Wannier functions are derived through a unitary transformation of the Bloch functions to localized functions. Specifically, we employ maximally localized Wannier functions (MLWFs), obtained by applying a "maximallocalization" criterion to produce a unique set of Wannier functions. The minimization of the spread of the MLWFs is based on the concepts of the modern theory of polarization. While the regular position operator can be used for finite systems, the spread function is derived in terms of a Berry phase under PBCs. MLWFs are localized in space around the Wannier centers r WC i . Adopting again a Γ-only formulation, the α component of the Wannier center is given by r WC αi = -