Topological coordination numbers and coordination reciprocity from electron-density distributions

Triangulated surface data sets of quantum theory of atoms in molecules (QTAIM) interatomic surfaces have been employed to calculate solid angles subtended at the nuclear positions by each diatomic contact surface. On this basis, topological effective coordination numbers were evaluated. This corresponds to a generalization of the established Voronoi-Dirichlet partitioning (VDP) based procedure. The topological coordination number (tCN) approach developed includes coordination reciprocity requirements necessary to extract coordination-consistent sub-coordination scenarios for identification of chemically meaningful coordination numbers. The ranking between different sub-coordination scenarios is accomplished by weighting functions derived from purely geometrical properties of square and semicircle areas. Exemplary cases analyzed using theoretical electron-density distributions span the range from the face centered cubic, body centered cubic, hexagonal close packed and diamond types of element structures, to rocksalt, CsCl and zincblende types of structures, to compounds of the TiNiSi structure type. An important difference compared with VDP-based coordination numbers arises from the natural inclusion of the effect of different atomic sizes in the tCN approach. Even in highly symmetrical element structures, differences between VDP and tCN results are obtained as an effect of atomic electron-density decay utilizing still available degrees of freedom in the crystal structure. Especially in the TiNiSi type of examples, the advantage of numerically ranking between different sub-coordination scenarios of similar importance emerges. Instead of being obliged to choose only one of them, a more precise characterization contains a listing of different scenarios with their relative weights and associated effective coordination numbers. This seems to be generally the more appropriate way to analyze atomic coordination, especially in more complex structures such as intermetallic phases, opening up its possible use as input for AI applications on structure-property relationships.