Inertial effects in the orientational relaxation of rodlike molecules in a uniaxial potential

The inertial rotational Brownian motion and dielectric relaxation of an assembly of noninteracting rodlike polar molecules in a uniaxial potential are studied. The infinite hierarchy of differential-recurrence relations for the equilibrium correlation functions is generated by averaging the governing inertial Langevin equation over its realizations in phase space. The solution of this hierarchy for the one-sided Fourier transforms of the relevant correlation functions is obtained using matrix continued fractions yielding the longitudinal dipole correlation function, the correlation time, and the complex polarizability, which are calculated for typical values of the model parameters. Pronounced inertial effects appear in these characteristics in the high-frequency region for low damping. The exact longitudinal correlation time is compared with the predictions of the Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)]. In the low temperature limit, the universal Mel'nikov and Meshkov formula for the inverse of the escape rate provides a good estimate of the longitudinal correlation time for all values of the dissipation including the very low damping, very high damping, and Kramers turnover regimes. Moreover, the low-frequency part of the spectra of the longitudinal correlation function may be approximated by a single Lorentzian with a halfwidth determined by this universal escape rate formula.