Quasi classical trajectory

In addition to the equations of motion, one needs to specify a procedure to evaluate the observables of interest. Within a quasi-classical trajectory approach, the expectation value of an observable A is given by Eq. (16). For example, the expression for the diabatic electronic population probability, which is defined as the expectation value of the electronic occupation operator, reads... 

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. 

Thermolysis of l,l-difluoro-2,3-diphenylcyclopropane in supercritical CO2 has allowed the rate of geometrical isomerization [i.e. cis-( 109) to /ra/M-(109)] and racemization [i.e. (/< )-( 109) to (S)-( 109)] to be determined from O2 dependence of the trapping rate of the postulated intermediate 1,3-biradical.246 Above 150 °C, the formation of 2,2-difluoroindane and its decomposition products is reported. A similar thermally induced equilibrating series of stereomutations has been observed with the analogous non-fluorinated cyclopropane in which rate constants and deuterium exchange isotope effects are reported.247 Theoretical studies of this isomerization have focused on classical248 and quasi-classical trajectories.249... 

This means that there is a crossover temperature defined by (1.7) (or hpc = 2it/o> ), above which tunneling switches off, because the quasi-classical trajectories that give the extremum to the integrand in (2.1) cease to exist. This change in the character of semiclassical motion is universal for barriers of sufficiently general shape. 

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. 

Obviously, purely quantum mechanical effects cannot be described when one replaces the time evolution by classical mechanics. Thus, the quasi-classical trajectory approach exhibits, e.g., the following deficiencies (i) zero-point energies are not conserved properly (they can, e.g., be converted to translational energy), (ii) quantum mechanical tunneling cannot be described. 

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].

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)].

Note that this average over phases is equivalent to the approach used in quasi-classical trajectory calculations for bimolecular reactions see, e.g., Fig. 4.1.2. 

Skouteris, D., Werner, H.-J., Aoiz, F. J., Banares, L., Castillo, J. F., Menendez, M., Balucani, N., Cartechini, L. and Casavecchia, P. (2001) Experimental and theoretical differential cross sections for the reactions Cl + H2/D2, J. Chem. Phys. 114, 10662-72 Aoiz, F. J., Banares, L., Castillo, J., Menendez, M., Skouteris, D. and Werner, H.-J. (2001) A quantum mechanical and quasi-classical trajectory study of the Cl + H2 reaction and its isotopic variants Dependence of the integral cross section on the collision energy and reagent rotation, J. Chem. Phys. 115, 2074-81. 

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... 

Quantum mechanical and quasi-classical trajectory study of the C( D) + H2 reaction dynamics, J. Chern. Phy.s. 118, 565-568. 

Frost, R.J. and Smith, l.W.M. (1987) Combining transition-state calculations with quasi-classical trajectory calculations. 2. Collinear collisions involving vibrationaUy excited reagents. Chem. Phys. Ill, 389-400. 

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

Figure 2.32 Distribution (for 100 CASSCF/3-21G combined quasi-classical trajectories) of the lifetimes (r) and torsion angles (ff) of the hop geometry of model chromophore Im.

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]. 

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