3.3.4  Nanotubes, conical and biconical fullerenes and their symmetry

For denoting different categories of symmetry groups, Bohm symbols will be used (Bohm and Dornberger-Schiff, 1966). In a symbol Gnst..., the first subscript n represents the maximal dimension of space in which the transformations of the symmetry group act, while the following subscripts st... represent the maximal dimensions of subspaces remaining invariant under the action of transformations of the symmetry group, that are properly included in each other. With regard to their symmetry, general fullerenes belong to the category of point groups G30. The category G30 consists of seven polyhedral symmetry groups without invariant planes or lines: [3,3] or Td, [3,3]+ or T, [3,4] or Oh, [3,4]+ or O, [3+,4] or Th, [3,5] or Ih, [3,5]+ or I, and from seven infinite classes of point symmetry groups with the invariant plane (and the line perpendicular to it in the invariant point):[q] or Cqv, [q]+ or Cq, [2+,2q+] or S2q, [2,q+] or Cqh, [2,q]+ or Dq, [2+,2q] or Dqd, [2,q] or Dqh, belonging to the subcategory G320 (Coxeter and Moser, 1980). The point symmetry groups G30 were mentioned before, considering symmetry of KLs (see page 46). For the groups of the subcategory G320, in the case of rotations of order q > 2, the invariant line (i.e., the rotation axis) may contain 0, 1 or 2 vertices of a general fullerene. According to this, among all general fullerenes with a geometrical symmetry group G belonging to G320, from the topological point of view we can distinguish, respectively, cylindrical fullerenes (nanotubes), conical and biconical ones. 

We easily conclude that for polyhedral 5/6 fullerenes G can be only [3,3] (Td), [3,3]+ (T), [3,5] (Ih), or [3,5]+ (I), since their topological structure (n5 = 12) is incompatible with the octahedral symmetry group [3,4] (Oh) or its polyhedral subgroups. In the case of nanotubes (or cylindrical fullerenes) we have infinite classes of 5/6 fullerenes with the geometrical symmetry group [2,q] (Dqh) and [2+,2q] (Dqd), and the same chemical symmetry. The first infinite class of cylindrical nanotubes with G = G' = D5h is obtained from a cylindrical 3/4/5 4-valent graph with two pentagonal bases, 10 triangular and 5(2k+1) quadrilateral faces (k = 0,1,2,...) and with the same symmetry group. By the vertex bifurcation preserving its symmetry, we obtain the infinite class of nanotubes C30, C50, C70,... where C70 is the first of them satisfying IPR. The geometrical structure of C70 admits different edge colorings (i.e., chemical isomers). Starting from arbitrarily two and reducing the length of digon chains we obtain different source links. The example of two different C70 isomers with the same geometrical structure and the same G and G¢, shows that for the exact recognition of fullerene isomers we need to know more then their geometrical and chemical symmetry (see the Subsection 3.2.3). In the same way, from 4-valent graphs with two hexagonal bases, 12 triangular and 6(2k+1) quadrilateral faces (k = 0,1,2,...) we obtain the infinite class of fullerenes C36, C60, C84,... with the symmetry group G = G' = D6h