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 π-molecular orbitals, energies, and symmetries of the cages that depend only on four indices m, n, p, and q. Leapfrog members of the series $(m-n=0\mathrm{}$ mod 3 and $p-q=0\mathrm{}$ 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 ${\mathrm{C}}_{12},$ ${\mathrm{C}}_{48},$ and ${\mathrm{C}}_{52}^{2+}$ 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.