First Order Methods for Geometric Optimization of Crystals: Theoretical Derivations

Crystal structures are important formations, bases of materials used in everyday life. A lot of effort is dedicated to the creation of new materials, starting from the study and combination of crystal structures' properties[1] through the crystals' geometric optimization. In the meantime, an equal amount of effort is dedicated to the study and development of methods employed in optimization problems; this, nevertheless, is carried out independently, in the context of computer science. There is a need, therefore, to acknowledge and highlight the overlap between the two fields, whilst creating a common language and establishing the theoretical foundations on which the optimization methods can become physically interpreted in the case of crystals. This article provides a detailed introduction to the theory behind the geometric optimization of crystals from the point of view of computer science, so as to facilitate interdisciplinary cooperation. Its target is to familiarize scientists from different fields with the problem, its parameters, and their derivations, in order to motivate their assistance in the creation, implementation, and application of new methods for the geometric optimization of crystal structures.