Reactive trajectories

We further comment that reactive trajectories that successfully pass over large barriers are straightforward to compute with the present approach, which is based on boundary conditions. The task is considerably more difficult with initial value formulation. 

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.

Reaction coordinates Reactive trajectories transition states, 43 transmission factor, 42 solution reactions, 46,90 Reaction coordinates, 41-44, 88,91 for enzymatic reactions, 215 reactive trajectories, see Reactive trajectories... 

Reactive trajectories, 43-44,45, 88,90-92,215 downhill trajectories, 90,91 velocity of, 90 Relaxation processes, 122 Relaxation times, 122 Reorganization energy, 92,227 Resonance integral, 10 Resonance structures, 58,143 for amide hydrolysis, 174,175 covalent bonding arrangement for, 84 for Cys-His proton transfer in papain, 141 for general acid catalysis, 160,161 for phosphodiester hydrolysis, 191-195,... 

Figure 21. Family of reactive trajectories in the ground adiabatic potential energy surface determined by Eq. (13). Crosses indicate the caustics. Taken from Ref. [29].

The stable and unstable manifolds are then described by I = 0, reactive trajectories by I > 0, and nonreactive trajectories by I < 0. 

An example for the predictive power of the TS trajectory is shown in Fig. 4. This figure shows a random instance of the TS trajectory (black) and a reactive trajectory (red) for the linearized Langevin equation (15) with N = 2 degrees of... 

The crucial ingredient in a reaction rate calculation is the identification of reactive trajectories. To this end, initial conditions sampled from Eq. (49) are propagated forward and backward to a time 7)nt. Those trajectories that begin on the reactant side of the barrier at t = — 7jnt and end on the product side at t = +T-mt are then regarded as (forward) reactive. The identification of reactive... 

Reactive trajectories can be identified a priori, without any numerical simulation, if the moving separatrices are used instead of the standard dividing surfaces. In relative coordinates, the portion of the barrier ensemble that is forward reactive can immediately be identified from Fig. 3 those trajectories are reactive whose initial velocity is so large that it lies above both the stable and unstable manifolds. Because the initial conditions in the ensemble (49) lie at Ax(0) = —xj, this reactivity criterion reads explicitly... 

T. Bartsch, T. Uzer, J. M. Moix, and R. Hernandez, Identifying reactive trajectories using a moving transition state, J. Chem. Phys. 124, 244310 (2006). 

In the transition path sampling method we are interested in trajectories that start in a certain region of configuration space, which we will call region si, and end in another region, 38. We call such trajectories reactive. Accordingly, we restrict the probability distribution from (7.3) to reactive trajectories only (see Fig. 7.2)... 

For determination of reaction probability and reaction cross section, a large number of collision trajectories have to be considered and appropriate averages over the initial conditions performed. The reaction probability is calculated for a specified initial relative velocity vR (i.e. initial relative kinetic energy), rotational state /, and impact parameter b. The reaction probability is the ratio of number of reactive trajectories to the total number trajectories, i.e. 

Consider a model PES for a collinear reaction, A + BC — AB + C, on which a typical reactive trajectory has been shown (Fig. 9.25). The motion along rAB would correspond to reactant translation/product vibrational and similarly motion along rBC would correspond to reactant vibration/product translation. 

A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-71]. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. 

R. A. Marcus It is certainly necessary to include all of the reactive trajectories, those that lead to immediate dissociation and those that do not. As Wigner pointed out, in effect, one needs to include all of the phase space occupied by the assumed transition state. Exclusion of any of these trajectories would include part of that phase space. Transition state theory is (classically) an upper bound to the rate since the trajectory may include parts of the TS phase space twice (multiple crossings of the transition state). 

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