Electronic Energy Bands in Metals

A method is presented for calculating the energy levels of a crystal. The method is based on finding a variational principle for the energy levels in terms of the Wannier functions instead of the usual Bloch wave functions. The variational principle does not give the energy $E\left(\mathrm{k}\right)$ directly, but if $E\left(\mathrm{k}\right)$ for a particular band is expanded in a Fourier series in k, then the variational principle gives the Fourier coefficients of $E\left(\mathrm{k}\right)$ in this expansion. The maxima and minima properties of the variational principle are investigated. The variational principle is properly modified for application to valence bands. The types of trial functions that will arise is discussed, and the method is applied to a one-dimensional crystal. Our results are compared with the results of the method of orthogonal plane waves for the same problem.