Homology groups, symmetry representations and polyhedral clusters

¹ Division of Quantum Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium
² Department of Chemistry, University of Exeter, Stocker Road, Exeter, EX4 4QD UK

Received: 8 August 1996
Accepted: 12 November 1996

Abstract

Physical properties of cages and clusters obey symmetry rules that are extensions of the celebrated Euler-Poincaré theorem on polyhedra. A connection is established between this result and a fundamental topological relationship in the theory of homology groups. The connection allows us to assign symmetry representations to physically relevant topological invariants. The results are illustrated by a derivation of the symmetries of the low-lying empty orbitals in leapfrog fullerenes.