The separation of the reaction coordinate in transition state theory: Regularity and dimensionality reduction resulting from local symmetry.

Abstract

Local symmetry in a transition state is defined as the absence of cross terms involving the reaction coordinate in the Taylor expansion about a saddle point of a potential energy surface (i.e., by the assumption ∂2V/∂R∂qj|* = 0). Then, the plane R = R* is, but in the immediate neighborhood of the saddle point only, a local plane of symmetry of the potential energy surface. When this decisive condition is met, together with a second requirement that the kinetic energy be diagonal, the reaction dynamics becomes extremely simple, even in a multidimensional system, because it is determined solely by the harmonic part of the potential, at least during the early stages of the evolution. The dimensionality of the system is then reduced, with the reaction coordinate neatly separated from the vibrational degrees of freedom. The latter are then spectators that can only interact among themselves. As a result of this local symmetry, the subset of reactive trajectories is, during an appreciable period of time, observed to form a bundle grouped around an average trajectory. The distance separating the centers of mass of the two dissociating fragments is the appropriate reaction coordinate. The dynamical reaction path, defined as the central curve of a reactive cylinder in phase space, can be derived in closed form as a surprisingly simple one-dimensional law of motion and can be said to derive from a 1D effective Hamiltonian. An alternative formulation of the problem is possible, in which bond lengths are adopted as internal coordinates, although the expression of the kinetic energy becomes much more complicated. Explicit conditions under which the reaction coordinate decouples from spectator modes can be stipulated.