Quantum/semiclassical approaches

It may be shown [8] that both semiclassical [83,84], and full quantum mechanical approaches [7,32,33,58,87] of anharmonic coupling have in common the assumption that the angular frequency of the fast mode depends linearly on the slow mode coordinate and thus may be written... 

On the other hand, in the semiclassical approaches [83-85,92-95], one uses Eqs. (126) and (127) and assumes that the fast mode obeys quantum mechanics whereas the slow one obeys classical mechanics. 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantum-semiclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of semiclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

Spin-spin coupling is observed in the NMR spectra of both solids and liquids, but its origin is quite different for the two phases. Solids will be considered first, using a qualitative, semiclassical approach, rather than the correct quantum-mechanical treatment. 

On the contrary, the semiclassical approach in the problem of the optical absorption is restricted to a great extent and the adequate description of the phonon-assisted optical bands with a complicated structure caused by the dynamic JTE cannot be done in the framework of this approach [13]. An expressive example is represented by the two-humped absorption band of A —> E <8> e transition. The dip of absorption curve for A —> E <8> e transition to zero has no physical meaning because of the invalidity of the semiclassical approximation for this spectral range due to essentially quantum nature of the density of the vibronic states in the conical intersection of the adiabatic surface. This result is peculiar for the resonance (optical) phenomena in JT systems full discussion of the condition of the applicability of the adiabatic approximation is given in Ref. [13]. 

The classical theory is a valuable complement of the quantum mechanical approaches. It is best suited for fast and direct photodissociation. Quantum mechanical effects, however, such as resonances or interferences inherently cannot be described by classical mechanics. The obvious extension is a semiclassical theory (Miller 1974, 1975) which incorporates the quantum mechanical superposition principle without the complexity of full quantum mechanical calculations. All ingredients are derived solely from classical trajectories. For an application in photodissociation see Gray and Child (1984). 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

Quantum and Semiclassical Approaches A. The Wigner Function and Weyl s Rule Quantum Scars in Phase-Space Quantizing the ARRKM Theory... 

In this chapter we have reviewed the development of unimolecular reaction rate theory for systems that exhibit deterministic chaos. Our attention is focused on a number of classical statistical theories developed in our group. These theories, applicable to two- or three-dimensional systems, have predicted reaction rate constants that are in good agreement with experimental data. We have also introduced some quantum and semiclassical approaches to unimolecular reaction rate theory and presented some interesting results on the quantum-classical difference in energy transport in classically chaotic systems. There exist numerous other studies that are not considered in this chapter but are of general interest to unimolecular reaction rate theory. 

Comparison of the results of quantum and wave-packet 539 calculations for the photodissociation of CH3I with results obtained with a simplified semiclassical approach Maximum entropy formalism applied to MPD. Applica- 540 tion to alkyl iodides... 

The theoretical description of the kinetics of electron transfer reactions starts fi om the pioneering work of Marcus [1] in his work the convenient expression for the free energy of activation was defined. However, the pre-exponential factor in the expression for the reaction rate constant was left undetermined in the framework of that classical (activate-complex formalism) and macroscopic theory. The more sophisticated, semiclassical or quantum-mechanical, approaches [37-41] avoid this inadequacy. Typically, they are based on the Franck-Condon principle, i.e., assuming the separation of the electronic and nuclear motions. The Franck-Condon principle... 

The first theoretical model of optical activity was proposed by Drude in 1896. It postulates that charged particles (i.e., electrons), if present in a dissymmetric environment, are constrained to move in a helical path. Optical activity was a physical consequence of the interaction between electromagnetic radiation and the helical electronic field. Early theoretical attempts to combine molecular geometric models, such as the tetrahedral carbon atom, with the physical model of Drude were based on the use of coupled oscillators and molecular polarizabilities to explain optical activity. All subsequent quantum mechanical approaches were, and still are, based on perturbation theory. Most theoretical treatments are really semiclassical because quantum theories require so many simplifications and assumptions that their practical applications are limited to the point that there is still no comprehensive theory that allows for the predetermination of the sign and magnitude of molecular optical activity. 

QUANTUM DYNAMICS OF LARGE CLUSTERS REDUCED DIMENSIONALITY AND SEMICLASSICAL APPROACH... 

The depth of understanding of V-V, V-T, and V-R, relaxation in atom-diatom collisions at low densities is profound. " The advent of state-to- state experiments and of quantum, semiclassical, and classical calculations has provided a wealth of information. Stochastic approaches, which are still under development for polyatomic sys-tems should mimic the essential features of thermally averaged atom-diatom energy transfer when applied to these simple systems. The friction is essentially the characteristic of kinetic energy relaxation. The energy diffusion equation of the energy probability density tr(E, t) is... 

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