Optimal Cluster Expansions: Predicting Alloy Thermodynamics with Desired Accuracy*

We show how to achieve a desired accuracy in first-principles prediction of alloy thermodynamics using an optimal cluster expansion [1], and discuss reliability of error estimates. We illustrate the scaling of computational complexity versus accuracy, and demonstrate that with controlled accuracy a reliably predicted phase-transition temperature converges to the experimental value. We also show there is a simple and rapid means to estimate transition temperatures using the optimal cluster expansion. [1] N.A. Zarkevich and D.D. Johnson, Phys. Rev. Lett. {\bf 92}, 255702 (2004).