Random-phase calculations of frequency-dependent polarizabilities and hyperpolarizabilities of long polyene chains

We apply the double-direct random-phase-approximation (DDRPA) method [Ågren et al., J. Chem. Phys. 98, 6417 (1993)] to calculate the static and dynamic polarizabilities and hyperpolarizabilities for a sequence of polyene chains. Like the direct self-consistent-field method, DDRPA is driven directly by the atomic-orbital integrals. It further uses iterative techniques based on direct linear transformations for solving the RPA eigenvalue equations and sets of linear equations. This allows applications to long chains, including ${\mathrm{C}}_{28}$ ${\mathrm{H}}_{30}$ in the present study. The calculated optical spectra, viz., excitation energies and transition moments, the polarizabilities, and the hyperpolarizabilities are in excellent agreement with available experimental data. Computations on the longer polyenes are facilitated by the weak basis-set dependency on chain length for the longitudinal components of the (hyper)polarizabilities. The length dependence of the dispersion is significant even at small frequencies, and is quite different for the various tensorial components and for the averaged static, Kerr, and electrical-field-induced second harmonic generation values. Some of the results can be rationalized by the observation that the most intensive ${}_{}{}^1{}_{}{}^{}$ ${\mathit{B}}_{\mathit{u}}$ transition also determines the band gap, and that the band gap converges very slowly with respect to chain length. The correlation length of the static polarizability is predicted to be about 40 unit cells, while for the static hyperpolarizability the prediction is well over 100.