The Leapfrog Rule and Electron Deficiency

The possibility of a general rule connecting electronic and geometric structure for fullerenes first emerged from Hiickel calculations on the icosahedral series described earlier, i.e., where C with n = (fi ij + f). The electronic configurations of the icosahedral fullerenes follow a remarkably simple rule  

Ih Ceo is the first of an infinite magic-number series of neutrals and / , C20 the first of a series with closed-shell dications but open-shell neutrals. Inspection of the two series shows a general geometrical relationship in that any open-shell icosahedral fullerene C with n = 60k + 20 = 20(3k + 1) canbe converted into a larger icosahedral fullerene C3 with 3n = 60(3k + 1) by a specific transformation, and all icosahedral closed-shell neutrals are so produced. The conversion operation is the so-called 

An alternative route to the same result is to take the dual first (in this case giving the dual of the dodecahedral parent, i.e., the icosahedron), and then truncate the result by shaving off a small pyramid from every vertex (in this case producing, of course, the truncated icosahedron).  

The leapfrog operation has a field of action much wider than the set of fullerene polyhedra it can be applied to other polyhedra, trivalent and nontrivalent, and in general to any map (graph embedded in a surface), and can be repeated an indefinite number of times to generate ever larger polyhedra. Trivalent polyhedra may be classified by their position in such series of repeated leapfroggings every trivalent 

The chemically significant fact about the leapfrog fullerenes is that, considered as neutral carbon frameworks Cy each has exactly 3 /2 bonding and 3n/2 antibonding 71 orbitals. As fullerenes exist for all n-20 + 2k(ki ), the leapfrog rule is thus °  

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