Magic Mathematical Relationships for Nanoclusters

Size and surface properties such as catalysis, optical quantum dot photoluminescense, and surface plasmon resonances depend on the coordination and chemistry of metal and semiconducting nanoclusters. Such coordination-dependent properties are quantified herein via “magic formulas” for the number of shells, n, in the cluster. We investigate face-centered cubic, body-centered cubic, simple cubic clusters, hexagonal close-packed clusters, and the diamond cubic structure as a function of the number of cluster shells, n. In addition, we examine the Platonic solids in the form of multi-shell clusters, for a total of 19 cluster types. The number of bonds and atoms and coordination numbers exhibit magic number characteristics versus n, as the size of the clusters increases. Starting with only the spatial coordinates, we create an adjacency and distance matrix that facilitates the calculation of topological indices, including the Wiener, hyper-Wiener, reverse Wiener, and Szeged indices. Some known topological formulas for some Platonic solids when n=1 are computationally verified. These indices have magic formulas for many of the clusters. The simple cubic structure is the least complex of our clusters as measured by the topological complexity derived from the information content of the vertex-degree distribution. The dispersion, or relative percentage of surface atoms, is measured quantitatively with respect to size and shape dependence for some types of clusters with catalytic applications.