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 ${\sum }_n〈r^2{〉}_n-〈\mathbf{r}{〉}_n^2$ 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 $U_{\mathrm{mn}}^{\left(\mathbf{k}\right)}$ 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.