Branching of trajectories

The remaining sections are devoted to a discussion of the various MQC methods. Among other issues, we consider the abihty of a method to (i) account for the branching of trajectories, (ii) accoimt for the electronic phase coherence, (iii) correctly describe the vibrational motion on coupled... 

Furthermore, a related and common criticism of the MFT method is that a mean-field approach cannot correctly describe the branching of wave packets at crossings of electronic states [67, 70, 82]. This is true for a single mean-field trajectory, but is not true for an ensemble of trajectories. In this context it may be stressed that an individual trajectory of an ensemble does not even possess a physical meaning—only the average does. 

Consider the example of condensed phase transitions between vibrational states, which have energies that are significantly drfferent compared with knT. The momentum on the initial surface before a hop and the final surface momentum after the hop are considerably drfferent for typical values of the initial momentum sampled Irom a canonical distribution. This causes the two branches of the combined trajectory to quickly diverge, and action for the combined trajectory to grow rapidly. The result is that the integrand converges very quickly as a function of x, particularly after the and Fj integrations have been performed. 

Fig. 5.10. The possibility of a stable stationary-state intersection on the middle branch of the g(a, 9) = 0 nullcline the nullclines are shown as broken curves, the solid curve gives the evolution of a typical trajectory towards the stable focal state.

For a same molecular ratio of aqueous NaY solutions (Y = OH, Cl), experimental data underlines specific effects of nascent OH radicals on transient UV and near-IR electronic configurations. Complex investigations of PHET reactions in the polarization CTTS well of aqueous CT and OH ions are in progress. We should wonder whether a change in the size of ionic radius (OH -1.76 A vs Cl" 2.35 A) or in the separation of the energy levels influence early branchings of ultrafast electronic trajectories. A key point of these studies is that the spectroscopic predictions of computed model-dependent analysis are compared to a direct identification of transient spectral bands, using a cooled Optical Multichannel Analyzer... 

The cases of hyperbolic-without-reflection and hyperbolic-with-reflection stability have to be distinguished. In both cases, the trajectories in the neighborhood of the periodic orbit trace out hyperbolic paths in the Poincare section, but if the stability is hyperbolic with reflection, the trajectories cross over between the branches of the hyperbola on each iteration. 

This is called Hopf bifurcation. Figure 10 (A-2) shows two Hopf bifurcation points with a branch of stable limit cycles connecting them. Figure 13 (A-2) shows a schematic diagram of the phase plane for this case when g = g. In this case a stable limit cycle surrounds an unstable focus and the behavior of the typical trajectories are as shown. Figure 11 (A-2) shows two Hopf bifurcation points in addition to a periodic limit point (PLP) and a branch of unstable limit cycles in addition to the stable limit cycles branch. 

Thus, we generally expect only very small values of x to contribute significantly to the transition probability. There are some exceptions to this. Population relaxation between degenerate or nearly degenerate vibrational states is an example of this, since the pre-hop and post-hop momenta are nearly the same and the two branches of the combined trajectory can separate quite slowly in this case.97 98... 

When such structural or static information is not sufficient (i.e., the excited state may not decay at the minimum of the conical intersection line, or the momentum developed on the excited state branch of the reaction coordinate may be sufficient to drive the ground state reactive trajectory along paths that are far from the ground state valleys), a dynamics treatment of the excited state/ ground state motion is required.53 54 These techniques also are illustrated in the next subsection. 

The presence of the momentum derivatives in J makes the action of this operator difficult to simulate, because it acts on all functions to its right. This will generate a branching tree of trajectories. This difficulty is avoided by making the momentum-jump approximation. To see how this approximation is obtained, the following change of variables is made ... 

Figure 12.9 depicts a comparison between classical trajectory results and exact close-coupling calculations for He--Cl2 and Ne- -Cl2, respectively. In both cases, the classical procedure reproduces the overall behavior of the final state distributions satisfactorily. Subtle details such as the weak undulations particularly for He are not reproduced, however. As shown by Gray and Wozny (1991), who treated the dissociation of van der Waals molecules in the time-dependent framework, the bimodality for He CI2 is the result of a quantum mechanical interference between two branches of the evolving wavepacket and therefore cannot be obtained in purely classical calculations. 

Consequently, the electronic transition probabilities can be interpreted as relative weights of the individual branches of a system point trajectory. The final weights of trajectories terminating in the product asymptote can then serve for calculating the reaction attributes in the same manner as the numbers of trajectories in case of adiabatic processes (7,49). 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

The other branch of the stable mani ld consists of a trajectory coming in from infinity. A computer-generated phase portrait (Figure 6.4,7) confirms our sketch. 

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