Zero-order pseudoatoms and the generalized pseudopotential theory

Abstract

The method of constructing localized $d$-basis states suggested by the author in a previous paper is reformulated and generalized. A zero-order pseudoatom, whose electron density represents a first approximation to the actual electron density within one atomic volume of the metal, is introduced to define both the core and the $d$ states which enter the generalized pseudopotential theory. In the most general case, the zero-order pseudoatom Hamiltonian includes an arbitrary localization potential for each state. Attention is focused on the case in which the localization potential is in the form of an infinite square well and is applied to the $d$ states of the transition-series elements. In the zero-order pseudoatom approach, matrix elements of the hybridization potential are conveniently divided into pseudoatom contributions, arising from the localization potential, and overlap contributions, arising from the overlap of $d$ states in the metal. In the limit of no overlap, the hybridization matrix element between a plane wave and a $d$ state is shown to have a simple analytic form for the case of an infinite-square-well localization potential. Applications are considered for 17 simple and $d$-band metals from the alkali, alkaline-earth, group-II B, and group-III A elements, with emphasis placed on the effect of hybridization on representative properties of the $d$-band metals. Hybridization is found to be generally small in the alkali and group-III A elements, but quite significant in the alkaline-earth and group-II B metals. Results for the alkaline-earth metals are similar to those obtained previously, but results for the other $d$-band metals are obtained for the first time. In zinc and cadmium it is found that the inclusion of hybridization can explain the high $\frac{c}{a}$ axial ratios observed in these metals.