3. Total energy as functionalof electron density

3.1 Expectation values

of one- and two-particle operators

We remember the separation of a Hamiltonian into one-particle and two-particle operatorsin Eq. (2.12). It can be shown that the expectation value of any one-particle operatorcan be expressed in terms of one-particle density matrix, that of two-particle operator –in terms of two-particle matrix etc. We consider an arbitrary single-particle operator

F =N∑

i=1

f(xi) , (3.1)

then its expectation value in a many-body state Ψ is

〈F 〉 =∫

dx1 . . . dxNΨ∗(x1 . . . xN )N∑

i=1

f(xi) Ψ(x1 . . . xn) =

=∫

dxN∑

i=1

f(x)[∫

dx2 . . . dxnΨ∗(y, x2 . . . xN)Ψ(x, x2 . . . xN )]

y=x.

In doing so, we used the fact that simultaneous interchange of any two arguments in both

Ψ∗ and Ψ does not change their product,

Ψ∗(x1 . . . xi . . . xN )Ψ(x1 . . . xi . . . xN) = Ψ∗(xi . . . x1 . . . , xN )Ψ(xi . . . x1 . . . xN) ,

and the integration parameter y different from x was introduced in Ψ∗ in order to ensurethat the operator f acts on Ψ only, not on Ψ∗. For a simple multiplication (e.g., fbeing an external potential) this is not important so that one can simplify in the lastformula [· · ·]y=x ⇒ ∫

dx2 . . . dxnΨ∗(x, x2 . . . xN )Ψ(x, x2 . . . xN ), but if, e.g., f = ∇2, thenit is essential to keep first arguments in Ψ∗ and Ψ formally different and set y = x afterthe integration. With the definition of the density matrix (2.20),

〈F 〉 =∫ [

f(x) γ(x; y)]

y=xdx , (3.2)

or, considering both f and γ as operators in the x-space,

〈F 〉 = tr (f γ) , (3.3)

For an operator f depending only on the coordinates of particles (like external electrostaticfield, or kinetic energy) the summation over spin components can be done in Eq. (3.2),resulting in

〈F 〉 =∫

dr[

f(r) γ(r; r′)]

r′=r. (3.4)

25

If the spin degrees of freedom matter for the one-particle operator in question, then itsexpectation values can be expressed via spin-dependent single-particle density matrices.The vector spin operator σ =

∑

α eασα is defined via cartesian unit vectors eα and Paulimatrices

σz =1

2

(

1 00 −1

)

, σx =1

2

(

0 11 0

)

, σy =1

2

(

0 −ii 0

)

, (3.5)

then the vector spin density will be

S(r) =N∑

i=1

σi δ(r − ri) , (3.6)

and the spin magnetization density

m(r) = 2 〈S(r)〉 = 2∑

ss′σs′sρss′(r) = 2 trs [σ γ(r)] (3.7)

with

ρss′(r) =

(

ρ++(r) ρ+−(r)ρ−+(r) ρ−−(r)

)

,

then

m(r) =

2 Re ρ−+(r)2 Im ρ−+(r)ρ++(r) − ρ−−(r)

. (3.8)

Similarly, for any two-particle operator V = 12

∑

i6=jv(xi, xj)

〈 V 〉 =1

2

∑

i6=j

∫

dx1 . . . dxNΨ∗(x1 . . . xN )v(xi, xj)Ψ(x1 . . . xN ) =

=1

2

∑

i6=j

∫

dx dy∫

Ψ∗(x, y, x3 . . . xN) v(x, y) Ψ(x, y, x3 . . . xN) dx3 . . . dxN =

(using the same argument as above, interchanging xi↔x, xj ↔y in both Ψ∗ and Ψ)

=∫

dx dy [v(x, y) γ2(x, y; x′, y′)]x′=x

y′=y

. (3.9)

Again, we introduced x′ independent from x and y′ independent from y in order to preventthat operator v(x, y) acts on the arguments in Ψ∗. However, if v(x, y) is simple multipli-cation (as is the case with Coulombic two-particle interaction), this is not necessary, soone can write down using Eq. (2.41)

〈 V 〉 =1

2

∫

dx dy v(x, y) ρ2(x, y) . (3.10)

Therefore the expectation value of any operator in a certain state of a many-electronsystem can be obtained through the knowledge of two functions, the single-particle densitymatrix γ(x, y) defined in Eq. (2.20) and the pair density ρ2(x, y) defined in Eq. (2.41).

26

3.2 Hohenberg–Kohn theorem

Of course the last statement applies to the total energy as the expectation value of theHamilton operator in a particular state. However, if we consider specifically the ground

state, the total energy must be extremal, that allows to formulate a more strong condition,as was done by Pierre Hohenberg and Walter Kohn in 19648:

In the ground state, the total energy is a functional of the electron density ρ(r).

That means actually that not only one doesn’t need to know density matrix and thepair density to determine the ground-state total energy, but the dependence of the latteron spin variables is not relevant as well. We’ll prove this ad absurdum, following theoriginal derivation of Hohenberg and Kohn. Underway, we’ll distinguish between what issometimes referred to as the first Hohenberg–Kohn theorem and the second one.

We start by specifying once again the many-electron Hamiltonian,

H[u] = − h2

2m

N∑

i=1

∇2i

︸ ︷︷ ︸

≡ T

+N∑

i=1

u(xi)

︸ ︷︷ ︸

≡ U

+1

2

N∑

i6=j

v(xi, vj)

︸ ︷︷ ︸

≡ V

.

u(xi) is an external potential, and in the following we assume we are free to vary it.Each external potential defines a corresponding ground-state wavefunction via solutionof the Schrodinger equation. Such wavefunctions are of course different for u1 and u2, ifu1(r) 6= u2(r) + const.. What about the particle density? It results from the integrationover N −1 variables in the wavefunction. It will turn out that still, there is no way toget ρ[u1] the same as ρ[u2] if the potentials differ by more than a constant. But first weassume that this is possible. Let Ψ1 be the ground-state wavefunction corresponding tothe external potential u1, then the corresponding total energy is:

E1 = 〈Ψ1|H[u1]|Ψ1〉 = 〈Ψ1|T +V |Ψ1〉 +∫

u1(r) ρ(r) dr .

This must be lower than the expectation value obtained with any other wavefunction Ψ2:

E1 < 〈Ψ2| H[u1] |Ψ2〉E1 < 〈Ψ2| H[u2] − u2 + u1 |Ψ2〉E1 < 〈Ψ2| H[u2] |Ψ2〉

︸ ︷︷ ︸

E2

+∫

dr [u1(r) − u2(r)] N∫

Ψ∗2(r2 . . . rN)Ψ2(r2 . . . rN) dr2 . . . drN

︸ ︷︷ ︸

= ρ(r)

.

On the other side, the ground-state energy E2 calculated with its “true” wavefunction Ψ2

must be lower than the expectation value of H[u2] calculated with any other wavefunction,including Ψ1:

E2 < 〈Ψ1| H[u2] |Ψ1〉E2 < 〈Ψ1| H[u1] − u1 + u2 |Ψ1〉E2 < 〈Ψ1| H[u1] |Ψ1〉

︸ ︷︷ ︸

E1

+∫

dr [u2(r) − u1(r)] N∫

Ψ∗1(r2 . . . rN)Ψ1(r2 . . . rN) dr2 . . . drN

︸ ︷︷ ︸

= ρ(r)

.

8Inhomogeneous Electron Gas, Phys. Rev. 136, B864 (1964)

27

Summing up two inequalities, we arrive at E1+E2 < E2+E1, that is a contradiction. Theorigin of this contradiction is that we assumed ρ(r) to be the same for two wavefunctionsgenerated by two different potentials u1 and u2. Hence this assumption was wrong, and theexternal potential uniquely determines the density. Since ρ[u1] 6= ρ[u2] for u1 6= u2+const,for any given ρ(r) there is at most one potential u(r) for which ρ(r) is the ground-statedensity.

At this point, one sometimes singles out as the first Hohenberg–Kohn theorem:

For isolated many-electron system, its ground-state one-electrondensity ρ(r) determines uniquely the external potential u(r)

and as the second Hohenberg–Kohn theorem:

The exact ground state energy E = E[Ψ0] of many-electron system with externalpotential u(r) is a functional of the associated ground-state electron density ρ0(r).

This second statement follows from the fact that since u(r) uniquely fixes H and hencethe many-particle ground state, the latter must be a unique functional of ρ(r):

E [ρ0(r)] = T [Ψ0] + V [Ψ0] +∫

dr u(r) ρ(r) . (3.11)

As a brief discussion, it is important to emphasize the following:– this is simply an existence theorem, which tells absolutely nothing about the actualdependence of the kinetic part T [Ψ0 [u[ρ(r)]]] and the electron-electron interaction partV [Ψ0 [u[ρ(r)]]] in the total energy on particle density. This is a serious challenge forpractical applications.– there is a problem of u-representability, i.e. a question of whether an arbitrary functionρ(r) can be mapped onto corresponding u(r). There has been a history of works aimedat more strict mathematical formulation.

3.3 Kohn–Sham equation

There are two essential problems of applying Eq. (3.11) to practical calculations: theconcretization of T [ρ] and V [ρ]. The Thomas–Fermi theory provides approximations forboth, but they are not accurate enough, particularly in what regards the kinetic energypart. Walter Kohn and Lu Sham in 19659 proposed a trick allowing to represent thekinetic energy better.

In a many-electron system without interaction (V switched off), the ground-statewavefunction (when non-degenerate) is a single Slater determinant. Then the kineticenergy can be easily evaluated:

H0 = T0 + u(r); Ψ0 =1√N !

det ‖ϕi(xk)‖ ;

T0[Ψ0] = 〈Ψ0| T |Ψ0〉 = − h2

2m

N∑

i=1

〈ϕi| ∇2i |ϕi〉 . (3.12)

9Phys. Rev. 140, A1133 (1965)

28

The particle density is ρ(r) =∑

isϕ∗

i (x)ϕi(x), as in the HF formalism, Eq. (2.19). With

the interaction actually present, the representation for the density can still be applied,understanding that ϕi(x) are not true one-particle wavefunctions anymore, but just anysupport functions, or pseudofunctions, of our convenience to represent the density. Thisis fully general. Further on, we approximate the kinetic energy of a true system by T0

constructed from ϕi according to Eq. (3.12). This introduces an error, which is furtheron attributed to the yet undefined part, which describes electron-electron interaction. thefunctions ϕi(x) are searched for using variational approach (instead of searching for ρ(x)directly, as in the Thomas-Fermi formalism). Once ϕi are found, we can immediatelyreconstruct the density. Specifically, the total energy from Eq. (3.11)

Etot = T [ρ] +e2

2

∫ ρ(x)ρ(x′)

|r − r′| dx dx′

︸ ︷︷ ︸

Coulomb interaction,including

self-interaction

+ EXC[ρ]

︸ ︷︷ ︸

corrections:exchange +correlation,

so good we can

︸ ︷︷ ︸

V [ρ]

+ e∫

u(x)ρ(x) dx (3.13)

will be substituted by

Etot = T0[ρ]

︸ ︷︷ ︸

kinetic energyof non-interacting

system of N particleswith the same density

+e2

2

∫ρ(x)ρ(x′)

|r − r′| dx dx′ + E ′XC[ρ]

︸ ︷︷ ︸

EXC+T [ρ]−T0[ρ]

+ e∫

u(x)ρ(x) dx . (3.14)

We apply the variational approach, searching for those pseudofunctions ϕi(x) which wouldminimize the total energy. From Eq. (3.12)

δT0

δϕ∗α(x)

= − h2

2m∇2ϕα(x) ;

δρ(x′)

δϕ∗α(x)

= ϕα(x) δ(x − x′) ;

δ

δϕ∗α(x)

[

1

2

∫ρ(x)ρ(x′)

|r − r′| dx dx′

]

=

[∫

ρ(x)

|r − r′| dx

]

ϕα(x) ;

hence

δ

δϕ∗α(x)

T0[ρ] +e2

2

∫ρ(x)ρ(x′)

|r − r′| dx dx′ + E ′XC[ρ] + e

∫

u(x)ρ(x) dx−

− εα

[∫

ρ(x) dr − N]

= 0 . (3.15)

where the Lagrange multipliers εα, as before, take care of maintaining the normalizationof the density in the course of variation. The variation of (yet unknown) XC-functional

29

in ϕ∗α can be reformulated via its variation in ρ:

δE′XC

δϕ∗α(x)

=∫

dx′ δE′XC

δρ(x′)

δρ(x′)

δϕ∗α(x)

︸ ︷︷ ︸

δ(x−x′) ϕα(x)

=δE′

XC

δρ(x)︸ ︷︷ ︸

XC-potential,VXC(x)

ϕα(x) .

Eq. (3.13) then reduces to[

− h2

2m∇2 + e2

∫

dx′ ρ(x′)

|r − r′| + u(x) + VXC(x) − εα

]

ϕα(x) = 0 . (3.16)

This is the system of equations resembling that of the HF method after introducingthe Slater averaging for exchange – the equations are linear in ϕα, which mix via thecontribution of all of them to the total density. The Slater averaged exchange potentialVX can be considered as an approximation to a more general VXC. VXC is a simple functionof r (and spin), but it depends on the distribution of density in the whole space.

However, the present derivation is more general; it does not impose any particularrestrictions on the wave function (we assumed for it a single-determinant form only in theabsence of electron-electron interaction, in order to construct T0).• This rearrangement puts everything yet unknown into VXC(x); the rest is explicitlydefined.• The equations give ϕα and then total density, but provide absolutely no informationabout the ground-state wavefunction (not even about the density matrix).• Everyting applies to the ground state only – that is nevertheless quite important, becauseproblems like equilibrium structure, magnetization, phonons, etc. fall into this category.• This approach essentially fixes the problems of the Thomas-Fermi method related topoor description of the kinetic energy.

The closed form for the ground-state total energy in the Kohn–Sham approach can beobtained from Eq. (3.14), using T0 as defined by Eq. (3.12):

T0[ρ] =N∑

i=1

(

− h2

2m

)∫

ϕ∗i (x)∇2ϕi(x) dx

from Eq. (3.16)

− h2

2m∇2ϕα(x) = εαϕα(x) −

[

u(x) + VXC(x) + e2∫

ρ(x′)

|r − r′|dx′

]

ϕα(x) ,

that results in

Etot =N∑

i=1

∫

ϕ∗i (x) εi ϕi(x) dx −

N∑

i=1

∫

ϕ∗i (x) VXC(x) ϕi(x)dx −

− e2N∑

i=1

∫

dxϕ∗i (x) ϕi(x)

∫ρ(x′)

|r− r′| dx′ +e2

2

∫ρ(x)ρ(x′)

|r− r′| dx dx′ + E ′XC[ρ] =

=N∑

i=1

εi −e2

2

∫ρ(x)ρ(x′)

|r − r′| dx dx′ −∫

VXC(x) ρ(x) dx + E ′XC[ρ] . (3.17)

30

εi is a set of N lowest eigenvalues of the Kohn–Sham equations, similarly to the aufbau

pronciple in the HF method.

3.4 Relation between exchange-correlation energy and

electron pair correlation function

We already discussed earlier that the exchange and correlation are essentially due to aCoulomb interaction of each electron with electron density missing in its environmet,so no wonder that EXC can be directly related to the pair density ρ2(x, y) definned byEq. (2.41). However, one must consider that E ′

XC incorporates as well the differencebetween the true kinetic energy T and that of non-interacting electron gas of the samedensity, T0. We model this situation by an adiabatic approximation: scale the electron-electron interaction in the Hamiltonian with λ, 0 ≤ λ ≤ 1, and assure that the densityρ(x) remains the same for all values of λ by adding a corresponding λ-dependent externalpotential uλ:

H = H0 + λW =∑

i

h(λ)i + λ

∑

ij

wij . (3.18)

For all λ along the part 0→1, there is a groud-state function Ψλ and total energy Eλ.Taking into account

∂

∂λ〈Ψλ|Ψλ〉 = 0 = 〈∂Ψλ

∂λ|Ψλ〉 + 〈Ψλ|

∂Ψλ

∂λ〉 ,

we get:

∂

∂λEλ =

⟨∂Ψλ

∂λ

∣∣∣H

∣∣∣Ψλ

⟩

+⟨

Ψλ

∣∣∣∂H∂λ

∣∣∣Ψλ

⟩

+⟨

Ψλ

∣∣∣H

∣∣∣∂Ψλ

∂λ

⟩

=

= Eλ

[⟨∂Ψλ

∂λ

∣∣∣Ψλ

⟩

+⟨

Ψλ

∣∣∣∂Ψλ

∂λ

⟩]

+⟨

Ψλ

∣∣∣∂H∂λ

∣∣∣Ψλ

⟩

=

=⟨

Ψλ

∣∣∣W

∣∣∣Ψλ

⟩

. (3.19)

This is the Hellmann–Feynman theorem. Then the total energy can be recovered as

E = E0 +

1∫

0

dλ 〈Ψλ|W |Ψλ〉 , (3.20)

Assuming that we switch on the interaction adiabatically, i.e. dλ/dt → 0 and the systemis everywere in the ground state, we specify the Hamiltonian as

H = H0 +∑

i

uλ(xi) +e2

2

∑

i6=j

λ

|ri − rj|;

dHdλ

=∑

i

duλ(xi)

dλ+

e2

2

∑

i6=j

1

|ri − rj |.

31

To calculate its expectation value, we use Eq. (3.2) and (3.10). Since both operators hereare simply numbers, one does not need to care about y→x in (3.2) and writes simply interms of single and pair density:

⟨

Ψλ

∣∣∣

∑

i

duλ(xi)

dλ

∣∣∣Ψλ

⟩

=∫

duλ(x)

dλρ(x) dx ; (3.21)

⟨

Ψλ

∣∣∣

∑

i6=j

1

|ri − rj |∣∣∣Ψλ

⟩

=∫ ρ(x, x′)

|r − r′| dx dx′ , (3.22)

We express the pair density in terms of pair correlation function g (2.42), which gets thedependennce on λ (the single-particle density is the same for all λ, but the correlationfunction and hence the pair density certainly not):

ρ(x, x′) = gλ(x, x′) ρ(x) ρ(x′) . (3.23)

dEλ

dλ=∫

ρ(x)∂uλ(x)

∂λdx +

e2

2

∫ρ(x)ρ(x′)

|r − r′| gλ(x, x′) dx dx′ .

Then

E − E0 =

1∫

0

dEλ

dλdλ =

∫

ρ(x) [uλ=1(x) − uλ=0(x)] dx +

+e2

2

∫ ρ(x) ρ(x′)

|r − r′| dx dx′

1∫

0

gλ(x, x′) dλ . (3.24)

Without interaction,

E0 = T0[ρ] +∫

ρ(x) uλ=0(x) dx ,

with full interaction (3.14)

E = T0[ρ] +e2

2

∫ρ(x)ρ(x′)

|r− r′| dx dx′ +∫

ρ(x) uλ=1(x) dx + E ′XC[ρ] ;

hence

E − E0 =e2

2

∫ ρ(x)ρ(x′)

|r − r′| dx dx′ +∫

ρ(x) [uλ=1(x) − uλ=0(x)] dx + E ′XC[ρ] ,

and comparing with (3.24) one gets:

E ′XC[ρ] =

e2

2

∫ρ(x)ρ(x′)

|r− r′| [g(x, x′) − 1] dx dx′ . (3.25)

with

g(x, x′) =

1∫

0

gλ(x, x′) dλ . (3.26)

Hennce the exact exchange-correlation energy is fully determined by the shape of theexchange-correlation hole (but, for the full range of interaction strengths).

32

3.5 EXC for homogeneous electron liquid

A large amount of useful information about general properties of EXC[ρ] has been obtainedin the course of simulating a simple benchmark system, homogeneous electron liquid. Thisis a system of interacting electrons with homogeneous density, compensated by equallyhomogeneous background positive charge.

We begin with some general considerations, in order to incorporate the effect of spinpolarization in the following treatment. We start from the exchange energy and thengeneralize the results over the correlation energy. Introducing explicitly spin variables,we get

EX = −e2

2

∫ ∫ ρ(+)X (r, r′) ρ(+)(r′)

|r − r′| dr dr′ − e2

2

∫ ∫ ρ(−)X (r, r′) ρ(−)(r′)

|r − r′| dr dr′ , (3.27)

since exchange interaction does not “mix” two spin components. Therefore

EX[ρ(+), ρ(−)] = EX[ρ(+)] + EX[ρ(−)] . (3.28)

Further on, we introduce exchange energy density εX(r) that satisfies

EX =∫

dr εX(r) ρ(r) (3.29)

without loss of generality.10 Comparing it with Eq. (3.27)

ρ(r) εX(r) = −e2

2

∫ρ

(+)X (r, r′) ρ(+)(r′)

|r − r′| dr′ − e2

2

∫ρ

(−)X (r, r′) ρ(−)(r′)

|r − r′| dr′ . (3.30)

For uniform electron gas of constant density ρ(+) and ρ(−), we may perform the integrationin Eq. (3.30) in analogy with (2.38). We emphasize that it doesn’t matter how the totaldensity is split into ρ(+) and ρ(−); the integration limit in k-space is fixed by the ρ valuein each spin channel separately. Therefore we use, instead of (2.37) where it was assumedρ(+) = ρ/2, explicitly

ρ(+)X (r, r′) = 9ρ(+)

[

j1(kF|r− r′|)kF|r− r′|

]2

;

instead of (2.35) we have

k3F

6π2= ρ(+) ; kF =

(

6π2ρ(+))1/3

.

The integration over r′′ = r′ − r, with fixed r, yields:

∫

9ρ(+)

[

j1(kF|r− r′|)kF|r− r′|

]2ρ(+)

|r− r′| dr′ = 9[

ρ(+)]2 · 4π

∞∫

0

r′′2dr′′

[

j1(kFr′′)

kFr′′

]21

r′′=

=36π

(6π2)2/3

[

ρ(+)]4/3

∞∫

0

(sin x − x cos x)2

x5dx

︸ ︷︷ ︸

1/4

= 3(

3

4π

)1/3 [

ρ(+)]4/3

. (3.31)

10this is not yet local density approximation as long as εX(r) depends on the density in all space, notjust at r.

33

Further on, we introduce spin polarization ζ(r)

ζ =ρ(+) − ρ(−)

ρwith ρ = ρ(+) + ρ(−) , (3.32)

whenceρ(+) =

ρ

2(1 + ζ), ρ(−) =

ρ

2(1 − ζ); , (3.33)

and exchange energy density, now dependent on spin polarization, becomes:

εX(r, ζ) = −3e2

2

(3

4π

)1/3 ρ1/3

24/3

[

(1 + ζ)4/3 + (1 − ζ)4/3]

. (3.34)

In the following, we’ll use instead of ρ the density parameter rs, that is the radius of asphere inclusing unit charge,

4π

3r3s ρ = 1 → rs =

(

3

4πρ

)1/3

. (3.35)

Note that rs is r-dependent for inhomogeneous density distribution. With this definition,

εX(r, ζ) = −3e2

4π

(9π

4

)1/3 (1 + ζ)4/3 + (1 − ζ)4/3

2rs. (3.36)

In what regards the dependence on the spin polarization, the exchange energy densitycan be looked at as an interpolation between limiting values of “paramagnetic” (ζ=0)and “ferromagnetic” (ζ=1) cases,

εX(r, 0) = −3e2

4π

(9π

4

)1/3

· 2

2rs

, εX(r, 1) = −3e2

4π

(9π

4

)1/3

· 24/3

2rs

,

so thatεX(r, ζ) = εX(r, 0) + [εX(r, 1) − εX(r, 0)] f(ζ), (3.37)

−1 0 10

1

f(ζ)

with

f(ζ) =(1 + ζ)4/3 + (1 − ζ)4/3 − 2

24/3 − 2(3.38)

known as the von Barth – Hedin interpolation function(see plot).

Von Barth and Hedin calculated11 the correlation energy of spin-polarized electronliquid in the lowest order in the random phase approximation for different values of ζand rs. It follows that the same interpolation function is accurate enough for correlationenergy density, so that one can use:

εXC(r, ζ) = εXC(r, 0) + [εXC(r, 1) − εXC(r, 0)] f(ζ) . (3.39)

11U. von Barth and L. Hedin, A local exchange-correlation potential for the spin polarized case: I,J. Phys. C: 5, 1629 (1972).

34

The exact results analysis of εXC(rs) are not so far known even for the simplest modelcase of free electron liquid, or jellium – a homogeneous distribution of interacting electronsat the background of uniformly smeared compensating positive charge. The analyticresults have been otained (up to certain acccuracy) only for limiting cases of low and highdensities. The high-density limit (rs → 0) in the paramagnetic case is given by

εXC(rs, 0) = − 3

4π

(9π

4

)1/3 1

rs

already described by Eq. (3.36)

+1 − ln 2

π2ln rs

W. Macke,Z. Naturforsch. 5a, 192 (1950)

+ B+ Crs ln rs

+ Drs

Carr + Maradudin,Phys. Rev. 133, A371 (1964)

+ . . .

The solution for the ferromagnetic case is known in the random phase approximation:12

εRPAXC (ρ, 1) =

1

2εRPAXC (24ρ, 0) . (3.40)

This scaling is consistent with Eq. (3.36) for the exchange energy density.

The low-density limit is essentially that of electron crystallization, i.e. their localiza-tion near well separated (and ordered) positions is space. The possibility of such behaviourat sufficiently low densities was first recognized by E. Wigner.13 Wigner compared dif-ferent crytsal lattices by their contributions to the total energy – essentially, Madelungterms evaluated by Ewald summation. The results obtained for several lattice types are:

−1.79186 r−1s Ry (body centered cubic)

−1.79172 (face centered cubic)−1.79168 (hexagonal close-packed)−1.760... (simple cubic)

As is seen, the bcc lattice seems to be the most probable candidate for the Wignercrystallization of jellium, but it is not strictly proven, and at least fcc and hcp latticesbecome competitive. The excitations from the “crystallized” state are electron latticevibrations, or “phonons”. With zero-point energy of such “phonons” taken into account,the perturbation expansion in the low-density limit is in powers of r−1/2

s and yields (in

12L. Hedin, Phys, Rev. 139, A796 (1965)13E. Wigner, Trans. Faraday Soc. 34, 678 (1938)

35

the paramagnetic case) for the energy in Ry per electron – see, e.g., Carr (1961): 14

E = −1.792

rs

Madelung energy of Wigner crystal

+2.66

r3/2s

zero-point energyof electron lattice vibrations

+ br2s

+ O

(

1r5/2s

)

perturbations tooscillator functions; b < 1

+ terms ∼ exp(−r1/2s )

Fig. 1 of Carr (1961). Energy of theelectron gas plotted against rs. Theopen circles are points given by theabove formula with anharmonic termsneglected. The filled circles were calcu-lated from the Gell-Mann and Brueck-ner equation [Phys. Rev. 106, 364(1957)]. In constructing the dashedline connecting the two end regions,consideration was given to the fact thatit must fall below the sum of Fermiplus exchange energy for plane waves.The open circle points below rs=6 havelower accuracy, since for these pointsthe exponential terms are appreciable.

The above figure shows the estimations of exchange-correlation energy of homogeneouselectron liquid obtained by perturbation expansion in two limiting cases. The solid curveis a free interpolation connecting the data is small rs and large rs regions. The regionof “relevant” rs values is 2 – 7 a.u., where none of limiting approximations works well.A number of calculations has been done by Hedin and Lundqvist, von Barth and Hedin,Gunnarsson and Lundqvist15 to bridge the region of intermediate rs for both paramag-netic and ferromagnetic cases. For example, the parametrization of the correlation energydensity by Hedin and Lundqvist16 (ζ=0 case), adopted also by von Barth and Hedin forthe ζ=1 case, uses the function

F (z) = (1 + z3) ln(

1 +1

z

)

+z

2− z2 − 1

3

14W. J. Carr, Jr., Energy, Specific Heat, and Magnetic Properties of the Low-Density Electron Gas,Phys. Rev. 122, 1437 (1961).

15O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and solids by the

spin-density-functional formalism, Phys. Rev. B 13, 4274 (1976).16L. Hedin and B. I. Lundqvist, Explicit local exchange-correlation potentials, J. Phys. C 4, 2064 (1971).

36

and

ε(rs, 0) = −CPMF(

rs

rPM

)

, ε(rs, 1) = −CFMF(

rs

rFM

)

,

with parameter values (rs in a.u., energy in Ry):

CPM = 0.0504; CFM = 0.0254; rPM = 30; rFM = 75.

This is consistent (within 1%) with the scaling relation (3.40) for the random phaseapproximation:

CFM =1

2CPM; rFM = 24/3rPM.

For more accurate parametrizations in the intermediate range of rs, it was extremely im-portant that (ultimately exact) results of Monte Carlo simulation became available due toCeperley and Alder.17 Their scheme was a search for “optimized” many-body wavefunc-tion and its corresponding energy in a diffusion process, starting from a trial wavefunctionand an arbitrary reference energy. The trial function was a product of two-body correla-tion factors (that remove singularities in the local energy as electrons approach each other)times a Slater determinant, constructed either from plane-wave states (for Fermi liquid),or from bcc-centered Gaussians (for the Wigner crystal). Paramagnetic or ferromagneticcase was fixed in advance.

With the Hamiltonian of usual form,

H =h2

2m

N∑

i=1

∇2i −

e2

2

∑

i6=j

1

|ri − rj|,

the parameter space R includes 3N spatial coordinates of the system of N electrons. Arandom walk in dR3N at time t is given by f(R, t) ·dR3N , f being the probability density.The value of f at t=0 is |ΨT(R)|, e.g., starting trial function. For large times,

f(R, t → ∞) = ΨT(R) ϕ0 exp[−t(Eref − E0)],

with ϕ0 as asymptotic (“true”) wavefunction and E0 – “true” eigenvalue. The diffusionequation for f(R, t) is

∂f

∂t=

h2

2m

[N∑

i=1

∇2i f −∇i

(

f ∇i ln |ΨT|2)]

−[ HΨT

ΨT︸ ︷︷ ︸

trialenergy

−Eref

]

f

and implies:– random diffusion;– drift induced by the trial “quantum force” ∇ ln |ΨT|2

(since ln |ΨT|2 has the meaning of energy);– branching with probability = (time step)×(Etrial − Eref).

17D. M. Ceperley and B. J. Alder, Ground State of the Electron Gas by a Stochastic Method,Phys. Rev. Lett. 45, 566 (1980).

37

For Etrial > Eref, the configuration is removed from the ensemble, for Etrial < Eref –duplicated in the ensemble.

The results of Ceperley and Alder are organized in a table of ground-state energiesfor several simulation objects (paramagnetic Fermi liquid, ferromagnetic Fermi liquid,Wigner crystal, and Bose liquid) over a range of rs. These results can be also representedas the following “phase diagram”:

Fig. 2 of Ceperley and Alder. The energy of the four phases studied relative to that of the lowest

boson state times r2s in rydbergs vs rs in Bohr radii. Below rs=160 the Bose fluid is the most

stable phase, while above, the Wigner crystal is most stable. The energies of the polarized and

unpolarized Fermi fluid are seen to intersect at rs=75. The polarized (Ferromagnetic) Fermi

fluid is stable between rs=75 and rs=100, the Fermi Wigner crystal above rs=100, and the

normal paramagnetic Fermi fluid below rs=75.

A number of analytical expressions interpolating the results by Ceperley and Alder hasbeen proposed, which are now widely in use (Perdew and Zunger 1981 and others).

3.6 Local density approximation

The expression used for the exchange-correlation energy, in analogy with Eq. (3.29),

EXC =∫

dr εXC(r) ρ(r) ,

is quite general, as long as we assume εXC to be a function of r, dependent on the wholedistribution of density, i.e., εXC[ρ(r)]. However, the benchmark studies we just discussedrefer to the case of homogeneous density and are not, strictly speaking, applicable to vary-ing ρ(r). The local density approximation (LDA) represents an ill-justified but reasonable

38

generalization, that is expected to work at least for the case of slowly varying density:

ELDAXC [ρ] =

∫

dr ρ(r) εXC

(

ρ(r), ζ(r))

. (3.41)

We’ll see that the accuracy of this approximation is better than could be a priori ex-pected. The exchange-correlation potential VXC(r) as introduced in by Eq. (3.16) can begeneralized for the spin-polarited case as follows:

δEXC

δϕ∗α(x)

=∫

δEXC

δρ(+)(x′)

δρ(+)(x′)

δϕ∗α(x)

︸ ︷︷ ︸

δ(x′−x)ϕα(x)

dx′ =δEXC

δρ(+)ϕα(x)

is ϕα contributes to, say, spin (+), so that

V(+)XC (r) =

∂

∂ρ(+)(r)

ρ(r) ε(

ρ(r), ζ(r))

.

There is no more integration in spatial coordinates since

∂

∂ρ(r)

∫

. . . dr′ =∫

∂ . . .

∂ρ(r)δ(r − r′) dr′ .

The differentiation in spin-resolved components of density, taking into account relations(3.32), reduces to

∂

∂ρ(+)=

∂

∂ρ

∂ρ

∂ρ(+)+

∂

∂ζ

∂ζ

∂ρ(+)=

∂

∂ρ+

1 − ζ

ρ

∂

∂ζ;

∂

∂ρ(−)=

∂

∂ρ− 1 + ζ

ρ

∂

∂ζ.

Then

V(±)XC (r) = εXC

(

ρ(r), ζ(r))

+ ρ(r)∂εXC

(

ρ(r), ζ(r))

∂ρ± [1 ∓ ζ(r)]

∂εXC

(

ρ(r), ζ(r))

∂ζ

≡ V LDAXC

(±)(

ρ(r), ζ(r))

. (3.42)

Since the energy won’t change in case of simultaneous reversal of all spins, εXC

(

ρ(r), ζ(r))

=

εXC

(

ρ(r),−ζ(r))

, and

∂εXC(ρ,−ζ)

∂ζ= −∂εXC(ρ, ζ)

∂ζ, hence V

(+)XC (ρ,−ζ) = V

(−)XC (ρ, ζ) .

Although seemingly poorly justified, the LDA works in practical calculations surprizinglywell. The reason is that the XC energy is the integral over the XC hole of density timesCoulomb interaction, see Eq. (3.25). The Coulomb interaction is spherically symmetricand leaves only spherically averaged, around r, part of ρXC(r, r′) to contribute to EXC.

39

The discussion to this effect can be found in the paper by Gunnarsson et al.18 and in thereview by Jones and Gunnarsson19 and can be illustrated by the following figures, whichshow the cuts of the exact exchange hole in the Ne atom in comparison with its localdensity approximation. The actual shape of the exchange hole is drastically differentfrom the approximated one (the true exchange hole in an atom has maximum at thenucleus whereas the averaged one is centered at the electron). However, the results ofaveraging over angles (with the center taken at the electron) are much closer for exactand LDA cases; moreover, only the radial integral of the angular-averaged function (timesdensity and the Coulomb factor) actually affects the exchange energy.

Fig. 5 of the Gunnarsson et al. (1979) paper. The exchange hole around an electron at r in

the neon atom shown as a function of distance from the electron along a line connecting the

electron and the nucleus. Full curves: the exact hole, dashed curves: the LDA result. Two

panels correspond to the electron at two different distances from the nucleus, 0.09 and 0.4 a.u.

Fig. 7 of the Gunnarsson et al. (1979) paper. Spherical average of the neon exchange hole

shown above.

18O. Gunnarsson, M. Jonson and B. I. Lundqvist, Description of exchange and correlation effects in

inhomogeneous electron systems, Phys. Rev. B 20, 3136 (1979).19R. O. Jones and O. Gunnarsson, The density functional formalism, its applications and prospects,

Rev. Mod. Phys. 61, 689 (1989).

40

Qualitatively, the success of the LDA even for such unhomogeneous densities as that inatoms can be traced to the fact that an important sum rule is by construction implementedin the LDA: the exchange-correlation hole is integrated to −1 everywhere in space, i.e.,for EXC[ρ] generally defined as

EXC[ρ] =e2

2

∫ρ(r) ρXC(r, r′ − r)

|r−r′| dr dr′ , (3.43)

∫

ρXC(r, r′−r) dr′ = −1 for all r . (3.44)

If one separates exchange-correlation hole into exact exchange part (as would follow fromthe HF treatment) and the correlation part as the rest, the corresponding sum rules wouldbe ∫

ρX(r, r′−r) dr′ = −1 ;∫

ρC(r, r′−r) dr′ = 0 .

The LDA substitutese2

2

∫ρXC(r, r′−r)

|r − r′| dr′ ⇒ εXC

(

ρ(r))

where the XC-energy density is some function of ρ(r) only. Because of this, density mayappear only as a prefactor to the integral over r′ and not in the integral over r′:

ρLDAXC (r, r′ − r) = ρ(r) [g(r, r′) − 1] ,

where g is the correlation function integrated over interaction strength, see Eq. (3.26).The performance of LDA in real systems with chemical bonding (solids, molecules) has

been addressed in large number of publications. When discussing ground-state properties,one can note that the equilibrium volume is usually within −5 − 0% of the experiment,i.e. almost always underestimated, that reflects the fact that the LDA somehow over-estimates the chemical bonding. Results for the bulk modulus, or generally on elasticconstants, vary from very good to hardly satisfactory; actually calculated elastic proper-ties are quite sensitive to a practical realization of DFT calculation (i.e., on the accuracyof representing a general-shape charge density). The same applies to the calculation ofphonon frequencies. For cohesive energies, the following table gives some idea about theaccuracy of LDA results. MJW (Moruzzi – Janak – Willliams) and VW (Vosko – Wilk)are two different “flavours”, i.e. different parametrizations of the LDA energy density.

Cohesive energy (in eV) calculated using the LDA

Li Na K Be Sr Al

LDA-MJW 1.65 1.12 0.90 3.97 1.89 3.84LDA-VW 1.74 1.21 0.96Exp. 1.66 1.26 0.94 3.33 1.70 3.34

Still, the error of LDA in reproducing ground-state properties is sometimes larger thanacceptable; there are some known errors in the equation of state (wrong crystal structurein the ground state of Fe etc.)

41

3.7 Self-interaction correction

One of shortcomings of LDA is the presence of spurious self-interaction. It leads to no-ticeable errors in light atoms (especially hydrogen) and in systems with strongly localizedstates. Let us consider the hydrogen atom,

ρ(+) = |φ(r)|2 ; ρ(−) = 0 ; ζ = 1 .

The exact expression for the XC energy would be:

EXC = −e2

2

∫ |φ(r)|2 |φ(r′)|2|r − r′| drdr′ ,

that exactly cancels Hartree energy (there is no electron density to interact with). Com-paring to the general expression (3.25), we see that this corresponds to vanishing cor-relation function, g(r; r′) = 0. In LDA, the XC energy is constructed implying somenon-zero correlation function. Explicitly for hydrogen atom, the electrostatic self-energy(e.g., Hartree term) of 8.5 eV is canceled by the XC energy of −8.1 eV to ≈95%.

A possible (not quite satisfactory) cure is to introduce explicit dependency on theindex of one-particle functions (Kohn–Sham orbitals) into the functional to be optimized.This procedure is called self-interaction correction (SIC):

ESIC = ELDA [ρ(r), ζ(r)]−∑

iσ

Θiσ , (3.45)

with Θiσ =e2

2

∫ρiσ(r) ρiσ(r

′)

|r − r′| dr dr′ + ELDA(

ρiσ(r), 1)

. (3.46)

Each correction term Θiσ involves the contribution to the density from the Kohn–Sham or-bital in question, which, if occupied, carries one electron and is hence fully spin-polarized:

ρiσ(r) =

|ϕiσ(r)|2 , ϕiσ occupied,0, otherwise

Similarly to the derivation of Kohn–Sham equations, Eq. (3.16), the variation of individualϕiσ yields:

[

− h2

2m∇2 + e2

∫ρ(x′)

|r − r′| dx + u(x) + V SICXC,α(x) − εα

]

ϕα(x) = 0 , (3.47)

where V SICXC,α(x), now taking place of VXC(x), is:

V SICXC,α(x) = VXC[ρ, ζ ](x) − VXC[ρα, 1](x) −

∫ρα(r′)

|r − r′| dr′ . (3.48)

In a sense, some similarity to the HF treatment is thus recovered (since there is no spuriousself-interaction in the HF), but VXC(x) may well include correlation effects beyond theexchange as well. The problems in the SIC scheme are the following:• The Hamiltonian is formally different for different Kohn–Sham functions, that results

42

in non-orthogonality of the orbitals;• The scheme is not invariant under a unitary transformation of the orbitals (i.e., differentresults may be obtained when using the basis of spherical harmonics or cubic harmonics).

Other useful extensions beyond the LDA can be sorted out into two groups. One groupof approximations deals with certain assumptions about the shape of the XC hole, using,for instance, appropriate model functions. The other group of approximations proceedswith treating inhomogeneities in a systematic way – either by Taylor expansion, or Fourierexpansion, of patially varying density. Let us consider several examples from both groups.

3.8 Averaged density approximation and

Weighted density approximation

These two approximations have been proposed and discussed in the above cited paper byGunnarsson – Jonson – Lundqvist (1979). The Averaged density approximation (ADA)uses, instead of density ρ at r only, an average of density over some vicinity of r. TheXC-density is then represented by

ρADAXC (r, r′−r′) = ρ(r)

1∫

0

[

g(

r−r′, λ, ρ(r))

− 1]

dλ , (3.49)

where g is some reasonable model for the pair correlation function. The averaging pro-cedure for ρ is not uniquely predetermined, and the freedom existing in its choice can beexploited to satisfy certain important criteria, like the sum rule (3.44) and consistencywith LDA results in the limit of constant density. Using such criteria, Gunnarsson emphetal. (1979) constructed the differential equation for the averaging function w that yields

ρ(r) =∫

w(

r−r′, ρ(r))

ρ(r′) dr′ ,

and solved this equation numerically. This approximation has not been widely used sofar.

The Weighted density approximation (WDA) uses another prescription for the XC-density,

ρWDAXC (r, r′−r′) = ρ(r′) G

(

|r−r′|, ρ(r))

(3.50)

that has the advantage of keeping a correct prefactor in the XC-density, compare Eqs. (3.25)and (3.43). ρ(r) is again an in some way averaged density, and G(r, ρ) is a model for thepair correlation function that doesn’t have to be exact; one is free to choose a convenientanalytical form which would guarantee the sum rule and correct asymptotics. A choiceby Gunnarsson and Jones20

G(r, ρ) = C(ρ)

1 − exp

[

−λ(ρ)

|r|5]

. (3.51)

20O. Gunnarsson and R. O. Jones, Phys. Scr. 21, 394 (1980); R. O. Jones and O. Gunnarsson, The

density functional formalism, its applications and prospects, Rev. Mod. Phys. 61, 689 (1989).

43

Two conditions,

ρ∫

G (|r|, ρ) dr = −1

and

ρe2

2

∫G (|r|, ρ)

|r| dr = εXC(ρ) ,

allow to determine C(ρ) and λ(ρ). The asymptotic of (3.51) is such that several importantlimiting cases are described correctly. The WDA can be constructed to be exact in thefollowing cases:(1) for homogeneous system;(2) for one-electron system, such as hydrogen atom (exact cancellation of the electronself-interaction);(3) for an atom, it provides correct asymptotic of the XC-energy far from nucleus,

εXC(r) = − e2

2r;

(4) far outside the crystal surface, the correct asymptotic of image potential is recovered:

εXC(z) = − e2

4z.

For comparison, the LDA holds only for (1), for (2) the cancellation is only partial, andthe criteria (3) and (4) are not satisfied.

Singh21 tested several flavours of WDA for ground-state properties of solids. Accordingto present-day experience, the WDA provides considerable improvement over LDA in thedescription of ground-state properties for essentially all systems. A selection of someresults by Singh is given in the following table.

Ground-state properties calculated within WDA by Singh (1993, 1998).a is lattice constant (in A), B - bulk modulus (in GPa)

C GaAs V KNbO3 SrO

a B a B a B a a

LDA 3.53 483 5.62 85 2.93 215 3.96 5.06WDA 3.57 404 5.74 62 3.00 145 4.02 5.16Exp. 3.57 442 5.65 75 3.02 157 4.02 5.16

21David J. Singh, Weighted-density-approximation ground-state studies of solids, Phys. Rev. B 48,14099 (1993); I. I. Mazin and D. J. Singh, Weighted Density Functionals for ferroelectric Materials,http://xxx...cond-mat/9801301

44

3.9 Inhomogeneities of density: k-space analysis and

gradient expansions

Another approach to inhomogeneities of particle density involves the Fourier transforma-tion and analysis of terms dependent on wavevector (of charge fluctuations). Below, wefollow the analysis of “inhomogeneous electron gas” in the original paper by Hohenbergand Kohn.

Consider the total energy functional as

E[ρ] =∫

v(r) ρ(r) dr +e2

2

∫ρ(r)ρ(r′)

|r − r′| dr dr′ + G[ρ] , (3.52)

where G[ρ] incorporates kinetic and exchange-correlation parts (the expression for therest is exact). We assume that spatial deviations of density from its mean value are small,

ρ(r) = ρ0 + ρ(r) , withρ(r)

ρ0

1 everywhere.

Moreover, the normalization (number of particles) remains constant, that leads to

∫

ρ(r) dr = 0 .

One can imagine the following formal expansion of G[ρ] in ρ as small parameter:

G[ρ] = G[ρ0] +∫

I(r) ρ(r) dr +

+∫

K(r, r′) ρ(r) ρ(r′) dr dr′ +

+∫

L(r, r′, r′′) ρ(r) ρ(r′) ρ(r′′) dr dr′ dr′′ + . . . (3.53)

The linear expansion coefficient must be zero, because the result cannot depend on theuniform shift ∆ of the coordinate system:

∫

I(r) ρ(r) dr =∫

I(r) ρ(r−∆)dr , hence∫

[I(r)−I(r+∆)] ρ(r) dr = 0 for any ∆;

therefore I(r) = const, and I∫

ρ(r) dr = 0, due to normalization of ρ(r).Similarly, the 2nd-order kernel K(r, r′) can only depend on |r− r′|, because the result

must be independent on the displacement of the ρ(r) distribution, as well as on rotationsabout the (r−r′) line. We skip the following terms in the expansion (3.53) for the moment,so

G[ρ] = G[ρ0] +∫

K(|r−r′|) ρ(r) ρ(r′) dr dr′ . (3.54)

We introduce the Fourier transformation of the quadratic kernel,

K(r − r′) =1

Ω

∑

k

K(k)e−ik(r−r′) (3.55)

45

and analyze the properties of K(k). For this, we introduce a small parameter λ in theFourier expansion of density:

ρ(r) = ρ0 +λ

Ω

∑

r

b(k) e−ikr . (3.56)

Hohenberg and Kohn argue that K(k) can be related (comparing terms with the samepowers of λ in the perturbation theory series for the interaction energy) to the electronicpolarizability α(k), or to dielectric constant ε(k) of the electronic liquid:

K(k) =2π

k2

[

1

α(k)− 1

]

=2π

k2

1

ε(k) − 1(3.57)

with ε(k) =E (primary field)

E − E · α︸ ︷︷ ︸

induced field

=1

1 − α(k).

The polarizability α(k) has the followinggeneral properties:

k → 0 : a(k) = 1 + c2 k2 + c4 k4 + ...

q → 2kF :dα

dk→ −∞;

q → ∞ : a(q) ∼ 1

q4.

Fig. 1 of Hohenberg and Kohn (1964).

Electronic polarizability α(q), for electron

density = 4 × 1022 cm−3.

For free electron gas, ε(k) is given by the Lindhard function, that leads for K(k):

K(k) =4π

k2· k2

8kF

[

1

2+

kF

2q

(

1 − k2

4kF

)

ln

∣∣∣∣∣

k + 2kF

k − 2kF

∣∣∣∣∣

]−1

. (3.58)

(∼ q2)

(const1 + const2 ·q4)

The prefactor 4π/k2 is the Fourier transform ofthe Coulomb interaction, and the rest – Fouriertransform of the correlation function. In orderto discuss exchange-correlation hole separately,we concentrate in the following on the exchange-

correlation local field factor

IXC(k) =k2

4πKXC(k) .

Fig. 2 of Hohenberg and Kohn (1964). Behavior of

the kernel K(q), for the same electron density.

46

Here, KXC(k) refers to the XC part of the kernel only, with the contribution due to kineticenergy of non-interacting particles subtracted.

For IXC(k), Monte Carlo results are available for several values of rc, as well as differ-ent analytical approximations. Actually, any model concretizing the exchange-correlationenergy density, be it LDA or different flavours of WDA, can be mapped onto a correspond-ing local-field XC factor. Examples have been given, e.g., by Mazin and Singh (1998).22

0.2

0.4

0.6

0.8

1

1.2

1.4

I xc(

q)

Gunnarsson-JonesPerdew-WangGritsenko et alLDAMC interpolation

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

I xc(

q)

Gunnarsson-JonesPerdew-WangGritsenko et alLDAMonte CarloMC interpolation

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4

I xc(

q)

q/kF

Gunnarsson-JonesPerdew-WangGritsenko et alLDAMonte CarloMC interpolation

Fig. 1 of Mazin and Singh (1998). Exchange-

correlation local field factor in the WDA by

Gunnarsson and Jones 1980 (Phys. Scr. 21, 394),

Gritsenko et al. 1993 (Chem. Phys. Lett. 205,

348), and derived from the homogeneous electron

gas pair correlation function, Perdew and Wang

1993 (Phys. Rev. B 46, 12947), as compared with

the Monte Carlo results and the interpolating

formula thereof, as given by Moroni, Ceperley and

Senatore 1995 (Phys. Rev. Lett. 75, 689). Den-

sities, from top to bottom, correspond to rs=1, 2, 5.

One can see that the LDA prescription forthe local field factor proceeds as KXC(0)/(4π) ·k2,and it is a quite good approximation if fluc-tuations of density are of long wavelength, k <2kF. At k = 2kF, the local-field factor, as thekernel K(k)) in general, experiences a kink, thatgives rise to Friedel oscillations of electron den-sity in the direct space. Beyond 2kF, IXC(k)increases ∼ k2 rather than tending to a con-stant; this is an indication that EXC includessome kinetic-energy part.

A different method of taking into account inhomogeneities of density is via gradientexpansions. One can construct the following expression for the energy density:

g[ρ] = g0(ρ) +3∑

i=1

gi(ρ)∇i ρ +3∑

ij

[

g(1,1)ij (ρ)∇iρ∇jρ + g

(2)ij ∇i∇jρ

]

+ . . . (3.59)

If the density is scaled as

ρ(r) = ρ′(r

r0

) ,

22I. I. Mazin and D. J. Singh, Nonlocal density functionals and the linear response of the homogeneous

electron gas, Phys. Rev. B 57, 6879 (1998)

47

the above series is an expansion in powers of (r−10 ). It does not strictly converge. However,

we’ll use it to sort out leading terms, inclusing gradients of density. The validity of thegradient expansion requires

|∇ρ|ρ

qF ;|∇i∇jρ||∇ρ| qF .

Making use of invariance relations under uniform displacements and rotations, one arrivesat the following form:

g[ρ] = g0(ρ) + g(2)2 (ρ)∇ρ · ∇ρ +

+[

g(2)4 (ρ)(∇2ρ)(∇2ρ) + g

(3)4 (ρ)(∇2ρ)(∇ρ · ∇ρ) + g

(4)4 (ρ)(∇ρ · ∇ρ)2

]

+ O(∇6i ) .

The expansion coefficients can be expressed via Taylor expansion coefficients of electronicpolarizability α(q).

A formally correct gradient expansion

EXC[ρ] =∫

ρ(r)εLDAXC

(

ρ(r))

dr +∫

BXC

(

ρ(r))

|∇ρ(r)|2dr + . . . (3.60)

meets certain problems in its practical implementation:– the terms of the 4th order and higher in the expansion of the exchange energy diverge;– the exchange potential diverges already in the 2d order;– in the 2d order, gradient corrections to exchange energy bring an improvement,

but gradient corrections to corellation energy yield worse results that the LDA.The last fault can be traced to the fact that the normalization of the XC-density (3.44),valid in the LDA, does not necessarily hold in a truncated gradient expansion (3.60).Particularly, it is violated in what regards the correlation hole (that must integrate tozero around any electron). Therefore, one has to impose necessary constrains, probablyusing additional gradient terms (nonlinear and not necessarily quadratic) beyond thoseincluded in the formal gradient expansion (3.60). These working schemes with explicitenonlinear dependence of the integrand on density and its gradients are refered to asdifferent flavours of generalized gradient approximation (GGA)

EGGAXC [ρ] =

∫

ρ(r)εGGAXC

(

ρ(r), |∇ρ(r)|,∇2ρ(r))

dr . (3.61)

The parametrization of εGGAXC is chosen so as to satisfy certain limiting cases, like the LDA

limit, scaling property EX[ρλ] = λEX[ρ] etc.An example of such arguing was given by Becke23 who introduced a now broadly

used (especially in quantum chemistry applications) form of a GGA approximation to theexchange energy. The correlation was explicitly left out in the Becke analysis, in order touse HF results as a benchmark for tuning this density functional scheme. The asymptoticproperties explicitly satisfied were the following: the Coulomb potential of the exchangecharge, i.e. of Fermi hole density far from the electron distribution

limr→∞

∫ρσ

X(r, r′)

|r− r′| dr′ = −1

r, (3.62)

23A. D. Becke, Density-functional exchange-energy approximation with correct asymptotic behavior,Phys. Rev. A 38, 3098 (1988)

48

and the exchange energy defined by Eq. (3.27) must behave according to this asymptotic.Morever, the asymptitic of the spin density is

limr→∞

ρσ(r) = e−aσr , (3.63)

where aσ is a constant related to the ionization potential of the system. The functionalform of the exchange energy proposed by Becke guarantees that the density satisfyingEq. (3.63) generates asymptotic potential as in Eq. (3.62):

E(Becke)X = ELDA

X − β∑

σ

∫

ρ4/3σ

x2σ

1 + 6βxσ sinh−1 xσ

dr (3.64)

where xσ is the dimensionless ratio

xσ =|∇ρσ|ρ

4/3σ

and β is an adjustable parameter, for which Becke proposed, based on least-square fit toHartree-Fock results over a variety of systems, a value of β=0.0042 a.u.

A more sophisticated form incorporating both exchange and correlation, that satis-fied more essential conditions, has been proposed in 1991 by Perdew and Wang24 forεGGAXC in Eq. (3.61), in terms of ρ and ζ of Eq. (3.32); rs as related to ρ by Eq. (3.35);

g = [(1+ζ)2/3+(1−ζ)2/3]/2; two differently scaled density gradients s = |∇ρ|/(2ρkF) and

t = |∇ρ|/(2gρ√

4kF/π), where kF = (3π2/ρ)1/3 is identical to definition of Eqs. (1.7) and

(2.35), and many tabulated numerical parameters:

ε(PW91)XC = εX + εC ;

εX = εLDAX (ρ, ζ)

[

1 + a1s sinh−1(a2s) + (a3 + a4 e−100s2

)s2

1 + a1s sinh−1(a2s) + a5s4

]

;

εC = εLDAC (ρ, ζ) + H(ρ, t) ,

H =g3β2

2αlog

[

1 +2α

β

t2 + At4

1 + At2 + A2t4

]

+ Cc0 [Cc(ρ) − Cc1] g3t2e

−100g4 4

πkF

t2,

A =2α

β

exp

[

−2αεLDAC (ρ, ζ)

g3β2

]

− 1

−1

,

Cc(ρ) = C1 +C2 + C3rs + C4r

2s

1 + C5rs + C6r2s + C7r3

s

. (3.65)

The inconvenience of having many adjustable parameters in a practical GGA scheme mo-tivated Perdew, Burke and Ernzerhof25 to drop several least important constraints in theformulation of GGA and present a somehow more simplified, as compared to Eq. (3.65),

24J. P. Perdew, in Electronic Structure of Solids ’91, edited by P. Ziesche and H. Eschrig (AcademieVerlag, Berlin, 1991), p. 11, reproduced in Perdew et al., Phys. Rev. B 46, 6671 (1992).

25John P. Perdew, Kieron Burke and Matthias Ernzerhof, Generalized Gradient Approximation Made

Simple, Phys. Rev. Lett. 77, 3865 (1996)

49

and simultaneously more physically transparent scheme. The effect of inhomogeneity ofρ is isolated in the “enhancement factor” FXC, modulating the local exchange:

E(PBE)XC

[

ρ(+), ρ(−)]

=∫

ρ(r) εunif.X (r)FXC(rs, ζ, s) dr . (3.66)

Whereas the correlation part of FXC(rs, ζ, s) is ex-pressable via the function H resembling (but ana-lytically more simple than) that of Eq. (3.65), theexchange part could be mapped onto

FX(s) = 1 + κ − κ

1 + µs2/κ,

where µ ' 0.21951 and κ = 0.804 are numericalconstants. The figure on the right shows thebehaviour of the “enhancement factor” for rele-vant values of rs and s in non-polarized and fullypolarized cases.

Fig. 1 of Perdew, Burke and Ernzerhof (1996).

Encancement factors FXC shownig GGA non-locality.

Solid curves denote the PBE results, open circles –

those of PW91.

An extensive literature exists on the performance of GGA in different systems, see,e.g., Dal Corso et al. (1996).26 Generally, GGA tends to correct to some extent theoverbonding that is typical in LDA results. The equilibrium volumes are typically largeraccording to calculations using GGA, and in many cases agreement with experimentimproves. In other cases, the GGA slightly overestimates the volume. Quite importantis the inclusion of GGA in the treatment of magnetic systems; one of the most knownexamples of LDA failures is the hierarchy of energies of different (magnetic and structural)phases of iron, where the LDA wrongly predicts fcc phase to be favourable over the bcc.The inclusion of gradient corrections restores the correct relation between these phases.

26Andrea Dal Corso, Alfredo Pasquarello, Alfonso Baldereschi and Robeto Car, Generalized-gradient

approximations to density-functional theory: A comparative study for atoms and solids, Phys. Rev. B 53,1180 (1996)

50