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Density Functionals for NonRelativistic

Coulomb Systems

John P Perdew and Stefan Kurth

Department of Physics and Quantum Theory GroupTulane University New Orleans LA USA

Outline

Introduction Quantum Mechanical ManyElectron Problem Summary of KohnSham SpinDensity Functional Theory

Wavefunction Theory Wavefunctions and Their Interpretation Wavefunctions for NonInteracting Electrons Wavefunction Variational Principle HellmannFeynman Theorem Virial Theorem

Denitions of Density Functionals Introduction to Density Functionals Density Variational Principle KohnSham NonInteracting System Exchange Energy and Correlation Energy CouplingConstant Integration

Formal Properties of Functionals Uniform Coordinate Scaling Local Lower Bounds Spin Scaling Relations Size Consistency Derivative Discontinuity

Uniform Electron Gas Kinetic Energy Exchange Energy Correlation Energy Linear Response Clumping and Adiabatic Connection

Local and SemiLocal Approximations

Local Spin Density Approximation

Gradient Expansion

History of Several Generalized Gradient Approximations

Construction of a GGA Made Simple

GGA Nonlocality Its Character Origins and EectsReferences


Introduction

Quantum Mechanical ManyElectron Problem

The material world of everyday experience as studied by chemistry and condensedmatter physics is built up from electrons and a few or at most a fewhundred kinds of nuclei The basic interaction is electrostatic or CoulombicAn electron at position r is attracted to a nucleus of charge Z at R by thepotential energy Zjr Rj a pair of electrons at r and r repel one anotherby the potential energy jr rj and two nuclei at R and R repel one anotheras Z ZjRRj The electrons must be described by quantum mechanics whilethe more massive nuclei can sometimes be regarded as classical particles Allof the electrons in the lighter elements and the chemically important valenceelectrons in most elements move at speeds much less than the speed of lightand so are nonrelativistic

In essence that is the simple story of practically everything But there isstill a long path from these general principles to theoretical prediction of thestructures and properties of atoms molecules and solids and eventually tothe design of new chemicals or materials If we restrict our focus to the important class of groundstate properties we can take a shortcut through densityfunctional theory

These lectures present an introduction to density functionals for nonrelativistic Coulomb systems The reader is assumed to have a working knowledgeof quantum mechanics at the level of oneparticle wavefunctions r Themanyelectron wavefunction r r rN is briey introduced here andthen replaced as basic variable by the electron density nr Various terms ofthe total energy are dened as functionals of the electron density and some formal properties of these functionals are discussed The most widelyused densityfunctionals the local spin density and generalized gradient approximations are then introduced and discussed At the end the reader should be preparedto approach the broad literature of quantum chemistry and condensedmatterphysics in which these density functionals are applied to predict diverse properties the shapes and sizes of molecules the crystal structures of solids bindingor atomization energies ionization energies and electron anities the heights ofenergy barriers to various processes static response functions vibrational frequencies of nuclei etc Moreover the readers approach will be an informed anddiscerning one based upon an understanding of where these functionals comefrom why they work and how they work

These lectures are intended to teach at the introductory level and not toserve as a comprehensive treatise The reader who wants more can go to severalexcellent general sources or to the original literature Atomic units inwhich all electromagnetic equations are written in cgs form and the fundamentalconstants h e and m are set to unity have been used throughout

Density Functionals for NonRelativistic Coulomb Systems

Summary of KohnSham SpinDensity Functional Theory

This introduction closes with a brief presentation of the KohnSham spindensity functional method the most widelyused method of electronicstructurecalculation in condensedmatter physics and one of the most widelyused methods in quantum chemistry We seek the groundstate total energy E and spindensities nr nr for a collection of N electrons interacting with one another and with an external potential vr due to the nuclei in most practicalcases These are found by the selfconsistent solution of an auxiliary ctitiousoneelectron Schrodinger equation

r vr un r vxcn n r

r r

nr X

jrj

Here or is the zcomponent of spin and stands for the set of remaining oneelectron quantum numbers The eective potential includes a classicalHartree potential

un r

Zdr

nr

jr rj

nr nr nr

and a multiplicative spindependent exchangecorrelation potentialvxcn n r which is a functional of the spin densities The step function in Eq ensures that all KohnSham spin orbitals with aresingly occupied and those with are empty The chemical potential ischosen to satisfy Z

drnr N

Because Eqs and are interlinked they can only be solved by iteration toselfconsistency

The total energy is

E Tsn n

Zdrnrvr U n Excn n

where

Tsn n X

X

h j rji

is the noninteracting kinetic energy a functional of the spin densities becauseas we shall see the external potential vr and hence the KohnSham orbitalsare functionals of the spin densities In our notation

h j Oji Zdrr

Or


The second term of Eq is the interaction of the electrons with the externalpotential The third term of Eq is the Hartree electrostatic selfrepulsion ofthe electron density

U n

Zdr

Zdr

nrnr

jr rj

The last term of Eq is the exchangecorrelation energy whose functionalderivative as explained later yields the exchangecorrelation potential

vxcn n r Excnr

Not displayed in Eq but needed for a system of electrons and nuclei is theelectrostatic repulsion among the nuclei Exc is dened to include everything elseomitted from the rst three terms of Eq

If the exact dependence of Exc upon n and n were known these equations would predict the exact groundstate energy and spindensities of a manyelectron system The forces on the nuclei and their equilibrium positions couldthen be found from ER

In practice the exchangecorrelation energy functional must be approximated The local spin density LSD approximation has long been popularin solid state physics

ELSDxc n n

Zdrnrexcnr nr

where excn n is the known exchangecorrelation energy per particlefor an electron gas of uniform spin densities n n More recently generalizedgradient approximations GGAs havebecome popular in quantum chemistry

EGGAxc n n

Zdrfn nrnrn

The input excn n to LSD is in principle unique since there is a possiblesystem in which n and n are constant and for which LSD is exact At least inthis sense there is no unique input fn nrnrn to GGA These lectureswill stress a conservative philosophy of approximation in which weconstruct a nearlyunique GGA with all the known correct formal features ofLSD plus others

The equations presented here are really all that we need to do a practicalcalculation for a manyelectron system They allow us to draw upon the intuitionand experience we have developed for oneparticle systems The manybody effects are in U n trivially and Excn n less trivially but we shall alsodevelop an intuitive appreciation for Exc

While Exc is often a relatively small fraction of the total energy of an atommolecule or solid minus the work needed to break up the system into separated


electrons and nuclei the contribution from Exc is typically about ! ormore of the chemical bonding or atomization energy the work needed to breakup the system into separated neutral atoms Exc is a kind of glue withoutwhich atoms would bond weakly if at all Thus accurate approximations toExc are essential to the whole enterprise of density functional theory Table shows the typical relative errors we nd from selfconsistent calculations withinthe LSD or GGA approximations of Eqs and Table shows the meanabsolute errors in the atomization energies of molecules when calculated byLSD by GGA and in the HartreeFock approximation HartreeFock treatsexchange exactly but neglects correlation completely While the HartreeFocktotal energy is an upper bound to the true groundstate total energy the LSDand GGA energies are not

Table Typical errors for atoms molecules and solids from selfconsistent KohnShamcalculations within the LSD and GGA approximations of Eqs and Note thatthere is typically some cancellation of errors between the exchange Ex and correlation

Ec contributions to Exc The energy barrier is the barrier to a chemical reactionthat arises at a highlybonded intermediate state

Property LSD GGA

Ex not negative enough Ec too negative bond length too short too longstructure overly favors close packing more correctenergy barrier too low too low

Table Mean absolute error of the atomization energies for molecules evaluatedby various approximations hartree eV From Ref

Approximation Mean absolute error eV

Unrestricted HartreeFock underbindingLSD overbindingGGA mostly overbindingDesired chemical accuracy

In most cases we are only interested in small totalenergy changes associatedwith rearrangements of the outer or valence electrons to which the inner orcore electrons of the atoms do not contribute In these cases we can replace eachcore by the pseudopotential it presents to the valence electrons and thenexpand the valenceelectron orbitals in an economical and convenient basis ofplane waves Pseudopotentials are routinely combined with density functionals


Although the most realistic pseudopotentials are nonlocal operators and notsimply local or multiplication operators and although density functional theoryin principle requires a local external potential this inconsistency does not seemto cause any practical diculties

There are empirical versions of LSD and GGA but these lectures will onlydiscuss nonempirical versions If every electronicstructure calculation were doneat least twice once with nonempirical LSD and once with nonempirical GGAthe results would be useful not only to those interested in the systems underconsideration but also to those interested in the development and understandingof density functionals

Wavefunction Theory

Wavefunctions and Their Interpretation

We begin with a brief review of oneparticle quantum mechanics An electronhas spin s

and zcomponent of spin

or

The Hamiltonian

or energy operator for one electron in the presence of an external potential vris

h r vr

The energy eigenstates r and eigenvalues are solutions of the timeindependent Schrodinger equation

hr r

and jr jdr is the probability to nd the electron with spin in volumeelement dr at r given that it is in energy eigenstate ThusX

Zdrjr j hji

Since h commutes with sz we can choose the to be eigenstates of sz ie wecan choose or as a oneelectron quantum number

The Hamiltonian for N electrons in the presence of an external potential vris

H

NXi

ri

NXi

vri

Xi

Xj i

jri rj j T Vext Vee

The electronelectron repulsion Vee sums over distinct pairs of dierent electronsThe states of welldened energy are the eigenstates of H

Hkr rNN Ekkr rNN


where k is a complete set of manyelectron quantum numbers we shall be interested mainly in the ground state or state of lowest energy the zerotemperatureequilibrium state for the electrons

Because electrons are fermions the only physical solutions of Eq arethose wavefunctions that are antisymmetric under exchange of two electronlabels i and j

r rii rjj rNN

r rjj rii rNN

There areN " distinct permutations of the labels N which by Eq allhave the same j j Thus N "jr rNN jdr drN is the probabilityto nd any electron with spin in volume element d

r etc and

N "

XN

Zdr

ZdrNN "jr rNN j

Zj j h ji

We dene the electron spin density nr so that nrdr is the probability

to nd an electron with spin in volume element dr at r We nd nr byintegrating over the coordinates and spins of the N other electrons ie

nr

N "X

N

Zdr

ZdrNN "jr r rNN j

NX

N

Zdr

ZdrN jr r rNN j

Equations and yield

X

Zdrnr N

Based on the probability interpretation of nr we might have expected therighthand side of Eq to be but that is wrong the sum of probabilitiesof all mutuallyexclusive events equals but nding an electron at r does notexclude the possibility of nding one at r except in a oneelectron systemEq shows that nrd

r is the average number of electrons of spin involume element dr Moreover the expectation value of the external potentialis

h Vexti h jNXi

vriji Zdrnrvr

with the electron density nr given by Eq


Wavefunctions for NonInteracting Electrons

As an important special case consider the Hamiltonian for N noninteractingelectrons

Hnon

NXi

ri vri

The eigenfunctions of the oneelectron problem of Eqs and are spinorbitals which can be used to construct the antisymmetric eigenfunctions ofHnon

Hnon Enon

Let i stand for ri i and construct the Slater determinant or antisymmetrizedproduct

pN "

XP

PPP N PN

where the quantum label i now includes the spin quantum number Here Pis any permutation of the labels N and P equals for an evenpermutation and for an odd permutation The total energy is

Enon N

and the density is given by the sum of jirj If any i equals any j inEq we nd which is not a normalizable wavefunction This is thePauli exclusion principle two or more noninteracting electrons may not occupythe same spin orbital

As an example consider the ground state for the noninteracting helium atomN The occupied spin orbitals are

r sr

r sr

and the electron Slater determinant is

p

r r r r

srsrp

which is symmetric in space but antisymmetric in spin whence the total spin isS

If several dierent Slater determinants yield the same noninteracting energyEnon then a linear combination of them will be another antisymmetric eigenstateof Hnon More generally the Slaterdeterminant eigenstates of Hnon dene acomplete orthonormal basis for expansion of the antisymmetric eigenstates ofH the interacting Hamiltonian of Eq


Wavefunction Variational Principle

The Schrodinger equation is equivalent to a wavefunction variational principle Extremize h j H ji subject to the constraint h ji ie set thefollowing rst variation to zero

nh j H jih ji

o

The ground state energy and wavefunction are found by minimizing the expression in curly brackets

The RayleighRitz method nds the extrema or the minimum in a restrictedspace of wavefunctions For example the HartreeFock approximation to thegroundstate wavefunction is the single Slater determinant that minimizeshj H jihji The congurationinteraction groundstate wavefunction isan energyminimizing linear combination of Slater determinants restricted tocertain kinds of excitations out of a reference determinant The Quantum MonteCarlo method typically employs a trial wavefunction which is a single Slater determinant times a Jastrow paircorrelation factor Those widelyused manyelectron wavefunction methods are both approximate and computationally demanding especially for large systems where density functional methods are distinctly more ecient

The unrestricted solution of Eq is equivalent by the method of Lagrangemultipliers to the unconstrained solution of

nh j H ji Eh ji

o

ie

h j H Eji Since is an arbitrary variation we recover the Schrodinger equation Every eigenstate of H is an extremum of h j H jih ji and vice versa

The wavefunction variational principle implies the HellmannFeynman andvirial theorems below and also implies the HohenbergKohn density functional variational principle to be presented later

HellmannFeynman Theorem

Often the Hamiltonian H depends upon a parameter and we want to knowhow the energy E depends upon this parameter For any normalized variationalsolution including in particular any eigenstate of H we dene

E hj Hji

ThendEd

d

dh j Hji

hjH

ji


The rst term of Eq vanishes by the variational principle and we nd theHellmannFeynman theorem

dEd

hjH

ji

Eq will be useful later for our understanding of Exc For now we shalluse Eq to derive the electrostatic force theorem Let ri be the positionof the ith electron and RI the position of the static nucleus I with atomicnumber ZI The Hamiltonian

H NXi

ri

Xi

XI

ZIjri RI j

Xi

Xj i

jri rj j

XI

XJ I

ZIZJjRI RJ j

depends parametrically upon the position RI so the force on nucleus I is

ERI

H

RI

Zdrnr

ZIrRIjrRI j

XJ I

ZIZJRI RJjRI RJ j

just as classical electrostatics would predict Eq can be used to nd theequilibrium geometries of a molecule or solid by varying all the RI until theenergy is a minimum and ERI Eq also forms the basis fora possible density functional molecular dynamics in which the nuclei moveunder these forces by Newtons second law In principle all we need for eitherapplication is an accurate electron density for each set of nuclear positions

Virial Theorem

The density scaling relations to be presented in section which constitute important constraints on the density functionals are rooted in the same wavefunctionscaling that will be used here to derive the virial theorem

Let r rN be any extremum of h j H ji over normalized wavefunctions ie any eigenstate or optimized restricted trial wavefunction where irrelevant spin variables have been suppressed For any scale parameter dene the uniformlyscaled wavefunction

r rN Nr rN

and observe thath ji h ji

The density corresponding to the scaled wavefunction is the scaled density

nr nr


which clearly conserves the electron number

Zdrnr

Zdrnr N

leads to densities nr that are higher on average and more contractedthan nr while produces densities that are lower and more expanded

Now consider what happens to h Hi h T V i under scaling By denitionof

d

dh j T V ji

But T is homogeneous of degree in r so

h j T ji h j T ji

and Eq becomes

h j T ji ddh j V ji

or

h T i hNXi

ri V

rii

If the potential energy V is homogeneous of degree n ie if

V ri rN nV ri rN

then

h j V ji nh j V ji

and Eq becomes simply

h j T ji nh j V ji

For example n for the Hamiltonian of Eq in the presence of a singlenucleus or more generally when the HellmannFeynman forces of Eq vanishfor the state


Denitions of Density Functionals

Introduction to Density Functionals

The manyelectron wavefunction r rNN contains a great deal ofinformation all we could ever have but more than we usually want Because itis a function of many variables it is not easy to calculate store apply or eventhink about Often we want no more than the total energy E and its changesor perhaps also the spin densities nr and nr for the ground state As weshall see we can formally replace by the observables n and n as the basicvariational objects

While a function is a rule which assigns a number fx to a number x afunctional is a rule which assigns a number F f to a function f For exampleh h j H ji is a functional of the trial wavefunction given the HamiltonianH U n of Eq is a functional of the density nr as is the local densityapproximation for the exchange energy

ELDAx n Ax

Zdrnr

The functional derivative Fnr tells us how the functional F n changesunder a small variation nr

F

Zdr

F

nr

nr

For example

ELDAx Ax

Zdr

nnr nr nr

o Ax

Zdr

nrnr

so

ELDAxnr

Ax

nr

Similarly

U n

nr un r

where the right hand side is given by Eq Functional derivatives of variousorders can be linked through the translational and rotational symmetries ofempty space


Density Variational Principle

We seek a density functional analog of Eq Instead of the original derivationof Hohenberg Kohn and Sham which was based upon reductio adabsurdum we follow the constrained search approach of Levy which isin some respects simpler and more constructive

Eq tells us that the ground state energy can be found by minimizingh j H ji over all normalized antisymmetric N particle wavefunctions

E min

h j H ji

We now separate the minimization of Eq into two steps First we considerall wavefunctions which yield a given density nr and minimize over thosewavefunctions

min

n

h j H ji min

n

h j T VeejiZdrvrnr

where we have exploited the fact that all wavefunctions that yield the same nralso yield the same h j Vextji Then we dene the universal functional

F n min

n

h j T Veeji hminn j T Veejminn i

where minn is that wavefunction which delivers the minimum for a given nFinally we minimize over all N electron densities nr

E minn

Ev n

minn

F n

Zdrvrnr

where of course vr is held xed during the minimization The minimizingdensity is then the groundstate density

The constraint of xed N can be handled formally through introduction ofa Lagrange multiplier

F n

Zdrvrnr

Zdrnr

which is equivalent to the Euler equation

F

nr vr

is to be adjusted until Eq is satised Eq shows that the externalpotential vr is uniquely determined by the ground state density or by any oneof them if the ground state is degenerate

The functional F n is dened via Eq for all densities nr which areN representable ie come from an antisymmetric N electron wavefunction


We shall discuss the extension from wavefunctions to ensembles in section The functional derivative Fnr is dened via Eq for all densities whichare vrepresentable ie come from antisymmetric N electron groundstatewavefunctions for some choice of external potential vr

This formal development requires only the total density of Eq and notthe separate spin densities nr and nr However it is clear how to get to aspindensity functional theory just replace the constraint of xed n in Eq and subsequent equations by that of xed n and n There are two practicalreasons to do so This extension is required when the external potential isspindependent ie vr vr as when an external magnetic eld couplesto the zcomponent of electron spin If this eld also couples to the currentdensity jr then we must resort to a currentdensity functional theory Even when vr is spinindependent we may be interested in the physical spinmagnetization eg in magnetic materials Even when neither nor applies our local and semilocal approximations Eqs and typicallywork better when we use n and n instead of n

KohnSham NonInteracting System

For a system of noninteracting electrons Vee of Eq vanishes so F n ofEq reduces to

Tsn min

n

h j T ji hminn j T jminn i

Although we can search over all antisymmetric N electron wavefunctions inEq the minimizing wavefunction minn for a given density will be a noninteracting wavefunction a single Slater determinant or a linear combination ofa few for some external potential Vs such that

Tsnr

vsr

as in Eq In Eq the KohnSham potential vsr is a functional ofnr If there were any dierence between and s the chemical potentials forinteracting and noninteracting systems of the same density it could be absorbedinto vsr We have assumed that nr is both interacting and noninteractingvrepresentable

Now we dene the exchangecorrelation energy Excn by

F n Tsn U n Excn

where U n is given by Eq The Euler equations and are consistentwith one another if and only if

vsr vr U n

nr

Excnr

Thus we have derived the KohnSham method of section


The KohnSham method treats Tsn exactly leaving only Excn to be approximated This makes good sense for several reasons Ts is typically avery large part of the energy while Exc is a smaller part Ts is largely responsible for density oscillations of the shell structure and Friedel types whichare accurately described by the KohnSham method E is somewhat bettersuited to the local and semilocal approximations than is Tsn for reasons tobe discussed later The price to be paid for these benets is the appearance oforbitals If we had a very accurate approximation for Ts directly in terms of nwe could dispense with the orbitals and solve the Euler equation directlyfor nr

The total energy of Eq may also be written as

E X

U nZdrnrvxcn r Excn

where the second and third terms on the righthandside simply remove contributions to the rst term which do not belong in the total energy The rst termon the right of Eq the noninteracting energy Enon is the only term thatappears in the semiempirical Huckel theory This rst term includes mostof the electronic shell structure eects which arise when Tsn is treated exactlybut not when Tsn is treated in a continuum model like the ThomasFermiapproximation or the gradient expansion

Exchange Energy and Correlation Energy

Excn is the sum of distinct exchange and correlation terms

Excn Exn Ecn

where Exn hminn j Veejminn i U n

When minn is a single Slater determinant Eq is just the usual Fock integralapplied to the KohnSham orbitals ie it diers from the HartreeFock exchangeenergy only to the extent that the KohnSham orbitals dier from the HartreeFock orbitals for a given system or density in the same way that Tsn diersfrom the HartreeFock kinetic energy We note that

hminn j T Veejminn i Tsn U n Exn

and that in the oneelectron Vee limit

Exn U n N The correlation energy is

Ecn F n fTsn U n Exng hminn j T Veejminn i hminn j T Veejminn i


Since minn is that wavefunction which yields density n and minimizes h T VeeiEq shows that

Ecn Since minn is that wavefunction which yields density n and minimizes h T iEq shows that Ecn is the sum of a positive kinetic energy piece and anegative potential energy piece These pieces of Ec contribute respectively tothe rst and second terms of the virial theorem Eq Clearly for any oneelectron system

Ecn N

Eqs and show that the exchangecorrelation energy of a oneelectron system simply cancels the spurious selfinteraction U n In the same waythe exchangecorrelation potential cancels the spurious selfinteraction in theKohnSham potential

Exnr

un r N

Ecnr

N

Thus

limr

Excnr

r

N

The extension of these oneelectron results to spindensity functional theory isstraightforward since a oneelectron system is fully spinpolarized

CouplingConstant Integration

The denitions and are formal ones and do not provide much intuitiveor physical insight into the exchange and correlation energies or much guidancefor the approximation of their density functionals These insights are providedby the couplingconstant integration to be derived below

Let us dene minn as that normalized antisymmetric wavefunction whichyields density nr and minimizes the expectation value of T Vee where wehave introduced a nonnegative coupling constant When minn is

minn

the interacting groundstate wavefunction for density n When minn isminn the noninteracting or KohnSham wavefunction for density n Varying at xed nr amounts to varying the external potential vr At vris the true external potential while at it is the KohnSham eectivepotential vsr We normally assume a smooth adiabatic connection betweenthe interacting and noninteracting ground states as is reduced from to

Now we write Eqs and as

Excn

hminn j T Veejminn i

hminn j T Veejminn i

U n

Z

dd

dhminn j T Veejminn i U n


The HellmannFeynman theorem of section allows us to simplify Eq to

Excn

Z

dhminn j Veejminn i U n

Eq looks like a potential energy the kinetic energy contribution to Exchas been subsumed by the couplingconstant integration We should rememberof course that only is real or physical The KohnSham system at and all the intermediate values of are convenient mathematical ctions

To make further progress we need to know how to evaluate the N electronexpectation value of a sum of onebody operators like T or a sum of twobodyoperators like Vee For this purpose we introduce oneelectron and twoelectron reduced density matrices

r r N

XNZ

dr

ZdrN

r r rNN r r rNN

r r NN

XN

Zdr

ZdrN jr r rNN j

From Eq nr r r

Clearly also

h T i X

Zdr

r r

r r

rr

h Veei

Zdr

Zdr

r r

jr rj

We interpret the positive number r rdrdr as the joint probability of

nding an electron in volume element dr at r and an electron in dr at r Bystandard probability theory this is the product of the probability of nding anelectron in dr nrdr and the conditional probability of nding an electronin dr given that there is one at r nr r

dr

r r nrnr r

By arguments similar to those used in section we interpret nr r as the

average density of electrons at r given that there is an electron at r Clearlythen Z

drnr r N

For the wavefunction minn we write

nr r nr nxcr r


an equation which denes nxcr r the density at r of the exchangecorrelation

hole about an electron at r Eqs and imply thatZdrnxcr r

which says that if an electron is denitely at r it is missing from the rest of thesystem

Because the Coulomb interaction u is singular as u jr rj theexchangecorrelation hole density has a cusp around u

u

Zdu

nxcr r u

u

nr nxcr r

whereRdu is an angular average This cusp vanishes when and

also in the fullyspinpolarized and lowdensity limits in which all other electronsare excluded from the position of a given electron nxcr r nr

We can now rewrite Eq as

Excn

Zdr

Zdr

nrnxcr r

jr rj

where

nxcr r

Z

dnxcr r

is the couplingconstant averaged hole density The exchangecorrelation energyis just the electrostatic interaction between each electron and the couplingconstantaveraged exchangecorrelation hole which surrounds it The hole iscreated by three eects selfinteraction correction a classical eect whichguarantees that an electron cannot interact with itself the Pauli exclusionprinciple which tends to keep two electrons with parallel spins apart in spaceand the Coulomb repulsion which tends to keep any two electrons apartin space Eects and are responsible for the exchange energy which ispresent even at while eect is responsible for the correlation energyand arises only for

If minn is a single Slater determinant as it typically is then the one andtwoelectron density matrices at can be constructed explicitly from theKohnSham spin orbitals r

r r

X

rr

r r nrnr nrnxr r

where

nxr r nxc r r

X

j r rjnr


is the exact exchangehole density Eq shows that

nxr r

so the exact exchange energy

Exn

Zdr

Zdr

nrnxr r

jr rj

is also negative and can be written as the sum of upspin and downspin contributions

Ex Ex E

x

Eq provides a sum rule for the exchange holeZdrnxr r

Eqs and show that the ontop exchange hole density is

nxr r nr n

r

nr

which is determined by just the local spin densities at position r suggestinga reason why local spin density approximations work better than local densityapproximations

The correlation hole density is dened by

nxcr r nxr r

ncr r

and satises the sum rule Zdrncr r

which says that Coulomb repulsion changes the shape of the hole but not itsintegral In fact this repulsion typically makes the hole deeper but more shortranged with a negative ontop correlation hole density

ncr r

The positivity of Eq is equivalent via Eqs and to the inequality

nxcr r nr

which asserts that the hole cannot take away electrons that werent there initiallyBy the sum rule the correlation hole density ncr r

must have positiveas well as negative contributions Moreover unlike the exchange hole densitynxr r

the exchangecorrelation hole density nxcr r can be positive


To better understand Exc we can simplify Eq to the realspace analysis

Excn N

Z

du uhnxcui

u

where

hnxcui N

Zdrnr

Zdu

nxcr r u

is the system and sphericalaverage of the couplingconstantaveraged hole density The sum rule of Eq becomes

Z

du uhnxcui

As u increases from hnxui rises analytically like hnx i Ou whilehncui rises like hnc i Ojuj as a consequence of the cusp of Eq Because of the constraint of Eq and because of the factor u in Eq Exc typically becomes more negative as the ontop hole density hnxcui getsmore negative

Formal Properties of Functionals

Uniform Coordinate Scaling

The more we know of the exact properties of the density functionals Excn andTsn the better we shall understand and be able to approximate these functionals We start with the behavior of the functionals under a uniform coordinatescaling of the density Eq

The Hartree electrostatic selfrepulsion of the electrons is known exactlyEq and has a simple coordinate scaling

U n

Zdr

Zdr

nrnr

jr rj

Zdr

Zdr

nrnr

jr rj U n

where r r and r r

Next consider the noninteracting kinetic energy of Eq Scaling all the

wavefunctions in the constrained search as in Eq will scale the density asin Eq and scale each kinetic energy expectation value as in Eq Thusthe constrained search for the unscaled density maps into the constrained searchfor the scaled density and

Tsn Tsn


We turn now to the exchange energy of Eq By the argument of the lastparagraph minn is the scaled version of

minn Since also

Veer rN Veer rN

and with the help of Eq we nd

Exn Exn

In the highdensity limit Tsn dominates U n and Exn Anexample would be an ion with a xed number of electrons N and a nuclearcharge Z which tends to innity in this limit the density and energy becomeessentially hydrogenic and the eects of U and Ex become relatively negligibleIn the lowdensity limit U n and Exn dominate Tsn

We can use coordinate scaling relations to x the form of a local densityapproximation

F n

Zdrfnr

If F n pF n then

Zdrf

nr

p

Zdrfnr

or fn pfn whence

fn np

For the exchange energy of Eq p so Eqs and implyEq For the noninteracting kinetic energy of Eq p so Eqs and imply the ThomasFermi approximation

Tn As

Zdrnr

U n of Eq is too strongly nonlocal for any local approximationWhile Tsn U n and Exn have simple scalings Ecn of Eq does

not This is because minn the wavefunction which via Eq yields the scaled

density nr and minimizes the expectation value of T Vee is not the scaledwavefunction Nminn r rN The scaled wavefunction yields nr

but minimizes the expectation value of T Vee and it is this latter expectationvalue which scales like under wavefunction scaling Thus

Ecn Ec n

where Ec n is the density functional for the correlation energy in a system for

which the electronelectron interaction is not Vee but Vee


To understand these results let us assume that the KohnSham noninteracting Hamiltonian has a nondegenerate ground state In the highdensity limit minn minimizes just h T i and reduces to minn Now we treat

Vee NXi

U

nri

Exnri

as a weak perturbation on the KohnSham noninteracting Hamiltonianand nd

Ec Xn

jhnjj ijE En

where the jni are the eigenfunctions of the KohnSham noninteracting Hamiltonian and j i is its ground state Both the numerator and the denominator ofEq scale like so

lim

Ecn constant

In the lowdensity limit minn minimizes just h Veei and Eq then showsthat

Ecn Dn Generally we have a scaling inequality

Ecn Ecn

Ecn Ecn

If we choose a density n we can plot Ecn versus and compare the result tothe straight line Ecn These two curves will drop away from zero as increasesfrom zero with dierent initial slopes then cross at The convex Ecn will then approach a negative constant as

Local Lower Bounds

Because of the importance of local and semilocal approximations like Eqs and bounds on the exact functionals are especially useful when the boundsare themselves local functionals

Lieb and Thirring have conjectured that Tsn is bounded from below bythe ThomasFermi functional

Tsn Tn where Tn is given by Eqs with

As


We have already established that

Exn Excn Exc n

where the nal term of Eq is the integrand Excn of the couplingconstantintegration of Eq

Excn hminn j Veejminn i U n

evaluated at the upper limit Lieb and Oxford have proved that

Exc n ELDAx n

where ELDAx n is the local density approximation for the exchange energyEq with

Ax

Spin Scaling Relations

Spin scaling relations can be used to convert density functionals into spindensityfunctionals

For example the noninteracting kinetic energy is the sum of the separatekinetic energies of the spinup and spindown electrons

Tsn n Tsn Ts n

The corresponding density functional appropriate to a spinunpolarized systemis

Tsn Tsn n Tsn

whence Tsn

Tsn and Eq becomes

Tsn n

Tsn

Tsn

Similarly Eq implies

Exn n

Exn

Exn

For example we can start with the local density approximations and then apply and to generate the corresponding local spin densityapproximations

Because two electrons of antiparallel spin repel one another Coulombicallymaking an important contribution to the correlation energy there is no simplespin scaling relation for Ec


Size Consistency

Common sense tells us that the total energy E and density nr for a system comprised of two wellseparated subsystems with energies E and E anddensities nr and nr must be E E E and nr nr nrApproximations which satisfy this expectation such as the LSD of Eq orthe GGA of Eq are properly size consistent Size consistency is notonly a principle of physics it is almost a principle of epistemology How couldwe analyze or understand complex systems if they could not be separated intosimpler components#

Density functionals which are not size consistent are to be avoided An example is the FermiAmaldi approximation for the exchange energy

EFAx n U nN where N is given by Eq which was constructed to satisfy Eq

Derivative Discontinuity

In section our density functionals were dened as constrained searches overwavefunctions Because all wavefunctions searched have the same electron number there is no way to make a numbernonconserving density variation nrThe functional derivatives are dened only up to an arbitrary constant whichhas no eect on Eq when

Rdrnr

To complete the denition of the functional derivatives and of the chemicalpotential we extend the constrained search from wavefunctions to ensembles An ensemble or mixed state is a set of wavefunctions or pure states andtheir respective probabilities By including wavefunctions with dierent electronnumbers in the same ensemble we can develop a density functional theory fornoninteger particle number Fractional particle numbers can arise in an opensystem that shares electrons with its environment and in which the electronnumber uctuates between integers

The upshot is that the groundstate energy EN varies linearly betweentwo adjacent integers and has a derivative discontinuity at each integer Thisdiscontinuity arises in part from the exchangecorrelation energy and entirelyso in cases for which the integer does not fall on the boundary of an electronicshell or subshell eg for N in the carbon atom but not for N in theneon atom

By Janaks theorem the highest partlyoccupied KohnSham eigenvalueHO equals EN and so changes discontinuously at an integerZ

HO

IZ Z N ZAZ Z N Z

where IZ is the rst ionization energy of the Zelectron system ie the leastenergy needed to remove an electron from this system and AZ is the electronanity of the Zelectron system ie AZ IZ If Z does not fall on the


boundary of an electronic shell or subshell all of the dierence between IZ andAZ must arise from a discontinous jump in the exchangecorrelation potentialExcnr as the electron number N crosses the integer Z

Since the asymptotic decay of the density of a nite system with Z electronsis controlled by IZ we can show that the exchangecorrelation potential tendsto zero as jrj

limjrj

Excnr

Z N Z

or more precisely

limjrj

Excnr

r

Z N Z

AsN increases through the integerZ Excnr jumps up by a positive additiveconstant With further increases in N above Z this constant vanishes rst atvery large jrj and then at smaller and smaller jrj until it is all gone in the limitwhere N approaches the integer Z from below

Simple continuum approximations to Excn n such as the LSD of Eq or the GGA of Eq miss much or all the derivative discontinuity and canat best average over it For example the highest occupied orbital energy fora neutral atom becomes approximately

IZ AZ the average of Eq

from the electrondecient and electronrich sides of neutrality We must neverforget when we make these approximations that we are tting a round peg intoa square hole The areas integrated properties of a circle and a square can bematched but their perimeters dierential properties will remain stubbornlydierent

Uniform Electron Gas

Kinetic Energy

Simple systems play an important paradigmatic role in science For examplethe hydrogen atom is a paradigm for all of atomic physics In the same waythe uniform electron gas is a paradigm for solidstate physics and also fordensity functional theory In this system the electron density nr is uniformor constant over space and thus the electron number is innite The negativecharge of the electrons is neutralized by a rigid uniform positive background Wecould imagine creating such a system by starting with a simple metal regardedas a perfect crystal of valence electrons and ions and then smearing out the ionsto make the uniform background of positive charge In fact the simple metalsodium is physically very much like a uniform electron gas

We begin by evaluating the noninteracting kinetic energy this section andexchange energy next section per electron for a spinunpolarized electron gasof uniform density n The corresponding energies for the spinpolarized case canthen be found from Eqs and


By symmetry the KohnSham potential vsr must be uniform or constantand we take it to be zero We impose boundary conditions within a cube ofvolume V ie we require that the orbitals repeat from one face of thecube to its opposite face Presumably any choice of boundary conditions wouldgive the same answer as V The KohnSham orbitals are then plane wavesexpik r

pV with momenta or wavevectors k and energies k The number

of orbitals of both spins in a volume dk of wavevector space is Vdkby an elementary geometrical argument

Let N nV be the number of electrons in volume V These electrons occupythe N lowest KohnSham spin orbitals ie those with k kF

N Xk

kF k V

Z kF

dk k V kF

where kF is called the Fermi wavevector The Fermi wavelength kF is theshortest de Broglie wavelength for the noninteracting electrons Clearly

n kF

rs

where we have introduced the Seitz radius rs the radius of a sphere which onaverage contains one electron

The kinetic energy of an orbital is k and the average kinetic energy perelectron is

tsn

N

Xk

kF kk

VN

Z kF

dk kk

kF

or of the Fermi energy In other notation

tsn

n

rs

All of this kinetic energy follows from the Pauli exclusion principle ie fromthe fermion character of the electron

Exchange Energy

To evaluate the exchange energy we need the KohnSham onematrix for electrons of spin as dened in Eq

r u r Xk

kF kexpik r upVexpik rpV

Z kF

dk kZ

dk

expik u

Z kF

dk ksinku

ku

kF

sinkFu kFu coskFukFu


The exchange hole density at distance u from an electron is by Eq

nxu j r u rj

n

which ranges from n at u where all other electrons of the same spinare excluded by the Pauli principle to like u as u The exchangeenergy per electron is

exn

Z

du unxu

kF

In other notation

exn

n

rs

Since the selfinteraction correction vanishes for the diuse orbitals of the uniform gas all of this exchange energy is due to the Pauli exclusion principle

Correlation Energy

Exact analytic expressions for ecn the correlation energy per electron of theuniform gas are known only in extreme limits The highdensity rs limitis also the weakcoupling limit in which

ecn c ln rs c crs ln rs crs rs from manybody perturbation theory The positive constants c and c

are known Eq does not quite tend to a constantwhen rs as Eq would suggest because the excited states of the noninteracting system lie arbitrarily close in energy to the ground state

The lowdensity rs limit is also the strong coupling limit in which theuniform uid phase is unstable against the formation of a closepacked Wignerlattice of localized electrons Because the energies of these two phases remainnearly degenerate as rs they have the same kind of dependence upon rs

ecn drs

d

rs

rs

The constants d and d in Eq can be estimated from the Madelungelectrostatic and zeropoint vibrational energies of the Wigner crystal respectively The estimate

d

can be found from the electrostatic energy of a neutral spherical cell just addthe electrostatic selfrepulsion rs of a sphere of uniform positive backgroundwith radius rs to the interaction rs between this background and the


electron at its center The origin of the rs term in Eq is also simple

Think of the potential energy of the electron at small distance u from the centerof the sphere as rs ku where k is a spring constant Since this potentialenergy must vanish for u rs we nd that k rs and thus the zeropointvibrational energy is

pkm rs

An expression which encompasses both limits and is

ecn c rs ln

crs rs r

s rs

where

cexp c

c

c

The coecients and are found bytting to accurate Quantum Monte Carlo correlation energies for rs and

The uniform electron gas is in equilibrium when the density n minimizes thetotal energy per electron ie when

ntsn exn ecn

This condition is met at rs close to the observed valence electron densityof sodium At any rs we have

Tsnr

nntsn

kF

Exnr

nnexn

kF

Eq with the parameters listed above provides a representation ofecn n for n n n other accurate representations are also available Eq with dierent parameters c c

can represent ecn n for n n andn the correlation energy per electron for a fully spinpolarized uniformgas But we shall need ecn n for arbitrary relative spin polarization

n nn n

which ranges from for an unpolarized system to for a fullyspinpolarizedsystem A useful interpolation formula based upon a study of the random phaseapproximation is

ecn n ecn cnf

f ecn ecnf

ecn cn O


where

f

In Eq cn is the correlation contribution to the spin stiness Roughlycn ecn ecn but more precisely cn can be parametrized inthe form of Eq with c c

For completeness we note that the spinscaling relations and imply that

exn n exn

tsn n tsn

The exchangehole density of Eq can also be spin scaled Expressions forthe exchange and correlation holes for arbitrary rs and are given in Ref

Linear Response

We now discuss the linear response of the spinunpolarized uniform electron gasto a weak static external potential vr This is a wellstudied problem and a practical one for the localpseudopotential description of a simple metal

Because the unperturbed system is homogeneous we nd that to rst orderin vr the electron density response is

nr

Zdrjr rjvr

where is a linear response function If

vr vq expiq r is a wave of wavevector q and small amplitude vq then Eq becomesnr nq expiq r where

nq qvq

and

q

Zdx expiq xjxj

is the Fourier transform of jrrj with respect to x rr In Eq thereal part of the complex exponential expi cos i sin is understood

By the KohnSham theorem we also have

nq sqvsq


where vsq is the change in the KohnSham eective oneelectron potential ofEq and

sq kF

F qkF

is the density response function for the noninteracting uniform electron gasThe Lindhard function

F x

xx

ln

x x

equals x x as x at x and x x asx dFdx diverges logarithmically as x

Besides vr the other contributions to vsr of Eq are

U

nr

Zdr

nr

jr rj

Excnr

Zdr

Excnrnr

nr

In other words

vsq vq

qnq

kFxcqnq

where the coecient of the rst nq is the Fourier transform of the Coulombinteraction jr rj and the coecient of the second nq is the Fouriertransform of Excnrnr

We rewrite Eq as

vsq vq

qGxcq nq

where

Gxcq xcq

q

kF

is the socalled localeld factor Then we insert Eq into Eq andnd

vsq vq

sq

where

sq q

Gxcqsq

In other words the density response function of the interacting uniform electrongas is

q sq

sq


These results are particularly simple in the longwavelength q limitin which xcq tends to a constant and

sq xcq kF

ksq

q

where

ks

kF

rs

is the inverse of the ThomasFermi screening length the characteristic distanceover which an external perturbation is screened out Eqs

and showthat a slowlyvarying external perturbation vq is strongly screened out bythe uniform electron gas leaving only a very weak KohnSham potential vsqEq shows that the response function q is weaker than sq by a factorqks

in the limit q In Eq

sq is a kind of dielectric function but it is not the standard

dielectric function q which predicts the response of the electrostatic potentialalone

vq

qnq

vq

q

By inserting Eq into Eq we nd

q

qq

It is only when we neglect exchange and correlation that we nd the simpleLindhard result

q sq Lq qsq xc

Neglecting correlation x is a numericallytabulated function of qkF withthe smallq expansion

xq

q

kF

q

kF

q

When correlation is included xcq depends upon rs as well as qkF in a waythat is known from Quantum Monte Carlo studies of the weaklyperturbeduniform gas

The secondorder change E in the total energy may be found from theHellmannFeynman theorem of Sect Replace vr by vr vr andnr by nr to nd

E

Z

d

Zdrnr

d

dvr

Z

d

Zdrn nrvr


Zdrnrvr

nqvq

Clumping and Adiabatic Connection

The uniform electron gas for rs provides a nice example of the adiabatic

connection discussed in Sect As the coupling constant turns on from to the ground state wavefunction evolves continuously from the KohnShamdeterminant of plane waves to the ground state of interacting electrons in thepresence of the external potential while the density remains xed One shouldof course regard the innite system as the innitevolume limit of a nite chunkof uniform background neutralized by electrons

The adiabatic connection between noninteracting and interacting uniformdensity ground states could be destroyed by any tendency of the density toclump A ctitious attractive interaction between electrons would yield such atendency Even in the absence of attractive interactions clumping appears inthe verylowdensity electron gas as a charge density wave or Wigner crystallization Then there is probably no external potential which will hold theinteracting system in a uniformdensity ground state but one can still nd theenergy of the uniform state by imposing density uniformity as a constraint on atrial interacting wavefunction

The uniform phase becomes unstable against a charge density wave of wavevector q and innitesimal amplitude when sq of Eq vanishes Thisinstability for q kF arises at low density as a consequence of exchange andcorrelation

Local and SemiLocal Approximations

Local Spin Density Approximation

The local spin density approximation LSD for the exchangecorrelation energyEq was proposed in the original work of Kohn and Sham and has provedto be remarkably accurate useful and hard to improve upon The generalizedgradient approximation GGA of Eq a kind of simple extension of LSD isnow more widely used in quantum chemistry but LSD remains the most popularway to do electronicstructure calculations in solid state physics Tables and provide a summary of typical errors for LSD and GGA while Tables and make this comparison for a few specic atoms and molecules The LSD isparametrized as in Sect while the GGA is the nonempirical one of PerdewBurke and Ernzerhof to be presented later

The LSD approximation to any energy component G is

GLSDn n

Zdrnrgnr nr


Table Exchangecorrelation energies of atoms in hartree

Atom LSD GGA Exact

H He Li

Be N Ne

Table Atomization energies of molecules in eV hartree eV From Ref

Molecule LSD GGA Exact

H CH NH HO CO O

where gn n is that energy component per particle in an electron gas withuniform spin densities n and n and nrd

r is the average number of electronsin volume element dr Sections provide the ingredients for TLSDs TELSDx and E

LSDc The functional derivative of Eq is

GLSD

nr

nn ngn n

By construction LSD is exact for a uniform density or more generally fora density that varies slowly over space More precisely LSD should be validwhen the length scale of the density variation is large in comparison with lengthscales set by the local density such as the Fermi wavelength kF or the screening length ks This condition is rarely satised in real electronic systems sowe must look elsewhere to understand why LSD works

We need to understand why LSD works for three reasons to justify LSD calculations to understand the physics and to develop improved density functionalapproximations Thus we will start with the good news about LSD proceed tothe mixed good$bad news and close with the bad news

LSD has many correct formal features It is exact for uniform densities andnearlyexact for slowlyvarying ones a feature that makes LSD well suited atleast to the description of the crystalline simple metals It satises the inequalities Ex Eq and Ec Eq the correct uniform coordinate


scaling of Ex Eq the correct spin scaling of Ex Eq the correctcoordinate scaling for Ec Eqs the correct lowdensity behavior of Ec Eq and the correct LiebOxford bound on Exc Eqs and LSD is properly sizeconsistent Sect

LSD provides a surprisingly good account of the linear response of the spinunpolarized uniform electron gas Sect Since

ELSDxcnrnr

r rnexcn

n

where r r is the Dirac delta function we nd

LSDxc q kF

nnecn

a constant independent of q which must be the exact q or slowlyvaryinglimit of xcq Figure of Ref shows that the exact xcq from a Quantum Monte Carlo calculation for rs is remarkably close to the LSD

prediction for q kF The same is true over the whole valenceelectron density

range rs and results from a strong cancellation between the nonlocali

ties of exchange and correlation Indeed the exact result for exchange neglectingcorrelation Eq is strongly qdependent or nonlocal The displayed terms

of Eq suce for q kF

Powerful reasons for the success of LSD are provided by the coupling constant integration of Sect Comparison of Eqs and reveals that theLSD approximations for the exchange and correlation holes of an inhomogeneoussystem are

nLSDx r r nunifx nr nr jr rj

nLSDc r r nunifc nr nr jr rj

where nunifxc n nu is the hole in an electron gas with uniform spin densitiesn and n Since the uniform gas is a possible physical system Eqs and obey the exact constraints of Eqs negativity of nx sum ruleon nx sum rule on nc and cusp condition

By Eq the LSD ontop exchange hole nLSDx r r is exact at least whenthe KohnSham wavefunction is a single Slater determinant The LSD ontopcorrelation hole nLSDc r r is not exact except in the highdensity lowdensity fully spinpolarized or slowlyvarying limit but it is often quite realistic By Eq its cusp is then also realistic

Because it satises all these constraints the LSD model for the systemspherically and couplingconstantaveraged hole of Eq

hnLSDxc ui

N

Zdrnrnunifxc nr nru

can be very physical Moreover the system average in Eq unweightsregions of space where LSD is expected to be least reliable such as near anucleus or in the evanescent tail of the electron density


Since correlation makes hnxcu i deeper and thus by Eq makeshnxcui more shortranged Exc can be more local than either Ex or Ec Inother words LSD often benets from a cancellation of errors between exchangeand correlation

Mixed good and bad news about LSD is the fact that selfconsistent LSDcalculations can break exact spin symmetries As an example consider stretchedH the hydrogen molecule N with a very large separation between thetwo nuclei The exact ground state is a spin singlet S with nr nr nr But the LSD ground state localizes all of the spinup density on one ofthe nuclei and all of the spindown density on the other Although or ratherbecause the LSD spin densities are wrong the LSD total energy is correctly thesum of the energies of two isolated hydrogen atoms so this symmetry breakingis by no means entirely a bad thing

The selfconsistent LSD ontop holedensity hnxc i hni is also right HeitlerLondon correlation ensues that twoelectrons are never found near one another or on the same nucleus at the sametime

Finally we present the bad news about LSD LSD does not incorporateknown inhomogeneity or gradient corrections to the exchangecorrelation holenear the electron Sect It does not satisfy the highdensity correlation scaling requirement of Eq but shows a ln divergence associatedwith the ln rs term of Eq LSD is not exact in the oneelectron limitie does not satisfy Eqs and Although the selfinteractionerror is small for the exchangecorrelation energy it is more substantial forthe exchangecorrelation potential and orbital eigenvalues As a continuumapproximation based as it is on the uniform electron gas and its continuousoneelectron energy spectrum LSD misses the derivative discontinuity of Sect Eectively LSD averages over the discontinuity so its highest occupied orbital energy for a Zelectron system is not Eq but HO IZ AZA second consequence is that LSD predicts an incorrect dissociation of a heteronuclear molecule or solid to fractionally charged fragments In LSD calculationsof atomization energies the dissociation products are constrained to be neutralatoms and not these unphysical fragments LSD does not guarantee satisfaction of Eq an inherently nonlocal constraint

The GGA to be derived in Sect will preserve all the good or mixedfeatures of LSD listed above while eliminating bad features and butnot Elimination of will probably require the construction ofExcn n from the KohnSham orbitals which are themselves highlynonlocalfunctionals of the density For example the selfinteraction correction to LSD eliminates most of the bad features and but not in an entirelysatisfactory way

Gradient Expansion

Gradient expansions which oer systematic corrections to LSD for electron densities that vary slowly over space might appear to be the natural next


step beyond LSD As we shall see they are not understanding why not will lightthe path to the generalized gradient approximations of Sect

As a rst measure of inhomogeneity we dene the reduced density gradient

s jrnjkFn

jrnj

n

jrrsj

which measures how fast and how much the density varies on the scale of thelocal Fermi wavelength kF For the energy of an atom molecule or solid the

range s is very important The range s is somewhat important

more so in atoms than in solids while s as in the exponential tail of thedensity is unimportant

Other measures of density inhomogeneity such as p rnkFn are alsopossible Note that s and p are small not only for a slow density variation but alsofor a density variation of small amplitude as in Sect The slowlyvaryinglimit is one in which ps is also small

Under the uniform density scaling of Eq sr sr sr Thefunctionals Tsn and Exn must scale as in Eqs and so their gradient expansions are

Tsn As

Zdrn s

Exn Ax

Zdrn s

Because there is no special direction in the uniform electron gas there can beno term linear in rn Moreover terms linear in rn can be recast as s termssince Z

drfnrn Zdr

f

n

jrnj

via integration by parts Neglecting the dotted terms in Eqs and which are fourth or higherorder in r amounts to the secondorder gradientexpansion which we call the gradient expansion approximation GEA

Correlation introduces a second length scale the screening length ks andthus another reduced density gradient

t jrnjksn

s

rs

In the highdensity rs limit the screening length ks rs is the onlyimportant length scale for the correlation hole The gradient expansion of thecorrelation energy is

Ecn

Zdrn

ecn nt

While ecn does not quite approach a constant as n n does


While the form of the gradient expansion is easy to guess the coecients canonly be calculated by hard work Start with the uniform electron gas in eitherits noninteracting TsEx or interacting Ec ground state and apply a weakexternal perturbation vsq expiq r or vq expiq r respectively Findthe linear response nq of the density and the secondorder response G ofthe energy component G of interest Use the linear response of the density asin Eqs or to express G entirely in terms of nq Finally expandG in powers of q observing that jrnj qjnqj and extract the gradientcoecient

In this way Kirzhnits found the gradient coecient for Ts

which respects the conjectured bound of Eq Sham found the coefcient of Ex

Sham

and Ma and Brueckner found the highdensity limit of n

MB

The weak density dependence of n is also known as is its spindependence Neglecting small r contributions the gradient coecients coecients ofjrnjn for both exchange and correlation at arbitrary relative spin polarization are found from those for through multiplication by

h

i

For exchange this is easily veried by applying the spinscaling relation ofEq to Eqs and

There is another interesting similarity between the gradient coecients forexchange and correlation Generalize the denition of t Eq to

t jrnjksn

s

rs

Then

MBnt Cxn

s

where

MB

Shams derivation of Eq starts with a screened Coulomb interaction u expu and takes the limit at the end of the calculation


Antoniewicz and Kleinman showed that the correct gradient coecient forthe unscreened Coulomb interaction is not Sham but

AK

It is believed that a similar orderof limits problem exists for in such a waythat the combination of Shams exchange coecient with the MaBrueckner correlation coecient yields the correct gradient expansion of Exc in the slowlyvarying highdensity limit

Numerical tests of these gradient expansions for atoms show that the secondorder gradient term provides a useful correction to the ThomasFermi or localdensity approximation for Ts and a modestly useful correction to the local density approximation for Ex but seriously worsens the local spin density results forEc and Exc In fact the GEA correlation energies are positive" The latter factwas pointed out in the original work of Ma and Brueckner who suggestedthe rst generalized gradient approximation as a remedy

The local spin density approximation to Exc which is the leading term ofthe gradient expansion provides rather realistic results for atoms moleculesand solids But the secondorder term which is the next systematic correctionfor slowlyvarying densities makes Exc worse

There are two answers to the seeming paradox of the previous paragraph Therst is that realistic electron densities are not very close to the slowlyvaryinglimit s ps t etc The second is this The LSD approximationto the exchangecorrelation hole is the hole of a possible physical system theuniform electron gas and so satises many exact constraints as discussed inSect The secondorder gradient expansion or GEA approximation to thehole is not and does not

The secondorder gradient expansion or GEA models are known for both theexchange hole nxr ru and the correlation hole ncr ru Theyappear to be more realistic than the corresponding LSD models at small u butfar less realistic at large u where several spurious features appear nxr ruGEAhas an undamped coskFu oscillation which violates the negativity constraintof Eq and integrates to Eq only with the help of a convergencefactor expu ncr ruGEA has a positive u tail and integratesnot to zero Eq but to a positive number s These spurious largeubehaviors are sampled by the long range of the Coulomb interaction u leadingto unsatisfactory energies for real systems

The gradient expansion for the exchange hole density is known to thirdorder in r and suggests the following interpretation of the gradient expansion When the density does not vary too rapidly over space eg in the weakpseudopotential description of a simple metal the addition of each successiveorder of the gradient expansion improves the description of the hole at small uwhile worsening it at large u The bad largeu behavior thwarts our expectationthat the hole will remain normalized to each order in r


The noninteracting kinetic energy Ts does not sample the spurious largeupart of the gradient expansion so its gradient expansion Eqs and works reasonably well even for realistic electron densities In fact we can useEq to show that

Tsn X

Zdr

r r

r r

rr

samples only the smallu part of the gradient expansion of the KohnSham oneelectron reduced density matrix while Exn of Eqs and also sampleslarge values of u The GEA for Tsn is in a sense its own GGA Moreoverthe sixthorder gradient expansion of Ts is also known it diverges for nitesystems but provides accurate monovacancy formation energies for jellium

The GEA form of Eqs and is a special case of the GGAform of Eq To nd the functional derivative note that

F

Zdrfn nrnrn

X

Zdr

f

nrnr

f

rnr rnr

X

Zdr

F

nrnr

Integration by parts gives

F

nr

f

nrr f

rnr

For example the functional derivative of the gradient term in the spinunpolarized highdensity limit is

nr

ZdrCxc

jrnrjn

Cxc

jrnrjn

rn

n

which involves second as well as rst derivatives of the density

The GEA for the linear response function xcq of Eq is found byinserting nr n nq expiq r into Eq and linearizing in nq

GEAxc q LSDxc Cxc

q

kF

For example the AntoniewiczKleinman gradient coecient for exchange ofEq inserted into Eqs and yields the q term of Eq


History of Several Generalized Gradient Approximations

In Ma and Brueckner derived the secondorder gradient expansionfor the correlation energy in the highdensity limit Eqs and Innumerical tests they found that it led to improperly positive correlation energiesfor atoms because of the large size of the positive gradient term As a remedythey proposed the rst GGA

EMBc n

Zdrnecn

MBt

necn

where was tted to known correlation energies Eq reduces toEqs and in the slowlyvarying t limit but provides a strictlynegative energy density which tends to zero as t In this respect itis strikingly like the nonempirical GGAs that were developed in or laterdiering from them mainly in the presence of an empirical parameter the absenceof a spindensity generalization and a less satisfactory highdensity limit

Under the uniform scaling of Eq nr nr we nd rsr rsr r r sr sr and tr tr Thus EMBc n tends to ELSDc n as and not to a negative constant as required byEq

In Langreth and Perdew explained the failure of the secondordergradient expansion GEA for Ec They made a complete wavevector analysis ofExc ie they replaced the Coulomb interaction u in Eq by its Fouriertransform and found

Excn N

Z

dkk

hnxcki

k

where

hnxcki Z

du uhnxcui sinkuku

is the Fourier transform of the system and sphericallyaveraged exchangecorrelation hole In Eq Exc is decomposed into contributions from dynamicdensity uctuations of various wavevectors k

The sum rule of Eq should emerge from Eq in the k limitsince sinxx as x and does so for the exchange energy at theGEA level But the k limit of nGEAc k turns out to be a positive numberproportional to t and not zero The reason seems to be that the GEA correlationhole is only a truncated expansion and not the exact hole for any physicalsystem so it can and does violate the sum rule

Langreth and Mehl proposed a GGA based upon the wavevectoranalysis of Eq They introduced a sharp cuto of the spurious smallkcontributions to EGEAc all contributions were set to zero for k kc f jrnnjwhere f is only semiempirical since f was estimated theoreticallyExtension of the LangrethMehl EGGAc beyond the random phase approximationwas made by Perdew in


The errors of the GEA for the exchange energy are best revealed in real spaceEq not in wavevector space Eq In Perdew showedthat the GEA for the exchange hole density nxr ru contains a spurious undamped coskFu oscillation as u which violates the negativity constraintof Eq and respects the sum rule of Eq only with the help of a convergence factor eg expu as This suggested that the required cutosshould be done in real space not in wavevector space The GEA hole densitynGEAx r ru was replaced by zero for all u where n

GEAx was positive and for all

u uxr where the cuto radius uxr was chosen to recover Eq Eq then provided a numericallydened GGA for Ex which turned out to be moreaccurate than either LSD or GEA In Perdew and Wang simplied thisGGA in two ways They replaced nGEAx r r u which depends upon bothrst and second derivatives of nr by %nGEAx r r u an equivalent expressionfound through integration by parts which depends only upon rnr Theresulting numerical GGA has the form

EGGAx n Ax

ZdrnFxs

which scales properly as in Eq The function Fxs was plotted and ttedby an analytic form The spinscaling relation was used to generate a spindensity generalization Perdew and Wang also coined the term generalizedgradient approximation

A parallel but more empirical line of GGA development arose in quantumchemistry around Becke showed that a GGA for Ex could beconstructed with the help of one or two parameters tted to exchange energies ofatoms and demonstrated numerically that these functionals could greatly reducethe LSD overestimate of atomization energies of molecules Lee Yang and Parr transformed the ColleSalvetti expression for the correlation energy froma functional of the KohnSham oneparticle density matrix into a functional ofthe density This functional contains one empirical parameter and works well inconjunction with Becke exchange for many atoms and molecules althoughit underestimates the correlation energy of the uniform electron gas by about afactor of two at valenceelectron densities

The realspace cuto of the GEA hole provides a powerful nonempirical wayto construct GGAs Since exchange and correlation should be treated in a balanced way there was a need to extend the realspace cuto construction from exchange to correlation with the help of a second cuto radius ucrchosen to satisfy Eq Without accurate formulas for the correlation hole ofthe uniform electron gas this extension had to wait until when it led to thePerdewWang PW GGA for Exc For most practical purposesPW is equivalent to the PerdewBurkeErnzerhof PBE GGA madesimple which will be derived presented and discussed in the next two sections


Construction of a GGA Made Simple

The PW GGA and its construction are simple in principle but complicated in practice by a mass of detail In Perdew Burke and Ernzerhof PBE showed how to construct essentially the same GGA in a muchsimpler form and with a much simpler derivation

Ideally an approximate density functional Excn n should have all of thefollowing features a nonempirical derivation since the principles of quantum mechanics are wellknown and sucient universality since in principleone functional should work for diverse systems atoms molecules solids withdierent bonding characters covalent ionic metallic hydrogen and van derWaals simplicity since this is our only hope for intuitive understandingand our best hope for practical calculation and accuracy enough to be useful in calculations for real systems

The LSD of Eq and the nonempirical GGA of Eq nicely balancethese desiderata Both are exact only for the electron gas of uniform density andrepresent controlled extrapolations away from the slowlyvarying limit unlikethe GEA of Sect which is an uncontrolled extrapolation LSD is a controlledextrapolation because even when applied to a density that varies rapidly overspace it preserves many features of the exact Exc as discussed in Sect LSDhas worked well in solid state applications for thirty years

Our conservative philosophy of GGA construction is to try to retain all thecorrect features of LSD while adding others In particular we retain the correct uniformgas limit for two reasons This is the only limit in which therestricted GGA form can be exact Natures data set includes the crystallinesimple metals like Na and Al The success of the stabilized jellium model rearms that the valence electrons in these systems are correlated very much asin a uniform gas Among the welter of possible conditions which could be imposed to construct a GGA the most natural and important are those respectedby LSD or by the realspace cuto construction of PW and these are theconditions chosen in the PBE derivation below The resulting GGA is onein which all parameters other than those in LSD are fundamental constants

We start by writing the correlation energy in the form

EGGAc n n

Zdrnecrs Hrs t

where the local density parameters rs and are dened in Eqs and and the reduced density gradient t in Eq The smallt behavior of nHshould be given by the lefthand side of Eq which emerges naturallyfrom the realspace cuto construction of PW In the opposite or t

limit we expect that H ecrs the correlation energy per electron of theuniform gas as it does in the PW construction or in the MaBrueckner GGAof Eq Finally under the uniform scaling of Eq to the highdensity limit Eq should tend to a negative constant as in Eq or in the numericallyconstructed PW This means that H must cancel thelogarithmic singularity of ec Eq in this limit


A simple function which meets these expectations is

H c ln

MBc

t

At

At At

where is given by Eq and

A MBc

exp ecrs c

We now check the required limits

t H c ln

MBc

t

MBt

t H c ln

MBcA

c lnexp

ecrs

c

ecrs

rs at xed s H c ln t c ln rs To a good approximation Eq can be generalized to

ecrs c ln rs c

which cancels the log singularity of Eq Under uniform density scaling to the highdensity limit we nd

EGGAc n cZdrn ln

s s

where s is dened by Eq a negative constant as required by Eq with

MBc

expcc

For a twoelectron ion of nuclear charge Z in the limit Z Eq is hartree and the exact value is Realistic results from Eq in the Z limit have also be found for ions with and electrons

Now we turn to the construction of a GGA for the exchange energy Becauseof the spinscaling relation we only need to construct EGGAx n whichmust be of the form of Eq To recover the good LSD description of thelinear response of the uniform gas Sect we choose the gradient coecient


for exchange to cancel that for correlation ie we take advantage of Eq to write

s Fxs s Then the gradient coecients for exchange and correlation will cancel for all rsand apart from small r contributions to EGGAx as discussed in the nextsection

The value of of Eq is times bigger than AK of Eq theproper gradient coecient for exchange in the slowlyvarying limit But thischoice can be justied in two other ways as well a It provides a decent tto the results of the realspace cuto construction of the PW exchangeenergy which does not recover AK in the slowlyvarying limit b It provides areasonable emulation of the exactexchange linear response function of Eq

over the important range of qkF but not of course in the limit q

where AK is neededFinally we want to satisfy the LiebOxford bound of Eqs and

which LSD respects We can achieve this and also recover the limit of Eq with the simple form

Fxs s

where is a constant less than or equal to Taking gives a GGAwhich is virtually identical to PW over the range of densities and reduced density gradients important in most real systems We shall complete the discussionof this paragraph in the next section

GGA Nonlocality Its Character Origins and Eects

A useful way to visualize and think about gradientcorrected nonlocality or tocompare one GGA with another is to write

EGGAxc n n Zdrn

crs

Fxcrs s

where c and crs exrs is the exchange energyper electron of a spinunpolarized uniform electron gas The enhancement factorFxcrs s shows the eects of correlation through its rs dependence spinpolarization and inhomogeneity or nonlocality s Fxc is the analog of in Slaters X method so its variation is bounded and plottable Figure shows Fxcrs s the enhancement factor for a spinunpolarized systemFigure shows Fxcrs s Fxcrs s the enhancement factorfor the spin polarization energy Roughly Fxcrs s Fxcrs s Fxcrs sFxcrs s The nonlocality is the sdependence and

FLSDxc rs s Fxcrs s


0 0.5 1 1.5 2 2.5 3

1

1.2

1.4

1.6

1.8

rs

rs

rs

rs

rs

rs

rs

s

Fxcrs s

Fig The enhancment factor Fxc of Eq for the GGA of Perdew Burke andErnzerhof as a function of the reduced density gradient s of Eq for The local density parameter rs and the relative spin polarization are dened inEqs

and respectively

is visualized as a set of horizontal straight lines coinciding with the GGA curvesin the limit s

Clearly the correlation energy of Eq can be written in the form ofEq To get the exchange energy into this form apply the spinscalingrelation to Eq then drop small rs contributions to nd

Fx s

Fx

s

Fx

s

h

i s


0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

rs

rs

rs

rs

rs

rs

rs

s

Fxcrs s Fxcrs s

Fig Same as Fig but for the dierence between the fully spinpolarized and unpolarized enhancement factors

NowFxcrs s Fx s Fcrs s

wherelimrs

Fcrs s

by Eqs and Thus the rs or highdensitylimit curve in eachgure is the exchangeonly enhancement factor Clearly Fx Fc andFx s by denition

The LiebOxford bound of Eq will be satised for all densities nr ifand only if

Fxcrs s For the PBE GGA of Eqs and this requires that

Fxs

or


as stated in Sect There is much to be seen and explained in Eq and Figs and

However the main qualitative features are simply stated When we make adensity variation in which rs decreases increases or s increases everywherewe nd that jExj increases and jEcExj decreases

To understand this pattern we note that the secondorder gradient expansion for the noninteracting kinetic energy Tsn n which is arguably itsown GGA can be written as

TGGAs n n

Zdrn

rs

G s

G s

h

i

s

using approximate spin scaling Eq plus neglect of r contributionsEqs and respect Eq and conrm our intuition based upon thePauli exclusion and uncertainty principles Under a density variation in which rsdecreases increases or s increases everywhere we nd that Tsn n increases

The rst eect of such an increase in Ts is an increase in jExj Ts and jExjare conjoint in the sense that both can be constructed from the occupiedKohnSham orbitals Eqs and With more kinetic energythese occupied orbitals will have shorter de Broglie wavelengths By the uncertainty principle they can then dig a more shortranged and deeper exchangehole with a more negative exchange energy Thus exchange turns on when wedecrease rs increase or increase s

The second eect of such an increase in Ts is to strengthen the KohnShamHamiltonian which holds noninteracting electrons at the spin densities nr andnr This makes the electronelectron repulsion of Eq a relatively weakerperturbation on the KohnSham problem and so reduces the ratio jEcExj Thuscorrelation turns o relative to exchange when we decrease rs increase orincrease s

We note in particular that Fxrs s increases while Fcrs s decreaseswith increasing s The nonlocalities of exchange and correlation are opposite

and tend to cancel for valenceelectron densities rs in the range

s The same remarkable cancellation occurs in the linear response

function for the uniform gas of Eq ie xcq LSDxc q xcq for qkF

The core electrons in any system and the valence electrons in solids sample

primarily the range s The highdensity core electrons see a strong

exchangelike nonlocality of Exc which provides an important correction to theLSD total energy But the valence electrons in solids see an almostcompletecancellation between the nonlocalities of exchange and correlation This helpsto explain why LSD has been so successful in solid state physics and why thesmall residue of GGA nonlocality in solids does not provide a universallybetterdescription than LSD


The valence electrons in atoms and molecules see s when s diverges

in the exponential tail of the density but the energeticallyimportant range is

s Figs and show that GGA nonlocality is important in this

range so GGA is almostalways better than LSD for atoms and molecules

For rs the residual GGA nonlocality is exchangelike ie exchange

and correlation together turn on stronger with increasing inhomogeneity It canthen be seen from Eq that gradient corrections will favor greater densityinhomogeneity and higher density Dening average density parameters hrsihi and hsi as in Ref we nd that gradient corrections favor changesdhsi

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