Fig. 14. Systematic numbering of (C84-D6h)[5,6]fullerene
In this fullerene the principal axis is the C6 axis passing through the midpoint of opposite and symmetry-equivalent hexagons (Fig. 14a). However, no contiguous helical numbering can be found starting from the six symmetry-equivalent atoms of one of these rings. Therefore, the six C2 axes must be considered. Three of these are symmetry-equivalent axes passing through the midpoint of opposite hexagons (one is shown by black and white dots in Fig. 14a); the remaining three symmetry-equivalent C2 axes pass through the midpoints of opposite bonds, and are therefore less preferred according to Fu-3.1.2. Either one of the former C2 axes can be used as the reference axis. Numbering can start in either one of the symmetry-equivalent hexagons. Because of symmetry, the atoms of this ring form two sets of atoms, indicated as a/a'/a"/a''', and b/b'. There are three pathways to be evaluated: a to b' to a'''; a to a' to b; and b' to a''' to a". Of these, the b' to a''' to a" pathway does not lead to a contiguous numbering, whereas the other two do. The numberings are shown in Fig. 14b (a to b' to a''') and in Fig. 14c (a to a' to b). Since both terminate at an atom belonging to the ring intersected by the axis, a selection cannot be made on the basis of Fu-3.1.3. However, inspection of the atom ranking according to Fu-3.1.4 shows that the numbering shown in Fig. 14b is preferred because it features a 6,6,6 rather than a 6,6,5 atom at position 2.
Fig. 14. Systematic numbering of (C84-D6h)[5,6]fullerene
Reference for this section
5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995