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Unique Truncated Cluster Expansions for Materials Design via Subspace Projection and Fractional Factorial Design

For alloy thermodynamics, we obtain unique, physical effective cluster interactions (ECI) from truncated cluster expansions (CE) via subspace-projection from a complete configurational Hilbert space; structures form a (sub)space spanned by a locally complete set of cluster functions. Subspace-projection is extended using Fractional Factorial Design with subspace "augmentation" to remove systematically the ECI linear dependencies due to excluded cluster functions - controlling convergence and bias error, with a dramatic reduction in the number of structural energies needed. No statistical fitting is required. We illustrate the formalism for a simple Hamiltonian and Ag-Au alloys using density-functional theory.