On the Hardness of Energy Minimisation for Crystal Structure Prediction

Abstract

Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem in computational chemistry. Due to the exponentially large search space, the problem has been referred in several materials-science papers as “NP-Hard” without any formal proof. This paper fills a gap in the literature providing the first set of formally proven NP-Hardness results for a variant of csp with various realistic constraints. In particular, this work focuses on the problem of removal: the goal is to find a substructure with minimal energy, by removing a subset of the ions from a given initial structure. The main contributions are NP-Hardness results for the csp removal problem, new embeddings of combinatorial graph problems into geometrical settings, and a more systematic exploration of the energy function to reveal the complexity of csp. These results contribute to the wider context of the analysis of computational problems for weighted graphs embedded into the 3-dimensional Euclidean space, where our NP-Hardness results holds for complete graphs with edges which are weighted proportional to the distance between the vertices.