Wannier function

The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes.

Although, like localized molecular orbitals, Wannier functions can be chosen in many different ways,[3] the original,[1] simplest, and most common definition in solid-state physics is as follows.

In practice, this is usually the maximally-localized set, in which the Wannier function ϕR is localized around the point R and rapidly goes to zero away from R. For the one-dimensional case, it has been proved by Kohn[6] that there is always a unique choice that gives these properties (subject to certain symmetries).

This consequently applies to any separable potential in higher dimensions; the general conditions are not established, and are the subject of ongoing research.

This approach is similar in spirit to the tight binding approximation, but in contrast allows for an exact description of bands in a certain energy range.