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: C₂₄₀, C₅₄₀, C₉₆₀, C₁₅₀₀, C₂₁₆₀, and C₂₉₄₀. 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 C₆₀ₙ² 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.