Frontier Orbitals in Leapfrog Fullerenes

Fullerenes are trivalent polyhedra of carbon, consisting of hexagons and pentagons. The following relations hold  

Since Fr = Ft x Fg, we could already simplify the face term to a form which precisely corresponds to the Clar representation  

This result is based on the C2v site-symmetry of an edge. The translation in this site has a radial t-component of ai symmetry, and two tangential itt-components of b + b2 symmetry. The fibre bundle will thus correspond to the induction of 

This is precisely the representation of the Fries bonds. We can thus compare the Fries and Clar structures in a general leapfrog, and find from Eq. (6.144)  

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. 

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