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 ∑_n〈r²〉_n-〈r〉_n² 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_mn^(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 C₂H₄, and LiCl will be presented.