Energetics and structure of toroidal forms of carbon

Total-energy calculations for 13 different fullerene clusters, including 8 toroidal fullerenes, are performed using a parametrized tight-binding functional, and in select cases, an all-electron first-principles local-density functional. The tori considered here are composed of graphitelike six-membered rings, with five-membered rings on the outer surface of the torus providing positive Gaussian curvature, and seven-membered rings on the inner toroidal surface providing negative Gaussian curvature. Comparisons are made with energies calculated from a Stillinger-Weber--type potential for graphite, which is found to be inadequate for addressing relative energetic stability among the tori. Two empirical strain relations are tested against our tight-binding energy calculations. A simple elastic theory is found to give reasonably good energy orderings. Charge-density calculations on the tori reveal that electron density is enhanced for atoms in five-membered rings, and depleted for atoms in seven-membered rings. First-principles total-energy calculations indicate that a single potassium atom is bound to the center of a C₁₄₄ torus by 0.5 eV. It is proposed that atoms with higher orbital angular momentum (e.g., Zr, etc.) may bond more tightly to the torus, and thus facilitate formation of C₁₂₀ or similar tori.