Maximally localized generalized Wannier functions for composite energy bands

We discuss a method for determining the optimally localized set of generalized Wannier functions associated with a set of Bloch bands in a crystalline solid. By ``generalized Wannier functions'' we mean a set of localized orthonormal orbitals spanning the same space as the specified set of Bloch bands. Although we minimize a functional that represents the total spread ${\ensuremath{\sum}}_{n}〈{r}^{2}{〉}_{n}\ensuremath{-}〈\mathbf{r}{〉}_{n}^{2}$ of the Wannier functions in real space, our method proceeds directly from the Bloch functions as represented on a mesh of $k$ points, and carries out the minimization in a space of unitary matrices ${U}_{\mathrm{mn}}^{(\mathbf{k})}$ describing the rotation among the Bloch bands at each $k$ point. The method is thus suitable for use in connection with conventional electronic-structure codes. The procedure also returns the total electric polarization as well as the location of each Wannier center. Sample results for Si, GaAs, molecular C${}_{2}$H${}_{4}$, and LiCl will be presented.