Periodic-Graph Approaches in Crystal Structure Prediction

The explosive growth in inorganic and organic materials chemistry has seen a great upsurge in the synthesis of crystalline materials with extended framework structures (zeolites, coordination polymers/coordination networks, metal–organic frameworks (MOFs), supramolecular architectures formed by hydrogen bonds and/or halogen bonds, etc.). There is a concomitant interest in simulating such materials and in designing new ones. In this respect, the role of new topological approaches in the modern crystallochemical analysis sharply increases compared to traditional geometrical methods that have been known for almost a century [1, 2]. As opposed to the geometrical model that represents the crystal structure as a set of points allocated in the space, the topological representation focuses on the main chemical property of crystalline substance – the system of chemical bonds. Since this system can be naturally described by an infinite periodic graph, the periodic-graph approaches compose the theoretical basis of the topological part of modern crystal chemistry. The history of these approaches is rather long, but only in the last two decades they have come into the limelight. Wells [3] was the first who thoroughly studied and classified different kinds of infinite periodic graph (net) and raised the question: what nets are important for crystal chemistry? This key question for successful prediction of possible topological motifs in crystals was being answered in two general ways initiated by Wells’ pioneer investigations. Firstly, mathematical basics for nets were developed in a number of works concerning quotient graph approach [4–10], special types of nets [11, 12], topological descriptors [13–16], and other topological properties related to nets, such as tiles and surfaces [17–19]. Secondly, net abundance and taxonomy were intensively explored in Refs. [20–26]. To solve the emerging problems, novel computer algorithms [5, 16, 19, 27, 28], program packages [6, 19, 27], and electronic databases [19, 27, 29] were developed that allowed to comprehensively analyze the topological motifs through hundred thousands of crystal structures. With these achievements, materials science and crystal chemistry comeup to a new level of their development, that is, characterized by deeper integration of mathematical methods, computer