Dynamic pathways to mediate reactions buried in thermal fluctuations. II. Numerical illustrations using a model...

The framework recently developed for the extraction of a dynamic reaction coordinate to mediate reactions buried in thermal fluctuation is examined with a model system. Numerical simulations are carried out for an underdamped Langevin equation with the Müller-Brown potential surface, which contains three wells and two saddles, and are compared to the prediction by the theory. Reaction probabilities for specific initial conditions of the system as well as their average over the Boltzmann distribution are investigated in the position space and in a space spanned by the position coordinates and the velocities of the system. The nonlinear couplings between the reactive and the nonreactive modes are shown to have significant effects on the reactivity in the model system. The magnitude and the direction of the nonlinear effect are different for the two saddles, which is found to be correctly reproduced by our theory. The whole position-velocity space of the model system is found to be divided into the two distinct regions: One is of mainly reactive (with reaction probability more than half) initial conditions and the other, the mainly nonreactive (with reaction probability less than half) ones. Our theory can actually assign their boundaries as the zero of the statistical average of the new reaction coordinate as an analytical functional of both the original position coordinates and velocities of the system (solute), as well as of the random force and the friction constants from the environment (solvent). The result validates the statement in the previous paper that the sign of the reaction coordinate thus extracted determines the fate of the reaction. Physical interpretation of the reactivity under thermal fluctuation that is naturally derived, thanks to the analyticity of the theoretical framework, is also exemplified for the model system.