Transition-state theory , barrier sampling

In both solvents, the variational transition state (associated with the free energy maximum) corresponds, within the numerical errors, to the dividing surface located at rc = 0. It has to be underlined that this fact is not a previous hypothesis (which would rather correspond to the Conventional Transition State Theory), but it arises, in this particular case, from the Umbrella Sampling calculations. However, there is no information about which is the location of the actual transition state structure in solution. Anyway, the definition of this saddle point has no relevance at all, because the Monte Carlo simulation provides directly the free energy barrier, the determination of the transition state structure requiring additional work and being unnecessary and unuseful. 

The key idea that supplements RRK theory is the transition state assumption. The transition state is assumed to be a point of no return. In other words, any trajectory that passes through the transition state in the forward direction will proceed to products without recrossing in the reverse direction. This assumption permits the identification of the reaction rate with the rate at which classical trajectories pass through the transition state. In combination with the ergodic approximation this means that the reaction rate coefficient can be calculated from the rate at which trajectories, sampled from a microcanonical ensemble in the reactants, cross the barrier, divided by the total number of states in the ensemble at the required energy. This quantity is conveniently formulated using the idea of phase space. 

We shall find it simpler in many of the examples of this section to consider the product descent off the barrier away from the transition state rather than the reactant climb toward the transition state. These two processes, in equilibrium systems, are time reversals of each other, for the forward and backward reaction pair. In what follows, we do not claim to present a complete catalog of all the studies that have been done in this area rather, we will give the interested reader a sample of the interaction between theory and simulation and provide some pointers to the literature. 

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