Nearly quantitative fitting of exciton energies and exciton binding energies of semiconducting single-walled carbon nanotubes (S-SWCNTs) within a $\pi$-electron model

We use to Pariser-Parr-Pople $\pi$-electron Hamiltonian, which has been widely used to understand the excitonic energy spectra of $\pi$-conjugated polymers, to determine the energy spectra of S-SWCNTs. For the Coulomb interaction parameters, we use the screened Ohno parameters that were successfully used before for the quantitative fitting of the excitonic energy spectrum of poly-(paraphenylenevinylene) \footnote{M. Chandross and S. Mazumdar, Phys. Rev. B {\bf 55}, 1497 (1997)}. In order to take into account of the curvature effects, we use a $\pi$-electron nearest neighbor hopping integral that is substantially smaller than the standard 2.4 eV. With these modified parameters we are able to fit the lowest two exciton energies of a very large number of S-SWCNTs. We make detailed comparisons between the calculated and experimental \footnote{R.B. Weisman and S.M. Bachilo, Nano Lett. {\bf 3}, 1235 (2003)}. We also obtain a binding energy of $\sim$ 0.4 eV for the lowest exciton of S-SWCNTs with diameters $\sim$ 0.8 nm, in agreement with experiments.