3.3. Fullerenes belonging to the Cs point group and having a contiguous helical numbering pathway according to rules Fu-3.3.1 to Fu-3.3.4.
The lack of proper rotation axes Cn (n>1) in these fullerenes requires an approach to helical pathway identification that is different from that employed for fullerenes having axial symmetry. Conceptually, however, the new approach should follow the previous one as much as possible. Accordingly, the following rules are adopted for fullerenes belonging to the Cs point group.
Rule Fu-3.3.1. Since there is no proper rotation axis, the plane of symmetry is used as the reference element. A search is made for a contiguous numbering helix that starts in a ring bisected by the plane of symmetry or along a bond lying in the plane.
Rule Fu-3.3.2. If there is a choice for the beginning of the numbering, rings bisected by the plane are preferred over bonds lying in the plane.
Rule Fu-220.127.116.11. If there is a choice among rings, a larger ring is preferred to a smaller one and, among rings of the same size, the preferred ring contains the highest-ranking atom at the first point of difference.
Rule Fu-18.104.22.168. If there is a choice among bonds lying in the plane, the preferred bond contains the higher number of higher ranking atoms.
Rule Fu-3.3.3. If there is still a choice among contiguous numberings, the preferred pathway should
It must be noted that in the approach used by CAS [ref 3] the plane of symmetry is used as a pivot plane. The spheroidal structure is then sliced into planes parallel to the pivot plane, constructed using specific rules. Either one of the two outer planes can be the reference plane. It should also be mentioned that the assignment of atoms to a given plane in the CAS procedure may depend on the geometry of the structure rather than the connectivity of atoms and can lead to ambiguities.
As a consequence, the numberings produced by CAS are always different from those described here. The following examples illustrate the use of the rules as recommended in this document.
3. A.L. Goodson, C.L. Gladys and D.E. Worst, J. Chem. Inf. Comp. Sci. 1995, 35, 969-978.