Nonreactive trajectories

A typical trajectory has nonzero values of both P and Q. It is part of neither the NHIM itself nor the NHIM s stable or unstable manifolds. As illustrated in Fig. la, these typical trajectories fall into four distinct classes. Some trajectories cross the barrier from the reactant side q < 0 to the product side q > 0 (reactive) or from the product side to the reactant side (backward reactive). Other trajectories approach the barrier from either the reactant or the product side but do not cross it. They return on the side from which they approached (nonreactive trajectories). The boundaries or separatrices between regions of reactive and nonreactive trajectories in phase space are formed by the stable and unstable manifolds of the NHIM. Thus once these manifolds are known, one can predict the fate of a trajectory that approaches the barrier with certainty, without having to follow the trajectory until it leaves the barrier region again. This predictive value of the invariant manifolds constitutes the power of the geometric approach to TST, and when we are discussing driven systems, we mainly strive to construct time-dependent analogues of these manifolds. 

A convenient quantitative characterization of the stable and unstable manifolds themselves as well as of reactive and nonreactive trajectories can be obtained by noting that the special form of the Hamiltonian in Eq. (5) allows one to separate the total energy into a sum of the energy of the reactive mode and the energies of the bath modes. All these partial energies are conserved. The value of the energy... 

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

In addition to describing the TS and the separatrices between reactive and nonreactive trajectories that are the central discovery of geometric TST, one can... 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

The most important of these manifolds for the purposes of TST are, as before, the surface given by AQi = APi that serves as a recrossing-free dividing surface and the stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories. The latter are given by AQi = 0 (stable manifolds) or APj = 0 (unstable manifolds), respectively. Together, they can be characterized as the zeros of the reactive-mode action... 

The diagnostic power of the time-dependent invariant manifolds Wu and Ws as separatrices between reactive and nonreactive trajectories was illustrated in Ref. 40 for the example of a driven Henon-Heiles system described by the Hamiltonian... 

We have outlined how the conceptual tools provided by geometric TST can be generalized to deterministically or stochastically driven systems. The center-piece of the construction is the TS trajectory, which plays the role of the saddle point in the autonomous setting. It carries invariant manifolds and a TST dividing surface, which thus become time-dependent themselves. Nevertheless, their functions remain the same as in autonomous TST there is a TST dividing surface that is locally free of recrossings and thus satisfies the fundamental requirement of TST. In addition, invariant manifolds separate reactive from nonreactive trajectories, and their knowledge enables one to predict the fate of a trajectory a priori. 

By this method, we have been able to study the repeller, in particular for the systems Hgl2 and CO2. Let us add that the stable and unstable manifolds play the very important role of separatrices between reacting and nonreacting trajectories [25],... 

Figure 4. (The color version is available from the authors.) The projection of the reactive and nonreactive trajectories into the q -p plane near q = p = 0.

The two trajectories shown in Fig. 15 are seen to be about identical until 530 fs. Figure 17, upper panel, shows the force acting along the CI2 bond (the force that will lead to dissociation) for the reactive and the nonreactive trajectories in Fig. 15 at 530 fs. For both trajectories the same Ar atoms exercise the force, but there is a large difference in the magnitude of the forces. This large difference is due to the quite small differences in the positions of the atoms, as shown in the bottom panel of Fig. 17. 

Fig. 17. Top panels The force (in reduced units) along the bond of the CI2 molecule, applied by the different rare gas atoms 150 fs after surface Impact at v = 5 km s vs. the (arbitrary) serial number of the laxe gas atoms, for the two trajectories shown in Fig. 15, Bottom panels The position of the CI2 molecule and the laxe gas atoms that applied a significant force along the CI2 bond, see top panels, for the reactive and nonreactive trajectories shown in Fig. 15. Note how small changes in the position of the atoms (Cl-Ar distance and Cl-Cl-Ar angle) cause a big change in the magnitude of the force applied by the rare gas atoms.

Figure 9. Histogram showing the distribution function of the scalar product of normalized initial momentum vectors for 21 reactive and 21 nonreactive trajectories for valine dipeptide conformational change.

Fig. 9, we show the distribution function of the scalar products of different (normalized) momentum vectors. The distribution for valine dipeptide is a combination of reactive and nonreactive trajectories. While the distribution is not exactly a Gaussian, it is not too far from it. 

The expressions for the ratios of the (local) partition functions and the joint probabilities can therefore be computed as averages over ensembles of reactive trajectories only. This is clearly an advantage in comparison to the usual trajectory calculations that include both reactive and nonreactive trajectories. The average above also converges quickly if the two Hamiltonians are not very different from each other. It is a similar gain to the calculations of relative free energies compared to absolute free energies. 

An important feature of classical transition state theory is that it is an upper hound to the correct result for any choice of the dividing surface. That is, since all reactive trajectories must cross the dividing surface, but all trajectories that cross it are not necessarily reactive (because they might recross it at a later time and be nonreactive), any error in the TST approximation, Eq. (12), is to count some nonreactive trajectories as reactive. Thus, while the exact rate expression does not depend on the choice of the dividing surface, the TST rate does, and by virtue of this bounding property the best choice of the dividing surface is the one which makes kTS, a minimum. This is the variational aspect of TST any parameters which specify the shape or location of the dividing surface are best chosen to minimize the TST rate (7). 

Figure 5.2 Projection of the dividing surface and reacting and nonreacting trajectories to the planes of the normal form coordinates. In the plane of the saddle coordinates, the projection of the dividing surface is the dark red diagonal line segment, which has = pi.

In the planes of the center coordinates, the projections of the dividing surface are the dark red discs. Forward and backward reactive trajectories (yellow and blue) project to the first and third quadrants in the plane of the saddle coordinates, respectively, and pass through the dividing surface. The red and green curves mark nonreactive trajectories on the reactant side (p, — q, > 0) and on the product side (pi — q, < 0) of the dividing surface, respectively. The turquoise regions indicate the projections of the energy surface. 

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