Quasiclassical average

As a first example, let us consider the time-dependent mean position of a normal mode xj of the system. In a mixed quantum-classical calculation, this observable is directly given by the quasiclassical average over the nuclear trajectories x t), that is. 

In a mixed quantum-classical simulation such as a mean-field-trajectory or a surface-hopping calculation, the population probability of the diabatic state v[/ t) is given as the quasiclassical average over the squared modulus of the diabatic electronic coefficients dk t) defined in Eq. (27). This yields... 

A practical forward-backward semiciassical dynamics (FBSD) methodology is reviewed in this Chapter. This uses a derivative identity to bring a Heisenberg operator into a form suitable for application of the forward-backward semiciassical approximation. The result is an attractive expression in which the semi-classical prefactor is eliminated and which can also be brought into the form of a quasiclassical average. Combined with suitable representations of the initial density, this version of FBSD provides an efficient tool for following time-dependent observables or correlation functions, offering an accurate description of the dynamics in polyatomic systems. 

Around a fixed energy E, the average reaction rate is given by the famous RRKM formula, which can be derived from both quasiclassical and quantal considerations [71, 72]. In the context of the Wigner matrix theory [133], the rate is given by the sum of the half-widths of all the open channels. The rate is thus the product of the number v(E) of open channels and the rate per channel fcchannei(E) = l/hnav(E), where h is the Planck constant. The average reaction rate is obtained as [134, 135]... 

When potential surfaces are available, quasiclassical trajectory calculations (first introduced by Karplus, et al.496) become possible. Such calculations are the theorist s analogue of experiments and have been quite successful in simulating molecular reactive collisions.497 Opacity functions, excitation functions, and thermally averaged rate coefficients may be computed using such treatments. Since initial conditions may be varied in these calculations, state-to-state cross sections can be obtained, and problems such as vibrational specificity in the energy release of an exoergic reaction and vibrational selectivity in the energy requirement of an endo-... 

Note that, as above, this classical-like dynamics is not without relation to the quasiclassical coherent state properties of the density operator involved in this average. 

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). 

The problem of an unphysical flow of ZPE is not a specific feature of the mapping approach, but represents a general flaw of quasiclassical trajectory methods. Numerous approaches have been proposed to fix the ZPE problem.They include a variety of active methods (i.e. the flow of ZPE is controlled and (if necessary) manipulated during the course of individual trajectories) and several passive methods which, for example, discard trajectories not satisfying predefined criteria. However, most of these techniques share the problem that they manipulate individual trajectories, whereas the conservation of ZPE should correspond to a virtue of the ensemble average of trajectories. 

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