Localization of Wannier Functions in Copper

The problem of constructing localized one-electron wave functions in metals having complicated band structures is considered with reference to copper. The extent to which the Wannier functions are localized depends critically on how the Bloch functions from which they are constructed are labeled with respect to the band index and on the choice of the phase factor associated with each Bloch function. A practical approach to the handling of these problems is described and calculations based on augmented-plane-wave (APW) Bloch functions for copper are reported. The resulting Wannier functions are found to be poorly localized. By relaxing the requirement of orthogonality, a set of pseudo-phase-factors can be determined so as to maximize the probability at the central site. The non-orthogonal "localized" functions which result have a definite symmetry within the central APW sphere, where they strongly resemble tight binding functions. However, their localization is generally not much better than that of the Wannier functions. The implications of our results for the Koster-Slater theory of dilute alloys and for recent developments in understanding the band structures of transition metals are briefly discussed.