kohn nobel

7/29/2019 Kohn Nobel

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Nobel Lecture: Electronic structure of matterwave functions

and density functionals*

W. Kohn

Department of Physics, University of California, Santa Barbara, California 93106

[S0034-6861(99)00505-X]

CONTENTS

I. Introduction 1253II. Schrodinger Wave FunctionsFew versus Many

Electrons 1255A. Few-electron systemsthe H2 molecule 1256B. Many electronsencountering an exponential

wall 1257C. Some meta-physical-chemical considerations 1257

III. Density-Functional TheoryBackground 1258IV. The Hohenberg-Kohn Formulation of Density-

Functional Theory 1259A. The density n(r) as the basic variable 1259

B. The Hohenberg-Kohn variational principle 1260C. The self-consistent Kohn-Sham equations 1260

V. Approximation for Exc n(r) : from Mathematics toPhysical Science 1262

A. The local-density approximation (LDA) 1262B. Beyond the local-density approximation 1263

VI. Generalizations and Quantitative Applications 1264A. Generalizations 1264B. Applications 1265

VII. Concluding Remarks 1265References 1265

I. INTRODUCTION

The citation for my share of the 1998 Nobel Prize inChemistry refers to the development of the density-functional theory. The initial work on Density-Functional Theory (DFT) was reported in two publica-tions: the first with Pierre Hohenberg (Hohenberg andKohn, 1964) and the next with Lu J. Sham (Kohn andSham, 1965). This was almost 40 years after E. Schro-dinger (1926) published his first epoch-making papermarking the beginning of wave mechanics. The Thomas-Fermi theory, the most rudimentary form of DFT, wasput forward shortly afterwards (Fermi, 1927; Thomas,1927) and received only modest attention.

There is an oral tradition that, shortly after Schrod-ingers equation for the electronic wave function hadbeen put forward and spectacularly validated for smallsystems like He and H2, P. M. Dirac declared that chem-istry had come to an endits content was entirely con-tained in that powerful equation. Too bad, he is said tohave added, that in almost all cases, this equation wasfar too complex to allow solution.

In the intervening more than six decades enormousprogress has been made in finding approximate solutionsof Schrodingers wave equation for systems with severalelectrons, decisively aided by modern electronic com-puters. The outstanding contributions of my Nobel Prizeco-winner John Pople are in this area. The main objec-tive of the present account is to explicate DFT, which isan alternative approach to the theory of electronic struc-ture, in which the electron density distribution n(r),rather than the many-electron wave function, plays acentral role. I felt that it would be useful to do this in acomparative context; hence the wording Wave Func-

tions and Density Functionals in the title.In my view DFT makes two kinds of contribution to

the science of multiparticle quantum systems, includingproblems of electronic structure of molecules and ofcondensed matter.

The first is in the area of fundamental understanding.Theoretical chemists and physicists, following the pathof the Schrodinger equation, have become accustomedto think in terms of a truncated Hilbert space of single-

particle orbitals. The spectacular advances achieved inthis way attest to the fruitfulness of this perspective.However, when high accuracy is required, so manySlater determinants are required (in some calculationsup to 109!) that comprehension becomes difficult. DFTprovides a complementary perspective. It focuses onquantities in the real, three-dimensional coordinatespace, principally on the electron density n(r). Otherquantities of great interest are: the exchange correlationhole density nxc (r,r) which describes how the presenceof an electron at the point r depletes the total density ofthe other electrons at the point r; and the linear re-sponse function, (r,r;), which describes the changeof total density at the point r due to a perturbing poten-tial at the point r, with frequency . These quantitiesare physical, independent of representation, and easilyvisualizable even for very large systems. Their under-

standing provides transparent and complementary in-sight into the nature of multiparticle systems.

The second contribution is practical. Traditional mul-tiparticle wave-function methods when applied to sys-tems of many particles encounter what I call an expo-nential wall when the number of atoms N exceeds acritical value which currently is in the neighborhood ofN010 (to within a factor of about 2) for a system with-out symmetries. A major improvement in the analyticaland/or computational aspects of these methods alongpresent lines will lead to only modest increases in N0 .Consequently problems requiring the simultaneous con-

*The 1998 Nobel Prize in Chemistry was shared by W. Kohnand John A. Pople. This lecture is the text of Professor Kohnsaddress on the occasion of the award.

1253Reviews of Modern Physics, Vol. 71, No. 5, October 1999 0034-6861/99/71(5)/1253(14)/$17.80 The Nobel Foundation 1998


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There follows a brief discussion of approximations forExc , which reflect nearsightedness, and other generalprinciples.

Sections IIIVI discuss applications of DFT to elec-tronic ground states, as well as a host of generalizationsto other electronic and nonelectronic systems.

Finally a few concluding remarks and speculations are

offered.

II. SCHRODINGER WAVE FUNCTIONSFEW VERSUS

MANY ELECTRONS

The foundation of the theory of electronic structure ofmatter is the nonrelativistic Schrodinger equation forthe many-electron wave function ,

2

2m j j2

j,l

Zle2

rjR l

12 jj

e2

rjrjE 0,

(2.1)

where rj are the positions of the electrons and R l, Zlthe positions and atomic numbers of the nuclei; , m ,and e are the conventional fundamental constants; andE is the energy. This equation reflects the Born-Oppenheimer approximation, in whichfor purposes ofstudying electron dynamicsthe much heavier nucleiare considered as fixed in space. This paper will deallargely with nondegenerate groundstates. The wave

function depends on the positions and spins of the Nelectrons but in this paper spins will generally not beexplicitly indicated. Thus

r1 ,r2 , . . . rN . (2.2)

The Pauli principle requires that

Pjj , (2.3)

where Pjj permutes the space and spin coordinates ofelectrons j and j. All physical properties of the elec-trons depend parametrically on the R l, in particular thedensity n(r) and total energy E, which play key roles inthis paper:

FIG. 2. The geometric structure of the clathrate Sr8Ga16Ge30 (Sr, red; Ga, blue; Ge, white) and its charge density in a planebisecting the centers of the cages. DFT calculations have shown that the Sr atoms are weakly bound and scatter phononseffectively, thereby lowering thermal conductivity. However, contrary to intuitive expectations, the Sr atoms do not donate

electrons to the frame and are practically neutral. Conductivity is due to electrons traveling through the frame, not through theone-dimensional Sr wires in the structure; there is thus little scattering of conduction electrons by Sr vibrations. For thesereasons, the compound is a metal with a large Seebeck coefficient (unlike ordinary metals). The calculation suggests that othercompounds of this type may be even better thermoelectrics (theory by N. P. Blake and H. Metiu, submitted for publication)[Color].

1255W. Kohn: Electronic structure of matter (Nobel Lecture)

Rev. Mod. Phys., Vol. 71, No. 5, October 1999


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n r n r;R1 , . . . RN , (2.4)

EE R1 , . . . RN , (2.5)

where N is the number of nuclei.

A. Few-electron systemsthe H2 molecule

The first demonstrations of the power of the Schro-dinger equation in chemistry were calculations of theproperties of the simplest multielectron molecule, H2:Its experimental binding energy2 and internuclear sepa-ration are

Exp: D4.75 eV, R0.740 . (2.6)

The earliest quantum-theoretical estimate was madeby Heitler and London in 1927, who used the ansatz

HLA H r1R 1 H r2R 2

H r1R 2 H r2R 1 0 , (2.7)where H(r1R1) is the orbital wave function of elec-tron 1 in its atomic ground state around a proton locatedat R1 , etc.; 0 denotes the spin singlet function; and A isthe normalization. The components of this wave func-tion describe two hydrogen atoms, at R1 and R2 , withspins pointing in opposite directions. The combinationsatisfies the reflection symmetry of the molecule and the

Pauli principle. The expectation value of the Hamil-tonian as a function of R R1R 2 was calculated. Itsminimum was found to occur at R0.87 , and the cal-culated dissociation energy was 3.14 eV, in semiquanti-tative agreement with experiment. However the errorswere far too great for the typical chemical requirements

of R 0.01 and D 0.1eV.An alternative ansatz, analogous to that adopted by

Bloch for crystal electrons, was made by Mullikan(1928):

BMmo l r1 mo l r2 0 , (2.8)

where

mo l r1 A H r1R 1 H r1R 2 , (2.9)

and A is the appropriate normalization constant. In thisfunction both electrons occupy the same molecular or-bital mo l(r). The spin function 0 is again the antisym-metric singlet function. The results obtained with this

function were R0.76 and D2.65 eV, again in semi-quantitative agreement with experiment.The Mullikan ansatz can be regarded as the simplest

version of a more general so-called Hartree-Fock ansatz,the Slater determinant

HF1

21/2Det m r1 1 m r2 2 , (2.10)

where m(r) is a general molecular orbital and and denote up and down spin functions. For given R R1R 2 , minimization with respect to m(r) of the expec-tation value of H leads to the nonlocal Hartree-Fock

2This is the observed dissociation energy plus the zero-pointenergy of 0.27 eV.

FIG. 3. Fully hydroxylated aluminum (0001) surface (red, O; blue, interior Al; Al; gray, H atoms; the green lines are H bonds).Each surface A atom in Al2O3 has been replaced by three H atoms. The figure represents a superposition of configurations in amolecular-dynamics simulation at regular intervals of 1 ps. These calculations help to understand the complex dynamics of wateradsorption on aluminum [K. C. Haas et al., Science 282, 265 (1998)] [Color].

1256 W. Kohn: Electronic structure of matter (Nobel Lecture)



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equations (Hartree, 1928; Fock, 1930) for the molecularorbital m(r), whose solution gives the following re-sults: R0.74, D3.63 eV.

The most complete early study of H2 was undertakenby James and Coolidge (1933). They made the very gen-eral variational ansatz

JC r1 ,r2 0 , (2.11)

where (r1 ,r2) is a general, normalized function of r1and r2 , symmetric under interchange of r1 and r2 andrespecting the spatial symmetries of the molecule. Thetrial function was written as depending on a numberof parameters, p1 ,p2 pM, so that for given R1R 2 ,the expectation value of the Schrodinger Hamiltonian in, an upper bound to the true ground-state energy, be-came a function of the parameters pj , EE(p1 , . . . ,pM). The calculations were made with Mup to 13. Minimization ofE(p1 , . . . ,pM) with respect tothe pj resulted in R0.740 and D4.70eV, in verygood agreement with experiment. More recent varia-tional calculations of the same general character give

theoretical results whose errors are estimated to bemuch smaller than experimental uncertainties, and othertheoretical corrections.

Before leaving the variational calculation for H2, wewant to make a very rough guesstimate of the numberof parameters M needed for a satisfactory result.

The number of continuous variables of(r1 ,r2) is 615, the reduction by 1 reflecting axial symmetry. Letus call the number of parameters per variable neededfor the desired accuracy p . Since a fractional accuracy ofO(102) is needed for the energy, implying a fractionalaccuracy of O(101) in , we guess that 3p10.Hence Mp535105102105.

By using symmetries and chemical and mathematicalinsights, this number can be significantly reduced. Suchrelatively modest numbers are very managable on to-days (and even yesterdays) computers.

It is thus not surprising that for sufficiently small mol-ecules, wave-function methods give excellent results.

B. Many electronsencountering an exponential wall

In the same spirit as our last guesstimates for H2,let us now consider a general molecule consisting ofNatoms with a total of N interacting electrons, where N10, say. We ignore symmetries and spin, which will not

affect our general conclusions. Reasoning as before, wesee that the number M of parameters required is

Mp3N, 3p10. (2.12)

The energy needs to be minimized in the space of theseM parameters. Call M the maximum value feasible withthe best available computer software and hardware; andN the corresponding maximum number of electrons.Then, from Eq. (2.12) we find

N13

ln M

lnp. (2.13)

Let us optimistically take M109 and p3. This givesthe shocking result

N13

90.48

6 ! . (2.14)

In practice, by being clever, one can do better thanthis, perhaps by one half order of magnitude, up to sayN20. But the exponential in Eq. (2.12) represents a

wall severely limiting N

.Let us turn this question around and ask what is the

needed M for N100. By Eq. (2.12) and taking p3 wefind

M330010150 ! . (2.15)

I cannot foresee an advance in computer sciencewhich can minimize a quantity in a space of 10150 dimen-sions. Of course, estimates like Eq. (2.15) are very roughand only their logarithm should be taken seriously. Butthe exponential wall is real and reflects the intercon-nectedness of(r1 , . . . ,rN) in the 3N-dimensional con-figuration space defined by all rj being inside the 3D

region containing the molecule.We conclude that traditional wave-function methods,which provide the required chemical accuracy, aregenerally limited to molecules with a small total numberof chemically active electrons, NO(10).

C. Some meta-physical-chemical considerations

The following remarks are related to a very old paperby one of my teachers, J. H. Van Vleck (1936), in whichhe discusses a problem with many-body wave functions,later referred to as the Van Vleck catastrophy.

I begin with a provocative statement. In general themany-electron wave function (r1 , . . . ,rN) for a systemof N electrons is not a legitimate scientific concept, whenNN0 , where N010

3.I will use two criteria for defining legitimacy: (a)

That can be calculated with sufficient accuracy and(b) that (r1 , . . . ,rN) can be recorded with sufficientaccuracy.Construction of an accurate approximation to .

Without leaving the context of wave functions, I shallcall the approximate wave function sufficiently accu-rate if

, 20.5, (2.16)

a rather liberal requirement (one could equally wellchoose 0.9 or 0.1).

Consider now the example ofNnonoverlapping iden-tical n-electron molecules with exact wave functionsl(r1 , . . . ,rn), and approximate wave functionsl(r1 , . . . ,rn). Let us take n10 and posit that a veryaccurate l can be calculated with

l, 1 where 102, (2.17)

again a liberal estimate.




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Then, for an N-molecule system with N102, andthus N103 electrons (,)(1)NeNe1

0.37, i.e., almost acceptable by the criterion (2.16).Note, however, that for N104, ( ,)e105105 so that , 23109(! )the exponentialwall is again there, in another form. For fully interactingsystems the situation is much worse and our estimate ofN010

3 is probably much too high.

Can this problem ever be overcome along presentlines of thought? I think not. Even if there were no com-putational limits, other physical effects, such as relativ-istic or radiative corrections which may be minor forsystems of small N exponentiate when N exceeds N0 .

(It is obvious that the estimates made above have onlylogarithmic validity.)Recording of(r1,r2, . . . ,rN).

Let us now assume that somehow we have obtainedan accurate approximation to , in the sense of Eq.(2.16), and wish to record it so it can be reproduced at alater time. How many bits are needed? Let us take q bitsper variable. Then the total number of bits is

Bq3N.

For q3, a very rough fit, and N103, B101500, aquite unrealistic number (the total number of baryons inthe accessible universe is estimated as 1080).

Having attempted to discredit the very-many-electronwave function (r1 , . . . ,rN), for many electrons I must,of course, recall two well-known facts: physically/chemically interesting quantities, like total energy E ,density n(r), pair correlation function g(r,r), etc., de-pend on only very few variables and, formally, can bethought of as obtained by tracing over all other vari-ables, e.g.,

n r N * r,r2 , . . . ,rN r,r2 , . . . ,rN

dr2 . . . drN;

and that some s which, by the criterion (1.16) arehopelessly bad for large N, can give respectable andeven very accurate results for these contracted quanti-ties. Of course not every bad trial will give good re-sults for these quantities, and the question of how onediscriminates the useful bad , from the vast majorityof useless bad s requires much further thought.This isue is related, I believe, to the concept of near-

sightedness which I have recently suggested (Kohn,1996).

In concluding this section I remark that DFT, whilederived from the N-particle Schrodinger equation, is fi-nally expressed entirely in terms of the density n(r), inthe Hohenberg-Kohn formulation, (Hohenberg andKohn, 1964) and in terms of n(r) and single-particlewave functions j(r), in the Kohn-Sham formulation(Kohn and Sham, 1965). This is why it has been mostuseful for systems of very many electrons where wave-function methods encounter and are stopped by the ex-ponential wall.

III. DENSITY-FUNCTIONAL THEORYBACKGROUND

In the fall of 1963 I was spending a sabbatical semes-ter at the Ecole Normale Superieure in the spacious of-fice of Philippe Nozieres. A few weeks after my arrivalPierre Hohenberg, also a visitor from the US joinedforces with me. Ever since my period at the CarnegieInstitute of Technology (19501959) I had been inter-ested in disordered metallic alloys, partly because of theexcellent metallurgy department and partly because ofthe interesting experimental program of Emerson Pugh,in Physics, on substitutional Cu alloys with the adjacentelements in the periodic table, such as CuxZn1x . Thesealloys were viewed in two rather contradictory ways: Asan average periodic crystal with nonintegral atomic num-ber ZxZ1(1x)Z2(Z129, Z230). This modelnicely explained the linear dependence of the electronicspecific heat on x . On the other hand the low-temperature resistance is roughly proportional to x(1x), reflecting the degree of disorder among the twoconstituents. While isolated Cu and Zn atoms are, ofcourse, neutral, in a CuZn alloy there is transfer ofcharge between Cu and Zn unit cells on account of theirchemical differences. The electrostatic interaction en-ergy of these charges is an important part of the totalenergy. Thus in considering the energetics of this systemthere was a natural emphasis on the electron densitydistribution n(r).

Now a very crude theory of electronic energy in termsof the electron density distribution, n(r), the Thomas-Fermi (TF) theory, had existed since the 1920s (Fermi,1927; Thomas, 1927). It was quite useful for describingsome qualitative trends, e.g., for totalenergies of atoms,but for questions of chemistry and materials science,which involve valence electrons, it was of almost no use;for example it did not lead to any chemical binding.However the theory had one feature which interestedme: It considered interacting electrons moving in an ex-ternal potential v(r), and provided a (highly over-simplified) one-to-one implicit relation between v(r)and the density distribution n(r):

n r vef f r3/2,

132

2m2

3/2

, (3.1)

vef f r v r n r rr dr, (3.2)where is the r-independent chemical potential; Eq.

(3.1) is based on the expressionn v 3/2 (3.3)

for the density of a uniform degenerate electron gas in aconstant external potential v; and the second term inEq. (3.2) is just the classically computed electrostatic po-tential times (1), generated by the electron density dis-tribution n(r). Since Eq. (3.1) ignores gradients ofvef f(r) it was clear that the theory would apply best forsystems of slowly varying density.

In subsequent years various refinements (gradient-,exchange-, and correlation corrections) were introduced,




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but the theory did not become significantly more usefulfor applications to the electronic structure of matter. Itwas clear that TF theory was a rough approximation tothe exact solution of the many-electron Schrodingerequation, but since TF theory was expressed in terms ofn(r) and Schrodinger theory in terms of(r1 , . . . ,rN),it was not clear how to establish a strict connection be-tween them.

This raised a general question in my mind: Is a com-plete, exact description of electron structure in terms ofn(r) possible in principle. A key question was whetherthe density n(r) completely characterized the system. Itwas true in TF theory, where n(r), substituted in Eq.(3.1), yields vef f(r) and, by Eq. (3.2), v(r) .In addition, n(r) also yields the total number of elec-trons by integration. Thus the physical system is com-pletely specified by n(r). It was also simple to checkthat the same was true for any one-particle system, aswell as for a weakly perturbed, interacting, uniform elec-tron gas

v r v0v1 r 1 , (3.4)

n r n0n 1 r , (3.5)

for which v1(r) can be explicitly calculated in terms ofn1(r) by means of the wave-number-dependent suscep-tibility of the uniform gas. This suggested the hypothesisthat a knowledge of the ground-state density of n (r) forany electronic system (with or without interactions)uniquely determines the system. This hypothesis becamethe starting point of modern DFT.

IV. THE HOHENBERG-KOHN FORMULATION OF

DENSITY-FUNCTIONAL THEORY

A. The density n(r) as the basic variable

The basic lemma of HK. The ground-state densityn(r) of a bound system of interacting electrons in someexternal potential v(r) determines this potentialuniquely (Hohenberg and Kohn, 1964).Remarks: (1) The term uniquely means here: up to anuninteresting additive constant. (2) In the case of a de-generate ground state, the lemma refers to any ground-state density n(r). (3) This lemma is mathematically rig-orous.

The proof is very simple. We present it for a nonde-generate ground state.

Let n(r) be the nondegenerate ground-state densityof N electrons in the potential v1(r), corresponding tothe ground state 1 , and the energy E1 . Then,

E1 1 ,H1 1

v1 r n r dr1 , TU 1, (4.1)where H1 is the total Hamiltonian corresponding to v1 ,and T and U are the kinetic and interaction energy op-erators. Now assume that there exists a second potential

v2(r), not equal to v1(r)constant, with ground state2 , necessarily e

i1 , which gives rise to the samen(r). Then

E2 v2 r n r dr 2 , TU 2. (4.2)Since 1 is assumed to be nondegenerate, the

Rayleigh-Ritz minimal principle for 1 gives the in-

equalityE1 2 ,H1 2

v1 r n r dr2 , TU 2E2 v1 r v2 r n r dr. (4.3)

Similarly,

E2 1 ,H 1 E1 v2 r v1 r1 n r dr, (4.4)

where we use since the nondegeneracy of2 was notassumed. Adding Eqs. (4.3) and (4.4) leads to the con-tradiction

E1E2E1E2 . (4.5)

We conclude by reductivo ad absurdum that the as-sumption of the existence of a second potential v2(r),which is unequal to v1(r)constant and gives the samen(r), must be wrong. The lemma is thus proved for anondegenerate ground state.

Since n(r) determines both N and v(r) (ignoring anirrelevant additive constant) it gives us the full H and Nfor the electronic system. Hence n(r) determines implic-

itly all properties derivable from H through the solutionof the time-independent or time-dependent Schrodingerequation (even in the presence of additional perturba-tions like electromagnetic fields), such as: the many-body eigenstates (0 )(r1 , . . . ,rN),

(1 )(r1 , . . . ,rN), . . . ,the two-particle Greens function G(r1t1 ,r2t2), the fre-quency dependent electric polarizability (), and so on.We repeat that all this information is implicit in n(r).

Remarks:

(1) The requirement of nondegeneracy can easily belifted (Kohn, 1985).

(2) Of course the lemma remains valid for the specialcase of noninteracting electrons.

(3) Lastly we come to the question whether any well-behaved positive function n(r), which integrates toa positive integer N, is a possible ground-state den-sity corresponding to some v(r). Such a density iscalled v-representable (VR). On the positive side itis easy to verify that, in powers of , any nearlyuniform, real density of the form n(r)non(q)e iq r is VR, and that for a single particleany normalized density n(r) (r) 2 is also VR.On the other hand Levy (1982) and Lieb (1982)have shown by an example which involves degener-ate ground states, that there do exist well-behaved




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densities which are notVR. The topology of the re-gions of v-representability in the abstract space ofall n(r) continues to be studied. But this issue has sofar not appeared as a limitation in practical applica-tions of DFT.

B. The Hohenberg-Kohn variational principle

The most important property of an electronic groundstate is its energy E . By wave-function methods E couldbe calculated either by direct approximate solution ofthe Schrodinger equation HE or from theRayleigh-Ritz minimal principle,

Emin ,H , (4.6)

where is a normalized trial function for the givennumber of electrons N.

The formulation of the minimal principle in terms oftrial densities n(r), rather than trial wave functions was first presented in Hohenberg and Kohn (1964).

Here we shall follow the more succinct derivation due toLevy (1982) and Lieb (1982), called the constrainedsearch method.

Every trial function corresponds to a trial densityn(r) obtained by integrating * over all variables ex-cept the first and multiplying by N. One may carry outthe minimization of Eq. (4.6) in two stages. First fix atrial n(r) and denote by n

the class of trial functionswith this n. We define the constrained energy minimum,with n(r) fixed, as

Ev

n r min n ,Hn

v r n

r drF n

r , (4.7)

where

F n r min n(r) , TU n(r)

. (4.8)

F n(r) requires no explicit knowledge of v(r). It is auniversal functional of the density n(r) (whether the lat-ter is VR or not). In the second step minimize Eq. (4.7)over all n,

Eminn(r)Ev n r

minn(r)

v r n r drF n r

. (4.9)

For a nondegenerate groundstate, the minimum is at-tained when n(r) is the ground-state density; and, forthe case of a degenerate ground state, when n(r) is anyone of the ground-state densities. The Hohenberg-Kohn(HK) minimum principle (4.9) may be considered as theformal exactification of Thomas-Fermi theory.

The formidable problem of finding the minimum of(,H) with respect to the 3N-dimensional trial func-tion has been transformed into the seemingly trivialproblem of finding the minimum of E

vn(r) with re-

spect to the three-dimensional trial function n(r).

Actually the definition (4.8) ofF n(r) leads us rightback to minimization with respect to 3 N-dimensionaltrial wave functions. Nevertheless significant formalprogress has been made: the strict formulation of theproblem of ground-state densities and energies entirelyin terms of the density distribution n(r) and of a well-defined but not explicitly known functional of the den-sity, F n(r) , which represents the sum of kinetic en-ergy and interaction energy (TU), associated with n[see Eq. (4.8)].

One could now easily rederive the Thomas-Fermi(TF) theory by making the approximations

T n r 310 kF2 n r dr, (4.10)

U12

n r n r rr

drdr, (4.11)

where kF(n) is the Fermi wave vector of a uniform elec-tron gas of density n and 310 kF

2 (n) is the mean kineticenergy per electron of such a gas. The expression for Uis the classical (or mean-field) approximation. Variouspreviously known corrections of TF theory for ex-change, correlation and density gradients could also beeasily rederived.

The main remaining error is due to the seriously inad-equate representation of the kinetic energy T by Eq.(4.10) or its gradient-corrected forms. This deficiency islargely remedied by the self-consistent, so-called Kohn-Sham equations, discussed in the following Sec. IV.C.

A second interesting class of systems n(r)n0n1(r), where n1(r)n0 , could also be treated usingthe n0-dependent density-density response functionK( rr ).

C. The self-consistent Kohn-Sham equations

Soon after the publication of the TF theory, Hartree(1928) proposed a set of self-consistent single-particleequations for the approximate description of the elec-tronic structure of atoms. The concept was physicallyvery simple. Every electron was regarded as moving inan effective single-particle potential

vH r Z

r n r

rr

dr, (4.12)

where the first term represents the potential due to anucleus of atomic number Z and the second the poten-tial due to the average electronic density distributionn(r) (the negative charge of the electron has been al-lowed for). Thus each electron obeys the single particleSchrodinger equation

12 2vH r j r jj r , (4.13)where j denotes both spatial as well as spin quantumnumbers. The mean density is given by




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n r j1

N

j r2, (4.14)

where, in the ground state, the sum runs over the Nlowest eigenvalues, to respect the Pauli exclusion prin-ciple. Equations (4.12)(4.14) are called the self-consistent Hartree equations. One may start from a firstapproximation for n(r), (e.g., from TF theory), con-

struct vH(r), solve Eq. (4.13) for the j ; and recalculaten(r) from Eq. (4.14), which should be the same as theinitial n(r). If it is not one iterates appropriately until itis.

In the winter of 1964, I returned from France to SanDiego, where I found my new post-doctoral fellow, LuSham. I knew that the Hartree equations describedatomic ground states much better than TF theory. Thedifference between them lay in the different treatmentsof the kinetic energy T [see Eqs. (4.10) and (4.13)]. I setourselves the task of extracting the Hartree equationsfrom the HK variational principle for the energy, Eqs.(4.9), (4.7), and (4.8), which I knew to be formally exact

and which therefore had to have the Hartree equationsand improvements in them. In fact it promised aHartree-like formulation, whichlike the HK minimalprinciplewould be formally exact.

The Hartree differential Eq. (4.13) had the form ofthe Schrodinger equation for noninteracting electronsmoving in the external potential vef f. Could we learnsomething useful from a DFT for noninteracting elec-trons moving in a given extenal potential v(r)? For sucha system, the HK variational principle takes the form

Ev(r) n v r n r drTs n r (4.15)

E , (4.16)

where assuming that n(r) is VR for noninteractingelectrons

Ts n r kinetic energy of the ground state of

noninteracting electrons with density

distribution n r . (4.17)

The Euler-Lagrange equations, embodying the fact thatthe expression (4.14) is stationary with respect to varia-tions of n(r) which leave the total number of electronsunchanged, is

Ev

n r n r v r

n rTs n r nn dr0, (4.18)

where n(r) is the exact ground-state density for v(r).Here is a Lagrange multiplyer to assure particle con-servation. Now in this soluble, noninteracting case weknow that the ground-state energy and density can beobtained by calculating the eigenfunctions j(r) and ei-genvalues j of noninteracting, single-particle equations

12 2v rj j r 0, (4.19)yielding

Ej1

N

j ; n r j1

N

j r2 (4.20)

(here j labels both orbital quantum numbers and spin

indices, 1).Returning now to the problem ofinteracting electrons,which had previously been addressed approximately bythe single-particle-like Hartree equations, we deliber-ately wrote the functional F n(r) of Eq. (4.8) in theform

F n r Ts n r 12

n r n r rr

drdr

Exc n r , (4.21)

where Ts n(r) is the kinetic energy functional for non-interacting electrons, Eq. (4.15). The last term,

Exc n(r) , the so-called exchange-correlation energy

functional is then defined by Eq. (4.21). The HK varia-tional principle for interacting electrons now takes theform

Ev

n r v r n r drTs n r

12

n r n r rr

drdrExc n r E.

(4.22)

The corresponding Euler-Lagrange equations, for agiven total number of electrons has the form

Ev

n r n r vef f r

n rTs n r n(r)n(r) dr0,

(4.23)

where

vef f r v r n r rr drvxc r (4.24)and

vxc r

n rExc n r n(r)n(r) . (4.25)

Now the form of Eq. (4.23) is identical to that of Eq.(4.18) for noninteracting particles moving in an effectiveexternal potential vef f instead of v(r), and so we con-clude that the minimizing density n(r) is given by solv-ing the single-particle equation

12 2vef f r j j r 0, (4.26)with




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n r j1

N

j r2, (4.27)

vef f r v r n r rr drvxc r , (4.28)where vxc (r) is the localexchange-correlation potential,depending functionally on the entire density distribution

n

(r) as given by Eq. (4.25). These self-consistent equa-tions are now called the Kohn-Sham (KS) equations.The ground-state energy is given by

Ej

jExc n r vxc r n r dv

12

n r n r rr

. (4.29)

If one neglects Exc and vxc altogether, the KS Eqs.(4.26)(4.29) reduce to the self-consistent Hartree equa-tions.

The KS theory may be regarded as the formal exacti-

fication of Hartree theory. With the exact Exc and vxc allmany-body effects are in principle included. Clearly thisdirects attention to the functional Exc n(r) . The prac-tical usefulness of ground-state DFT depends entirely onwhether approximations for the functional Exc n(r)could be found which are at the same time sufficientlysimple and sufficiently accurate. The next section brieflydescribes the development and current status of suchapproximations.

Remarks:

(1) The exact effective single-particle potential vef f(r)of KS theory, Eq. (4.28) can be regarded as thatunique, fictitious external potential which leads, for

noninteracting particles, to the same physical den-sity n(r) as that of the interacting electrons in thephysical external potential v(r). Thus if the physicaldensity n(r) is independently known (from experi-ment orfor small systemsfrom accurate, wave-function-based calculations) vef f(r) and hence alsovxc (r) can be directly obtained from the densityn(r) (Wang and Parr, 1993).

(2) Because of their linkage to the exact physical den-sity n(r), the KS single particle wave functionsj(r) may be considered as density optimal,while, of course, the Hartree-Fock HF wave func-tions j

HF(r) are total-energy optimal in the sense

that their normalized determinant leads to the low-est ground-state energy attainable with a single de-terminant. Since the advent of DFT the term ex-change energy is often used for the exchangeenergy computed with the exact KS j , and not withthe HF j

HF (for the uniform electron gas the twodefinitions agree; typically the differences aresmall).

(3) Neither the exact KS wave functions j nor energiesj have any known, directly observable meaning, ex-cept for (a) the connection (4.27) between the jand the true, physical density n(r) and (b) the fact

that the magnitude of the highest occupied j , rela-tive to the vacum equals the ionization energy(Almbladh and Barth, 1985).

In concluding this section we remark that most practicalapplications of DFT use the KS equations, rather thanthe generally less accurate HK formulation.

V. APPROXIMATION FOR Exc n(r) : FROMMATHEMATICS TO PHYSICAL SCIENCE

So far DFT has been presented as a formal math-ematical framework for viewing electronic structurefrom the perspective of the electron density n(r). Thismathematical framework has been motivated by physi-cal considerations, but to make concrete use of it werequire effective approximations for F n(r) in the HKformulation, and for Exc n(r) in the KS formulation.These approximations reflect the physics of electronic

structure and come from outside of DFT. In this accountI limit myself to the much more extensively used func-

tional Exc .The most important approximations for Exc n(r)have a quasilocal form. As will be discussed in Sec. V.B,Exc n(r) can be written in the form

Exc n r exc r; n r n r dr, (5.1)where exc (r; n(r) ) represents an exchange-correlation(xc) energy/particle at the point r, which is a functionalof the density distribution n(r). It depends primarily onthe density n(r) at points r near r, where near is amicroscopic distance such as the local Fermi wavelengthF(r) 3

2n(r) 1/3 or TF screening length, typicallyof similar magnitude. The general form of Eq. (5.1), rep-resenting the total Exc as an integral over all space of asuitable integrand, is similar to the treatment of kineticenergy in Thomas-Fermi theory, Eq. (4.10). All compo-nents of the KS energy, except the long-range classicalCoulomb interaction, can be expressed in terms of theone- and two-particle density matrices of the interactingand noninteracting system G1(r1 ;r1), G2(r1 ,r2 ;r1 ,r2)and G1

0(r1 ;r1), G20(r1r2 ;r1r2), all corresponding to and

uniquely defined by the same physical n(r); their calcu-lation involves these Greens functions primarily for ar-guments, such as (r1 ,r1) and (r1 ,r2 ;r1r2), which are mi-croscopically close to one another; furthermore, forgiven r

1, these Greens functions depend only on the

form ofn(r

) for r

near r1the property of nearsight-edness previously mentioned (Kohn, 1996). This leadsimmediately to the form (5.1) for Exc n(r) , where excis a nearsighted functional of n(r).

We now briefly discuss several implementations ofthis quasilocal approach.

A. The local-density approximation (LDA)

The simplest, and at the same time remarkably ser-viceable, approximation for Exc n(r) is the so-calledlocal-density approximation (LDA),




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ExcLD A excn r n r dr, (5.2)

where exc (n) is the exchange-correlation energy perparticle of a uniform electron gas of density n (Kohnand Sham, 1965). The exchange part is elementary andgiven, in atomic units, by

ex n 0.458

rs, (5.3)

where rs , is the radius of a sphere containing one elec-tron and given by (4/3)rs

3n1. The correlation part

was first estimated by E. P. Wigner (1938):

c n 0.44

rs7.8, (5.4)

and more recently with a high precision of about 1%by D. M. Ceperley (1978); Ceperley and Alder, (1980)using Monte Carlo methods.

Remarks:

(1) The LDA, obviously exact for a uniform electron

gas, was a priori expected to be useful only for den-sities varying slowly on the scales of the local Fermiwavelength F and TF wavelength, TF . In atomicsystems these conditions are rarely well satisfied andvery often seriously violated. Nevertheless the LDAhas been found to give extremely useful results formost applications. This has been at least partly ra-tionalized by the observation that the LDA satisfiesa sum rule which expresses the normalization of theexchange-correlation hole. In other words, giventhat an electron is at r, the conditional electron den-sity n(r;r) of the other electrons is depleted near rin comparison with the average density n(r) by the

hole distribution nh(r

;r) which integrates to 1.(2) The solution of the KS equation in the LDA is mini-mally more difficult than the solution of the Hartreeequation and very much easier than the solution ofthe HF equations. Its accuracy for the exchange en-ergy is typically within O (10%), while the normallymuch smaller correlation energy is generally overes-timated by up to a factor of 2. The two errors typi-cally cancel partially.

(3) Experience has shown that the LDA gives ioniza-tion energies of atoms, dissociation energies of mol-ecules and cohesive energies with a fair accuracy oftypically 1020%. However the LDA gives bondlengths and thus the geometries of molecules andsolids typically with an astonishing accuracy of1%.

(4) The LDA (and the LSDA, its extension to systemwith unpaired spins) can fail in systems, like heavyfermion systems, so dominated by electron-electroninteraction effects that they lack any resemblance tononinteracting electron gases.

B. Beyond the local-density approximation

The LDA is the mother of almost all approxima-tions currently in use in DFT. To discuss more accurate

approximations we now introduce the concept of the av-erage xc hole distribution around a given point r. The

physical xc hole is given by

nxc r,r g r,r n r , (5.5)

where g(r,r) is the conditional density at r given thatone electron is at r. It describes the hole dug into theaverage density n(r) by the electron at r. This hole isnormalized

nxc r,r dr1, (5.6)which reflects a total screening of the electron at r,and is localized due to the combined effect of the Pauliprinciple and the electron-electron interaction. Ofcourse, like everything else, it is a functional of the den-sity distribution n(r). To define the average xc hole oneintroduces a fictitious Hamiltonian H for the manybody system, 01, which differs from the physicalHamiltonian, H1 , by the two replacements

e2

rirj

e2

rirj, (5.7)

v r v r , (5.8)

where the fictitious v(r) is so chosen that for all inthe interval (0, 1) the corresponding density equals thephysical density, n(r):

n r n1 r n r . (5.9)

The procedure (5.2), (5.3) represents an interpolationbetween the KS system (0) and the physical system(1) in which n1(r)n(r). The average xc hole den-sity n(r,r) is then defined as

nxc r,r

01

dnxc r,r; . (5.10)

Its significance stems from the exact result, proved in-dependently in three important publications (Harris andJones, 1974; Langreth and Perdew, 1975; Gunnarson andLundquist, 1976), that

Exc12 drdr

n r nxc r,r rr

. (5.11)

An equivalent expression is (Kohn and Mattsson,1998)

Exc1

2 drn r Rxc

1r, n r , (5.12)

where

Rxc1r,n r drnxcr,r n r rr , (5.13)

is the moment of degree (1) ofnxc (r,r), i.e., minusthe inverse radius of the -averaged xc hole. Compari-son of Eqs. (5.12) and (5.1) gives the very physical, for-mally exact relation

excr; n r 12

Rxc1r; n r . (5.14)




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Gradient expansion and generalized gradient approxima-tion.

Since Rxc1(r) is a functional of n(r), expected to be

(predominantly) shortsighted, we can formally expandn(r) around the point r which we take to be the origin:

n rnn iri12

n ij rirj . . . , (5.15)

where nn(0), n i in(r) r0 , etc., and then considerRxc (r) as a function of the coefficients n ,n i ,n ij , . . . . Or-dering in powers of the differential operators and re-specting the scalar nature of Rxc

1 gives

Rxc1 r F0n r F21n r

2n r F22n r

in r in r . . . . (5.16)

When this is substituted into Eq. (5.12) for Exc it leads(after an integration by parts) to the gradient expansion

ExcExcLD A

G 2 n n

2dr

G 4 n 2n 2 dr . . . , (5.17)where G2(n) is a universal functonal of n (Kohn andSham, 1965). In application to real systems this expan-sion has generally been disappointing, indeed has oftenworsened the results of the LDA.

The series (5.15) can however be formally resummedto result in the following sequence:

Exc0 n r n r dr LDA , (5.18)

Exc(1 ) f

(1 )n r , n r n r dr GGA , (5.19)

Exc(2 ) f(2 )n r , n r 2n r . (5.20)

Exc0 is the (LDA), requiring the independently calcu-

lated function of one variable, xn . Exc(1 ) , the so-called

generalized gradient approximation (GGA) requires theindependently calculated function of two variables, xn, y n , etc.

Thanks to much thoughtful work, important progresshas been made in deriving successful GGAs of the form(5.19). Their contruction has made use of sum rules,

general scaling properties, asymptotic behavior of effec-tive potentials, and densities in the tail regions of atomsand their aggregates. In addition, A. Becke in his workon GGAs introduced some numerical fitting parameterswhich he determined by optimizing the accuracy of at-omization energies of standard sets of molecules. Thissubject was recently reviewed (Perdew and Kurth,1998). We mention here some of the leading contribu-tors: A. D. Becke, D. C. Langreth, M. Levy, R. G. Parr,J. P. Perdew, C. Lee, and W. Yang.

In another approach A. Becke introduced a successfulhybrid method:

Exchy bEx

KS 1 Exc

GG A , (5.21)

where ExKS is the exchange energy calculated with the

exact KS wave functions, ExcGG A is an appropriate GGA,

and is a fitting parameter (Becke, 1996). The form ofthis linear interpolation can be rationalized by the-integration in Eq. (5.10), with the lower limit corre-sponding to pure exchange.

Use of GGAs and hybrid approximations instead of

the LDA has reduced errors of atomization energies ofstandard sets of small molecules, consisting of light at-oms, by factors of typically 35. The remaining errorsare typically (2 3) kg moles per atom, about twice ashigh as for the best current wave-function methods. Thisimproved accuracy, together with the previously empha-sized capability of DFT to deal with systems of verymany atoms, has, over a period of relatively few yearsbeginning about 1990, made DFT a significant compo-nent of quantum chemistry.

For other kinds of improvements of the LDA, includ-ing the weighted density approximation (WDA) andself-interaction corrections (SIC), we refer the reader to

the literature, e.g., Perdew and Kurth (1998).Before closing this section I remark that the treat-

ments of xc effects in the LDA and all of its improve-ments, mentioned above, is completely inappropriate forall those systems or subsystems for which the startingpoint of an electron gas of slowly varying density n(r) isfundamentally incorrect. Examples are (a) the electronicWigner crystal; (b) Van der Waals (or polarization) en-ergies between nonoverlapping subsystems; (c) the elec-tronic tails evanescing into the vacuum near the surfacesof bounded electronic systems. However, this does notpreclude that DFT with appropriate, different approxi-mations can successfully deal with such problems (see

Sec. VIII).

VI. GENERALIZATIONS AND QUANTITATIVE

APPLICATIONS

While DFT for nondegenerate, nonmagnetic systemshas continued to progress over the last several decades,the DFT paradigm was also greatly extended and gener-alized in several directions. The purpose of this sectionis to give the briefest mention of these developments.For further details we refer to two monographs (Parrand Yang, 1989; Dreizler and Gross, 1990) and a recentset of lecture notes (Vosko, Perdew, and MacDonald,1975).

A. Generalizations

a. Spin DFT for spin polarized systems: v(r), Bz(r);n(r) , n(r)n(r) .

b. Degenerate ground states: v(r); ,n(r);1, . . . M; E0 .

c. Multicomponent systems (electron hole droplets,nuclei): v(r); n(r); E0 .

d. Ensemble DFT for M degenerate ground states:v(r); n(r)(M1Tr n(r); E0 .




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e. Free energy at finite temperatures T: v(r); n(r),(grand potential).

f. Superconductors with electronic pairing mecha-nisms: v(r), (r) (gap function); nnorm(r), nsu pe r(r),E0 .

g. M excited states, equiensembles v(r), n(r)M1 1

Mnm(r), EM1

1MEm .

h. Relativistic electrons.i. Current-density functional theory, diamagnetism:

v(r), curlA(r); n(r), curlj(r); E0 .j. Time-dependent phenomena: v(r, t); n(r,t), and

excited states v(r)eit; n(r)eit; EjE i.k. Bosons (instead of fermions) v(r); n(r); E0 .l. Combination of DFT with molecular-dynamics or

Monte Carlo methods (especially for determinations ofstructures).

This incomplete list is only intended to give a generalsense of the great diversity of contexts in which the basicconcept of DFT has been found useful.

B. Applications

To do any kind of justice to the many thousands ofapplications of DFT to physical and chemical systems isentirely impossible within the framework of this lecture.So I will, quite arbitrarily, choose one example, the spinsusceptibility of the alkali metals (Table I; Vosko, Per-dew, and MacDonald, 1975).

This is an early, completely parameter-free calcula-tion. It uses only the independently calculated externalpseudopotential v(r) and the exchange-correlation en-ergy of a spatially uniform, magnetized electron gas (theso-called local spin density approximation, LSDA). Theonly input specific to each metal is the atomic numberZ. Note how accurately the theoretical results agree

with the rather irregular sequence of experimental data.The deviations of the ratio (/0) from 1, are due, incomparable degree, to the combination of the effects ofthe nonuniform, periodic potentials and the electron-electron interactions.

Of course these metals have, over most of space, fairlyuniform densities, which makes them favorable test-cases for local spin density calculations. For other classesof systems and their properties the accuracies can beconsiderably poorer, with the exception of the alreadymentioned very accurate results for structures, typicallya 1% error (which is still somewhat astonishing to me).

Inclusion of gradient corrections and/or hybridschemes have improved calculated energies for largeclasses of chemical applications by typically almost anorder of magnitude; in physical applications the im-provement is usually less dramatic. Accuracies of geo-metric parameters remain at the 1% level.

VII. CONCLUDING REMARKS

DFT has now been widely accepted by both physicistsand chemists. For periodic solids it is sometimes referredto as the standard model. In chemistry DFT comple-ments traditional wave-function-based methods particu-larly for systems with very many atoms (10).

In both these fields DFT has been very useful for sur-veying large classes of systems and often has yieldedimportant practical results. In cases where DFT cur-rently works still rather poorly (e.g., long-range polar-ization energies; regions of evanescent electron densi-ties; partially filled electronic shells; reaction barriers) it

often provides clues of how our present understandingof electronic structure in real space coordinates needs tobe modified.

Looking into the future I expect that wave-function-based and density-based theories, will in complementaryways, continue not only to give us quantitatively moreaccurate results, but also contribute to a better physical/chemical understanding of the electronic structure ofmatter.

REFERENCES

Almbladh, C. O., and U. von Barth, 1985, Phys. Rev. B 31,3231.Becke, A. D., 1996, J. Chem. Phys. 104, 1040.Blake, N. P., and H. Metiu, submitted for publication.Ceperley, D. M., 1978, Phys. Rev. B 18, 3126.Ceperly, D. M., and B. J. Alder, 1980, Phys. Rev. Lett. 45, 566.Dreizler, R. M., and E. K. V. Gross, 1990, Density Functional

Theory (Springer, Berlin).Fermi, E., 1927, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat.

Rend. 6, 602.Fock, F., 1930, Z. Phys. 61, 126.Gunnarsson, O., and B. I. Lundquist, 1976, Phys. Rev. B 13,

4274.Harris, J., and R. O. Jones, 1974, J. Phys. F 4, 1170.

Hartree, D. R., 1928, Proc. Cambridge Philos. Soc. 24, 89.Haas, K. C., W. F. Schneider, A. Curioni, and W. Andreoni,1998, Science 282, 265.

Heitler, W., and F. London, 1927, Z. Phys. 44, 455.Hohenberg, P., and W. Kohn, 1964, Phys. Rev. 136, B864.James, H. M., and A. S. Coolidge, 1933, J. Chem. Phys. 1, 825.Kohn, W., 1985, in Proceedings of the International School of

Physics, Enrico Fermi, Course LXXXIX, p. 4.Kohn, W., 1996, Phys. Rev. Lett. 76, 3168.Kohn, W., and A. E. Mattsson, 1998, Phys. Rev. Lett. 81, 16.Kohn, W., and L. J. Sham, 1965, Phys. Rev. 140, A1133.Langreth, D. C., and J. P. Perdew, 1975, Solid State Commun.17, 1425.

TABLE I. Spin susceptibility of the alkali metals.

Metal

/0a

Variational theory Experiment

Li 2.66 2.57Na 1.62 1.65K 1.79 1.70Rb 1.78 1.72

Cs 2.20 2.24

a0 is the Pauli susceptibility of a free-electron gas.




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Levy, M., 1982, Phys. Rev. A 26, 1200.Lieb, E., 1982, in Physics as Natural Philosophy: Essays in

Honor of Laszlo Tisza on His 75th Birthday, edited by A.Shimony and H. Feshbach (MIT, Cambridge, MA) p. 111.

Mullikan, R. S., 1928, Phys. Rev. 32, 186.Nusterer, E. P., P. Bloechl, and Schwarz, K., 1996, Angew.

Chem. 35, 175.Parr R. G., and W. Yang, 1989, Density Functional Theory ofAtoms and Molecules (Oxford University Press, Oxford).

Perdew J. P., and S. Kurth, 1998, in Density Functionals:

Theory and Applications, edited by D. Joubert (Springer,Berlin), p. 8.

Schroedinger, E., 1926, Ann. Phys. (Leipzig) 79, 361.Thomas, L. H., 1927, Proc. Cambridge Philos. Soc. 23, 542.Van Vleck, J. H., 1936, Phys. Rev. 49, 232.Vosko, S. H., J. P. Perdew, and A. H. MacDonald, 1975, Phys.

Rev. Lett. 35, 1725.Wang, Y., and R. Parr, 1993, Phys. Rev. A 47, 1591.

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kohn nobel

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