Computational Physics

References 119

charge from the system. In practice this means that the highest level (which is level

N) is not fully occupied. We usually calculate the density according to

n(r)=

∑N

i= 1

fi|ψN(r)|^2.

The change in the density is realised by reducing the valuefNslightly:

fN→fN−δfN.

This induces a change in the density

δn(r)=δfN|ψN(r)|^2.

The total energy is calculated according to:

E(N)=

∑N

i= 1

fiεi−

∫

1

2

d^3 rd^3 r′

n(r)n(r′)

|r−r′|

−

∫

d^3 rn(r)Vxc(r)+Exc[n].

The levelsεiarise from taking the matrix elements〈ψi|H(N)|ψi〉. As a result of the

change in density, both the Hamiltonian occurring in these matrix elements and the

remaining terms in the energy expression change.

We therefore have three contributions to the change in the total energy. First, the

factorfNin the sum over the energy levels changes; second, the potential for which

the levels are calculated changes slightly; and third, the correction terms in the

expression for the energy change.

Show that, to linear order inδn(r), the combined effect of the change in the

Hamiltonian matrix elements is precisely compensated by the change in the

remaining terms in the energy expression so that we obtain

E(N)−E(N−δfN)=εNδfN

Hint: the change in the exchange correlation energyExc[n]is given by the expression

δExc[n]=

∫

d^3 rδn(r)

δExc[n]

δn(r)

=

∫

d^3 rVxc[n](r)δn(r).

This proves Janak’s theorem [25].

References

[1] C. Pisany, R. Dovea, and C. Roetti,Hartree–Fock Ab-initio Treatment of Crystalline Systems.
Berlin, Springer, 1988.
[2] P. Hohenberg and W. Kohn, ‘Inhomogeneous electron gas,’Phys. Rev., 136 (1964), B864–71.
[3] W. Kohn and L. J. Sham, ‘Self-consistent equations including exchange and correlation effects,’
Phys. Rev., 140 (1965), A1133.
[4] R. O. Jones and O. Gunnarsson, ‘The density functional formalism, its applications and
prospects,’Rev. Mod. Phys., 61 (1989), 689–746.
[5] S. Lundqvist and N. March,Theory of the Inhomogeneous Electron Gas. New York, Plenum,
1983.