3.2. Fullerenes having at least one symmetry axis (C_n, n>1) but no contiguous helical pathway determined on the basis of rules Fu-3.1.1 to Fu-3.1.4

Contents

3.2. Fullerenes having at least one symmetry axis (C_n, n>1) but no contiguous helical pathway determined on the basis of rules Fu-3.1.1 to Fu-3.1.4
   Rule Fu-3.2.1
   Rule Fu-3.2.2
   Rule Fu-3.2.3
   Rule Fu-3.2.4

Fullerenes with at least one symmetry axis C_n, n>1 exist for which a contiguous helical pathway for numbering cannot be found using rules Fu-3.1.1 to Fu-3.1.4. The problem of numbering discountiguity in these fullerenes has already been tackled by previous publications [1,3], and solutions were proposed using different and sometimes non-clearly established procedures. To devise unique numbering for these fullerenes in a clear and relatively straightforward way, it was decided to establish a modified set of rules adopting an approach similar to that used in the case of fullerenes with contiguous numbering.

If no contiguous helical numbering can be found in a fullerene with at least one rotational axis of symmetry C_n, n>1 by application of rules Fu-3.1.1 to Fu-3.1.4, discontiguous numberings made up of two or more contiguous segments are examined, with the first of their segments being built around a proper rotation axis in complete analogy to the procedure employed in 3.1 to determine a contiguous pathway. If a preferred first segment is obtained based on rules Fu-3.2.1 to Fu-3.2.3 (see below), its follow-up segment(s) is (are) constructed based on rule Fu-3.2.4 (see below) to complete the preferred numbering. If there is still a choice after comparison of different first segments, subsequent segments are constructed (Fu-3.2.4) and compared (Fu-3.2.1 to Fu-3.2.3) at each level n until a preference becomes apparent.

Rule Fu-3.2.1 (cf. Fu-3.1.1):

1. The nth discontiguity must occur as late as possible in the numbering.
2. Among different pathways having the nth discontiguity at the same highest locant, the preferred numbering pathway is built around the proper rotation axis of higher order.

Rule Fu-3.2.3 (cf. Fu-3.1.4; for practical reasons, no rules analogous to Fu-3.1.3 is applied to segments of discontiguous numberings): When there is still a choice, the nth segments are compared according to rule Fu-3.1.4.

Once a preferred segment is identified based on rules Fu-3.2.1 to Fu-3.2.3 the numbering of the fullerene is completed by construction of the subsequent segment(s). The starting point of a new segment is defined according to rule Fu-3.2.4a (see below), and numbering within a segment is continued according to rule Fu-3.2.4b.

Rule Fu-3.2.4: After each discontiguity,

1. the next atom to be numbered must be directly connected to one of the previously numbered atoms; among the latter, the atom with the highest locant is preferred.
2. numbering continues by the shortest path, in terms of number of bonds, to reach an unnumbered atom that is directly connected to a numbered atom with the lowest locant already assigned. (In this way the sense of the numbering helix should be maintained and its pitch kept as small as possible.)

• 3.2.1. (C₂₄-O_h)[3,8]fullerene
• 3.2.2. (C₄₈-O_h)[4,6,8]fullerene
• 3.2.3. (C₃₂-O_h)[4,6]fullerene
• 3.2.4. (C₃₄-C_2v)[4,6]fullerene
• 3.2.5. (C₆₀-D_6h)[5,6]fullerene
• 3.2.6. (C₇₀-C_2v)[5,6]fullerene
• 3.2.7. (C₈₀-I_h)[5,6]fullerene
• 3.2.8. (C₈₂-C_3v)[5,6]fullerene
• 3.2.9. (C₈₄-D_2d)[5,6]fullerene
• 3.2.10. (C₁₁₂-T_d)[5,6]fullerene
• 3.2.11. (C₁₂₀-I_h)[4,6,10]fullerene