Localized Gaussian Type Orbital−Periodic Boundary Condition−Density Functional Theory Study of Infinite-Length Single-Walled Carbon Nanotubes with Various Tubular Diameters^†

The detailed geometrical structures of zigzag and armchair type single-walled carbon nanotubes (SWCNTs) with infinite tubular length were investigated using localized Gaussian type orbital−periodic boundary condition−density functional theory (LGTO−PBC−DFT) method. The structures of (n, 0) zigzag SWCNTs were optimized for n = 5−21, (n, n) armchair SWCNTs for n = 3−12. For comparison, the optimized geometry of a two-dimensional graphite sheet was also calculated. It was found that the optimized structures of the SWCNTs showed two C−C bond lengths that decrease with an increase in the tubular diameter. More specifically, the two bond lengths converged with those found in the two-dimensional graphite sheet. We also found a degeneracy in the highest occupied crystal orbitals if identical bond lengths were employed for the zigzag SWCNTs and the two-dimensional graphite sheet. This implies that the two different bond lengths found in the zigzag SWCNTs and the two-dimensional graphite sheet are probably due to the Jahn−Teller effect. The armchair SWCNTs show two slightly different bond lengths if the diameter is less than 12 Å; otherwise they are almost identical, approaching the longer bond length of the two-dimensional graphite sheet. This can be due to the fact that the armchair SWCNTs do not have degeneracy in occupied crystal orbitals for identical C−C bond lengths. The crossing point of the conducting and valence bands of each armchair SWCNT were also calculated and show a diameter dependence in which the deviation from 2π/3a decreases as diameter increases.