Leapfrog transformation

Notably the leapfrog transformation may be applied to stmctures much more general 243,248-250 than fullerenes. 

The leapfrog transformation may be viewed as a representative member of a whole algebra of transformations. To see this let us denote the capping and dualing transformations of (a) and (b) above as C and D, respectively, while the vertex truncation process of (b ) above is denoted by T. Then the equivalence between the two formulations for the leapfrog transformation may be written as... 

Figure 25. The dual-truncation process for leapfrog transformation.

Sah has described a yet more general way to extend the leapfrog transformation to a whole sequence of possibilities. These transformations may be represented within our algebra as DT( where a and b are integers such that a > 0 and a > > 0. The particularities of the new transformation are conveniently described in terms of the triangular lattice acts on the dual polyhedron to replace each triangular face by... 

Diu03] M. Diudea, Capra — a leapfrog related map transformation, Studia Universitatis Babes-Bolyai, Chemia 48 (2003) 3-21. 

Figure 4. Transformations of structural components (faces of two types, vertices and edges) of a fullerene under the leapfrog operation. The double bonds indicate those edges of the leapfrog that derive from the parent, and give a consistent Fries Kekul6 pattern with the maximal number of "benzenoid" rings and therefore the maximal stability on a localized model of n-electronic structure,...

The Clar structure thus has six extra bonding orbitals as compared with the Fries structure. When both bonding schemes are correlated, as illustrated in Fig. 6.11, this sextet must correlate with the anti-bonding half of the Fries stmcture. It will thus be placed on top of the Clar band, and actually be nearly non-bonding, forming six low-lying virtual orbitals, which explains the electron deficiency of the leapfrog fullerenes. Moreover, as the derivation shows, they transform exactly as rotations and translations. 

---