Hierarchical topological analysis of crystal structures: the skeletal net concept

Topological analysis of crystal structures faces the problem of the `correct' or the `best' assignment of bonds to atoms, which is often ambiguous. A hierarchical scheme is used where any crystal structure is described as a set of topological representations, each of which corresponds to a particular assignment of bonds encoded by a periodic net. In this set, two limiting nets are distinguished, complete and skeletal, which contain, respectively, all possible bonds and the minimal number of bonds required to keep the structure periodicity. Special attention is paid to the skeletal net since it describes the connectivity of a crystal structure in the simplest way, thus enabling one to find unobvious relations between crystalline substances of different composition and architecture. The tools for the automated hierarchical topological analysis have been implemented in the program package ToposPro. Examples, which illustrate the advantages of such analysis, are considered for a number of classes of crystalline substances: elements, intermetallics, ionic and coordination compounds, and molecular crystals. General provisions of the application of the skeletal net concept are also discussed.