Crossing the dividing surface of transition state theory. I. Underlying symmetries and motion coordination in...

The objective of the present paper is to show the existence of motion coordination among a bundle of trajectories crossing a saddle point region in the forward direction. For zero total angular momentum, no matter how complicated the anharmonic part of the potential energy function, classical dynamics in the vicinity of a transition state is constrained by symmetry properties. Trajectories that all cross the plane R = R* at time t = 0 (where R* denotes the position of the saddle point) with the same positive translational momentum P(R*) can be partitioned into two sets, denoted "gerade" and "ungerade," which coordinate their motions. Both sets have very close average equations of motion. This coordination improves tremendously rapidly as the number of degrees of freedom increases. This property can be traced back to the existence of time-dependent constants of the motion.