Nanomaterials are interesting due to their unexpected and unique properties arising from phenomena occurring at the so-called mesoscale (that is, between single atoms and bulk solids). Among nanomaterials, one may distinguish quantum dots, which are highly crystalline nanocrystals with sizes up to c.a. 10 nm. Due to the quantum confinement effect, quantum dots exhibit extraordinary electronic and optical properties and may be utilized in photocatalysis. Semiconducting quantum dots may absorb photons, which results in the excitation of electrons from valence to conducting bands. Excited electrons in the conducting band and positive holes in the valence band may interact with chemical molecules (e.g., with water molecules), forming highly reactive radicals. Consequently, quantum dots may be utilized in advanced oxidation processes based on the action of light (i.e., photo-based advanced oxidation processes). Furthermore, quantum dots have advantages, such as having a tunable energy band gap and relative cost-effectiveness. Advanced oxidation processes are very important in the context of the constantly increasing pollution of the natural environment. Contaminants of emerging concern, such as pesticides, endocrine-disrupting compounds, and flame retardants, are still being detected in naturally present water. Such compounds may be degraded using advanced oxidation processes, utilizing quantum dots as photocatalysts. However, many operational parameters (such as quantum dots’ properties, including the means of their preparation) influence the efficiency of such processes; thus, detailed studies are being conducted.

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).