Dynamics of Chemical Reaction around a Saddle Point: What Divides Reacting and Non-Reacting Trajectories?
In the transition state theory, the reaction rate is evaluated by the flux through a dividing surface in the phase space that separates the reactant and the product regions. This is validated by the non-recrossing assumption that any trajectory having crossed the dividing surface will not return to the surface again, so that the entire flux crossing the surface in the positive direction is regarded as eventually going to the product side. It is usually thought that this assumption generally breaks down in multi-dimensional reaction systems because the couplings among multiple degrees of freedom cause complicated motion of the system, resulting in trajectories going back and forth several times in the saddle region. Here it is shown, under moderate assumptions, that this recrossing problem is solved and that a non-recrossing dividing surface can be taken even under the existence of coupling by cleverly choice of a coordinate system. The coordinate is introduced through a special coordinate transformation to cancel out the couplings terms, simplifying the dynamics by reducing the system onto essentially one-dimensional motion along the newly defined reaction coordinate. The analyses of the reaction dynamics and the calculation of the rate constants are presented using this coordinate transformation.