Trajectories, semiclassical

The techniques outlined above provide information on the structure and accessibility of the photochemical reaction paths. As mentioned, this information is structural (i.e., nondynamical) and provides insight into the mechanism of photoproduct formation from vibrationally cold excited state reactants such as those encountered in many experiments where slow excited state motion or/and thermal equilibration is possible (in cool jets, in cold matrices, and in solution). 

In many cases, such structural or static information is not sufficient. The excited state may not decay at the point where the excited state path (MEP) intersects the n - 2 hyperline. Alternatively, the momentum developed on the excited state branch of the reaction coordinate may be sufficient to drive the ground state reactive trajectory along paths that are far from the ground state valleys. In such cases, a dynamics treatment of the excited state/ground state motion is required for mechanistic investigations. Furthermore a dynamics treatment is required to gain information of the time scales and quantum yields of the reaction. 

We now briefly describe the way in which the nonadiabatic event (surface hop) can be described in on the fly dynamics methods. We can represent the time-dependent wavefunction in the Cl space as a vector  

The problem of combining dynamics with quantum chemistry is at the forefront of current research. The best method to be used will emerge as a compromise between quantum mechanical rigor and computational feasibility.99 

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A Warshel. Dynamics of reactions m polar solvents. Semiclassical trajectory studies of electron-transfer and proton-transfer reactions. J Phys Chem 86 2218-2224, 1982. 

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. 

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. 

Studies of vibrational and/or rotational energy transfer which have utilized photochemical techniques are collected in Table 14. Dzelzkalns and Kaufman have continued a series of investigations dealing with the relaxation of vibrationally excited diatomic molecules. A fast-flow chemiluminescence technique was used to measure relaxation rate constants for HF(n = 1—4) in the presence of HF, CO2, and N2. These data were combined with earlier measurements for HF(n = 5—7) and discussed in terms of V-V or V-T/R transfer. The fraction of V-V transfer in HF(n = 2— 7) + HF(u = 0) relaxation has been determined and compared with a recent measurement for HF(n = 2) and with the results of semiclassical trajectory calcula-... 

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. 

Warshel, A. and Karplus, M. (1975) Semiclassical Trajectory Approach to Photoisomerization, Chem. Phys. Letts. 32, 11—17. 

Warshel, A. and Hwang, J.-K. (1986). Simulation of the dynamics of electron transfer reactions in polar solvents semiclassical trajectories and dispersed polaron approaches. 

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. 

Sample CASSCF left) and CASPT2 righO semiclassical trajectories run on the lowest-energy (Tm ) state of guanine from the FC region... 

A. Warshel and M. Karplus, Chem. Phys. Lett., 32, 11 (1975). Semiclassical Trajectory... 

S. M. Colwell and N. G. Handy, Vibrational energies of triatomic molecules using a semiclassical trajectory method. Mol. Phys. 

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