Method of classical trajectories

The method of classical trajectories has first received wide acceptance for studying the dynamics of electronically adiabatic elementary reactions. In the fiamewoik of this method, the dynamics of particles involved in elementary processes is smdied by the solution of classical equations of motion (for example, the Hamilton equations). For the generalized coordinates Rj (Ri, R2, 

Equations (2.53) are solved after the Ri and values were specified at the initial moment. The solution gives / i(t) and P t), which describe the state of the system at any moment. 

The method of classical trajectories provides a very clear picture of elementary processes. The totality of the R coordinates is named the configuration space of the system and the totality of the R coordinates and P pulses is named the phase space. 

At any moment the state of the system is described by a point in the phase space. This point is named the representative point. The totality of coordinates of the representative point gives the trajectory in the configuration or phase space, which, under certain initial conditions, parametrically specifies the behavior of the system in time. Knowing the trajectory of the system in the configuration space, we can go to the trajectory of motion over the potential energy surface. 

The probability of branching of the elementary process to the a channel Pet) depends on many initial conditions, including the relative velocity v, quantum states of reactants a, quantum states of products cf, and impact parameter b. For the random choice of die initial conditions, taking into account the corresponding distribution functions, it is expressed by a rather sinq)le formula 

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Collision rate calculated by the method of classical trajectories in Ref. 48. 

Kuznetsov, A.M. and Kharkats, Y.I. (1977) Method of classical trajectories in theory of bridge-type electron-transfer reactions. Soviet Electrochemistry, 13, 1283-1288. 

Before we start with quantum description, let us recollect the classical solutions which will be used later in the method of classical trajectories to study some quantum properties of the fields. Equations (56) are valid also for classical fields after replacing the field operators a and b by the c-number field amplitudes a and p, which are generally complex numbers. They can be derived from the Maxwell equations in the slowly varying amplitude approximation [1] and have the form. 

The <2 functions are very wide, thus no linearization of the quantum problem is possible and no pure quantum technique can be used for estimation of the observed values Ff 1.50 and F 0.83. However, good quantitative explanation of these numerical values can be obtained by the method of classical trajectories as will be shown in Section n.B. 

The answer to our question concerning the origin of sub-Poissonian behavior can be found by the method of classical trajectories. The method is very general. 

The method of classical trajectories can be used not only numerically (Figs. 5 and 6) but also analytically in special cases. For example, the evolution of... 

The results of Section II.C can be used in the method of classical trajectories in analogy with the technique described in Section II. B. We need to express the trajectories in their dependence on small noise parameters x and y. The integrals of motion, given by (38), can be expressed in a form of corrections in... 

We have shown, in agreement with the results presented in Ref. 21, that the method of classical trajectories gives very good predictions in the case of strong-field interactions (i.e., for the photon numbers larger than 10). The calculation speed of the method does not depend on numbers of interacting... 

In our computational procedure for investigating the dynamics of unimolecular reactions by Jhe method of classical trajectories, we can use our initialization routines to select hot rotational energy distributions... 

The small factor in reaction probabihty (6-13) is a pre-exponential one, whereas the O -factor characterizing the efficiency of vibrational energy in the elementary process is high and close to unity (a = 1). A detailed calculation of the adiabatic elementary reaction in the framework of the method of classical trajectories gives a 0.5 (Levitsky Polak,... 

Levitsky, A. A. (1978b), Investigation of Elementary Reaction O -k N2 Using Method of Classical Trajectories, in New Aspects of Petrochemical Synthesis, Institute of Petrochemical Synthesis of USSR Academy of Sciences, Moscow. 

Classical Trajectory Calculations.—Classical mechanics is often assumed to be a good approximation for the motion of nuclei on adiabatic electronic potential-energy surfaces The method of classical trajectory calculations as applied to chemical kinetics has been reviewed. It is now quite standard for ect... 

Recently new methods for the dynamical description of elementary processes of charge transfer in a condensed phase have been developed, which take into account the interaction of a dynamical subsystem with a thermostat (the role of which is played by a part of the medium). The method of classical trajectories is one of these. Below are given the results of calculations using this method for a simple model which allows us to... 

Figure 1.3 HD rotational and vibrational state distributions measured for the H + D2 reaction at a collision energy of 1.3 eV. The energy is determined by the recoil energy of the H atom in the photodissociation of HI at a wavelength where it dissociates primarily to ground state I atoms. The experimental results shown [adapted from D. P. Gerrlty and J. J. Valentini, J. Chem. Phys. 81, 1298, (1984) and Valentin and Phillips (1989)] used CARS spectroscopy to determine the state of HD. E. E. Marinero, C. T. Rettner, and R. N. Zare, J. Chem. Phys. 80,4142 (1984) used resonance enhanced multiphoton ionization, REMPI, for this purpose. The figure also shows curves. Those on the left are the so-called linear surprisal representation, see Section 6.4. The plot on the right shows the same experimental data on a logarithmic scale. The curves [adapted from N. C. Blais and D. G. Truhlar, J. Chem. Phys. 83, 2201 (1985)] are a dynamical computation by the method of classical trajectories. Section 5.2.

J. Chem. Phys. 107, 2351 (1997)]. We expect that when the O atom runs head-on into D2, OD will scatter backwards. Dynamical computations (dashed line, QCT) by the method of classical trajectories. Chapter 5, verify that the abstraction reaction contributes primarily in the backwards direction and that the OD formed by the insertion reaction indeed shows no forwards-backwards preference with respect to the incident O atom. This is further discussed in Chapters 4 and 10. 

An essential difference between classical and quantal mechanics is the number of initial conditions that need to be specified if tiie initial state is to be fully defined. In classical mechanics one must specify botii tiie position and the momentum for each degree of fi eedom. In quantum mechanics the uncertainty principle implies tiiat if, say, tiie momentum is well specified, tiie value of the position can be anywhere witiiin its possible range. Since molecules are inherently quantal, a complete specification of initial conditions for a colhsion in a system of n degrees of fi eedom consists of n quantum numbers. In contrast, a classical trajectory for tiie system requires In initial conditions. The method of classical trajectories mimics this quantal aspect by running many classical trajectories where, of the In initial conditions, n are held constant (tiie same n tiiat correspond to the quantal case) while the other n are allowed to vary. The final outcome is determined by averaging over those initial conditions tiiat are varied. We refer to these initial conditions that need to be inherently averaged over as the phases. 

The Monte Carlo sampling of initial conditions in the method of classical trajectories is illustrated here for the particular problem of computing the reaction cross-section ctr given in terms of the opacity function P b) by, Eq. (3.14),... 

The theoretical approach that so far has been most effective in describing the dynamics of adsorption esorption and of reactive gas surface colhsions is based on the method of classical trajectories. The essence of the problem is to provide a tractable yet reahstic approach to the coupling of the molecular and surface (and bulk) degrees of freedom. In principle, one can introduce a (often, semi-empirical) potential energy, which is a function of the positions of all atoms, both those of the molecule and those of the surface. The classical equations of motion can then be solved. Since each atom of the solid is interacting with its neighbors, the number of coupled differential equations that need to be solved in... 

As in the liquid, evaluating the two additional terms exactly is as difficult as the original problem. However, because of their physical interpretation it is possible to provide simple yet realistic approximations for them. In practice, one sometimes solves the equations of motion not only for the molecular degrees of freedom but also for those surface atoms to which they are directly coupled. Only the rest of the solid is averaged over. This structureless, pillow-like description of the environment has enabled the method of classical trajectories to be applied not only to reactions at the surface but also in solution. Unique to the surface is the need to allow for electron-hole pair excitations. 

When the reaction is considered by the method of classical trajectories, the system is described by equation (2.55) where the Hamilton function is specified by expression (2.56). Then the reaction can be considered as the motion of the representative point in the phase space F. 

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