Classical Collision Dynamics

The classical description is quite different from the quantum. In classical dynamics we describe the coordinates and momenta simultaneously as a function of time and can follow the path of the system as it goes from reactants to products during the collision. These paths, called trajectories, provide a fnotion picture of collision process. The results of any real collision can be represented by computing a large number of trajectories to obtain distribution of post-collisions properties of interest (e.g. energy or angular distribution). In fact, the trajectory calculation means the transformation of one distribution function (reagent distribution, pre-collision) into another (product distribution, post-collision), which is determined by PE function. 

Two-particle collisions are often adequately described using the ideas of classical dynamics. The quantity, do, can be formally obtained from the following well-known equations ... 

The integrand decreases as an inverse cube of the distance separating the molecule from the charged particle (i.e., as Rq3), meaning that the interaction is important only at distances about the distance of closest approach. Using the classical dynamics of collisions,148 we find the... 

Thus as a starting point for understanding the bombardment process we have developed a classical dynamics procedure to model the motion of atomic nuclei. The predictions of the classical model for the observables can be compared to the data from sputtering, spectrometry (SIMS), fast atom bombardment mass spectrometry (FABMS), and plasma desorption mass spectrometry (PDMS) experiments. In the circumstances where there is favorable agreement between the results from the classical model and experimental data It can be concluded that collision cascades are Important. The classical model then can be used to look at the microscopic processes which are not accessible from experiments In order to give us further insight into the ejection mechanisms. 

Porter, R.N. and Raff, L.M. (1976). Classical trajectory methods in molecular collisions, in Dynamics of Molecular Collisions, Part B, ed. W.H. Miller (Plenum Press, New York). 

A theoretical determination of the rate constant for a chemical reaction requires a calculation of the reaction cross-section based on the dynamics of the collision process between the reactant molecules. We shall develop a general relation, based on classical dynamics, between reaction probabilities that can be extracted from the dynamics of the collision process and the phenomenological reaction cross-section introduced in Chapter 2. That is, we give a recipe for how to calculate the reaction cross-section in accord with the general definition in Eq. (2.7). 

The collision is described here by classical dynamics, and we assume that the motion takes place in a spherically symmetric potential U(r). It is well known that the relative motion of the atoms is equivalent to the motion of a particle with the reduced mass p, in an effective one-dimensional potential given by... 

Perera and Amar (1989) found more detailed support for the structural control of caging in classical dynamics calculations on a model of Br2 in large clusters of Ar and C02. The dissociation channel was found to become closed, as a function of cluster size, between 11 and 12 C02 molecules in the BrJ(C02)M clusters, correlating with the appearance of double-capped minimum energy structures. This correlation was found in the Br2 Ar clusters as well. Collisions between a vibrating diatomic molecule in a cluster and the solvent particles may cause V-T energy transfer and rapid evaporation of the cluster. 

Recent advances have resulted from the development of more powerful experimental methods and because the classical collision dynamics can now be calculated fully using high-speed computers. By applying Monte Carlo techniques to the selection of starting conditions for trajectory calculations, a reaction can be simulated with a sample very much smaller than the number of reactive encounters that must necessarily occur in any kinetic experiment, and models for reaction can therefore be tested. The remainder of this introduction is devoted to a simple explanation of the classical dynamics of collisions, a description of the parameters needed to define them, and the relationship between these and the rate coefficient for a reaction [9]. 

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],... 

A more complete treatment of the classical dynamics of a system containing an ion and a rotating polar molecule involves the numerical solution of the equation of motion (trajectory calculation) by Dugan et al. [64—68]. In their treatment, a capture collision is defined by an ion trajectory that penetrates to within a certain value of r. Their results also show that the locking in of the dipole is not likely to occur because of the conservation of angular momentum. 

The basic theoretical concepts describing the interaction of a sufficiently massive and energetic particle with a surface are the binary collision (BC) model and the molecular or classical dynamics (MD) model. 

Fig. 2. Average rotational energy transfer (ARET) for different collision energies. Solid line quantum mechanical wave packet propagation using the MCTDH method (from Ref. [22]) dashed line MQCB method (equation (47)) dotted line classical dynamics.

We recall, however, that the values of and X can be con-sidered as tunneling corrections to the simple collision theory and the Eyring s activated complex theory, respectively, only if the classical dynamical effects are negligible. Otherwise, the corresponding... 

Consider the specific case of NaBrKCl for which theoretical, classical dynamics studies are available at energies where two dissociative channels are energetically accessible. (Questions as to the validity of the classical picture are relegated to later sections.) This system possesses attractive forces between the atoms such that the bound NaBrKCl species lies at an energy of approximately 40 kcal/mol below NaBr - - KCl or NaCl -I- KBr. Specifically, consider the case where energized NaBrKCl is formed by the collision... 

The amplitude is generally a complex number. Computing it is the business of the Schrodinger equation. The essential point is that when a final outcome can be reached in several different ways, the amplitude to realize the outcome is the sum (= superposition) of the amplitudes for the different routes. For molecular collisions, where classical dynamics is often a realistic description, it is a useful... 

The general understanding of molecular dynamics rests mainly upon classical mechanics this holds true for full bimolecular collisions (see Trajectory Simulations of Molecular Collisions Classical Treatment) as well as half-collisions, i.e., the dissociation of a parent molecule into different products. The classical picture of photodissociation closely resembles the time-dependent picture the electronic transition from the ground to the excited electronic state is assumed to take place instantaneously so that the internal coordinates (Qi) and corresponding momenta (/, ) of the parent molecule remain unchanged during the excitation step (vertical transition). After the molecule is promoted to the PES of the upper state it starts to move subject to the classical equations of motion (Hamilton s equations)... 

Classical Dynamics of Nonequilibrium Processes in Fluids Integrating the Classical Equations of Motion Control of Microworld Chemical and Physical Processes Mixed Quantum-Classical Methods Multiphoton Excitation Non-adiabatic Derivative Couplings Photochemistry Rates of Chemical Reactions Reactive Scattering of Polyatomic Molecules Spectroscopy Computational Methods State to State Reactive Scattering Statistical Adiabatic Channel Models Time-dependent Multiconfigurational Hartree Method Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Valence Bond Curve Crossing Models Vibrational Energy Level Calculations Vibronic Dynamics in Polyatomic Molecules Wave Packets. 

Classical Trajectory Simulations Final Conditions Mixed Quantum-Classical Methods Rates of Chemical Reactions State to State Reactive Scattering Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Wave Packets. 

Interaction potentials are introduced for general electronic representations and are discussed in the adiabatic representation to emphasize the different nature of long-range and short-range interactions, particularly for polyatomic systems. The size of a target, i.e., the number of atoms in it, is shown to influence how one approaches the collision dynamics. We also describe areas of recent progress in quantal, semiclassical, and classical dynamics. 

Classical dynamics has been used extensively over the past twenty years to aid in our microscopic understanding of chemical reactions and properites of matter. As our experience has grown, the complexity of the systems studied has expanded from simple atom-diatom collisions and hard sphere liquids to more complicated gas-phase reactants (see the chapters by Schatz and Eigersma in this book) and more realistic liquids. Dynamics calculations allow the determination of average experimental quantities, and at the same time, they give physical insight into the microscopic mechanisms. Results of the calculations are very visual, allowing one to picture the motion of particles. The variety of applications of classical dynamics in chemistry is evidenced by the contributions to this volume. 

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