Computational Physics

140 Solving the Schrödinger equation in periodic solids

Γ XW L Γ K

k

E

0.0

0.1

0.2

0.3

Figure 6.10. Band structure of fcc copper. The Fermi energy is shown as a

horizontal dashed line.

quotientR′l(R)/Rl(R)as obtained from this solution in (6.34). Our program will
not be self-consistent as we shall use a reasonable one-electron potential.^2
It is best to use some numerical routine for calculating the determinant. If such
a routine is not available, you can bring your matrix to an upper-triangular form
as described in Appendix A8 and multiply the diagonal elements of the resulting
upper triangular matrix to obtain the determinant.
If you have a routine at your disposal which can calculate the determinant
for eachk-vector and for any energy, the last step is to calculate the eigenval-
ues (energies) at somek-point. This is a very difficult step and you are advised
not to put too much effort into finding an optimal solution for this. The prob-
lem is that often the determinant may not change sign at a doubly degenerate
level, and energy levels may be extremely close. Finally, changes of sign may
occur across a singularity. A highly inefficient but fool-proof method is to cal-
culate the determinant for a large amount of closely spaced energies containing
the relevant part of the spectrum (to be read off fromFigure 6.10) and then scan
the results for changes of sign or near-zeroes. It is certainly not advisable to try
writing a routine which finds all the energy eigenvalues automatically using clever
root-finding algorithms.

(^2) The potential can be found onwww.cambridge.org/9780521833469. There are in fact two files in which
the potential is given on different grids: the first one is a uniform grid and the second an exponential grid
considered in Problem 5.1. Details concerning the integration of the Schrödinger equation on the latter are to be
found in this problem.