Diffusion along the Splitting/Commitment Probability Reaction Coordinate

The splitting or commitment probabilities of states in the region of configuration space that separates reactants and products play an important role in the theory of chemical reactions. Assuming that the splitting probability changes more slowly than any other coordinate, we project multidimensional diffusive dynamics onto it. The resulting one-dimensional diffusion equation is not exact because the assumed separation of time scales does not hold in general. Nevertheless, this equation has the remarkable property that it always predicts the exact value of the number of transitions between reactants and products per unit time at equilibrium and hence the exact reaction rate. In the special case of two deep basins separated by a harmonic saddle, this equation is equivalent to the one that describes diffusion along a coordinate perpendicular to the transition state, defined as the surface starting from which reactants and products are reached with equal probability.