Long time classical trajectories

Our chemical experiences suggest that differential equations seem to be something stable, and by that we mean that, if there is a small change in one of the conditions, either initial concentrations or rate constants, we expect small changes in the outcomes as well. The classical example for a stable system is our solar system of planets orbiting the sun. Their trajectories are defined by their masses and initial location and velocity, all of which are the initial parameters of a relatively simple system of differential equations. As we all know, the system is very stable and we can predict the trajectories with an incredible precision, e.g. the eclipses and even the returns of comets. For a long time, humanity believed that the whole universe behaves in a similarly predictable way, of course much more complex but still essentially predictable. Descartes was the first to formally propose such a point of view. 

Equation (369) indicates that to obtain the semiclassical reaction rate constant k T) one needs to carry out the multidimensional phase-space average for a sufficiently long time. This is far from trivial, since the integrand in Eq. (369) is highly oscillatory due to quantum interference effects between the sampling classical trajectories. The use of some filtering methods to dampen the oscillations in the integrand may improve the accuracy of the semiclassical calculation. 

Cylindrical Manifolds. There is one big advantage of looking at 2-DOF TS in phase space It puts emphasis on the existence of the tubes that determine the transport of classical probability in phase space. Existence of those tubes has been known for a long time [48]. These tubes are the set of trajectories that constitute the stable/unstable manifolds of PODS. Locally, in the vicinity of P, they immediatly generalize to higher dimensions. They are constructed as follows ... 

One-electron atoms subjected to a time-dependent external field provide physically realistic examples of scattering systems with chaotic classical dynamics. Recent work on atoms subjected to a sinusoidal external field or to a periodic sequence of instantaneous kicks is reviewed with the aim of exposing similarities and differences to frequently studied abstract model systems. Particular attention is paid to the fractal structure of the set of trapped unstable trajectories and to the long time behavior of survival probabilities which determine the ionization rates of the atoms. Corresponding results for unperturbed two-electron atoms are discussed. 

Exponential divergence in systems that are chaotic prevents accurate long-time trajectory calculations of their dynamics. That is, numerical errors18 propagate exponentially during the dynamics so that accuracy beyond 100 characteristic periods of motion is extraordinarily difficult to achieve thus, accurate long-time dynamics is essentially uncomputable for chaotic classical systems. This serves as additional... 

The coupling functions 1 and still depend on the molecular vibrational and rotational degrees of freedom as well as the relative molecule-perturber separation, R. Since the experiments imply that the physical origin of the collision-induced intersystem crossing resides in long-range attractive interactions, we may adopt a semiclassical approximation where the quantum-mechanical variables for the relative translation is replaced by a classical trajectory, R(l), for the relative molecule-perturber motion. The internal dynamics is then influenced by the time-dependent interactions f s[ (0] and Fj-j-fR(r)], which are still functions of molecular rotational and vibrational variables. For simplicity and for illustrative purposes we consider only the pair of coupled levels S and T and a pure triplet level T, which represents the molecular state after the collision. Note T may differ in rotational and/or vibrational quantum... 

Classical trajectory calculations for the reaction H2 + I2 HI + HI and its reverse have been carried out for two potential energy surfaces. Such calculations are not easy to perform because of the large number of possible states of reactant species, the mismatch of the masses of H and I atoms, and the low probability of reaction. The light H atoms require small time steps to avoid time-step error and the heavy I atoms require a long time for movement. The probability of reaction of two HI molecules, randomly selected from pairs at 700 K with sufficient total energy to react, is about 1 in 10 [15]. In a thermal system collisions of H2 with I2 (or 21) which result in reaction to form HI + HI are indeed rare events. 

The Bohmian quantum-classical and QMF approaches have been tested with a model application designed to simulate the interaction of an oxygen molecule with a platinum surface [13,21,22,91]. With trajectory branching the Bohmian quantum-classical method recovers the correct asymptotic behavior of the scattering probability of the quantum subsystem. The QMF approach shows improvement in both the short and long time scattering probabilities. The improvement is achieved due to the proper treatment of ZPE. 

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