Recrossing of the transition state

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

If there are recrossings of the transition state, this will cause the positive contribution to kit) to be somewhat less than 1 and the negative contribution to be somewhat greater... 

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. 

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 activation free energy AA can be used to compute the TST approximation of the rate constant = Ce, where C is the preexponential factor. Because not every trajectory that reaches the transition state ends up as products, the actual rate is reduced by a factor k (the transmission coefficient) as described earlier. The transmission coefficient can be calculated using the reactive flux correlation function method. " " " Starting from an equilibrated ensemble of the solute molecules constrained to the transition state ( = 0), random velocities in the direction of the reaction coordinate are assigned from a flux-weighted Maxwell-Boltzmann distribution, and the constraint is released. The value of the reaction coordinate is followed dynamically until the solvent-induced recrossings of the transition state cease (in less than 0.1 ps). The normalized flux correlation function can be calculated using " ... 

Chemical dynamics simulations of the gas phase 5 2 reactions of methyl halides have been carried out at many different levels of theory and compared with experimental measurements and predictions based on transition state theory and RRKM (Rice-Ramsperger-Kassel-Marcus) theory. Although many 5 2 reactions occur by the traditional pre-reaction complex, transition state, post-reaction complex mechanism, three additional non-statistical mechanisms were detected when the F -CH3-I reaction was analysed at an atomic level (i) a direct rebound mechanism where F attacks the backside of the carbon and CH3-F separates (bounces off) from the iodine ion, (ii) a direct stripping mechanism where F approaches CH3-I from the side and strips away the CH3 group, and (iii) an indirect reaction where the pre-reaction complex activates the C-I bond causing a CH3-I rotation and then the 5 2 reaction. The presence of these processes demonstrate that three non-statistical effects, (i) recrossing of the transition state is important, (ii) the transfer of the translational energy from the reactants into the rotational and vibrational modes of the substrate is inefficient, and (iii) there is... 

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. 

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 Canonical Variational Theory [39] is an extension of the Transition State Theory (TST) [40,41]. This theory minimizes the errors due to recrossing trajectories [42-44] by moving the dividing surface along the minimum energy path (MEP) so as to minimize the rate. The reaction coordinate (s) is defined as the distance... 

Of course, TST is sometimes incorrect even in gases (see, for example, the well known breakdown of TST in its standard form exhibited by activated unimolecular reactions) in a solvent, this approach can fail due to different reasons, such as the retarding effects or collisionally induced recrossing. All these sources of breakdown of the Transition State Theory have... 

The density of reactive states p( ) defined by Eq. (6) is the quantum mechanical analogue of the transition state theory p ( ) of Eq. (14). Transition state theory with quantum effects on the reaction coordinate motion and recrossing predicts that the CRP will increase in smooth steps of height kt at each energy level of the transition state and that p( ) will be a sum of bell-shaped curves, each centered at an energy E. We have found clear evidence for this prediction in the densities of reactive states p(E) that we have calculated by accurate quantum dynamics. 

In the usual formulation [79, 81a, 89], the slow mode is diffusive with diverging average velocity. However, due to the presence of the strong sink there are virtually no recrossings of the transition point and once the system has overcome the barrier, it may be considered to be in the product state. Therefore, instead of diffusive we may search for a collective quasiballistic (ballistic within the transition region) slow mode by discarding a high-frequency part of the relaxation spectrum. The simplest way to do so is just to introduce a certain cutoff frequency X, and the slow mode will correspond to the part of the spectrum J(co) with co < A, that is. 

Figure 7.1 Schematic illustrations for the concept of transition state, (a) The transition state is a dividing surface between the reactant and the product regions in the phase space, which any reacting trajectory crosses only once and any non-reacting trajectory does not cross, (b) Illustration of recrossing trajectories. Such recrossings are prohibited by the definitions of the transition state.

We will first give an overview of the issues involved via a brief description of the Transition State Theory and the dynamic Grote-Hynes Theory, as developed for charge transfer reactions in solution by van der Zwan and Hynes.This will introduce the ideas of equilibrium and nonequilibrium solvation, friction and barrier recrossing. We then indicate some of the consequences and predictions for the Sfjl and Sfj2 reaction types. 

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