Computational Physics

5.3 Exchange and correlation: a closer look 99

the full many-body wave function into account: it is the exchange-only part of the
correction.
There is another fruitful way of looking at the exchange correlation term, which
is related to the discussion in Section 5.1.2. There we considered the probability
density for finding the particles 1 and 2 with coordinatesxandx′respectively:

P(x,x′)=

∫

|(x,x′,x 3 ,...,xN)|^2 dx 3 ...dxN. (5.38)

We now use this definition for a general wave function (not necessarily a Slater
determinant).
The single-particle density is given as

n(x)=N

∫

|(x,x 2 ,...,xN)|^2 dx 2 ...dxN. (5.39)

Integrating the single-particle density gives the number of particles:
∫
n(x)dx=N. (5.40)

From the definition ofn(x)we immediately see that

N

∫

P(x,x′)dx′=n(x). (5.41)

The reason for introducing these quantities is that they give insight in the exchange
and correlation energies. To see this, consider the Coulomb energy:

Ec=

N(N− 1 )

2

∫

P(x,x′)

1

|r−r′|

dxdx′ (5.42)

(the prefactor counts the number of particle pairs). We now define theexchange
correlation hole,nxc(x,x′), through

N(N− 1 )P(x,x′)=n(x)n(x′)+n(x)nxc(x,x′). (5.43)

The exchange correlation hole indicates how the actual distribution of a second
electron, given a first electron atx, deviates from the average density. Then we can
write

Ec=EH+

1

2

∫

n(x)nxc(x,x′)

1

|r−r′|

dxdx′. (5.44)

Note that the second term can be identified with the exchange correlation energy.
The most important consequence of this is that we can derive some properties
of the exchange correlation hole, which any exchange correlation energy should
satisfy. The first of these properties follows from the normalisation ofP:
∫
P(x,x′)dxdx′= 1 (5.45)