Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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296 Part B Atoms

cal intuition and the mathematically rigorous results

obtained in the 1970’s and 1980’s. The older liter-

ature was reviewed by Gombás [20.18, 19] and by

March [20.20].

An outgrowth of Thomas–Fermi theory is the gen-

eral density functional theory initiated by Hohenberg

and Kohn [20.21] and by Kohn and Sham [20.22], as

discussed in Sect. 20.3 of this chapter.

20.1 Thomas–Fermi Theory

20.1.1 Thomas–Fermi Theory

In a D-dimensional Euclidean space, the expectation

value of the electronic kinetic energy operator in a quan-

tum state ψ can be approximated by

2π

D+ 2



2Ω



2/D



(D+2)/D

(r) d

r ,

(20.1)

where

Ω

= Dπ

D/2

/Γ (1 + D/2) (20.2)

is the surface area of a unit hypersphere in D dimen-

sions [20.17, p. 176]. These expressions can easily be

derived by considering the energy levels of a system

of a large number of noninteracting fermions conﬁned

to a D-dimensional box. Specialization to the physic-

ally interesting case of D = 3 yields the well-known

expression

2π



2Ω



2/3



5/3

(r) d r , (20.3)

where

Ω

= 3π

3/2

/Γ (1 + 3/2) = 4π. (20.4)

The electron–nucleus attraction energy in a three-

dimensional space is given exactly by



ρ(r)V(r) d r ,

(20.5)

where V(r) is the Coulomb potential due to

a single nucleus (V(r) =−Z/r)ortoseveralnu-

clei [V(r) =−



/|r − R

|]. The electron–electron

Coulomb repulsion energy in a three-dimensional space

is approximated by



ρ(r)ρ(r



)

|r− r



d r d r



, (20.6)

which tends to overestimate the actual repulsion energy

because it includes the self-energy of the densities of

individual electrons. This is, however, a higher-order

effect for a system with a large number of electrons con-

centrated in a small region of space. As was suggested

by Fermi and Amaldi [20.23], this overestimation can be

approximately corrected for an atom with N electrons

by multiplying this term by the ratio of the number of

ordered pairs of different electrons to the total number

of ordered pairs

N(N − 1)

= 1 −

(20.7)

This is approximately correct for an atom, with many

electrons concentrated close together, but it would still

be an overestimate for a diffuse system, such as one

composed of N electrons and N protons separated by

large distances of O(R), for which the ground-state

electron–electron repulsion term should be proportional

N(N − 1)/R rather than to N times a constant of

O(1). For this reason the Fermi–Amaldi correction,

which complicates the mathematical analysis with-

out eliminating the unphysical overestimation of the

electron–electron repulsion term, is not usually in-

cluded. It is evident that the treatment of both the

electronic kinetic energy term and the electron–electron

repulsion energy term depends on the assumption that

the number N of electrons (actually, the number of elec-

trons peratom) is large. Hence the Thomas–Fermi model

is sometimes called the statistical model of an atom.

The three contributions to the total energy are now

added together and one seeks to minimize their sum, the

Lenz functional [20.6]

E[ρ]=

2π



2Ω



2/3



5/3

(r) d r



ρ(r)V(r) d r +



ρ(r)ρ(r



)

|r− r



d r d r



(20.8)

over all admissible densities ρ. The mathematical ques-

tion now arises: what is an admissible density? The

answer was provided by Lieb and Simon [20.9, 10]:

a density for which both



ρ(r) d r ,

(20.9)

Part B 20.1

Thomas–Fermi and Other Density-Functional Theories 20.1 Thomas–Fermi Theory 297

the total number of electrons, and



5/3

(r) d r , (20.10)

which is proportional to the estimate of their kinetic

energy, are ﬁnite, automatically yields ﬁnite values of

the other terms in the expression for the energy. As Lieb

and Simon proved, the minimization of this functional

over all such densities yields a well-determined result.

Carrying out the variation of E[ρ] with respect to ρ

yields the Euler–Lagrange equation

0 =

2π



2Ω



2/3

(r)

+ V(r) +



ρ(r



)

|r− r



d r



. (20.11)

The sum of the last two terms is of course the negative

of the total electrostatic potential φ(r), so one sees that

in Thomas–Fermi theory the density is proportional to

the 3/2-power of the potential. To simplify subsequent

manipulations, let

2π



2Ω



2/3



3π



2/3

= γ

(20.12)

so that

2/3

(r) = φ(r). (20.13)

By Poisson’s theorem,

−∇

φ = 4π





(3)

(r− R

) − ρ(r)



(20.14)

and from (20.13) one has ρ = γ

−3/2

3/2

, so from the

integral equation for the electronic density ρ one obtains

the differential equation

−∇

φ = 4π





(3)

(r− R

) − γ

−3/2

3/2



(20.15)

for the potential φ. In the case of a single nucleus, the

usual separation of variables in spherical polar coordi-

nates yields for φ the ordinary differential equation

(rφ) = 4πγ

−3/2

3/2

, (20.16)

whose similarity to Emden’s equation, which Eddington

had used to study the internal constitution of stars, was

recognized by Milne [20.24]. The numerical solution

of this equation with the appropriate boundary condi-

tions at r = 0andr =∞was extensively discussed by

Baker [20.25], and accurate solutions tabulated by Tal

and Levy [20.26]. The numerical solution determines

that the total energy of a neutral atom is

E =−3.678 745 21 ... γ

−1

7/3

=−1.537 490 24 ... Z

7/3

Ry . (20.17)

Another possibility is to do the constrained miniza-

tion over all densities which obey



ρ(r) d r = λ,

(20.18)

where λ is the number of electrons, which for purposes of

mathematical analysis is allowed to be nonintegral. One

then introduces a Lagrange multiplier −µ,thechemical

potential, to correspond with this constraint, and thereby

obtains the Euler–Lagrange equation

0 =

2π



2Ω



2/3

(r)

+ V(r) +



ρ(r



)

|r− r



d r



+ µ, (20.19)

which holds wherever ρ is positive. As was shown

by Lieb and Simon [20.9, 10], this procedure too is

well-deﬁned. The analogue of (20.13), the relationship

between the density and the electrostatic potential for

the neutral atom, is now

2/3

(r) =[φ(r) − µ]

, (20.20)

where [ f ]

= max( f, 0). The corresponding differen-

tial equation for the potential φ is

−∇

φ = 4π





(3)

(r− R

)

−γ

−3/2

[φ(r) − µ]

3/2



(20.21)

Lieb and Simon rigorously proved a large number

of results concerning the solution of the Thomas–Fermi

model. When V(r) is a sum of Coulomb potentials aris-

ing from a set of nuclei of positive charges Z

, with



= Z, then the energy E(λ) is a continuous, mono-

tonically decreasing function of λ for 0 ≤ λ ≤ Z,and

its derivative dE/dλ is the chemical potential −µ(λ),

which vanishes at λ = Z.Forλ in this range, there is

a unique minimizing density ρ, whereas for λ>Z there

Part B 20.1

298 Part B Atoms

is no unique minimizing ρ, since one can place arbitrar-

ily large clumps of charge with arbitrarily low energy

arbitrarily far away from the nuclei. In the atomic case,

with a single nucleus, ρ(r) is a spherically symmetric

monotonically decreasing function of r.

Moreover, for an atom or atomic ion, the Thomas–

Fermi density obeys the virial theorem

2T =−V=−2E ,

(20.22)

and for a neutral atom the electronic kinetic en-

ergy, electron–nucleus attraction energy, and electron–

electron repulsion energy terms in the expression for the

total energy satisfy the ratios 3 :−7 : 1.

It is straightforward to examine the behavior of the

Thomas–Fermi density ρ in the limit as either r → 0

or r →∞.Forlarger, the electron density vanishes

identically outside a sphere of ﬁnite radius for a positive

ion. For a neutral atom, the ordinary differential equation

(20.16) for the potential φ can be analyzed to show that

φ(r)  γ

(3γ

/π)

−4

, (20.23)

from which it follows that

ρ(r)  (3γ

/π)

−6

, (20.24)

independent of Z.ThisimpliesthatasZ →∞, a neutral

atom described by the Thomas–Fermi model has a ﬁnite

size deﬁned in terms of a radius within which all but

a ﬁxed amount of electronic probability density is lo-

cated. For example, if one deﬁnes the size of an atom as

that value of r

for which



|r|≥r

ρ(r) d r =

(20.25)

one ﬁnds that in the large-Z limit



8π



1/3

3γ

(20.26)

In atomic units, γ

(3π

)

2/3

,and

= (9π)

2/3

 9.3 a

, (20.27)

which is about what one would expect for a ‘real’ non-

relativistic atom with a large nuclear charge Z.Onthe

other hand, the characteristic distance scale in Thomas–

Fermi theory, deﬁned as the ‘average’ value of r,orin

terms of a radius within which a ﬁxed fraction of elec-

tronic probability density is located, is proportional to

−1/3

, which shrinks to 0 as Z →∞. The resolution

of this paradox is that outside the typical ‘core’ scale of

distance set by Z

−1/3

, within which most of the electron

density is located, there resides in the ‘mantle’ region

a fraction of electrons proportional to Z

2/3

/Z = Z

−1/3

and almost all of these are concentrated within a sphere

of radius of about 10 a

Moving deeper into the core and approaching the

nucleus, the −Z/r singularity in the electron–nucleus

Coulomb potential dominates the smeared-out electron–

electron potential, so one readily ﬁnds that

ρ(r) 





3/2

. (20.28)

This singularity is integrable but unphysical, since it

arises from the approximation of the local kinetic en-

ergy by ρ

5/3

, which breaks down where ρ is rapidly

varying on a length scale proportional to 1/Z. In a ‘real’

nonrelativistic heavy atom governed by the Schrödinger

equation, the actual electron density at the nucleus is

ﬁnite, being proportional to Z

. This unphysical singu-

larity in the electron density in Thomas–Fermi theory

can be eliminated by adding a gradient correction to the

Thomas–Fermi kinetic energy term.

20.1.2 Thomas–Fermi–von Weizsäcker

Theory

The semiclassical approximation (20.3) for the quantum

kinetic energy in terms of a power of the density is

capable of improvement, particularly in regions of space

where the density is rapidly varying. The incorporation

of such corrections leads to a gradient expansion for the

kinetic energy [20.27]. The leading correction is of the

form







∇ρ

1/2



(r)



d r . (20.29)

Addition of such a term to the Thomas–Fermi expression

for the kinetic energy yields a theory which avoids many

of the unphysical features of ordinary Thomas–Fermi

theory at very short and moderately large distances. The

more important points, as rigorously proved in Lieb’s

review article [20.15, 16], are as follows. The lead-

ing features of the energy are unchanged; for large Z

the energy E(Z) of a neutral atom or atomic ion is

still proportional to Z

7/3

, but now there enter higher-

order corrections arising from the gradient terms of

order Z

7/3

−1/3

= Z

and higher powers of Z

−1/3

.The

maximum number of electrons which can be bound by

an atom of nuclear charge Z is no longer exactly Z,

but a slightly larger number; thus Thomas–Fermi–von

Weizsäcker theory allows for the formation of nega-

tively charged atomic ions. It was further proven by

Part B 20.1

Thomas–Fermi and Other Density-Functional Theories 20.1 Thomas–Fermi Theory 299

Benguria and Lieb [20.28] that in the Thomas–Fermi–

von Weizsäcker model a neutral atom can bind at most

one extra electron, and that a neutral molecule can bind

at most as many extra electrons as it has nuclei.

The effect on the electronic density ρ is more pro-

found. While the general shape and properties of ρ in the

‘core’ and ‘mantle’ regions is unchanged, the fact that

∇ρ

1/2

need not, and in general does not, vanish when

ρ vanishes on some surface implies that for a positive ion

ρ no longer vanishes outside of a sphere, as it does in the

case of Thomas–Fermi theory, but instead extends over

all space. For positive ions, neutral atoms, and negative

ions alike, ‘differential inequality’ techniques [20.29]

can be used to show that ρ(r) decays exponentially for

large r, with the constant in the exponential proportional

to µ

1/2

(λ).Forsmallr, the gradient terms dominate the

energy expression, so one ﬁnds that the electronic dens-

ity no longer diverges as r → 0, but instead tends to

a ﬁnite limit, with a ﬁrst derivative which obeys a rela-

tion analogous to the Kato cusp condition [20.30](see

Sect. 11.1.1).

The study of molecules within the Thomas–Fermi–

von Weizsäcker model involves several subtleties and

pitfalls, which can lead to physical absurdities. Since

two neutral atoms with different nuclear charges will in

general have different chemical potentials, a pair of such

atoms placed a long distance apart will spontaneously

ionize, with a small amount of electric charge being

transferred from one to the other until the chemical po-

tentials of the positively charged ion and the negatively

charged ion become equal. The result is a long-range

Coulomb attraction between them [20.31]. This phe-

nomenon does not occur in the real world, since the

amount of electric charge which can be transferred is

quantized in units of −e, and it is empirically true

that the smallest atomic ionization potential exceeds the

largest atomic electron afﬁnity. For two neutral atoms

with the same nuclear charges, the situation is more

subtle. Nonetheless, a careful analysis shows that in this

case too, though no spontaneous ionization occurs, there

is a long-range attractive interaction between them aris-

ing from the overlap of the exponentially small tails of

the electron clouds. Since electron correlation is not in-

cluded in this model, it could not be expected to describe

attractive van der Waals forces.

In summary, the Thomas–Fermi–von Weizsäcker

model yields a more realistic picture of a single atom

than does the Thomas–Fermi model. However, it does

not provide a useful picture for understanding the inter-

action between atoms at large distances. These kinds of

unphysical features provide a glimpse into the compli-

cated nature of the universal density functional, which

must include terms which rigorously suppress an un-

physical feature like spontaneous ionization of a distant

pair of heteronuclear atoms [20.32, 33]. It is evident

from the mathematical properties of Thomas–Fermi–

von Weizsäcker theory and related models that a density

functional which ‘ﬁxes up’ the Thomas–Fermi expres-

sion simply by adding a few gradient terms and/or simple

exchange terms and the like must still differ in important

ways from the universal density functional, particularly

for properties of extended systems.

20.1.3 Thomas–Fermi–Dirac Theory

The effect of the exchange of electrons can be ap-

proximated, following Dirac [20.7], by including in the

Thomas–Fermi energy functional an expression of the

form

−

4π



3π



4/3



4/3

(r) d r . (20.30)

Minimization of the resulting Thomas–Fermi–Dirac en-

ergy functional over all admissible densities ρ whose

integral is λ yields a well-deﬁned E(λ), which has the

correct behavior for λ ≤ Z, and it has been shown that

for an atom the exchange correction to the energy is of

order Z

5/3

. However, this model exhibits unphysical be-

havior for λ>Z, because one can obtain a completely

artiﬁcial lowering of the energy by placing many small

clumps of electrons a large distance from the nucleus, for

which the negative



4/3

d r term dominates the energy

expression [20.15, 16]. At the conclusion of his original

article, Dirac clearly stated that the correction he had

derived, although giving a better approximation in the

interior of an atom, gives “a meaningless result for the

outside of the atom” [20.7]. It is therefore clear that any

physically reasonable theory must somehow profoundly

modify this correction in the region where the electronic

density is very small.

20.1.4 Thomas–Fermi–von

Weizsäcker–Dirac Theory

One can also include the Dirac exchange correction in

the Thomas–Fermi–von Weizsäcker energy functional.

In this case, however, the mathematical foundations of

the theory are still incomplete ([20.15, 16, pp. 638–9]).

Nonetheless, it is clear that this theory too suffers from

the unphysical lowering of the energy by small clumps

of electrons at large distances from the nucleus.

In summary, one can say that the inclusion of Dirac’s

exchange correction in its most straightforward form

Part B 20.1

300 Part B Atoms

leads to an improvement of energies for positive ions or

neutral atoms, but to unphysical behavior for systems

where the charge of the electrons exceeds the nuclear

charge, in line with Dirac’s own observations on the

limitations of his correction [20.7].Weseehereagain

a manifestation of how complicated must be the behavior

of the true universal density functional.

20.1.5 Thomas–Fermi Theory

with Different Spin Densities

As was remarked by Lieb and Simon [20.10], it is possi-

ble to consider a variant of Thomas–Fermi theory with

a pair of spin densities ρ

and ρ

for the spin-up and

spin-down electrons, with the two adding together to

produce the total electronic density ρ. This theory has

been rigorously formulated and analyzed by Goldstein

and Rieder [20.34]. Because the problem is nonlinear,

the mathematical complications are substantial, and the

theory is not a trivial extension of ordinary Thomas–

Fermi theory. Goldstein and Rieder ﬁrst considered the

case where the total number of electrons of each type of

spin is speciﬁed in advance [20.35]. There is no math-

ematical obstacle to constructing such a spin-polarized

Thomas–Fermi theory, but it does not yield the kind of

spontaneous spin-polarization that one observes in the

ground states of many real quantum mechanical atoms

and molecules, which is not surprising in view of the fact

that such spin-polarization in accord with Hund’s ﬁrst

rule arises from exchange and correlation effects not in-

cluded in this simple functional. However, in the case

where the electronic spins (but not their currents) are

coupled to an external magnetic ﬁeld, the ground state

is naturally spin-polarized [20.34].

20.2 Nonrelativistic Energies of Heavy Atoms

Thomas–Fermi theory suggests that (20.17) provides

the leading term in a power series expansion for

the nonrelativistic energy of a neutral atom of the

form

E(Z) =−



7/3

+ c

6/3

+ c

5/3

+···



(20.31)

with c

= 1.537 490 24 ...Ry, c

=−1Ry, and c



0.5398 Ry. The c

term was ﬁrst calculated by

Scott [20.36] from the observation that it arises from

the energy of the innermost electrons for which

the electron–electron interaction can be neglected.

The difference between the exact and Thomas–Fermi

energies for this case of noninteracting electrons

yields the correct c

[20.17, 37]. A mathemati-

cally nonrigorous but physically insightful justiﬁcation

of the Scott correction was provided in 1980 by

Schwinger [20.38]. This result has now been rigorously

proved, with upper and lower bounds coincid-

ing [20.39–43].

The c

term is much more subtle, since it arises from

a combination of effects from the exchange interaction

and from the bulk motion of electrons in the Thomas–

Fermi potential. A general analytic procedure devised

by Schwinger [20.44] yields the above value, in good

agreement with a much earlier estimate by March and

Plaskett [20.45].

A numerical check of these results, based on a ﬁt

to Hartree–Fock calculations for Z up to 290 with

correlation corrections, yielded the values [20.46]

= 0.55 ± 0.02 Ry and c

 0. It seems likely that,

because of shell structure, the terms c

and beyond have

an oscillatory dependence on Z [20.47]. The oscillatory

structure and other reﬁnements of Thomas–Fermi the-

ory are considered in a series of papers by Englert and

Schwinger [20.48–50].

Iantchenko, Lieb,andSiedentop [20.51]have

proven Lieb’s ‘strong Scott conjecture’ that for small

r, the rescaled density for the exact quantum sys-

tem converges to the sum of the densities of the

bound noninteracting hydrogenic orbitals; the prop-

erties of this function were explored by Heilman

and Lieb [20.52]. Fefferman and Seco [20.53]have

rigorously proved the correctness of Schwinger’s pro-

cedure for calculating not just the O(Z

6/3

) Scott

correction but also the O(Z

5/3

) exchange term.

Their full proof includes a demonstration that the

Hartree–Fock energy agrees with the exact quan-

tum energy through O(Z

5/3

), with an error of

smaller order [20.54]. Numerous auxiliary theorems

and lemmas are published in [20.55–58]. Progress

toward obtaining higher-order oscillatory terms is de-

scribed in [20.59–63]. The analytical evaluation of

accurate approximations to the energy of a heavy

atom, or at least of the contributions to that en-

ergy of all but the few outermost electronic orbitals,

would be of particularly great value if it led to the

construction of more accurate and better justiﬁed pseu-

dopotentials [20.64–67] for describing the valence

orbitals.

Part B 20.2

Thomas–Fermi and Other Density-Functional Theories 20.3 General Density Functional Theory 301

20.3 General Density Functional Theory

The literature on general density functional theory

and its applications is enormous, so any bibliography

must be selective. The reader interested in learning

more could begin by consulting a number of review

articles [20.68–73], collections of articles [20.74–76],

and conference proceedings [20.77–87], and the recent

textbooks by Parr and Yang [20.88] and by Dreizler and

Gross [20.89].

20.3.1 The Hohenberg–Kohn Theorem

for the One-Electron Density

In 1964 Hohenberg and Kohn [20.21] argued that there

exists a universal density functional F[ρ], independent

of the external potential V(r), such that minimization of

the sum

F[ρ]+



ρ(r)V(r) d r ,

(20.32)

subject to the constraint



ρ(r) d r = N (a positive integer) ,

(20.33)

yields the ground state energy of a quantum-mechanical

N-electron system moving in this external potential.

However, Hohenberg and Kohn’s ‘theorem’ is like

a mathematical ‘existence theorem’; no procedure exists

to calculate explicitly this unknown universal functional,

which surely is extremely complicated if it can be written

down at all in closed form. (E. Bright Wilson, however,

deﬁned it, implicitly and whimsically, as follows: “Take

the ground-state density and integrate it to ﬁnd the to-

tal number of electrons. Find the cusps in the density

to locate all nuclei, and then use the cusp condition –

that the radial derivative of the density at the cusp is

minus twice the nuclear charge density at each cusp –

to determine the charges on each nucleus. Finally solve

Schrödinger’s equation for the ground-state density or

any other property that is desired” (paraphrased by B. I.

Dunlap, in [20.83, p. 3], from J. W. D. Connolly).)

Moreover, Hohenberg and Kohn glossed over two

problems: it is not clear a priori that every well-

behaved ρ is derivable from a well-behaved properly

antisymmetric many-electron wave function (the so-

called n-representability problem, since n was used

by Hohenberg and Kohn to represent the density of

electrons), and it is also not clear a priori that every

well-behaved density ρ can be derived from a quantum-

mechanical many-electron wave function ψ which is the

properly antisymmetric ground-state wave function for

a system of electrons moving in some external poten-

tial V(r) (the so-called v-representability problem, since

Hohenberg and Kohn used v in place of V).

The n-representability problem was solved by

Gilbert [20.90] and by Harriman [20.91], who gave

a prescription for starting from an arbitrary well-

behaved ρ and from it constructing a many-electron

wave function ψ which generated that ρ [20.33,92]. The

v-representability problem is much more formidable,

as demonstrated by the discovery that there are well-

behaved densities ρ which are not the ground-state

densities for any fermionic system in an external po-

tential V [20.92, 93]. Following Percus’ deﬁnition of

a universal kinetic energy functional for independent

fermion systems [20.94], Levy [20.95] proposed to cir-

cumvent this v-representability problem by modifying

the deﬁnition of F[ρ] so that instead of being deﬁned in

terms of densities which might not be v-representable,

it is deﬁned as

F[ρ]=min

[

ψ, (T + V)ψ

]

(20.34)

with the minimum being taken over all properly anti-

symmetric normalized ψ’s which yield that ρ.

A great deal of effort has been devoted to trying

to ﬁnd approximate representations of the universal

functional F[ρ]. One route is mathematical, and fea-

tures a careful exploration of the abstract properties

which F[ρ] must have. Another route is numerical,

and can be characterized as involving the guessing of

some ansatz with a general resemblance to Thomas–

Fermi–von Weizsäcker–Dirac theory, with some ﬂexible

parameters which are determined by least-squares ﬁtting

of the energies resulting from insertion of Hartree–

Fock densities into the trial functional to theoretical

Hartree–Fock energies, or the like. If, however, the ba-

sic ansatz exhibits unphysical features in the case of

negatively charged ions or heteronuclear molecules, it

is not likely that the optimization of parameters in

that ansatz will get one closer to the true universal

density functional. In the opinion of this writer, signif-

icant progress in density functional theory based solely

upon the one-electron density is likely to require a ma-

jor revolution in our mathematical understanding of

this ﬁeld, with a useful procedure made explicit for

constructing progressively better approximations to the

universal density functional, which, like π or other

transcendental numbers, probably will never be written

down exactly in closed form. Moreover, the numeri-

cal solution of the highly nonlinear Euler–Lagrange

Part B 20.3

302 Part B Atoms

equations for a very complicated density functional is

likely to require large amounts of computer time, as

well as problems with landing in local minima of the

energy.

20.3.2 The Kohn–Sham Method

for Including Exchange

and Correlation Corrections

Density functional theory posed solely in terms

of the one-electron density and based upon the

Hohenberg–Kohn variational principle provides no gen-

eral procedure for accurately calculating relatively small

energy differences such as excitation energies, ioniza-

tion potentials, electron afﬁnities, or the binding energies

of molecules. There is, however, a powerful method

inspired by the Hohenberg–Kohn variational principle,

which has been used with great success in the calculation

of such quantities. This is the Kohn–Sham variational

method [20.22].

The key idea in the Kohn–Sham variational method

is to replace the nonlocal exchange term in the Hartree–

Fock equations with an exchange-correlation potential,

which at least in principle can be used to determine

energies exactly. The oldest, simplest, and most com-

mon ansatz used for the exchange-correlation potential

involves the local density approximation (LDA), in

which one assumes that the exchange-correlation po-

tential for the actual system under study has the same

functional form as does the exchange-correlation poten-

tial for a uniform interacting gas of electrons. If the

density is not too small or not too rapidly varying,

the exchange part of this potential can be approxi-

mated by ρ

1/3

, which appears in Dirac’s ﬁrst-order

approximation for exchange energies, with a system-

atic procedure for deriving higher-order corrections in

a gradient expansion. The correlation part of this po-

tential is accurately known from Ceperley and Alder’s

quantum Monte Carlo calculation of the properties

of the uniform electron gas [20.96]. One therefore

retains the important features of the quantum the-

ory based on wave functions, with a determinantal

approximation to ψ, while approximately including

exchange and correlation effects through a simply com-

putable effective potential. Higher corrections, which

are important for quantitative accuracy, can be incor-

porated by taking account of the variation of ρ by

means of a gradient expansion [20.27] involving ∇ρ

and higher derivatives [20.89, Chapt. 7], thus yielding

a generalized gradient approximation (GGA)forthe

exchange-correlation potential.

The Kohn–Sham procedure has become the back-

bone for the vast majority of accurate calculations of

the electronic structure of solids [20.72, 86]. In the

1990’s, motivated by Becke’s work on constructing sim-

ple gradient-corrected exchange potentials [20.97–103],

and incorporating the Lee, Yang, and Parr (LY P ) expres-

sion for the correlation potential [20.104] derived from

Colleand Salvetti’s correlation-energy formula [20.105–

107], the Kohn–Sham method is ﬁnding increasing

application in efﬁciently estimating relatively small en-

ergy differences of relevance to chemistry [20.108–110]

(However, Becke’s gradient-corrected exchange poten-

tial does not have the correct 1/r behavior at large r,

as was observed by several authors [20.111–113]). For

deﬁnitive results, however, one must still resort to an ab

initio theory which at least in principle converges toward

the correct result.

The generation of improved generalized gradient

approximations has recently become a growth indus-

try, with increasingly many proposals of increasingly

greater complexity [20.97, 104, 113–125]. Inevitably,

some expressions work better for some properties than

for others. It is found that usually most of the errors

in the long-range tails of the exchange and correla-

tion potentials tend to cancel each other, thus leading

to better overall energies than one could reasonably

expect [20.126]. Under these circumstances, it is im-

portant to have benchmarks for testing the accuracy

of the various approximations. Such comparisons have

been carried out for two important sets of two-electron

systems [20.127–129]:

1. a pair of electrons moving in harmonic potential

wells and coupled by the Coulomb repulsion, which

yields an exactly solvable system;

2. helium-like ions of variable nuclear charge Z,

for which extremely accurate energies and wave

functions are available which take account of the

behavior of the exact but unknown wave function in

the vicinity of all two-particle coalescences and the

three-particle coalescence.

The results indicate that the approximate exchange-

correlation potentials differ quite considerably from the

true exchange-correlation potentials, thus indicating the

need for further analytical work in understanding how

to design accurate exchange-correlation potentials, and

for devising tests of exchange-correlation potentials for

larger atoms and for molecules.

Another important way of testing the validity of

various approximate exchange and correlation po-

tentials is checking whether they obey inequalities

Part B 20.3

Thomas–Fermi and Other Density-Functional Theories 20.4 Recent Developments 303

imposed by such general properties as scaling and the

Hellmann–Feynman theorem. Such general tests have

been devised by Levy and his co-workers [20.130–

139], who have found that many of the commonly

used approximate potentials violate general inequal-

ities which must be obeyed by the exact potential.

These abstract results are helpful in designing poten-

tials which should be better approximations to the true

potential.

20.3.3 Density Functional Theory

for Excited States

The Hohenberg–Kohn theorem and the Kohn–Sham

method were originally formulated in terms of the

ground electronic state. These techniques can be

extended to calculate the ground state of a given sym-

metry [20.140], but that leaves unresolved the issue

of using density functional theory to calculate the en-

ergies of excited states for a given symmetry. Using

the Rayleigh–Ritz principle for ensembles, general ab-

stract procedures for generalizing density functional

theory to excited state calculations have been formulated

by Theophilou [20.141] and by several other work-

ers [20.142–151]. Unfortunately, the errors typically

seem to be much larger than for ground-state density

functional theory.

20.3.4 Relativistic and Quantum Field

Theoretic Density Functional Theory

At a formal level, one can discuss the development of

density functional theory for a relativistic system of elec-

trons. For an overview of this challenging subject, see

the discussions by Dreizler and Gross [20.89, Chapt. 8]

and by Dreizler [20.152]. Much of the formalism carries

over, but no good way has yet been found of incorporat-

ing vacuum polarization corrections.

20.4 Recent Developments

During the last eight years there has continued to be ex-

ponentially growing interest in applications of density

functional theory of the Kohn-Sham variety to atoms

and molecules, especially those of chemical relevance

which are too large for accurate ab initio electronic

structure calculations. The awarding of the 1998 No-

bel Prize in Chemistry to Walter Kohn and John A.

Pople recognised their individual contributions to this

increasingly important ﬁeld. Their Nobel lectures were

published the following year in the Reviews of Modern

Physics [20.153, 154].

Since a comprehensive summary of the wide-

ranging developments in density functional theory

during the past decade is not feasible within the lim-

ited space available for this supplementary section, I

will brieﬂy cite some of the most extensive surveys of

various aspects of this ﬁeld that have appeared since

1995.

Many aspects of density functional theory were re-

viewed in four consecutive volumes of Topics inCurrent

Chemistry published in 1996 [20.155], and in 1999 an

entire volume of Advances in Quantum Chemistry was

devoted to density functional theory [20.156]. This has

also been the subject of several conference proceed-

ings [20.157–159] and introductory textbooks [20.160,

161]. Developments in time-dependent density func-

tional theory for chemical systems have been surveyed

in two very recent review articles [20.162, 163].

Although the locality of DFT was proved for a large

class of functionals [20.164–166], this issue has come

under recent dispute. The question that has been raised is

whether there exists an exact Thomas-Fermi model for

non-interacting electrons. If such an exact model does

not exist, as it is a direct consequence of the Hohenberg-

Kohn theorem, then DFT would be incomplete.

Nesbet [20.167–171] has argued that such a theory

would be inconsistent with the Pauli exclusion princi-

ple for atoms of more than two electrons (or for a two

electron atom where both electrons are in the same spin

state). The contention is that if only the total electron

density were normalized (which corresponds to only one

Lagrange multiplier), as in the TF model, then no shell

structure can exist; hence such a system would violate

the exclusion principle.

A counter-example has recently been constructed

by Lindgren and Salomonson [20.172] showing that

shell structure can indeed be generated through a sin-

gle Lagrange multiplier. In addition, they have veriﬁed

numerically that a local Kohn-Sham potential can repro-

duce to high accuracy the many-body electron density

and the 2s eigenvalue for the 1s2s

S state of neutral

helium.

Part B 20.4

304 Part B Atoms

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