Computational Physics

100 Density functional theory

which follows directly from the normalisation of the wave function. Furthermore,
∫
n(x)dx=N (5.46)

for the same reason. IntegratingEq. (5.43)now over dx′(this actually denotes an
integration overr′and a sum overs′), we obtain the result

(N− 1 )n(x)=Nn(x)+n(x)

∫

nxc(x,x′)dx′; (5.47)

in other words, ∫

nxc(x,x′)dx′=−1. (5.48)

Realising that the second term inEq. (5.44)is the exchange correlation correction to
the Coulomb energy, we see that this correction can be described in terms of a charge
distribution which carries a positive unit charge: this is the exchange correlation
holenxc.
Now let us return to the Hartree–Fock approximation. There we considered a
Slater determinant containing all exchange effects. If we apply the above analysis
to a Slater determinant we obtain an exchange hole (Coulomb correlations are
absent in this case) which integrates up to a charge−1 (that is, a positive hole),
irrespective of the strengthλof the Coulomb interaction. Therefore we conclude
that the exchange hole adds up to−1 and, supposing that the exchange correlation
hole is the sum of an exchange and a correlation contribution, the correlation hole
must add up to 0.
Let us summarise the results obtained so far. The first is that we can remove the
exchange and correlation contribution to the kinetic energy from the description
by applying the adiabatic connection formula. The price we have to pay is that
we have to integrate the Coulombic term due to exchange and correlation over the
interaction strengthλ. The second result is that this contribution can be described
in terms of an exchange and a correlation hole, the first of which integrates up to− 1
and the second integrates to 0.

5.3.2 The generalised gradient approximation

We can now understand the success of LDA: the exchange and correlation holes are
taken from very accurate quantum Monte Carlo results for the homogeneous elec-
tron gas and therefore they satisfy the two normalisation conditions for exchange
and correlation just described.
We can now also describe how a gradient expansion can be constructed: we must
take into account isotropy conditions and then make sure that the exchange and the
correlation hole satisfy their respective normalisation conditions. This scheme has