Electronic structure of transition metals. III.d-band resonance and Regge-pole theory

This is the third paper in our current series on the electronic structure and properties of the transition metals in the pseudopotential approximation. Here we review, reformulate, and generalize the resonance model of $s-d$ hybridization in one-electron energy bands so as to be applicable to scattering processes at arbitrary energies different from the eigenvalues of the one-electron wave equation in a crystal. This is achieved by formulating the resonance model in two ways, first in the framework of the one-electron energy-band secular equation given by the Korringa-Kohn-Rostoker (KKR) method in Ziman's "plane-wave representation," and second as a scattering problem in terms of standard partial-wave phase-shift analysis. The former leads to a real resonance energy, say $E_d$, while the latter leads to a complex resonance energy, say $E_d+\frac{1}{2}\mathrm{iW}$ where $W$ is a measure of $s-d$ interaction which also determines the $d$ bandwidth. To relate $E_d$ and $W$, and hence determine $W$ in the framework of the KKR or any other valid transition-metal pseudopotential form factor, the scattering problem is reformulated in complex angular momentum representation using the Regge-pole theory in nonrelativistic potential scattering; the result, $W\propto {\left(\frac{\mathrm{dE}}{\mathrm{dl}}\right)}_{l=2\mathrm{}}$, is then interpreted in the context of the transition-metal-model-potential form factor discussed in the first two papers of this series and the quantitative predictions are compared with the results of augmented-plane-wave one-electron energy-band calculation and those of the renormalized-atom method, and with recent photoemission data. The agreement between the various results is systematically good for transition metals of the $3d$, $4d$, and $5d$ series, as far as the results are known.