Electronic structure of single- and multiple-shell carbon fullerenes

We study the electronic states of giant single-shell and the recently discovered nested multiple-shell carbon fullerenes within the tight-binding approximation. We use two different approaches, one based on iterations and the other on symmetry, to obtain the π-state energy spectra of large fullerene cages: ${\mathrm{C}}_{240}$, ${\mathrm{C}}_{540}$, ${\mathrm{C}}_{960}$, ${\mathrm{C}}_{1500}$, ${\mathrm{C}}_{2160}$, and ${\mathrm{C}}_{2940}$. Our iteration technique reduces the size of the problem by more than one order of magnitude (factors of ∼12 and 20), while the symmetry-based approach reduces it by a factor of 10. We also find formulas for the highest occupied and lowest unoccupied molecular orbital energies of ${\mathrm{C}}_{60\mathit{n}}^2$ fullerenes as a function of n, demonstrating a tendency towards a metallic regime for increasing n. For multiple-shell fullerenes, we analytically obtain the eigenvalues of the intershell interaction.