s−dInteraction in Transition Metals

It has been shown recently that the calculated electronic band structure of transition metals may be represented quite precisely by a model Hamiltonian with a nearly-free electron $\mathrm{sp}$ band crossing and hybridizing with a tight-binding $d$ band. The form of this model Hamiltonian is now exhibited as a transformation of the fundamental Korringa-Kohn-Rostoker formulation of the energy-band problem. The hydridization appears as a resonance between a localized $d$ state and the plane-wave band. In addition to throwing light on some details of the model Hamiltonian, the derivation serves to express some of its constituents in terms of simple atomic quantities. In particular, the structure and width of the $d$ band and the hybridization are all related to one intra-atomic matrix element. The $d$ band apears as an interference between resonances, its shape determined by purely geometrical structure constants and its width by the width of the resonance. These relations are demonstrated quantitatively for the band structure of copper.