Computational Physics

106 Density functional theory

is given as the integral of the induced charge n(r′′|r)overr, weighted by|ψ(r)|^2.
Putting these results back into Eq. (5.56) leads to the formal expression

E=

1

2

∫

δ(r−r′)[w(r,r′)−v(r,r′)]|ψ(r)|^2 d^3 rd^3 r′. (5.58)

If we take the functional derivative of this expression with respect toψ(r),we
obtain a term

Vψ(r)=

1

2

∫

δ(r−r′)[w(r,r′)−v(r,r′)]d^3 r′ψ(r) (5.59)

in the one-particle Schrödinger equation. Exchange is already treated correctly, so
in the exchange term, we can simply replace the bare Coulomb interaction by the
screened interaction. We see that the correction boils down to taking into account
the COulomb Hole and Screened EXchange – hence the name COHSEX.
Now there is still something missing: we do not know the screened interaction
potentialw(r,r′). This can however be found in a so-called random phase approx-
imation (RPA) scheme, which is based on perturbation theory. It works as follows.
We place a test charge at positionr. As we have seen above, this test charge gener-
ates a change nof the resident charge distribution, and the bare potentialv(r,r′)is
replaced by the screened potentialw(r,r′). The relation between the two is usually
formulated in terms of the dielectric constant. This is defined as:

v(r,r′)=

∫

ε(r′,r′′)w(r,r′′)d^3 r′′. (5.60)

We can therefore write for the screening correction

w(r,r′)−v(r,r′)=

∫

[δ(r′,r′′)−ε(r′,r′′)]w(r,r′′)d^3 r′′

=

∫

n(r′′|r)v(r′,r′′)d^3 r′′, (5.61)

where n(r′′|r)denotes a change of the density atr′′due to a unit point charge
being placed atr.
Now we view the effect of this point charge atras a perturbing potentialw. The
lowest order correction to the occupied orbitaljin stationary perturbation theory is
given by[31]

ψj(r′)=

∑

kunocc.

〈ψk|w|ψj〉

Ej−Ek

ψk(r′). (5.62)