Approximate solutions to the cluster variation free energies by the variable basis cluster expansion

In this paper we obtain approximate solutions to the Cluster Variation free energy by carrying out a cluster expansion of the probabilities appearing in the free energy functional in terms of concentration-dependent basis functions, and by truncating the expansion at different cluster levels prior to minimization. We show that a significant improvement over the Bragg-Williams approximation can be achieved by truncating the expansion of the cluster probabilities at relatively small clusters, thus dramatically reducing the number of equations that need to be solved in order to minimize the free energy. Furthermore, the free energy functional in the Cluster Variation Method offers a well-controlled case study to infer the effects of truncating the expansion of the energy of alloy formation in the commonly used Cluster Expansion method, versus the effects of truncating the expansion when using a concentration-dependent basis. Examples of the approach are given for simple Ising models for fcc- and bcc-based prototype alloy systems.