The physics behind the family behavior of optical transition energies in single-wall carbon nanotubes

Experimental optical spectroscopy studies of single-wall carbon nanotubes (SWNTs) revealed $2n{+}m{=}$constant family patterns in the electronic transition energies. Meanwhile, the family behavior remained unexplained within the simple tight-binding approximation that has been commonly used for calculations of the SWNT band structure. We here present calculations for the optical transition energies in SWNTs using an extended tight- binding approximation which allows optimization of C-C bond lengths and bond angles along with the many-body corrections reported in the literature. Our calculations closely reproduce the experimentally observed family behavior, and find that the family behavior can be attributed to the collective effect of curvature-induced rehybridization, long-range atomic interactions, geometrical structure relaxation, and many-body interactions. Our calculations clarify controversial results concerning the magnitude of quasiparticle corrections and exciton binding energies in SWNTs.