in the results thus obtained. There arecertainly theoreticians who would be prone to scorn thedescribed procedure but we find such attitudes unwise.First of all, the semi-phenomenological procedure willenable us to treat more complicated systems with muchgreater ease as compared to ab initio methods. And wecould even attack systems which presently are beyondcomputational feasibility and we could concentrate onother physical aspects like, e.g., molecular reactions.Secondly, we would have learned something about thefunctional Exc)n*: For instance, if our chosen functionalform contains only the density and its first spatialderivatives we would know that it must be possible tocast a sophisticated ab initio theory in a form containingthe same basic ingredients.When we, after this philosophical digression, return to

Becke’s work we must stress that a considerable amount ofphysical insight is needed in order to invent a reasonablefunctional form for Exc)n*: A better functional form leadsto fewer adjustable parameters and Becke’s functionals docontain few parameters. He started with the observationthat when density gradients become large, gradientcorrections are irrelevant. Concentrating on the exchangeenergy, Becke [64] designed a functional which, at smallgradients, amounts to a normal second-order gradientcorrection with an adjustable coefficient and whichbecomes an adjustable constant at large gradients. As amatter of fact most of Becke’s gradient corrections forexchange can be expressed in terms of an enhancementfactor Fx%s& times the LDA or LDX as discussed in Section5.4 in connection with the Perdew–Wang scheme. The firstone [64] used

Fx%s& " 1$ )s2

1$ *s2:

The parameters ) and * were adjusted so that thisparticular form gave an accurate fit to the exchangeenergies of a large number of atoms%) " 0:2351; * " 0:2431&: To this exchange energy mustbe added some approximation for the correlation energyand in his first semi-empirical approach toward molecularenergies Becke chose one due to Stoll et al. [107] Thisapproximation is much superior to the LDA for correla-tion energies of finite systems but is has the wrong slowlyvarying limit. The resulting atomization energies of the firstrow dimers can be studied in Table XV. The maximumerror is 0.8 eV and the average error for the seven moleculesis 0.3 eV. The corresponding figures for the LDA are 2.4 eV

and 1.2 eV. Considering the simplicity of the approxima-tion this represents an extraordinary achievement whichonly recently has been surpassed by a rather specializedGGA based on many adjustable parameters.

Based on ideas concerning the behavior of exchangeenergies in strongly inhomogeneous systems (large gradi-ents) Becke [96] later proposed a minor modification of theabove Fx amounting to raising the denominator to the 4/5power and changing ) and * but, to our knowledge, thisfunctional has never been tested in molecules. This iscertainly not true about the next gradient correction for theexchange energy by Becke [11] which reads

Fx%s& " 1$ )s2

1$ %2:25=#& )s arc sin h%*s&:

Here, * is a well defined constant equal to * " 2%6#2&1=3 (aconversion between Becke’s x " jr+j , +#4=3 and Perdew’ss " jrnj=%2kFn&) and only ) is an adjustable parameterwhich is determined by a fit to atomic exchange energies. Avalue of ) " 0:2743 gives an average exchange-energy errorof only 0.11% for six noble-gas atoms—a truly remarkableresult considering the simplicity of the above ansatz. Thechosen form for F might appear peculiar but it isconstructed to give the correct inverse-distance dependenceof the exchange-energy density far outside a finite system.In this region, the density decays as e#l , r and it is notdifficult to see from the properties of the inverse sinehyperbolic function that this leads to a contribution of theform

# 1

2

!

r>R

n%r&r

d3r

to Ex)n* at large R: In fact, this particular exchange-energyfunctional is presently the only available functional withthis correct behavior. But we are a little doubtful about thevalue of this property since the energy contributions fromthe large-R regions are small due to the very small densityin these regions. Nevertheless, this ‘‘ansatz’’ by Becke hasproven extremely successful. Becke [106] has tested thefunctional by calculating the atomization energies of the 55molecules of the so-called Gaussian-1 data base of Pople etal. [12,108] For this test, Becke used the normal LDA forthe correlation energy—an approximation known to over-estimate the correlation energy by a factor two to three.One can only speculate why Becke refrained from addingone of the existing gradient-type correlation corrections tothe LDA. In any case, his results were astonishingly good.The average absolute error was only 0.16 eV (3.7 kcal/mol)and the maximum error was 0.48 eV. The correspondingerrors of the LDA are 1.6 eV and 3.7 eV, i.e., an order ofmagnitude larger. Since there is a large overestimate of thecorrelation energies in the test, Becke’s exchange functionalmust, somehow, make up for the difference. It is aninteresting task for future research to understand how thiscancellation of errors comes about. The Cl-oriented abinitio results for the same set of molecules have averageerrors of the order of 1 kcal/mol [12,108] and we clearly seethat DF methods are approaching the same quality butwith much less effort.



In other tests on atoms and molecules, Becke found thatthe tested functional had an unsatisfactory performance inthe case of electron non-conserving processes like, e.g.,ionization. Becke [109] then decided to add to his previousfunctional the, at the time, most recent gradient correctionby Perdew et al. [97] (PW91C) for the correlation energy.The approximation for the exchange energy remained thesame—the arcsinh-ansatz. Becke also enlarged his database by including many ionization potentials and severalproton affinities and redid the test, again with a veryimpressive result [109]. The average absolute error and themaximum error for the atomization energies of themolecules of the Gaussian-1 data base was found to be0.25 eV and 0.8 eV respectively. Thus, an ‘‘improved’’description of correlation effects results in slightly largerbinding-energy errors but there is a big gain as far asionization potentials are concerned. These show an averageerror of only 0.15 eV and a maximum error of 0.44 eV. Thecorresponding numbers for the LDA are 0.23 eV and0.62 eV for 42 ionization potentials.These results represent, on the average, close to an order

of magnitude improvement over the LDA and this isindeed encouraging. We are closer to achieving ‘‘chemical’’accuracy but there is still some way to go. Occasionalerrors of the order of 0.8 eV cannot be tolerated. The Beckefunctionals are specially designed for finite systems (atomsand molecules) and they have the wrong slowly varyinglimit. What this would mean for solids or larger moleculesis hard to say. Since only the gradient terms are in error, wewould not expect the Becke functional to be any worsethan the LDA in extended systems. From a theoreticalpoint of view, however, we would prefer a functional withall the correct limits built in but no such functional canpresently overshadow the one by Becke in smallermolecules.As mentioned several times in these notes, exchange

energies are much larger than correlation energies even inextended systems like metals. This immediately suggeststhat one should treat exchange exactly and, e.g., use aGGA for only the correlation energy. So far, attempts in inthis direction have failed as can be seen in Table XVI. In1993, Becke [110] suggested a hybrid method in which halfof the exchange energy is treated exactly leaving the otherhalf plus the correlation contribution to approximations ofthe GGA-type. Theoretical support for this scheme canagain be obtained from the basic formula of Section 2.3 forthe xc-energy sometimes referred to as the ‘‘adiabaticconnection’’ formula. Using the bare LDA for thecorrelation contributions Becke found that this newscheme had an accuracy similar to his previous functionalsfor the atomization energies of the Gaussian-1 data base.In one further step Becke [111] included some of thecorrelation correction of Perdew et al. (PW91C) anddecided to treat the amount of exact exchange as a fittingparameter. His ‘‘hybrid’’ functional can be summarized as

Exc)n* "ELDxc )n* $ ao , %Ex)n* # ELD

x )n*&

$ ax ,%EB88x $ ac ,%EPW91

c

where ao; ax; and ac are semi empirical coefficients to bedetermined by an appropriate fit to experimental data. Thesymbol % signifies the gradient part of the corresponding

approximation, B88 refers to the Becke [11] gradientcorrection for exchange (the arcsinh-ansatz), and PW91refers to the mentioned gradient corrections for thecorrelation energy by Perdew et al. [97] The functionalEx)n* is the exact exchange-energy functional, i.e., theHartree-Fock exchange-energy expression, here evaluatedusing the DF orbitals, and ELD

x is the LDA version of onlythe exchange energy. The functional defined by the aboveequation is, in the literature, referred to as the B3PW91functional where the ‘‘3’’ referes to the term ‘‘thermo-chemistry III’’ appearing in the title of Ref. [111]. Theparameters ao; ax; and ac of the B3PW91 were determinedby fitting to the Gaussian-1 data base including also 42ionization energies and 8 proton affinities and the resultingoptimum parameters were found to be ao " 0:20;ax " 0:72; and ac " 0:81: These values resulted in anaverage absolute error of only 0.10 eV (2.4 kcal/mol) for theatomization energies with a maximum error of 0.33 eV.These results are now about a factor of two away from theproclaimed goal of chemical accuracy meaning errors ofthe order of 1 kcal/mol.

Results like those just discussed had a strong impact onthe community of quantum chemists and in the beginningof the nineties the use of DF methods started to spreadrapidly within this community. [112,113] New gradientcorrections for the correlation energy were constructed byBecke [114] and by Lee, Yang, and Parr (LYP) [115]. Boththese functionals have been tested on small molecules byMiehlich et al. [112] Johnsson and coworkers [113] havealso applied the LYP correlation functional in conjunctionwith the Becke gradient correction for exchange in anextensive test on a large number of smaller molecules. Theconclusion from these tests is that the performance of theBecke correlation functional, the LYP functional and thepreviously discussed PW91 functional for correlationenergies is rather similar.

The most widespread functional among the chemists isprobably the one called B3LYP. It is a again a so calledhybrid functional meaning that some portion of exactexchange is mixed into the expression for the exchange-correlation energy. It actually looks almost identical to theBecke functional B3PW91 displayed above except for thefact that the gradient part of the correlation energy isreplaced by the LYP expression. [115] Using the para-maters (ao; ax; and ac) optimized by Becke also in theB3LYP functional, Curtiss and coworkers [116–118] havepeformed extensive tests of the latter as applied to manyproperties of a wide range of molecules. In molecularapplications, the B3LYP rather consistently outperformsmost other density functionals allthogh the difference to,e.g., the B3PW91 is marginal.

The tests reveal, however, that results deteriorate withthe size of the molecules. This should, however, come as nosurprise since the the well known anomalies caused inextended systems by an unscreened exchange is expected toruin the results of the hybrid functionals in solids andcompromise their performance in larger molecules. Still thesuccess of the hybrid functionals B3LYP and B3PW91indicate that some sort of non-local exchange should beincorporated into future more accurate functionals—non-localities above those that can be accounted for by meregardients.

32 U. von Barth


5.6. The Meta-GGA

As we have seen above, generalized gradient approxima-tions (GGA:s) constructed from only the density and itsfirst spatial gradient are still some distance away fromchemical accuracy (1 kcal/mol). And the efforts to con-struct GGA:s range from ab initio and parameter-freetheory to advanced fitting schemes involving a largenumber of parameters. The situation is summarized by avery nice argument due to Philipsen and Baerends [120].They first observe that the binding energies of moleculesand solids are dominated by exchange effects and then goon to write the exchange energy like we did in ourdiscussion of the Perdew–Wang scheme in Section 5.4. Onecan write all GGA:s involving only the density and its firstgradient in the form

Ex)n* "!n%r&"x%%n%r&&Fx%s%r&& d3r

where "x%n& is the exchange energy per electron of thehomogeneous electron gas of density n and s%r& is thereduced dimensionless density gradient as defined inSection 5.4. Different GGA:s only differ in their choiceof enhancement factor Fx%s& where Fx " 1 corresponds tothe LDX, i.e., to the exchange-only version of the LDA.Finally, their investigations show that the binding energiesoriginate in regions of space where the reduced density slies in the interval 0:5 < s < 2:5: Regions of very smallgradients—the slowly varying regime—or of large gradi-ents—the asymptotic region outside atoms—contributelittle to the binding energies. In the important interval fors; most GGA:s have a rather smoothly increasingenhancement factor and if there were such a thing as an‘‘exact Fx’’, it would certainly be very easy to model thatwith a few parameters. On the other hand, distorting Fx toobtain exact exchange energies for a few systems wouldcertainly worsen the results for others. We can thus safelyconclude that the whole idea of modeling exchangeenergies by the density and its first gradient is of limitedvalidity. Moreover, given the failure of extensive fittingprocedures in reaching chemical accuracy, we haveprobably already reached the limits of this approach.In view of the discussion in the previous paragraph, it

appears rather strange that there exist so few attempts atincorporating also more complicated but still localdependencies on the density like, e.g., the Laplacian ofthe density or the kinetic-energy density. Suggestions inthis direction have not been lacking [47,93,121] and somequantum chemists have included the Laplacian in theirattempts to fit chemical data to results from Meta-GGA:s[122–124]. From a fundamental point of view, gradientcorrections to the LDA involving the Laplacian originatein second-order response theory about which little isknown. And from a more pragmatical point of view thefitted corrections based on the Laplacian did not lead to abreakthrough in DFT although the results from theseattempts clearly demonstrated the advantage of includingalso the Laplacian when constructing phenomenologicalgradient corrections [122]. The theoretical situationimproved when the coefficient involving the Laplacian tolowest order was calculated by Svendsen and von Barth[47] in the Exchange-Only approximation (EXO). This

inspired Perdew and coworkers [53] to apply their brand-marked philosophy of designing gradient corrections byincorporating as many known exact results as possible intothe construction. In addition to the Laplacian, the resultingMeta-GGA had the kinetic-energy density as an extraingredient. But the latter was expanded in gradients up toand including the Laplacian of the density thereby allowingPerdew et al. to express the Laplacian in the kinetic-energydensity. In this way, they could substitute the latter for theformer. The considerable advantage of this procedurestems from the fact that the kinetic-energy density, asopposed to the Laplacian, is well behaved both close to thenuclei and in the outskirts of finite or semi infinite systems.The result is a Meta-GGA with the correct expansion in theslowly varying regime but without the known singularitiesof the Laplacian. In the former respect, their Meta-GGA isactually superior to their previous and widely used GGA—the PBE [61].

Several tests of this Meta-GGA in variety of systemsranging from localized to extended have revealed [39,53] asubstantial improvement over, e.g., the PBE [61]. Theresults for the atomization energies of a number ofmolecules can be seen in Table XVII.

We see from Table XVII the relatively better ability ofthe Meta-GGA, as compared to the PBE, to distinguishbetween molecules having double and triple bonds. Theresults of the Meta-GGA are now comparable to thoseobtained by Becke [110,111] in his procedure to include acertain suitably fitted percentage of the exact exchangeenergy into his functionals as discussed in Section 5.5. Andthis has been achieved through the use of fundamentaltheory. It should also be remembered that Becke’sfunctionals are designed for finite systems in particularand do not perform well in extended systems. The sameremark can be made about most of those functionalscontaining a large number of parameters and fitted to an

Table XVII. Atomization energies (in kcal/mole) calculatedusing the LDA, the PBE-GGA [61], and the Meta-GGA[53] of Perdew and coworkers compared to experimentaldata (Exp.) stripped of the energy of the zero-point motion.

Molecule LDA PBE-GGA Meta-GGA Exp.

H2 113.3 104.6 114.5 109.5LiH 61.1 53.5 58.4 57.8CH4 462.6 419.8 421.1 419.3NH3 337.3 301.7 298.8 297.4OH 124.2 109.8 107.8 106.4H2O 266.6 234.2 230.1 232.2HF 162.3 142.0 138.7 140.8Li2 23.8 19.9 22.5 24.4LiF 156.1 138.6 128.0 138.9Be2 12.8 9.8 4.5 3.0C2H2 460.3 414.9 401.2 405.4C2H4 632.7 571.5 561.5 562.6HCN 360.8 326.1 311.8 311.9CO 298.9 268.8 256.0 259.3N2 266.9 243.2 229.2 228.5NO 198.4 171.9 158.5 152.9O2 174.9 143.7 131.4 120.5F2 78.2 53.4 43.2 38.5P2 143.0 121.1 117.8 117.3Cl2 82.9 65.1 59.4 58.0mean abs. error 31.7 7.9 3.1 –



extensive body of chemical data, even though some of theseoutperform the present Meta-GGA in molecules [122].The performance of the Meta-GGA has also been tested

in calculations of the surface energies of metallic systems[39,53], again with very encouraging results. Part of thissuccess is certainly due to the present Meta-GGA beingcorrect in the slowly varying limit, a property not shared bymost other GGA:s.Still, as we also see from Table XVII, errors in

atomization energies of molecules can easily amount tohalf an eV or more. This is actually a factor of ten worsethan our proclaimed goal of so called ‘chemical accuracy’.Neither have any of the semi-empirical functionals beenable to consistently achieve this goal. In my opinion, thehope of capturing the complexity of the full static many-body problem in terms of only the density and a few of itsgradients, is futile. Nevertheless, I recommend the Meta-GGA described above as the standard approximation to beused in all DFT calculations of substances, finite as well asinfinite, about which little is known before hand.

6. Real space methods

My comments throughout the sections on the GGA:sprobably convey the message that we are at ways end as faras these methods are concerned. And this was actually myintention because this is my present working hypothesis.The fitting schemes involving the density and the first-ordergradient and being constructed by means of neuralnetworks give average errors around 2 kcal/mol butoccasional errors ten times as large in localized systems.Including also the Laplacian in the fitting procedures mightreduce these errors further. Judging from the discussed (seeSection 5.6) experience from the Meta-GGA—maybedown to average errors of -1:0–1:5 kcal=mol and occa-sional errors of the order of 10 kcal/mol. Adding evenhigher-order gradients is likely to result in successivelysmaller improvements.Given the extreme computational advantage in using DF

methods and the need for higher accuracy, how do weproceed? I do certainly not have the answer to this questionbut there is another route which has not been followeduntil the end. In Section 2.3, we discussed the LDA fromthe point of view of modeling the exchange-correlation (xc)hole in real space and we will now explore furtherpossibilities along this track.

6.1. Early attempts

The construction of the LDA was described formally inSection 2.3 but we might gain some understanding bydressing the construction in words. We might recall thatthe xc energy is the total energy associated with eachelectron interacting with its xc hole. We obtained the LDAby cutting the xc hole around each electron out of aconstant density given by the value n of the density at thatelectron. Moreover, the hole contained precisely oneelectron and had a spherical shape and an extent givenby the rs-value %4#r3s n " 3& at the reference electron. Whenthe density varies rather rapidly from point to point itwould seem more appropriate to dig the hole out of someeffective density "nn obtained by averaging the correctdensity around the reference electron. Such a procedure

was suggested by Gunnarsson, Jonson and Lundqvist(GJL) [54] who also suggested that the extent of the hole begoverned by the same average density "nn: The resultingexpression for the xc energy then becomes

EADxc )n* "

!n%r&"xc% "nn%r&& d3r

where the average density "nn is obtained from

"nn%r& "!W%r# r0& n %r0& d3r0

with some appropriate weight function W%r& obeying thesum rule!W%r& d3r " 1

dictated by particle conservation. Here, as before, "xc%n& isthe xc energy per electron of the homogeneous electron gas.As in the case of the LDA, this means that the pair-correlation function is still modeled by that of the gas. Theweighting function W%r& can certainly be chosen in manydifferent ways but, in analogy with the construction ofgradient expansions, GJL suggested to choose W such thatthe new theory exactly reproduces the static linear densityresponse function of the homogeneous electron gas. Theyshowed that this requirement actually leads to a uniquedetermination of the weight function W provided someextra minor conditions of physical origin are imposed.Consequently, the weight function W becomes a uniqueproperty of the homogeneous gas. It is a function of thelength r of the radius vector and of the homogeneousdensity n which can be tabulated once and for all. In orderto obtain the simple formula for the xc energy shownabove, the density argument in the weight function W mustagain be chosen to be the resulting average density "nn%r& atthe position %r& of the reference electron.

Before discussing the numerical consequences of the newscheme, which by GJL was referred to as the AverageDensity (AD) approximation, we will also introduce theWeighted Density (WD) approximation by the sameauthors [54]. This approximation was independentlysuggested by Alonso and Girifalco [125] and GJL calledit the Weighted Density (WD) approximation. This schemeis rather similar in spirit to the AD approximation. The xchole is still modeled by means of the pair-correlationfunction of the homogeneous gas but the density pre-factoris the correct pre-factor n%r0& of the exact expression for thexc energy. Thus the xc hole is dug out of the true density ofthe inhomogeneous system and the requirement that thehole must contain precisely one electron is, for eachreference electron at r; guaranteed by choosing anappropriate density argument in the pair-correlationfunction of the gas. The expression for the xc energyEWDxc in the WD approximation becomes

EWDxc )n* " 1

2

!n%r&n%r 0&f ~ggh%r# r 0; "nn%r&& # 1gv%r# r 0& d3r d3r 0

where the density "nn; for each point r in space, has to bedetermined such that!n%r 0&f ~ggh%r# r 0; "nn%r&& # 1g d3r 0 " #1:

34 U. von Barth


From these expressions we immediately notice theirasymmetry with respect to their spatial arguments r andr0—in contrast to the symmetry displayed by the exactexpression for the xc energy. The asymmetry is also buildinto the AD approximation and we remind the reader (seeSection 2.3) that also the LDA suffers from the sameailment. It might not mean so much as far as the resultingvalue for the energy is concerned since the value of theintegral remains the same if we replace r by r0 in the densityargument of the pair-correlation functional. But it makes abig difference when we later want to construct the resultingxc potential by taking a functional derivative of the xcenergy with respect to the density.Both the AD and the WD schemes certainly have more

physically correct features added to them as compared tothe LDA and we would thus expect them to outperform thelatter. The WD approximation was first tested by Alonsoand Girifalco in a calculation of the exchange energies of anumber of atoms. Much to their liking they found a largereduction of the - 10% exchange-energy errors one is usedto from the LDA. Unfortunately, the WD approximationover corrected the LDA exchange energies. When GJLlater included also the correlation effects they found thatthe WD scheme destroys the nice cancellation occurringwithin the LDA between the errors coming from exchangeand correlation. As a result the atomic xc energies of theWD scheme are actually worse than those of the LDA.Also the AD scheme gives reasonable atomic exchangeenergies—better than those of the LDA but somewhatworse than those of the WD scheme. Unfortunately, theAD approximation gives a very bad description of theatomic correlation energies (too small in magnitude).Nevertheless, the total xc energies are, on the average, afactor of two better than those of the LDA. Consideringthe extra effort and superior physical reasoning involved inthe construction of the AD and WD schemes, the atomictest results are a clear disappointment. Using a techniquecalled shell partitioning, GJL managed to improveconsiderably on the atomic results of both the AD andthe WD approximations. But this theory is very inelegantand very much dependent on the system under study. Theconcept of shells is rather well established in atoms butbecome ill-defined in pseudo solids and we do not wish topursue this line of work any further.GJL also tested both the AD and the WD schemes in the

calculation of surface energies. Unfortunately, the resultswere very unphysical, a fact which they blamed on thedifference between the exact xc holes and of the approx-imate ones extending too far into the bulk regions. Boththe AD and the WD schemes give, however, a much betterdescription of the surface region as compared to the LDAwith its exponentially decaying xc potential. The xcpotentials of both schemes have an image-like behavioralthough the coefficients are wrong. Likewise, bothschemes give a one-over-distance dependence of the xcpotentials outside any finite system—in contrast to theexponentially decaying potential of the LDA. Like in thecase of the surface, the coefficients are, however, notcorrect. In my opinion, this failure is in both casesconnected to the asymmetry of the expressions for the xcenergies in the two schemes (see the discussion above).

In a later publication [126], Gunnarsson and Jonesabandoned the idea of trying to model the xc hole bymeans of the pair-correlation function of the homogeneousgas. Instead, they choose a very localized function of theform

~gg%r; r 0& " go%jr# r 0j; "nn%r&&

where

go%r; n& " 1# A%n&f1# exp)#l5%n&=r 5*g:

By requiring this ansatz to give the correct xc energy andnormalization of the xc hole for the homogeneous electrongas, the two parameters A and l become unique functionsof the density n: As in the WD scheme, the density "nn%r& isthen determined for each position r of the referenceelectron, by requiring the xc hole of the inhomogeneoussystem to contain precisely one electron. The power five onthe distance dependence in the exponent is actually chosensuch that the resulting approximation gives an image-likebehavior with the correct coefficient outside a metallicsurface. We notice again that this scheme lacks thesymmetry discussed above and is thus likely to show thesame inconsistency between energy density and potentialthat we noted in all the previous cases—be it the LDA orthe AD or WD approximations.

The new scheme gives atomic xc energies which arealmost an order of magnitude more accurate than those ofthe LDA—with errors typically less than 1%. And with themore rapid decay of the xc hole and the correct imagebehavior outside metallic surfaces, the new scheme stands achance of producing reasonable surface energies. As far aswe know, however, this has never been tested. But it hasbeen tested on atomic ionization potentials where itactually does worse than the LDA. Notice that ionizationpotentials are differences between two ground-state ener-gies and therefore obtainable from two DF calculations.The fact that the new scheme does not do well for aproperty mainly associated with the more slowly varyingvalence electrons suggest that the model is much toosimplified and contains too little physics.

6.2. The screened exchange model

The relatively bad results, at least in comparison with thesuccesses of the GGA:s, obtained from the early attemptsto model exchange and correlation in real space probablycaused these methods to fall into disrepute. An additionaldiscouraging factor is that the real-space methods invari-ably seem to require more computational work in order toobtain the energy and, particularly, the potential. As wehave argued many times previously in these notes, the latterobjection is not a serious one since we can always rely onthe variational property of the energy functional tocalculate the total energy of any new functional byapplying it to the self-consistent density of a GGA oreven an LDA calculation.

We are, however, not prone to giving up easily and mypostdoc Robert van Leeuwen and I decided to haveanother go at it in 1996 [127]. In the spirit of John Perdew,we decided to try and build in as many known correctproperties as we could without making the theory



inapplicable to real systems of interest. Here is a list of theproperties which guided or most recent construction:

. Exchange energies are a factor of four to thirty largerthan correlation energies where the former factor refersto metallic sodium and the latter to many atoms.

. In many atoms, the term structure, i.e. the energies oflow lying excitations of definite symmetry, can beaccurately predicted on the basis of simple term-dependent Hartree–Fock (HF) theory provided theSlater Coulomb %Fk& and exchange %Gk& integrals arereduced by 25% in value [75]. This means that the effectsof different symmetries are reasonably well described byHF theory.

. In larger systems, the effect of correlations is, amongother things, to reduce the range of the xc hole. In anextended system the pair-correlation function of HFtheory decays as an oscillating function over distance tothe third power. When correlation effects are includedthe decay changes to one over distance to the fifth power.

. In Section 4.5 we discussed the shortcomings of the LDArelative to HF theory in describing the so called s-ptransfer energies. On this basis we argued that the nodalstructure of the one-electron orbitals should be includedin the construction of an accurate functional for the xcenergy.

. In the prototype molecule H2; HF theory gives anexchange hole the shape of which is independent of theposition of the reference electron and which puts half anelectron on each atom when they are pulled apart. Inreality, the xc hole is relatively independent of theposition of the reference electron if this is close to one ofthe protons. But at larger proton separation, the xc holehas almost all its weight on the atom where the referenceelectron sits. It seems impossible to model this behaviorusing any function of r# r0 reminiscent of the electrongas.

These considerations resulted in the following model forthe xc functional which we coined the Screened-ExchangeModel (SEM):

ESEMxc )n* " #1

4

!jn%r; r0&j2A%r&A%r0&B%r# r0& d3r d3r0

where the quantity n%r; r0& is the one-particle density matrixof Kohn–Sham orbitals,

n%r; r0& " 2Xocc

k

&k%r&&!k%r

0&:

The correlation factor A ,A ,B has two ingredients:

(i) The electron-gas-like component B serves the purposeof contracting the exact exchange hole in a mannersimilar to this effect in the electron gas with aneffective density given by the geometric mean of thedensities at r and r0 Thus, we choose

B%r; n& " 1$ a%n&r=rs $ b%n&%r=rs&2. /

e#c%n&r=rs

with an effective density given by (as usual 4#r3s n " 3)

n " "nn "+++++++++++++++++n%r&n%r0&

p:

Two of the density-dependent parameters a; b; and care chosen such that the model becomes exact for theelectron gas (xc energy and sum rule), and the thirdcan, e.g., be determined such that the theory gives theexact total energy for the He atom.

(ii) The factor A is based on a new idea and is what allowsthe xc hole to move to one side of a dimer when thereference electron goes there. It is determined by thesum rule prescribing one unit of charge in the xc hole:

A%r&!jn%r; r0&j2A%r0&B%jr# r0j; "nn%r; r0&& d3r0 " 2n%r&:

Consequently, the correlation factor A%r& is deter-mined by by an integral equation which has to besolved at each point in space. Fortunately, the factorA%r& varies slowly between one and two and fewiterations are usually needed for determining thisfactor.

In comparison to previous models this one has a number ofimprovements:

. It contains the exact exchange energy.

. It is fully symmetric in the arguments r and r0:

. It gives the correct energy densities and potentialsoutside finite systems and outside metallic surfaces.

. It gives a correct description of the dissociation of theHydrogen molecule.

. It is exact in the slowly varying limit.

Unfortunately, we have not had the time to pursue thisresearch much further but the model definitely appears tohold a lot of promise. In Fig. 1 and Fig. 2 we compare thecorrelation hole and the xc hole of the H2 molecule at itsequilibrium distance (-1.4 Bohr) to the correspondingquantities from the Screened Exchange Model. In bothcases we show a plane through the molecular axis and thereference electron is on that axis and close to the Hydrogenatom to the right in the figures (at -0.7 Bohr). As seen, the

Fig. 1. A comparison of the true correlation hole of the H2 molecule at theequilibrium distance to that (dotted) of the Screened Exchange model. Thereference electron is close to the hydrogen atom to the right (at -0.7Bohr).

36 U. von Barth


screened-exchange model gives a very realistic descriptionof the exact xc hole in the H2 molecule.One might also wonder if this model would stand a

chance at describing also the very interesting van-der-Waal’s (vdW) effects. After all, it seems to have severalcorrect long-range features build into it. Our gut feeling atthe moment is that the model is too simple minded fordealing with these extremely long-ranged effects. For sucheffects we refer to recent work by Langreth, Lundqvist andcollaborators (see e.g. Ref. [128] and references therein).Their theory is, however, especially designed for treatingthe vdW effects and, with regard to correlations, it does notcover the whole range of distances from the vdW regioninto the interior of atoms.

7. Conclusions

We will end these lecture notes by voicing our prejudicesconcerning the present remaining problems within staticDFT.

. The band-gap problem.The nature of correlations and also of the gradient partof the exchange energy is drastically altered by theintroduction of a band gap. This represents an extremelynon-local effect which we can never hope to captureusing, e.g., gradients. It must be explicitly fed into anyvery accurate functional for the xc energy [129]. Mostlikely also into the correlation factor of the Screened-Exchange Model of Section 6.2.

. The symmetry problem.In finite systems, a difference in the symmetry of twostates can have a strong effect on their energies whiletheir densities can be very similar. None of the existingGGA:s or Meta-GGA:s have this effect built into them.This points toward orbital-dependent functionals.

. The long-range problem.There are two kinds of long-range problems—the van-der-Waal’s problem and the extremely long-rangeproblem associated with, e.g., the build up of a staticpolarization in an extended system. As in the case ofband gaps, the latter problem might be solvable by

feeding the polarization directly into the xc functional.There is, however, an interesting new approach in whichthis problem can be dealt with through the use of localapproximations but within current density-functional(CDFT) theory rather than within standard DFT [130].

There is so far no unified approach to the standardcorrelation problem in solids and molecules and thetreatment of the van-der-Waal’s interactions. Such anapproach is more likely to be found among orbital-dependent functionals than among the GGA:s.

. The near degeneracy problem.This problem was discussed in Section 4.5 and iscertainly outside any approach based on gradients. Itcould, perhaps be dealt with through models of the xchole based on several determinants. But we do, at themoment, not know how to make this into a systemindependent approach. After all, a few determinantswithin one set of orbitals is an infinity of determinants inanother.

. The surface problem.We are here referring to the problem of finding the xcenergies coming from the surfaces of solids, from low-density regions in solids, or from the outskirts of atomsand molecules. This is the problem to which the GGA:sor the Meta-GGA:s are best suited and these areprobably already accurate enough provided we do notrequire more of them than just that.

. The core problem.This is a very difficult problem within DFT whichfortunately is rather irrelevant. We deal with thisproblem by either relying on a cancellation of errors orthrough the use of pseudo potentials. A word of caution,however. We are soon moving into the area of non-localxc functionals. These are non-local in the sense that, e.g.,the xc energy density depends not only on the local valueof the density and its different gradients but rather onsome integrated property of the density. It is then not atall clear that such a theory can be pseudized to a veryhigh degree of accuracy.

Thank you for your attention!

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1494 (1979).

Fig. 2. A comparison of the true exchange-correlation hole of the H2

molecule at equilibrium distance to that (dotted) of the Screened ExchangeModel. The reference electron is on the molecular axis close to thehydrogen atom to the right (at -0.7 Bohr).



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129. I tried to sell this idea to Andreas Savin during the QuantumChemistry Meeting in Montreal in 1985 but we never got theopportunity to collaborate on this exciting topic. Savin, JoeKrieger, and collaborators have done some work along theselines on atoms in: J. B. Krieger, J. Chen, G. J. Iafrate, and A.Savin, ‘‘Construction of an Accurate Self-Interaction CorrectedCorrelation Energy Functional Based on an Electron Gas with aBand Gap,’’ p. 463, in the Proceedings of Electron Correlation andMaterial Properties (Edited By A. Gonis and N. Kioussis),(Plenum, New York 1999).

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