Calculation of Classical Trajectories

A chemical reaction cannot be described by only two particles interacting. A minimum of three particles is necessary. Unfortunately, for such three-bodied systems it is not possible to obtain analytical solutions and it becomes necessary to use numerical methods. Among these, the Monte Carlo method assumes particular relevance for the calculation of classical trajectories using random sampling. 

By knowing the PES for a chemical reaction it is possible to solve the classical equations of motion for a collision on this surface. In the context of classical mechanics, the movement of particles is described by Newton s second law  

The simplest example of a classical trajectory is the collinear collision complex A + BC, for which there are only two independent coordinates the A-B distance (rj) and the B-C distance (rj). However, even in such a simple system, a complication is introduced by transforming from internal molecular coordinates to Cartesian space. This complication can be appreciated by writing the complete Hamiltonian of the system  

The fundamental parameter that is obtained in these calculations is the probability of reaction for any given relative initial velocity (i.e. initial reactive kinetic energy), roto- 

If we want to make a comparison between these ctoss sections and the experimental values for the reaction in thermal equilibrium, it is necessary to make first an average over the initial states (reactants) to obtain an expression that gives the rate of appearance of the product in a determined state, and then sum over all the states of the products to obtain the total rate of appearance of products. This procedure constitutes the basis of molecular dynamics. Normally, it is assumed that the molecular velocities are described by a MaxweU-Boltzman distribution, and the specific rate constant is expressed relative to the quantum state of BC not in terms of the initial relative velocity v , but in terms of the relative translational energy, = l /iA-Bc( r°)  

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Olender and Elber, 1996] Olender, R., and Elber, R. Calculation of classical trajectories with a very large time step Formalism and numerical examples. J. Chem. Phys. 105 (1996) 9299-9315... 

Wall, Hiller, and Mazur [300, 301] first used a computer to integrate the classical motion equations for a system of three atoms, and in the 1960s the technique was developed by Blais and Bunker [48, 302-306], and by Karplus [19, 20, 72, 307-311] and Polanyi [71,73, 74, 267, 312, 313] and their coworkers. Recently calculations have been performed on systems simulating abstraction reactions involving more than three atoms [314] and four-center reactions involving four atoms, that is, AB + CD - AC + BD [315-317]. Here we present first a general survey of the Monte Carlo calculations of classical trajectories and then a brief review of some of the results of these calculations. Emphasis is placed on data for reactions that have been studied experimentally and have been mentioned earlier in this chapter. 

Calculation of Classical Trajectories with Boundary Value Formulation... 

Ron Elber Calculation of Classical Trajectories with Boundary Value Formulation, Lect. Notes Phys. 703, 435-451 (2006)... 

Calculation of classical trajectories of molecular systems is a powerful tool for stud3dng the thermod3mamics and the kinetics of biophysical problems. Indeed the availability of numerous computer packages and journals devoted to such simulations suggests that these studies are extremely popular, and are influencing many fields. It is therefore no surprise that a number of chapters in this book are devoted to the use of classical d3mamics in condensed matter physics. 

There has been some work on semi-classical quantization of the linear Jahn-Teller Hamiltonian (2.2) [41,42]. The quantization scheme which bears the closest relation to the present wave packet treatment involved the calculation of classical trajectories while slowly turning on the nonadiabatic interaction [41(a)]. The main emphasis of that work was the development of a method for obtaining energy levels in molecules with nonadiabatic dynamics which might then be applied to larger multimode systems. 

The currently most important technique of the dynamic approach is based on the calculation of classical trajectories. In such an approximation, nuclei of a chemical system in question are treated as classical particles moving under forces defined by the PES. The trajectories represent the solutions to the Hamiltonian (or Lagrangian) of Eq. (1.27). 

R. Olender and R. Elber,/. Chem. Phys., 105,9299 (1996). Calculation of Classical Trajectories... 

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