Merging quadratic programming with kernel smoothing for automated cluster expansions of complex lattice Hamiltonians

Abstract

We present a general outline for automating cluster expansions of configurational energetics in systems with crystallographic order and well defined space group symmetry. The method presented herein combines constrained optimization techniques of positive-definitive quadratic forms with the mathematical tool of Tikhonov regularization (kernel smoothing) for automated expansions of an arbitrary general physical property without compromising the underlying physics. Throughout the thesis we treat formation energy as the fundamental physical observable to expand on since the predominant application of cluster expansions is the extraction of robust approximations for configurational energetics in alloys and oxides. We therefore present the implementational aspects of the novel algorithmic route on a challenging material system NaxCoO2 and reconstruct the corresponding GGA ground state line with arbitrary precision in the formation energy-configuration space. The mathematical arguments and proofs, although discussed for cases with arbitrary spin assignments and multiple candidate species for single site occupancy, are eventually formulated and illustrated for binary systems. Various numerical challanges and the way they are resolved in the framework of kernel smoothing are addressed in detail as well. However, the applicability of the procedure described herein is more universal and can be tailored to probe different observables without resorting to modifications in the algorithmic implementation or the fundemantal mathematical construction. The effectiveness in recovering correct physics shall than be solely tied to the presence of superposable nature (of the physical property of interest) of local atomic configurations or lackthereof.