Renormalized interactions in truncated cluster expansions

It is shown that truncating the cluster expansion of the energy of alloys gives rise to renormalized effective cluster interactions that are explicit functions of the configurational variables. Such a dependence of the renormalized cluster interactions is in addition to their dependence on the volume of the alloy and on other structural parameters. The physical picture that emerges is different from the commonly used representation of the configurational energy by means of a generalized Ising-like model, which follows from the assumption that the contributions of the effective cluster interactions can be neglected beyond a relatively small cluster size. The physical picture is one in which the sum of the effective interactions contributes over long distances but the expected ``near-sightedness'' of the energy is preserved by the renormalized interactions. Furthermore, the cluster expansion is implemented by simultaneously fitting the volume- and configuration-dependent energy function to the zero-pressure values of the energies of formation, volumes, bulk moduli, and pressure derivatives of the bulk modulus of a set of ordered compounds. As an example of this formulation of the cluster expansion, we apply the methodology to the Cu-Au system for different types of cell-internal and cell-external relaxations.