Computational Physics

128 Solving the Schrödinger equation in periodic solids

k

Γ XW L Γ K

Energy (atomic units) 0.0

0.1

0.2

0.3

0.4

–0.1

0.5

Figure 6.3. Band structure of aluminium. Also shown (open squares) is the free

electron result.X,Wetc. are special points in the Brillouin zone (seeSection 6.5).

The coefficientsCp(k)can be found from the variational principle. Applying the
techniques ofChapter 3, we see that we must solve the generalised eigenvalue
problem

HC(k)=ESC(k) (6.15)

for the vectorC(k)with componentsCp(k). The matrix elements of the Hamiltonian
Hand of the overlap matrixS(which depend onk) are given by

Hpq=〈φp,k|H|φq,k〉and (6.16a)

Spq=〈φp,k|φq,k〉. (6.16b)

Writing out the matrix elements using (6.13) and the lattice periodicity, we obtain
the following expressions:

Hpq=

∑

R

eik·R

∫

d^3 rup(r−R)Huq(r) (6.17a)

Spq=

∑

R

eik·R

∫

d^3 rup(r−R)Suq(r). (6.17b)

As the statesup(r)are rather strongly localised near the nuclei, they will have
virtually no overlap when centred on atoms lying far apart. This restricts the sums
in (6.17) to only the first few shells of neighbouring atoms, sometimes only nearest