Chemical applications of topology and group theory

Abstract

This paper characterizesforbidden polyhedra, which are polyhedra with fewer than 9 vertices which cannot be formed using only the 9s,p, andd atomic orbitals. In this connection polyhedra are of particular interest if their symmetry groups are direct product groups of the typeR × C′ _s in whichR is a group containing only proper rotations andC′ _s is eitherC _s orC _i in which the non-identity element is an inversion center or a reflection plane which is called theprimary plane of the groupR ×C′ _s . Using this terminology polyhedra of the following types are shown always to be forbidden polyhedra: (1) Polyhedra having 8 vertices, such direct product symmetry point groups, and either an inversion center or aprimary plane fixing either 0 or 6 vertices; (2) Polyhedra having a 6-fold or higherC _n rotation axis. However, these conditions are not necessary for a polyhedron to be forbidden since in addition to one 7-vertex polyhedron and ten 8-vertex polyhedra satisfying one or both of the above conditions there are two forbiddenC _3v 8-vertex polyhedra which satisfy neither of the above conditions.