Recrossing dynamics

E. Poliak and P. Talkner, Transition-state recrossing dynamics in activated rate processes, Phys. Rev. E 51, 1868 (1995). 

Figure 3.2 Cumulative reaction probability for the coUinear H+H2 reaction. The circles are the J —matrix propagation results (summed over final states) by Bondi et aL, and the line is the power series Green s function results. Excellent agreement is obtained over the entire energy range. The staircase structure is evident, with N E) = 1 when the is one open channel, and = 1.6 with two open channels. The non-monotonic increase indicates recrossing dynamics, and the peak at = 0.87 eV indicates a collision complex for H3.

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. 

Conical intersections can be broadly classified in two topological types peaked and sloped [189]. These are sketched in Figure 6. The peaked case is the classical theoretical model from Jahn-Teller and other systems where the minima in the lower surface are either side of the intersection point. As indicated, the dynamics of a system through such an intersection would be expected to move fast from the upper to lower adiabatic surfaces, and not return. In contrast, the sloped form occurs when both states have minima that lie on the same side of the intersection. Here, after crossing from the upper to lower surfaces, recrossing is very likely before relaxation to the ground-state minimum can occur. 

Because in an autonomous system many of the invariant manifolds that are found in the linear approximation do not remain intact in the presence of nonlinearities, one should expect the same in the time-dependent case. In particular, the separation of the bath modes will not persist but will give way to irregular dynamics within the center manifold. At the same time, one can hope to separate the reactive mode from the bath modes and in this way to find the recrossing-free dividing surfaces and the separatrices that are of importance to TST. As was shown in Ref. 40, this separation can indeed be achieved through a generalization of the normal form procedure that was used earlier to treat autonomous systems [34]. 

E. Dynamical Model for Sn2 Substitution and Central Barrier Recrossing. 152... 

The dynamical model described in Figure 9 indicates that the trajectories may recross the central barrier several times if the Cintra R Cintra p transition is faster... 

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. 

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