Quasi-classical trajectory results

Figure 3.11 Orientation dependence of the cross-section for the reaction H + D2(v= /= 0) HD + D at the two indicated values of the collision energy Ej- The ordinate is dff R/dcos y = 2ctr(cos y). The solid curves were calculated from the angle-dependent line-of-centers model, Eq. (3.34), and the (open and filled) points represent dynamical computations (these are quasi-classical trajectory results that have statistical error bars as discussed in Chapter 5) on the ab initio potential surface referred to in Figure 3.10 [adapted from N. C. Blais, R. B. Bernstein, and R. D. Levine, J. Phys. Chem. 89, 20 (1985)].

Figure 24. Diabatic (left) and adiabatic (right) population probabilities of the C (full line), B (dotted line), and X (dashed line) electronic states as obtained for Model II, which represents a three-state five-mode model of the benzene cation. Shown are (A) exact quantum calculations of Ref. 180 mean-field trajectory results [panels (B),(E)] and quasi-classical mapping results including the full [panels (C),(F)] and 60% [panels (D),(G)] of the electronic zero-point energy, respectively.

Figure 27. Diabatic (a) and adiabatic (b) population probabilities for Model IVb. Shown are exact quantum results (thick full lines), mean-field-trajectory results (upper thin full line), quasi-classical mapping results including the full zero-point energy (i.e. y = 1, lower thin full line), as well as ZPE-corrected mapping results corresponding to y = 0.6 (dashed line) and y = 0.8 (dotted line), respectively.

Theoretically, Nitzamov et al. [6] studied the rate constants and energy partitioning for the OH -I- HBr reaction via quasi-classical trajectories. Liu and coworkers [9] predicted that the OH -I- HBr reaction has a small barrier (<1.0 kcal/mol, from CCSD(T)/6-31 H-G(2df,2p) singlepoint energies), and their theoretical rate constants are comparable with the experimental results [4, 11-14]. 

Several of the runs were carried out expressly to make comparisons with other work, either quantum mechanical calculations or quasi-classical trajectories. These runs are generally for low-internal-energy states of H2, whereas the major part of the work reported here is for high-internal-energy states. Nevertheless, some interesting comparisons can be made and we can also make checks for consistency. For the results presented in this section, the final states of the trajectories were assigned by the histogram method. 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

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. 

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).

Figure 4 Quasi-classical opacity function P(p), defined as the fraction of reactive trajectories for a given impact parameter, p (solid line). Also plotted is Krei, the component of the relative incident-target H atom kinetic energy parallel to the surface, following a non-reactive collision (dotted line). The results correspond to H-on-D for the flat-surface potential described in the text.

The accuracy of trajectory calculations have been examined by comparing the results of exact quantum and quasi-classical calculations [143], The most difficult problem lies in the selection of a method for quantising the continuous classical product energy distributions. There is no formal justification for such a procedure, but it enables comparison with experimental vibrational and rotational distributions. No single method appears to be suitable for all systems. 

PESs, and (iv) obey the principle of microreversibility. Section 3 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 Sec. 4, which discusses various motivations of the ansatz as well as several variants of the implementation. Section 5 gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. The mapping formalism, its relation to other formulations, and its quasi-classical implementation is introduced in Sec. 6. Section 7 is concerned with the semiclassical description of nonadiabatic quantum mechanics. Section 8 summarizes our results and concludes with some general remarks. 

It is necessary to determine vibrational semiclassical eigenvalues to determine initial coordinates for a quasi-classical calculation. This requires finding molecular coordinates and momenta such that the resulting good actions are integer multiplies of h. Usually, the harmonic actions can be used to define initial conditions that are approximately correct, and then the ratio of desired to calculated actions is used to scale the coordinates and momenta until the calculated actions are equal to the desired actions within some tolerance. Once the semiclassical eigenvalue has been determined, it is necessary to calculate molecular coordinates and momenta that can be used as initial conditions for colhsion simulations. These can be determined from the Fourier representation, or one can save coordinates and momenta from the trajectory that is used to determine the semiclassical good actions. 

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