Electronic stability of fullerenes: eigenvalue theorems for leapfrog carbon clusters

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Abstract
A leapfrog carbon cluster can be constructed geometrically by omnicapping and dualising a polyhedral parent cluster with one third the number of carbon atoms. In this paper the Hückel molecular orbital energy levels of leapfrog fullerenes, and other related leapfrog carbon clusters, are investigated analytically using graph theory. It is proved in particular that all trivalent leapfrog clusters have antibonding lowest unoccupied molecular orbitals (LUMOs), and that all trivalent leapfrog clusters containing at least one ring of r≠ 3k atoms have bonding highest occupied molecular orbitals (HOMOs). More generally, all trivalent leapfrog clusters have bonding or non-bonding HOMOs. A chemically significant corollary is that all trivalent leapfrog clusters, irespective of their ring counts, are closed shell.

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