Variational Methods for Periodic Lattices

The problem of finding the propagating solutions of the Schr\"odinger equation in periodic lattices is formulated as a variational principle. This may be used as a starting point to establish the general properties of bands. Furthermore it is shown that by introducing various approximations into the variational principle, the chief existing approximation methods can all be derived from it. Improvements of these methods are suggested. Numerical illustrations are presented and the possibilities of the variational method for more accurate calculations of the energy bands of solids are discussed.