Quasiclassical approximation

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3.8.13 (and/or the equivalent time integral of the flux-flux correlation fiinction). While this approach has to date only been applied to problems with simplified hannonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a fiirther classical or quasiclassical approximation. 

Our discussion of solitons and breathers has been purely classical, since we implied the existence of excitations corresponding to the classical solutions (7.71) and (7.75). To see this correspondence it is natural to invoke the quasiclassical approximation. Dashen et al. [1975] quantized the sine-Gordon problem and showed that in the WKB approximation for a discrete chain with periodic boundary condition the soliton mass is renormalized to... 

E.I.Dashevskaya and E.E.Nikitin, Quasiclassical approximation in the theoiy of scattering of polarized atoms, in Atomic Physics Methods in Modem Research, Lecture Notes in Physics, v. 499, eds K. Jungmann et al., Heidelberg, Springer 1997, p.185... 

In adopting the quasiclassical approximation, we use the important property of the VP dynamics that VP transition probability is induced by the interaction close to the inner turning point of the LF oscillator. Then, the expression for the rate constant (e,Ac) can be split in two factors, the classical frequency v(e) and the quasiclassical probability / (c,Ac) for VP transition per one excursion of the LF oscillator to the repulsive part of the potential. 

If the reaction is adiabatic ( y >>1) in such a way that 12 , the transition probability in the low energy range is given in a quasiclassical approximation by the formula (178b.II), which applies equally well to arbitrary continuous potential energy barriers. 

For reactions in a condensed phase (solid or liquid solution) it is often convenient to. .consider the reactants in the initial and final states at a fixed finite separation, as making small vibrations around the positions of minimum potential energy due to the interactions with the medium. Therefore, the quasiclassical approximation for this motion, assumed to correspond to the reaction coordinate, may be used for the same reason as for a unimolecular gas phase reaction which occurs at high presure. 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Sec. 7), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasiclassical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore... 

All approaches for the description of nonadiabatic dynamics discussed so far invoke simple quasiclassical approximations to treat the dynamics of the nuclear degrees of freedom. As a consequence, these methods are in general not able to describe processes or observables for which quantum effects of the nuclear degrees of freedom are important. Such processes include nuclear tunneling, interference effects in wave-packet dynamics as well as the conservation of zero-point energy. In contrast to purely classical approximations, semiclassical methods are in principle capable of describing quantum effects. 

In the present article we will concentrate on a set of gas phase elementary reactions (1-3), which play a central role in combustion [17] and atmospheric chemistry [18], and which have in recent years become important benchmark systems for comparison between the results of experimental reaction dynamics studies and quasiclassical, approximate and - most recently - exact quantum mechanical reactive scattering calculations. 

The first two types of the inverse problem will be considered in Chap.l5 within a quasiclassical approximation. 

As has been shown in Part I, the quasiclassical approximation leads to the separation of dynamical and statistical effects. This fact explains the significant advantage of this approach for the solution of inverse scattering problem. 

Quasiclassical approximation for the amplitude Ta,An results in the integral representation (3.2.5) involving the classical action increment. On the base of this representation and conventional expansion (Gel fand and Shilov 1959)... 

As a simple illustration of this technique consider the case of high frequency instanton trajectory (but ibarrier height V ). Then the quasiclassical approximation can be invoked to solve eq. (4.15a), which yields for A... 

The standard approaches usually use the quasiclassical approximation, that is, the initial conditions are selected in accordance with quantum mechanics and then the evolution of the phase space points is treated purely classically. The quasiclassical approximation thus requires that the trajectories are initiated in the semiclassical eigenstates of molecules. This is easily accomplished for diatomic molecules. For example, approximate analytical solutions for the rotating Morse oscillator have been derived which allow for straightforward selections of initial conditions. If the molecular vibrations are such that the harmonic approximation can be made, then the required analytical relationship is even simpler. ... 

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