Quasi-classical trajectories method

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. 

Table 6.3 A comparison of different theoretical approaches to the evaluation of the thermal rate constant for the F + H2 —> HF + H reaction at T = 300 K. TST is transition-state theory (Example 6.2), QCT is the quasi-classical trajectory method [Chem. Phys. Lett. 254, 341 (1996)], and QM is (exact) quantum mechanics [J. Phys. Chem. 102, 341 (1998)].

Chaos and longer time evolution of the quasi-classical trajectory method... 

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... 

Dynamics Method Fully Quantum Quasi-Classical Trajectory Surface Hop Semi-Classical (Ehrenfest) Molecular Mechanics... 

In the full-quantum dynamics method, the distribution of nuclear positions is accounted for in nuclear wavepacket form, that is, by a function that defines the distribution of momenta of each atom and the distribution of the position in the space of each atom. In classical and semi-classical or quasi-classical dynamics methods, the wavepacket distribution is emulated by a swarm of trajectories. We now briefly discuss how sampling can generate this swarm. 

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. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

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).

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. 

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. 

Force-field methods form the basis of molecular dynamics. They use a parameterised quasi-classical description of interatomic forces to model the trajectory of systems typically composed of hundreds or even thousands of atoms. One good feature of these types of calculations is that with large systems the computational effort increases linearly with the size of the problem. This means that increased computational power allows considerably larger systems to be studied. Further gains can also be made by using parallel processors since energy calculations in molecular dynamics simulations are inherently parallel. 

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