Semiclassical model dynamics

Two parameters that are accessible to the experimentalist have been the effect of the interchange of a deuteron for a proton upon the dynamics of transfer and the effect of temperature variation upon the kinetics of proton-deuteron transfer [49]. As previously mentioned, the semiclassical model has been employed in the rationalization of kinetic deuteron isotope effects that exceed the factor of 7.0, the maximum predicted by the classical model [5]. However, the full quantum model also allows for the wide range in the kinetic deuteron isotope effect, the range of which overlaps that predicted by the semiclassical model [53]. Thus, the kinetic deuteron isotope effect in and of itself cannot be used to distinguish between the two models. 

Many kinds of molecular systems pumped by a strong laser light show chaotic dynamics. Indeed, in a semiclassical model of a multiphoton excitation on molecular vibration, chaos was discovered by Ackerhalt et al. [85] and theoretically and numerically investigated in detail [86,87]. Moreover, the equations of motion that describe a rotating molecule in a laser field can exhibit a chaotic behavior and have been applied in the classical case of a rigid-rotator approximation [87,88]. 

On the simplest level, one can consider a semiclassical model of limited motions of various parts of the molecule relative to one another. Within such approximation, the dynamic shape variations due to internal motions, for example, those due to vibrations, can be modeled by an infinite family of geometrical arrangements. Within this approach, we consider a family of shapes occurring for these arrangements and study the common, invariant topological features. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Although in the semiclassical model the only dynamical variables are those of the molecule, and the extended Hilbert space. if = M M and the Floquet Hamiltonian K can be thought as only mathematically convenient techniques to analyze the dynamics, it was clear from the first work of Shirley [1] that the enlarged Hilbert space should be related to photons. This relation was made explicit by Bialynicki-Birula and co-workers [7,8] and completed in [9]. The construction starts with a quantized photon field in a cavity of finite volume in interaction with the molecule. The limit of infinite volume with constant photon density leads to the Floquet Hamiltonian, which describes the interaction of the molecule with a quantized laser field propagating in free space. The construction presented below is taken from Ref. 9, where further details and mathematical precisions can be found. 

In conclusion to this section, band-shape analysis of vibrational spectra and ground state splitting observed with INS demonstrate that proton transfer dynamics are quantal in nature, even at room temperature. Semiclassical models are not relevant. The dramatic failure of quantum chemistry to account for the observed dynamics should be regarded as one of the major unsolved theoretical problems at the present time. 

This paper reviews this classical S-matrix theory, i.e. the semiclassical theory of inelastic and reactive scattering which combines exact classical mechanics (i.e. numerically computed trajectories) with the quantum principle of superposition. It is always possible, and in some applications may even be desirable, to apply the basic semiclassical model with approximate dynamics Cross7 has discussed the simplifications that result in classical S-matrix theory if one treats the dynamics within the sudden approximation, for example, and shown how this relates to some of his earlier work8 on inelastic scattering. For the most part, however, this review will emphasize the use of exact classical dynamics and avoid discussion of various dynamical models and approximations, the reason being to focus on the nature and validity of the basic semiclassical idea itself, i.e., classical dynamics plus quantum superposition. Actually, all quantum effects—being a direct result of the superposition of probability amplitudes—are contained (at least qualitatively) within the semiclassical model, and the primary question to be answered regards the quantitative accuracy of the description. 

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

Considering the semiclassical description of nonadiabatic dynamics, only the mapping approach [99, 100] and the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller [112] appear to be amenable to a numerical treatment via an initial-value representation [114, 116, 117, 121, 122]. Other semiclassical formulations such as Pechukas path-integral formulation [45] and the various connection... 

The classical-path approximation introduced above is common to most MQC formulations and describes the reaction of the quantum DoF to the dynamics of the classical DoF. The back-reaction of the quantum DoF onto the dynamics of the classical DoF, on the other hand, may be described in different ways. In the mean-field trajectory (MFT) method (which is sometimes also called Ehrenfest model, self-consistent classical-path method, or semiclassical time-dependent self-consistent-field method) considered in this section, the classical force F = pj acting on the nuclear DoF xj is given as an average over the quantum DoF... 

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