Magic Numbers in Polygonal and Polyhedral Clusters

Certain values for the nuclearity of clusters tend to occur more frequently than others. This paper gives a mathematical derivation of the preferred values of nuclearity (the “magic numbers”) corresponding to various symmetrical, close-packed polygons (in two dimensions) and polyhedra (in three dimensions). Simple general formulas are obtained for the numbers of atoms in these clusters. Explicit coordinates are given for the atoms. Application of these atom-counting results to electron counting of close-packed high-nuclearity metal clusters is illustrated.