Mathematical Modeling of Properties and Structures of Crystals: From Quantum Approach to Machine Learning

The crystalline state of matter serves as a reference point in the context of studies of properties of a variety of chemical compounds. This is due to the fact that prepared crystalline solids of practically useful materials (inorganic or organic) may be utilized for the thorough characterization of important properties such as (among others) energy bandgap, light absorption, thermal and electric conductivity, and magnetic properties. For that reason it is important to develop mathematical descriptions (models) of properties and structures of crystals. They may be used for the interpretation of experimental data and, as well, for predictions of properties of novel, unknown compounds (i.e., the design of novel compounds for practical applications such as photovoltaics, catalysis, electronic devices, etc.). The aim of this article is to review the most important mathematical models of crystal structures and properties that vary, among others, from quantum models (e.g., density functional theory, DFT), through models of discrete mathematics (e.g., cellular automata, CA), to machine learning (e.g., artificial neural networks, ANNs).