Computational Physics

6.3 Approximations 127

Energy

k

Figure 6.2. Nearly free electron spectrum for a periodic potential in one dimension.

consisting of reciprocal pointsqwhich satisfy

|q|=|K−q|, (6.12)

whereKis a reciprocal lattice vector. At a Bragg plane, a band gap of size 2|Vq|
opens up.Figure (6.2)gives the resulting bands for the one-dimensional case.
Figure (6.3)shows how well the bands in aluminium resemble the free electron
bands.

6.3.2 The tight-binding approximation

The tight-binding (TB) approximation will be discussed in more detail as it is an
important way for performing electronic structure calculations with many atoms
in the unit cell. It naturally comes about when considering states which are tightly
bound to the nuclei. The method is essentially a linear combination of atomic
orbitals (LCAO) type of approach, in which the atomic states are used as basis
orbitals. Let us denote these states, which are assumed to be available from some
atomic electronic structure calculation, byup(r−R), in which the indexplabels the
levels of an atom located atR. From these states, we build Bloch basis functions as

φp,k(r)=

1

√

N

∑

R

eik·Rup(r−R), (6.13)

and a general Bloch state is a linear combination of these:

φk(r)=

∑

p

Cp(k)φp,k(r). (6.14)