Recrossing assumption

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

Note that when f = 1 we find that the assumptions of TST are met and K = 1. As the number of recrossings of the transition state increases, both P and K decrease. 

The second major assumption related to the dynamics is that recrossings of the saddle point are absent, i.e., the classical transmission probability is given by... 

The motion in the reaction coordinate Q is, like in gas-phase transition-state theory, described as a free translational motion in a very narrow range of the reaction coordinate at the transition state, that is, for Q = 0 hence the subscript trans on the Hamiltonian. The potential may be considered to be constant and with zero slope in the direction of the reaction coordinate (that is, zero force in that direction) at the transition state. The central assumption in the theory is now that the flow about the transition state is given solely by the free motion at the transition state with no recrossings. So when we associate a free translational motion with that coordinate, it does not mean that the interaction potential energy is independent of the reaction coordinate, but rather that it has been set to its value at the transition state, Q j = 0, because we only consider the motion at that point. The Hamiltonian HXlans accordingly only depends on Px, as for a free translational motion, so... 

Note that the integration over all positive Pf leading from the reactant part of phase space to the product part comes from the assumption of no recrossing of the transition state. Integration over all momenta // in the first line of Eq. (10.33) gives... 

All the theories require some assumptions to be made about the nature of some critical reaction intermediate which may be an activated complex located at a barrier in the potential surface or at a barrier formed in the long-range attractive potential by the orbital angular momentum of the reactants or product. The determination of the correct location for the critical transition state is a major problem in applying statistical theories to chemical reactions. The underlying assumption of statistical theories is that once the transition state is passed in the direction of reaction products, it is not recrossed. The nature of the transition state determines... 

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. 

The key further assumption of RRKM theory is that there is no recrossing of the dividing surface. In other words, it is assumed that all trajectories passing through the transition state in the direction of products continue directly on to products. With this assumption, the reactivity function is approximated by a step function in the velocity through the dividing surface, yielding... 

In all cases, TST provides a relatively simple method for the prediction of the rate constant. However, one must always keep in mind the assumption of TST, especially the short-time and positive momentum criteria, which prohibit recrossing of the saddle point. The fundamental idea that the dynamical process follows a reaction path must also be critically examined. 

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