Longitudinal and transverse hyperpolarizabilities of carbon nanotubes: a computational investigation through the coupled-perturbed Hartree–Fock/Kohn–Sham scheme

Abstract

Static electronic polarizability $\alpha$ and second hyperpolarizability $\gamma$ of semiconducting and conducting carbon nanotubes with radius up to 7.5 Å are evaluated using the coupled-perturbed Hartree–Fock/Kohn–Sham scheme, as implemented in the periodic CRYSTAL14 code, and a split-valence basis set. Two density functionals, namely LDA (pure local) and B3LYP (hybrid), and the Hartree–Fock Hamiltonian are compared. A few PBE (gradient corrected) density functional data are also produced for comparison with previous calculations. Convergence of both longitudinal (L) and transverse (T) components is documented. It is shown how the second hyperpolarizability depends critically on the computational conditions, the more so the larger the radius of the nanotube (and thus the smaller the energy gap). The longitudinal component is sensibly affected by the truncation of the exact exchange series (HF and B3LYP), which must include electron–electron interactions at a distance up to 100 Å in order to have $\gamma _L$ converged to better than 1 %. The transverse $\gamma _T$ component of conducting tubes critically depends on the number of k points in reciprocal space: at least 900 k points are required to converge better than 1 % at the LDA level. Coupled-perturbed results are compared to uncoupled values obtained from a sum-over-states (SOS) approach. The difference between the two is particularly important along the transverse direction and when pure DFT functionals are used: the coupled-perturbed correction can shrink the SOS value by several hundreds times. The ratio LDA/HF is roughly constant around 2 for $\alpha _L$; it ranges between 25 and 60 for $\gamma _L$. As regards the convergence with the nanotube radius, the $R^2$ law is confirmed for $\alpha _L$ and $\alpha _T$ (normalized for the cell parameter) at all levels of theory. For the second hyperpolarizabilities $\gamma _L$ and $\gamma _T$, a clear $R^5$ dependence is observed.