Violation of the Condon Approximation in Semiconducting Carbon Nanotubes

Abstract

The Condon approximation is widely applied in molecular and condensed matter spectroscopy and states that electronic transition dipoles are independent of nuclear positions. This approximation is related to the Franck-Condon principle, which in its simplest form holds that electronic transitions are instantaneous on the time scale of nuclear motion. The Condon approximation leads to a long-held assumption in Raman spectroscopy of carbon nanotubes: intensities arising from resonance with incident and scattered photons are equal. Direct testing of this assumption has not been possible due to the lack of homogeneous populations of specific carbon nanotube chiralities. Here, we present the first complete Raman excitation profiles (REPs) for the nanotube G band for 10 pure semiconducting chiralities. In contrast to expectations, a strong asymmetry is observed in the REPs for all chiralities, with the scattered resonance always appearing weaker than the incident resonance. The observed behavior results from violation of the Condon approximation and originates in changes in the electronic transition dipole due to nuclear motion (non-Condon effect), as confirmed by our quantum chemical calculations. The agreement of our calculations with the experimental REP asymmetries and observed trends in family dependence indicates the behavior is intrinsic.