Effective cluster interactions from cluster-variation formalism. I

States of (partial) order in binary-alloy systems are described by means of expansions in a complete set of orthogonal cluster functions. The expectation value of the energy of such systems can then be expressed as a bilinear form in multisite correlation variables and effective cluster interactions (ECIs), as originally proposed by Sanchez, Ducastelle, and Gratias [Physica A 128, 334 (1984)]. It is shown that ECIs are defined as interchange energies averaged over all atomic configurations with a fixed concentration or over all configurations and concentrations, depending on the orthogonal expansion considered. The former averaging process leads to concentration-dependent ECIs, the latter to concentration-independent ECIs. From their formal definitions, certain relations will be shown to hold between the interactions derived in the two averaging schemes in the thermodynamic limit. In particular, an interesting convergence criterion is established for the concentration-dependent ECIs.