Ab initio calculation of nonlinear optical properties for chiral carbon nanotubes. Second harmonic generation and dc-Pockels effect

Although the static frst hyperpolarizability 훽 of graphene and chiral carbon nanotubes (CNTs) vanishes by symmetry, that is no longer true for frequency-dependent felds in which case there is a unique nonzero component. We evaluate that component for the dc-Pockels efect and second harmonic generation of several CNTs by means of the coupled perturbed Kohn–Sham method with the CAM-B3LYP functional as implemented in the CRYSTAL code. Both electronic and vibrational contributions are considered. Of particular interest is the frequency dependence in the region below the frst resonance. Our results for the role of band gap, type of chirality, CNT radius and efect of orbital relaxation are analyzed using the conventional sum-over-states formulation in conjunction with the calculated band structure and density of states.