Localized description of superconductivity. II. Strong-coupling formulation

A localized description of superconductivity in narrow-band metals, proposed by Appel and Kohn, is extended to the strong-coupling case. Equations similar, but not identical, to the Eliashberg equations are obtained. These equations are solved numerically and the results are represented by a semiphenomenological formula for $T_c$. A microscopic calculation of the electron-phonon coupling constant in the localized representation, based on measured phonon frequency-distribution curves and augmented-plane-wave band calculations, is described. Results for V, Nb, and Ta yield values of $\lambda$ within a factor of 2 from the expected ones, without any adjustable parameter.