Localized description of superconductivity in narrow-band metals

A localized description of superconductivity in narrow-band metals, proposed by Appel and Kohn, is applied to the calculation of the phonon-exchange vertex in three transition metals. The essence of this description is the formulation of the equation for the vertex function in the Wannier representation and the assumption of a contact interaction, in which the two electrons in a Cooper pair scatter together from site to site. The value of the vertex function at ω=0 plays the same role in determining Tc as the coupling parameter λ in the BCS--- or McMillan---theory. The phonon-exchange vertex is calculated for V, Nb, and Ta, assuming that the d electrons occupy the threefold degenerate Γ^'₂₅ band. The calculation makes use of tabulated atomic wave functions and potentials, and of measured phonon frequency distributions. The electron-phonon interaction is described within the so-called modified tight-binding approximation, in which the zeroth-order electron wave functions follow adiabatically the vibrating ions. The values of the phonon-exchange vertex at ω=0 are compared with λ given by McMillan and correct orders of magnitude are obtained with reasonable screening of the atomic potentials. The validity of the contact approximation is checked numerically.