Quasi-classical approximation

After having discussed the approximate quasi-classical dynamics, we return (see Section 1.1) now to exact quantum dynamics.9 The Schrodinger equation for motion of the atomic nuclei is given by Eq. (1.10) ... 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

The breakdown of the SH scheme in the case of classically forbidden electronic transitions should not come as a surprise, but is a consequence of the rather simplifying assumptions [i.e., Eqs. (37) and (43)] underlying the SH model. On a semiclassical level, classically forbidden transitions may approximately be described within an initial-value representation (see Section VIII) or by introducing complex-valued trajectories [55]. On the quasi-classical... 

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. 

To perform a PO analysis of nonadiabatic quantum dynamics, we employ a quasi-classical approximation that expresses time-dependent quantities of a vibronically coupled system in terms of the vibronic POs of the system [123]. Considering the quasi-classical expression (16) for the time-dependent expectation value of an observable A, this approximation assumes that the integrable islands in phase space represent the most significant contributions to the dynamics of the observables considered [236]. As a consequence, the short-time dynamics of the system is determined by its shortest POs and can be approximated by a time average over these orbits. Denoting the A th PO with period 7 by qk t),Pk t) we obtain [123]... 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

All approaches for the description of nonadiabatic dynamics discussed so far have used the simple quasi-classical approximation (16) to describe the dynamics of the nuclear degrees of freedom. As a consequence, these methods are in general not able to account for 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, and the conservation of zero-point energy. In contrast to quasi-classical approximations, semiclassical methods take into account the phase exp iSi/h) of a classical trajectory and are therefore capable—at least in principle—of describing quantum effects. 

In contrast to the quasi-classical approaches discussed in the previous chapters of this review, Eq. (114) represents a description of nonadiabatic dynamics which is semiclassically exact in the sense that it requires only the basic semiclassical Van Vleck-Gutzwiller approximation [3] to the quantum propagator. Therefore, it allows the description of electronic and nuclear quantum effects. 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

C. The Quasi-Classical Approximation The Impact Parameter Method... 

Thus, in order to determine the scattering cross sections we must find the wavefunctions of the system after the scattering for a known interaction potential. This is a very complicated problem in the case of many-electron systems and can be solved only with various approximate methods. We will only briefly discuss the results obtained in the Born approximation and in the quasi-classical impact parameter method. A detailed discussion of various approximate methods can be found in special monographs (e.g. in Refs. 104 and 107) or in reviews (see Refs. 105, 108-112). 

As a simple illustration of this technique, consider the case of high-frequency co, viz., co 2da),/dT < 1 for the instanton trajectory (but ,/2 is still small compared to the total barrier height V ). Then the quasi-classical approximation can be invoked to solve Eq. (4.15), which yields for A... 

For practical reasons, a quasi-classical approximation to the quantum dynamics described by Eq. (1.10) is often sought. In the quasi-classical trajectory approach (discussed in Section 4.1) only one aspect of the quantum nature of the process is incorporated in the calculation the initial conditions for the trajectories are sampled in accord with the quantized vibrational and rotational energy levels of the reactants. 

The harmonic approximation is unrealistic in a dynamical description of the dissociation dynamics, because anharmonic potential energy terms will play an important role in the large amplitude motion associated with dissociation. An accurate potential energy surface must be used in order to obtain a realistic dynamical description of the dissociation process and, as in the quasi-classical approach for bimolecular collisions, a numerical solution of the classical equations of motion is required [2]. 

Fig. 8.1.1 An illustration of the relations between the rate constants, k(T) and k(E), and the reaction probability P as obtained from either quantum mechanics, quasi-classical mechanics, or various assumptions (approximations) for the reaction dynamics.

It should be noted that the expression (11) for Coulomb s potential is only valid in the asymptotic region, far from the donor and acceptor. At the small distance from the charge, the dielectric permittivity does not weaken the electric field. Therefore the approximate expression for Green s function in this case can be obtained by combining its exact expression in Coulomb s field [6] and the quasi-classic approach [2,4] ... 

A further analytical approximation to Eq. (369), proposed by Miller and coworkers [84-86], demonstrates how the above semiclassical reaction rate theory approaches a quasi-classical reaction rate theory. Specifically, consider the... 

The large well depth in the ground electronic state ( 7.2 e ) and the high exoergicity (1.9 e ) makes particularly difficult an exact QM study of the dynamics of this reaction. Theoretical studies often used the quasi-classical trajectory (QCT) method [12. 16. 18, 21]. Only a few quantum-mechanical (QM) studies have been reported. They are exact for the total angular momentum J = 0 but approximate for higher J. Total reaction probability has been calculated with a... 

The overlap integral J securing the carrier tunneling through the barrier between the two neighboring polaron wells in the quasi-classical approximation has the following appearance ... 

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