Family behavior of the optical transition energies in single-wall carbon nanotubes of smaller diameters

Using the extended tight-binding model that allows bond lengths and angles to vary, the optical transition energies $E_{ii}$ in single-wall carbon nanotubes are calculated as a function of inverse tube diameter. After geometrical structure optimization, the $2n+m=\text{constant}$ family behavior observed in photoluminescence (PL) experiments is obtained, and detailed agreement between the calculations and PL experiments is achieved after including many-body corrections.