Computational Physics

6.7 The pseudopotential method 149
If we now consider an energy-dependent pseudopotentialVpswith eigenstateφ,
we obtain, aside from the surface integral on the right hand side, an integral over
the energy derivative ofVps:

∫

core

d^3 r|φ(r)|^2 =−

1

2

∫

d R^2 cφ(r)

∂^2 φ(r)

∂r∂E

∣∣

∣∣

r=Rc

+

∫

core

d^3 r

∂V(r,E)
∂E
|φ(r)|^2.
(6.60)
The first term on the right hand side is equal to the right hand side in (6.59) if we
fix the amplitude ofφto be equal to that ofψatRc. Therefore, if both solutions
have the same amount of charge inside the core region, the second term on the
right hand side must vanish, which implies that the pseudopotential is independent
of energy. This means that if we have solved the orthogonality hole problem, we
have obtained an energy-independent pseudopotential, so that we have solved two
problems at once.
Bachelet, Hamann and Schlüter [23] have constructed accurate norm-conserving
pseudopotentials and we refer to their paper and to the review article by Pickett [21]
for further details. Goedecker, Teter and Hutter [24] have developed a particularly
convenient type of pseudopotential which, being based on Gaussian functions,
allows for analytic Fourier transforms. We shall use this pseudopotential in the
following section in the construction of a fully self-consistent pseudopotential
program.

6.7.3 Building a self-consistent pseudopotential program

The construction of a fully self-consistent pseudopotential program is quite elabor-
ate – we shall therefore restrict ourselves to the case of a cubic unit cell and instead
of summing over all points in the Brillouin zone, we shall only consider the-point
(i.e. reciprocal vectorK= 00 0). This restriction is often applied when dealing with
molecules: the cell is taken big enough to ensure that the electron density vanishes
near the cell boundary. This choice therefore renders our program more suitable
for molecular systems or clusters than for periodic solids. However, the method for
periodic systems uses very similar techniques, and the interested reader is invited
to extend his or her program to that case.
It is important to build up the program in a step by step fashion and check each
step very carefully. The steps are described below. We closely follow the setup of
the CPMD program, described in the review paper by Marx and Hutter[25]. For
more details concerning the program and further background, that paper should be
consulted.
We start with some remarks concerning definitions and conventions relating
to Fourier transforms and choice of basis functions. For simplicity, we restrict