Nonlinearly coupled generalized Fokker-Planck equation for rotational relaxation

We have investigated the stochastic dynamics of a one dimensional rotor with ${\mathit{C}}_3$ symmetry and zero barrier to rotation about the symmetry axis. Angular momentum correlation functions derived from various stochastic dynamics models were compared to the corresponding correlation functions obtained from a molecular dynamics simulation [G. Widmalm, R. W. Pastor, and T. E. Bull, J. Chem. Phys. 94, 4097 (1991)]. None of the existing classical models agrees with the simulation; and we have shown in general that no linearly coupled generalized Langevin equation with Gaussian random noise can reproduce the simulation results. A quantum stochastic dynamics model [T. E. Bull, Chem. Phys. 143, 381 (1990)] extrapolated to the classical limit does, however, agree with the computer simulations. But this model is limited to very small molecules because the matrices involved become prohibitively large for even moderately sized molecules. In order to address some of these limitations, we have constructed a nonlinearly coupled rotor-bath model for the rotor. The form of the nonlinear coupling between the rotor and bath is determined by the symmetry of the rotor. A classical nonlinearly coupled generalized Langevin equation and its corresponding nonlinearly coupled Fokker-Planck equation were derived from this microscopic rotor-bath model using the projection operator formalism. In the limit of white noise, these equations reduce to the standard equations derived with linear coupling. With colored noise, however, the linearly and nonlinearly coupled equations are distinct. Angular momentum correlation functions calculated with this nonlinearly coupled Fokker-Planck equation are in excellent agreement with the simulations both in terms of the short time Gaussian decay and long time exponential tail and in terms of the magnitudes of the correlation functions. Collision operators derived from this model should therefore provide a more accurate connection between experimentally measured quantities and the underlying microscopic dynamics.