Systematics of bonding in non-icosahedral carbon clusters

General formulas are presented for the vertex numbers, , of pentagon+hexagon polyhedra of icosahedral, tetrahedral or dihedral symmetries. Criteria for uniqueness of representation, isomer counts and grouping of pentagons are established. All polyhedra with 256 vertices or less and belonging to T, D ₅, D₆or their supergroups are listed. With the addition of C₃ to the dihedral and higher groups, at least one pentagon+hexagon cluster is found for all even 20 except for = 22 which is unrealisable in any symmetry, and = 46 (for which a C₃ polyhedron exists). Carbon clusters with closed electronic shells are shown to be generated by a geometrical leapfrog procedure: for all = 60+6k (where k is zero or greater than one) at least one closed shell structure is predicted. In dihedral symmetry closed shells also exist for some other values of . Separation of the 12 pentagonal faces is not sufficient to ensure a closed electronic shell but appears to be a necessary condition in dihedral or tetrahedral symmetry.