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The cluster decomposition of the configurational energy of multicomponent alloys

Lattice models parameterized using first-principles calculations constitute an effective framework to simulate the thermodynamic behavior of physical systems. The cluster expansion method is a flexible lattice-based method used extensively in the study of multicomponent alloys. Yet despite its prevalent use, a well-defined understanding of expansion terms has remained elusive. In this letter, we introduce the cluster decomposition as a unique and basis-agnostic decomposition of any general function of the atomic configuration in a crystal. We demonstrate that cluster expansions constructed from arbitrary orthonormal basis sets are all representations of the same cluster decomposition. We show how the norms of expansion coefficients associated with the same crystallographic orbit are invariant to changes between orthonormal bases. Based on its uniqueness and orthogonality properties, we identify the cluster decomposition as an invariant ANOVA decomposition. We leverage these results to illustrate how functional analysis of variance and sensitivity analysis can be used to directly interpret interactions among species and gain insight into computed thermodynamic properties. The work we present in this letter opens new directions for parameter estimation, interpretation, and use of applied lattice models on well-established mathematical and statistical grounds.