Combinatorial and algorithmic complexity of crystal structures

Abstract

Research subject. Numeral indexes describing the complexity of the system of contacts between structural units in crystal structures. Aim. Development of a complexity index for the system of contacts between periodic structural units based on the indices available for those between structural units in island (molecular) structures. Materials and methods. Structural data were selected from the COD, AMCSD, and CSD crystallographic databases. The system of contacts in the structures was analyzed by the Voronoi–Dirichlet polyhedra (VDP) method in the ToposPro software package. Results. The method of topological analysis of the system of contacts in molecular crystals was adapted to all heterodesmic crystal structures and tested on the structures of compounds of several classes. Target complexity indices were developed. Conclusions. Networks of contacts between periodic structural units are low-dimensional. A generalized structural class for such networks can be derived from the original crystal structure data. The algorithmic complexity of heterodesmic structures is subadditive, in contrast to superadditive combinatorial complexity. For the first time, the number of bearing contacts was calculated between periodic structural units, reflecting the algorithmic complexity of the structure at the appropriate level of structural description.