Trajectory methods

Hase W L (ed) 1998 Comparisons of Classical and Quantum Dynamics (Adv. in Classical Trajectory Methods III) (Greenwich, CT JAI Press)... 

W L Hase (ed) 1992 Advances in Classical Trajectory Methods. 1. Intramolecular and Nonlinear Dynamics (London JAI)... 

Approaches used to model ozone formation include box, gradient transfer, and trajectoty methods. Another method, the particle-in-cell method, advects centers of mass (that have a specific mass assigned) with an effective velocity that includes both transport and dispersion over each time step. Chemistry is calculated using the total mass within each grid cell at the end of each time step. This method has the advantage of avoiding both the numerical diffusion of some gradient transfer methods and the distortion due to wind shear of some trajectory methods. 

At first sight, the easiest approach is to fit a set of points near the saddle point to some analytical expression. Derivatives of the fitted function can then be used to locate the saddle point. This method has been well used for small molecules (see Sana, 1981). An accurate fit to a large portion of the potential energy surface is also needed for the study of reaction dynamics by classical or semi-classical trajectory methods. 

A time-independent quantum mechanical study by Engel and coworkers (1985) gave qualitatively the same results as the time dependent analog and the quassi classical trajectory method. A quantum theory for probing transition-state absorption/emission has also been developed by Lee et al. (1989). 

One formalism which has been extensively used with classical trajectory methods to study gas-phase reactions has been the London-Eyring-Polanyi-Sato (LEPS) method . This is a semiempirical technique for generating potential energy surfaces which incorporates two-body interactions into a valence bond scheme. The combination of interactions for diatomic molecules in this formalism results in a many-body potential which displays correct asymptotic behavior, and which contains barriers for reaction. For the case of a diatomic molecule reacting with a surface, the surface is treated as one body of a three-body reaction, and so the two-body terms are composed of two atom-surface interactions and a gas-phase atom-atom potential. The LEPS formalism then introduces adjustable potential energy barriers into molecule-surface reactions. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

Figure 3. Time-dependent simulations of the nonadiabatic photoisomerization dynamics exhibited by Model III, comparing results of the mean-field-trajectory method (dashed lines), the surface-hopping approach (thin lines), and exact quanmm calculations (full lines). Shown are the population probabilities of the initially prepared (a) adiabatic and (b) diabatic electronic state, respectively, as well as (c) the probability Pdsit) that the sytem remains in the initially prepared cis conformation.

III. MEAN-FIELD TRAJECTORY METHOD A. Classical-Path Approximation... 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

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. 

Let us first consider the relation to the mean-field trajectory method discussed in Section III. To make contact to the classical limit of the mapping formalism, we express the complex electronic variables imaginary parts, that is, mean-field Hamiltonian function which may be defined as... 

The problem of an unphysical flow of ZPE is not a specific feature of the mapping approach, but represents a general flaw of quasi-classical trajectory methods. Numerous approaches have been proposed to fix the ZPE problem [223]. 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 that, 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. 

Figure 19. Time-dependent (a) diabatic and (b) adiabatic electronic excited-state populations and (c) vibrational mean positions as obtained for Model 1. Shown are results of the mean-field trajectory method (dotted lines), the quasi-classical mapping approach (thin full lines), and exact quantum calculations (thick full lines).

A value of y = 1 corresponds to the original mapping formulation, which takes into account the full amount of ZPE. If, on the other hand, all electronic ZPE is neglected (i.e., y = 0), the mapping approach becomes equivalent to the mean-field trajectory method. 

W. L. Hase, Advances in Classical Trajectory Methods, Vol. 1, Jai Press, London, 1992. 

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