Computational Physics

146 Solving the Schrödinger equation in periodic solids

d

x

y

z

Figure 6.12. The diamond structure.

There are numerous review articles on the pseudopotenial method and readers who
are interested in the subject are referred to those by Heine [10], Brust [20] and
Pickett [21].

6.7.1 A pseudopotential band structure program for silicon

In this section, the construction of a pseudopotential program for silicon is
described. For details, the review by Brust [20] and the paper by Chelikowsky
and Cohen [22] may be consulted.
Silicon is considered here in the diamond structure which is a fcc lattice with,
at each lattice point, two atoms at relative positions± 1 / 8 (a 1 +a 2 +a 3 )(see
Figure 6.12). We have already described the fcc crystal structure and the special
points in the first Brillouin zone inSection 6.5. The lattice constant is 5.43a 0. The
pseudopotential is given in the convenient form of a few Fourier components (see
above). We restrict the number of coefficients by assuming the pseudopotential to
be a repetition of spherically symmetric potentials in cells surrounding the atoms,
which leads to the following form of the Fourier components of the pseudopotential
Vpsarising from a single atom per cell:

Vps(at)(K)=

1

Vcell

∫

cell

d^3 rVps(at)(r)e−iK·r. (6.53)

The Fourier components depend only on the length of the wave vectorK, and this
property reduces the number of independent Fourier coefficients.
Another reduction comes about when calculating the Fourier compon-
ent of the sum of the potentials arising from the two atoms at positions