Trajectories fractional dynamics

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

The main difference between these stretched-bipyramidalized conical intersections in rings and substituted ethylenes is the process by which they are reached. As already discussed before (Section 8.4), dynamics calculations [38, 66, 90] showed that an important fraction of trajectories of polar substituted ethylenes undergoes stretching and bipyramidalization in the beginning of the time evolution. Nevertheless, in rings the stretched-bipyramidalized configuration cannot be reached by the direct activation of these modes, but it is obtained indirectly as a consequence of the torsional motion around specific bonds. Despite the fact... 

The first step in the dynamics, the S Sj deactivation, is completed in only 52 1 fs, presenting the expected monoexponential decay profile, as can be seen in Figure 8-1 la. This figure shows the fraction of trajectories in each state between 0 and 200 fs for the 35 trajectories computed. Between 20 and 30 fs and again at 37 fs it is possible to observe a revival of the S2 state occupation. The fraction of trajectories in Sj is not shown in the figure for sake of clarity. It is just complementary to the fraction of trajectories in S2. Thus, a revival in S2 is companied by a decrease in Sj occupation. The revivals in the S2 occupation occur when the total number of... 

Brownian dynamics by generating trajectories starting from r = b and terminating them when the molecule collides with the reaction surface or the truncation surface at r = q is the fraction of molecules that collide with the... 

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