No-recrossing assumption

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... 

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 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 form 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, with no recrossing, is a fundamental assumption of transition state theory [21], Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

The factor a is included to account for all indistinguishable ways the reactants can approach each other. Notice that due to the assumption of no recrossing of the barrier once a transition state is formed, the rate constant does not depend on the products. 

To calculate rate constants for bimolecular systems, Eqs (7.19) and (7.20) are used. An equivalent theory can be derived for unimolecular reactions. In a unimolecular reaction only one reactant forms a transition state through radiation or in an apparent unimolecular reaction through collision, which results in product generation. Applying similar assumptions as above (classical motion when crossing the barrier, no recrossing events, and thermal equilibrium) leads to the canonical rate constant, given by... 

Actually, the TST expression only holds if all molecules that pass from the reactant over the TS go on to the product, the ideal situation to the ion-molecule barrierless association reaction. The TST assumption is that no recrossing occurs for a given temperature, i.e., all molecules passing through the dividing surface will go on to form the product. 

One fundamental assumption in classical transition state theory is that of no recrossing over the transition state. The advantage, of the RP theory is therefore that since it provides a hamiltonian it can be used in dynamical calculations thereby incorporating the effect of recrossing. However, it is also possible to use the hamiltonian for an estimate of the transmission factor, i.e. the correction to transition state theory from recrossing of the trajectories. An additional correction factor comes from quantum tunneling (see below). Considering the reaction rate constant it may be expressed as... 

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. 

A number of MD studies on various unimolecular reactions over the years have shown that there can sometimes be large discrepancies (an order of magnitude or more) between reaction rates obtained from molecular dynamics simulations and those predicted by classical RRKM theory. RRKM theory contains certain assumptions about the nature of prereactive and postreactive molecular dynamics it assumes that all prereactive motion is statistical, that all trajectories will eventually react, and that no trajectory will ever recross the transition state to reform reactants. These assumptions are apparently not always valid otherwise, why would there be discrepancies between trajectory studies and RRKM theory Understanding the reasons for the discrepancies may therefore help us learn something new and interesting about reaction dynamics. 

Trajectory computations can validate the assumption of no return. When the available energy is only barely above the barrier, which is typically the case for thermal reactants and chemical-sized barriers, a crossing trajectory seldom returns. This is because the potential on the opposite side of the barrier falls quite steeply so that the trajectory is accelerated toward the products. It is only at higher energies that recrossing becomes more common. ... 

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