Electronic structure of polyhedral carbon cages consisting of hexagons and triangles

An infinite series of (3, 6) cages is defined by trivalent carbon polyhedra composed of hexagonal and four triangular rings. A zone-folding construction is applied to the graphene band structure to yield explicit expressions for the pi-molecular orbitals, energies, and symmetries of the cages that depend only on four indices m, n, p, and q. Leapfrog members of the series (m-n=0 mod 3 and p-q=0 mod 3) have closed shells in a neutral form with two filled nonbonding orbitals; all others have closed shells as dications. Quantum chemical calculations on C-12,C-48, and C-52(2+) confirm this result. Embedding relationships are proved for the spectra of (3, 6) cages related by inflation transformations corresponding to stretching and rotation of the polyhedral net.