Quoting John McCarthy <jmc@steam.Stanford.EDU>:
As I remember from a quantum mechanics course I took in the 1940s, there is a way of estimating energy levels that does not solve the wave equation. I learned it only for discrete energy levels, but I'd suppose it applies to bands.
It is based on the fact that the energy level is a minimum of an energy integral over all functions satisfying certain conditions. The method consists of representing a hypothetical wave function by a linear combination of terms satisfying boundary conditions. You then choose coefficients to minimize the energy which is a sum of products. As I recall it was used to estimate the energy levels of the helium atom and the ionized hydrogen molecule. You get an upper bound, and sometimes it was quite close to the experimental value.
I'd guess that this is the Rayleigh-Ritz principle. In terms of matrices, XtMX/XtX is less than the largest eigenvalue, so vary X and pick the biggest result; a more extensive search gives better results. Usually you pick a unit X and look at XtMX and the inequality reverses and you have lowering upper bounds. Think of looking for the longest semiaxis on an ellipsoid. A variant is the Courant minimax principle, which says that the minumum over hyperplanes of the longest semiaxis in the hyperplane is an eigen vector, whose eigenvalue is the semiaxis. As I recall, these calculations are harder to manage; you are looking for saddle points on the ellipsoid, in terms of distance from the center. Use positive definite matrices so as not to worry about hyperboloids, but I'd say the critical pointology is the same. In terms of band theory, there is a continuum of eigenfuctions, typically non-normalizable, but I wouldn't say that there isn't a relevant variational principle. Moreover, you are looking for band edges, not wave functions in the interior of the band; these have distinguishing characteristics which may help searching for them. But I don't recall the details, just sitting here. It has to do with the trace being exactly 1 or -1; for -1 you get subharmonics (in vibration theory). - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos