Transitions surface states

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

Figure 1 Double well potential for a generic conformational transition showing the regions of reactant and product states separated by the transition state surface.

Figure 2 A typical trajectory satisfying the assumptions of transition state theory. The reactive trajectory crosses the transition state surface once and only once on its way from activated reactant to deactivated product.

We assume that when the activated reactants cross the transition state a fraction P are deactivated as product and the remaining fraction 1 — f recross the transition state surface [8,24]. If each fraction has roughly the same distribution of momenta as the original fraction, we can say that of the fraction 1 — f that recross, P( — P) will be deactivated in the reactant well and the remaining (1 — P)- will recross the transition state into the... 

But this is not the whole story We not only need to know that a trajectory that crosses the transition state surface is eventually deactivated as product, we also need to know whether it originated from the reactant well A trajectory that originates from the product well and ends up as product won t contribute to the forward rate of reaction. Some of the trajectories did originate as product. We need to find that fraction and subtract it. 

The stereoselectivity of some Diels-Alder reactions was also strongly affected in water.26 At low concentrations, in which both components were completely dissolved, the reaction of cyclopentadiene with butenone gave a 21.4 1 ratio of endo/exo products when they were stirred at 0.15 M concentration in water, compared to only a 3.85 1 ratio in excess cyclopentadiene and an 8.5 1 ratio with ethanol as the solvent. Aqueous detergent solution had no effect on the product ratio. The stereochemical changes were explained by the need to minimize the transition-state surface area in water solution, thus favoring the more compact endo stereochemistry. The results are also consistent with the effect of polar media on the ratio.27... 

Transition state strain energies Estr and transition state-surface interaction energies Eint- Diefenbach et al. (88). 

The effects of intrafragment modes on the D/A reactions arc similar in both the gas and liquid phases. The V-T and/or R-T energy transfer is found to excite the reactive T mode or gives rise to dissociation, and T-V and/or T-R energy transfer often causes association. In addition to these mechanisms, the D/A reaction is found to occur by an energy transfer between reactant relative translational mode and solvent modes. The reaction rate is determined by the flux which cross the transition state surface Et=0, and this flux arises essentially by energy transfers between the reactive T mode and V, R, and S modes of reactant and solvent. 

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. 

This work has indicated that VTST may be a useful method for calculating solution rate constants and thereby helping to understand solution reaction dynamics through analysis of the solution transition state surface. However, for this to occur, a simple prescription must be developed for applying VTST to solution systems where full molecular dynamics can be performed, much in the way that Grote-Hynes theory has been used to understand a wide variety of molecular dynamics results. Work by Hahn and Pollak o is proceeding in this direction through the application of VTST to the Cl -I- CHjCI Sn2 reaction in aqueous solution discussed elsewhere in this review.(A closely related study is the VTST work by Tucker and Truhlar on the mono- and dihydrated versions of this S 2 reaction.)... 

Figure 1.16. Two illustrative two dimensional free energies v> q, q ) depending on two variables q and q, and their corresponding reduced free energy functions w q) = —k Thi dq exp[ —j3w( , )] In both cases, w(q) has the same bistable form, but in (a) the coordinate q is a reasonable reaction coordinate, as the transition state surface coincides with q = q. In (6), on the other hand, q is not at a reasonable reaction coordinate. The orthogonal variable q" is crucial for the mechanism of T -> S transitions, and the maximum in w(q) at q = q does not coincide with the transition state surface. The trajectories initiated at configurations with q(r) = q and all ending in B illustrate this.

Here, the rate constant is the product of the mean velocity of crossing the transition-state surface, < dd/dt >Ts, times a probability of being at that surface. Tliis latter quantity depends not only upon the probability p(8 ) of being at the transition state (near the top of the barrier that separates reactants from products) but also upon the width of the reactant well for broader wells, the integral in the denominator of Equation 2.4 will be larger and the rate constant smaller. The mean velocities can be calculated from several theories or directly from dynamical simulations. In addition, corrections to the transition state theory estimate given in Equation 2.4 can also be made from dynamical simulations, but these considerations are beyond the scope of this letter. 

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