Accurate polarization within a unified Wannier function formalism

We present an alternative formalism for calculating the maximally localized Wannier functions in crystalline solids, obtaining an expression which is extremely simple and general. In particular, our scheme is exactly invariant under Brillouin zone folding, and therefore it extends trivially to the $\Gamma$-point case. We study the convergence properties of the Wannier functions, their quadratic spread and centers as obtained by our simplified technique. We show how this convergence can be drastically improved by a simple and inexpensive “refinement” step, which allows for very efficient and accurate calculations of the polarization in zero external field.