Cluster expansions of multicomponent ionic materials: Formalism and methodology

The cluster expansion (CE) method has seen continuous and increasing use in the study of configuration-dependent properties of crystalline materials. The original development of the CE method along with the underlying mathematical formalism and assumptions was focused on the study of metallic alloys. Since then the methodology has been actively and successfully used in the study of ionic materials as well. In this work, we present a cohesive reformulation of the mathematical formalism underlying the CE method based on a synthesis of its original formulation, several additions and extensions that have been proposed since, and a revised representation of its constituent mathematical objects. We then proceed to describe some of the formal implications of using the methodology for charge-neutral configurations in ionic systems. In particular, we discuss the reduction of the size of configuration spaces and the resulting linear dependencies that arise among correlation functions that span the larger unconstrained configuration space. Additionally, we explore the effects of long-range electrostatic interactions. We also demonstrate how the previously proposed use of a point electrostatic term successfully accounts for the majority of the longer-range electrostatic interactions, and leaves the cluster expansion terms to capture mostly short-range interactions. Finally, we present and discuss a variety of recently developed methodologies, including training structure selection, oxidation state assignment, structure mapping, and regression algorithms, that are necessary to address these formal mathematical notions for a practical implementation of the CE method in the study of multicomponent ionic materials.