Quasi-Classical Calculations

As mentioned in Sec.3.II., the initial conditions for integration of the classical equations of motion (14.11) may be chosen in such a way that they correspond to quantized values of the molecular vibration and rotation energies. The shortcoming of this semiclassical approach is that it does not satisfy the principle of detailed equilibrium, which states that the probability of forward transition (i— -f) equals the probability of reverse transition (f——i). This is easily seen if one introduces the action-angle variables /105/ 

Since an exact quantum-mechanical description of the collisions is possible only for very simple systems, it is clear that approximate methods, combining the simplicity of classical calculations with the possibility of taking into account quantum effects, will be very useful for the molecular collision theory. A promising method of this kind is the classical S-matrix method developed by MILLER /IO6/. It represents a quasi-classical approach which consists of passing to the classical limit of the scattering operator defined by (21.11). 

An expression for the matrix element (134.11) can be derived using the FEYNMAN path integral formulation of quantum mechanics /107/ which yields in the classical limit (h— 0) 

L a 2T - H being the Lagrange function (H is the classical Hamilton function). Summation in (135.II) is made over all classical trajectories for which the boundary conditions x(t- ) = x and x(t2) = X2 

Considering, for example, the simple case of the colinear collision A + BC, one introduces the distance R between atom A and the center-off-mass of molecule BC, the internuclear separation r in the molecule BC and the corresponding momenta p and p. The wave 

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Note that the energy dependence of the reaction cross-section predicted by the model in Section 4.2.2 (Fig. 4.1.5) is in rough agreement with the quasi-classical calculations. 

Let us then go on and describe how the quasi-classical calculations leading to these results are done. We begin with the Hamiltonian for a system of N atoms ... 

Figure 7 Product HD rotational state distributions, from the experiments of Ref. [16] (open squares) and from the quasi-classical calculations (filled squares).

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. 

D. A. V. Kliner, K. D. Rinen, and R. N. Zare, The D + H2 reaction. Comparison of experimental with quantum-mechanical and quasi-classical calculations, Chem. Phys. Lett. 166 107 (1990). 

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. 

To the best of our knowledge, this is the first time that a stepwise internal-conversion process has been treated by a microscopic quantum calculation. Subsequently, Stock performed quasi-classical calculations, employing the classical-electron-analog Hamiltonian of Meyer and Miller, which made it possible to include all vibrational modes. This calculation predicts an even somewhat fa.ster C - B - X decay. In any case, the above findings are fully consistent with the experimental upper limit on the C fluorescence quantum yield. They should also provide the explanation of the final steps in the nonradiative decay chain of the D and E states of CeHe" ", because low-energy conical intersections between... 

Fig. 8. Total reaction probability for collinear LiF + H -> HF + Li (solid curve) and collinear LiF + D DF -f Li (dashed curve) as a function of initial relative translational energy from a quasi-classical calculation. LiF is initially in its ground vibrational state. The curves are for a laser power such that Uqi O 0.01 eV and a laser frequency = 6.2 eV.

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

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)... 

That is, the classical DoF propagate according to a mean-field potential, the value of which is weighted by the instantaneous populations of the different quantum states. A MFT calculation thus consists of the self-consistent solution of the time-dependent Schrodinger equation (28) for the quantum DoF and Newton s equation (32) for the classical DoF. To represent the initial state (15) of the molecular system, the electronic DoF dk Q) as well as the nuclear DoF xj Q) and Pj 0) are sampled from a quasi-classical phase-space distribution [23, 24, 26]. 

The quasi-classical SH model employs the simple and physically appealing picture in which a molecular system always evolves on a single adiabatic potential-energy surface (PES). When the trajectory reaches an intersection of the electronic PESs, the transition probability pk t to the other PES is calculated... 

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. 

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 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.

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

To study to what extent the mapping approach is able to reproduce the quantum results of Model 111, Eigs. 34 and 35 show the quasi-classical probability densities P (cp,f) for the two cases. The classical calculation for E = 0 is seen to accurately match the initial decay of the quantum-mechanical... 

A significant limitation in both the adiabatic and non-adiabatic dynamics discussed above is that they are only 3D. Recently quasi-classical adiabatic dynamics calculations of S on a 6D DFT PES have shown that the predominant reason that 5 <<1... 

There has been a long history in theoretical efforts to understand H + H/Cu(lll) and its isotopic analogs because it represents the best studied prototype of an ER/HA reaction. These have evolved from simple 2D collinear quantum dynamics on model PES [386] to 6D quasi-classical dynamics on PES fit to DFT calculations [380,387,388], and even attempts to include lattice motion on ER/HA reactions [389]. These studies show that there is little reflection of incident H because of the deep well and energy scrambling upon impact, i.e., a % 1. Although some of the... 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

For practical reasons, a quasi-classical approximation to the quantum dynamics described by Eq. (1.10) is often sought. In the quasi-classical trajectory approach (discussed in Section 4.1) only one aspect of the quantum nature of the process is incorporated in the calculation the initial conditions for the trajectories are sampled in accord with the quantized vibrational and rotational energy levels of the reactants. 

Fig. 4.1.2 Harmonic oscillator with the energy E = p2/(2m) + (1/2)kq2 (which is the equation for an ellipse in the (q,p)-space). In the quasi-classical trajectory approach, E is chosen as one of the quantum energies, and all points on the ellipse may be chosen as initial conditions in a calculation, i.e., corresponding to all phases a [0, 27r].

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