Isomers of the toroidal forms of graphitic carbon

Eight isomers of the toroidal cage form of graphitic carbon C₂₄₀ and C₂₅₀ that contain pentagonal chains composed of hexagons of carbon atoms are proposed. The separation between the parallel fringes in these structures is close to that observed experimentally for nanometer-scaled carbon tubes (0.34 nm). The isomers consist of pairs of pentagons and heptagons with different orientation and position along with hexagons of carbon atoms. The cohesive energies and thermal stabilities of the isomers are studied by molecular dynamics. The proposed structures are found to be thermodynamically stable, although their cohesive energies are slightly higher than the spherical fullerene C₂₄₀. Furthermore, the electronic states for the isomers are calculated by molecular-orbital calculations. Highest-occupied-molecular-orbitals--lowest-unoccupied-molecular-orbitals gaps for the tori obtained by simple H\"uckel-type calculations indicate that toroidal structures tend to be chemically stable when the pentagons and heptagons are radially aligned, i.e., the inner and outer hexagon tubes have the same orientation.