Computational Physics

5.3 Exchange and correlation: a closer look 97

5.3 Exchange and correlation: a closer look

5.3.1 The adiabatic theorem and the normalisation conditions

In this section we consider exchange and correlation in more detail. We shall take
into account the spin as well as the spatial coordinates. All spin-space coordinates
(r 1 ,s 1 ;...rN,sN)are denoted byX. Let us first consider the exact energy-functional
(of the spin-orbitals):

Eexact=

∫

AS∗ (X)

(

−

1

2

∑

i

∇i^2 +Vext+Vee

)

AS(X)dX. (5.26)

Here,AS(X)is a wave function which is antisymmetric in thexi=(ri,si),but
not necessarily a Slater determinant. We compare the exact energy with the Kohn–
Sham functional (which should also be exact for the correct exchange-correlation
functional):

EKS=−

∑

k

∫

ψk∗(x)

1

2

∇k^2 ψk(x)dx+

∑∫

n(r)Vext(r)dx

+

1

2

∫

n(r)

1

|r−r′|

n(r′)d^3 rd^3 r′+Exc[n]. (5.27)

The terms related toVext(r)are the same in both cases: the exchange and correlation
termExcmakes up for the difference in the kinetic energies and the difference
between the exact Coulomb interaction and the Hartree approximation in the Kohn–
Sham scheme.
We now try to connect the exact form to the Kohn–Sham picture in order to
pinpoint this difference better. This is done in theadiabatic connectionprocedure,
which works as follows. We first introduce a tunable electron–electron interaction

Vc,λ=

∑

ij

λ

|ri−rj′|

=λVc, (5.28)

where the subscript C stands for Coulomb and whereVcis identified withVc,λ= 1.
Just as we did in Section 5.1.2, we split the many-body Hamiltonian into that of
an homogeneous electron gas with interactionVλand the external potential:

Hλ=H0,λ+

∑

i

Vext(ri)=(T+Vc,λ)+

∑

i

Vext(ri), (5.29)

and note that forfixeddensitiesn(r), the last term will always give the same con-
tribution to the energy. Indeed, we minimise this Hamiltonian for such a fixed
density:

Eλ[n]=min

ψ|n

〈|H0,λ|〉+

∫

Vext(r)n(r)d^3 r=Fλ[n]+

∫

Vext(r)n(r)d^3 r,

(5.30)