Orthogonalization Procedures and the Localization of Wannier Functions

The method of "symmetric orthonormalization" is shown to have a least-squares property: it constructs those unique orthonormal functions which minimize the sum of squared distances (in Hilbert space) between each initial function and a corresponding function of the orthonormal set. The localization of Wannier functions is a consequence of this property, since they can be obtained from localized atomic orbitals by symmetric orthonormalization. The theorem further implies an optimal resemblance of Wannier functions to atomic orbitals.