Calculating properties with the coherent-potential approximation

It is demonstrated that the expression that has hitherto been used for calculating the Bloch spectral-density function $A^B(E,\overrightarrow{\mathrm{k}})$ in the Korringa-Kohn-Rostoker coherent-potential-approximation theory of alloys leads to manifestly unphysical results. No manipulation of the expression can eliminate this behavior. We develop an averaged Green's-function formulation and from it derive a new expression for $A^B(E,\overrightarrow{\mathrm{k}})$ which does not contain unphysical features. The earlier expression for $A^B(E,\overrightarrow{\mathrm{k}})$ was suggested as plausible on the basis that it is a spectral decomposition of the Lloyd formula. Expressions for many other properties of alloys have been obtained by manipulations of the Lloyd formula, and it is now clear that all such expressions must be considered suspect. It is shown by numerical and algebraic comparisons that some of the expressions obtained in this way are equivalent to the ones obtained from a Green's function, while others are not. In addition to studying these questions, the averaged Green's-function formulation developed in this paper is shown to furnish an interesting new way to approach many problems in alloy theory. The method is described in such a way that the aspects of the formulation that arise from the single-site approximation can be distinguished from those that depend on a specific choice for the effective scatterer.