Quasiclassical method

Interestingly, for the lower temperature case of 3 =8, the CMD method is in much better agreement with the exact result. In contrast, the classical result does not show any low temperature coherent behavior. The more accurate low temperature CMD result also suggests that CMD should not be labeled a quasiclassical method because the results actually improve in the more quantum limit for this system. The improvement of these results over the higher temperature case can be understood through an examination of the effective centroid potential. The degree of nonlinearity in the centroid potential is less at low temperature, so the correlation function dephases less. 

Thus, as long as the interference phenomena are suppressed, the quasiclassical method yields a reasonable qualitative view of the dynamics of elementary processes. We will adopt this view in the forthcoming discussions in the next section it will be shown that the quasiclassical trajectory method can be generalized to treat a large class of electronically nonadiabatic processes. 

The harmonic oscillator model (V = kx2H) for a diatom is an instructive example. Following the spirit of the quasiclassical method, once the vibrational quantum number is determined the oscillator is assigned the energy corresponding to the quantum mechanical energy eigenvalue (eq. (29)). Newton s equation for the harmonic oscillator is... 

It should be noted that the adiabatic approximation may be violated during the passage of the region x x. if the nuclear velocity dx/dt is high. However, this approximation is certainly valid outside it, i.e., in the initial and final states of the system. Under these conditions can be evaluated using the quasiclassical method of... 

After that each of the integrals is evaluated within the quasiclassical method separately before and after fixing point t = 0. The influence functional F[F, F ] at each trajectory branch is considered as a pre-exponential factor and is replaced with its value at the classical trajectory, obtained by the variation of S — S. In this case the classical trajectories r t),r (t) aie real and coincide, but they have jumps in z and P at < = 0 (while the trajectories of the generalized eikonal method have jumps only in p). 

The basic technology of classical trajectory applications to atom-diatom and similar systems has been described and reviewed numerous times. As long as classical microcanonical or canonical ensembles are used to generate and analyze initial and final conditions, it is not difficult to apply classical trajectory methods to polyatomic collision systems as well. The implementation of quasiclassical methods to determine state-resolved cross sections and rate constants in polyatomic systems is, however, much more complicated, and attention has been given to this only recently. The... 

Quasiclassical trajectory calculations are the method of choice for determining the dynamics of intramolecular vibrational energy redistribution leading to a chemical reaction. If this information is desired, an accurate reaction rate can be obtained at little extra expense. 

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 Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical 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 (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

The zeta function methods have proved to be extremely powerful to obtain the resonances of classical scattering systems, which give the quasiclassical reaction rates [61]. In transport processes, the classical resonances give the dispersion relations that characterize the relaxation of hydrodynamic modes [64], These results bring about a new understanding of the problem of irreversibility at the classical level, as discussed elsewhere [64],... 

The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65, 66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 68]. 

The ion-neutral reaction that has received the greatest attention from a theoretical viewpoint is the H2+ -He process. This is because of the relative simplicity of this reaction (a three-electron system), which facilitates accurate theoretical calculations and also to the fact that a wealth of accurate experimental data has been obtained for this interaction. Several different theoretical approaches have been applied to the H2+He reaction, as indicated by the summary presented in Table VI. Most of these have treated the particle-transfer channel only, and few have considered the CID channel. Various theoretical methods applicable to ion-neutral interactions are discussed in the following sections. For the HeH2+ system, calculations using quasiclassical trajectory methods, employing an ab initio potential surface, have been shown to yield results that are in good agreement with the experimental results. 

Direct dynamics calculations were carried out with quasiclassical normalmode sampling from a canonical ensemble at 923 K (the experimental reaction temperature). Simulations initiated at the vicinity of TS for rearrangement of carbene 13 to 14 via oxirene 12, and 300 trajectories were obtained at DFT methods. The preliminary results reported in the manuscript showed that preferred formation of 15a over 15b by the ratio of 1.8 7.6 depends on the method used. The results were qualitatively consistent with the value of 2.5 deduced from the experiment. The non-unity ratio likely arises from the situation that two methyl groups in 14 are dynamically unequal on the carbene formation process. 

Ion-molecule association is seemingly well suited for the application of the quasiclassical trajectory (QCT) method (Porter and Raff 1976 Raff and Thompson 1985 Truhlar and Muckerman 1979). Since there is no potential barrier and the centrifugal potential is broad, quantum mechanical tunneling is typically unimportant. Energy transfer from relative translational to vibrational and/or rotational motions of the complex should be reasonably classical because of the... 

There has recently been an upsurge of interest in the classical approach as a complement to full quantum dynamics. In many cases, it is found that the agreement between classical and quantum results for the dissociation probability is acceptable if not perfect [55-60], This is illustrated in Fig. 7, which shows the quantum dissociation probability computed for the H2/Cu(l 0 0) system compared to classical and quasiclassical results [57]. In quasiclassical calculations, we use classical methods to... 

The quasiclassical trajectory method disregards completely the quantum phenomenon of superposition (13,18,19) consequently, the method fails in treating the reaction features connected with the interference effects such as rainbow or Stueckelberg-type oscillations in the state-to-state differential cross sections (13,17,28). When, however, more averaged characteristics are dealt with (then the interference is quenched), the quasiclassical trajectory method turns out to be a relatively universal and powerful theoretical tool. Total cross-sections (detailed rate constants) of a large variety of microscopic systems can be obtained in a semiquantitative agreement with experiment (6). 

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