Topological relations between three-dimensional periodic nets. I. Uninodal nets.

A method is proposed to search for topological relations between periodic nets. The method is based on a sequence of steps of decreasing the node degree and symmetry of the initial net (supernet) and, as a result, gives all its subnets. It is implemented into the program package TOPOS which automatically constructs the net relation graph (NRG) for a given set of initial nets. The method is used to find all supernet-subnet relations for 924 initial 4-12-coordinated uninodal nets. The resulting NRG consists of 6528 3-12-coordinated uninodal nets; 5278 of them have topologies not described earlier. It is shown that many NRG properties are useful in crystal chemistry. In particular, a path between NRG nodes corresponds to a sequence of transformations that relate the nets, the adjacency sequence of a NRG node may be used as a criterion for crystallochemical 'significance' of the corresponding net. Many well known net topologies are found to have a large number of relations with other topologies that cause their special place in the NRG and crystallochemical 'significance'. The peculiarities of the proposed approach are illustrated by examples of the nets often occurring in crystal structures.