134 Solving the Schrödinger equation in periodic solids
V(r)Ψ(r)Figure 6.7. Valence state and Coulomb potential in a crystal.becomes small. Second, the potential is approximately spherically symmetric near
the nuclei, whereas at larger distances the crystal symmetry dominates.
Consider the valence state shown in Figure 6.7, together with the potential. Close
to the nucleus, the valence state feels the strong spherically symmetric Coulomb
potential and it will oscillate rapidly in this region. In between two nuclei, the
potential is relatively weak and the orbital will oscillate slowly. The shape of this
valence function can also be explained in a different way. Suppose that close to
the nucleus, where the potential is essentially spherically symmetric, the valence
wave function has s-symmetry (angular momentum quantum numberl=0), i.e.
no angular dependence. There might also be core states with the same symmetry.
As the states must be mutually orthogonal, the valence states must oscillate rapidly
in order to be orthogonal to the lower states. A good basis set must be able to
approximate such a shape using a limited number of basis functions.
For each Bloch wave vector, we need a basis set satisfying the appropriate Bloch
condition. The most convenient Bloch basis set consists of plane waves:
ψkPW+K(r)=exp[i(k+K)·r]. (6.25)For a fixed Bloch vectorkin the first Brillouin zone, each reciprocal lattice vector
Kdefines a Bloch basis function fork. If we take a sufficient number of such basis
functions into account, we can match any Bloch function for a Bloch vectork.
However, Figure 6.7 suggests that we would need a huge number of plane waves
to match our Bloch states because of the rapid oscillations near the nuclei. This is
indeed the case: the classic example is aluminium for which it is estimated that 10^6
plane waves are necessary to describe the valence states properly[10]. Although
plane waves allow efficient numerical techniques to be used, this number is very high
compared with other basis sets, which yield satisfactory results with of the order of
only 100 functions. Plane waves can only be used after cleverly transforming away
the rapid oscillations near the nuclei, as in pseudopotential methods (Section 6.7).