The reaction rate constant in a system with localized trajectories in the transition region: Classical and quantum dynamics

We examine here a model for a chemical reaction which has a large number of trajectories localized in the transition region to find out how they affect reaction dynamics and the rate constant. One issue is whether the incessant crossing and recrossing of the dividing surface by the trapped trajectories impairs our ability to define a rate constant. To study this we rewrote the correlation function formula for the rate constant in a form that allows us to follow the crossing and recrossing of the dividing surface in time. We find that due to a ''dephasing'' process the correlation function converges towards a well-defined rate constant, even though the recrossing of the dividing surface by the bound trajectories is still going on. It is well known that error in the transition state approximation is caused by recrossings. In view of this, it is surprising that the variational transition state theory applied to our model picks as the best dividing surface one that leads to an infinite number of recrossings. The transition state rate constant for another dividing surface, which has only a few recrossings, has large errors. We show that the variational transition state theory improves the value of the rate constant by diminishing the contribution from trajectories that recross an odd number of times, relative to those that undergo an even number of recrossings; it does not, and need not, diminish the total number of recrossings. Quantum calculations show that the bound trajectories correspond to high energy, long lived resonances. In spite of this change in the physical description, the evolution of the quantum correlation functions follows closely that of the classical ones. (C) 1997 American Institute of Physics.