Wannier Functions and the Phases of the Bloch Functions

Abstract

We discuss the problem of what phases should be attributed to the Bloch functions in order to obtain Wannier functions with minimal widths. An exact solution to this problem is found by means of the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ perturbation formalism. We also consider localized crystal functions of a more general kind. These functions are found to obey a Schr\"odinger equation in reciprocal space, and may have some important assets when compared to the Wannier functions.