Quasiclassical trajectory approach

In addition to the equations of motion one needs to specily a procedure to evaluate the observables of interest. Within a quasiclassical trajectory approach, the expectation value of an observable A is given by Eq. (13). For example, the expression for the electronic population probability, which is defined as the expectation value of the electronic occupation operator, reads... 

This account highlights some of the recent developments in methods and current applications of classical trajectory simulations of molecular collisions. It has been more than 60 years since the first classical trajectory calculation was attempted on a mechanical calculator and almost 40 years since the first ensembles of trajectories were calculated on a digital computer and, even though the real world is quantum mechanical, classical trajectory simulations continue to play a crucial role in studies of chemical dynamics. The classical approximation, fortunately, is valid for many processes and conditions of interest to chemists, and the Monte Carlo quasiclassical trajectory approach is relatively simple and straightforward to apply. The work reviewed here clearly illustrates that it continues to be used productively, creatively, and widely, to study the details of chemical processes. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

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. 

Most of our present understanding of the dynamics and of the collisional mechanisms of elementary chemical reactions comes from classical approaches, from simple classical models, and from quasiclassical trajectory studies. More recently, quantum mechanical results on the dynamics of directmode reactions have become available. 

Some useful reviews of quasiclassical and semiclassical dynamics include D. G. Truhlar and J. T. Muckerman, in Atom-Molecule Collision Theory. R. B. Bernstein, Ed., Plenum Press, New York, 1979, pp. 505-566. Reactive Scattering Cross Sections. 111. Quasiclassical and Semiclassical Methods. L. M. Raff and D. L. Thompson, in Theory of (%emical Reaction Dynamics, M. Baer, Ed., CRC Press, Boca Raton, FL, 1986, Vol. Ill, pp. 1—121. The Classical Trajectory Approach to Reactive Scattering. M. S. Child, in Theory of Chemical Reaction Dynamics, M. Baer, Ed., CRC Press, Boca Raton, FL, 1986, Vol. Ill, pp. 247-279. Semiclassical Reactive Scattering. 

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. 

Pig. 8. Time-dependent (a) diabatic and (b) adiabatic electronic excited-state populations and (c) vibrational mean positions as obtained for Model I. Shown are results of the mean-field trajectory method (dotted lines), the quasiclassical mapping approach (thin full lines), and exact quantum calculations (thick full lines). 

The approach used for the present study is quasiclassical trajectory (QCT) calculations based on global potential surfaces that have been derived from high quality ab initio calculations. QCT calculations have been used in the past to study mode-specific reactivity in H + HOD and related reactions [4,7]. Correspondence of the results with quantum dynamics calculations [6] has been good at the qualitative level, and sometimes even quantitative comparisons have been obtained. There have been quantum dynamics studies of H + HCN in the past [18-20], but these studies used reduced dimensionality approximations that did not allow for formation of H2CN intermediate complexes. Hiere have not been previous QCT studies of either H + HCN or H + N2O, but Bradley et al [14] have studied the NH + NO reaction using the same potential surface that we now use to study H + N2O. 

H + H2 Quantum mechanical probabilities using the time-dependent wavepacket approach, J. Chem. Phys. 69 5064 (1978) J. C. Gray, G. A. Fraser, D. G. Truhlar, and K. C. Kulander, Quasiclassical trajectory and quantal wavepacket calculations for vibrational energy transfer at energies above the dissociation threshold,... 

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. 

However, not all space curves singly connecting reactant and product asymptotes correspond to a realistic time evolution describing an elementary process. Such evolution is determined by the equations of motion within the quasiclassical approach the space curves can be interpreted as system point trajectories in M with their end points located in the reactant and product asymptotes 43,44), a trajectory Q — Q(t) is then determined by the classical equations of motion [i.e., within some of the equivalent formulations of classical mechanics tantamount to Eq. (9)]. 

The way in which reactant change into products in an elementary process will be regarded as the microscopic collision mechanism of the elementary process in question. It is determined within the quasiclassical approach by the characteristics of the system point trajectories. 

Although each set of boundary conditions defines a unique trajectory, not all of the IF quantities at each of the end points can be controlled in an experiment (4,47,48) these quantities usually have random distributions (impact parameter, vibrational phase, etc.). Consequently, it is useful to choose the values of the uncontrollable boundary conditions randomly. For a sufficiently high number of the randomly chosen values (on a relevant interval), all boundary conditions are included and the resulting set of trajectories (related to an elementary process) represents a dynamical picture of the elementary process within the quasiclassical approach (6,44). 

The CMD method is based on the propagation of quasiclassical centroid trajectories q t) derived from the mean force on the centroid as a function of position [cf. Eqs. (3.11) and (3.12)]. This method, combined with Eq. (3.13), generally provides an accurate representation of the exact quantum real-time position or velocity correlation functions. There is a significant theoretical question, however, on how CMD might be used to compute time correlation functions of the form (A(f)5(0)), where A and B are general quantum operators [4,5]. Two approaches to this problem will now be described. 

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