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 (Formula presented) of the Wannier functions in real space, our method proceeds directly from the Bloch functions as represented on a mesh of (Formula presented) points, and carries out the minimization in a space of unitary matrices (Formula presented) describing the rotation among the Bloch bands at each (Formula presented) 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(Formula presented)H(Formula presented), and LiCl will be presented.