Reactant region

One way to overcome this problem is to start by setting up the ensemble of trajectories (or wavepacket) at the transition state. If these bajectories are then run back in time into the reactants region, they can be used to set up the distribution of initial conditions that reach the barrier. These can then be run forward to completion, that is, into the products, and by using transition state theory a reaction rate obtained [145]. These ideas have also been recently extended to non-adiabatic systems [146]. 

To evaluate the rate constant in a more rigorous way (as is done in detail in Ref. 1), let us consider first the average behavior of many reacting systems with a one-dimensional surface of the type described in Fig. 2.1. We will try to determine what fraction of the systems that pass the reactant region toward the barrier would react. The number of systems in a path length Ax, which have a momentum between p and p + Ap, is given by (see Ref. lb)... 

FIGURE 2.2. A schematic description of the evaluation of the transmission factor F. The figure describes three trajectories that reach the transition state region (in reality we will need many more trajectories for meaningful statistics). Two of our trajectories continue to the product region XP, while one trajectory crosses the line where X = X (the dashed line) but then bounces back to the reactants region XR. Thus, the transmission factor for this case is 2/3. 

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

Here, h (z) is the characteristic function for the reactant region si, equal to unity if state z is in this region and that vanishes otherwise. The characteristic function h.A/-) for the product region 38 is defined similarly. The factor... 

The use of a Type 1 mass spectrometer with a collision region may allow a reaction time on the order of milliseconds. Often the term single collision is used, which depends on the pressure of the reactant molecules, and in many studies the residence time in the reactant region is not specified. 

Note that the activation energy is composed of two components, consisting of the energies associated with the reactant region and the first zone of the transition region ... 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do munerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

Figure 7 Application of the dimer method to a two-dimensional test problem. Three different starting points are generated in the reactant region by taking extrema along a high temperature dynamical trajectory. From each one of these, the dimer isjirst translated only in the direction of the lowest mode, but once the dimer is out of the convex region a full optimization of the effective force is carried oat at each step (thus the kink in two of the paths). Each one of the three starting p>oints leads to a different saddle point in this case.

Figure 4.27. Definition of the vicinal region of the transition state and the perpendicular forward velocity. Here R designates the reactant region and P the product region.

It is assumed in transition state theory that configurations, which are found in the transition state and have a velocity towards the product region will eventually end up in the product region. This means that cases where the supposed product crosses back into the reactant region are miscounted (see Figure 4.28). 

By performing a normal mode analysis in the initial state and in the saddle point, it is then possible to obtain the harmonic expansion of the potential in the reactant region ... 

By virtue of the known relation (eq. 20) between p and p, it establishes a calorimetric path connecting the reactant region with the bottleneck, allowing the first factor in eq. 7 to be calculated. 

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