Semiclassical trajectory calculations

In order to investigate which model for the behavior of 8 is closer to being correct, our group provided Carpenter with an analytical expression, fit to our PES, so that Carpenter could perform semiclassical trajectory calculations on our PES. At the same time. Doubleday and Hase undertook direct dynamics calculations. As discussed in Chapter 21 in this volume, in the latter type of trajectory calculation the forces acting on a molecule at different points on a PES are found by performing electronic structure calculations. For this purpose. Doubleday and Hase used a reparameterized version of AMI that provided a PES, similar to those calculated by Getty and by Baldwin, Yamaguchi, and Schaefer. 

Kdppel, 1990), or a more approximate semiclassical trajectory calculation (Herman, 1984). In systems of interest to the organic photochemist, sinmlta-neous loss of vibrational energy to the solvent would also have to be included, and reliable calculations of quantum yields are not yet possible. It is perhaps useful to provide a simplified description in terms of classical trajectories for the simplest case in which the molecule goes through the bottom of the funnel, that is, the lowest energy point in the conical intersection space. 

While chemical reactivity is normally discussed in terms of transition state theory, corresponding to motion on an adiabatic potential energy surface, a complementary theory has been formulated (see the review by Metiu et in which the passage from reactants to products is formulated in terms of a transition between two diabatic surfaces. We shall refer to this as diabatic transition structure theory. This approach is a very natural one to use in semiclassical trajectory calculations (see, for example. Refs. 40) and, as we shaU presently discuss, enables a simple interpretation of transition structure. 

Cross sections can be predicted from semiclassical trajectory calculations, in which equations of motion are... 

So as to calculate electron transfer reaction rates in polar media,a number of effective approaches have been evolved based on model forms of the PES and on various models for evaluating the medium polarization and the energy of solvent reorganization [2,17,18,20,23-26]. Recently this analysis has been raised to the level of semiclassical trajectory calculations of the dynamics of above reactions [27]. At the same time, there are so far no sufficiently complete quantum mechanical calculations of the PES of the electron transfer reactions on account of complexities of the calculation involving extension of the basis sets and careful assessment of resolvation of the reactants in the development of the reaction process. 

Let us consider first the in vacuo cases. Dynamical aspects of the reaction in vacuo may be recovered by resorting to calculations of semiclassical trajectories. A cluster of independent representative points, with accurately selected classical initial conditions, are allowed to perform trajectories according to classical mechanics. The reaction path, which is a static semiclassical concept (the best path for a representative point with infinitely slow motion), is replaced by descriptions of the density of trajectories. A widely employed approach to obtain dynamical information (reaction rate coefficients) is based on modern versions of the Transition State Theory (TST) whose original formulation dates back to 1935. Much work has been done to extend and refine the original TST. 

Considerable use continues to be made of classical trajectory calculations in relating the experimentally determined attributes of electronically adiabatic reactions to the features in the potential energy surface that determine these properties. However, over the past 3 or 4 years, considerable progress has been made with semiclassical and quantum mechanical calculations with the result that it is now possible to predict with some degree of confidence the situations in which a purely classical approach to the collision dynamics will give acceptable results. Application of the semiclassical method, which utilises classical dynamics plus the superposition of probability amplitudes [456], has been pioneered by Marcus [457-466] and by Miller [456, 467-476],... 

From a practical point of view the classical method using directly the probability density function is not convenient, and it is computationally preferable to use an approach that involves trajectory calculations. A derivation of such formulation can be made by starting from the quantum-mechanical TDSCF, and using semiclassical (gaussian) wavepackets. Here we merely quote the final result. In analogy to (62), the single-mode classical SCF potentials are given by... 

Finally, I would like to discuss briefly the bimodal structure that has been observed recently in product vibrational state distributions in three-dimensional classical trajectory calculations.39 It is illustrative to see how this arises even in the simplest situation, the non-reactive collinear model. Within a completely classical framework, neglecting semiclassical interference terms, the n, - n2 vibrational transition probability of (54) is... 

J. Bowman, B. Gazdy, and Q. Sun, A. method to constrain vibrational energy in quasiclass-ical trajectory calculations, J. Chem. Phys. 91 2859 (1989) R. Alimi, A. Garcia-Vela, and R. B. Gerber, A remedy for zero-point energy problems in classical trajectories a combined semiclassical/classical molecular dynamics algorithm, J. Chem. Phys. 96 2034 (1992). 

A consideration of these results shows, first of all, that both the simple collision and the activated complex theories require considerable corrections in the whole temperature range investigated, not only in a quantum-mechanical, but also in a semiclassical treatment based on quasiclassical trajectory calculations. 

Classical trajectories are the only feasible means to explicitly treat all atoms in a dynamical study of a unimolecular reaction. Trajectories have been used extensively to interpret A + BC bimole-cular reactions and a considerable amount of literature exists with respect to these studies. Excitation functions, scattering angles, product energy distributions, and other dynamical properties are usually quantitatively determined by the trajectory calculations. The semiclassical studies of Marcus and Miller have in general confirmed the accuracy of classical trajectories in calculating dynamical properties for bimolecular reactions. However, the trajectories do not describe quantum mechanical effects such as interferences, tunneling, and nonadiabatic electronic transitions. 

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