Transition state theory dynamic recrossings

It should be emphasized that these dynamical effects can lead to significant corrections to conventional transition-state theory where recrossings are neglected. However,... 

In transition state theory, dynamic effects are included approximately by including a transmission coefficient in the rate expression [9]. This lowers the rate from its ideal maximum TS theory value, and should account for barrier recrossing by trajectories that reach the TS (activated complex) region but do not successfully cross to products (as all trajectories reaching this point are assumed to do in TS theory). The transmission coefficient can be calculated by activated molecular dynamics techniques, in which molecular dynamics trajectories are started from close to the TS and their progress monitored to find the velocity at which the barrier is crossed and the proportion that go on to react successfully [9,26,180]. It is not possible to study activated processes by standard molecular dynamics because barrier crossing events occur so rarely. One reason for the... 

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... 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

Beyond Transition State Theory (and, therefore, beyond Monte Carlo simulations) dynamical effects coming from recrossings should be introduced. Furthermore, additional quantum mechanical aspects, like tunneling, should be taken into account in some chemical reactions. 

Most modern investigations of the effects of a solvent on the rate constant, where dynamical fluctuations are included, are based on a classical paper by Kramers from 1940 [1], His theory is based on the transition-state theory approach where we have identified the reaction coordinate as the normal mode of the activated complex that has an imaginary frequency. In ordinary transition-state theory, we assume that the motion in that coordinate is like a free translational motion with no recrossings. This... 

First of all, liquid-phase studies generally do not obtain data which allows static and dynamic solvent effects to be separated [96,97], Static solvent effects produce changes in activation barriers. Dynamic solvent effects induce barrier recrossing and can lead to modification of rate constants without changing the barrier height. Dynamic solvent effects are temperature and viscosity dependent. In some cases they can cause a breakdown in transition state theory [96]. 

Inclusion of dynamical effects allows calculation of corrections to simple Transition State Theory, often described by a transmission coefficient k to be multiplied with the TST rate constant (Section 12.1), or used in connection with variational TST (Section 12.3). Classical dynamics allow corrections due to recrossing to be calculated, while a quantum treatment is necessary for including tunnelling effects. Owing to the stringent... 

In the standard Transition State Theory (TST) the general reaction is viewed as a passage over a mean potential barrier located between reactant and product potential wells. It is often stated that TST is exact under a certain dynamical condition, i.e. a trajectory initially crossing the barrier top or transition state surface S from the side of reactants to the side of products must proceed to products without recrossing S. This no-recrossing condition makes the TST rate constant an upper limit. 

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 this spirit, we will also briefly describe the basis for some of the microscopic kinetic theories of unimolecular reaction rates that have arisen from nonlinear dynamics. Unlike the classical versions of Rice-Ramsperger-Kassel-Marcus (RRKM) theory and transition state theory, these theories explicitly take into account nonstatistical dynamical effects such as barrier recrossing, quasiperiodic trapping (both of which generally slow down the reaction rate), and other interesting effects. The implications for quantum dynamics are currently an active area of investigation. 

For a reaction with a defined transition state and without recrossing, reaction rate can be well approximated by many methods. For such reaction, we can assume that there is a dynamics bottleneck located at the transition state (conventional transition state theory, TST) or at a generalized transition state obtained by a canonical (CTV) or microcanonical (/zVT) criterion. In the later cases, the dividing surface is optimized variationally to minimize the recrossing. Evans first proposed to place the transition state at the location that maximizes the free energy of activation which provides a key conceptual framework for modern variational transition state theory [33]. However, recrossing always possibly exists and only a full-dimensional reactive scattering dynamics calculations are able to provide us the exact rate constant on a defined PES. Eor a detailed discussion, one may refer to the reviews by Truhlar et al. [38,136]. 

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. 

Third, there is a time scale for the dying out of the recrossings which lead to the reduction of the rate below that predicted by transition state theory. This flux-flux correlation time scale we find in the reactions we have so far studied to be a few tens of femtoseconds. One can hope to discover, again by molecular dynamics simulation, the microscopic cause and nature of these recrossings. It... 

---