Energy splitting between bright and dark excitons in carbon nanotubes

We have calculated the bright-dark energy splitting in semiconducting zigzag carbon nanotubes in a simple tight-binding model for π-electrons. The short-range cut-off of the Coulomb interaction is used as a fitting parameter. We have found that the splitting depends not only on the type of transition, E11 or E22, but also on the structure, or the index, of carbon nanotube, and that family pattern appears in the cut-off parameter dependence of the splitting.