Transport properties of random and nonrandom substitutionally disordered alloys. II. New cluster formulation of the ac conductivity and numerical applications

A cluster-field theory providing a rigorous foundation for dealing with cluster calculations in ordered and disordered materials is presented. The theory is based on a special matrix (given in the text) which plays the role of intracluster interactions and allows the formulation of a general cluster perturbation theory. The use of this cluster theory is demonstrated by deriving a cluster formula for the ac conductivity of general physical systems with specific attention paid to substitutionally disordered alloys. The formalism is general enough to encompass Hamiltonians of various and diverse kinds, such as those of tight-binding and muffin-tin character, and allows us to obtain previously established results in a straightforward and unified manner. In addition it allows, by utilizing the Ward identity, the rigorous expression of vertex corrections in terms of cluster quantities and makes possible the setting up of controlled approximation schemes for their evaluation. The usefulness of the method is illustrated by means of calculations of the ac conductivity of one-dimensional disordered model systems.