Cluster expansions and the configurational energy of alloys

A formal description of configurational functions in alloys is presented. The approach introduces an infinity of orthogonal and complete basis sets in the configurational space of finite clusters, and leads naturally to generalized cluster expansions. In the present approach, the orthogonality of the basis functions is defined with respect to the scalar product given in terms of unrestricted sums over all possible configurations of the cluster. In the thermodynamic limit, this definition of the scalar product corresponds to sums in the canonical ensemble and, as such, the basis are well adapted to describe functions at any fixed concentration. A general relation between the expansion coefficients in different basis is derived.