Phase space barriers

Figure 3. (The color version is available from the authors.) Two electron trajectories, so close in phase space that they appear as one, approach the ionization saddle [6] from the top right. After some complex dynamics at the saddle point, one reacts (by crossing the TS at the dot in the upper left-hand corner of the bundle) and goes off to the top left, and the nonreactive one returns to the bottom right. We engineered this outcome by selecting their initial conditions on opposite sides of the impenetrable phase-space barrier (one of the scissors [20]) between ionizing and nonionizing hypervolumes.

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

Rosenstock (55) pointed out that the initial formulation of the theory failed to consider the effect of angular momentum on the decomposition of the complex. The products of reaction must surmount a potential barrier in order to separate, which is exactly analogous to the potential barrier to complex formation. Such considerations are implicit in the phase space theory of Light and co-workers (34, 36, 37). These restrictions limit the population of a given output channel of the reaction com-... 

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. 

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. 

S. Wiggins, L. Wiesenfeld, C. Jaffe, and T. Uzer, Impenetrable barriers in phase-space, Phys. Rev. Lett. 86, 5478 (2001). 

At a higher temperature T = 11.0, for flow rates near the transition rate c, the free-energy barrier between the coiled and stretched conformation is much lower than that for T = 9.0. The chain can therefore explore the phase space and jump back and forth from the coiled to the stretched state. Similar behavior has already been observed in [59] and [60]. Figure 27 illustrates this feature. 

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