Exponential Decay Properties of Wannier Functions and Related Quantities

The spatial decay properties of Wannier functions and related quantities have been investigated using analytical and numerical methods. We find that the form of the decay is a power law times an exponential, with a particular power-law exponent that is universal for each kind of quantity. In one dimension we find an exponent of $-3/4\mathrm{}$ for Wannier functions, $-1/2\mathrm{}$ for the density matrix and for energy matrix elements, and $-1/2\mathrm{}$ or $-3/2\mathrm{}$ for different constructions of nonorthonormal Wannier-like functions.