Recrossing trajectories

For many chemical reactions with high sharp barriers, the required time dependent friction on the reactive coordinate can be usefully approximated as the tcf of the force with the reacting solute fixed at the transition state. That is to say, no motion of the reactive solute is permitted in the evaluation of (2.3). This restriction has its rationale in the physical idea [1,2] that recrossing trajectories which influence the rate and the transmission coefficient occur on a quite short time scale. The results of many MD simulations for a very wide variety of different reaction types [3-12] show that this condition is satisfied it can be valid even where it is most suspect, i.e., for low barrier reactions of the ion pair interconversion class [6],... 

Figure 2. The projections of a reactive, recrossing trajectory onto the configurational space and the phase space (see text for details).

Figure 3. The projection of a nonreactive recrossing trajectory onto the configurational...

Figure 5. The distributions of the recrossing trajectories over configurational surface S qi = 0) at time t = 0 on the phase-space planes (pf (p,q), (p,q)) at E = 0.5e, where most modes are strongly chaotic—except 4i(p,q). (a) First and (b) second orders The circle and triangle symbols denote the system trajectories having negative and positive incident momenta p (t = 0) on the S(qi = 0), and the open and filled symbols denote those whose final states were predicted correctly and falsely by Eq. (11), respectively [45].

Figure 14. A schematic picture of how a reactive recrossing trajectory passing througji a naive TS, = 0), on a double-well Hamiltonian, Eq. (21), is rotated away to a single crossing through a...

Being a repulsive PODS—a fuUy nonlinearly determined NHIM— the ubiquitous problem of recrossings (trajectory t3 in Fig. 9 see Ref. 6) is totally avoided. 

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... 

Figure 2.2. A representative saddle-recrossing trajectory at = 0.05, and 0.5e over the dividing surface Siq = 0), projected onto the q, plane and the PES contour plot in this plane. The window in this figure is scaled to —0.01 < 0.01 and —0.3 q 2 0.2. The...

An even more striking consequence of the LCPT transformation appears in the behavior of the reactive degrees of freedom. Figure 2.7 shows the projections of the recrossing trajectories onto the (5i (p, q), 5f(p,q)). The abscissas in the figure correspond to a reaction coordinate, that is, coj e g, and the ordinates, to the nonreactive coordinates, that is, in each... 

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.

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