Extension of Euler's theorem to symmetry properties of polyhedra

Abstract

POLYHEDRAL cages and clusters are widespread in chemistry. Examples of fully triangulated polyhedra (deltahedra) are the skeletons of closo-boranes B_nH²⁻_n, many heteroboranes and transition-metal carbonyls^1. Three-connected cages occur for carbon^2,3 and in zeolites¹. The numbers v, f and e of vertices, faces and edges of a convex polyhedron are related by Euler's theorem^4,5 v+f=e + 2. Here we show that within the point group of the polyhedron the symmetries spanned by the sets of vertices, faces and edges are also related. We prove a general theorem relating these symmetries for convex polyhedra, and give further relations specific to deltahedra and 3-connected polyhedra. The latter extensions of Euler's theorem to point-group characters allow us to generate complete sets of internal vibrational coordinates from bond stretches for deltahedra, and to classify, from symmetry properties alone, the bonding or antibonding nature of molecular orbitals of 3-connected cages.