A method for hierarchical comparative analysis of crystal structures

A geometrical-topological description of crystal structure as a three-dimensional graph with coloured nodes, weighted and coloured edges is used to generate a hierarchical sequence of the structure representations. The solid angles of Voronoi-Dirichlet polyhedra of atoms are used as the edge weights and the nodes and edges are coloured according to chemical reasons. Two operations are defined to derive the representations: contracting an atom to other atoms keeping the local connectivity, and removing an atom together with all its bonds. The atoms of the crystal structure are called origin, removed, contracted or target according to their roles in the operations. Each structure representation is described as a labelled quotient graph and determined by (i) colours of the graph nodes and edges, (ii) some level for edge weights, and (iii) an arrangement of atoms according to their roles. The computer enumeration and topological comparative analysis of all representations for crystal structures of any composition and complexity are implemented into the TOPOS program package. The advantages of the method are shown by the analysis of typical inorganic compounds and a molecular packing.