Classical dynamics, validity

The highly excited and reactive dynamics, the details of which have been made accessible by recently developed experimental techniques, are characterized by transitions between classically regular and chaotic regimes. Now molecular spectroscopy has traditionally relied on perturbation expansions to characterize molecular energy spectra, but such expansions may not be valid if the corresponding classical dynamics turns out to be chaotic. This leads us to a reconsideration of such perturbation techniques and provides the starting point for our discussion. From there, we will proceed to discuss the Gutzwiller trace formula, which provides a semiclassical description of classically chaotic systems. 

On the methodological side, it is gratifying to note, once these computer time consuming CP-MD calculations have been performed, that rapid classical dynamics AMBER calculations on the 18C6-H20 hydrate essentially yield similar qualitative conclusions instability of the Q hydrate, and "waltzing dynamics" of H20 over the D3d crown, which also validates early and subsequent force field studies, also pointing out the limitations of static views, even obtained by sophisticated quantum calculations. 

Thus, we see that in order to obtain the mean field equations of motion, the density matrix of the entire system is assumed to factor into a product of subsystem and environmental contributions with neglect of correlations. The quantum dynamics then evolves as a pure state wave function depending on the coordinates evolving in the mean field generated by the quantum density. As we have seen in the previous sections, these approximations are not valid and no simple representation of the quantum-classical dynamics is possible in terms of single effective trajectories. Consequently, in contrast to claims made in the literature [54], quantum-classical Liouville dynamics is not equivalent to mean field dynamics. 

There is another more direct way of calculating the rate constant k(T), i.e., it is possible to bypass the calculation of the complete state-to-state reaction probabilities, S m(E) 2, or cross-sections prior to the evaluation of the rate constant. The formulation is based on the concept of reactive flux. We start with a version of this formulation based on classical dynamics and, in a subsequent section, we continue with the quantum mechanical version. It will become apparent in the next section that the classical version is valid not only in the gas phase, but in fact in any phase, that is, the foundation for condensed-phase applications will also be provided. 

The observations for 9-11 described in the foregoing confirm that the validity of the DQR approach is not restricted to the cryogenic range. For the first time, wave-like properties of such a massive object as the methyl group were detected in its environment-induced dynamics at ambient temperatures. For the objects subject to the Pauli principle, the common practice to use the term stochastic dynamics as a synonym for classical dynamics need not be valid even when the relevant behaviour under ambient conditions is referred to. 

This paper reviews this classical S-matrix theory, i.e. the semiclassical theory of inelastic and reactive scattering which combines exact classical mechanics (i.e. numerically computed trajectories) with the quantum principle of superposition. It is always possible, and in some applications may even be desirable, to apply the basic semiclassical model with approximate dynamics Cross7 has discussed the simplifications that result in classical S-matrix theory if one treats the dynamics within the sudden approximation, for example, and shown how this relates to some of his earlier work8 on inelastic scattering. For the most part, however, this review will emphasize the use of exact classical dynamics and avoid discussion of various dynamical models and approximations, the reason being to focus on the nature and validity of the basic semiclassical idea itself, i.e., classical dynamics plus quantum superposition. Actually, all quantum effects—being a direct result of the superposition of probability amplitudes—are contained (at least qualitatively) within the semiclassical model, and the primary question to be answered regards the quantitative accuracy of the description. 

Consider the specific case of NaBrKCl for which theoretical, classical dynamics studies are available at energies where two dissociative channels are energetically accessible. (Questions as to the validity of the classical picture are relegated to later sections.) This system possesses attractive forces between the atoms such that the bound NaBrKCl species lies at an energy of approximately 40 kcal/mol below NaBr - - KCl or NaCl -I- KBr. Specifically, consider the case where energized NaBrKCl is formed by the collision... 

Bartell and co-workers have made significant progress by combining electron diffraction studies from beams of molecular clusters with molecular dynamics simulations [14, 51, 52]. Due to their small volumes, deep supercoolings can be attained in cluster beams however, the temperature is not easily controlled. The rapid nucleation that ensues can produce new phases not observed in the bulk [14]. Despite the concern about the appropriateness of the classic model for small clusters, its application appears to be valid in several cases [51]. 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

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