Computational Physics

118 Density functional theory

Table 5.4.Parameters for correlation energy

Unpolarised Polarised

A 0.0311 0.01555

B −0.048 −0.0269

C 0.0020 0.0014

D −0.0116 −0.0108

γ −0.1423 −0.0843

β 1 1.0529 1.3981

β 2 0.3334 0.2611

(b) [C] A parametrised form ofεcis given by the following expressions. Forrs≥ 1

we have

εc=γ/( 1 +β 1

√

rs+β 2 rs)

and forrs> 1

εc=Alnrs+B+Crslnrs+Drs.

From this, we obtain the following expressions for the correlation potential:

Vc(rs)=εc

1 + 7 / 6 β 1 √rs+β 2 rs

1 +β 1

√

rs+β 2 rs

forrs≥1 and

Vc(rs)=Alnrs+B−A/ 3 +

2

3

Crslnrs

+( 2 D−C)rs/3.

The values of the parametersA,Betc. depend on whether we are dealing with the

polarised (all spins samezcomponent) or unpolarised case. For both cases, the

values are given inTable 5.4.

Use this parametrisation in your helium density functional theory program

(unpolarised). You should find an energyE=−2.83 atomic units,to be compared

with−2.72 without this correction.

(c) [C] Use the polarised parametrisation for the hydrogen program of the previous

problem. You should find an energyE=−0.478 a.u.

(d) [C] It is also possible to combine the self-energy correction with the correlation

energy. You should consult the paper by Perdew and Zunger[9], if you intend to

do this. This results in an energyE=−2.918 a.u., which is only 0.015 a.u. off the

experimental value.

5.4 In this problem, we consider a generalisation of Koopman’s theorem (see
Section 4.5.3) to the density functional formalism. To this end, we consider the
spectrum{εi}and the corresponding eigenstates of the Kohn–Sham Hamiltonian.
We consider the chemical potential, which is found by removing a small amount of