Computational Physics

144 Solving the Schrödinger equation in periodic solids

V

V

1

2

2

1

r

V

ψ

ψ

Figure 6.11. The principle of the pseudopotential. The wave functions of the full

potential( 1 )and of the pseudopotential( 2 )are equal beyond some radius.

6.7 The pseudopotential method

We have already seen that the main problem in calculating band structures is the
deep Coulomb potential giving rise to rapid oscillations close to the nuclei. In the
pseudopotential method this problem is cleverly transformed away by choosing a
potential which is weak. It is not immediately obvious that this is possible: after
all, the solutions of the problem with a weak and with a deep potential can hardly
describe the same system! The point is that the pseudopotential does not aim at
describing accurately what happens in the core region, but it focuses on the valence
region. A weak potential might give results that outside the core region are the same
as those of the full potential.
In order to obtain a better understanding of this, we must return to Chapter 2,
where the concept of phase shift was discussed. The phase shift uniquely determines
the scattering properties of a potential – indeed, we seek a weak pseudopotential
that scatters the valence electrons in the same way as the full potential, so that the
solution beyond the core region is the same for both potentials. An important point
is that we can add an integer timesπto the phase shift without changing the solution
outside the core region, and there exist therefore many different potentials yielding
the same valence wave function. To put it another way: if we make the potential
within the core region deeper and deeper, the phase shift will increase steadily,
but an increase byπdoes not affect the solution outside. The pseudopotential is a
weak potential which gives the same phase shift (moduloπ) as the full potential
and hence the same solution outside the core region. The principle is shown in