Computational Physics

148 Solving the Schrödinger equation in periodic solids

Γ XW L Γ K

k

E

0.0

0.5

Figure 6.13. Band structure of silicon.

6.7.2 Accurate energy-independent pseudopotentials

Suppose we have a pseudopotential that gives exactly the same phase shift as the
full potential. In the valence region, the wave functions have the same shape as
for the full potential, but their normalisation may differ: the wave functions in
the valence region for the two potentials are the same only up to a scaling factor.
The point is that if two normalised wave functions differ within the core region
while being similar (up to a multiplication constant) in the valence region, their
respective charges will be distributed differently among core and valence regions.
The resulting charge difference is calledorthogonality holeand one should correct
for it, for example by rescaling the full pseudo-wave function.
It turns out that the normalisation of the states is related to energy dependence of
the pseudopotential. It can be shown (see Problem 6.1) that for thefullpotential, the
charge inside a sphere around the nucleus with radiusRccarried by the solutionψ
of the Schrödinger equation evaluated at energyEis related to the energy derivative
of the wave function atRc:

∫

core

d^3 r|ψ(r)|^2 =−

1

2

∫

d R^2 cψ(r)

∂^2 ψ(r)

∂r∂E

∣∣

∣∣

r=Rc

(6.59)

where the integration is carried out over the spherical angles, d =d cosθdφ. This
is another instance of the relation between norm and energy derivative which was
previously encountered in connection with the energy derivatives of the solution of
the radial equation in the LAPW wave functions inSection 6.6.