Toroidal Fullerenes

Here, the topological nature of the tori will be discussed briefly. Figure 1 shows the five possible prototypes of toroidal forms that are considered to be related to fullerenes. These structures are classified by the ratios of the inner and outer diameters r, and r, and the height of the torus, h. (Note that is larger than / ,) As depicted in Fig. 1, if r, = r, and h r, and h = — r,) then the toroidal forms are of type... 

Because the cohesive energy of the fullerene Cyo is —7.29 eV/atom and that of the graphite sheet is —7.44 eV/atom, the toroidal forms (except torus C192) are energetically stable (see Fig. 5). Finite temperature molecular-dynamics simulations show that all tori (except torus Cm2) are thermodynamically stable. 

Spherical and toroidal fullerenes have an extensive chemical literature, and Klein bottle polyhexes have been considered, for example, in [Kir97, KlZh97]. 

Figure 3.1 Smallest spherical, toroidal, Klein bottle and projective fullerenes. The first column lists the graphs drawn m the plane, the second the map on the appropriate surface and the third the dual on the same surface. The examples are (a) Dodecahedron (dual Icosahedron), (b) the Heawood graph (dual Ky), (c) a smallest Klein bottle polyhex (dual 3,3,3), and (d) the Petersen graph (dual Ke).

At least one toroidal polyhex that is cell-complex exists for all numbers of vertices v > 14. The unique cell-complex toroidal fullerene at v = 14 is a realization of the Heawood graph. It is GC2,i(hexagon) in terms of Goldberg-Coxeter construction and is the dual of K7, which itself realizes the 7-color map on the torus. This map and its dual are shown in Figure 3.1. 

Kir94] E. C. Kirby, On toroidal azulenoids and other shapes of fullerene cage, Fullerene Science and Technology 2-4 (1994) 395-404. 

Kir97] E. C. Kirby, Recent works on toroidal and other exotic fullerene structures, chapter 8 in From Chemical Topology to 3-Dimensional Geometry, ed. by A.T. Balaban, Plenum Press, 1997,263-296. 

Such a molecule can be stabilized by a system of delocalized Ji-electrons, which is closed into a toroid of 10 aromatic rings. Reactive sites are four CH groups, which are at the ends of this molecular tube. Such substances belong apparently to a new class of organic compounds, which is intermediate between planar polycyclic aromatic hydrocarbons and three-dimensional fullerenes, nanotubes. Quantum-chemical calculations of the electronic and spatial structure of C32H8 and some other molecules indicate that they have an increased reactivity and semiconductor properties. 

Cash, G.G. (1998). A Simple Means of Computing the Kekule Structure Count for Toroidal Polyhex Fullerenes. J.Chem.lnf.Comput.Sci., 38, 58-61. 

Almost any divalent polyhedron, spherical, toroidal, or of higher genus, is at least conceivably a candidate for an sp carbon framework if reasonable bond lengths and angles can be achieved, and so there has naturally been some discussion in the literature of extended definitions of fullerenes (e.g.. Refs. 8-17). Exotic topologies are dealt with elsewhere in the present volume in the chapters by Kirby and Klein, but even within the class of pseudospherical polyhedra other face sizes are possible. The Euler formula for an n-vertex divalent spherical polyhedron is... 

Recent Work on Toroidal and Other Exotic Fullerene Structures... 

Critics of such exotic fullerene work can still, correctly, point out that there is as yet no evidence for the existence of toroidal forms of carbon, nor any very obvious synthetic method to try. However, much the same was said about buckminsterfullerene before 1990 the fact that carbon tubules exist, and that more than one calculation method has shown that certain toroidal structures might have reasonable stability, does encourage some optimism. 

Figure S. Two stereoisomeric (but isomorphic) toroidal fullerenes formed b> gluing pairs of sides in different order in (a) the rectangle is folded down into the plane of the paper and glued top to bottom in (b) the rectangle is rotated 90 anticlockwise first. Both cylinders are then bent outwards from the paper before joining to form a torus. The two stereoisomers can be Interchanged by a formal topological deformation process, but not by any process that is physically plausible. [This illustration is an adaptation of one used by Klein (Ref. 20a). Copyright 1994 American Chemical Society.)...

Figure 10. An example of pentagon and heptagon formation, arising from the connecting of polyhex sheets that are oriented in different planes, during the construction of toroidal fullerenes (see Figure 9). (From Ref. 25.)...

Table . A Selection of Variously Sized Toroidal Fullerenes with Five-, Six-, and Seven-Membered Rings, and Their HOMO-LUMO Values (P units) Compared with C, ...

SOME PARTICULAR CLASSES OF STRUCTURE 8.9.1. Toroidal Fullerenes... 

Molecular mechanics calculations have been carried out on several toroidal structures, including both a toroidal polyhex, and certain mixed systems containing 5-, 6-, and 7-ring sizes. Not surprisingly, suitable mixed systems are more favorable with regard to mechanical strain, and the energy of formation becomes comparable to values for medium-size fullerenes. 

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