When topology meets geometry: topological motifs and uniformity of atomic sublattices in inorganic crystals†

We have analyzed topological motifs of whole crystal structures and unconnected non-metal sublattices in 967 borides, 721 carbides, 899 nitrides and 927 silicides of metals within the periodic net model. In addition, we have classified the topological types of connected non-metal motifs in 333 carbides and 86 nitrides containing N–N or C–C bonds. The topology of the whole structure was found to be essentially predetermined by the sublattice motif; however, there were exceptions, which were caused by geometrical distortions of the structure. In contrast, sublattices of the same type can exist in different structural and topological types. We have proposed an integral criterion 〈G₃〉 of the sublattice distortion, which can also be considered as a measure of the sublattice uniformity. In most cases, the sublattices of non-metal atoms possess high uniformity, which reflects the tendency of the negatively charged atoms to be placed as distant as possible from each other. The sublattices, which have high symmetry, simple topology, and high uniformity in the most symmetrical embedding, were found to be the most suitable for the realization in different structures. The low uniformity of a sublattice indicates either strong interatomic interactions in the sublattice or its subordinate structural role.