Computational Physics

Exercises 117

5.2 [C] The Hartree energy

EH=

1

2

∫
d^3 rd^3 r′
n(r′)n(r)
|r−r′|
overestimates the classical electrostatic energy of the electrons because it includes
interactions of the electrons with themselves – these are the so-called
self-interactions. In Hartree–Fock theory, this spurious effect is cancelled by the
exchange energy. In density functional theory, the exchange correlation energy does
not ensure this cancellation a priori and we can only hope that it cancels the
self-interaction as much as possible. To see to what extent the exchange correlation
potential succeeds in doing so, we consider the hydrogen atom in DFT (of course,
DFT was designed for many-electron systems, but its use is not a priori restricted to
systems containing more than one electron). In the hydrogen atom, we find a
nonvanishing Hartree and exchange correlation energy, which can easily be evaluated
with our DFT program for helium.
Change the nuclear charge back toZ=1 and make sure that the density used in
the Hartree and exchange correlation energies is evaluated for the single electron (i.e.
not multiplied by 2 as in the helium case). Evaluate both energies for the exact
solution of the hydrogen atom.
You should find that the exchange correlation energy compensates about 80% of
the self-interaction. For better exchange correlation energies, a value of 96% can be
found–seethefollowingproblemandRef. [9].SeealsoRef. [46] for more examples.
5.3 [C] The Slater exchange potential

Vx(r)=−

(

3

π

) 1 / 3

n^1 /^3 (r)

is based on the exchange energy in a homogeneous electron gas[47]. It is quite a crude

approximation, and a refinement can be made using quantum Monte Carlo results

obtained by Ceperley and Alder [9, 11, 45]. This leads to a parametrised correlation

energy which should beadded tothe Slater term given above. The parametrisation is

given in terms of the parameterrswhich is related to the densitynaccording to

n=

3

4 πrs^3

.

The parametrisation is split into two parts:rs≥1 andrs<1. We need an expression

for the correlation energy parameterεcdefined by

Ec=

∫

d^3 rn(r)εc(n)n(r).

(a) Show that from this an expression for the correlation potentialVccan be derived

according to

Vc(rs)=

(

1 −

rs

3

d

drs

)

εc(rs).