Monte carlo trajectory studies

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

The results of a complete Monte Carlo trajectory study are those properties of a reaction and its chemical products that have already been mentioned at various points in this chapter ... 

Recently, Monte Carlo trajectory studies have been performed [325, 326] on systems with appreciable energy barriers, with a view to discovering whether excitation of particular degrees of freedom in the reagents promotes reaction. For three-atom reactions, if the barrier lies in the exit valley, vibrational (rather than translational or rotational) excitation can be used most effectively for surmounting the barrier. Conversely, if the barrier is in the entry valley, it is most easily surmounted if energy is located in the relative translation of the products rather than in vibration. For appreciably endothermic reactions, the barrier is very likely to be in the exit valley [323], and the conclusion that vibrational excitation will considerably assist the occurrence of such reactions is supported by calculations based on the applications of microscopic reversibility to the detailed rate coefficients for several exothermic reactions [327, 227]. It appears that similar criteria apply to four-center reactions of the AB + CD - AC + BD type [317], and the effect of vibrational excitation on the rate of such reactions has been investigated [316,317]. 

In several Monte Carlo trajectory studies, attempts have been made to discover the nature of the potential that operates in reactions of alkali metal atoms which produce alkali halides. The results of the calculations are compared to the particularly detailed information provided by crossed-molecular-beam experiments. The reaction... 

Classical complex formation such as outlined above has been observed in a number of classical Monte Carlo trajectory studies,45 and Brumer and Karplus46 have recently reported an extensive study of alkali halide-alkali halide reactions which involve long-lived collision complexes. These purely classical studies cannot, of course, describe the resonance structure in the energy dependence of scattering properties, but rather give an average energy dependence the resonance structure, a quantum effect, is described only by a theory which contains the quantum principle of superposition. 

Of course, many investigations fall somewhere between the two limiting types. Thus, a potential may be chosen on limited experimental information and a Monte Carlo trajectory study carried out to predict the values of quantities that have not been observed experimentally. These results should not be accepted unreservedly, since the collision dynamics are determined by the form of the assumed potential. Therefore, the evidence on which the potential was selected should be carefully scrutinized. Recent calculations which were designed to provide information about the relaxation of HF by H atoms, i.e. 

N. C. Blais and D. G. Truhlar, Monte Carlo trajectory study of Ar + H2 collisions, I, Potential energy surface and cross sections for dissociation, recombination, and inelastic scattering, J. Chem. Phys. 65 5335 (1976). 

The techniques and applications of the Monte Carlo trajectory method have been explained fully elsewhere. The object in the next few paragraphs is simply to highlight the results which are specially relevant to the subject of selective energy consumption. Trajectory studies that relate to particular systems are referred to later when the corresponding experimental data are reviewed. 

A sequence of successive configurations from a Monte Carlo simulation constitutes a trajectory in phase space with HyperChem, this trajectory may be saved and played back in the same way as a dynamics trajectory. With appropriate choices of setup parameters, the Monte Carlo method may achieve equilibration more rapidly than molecular dynamics. For some systems, then, Monte Carlo provides a more direct route to equilibrium structural and thermodynamic properties. However, these calculations can be quite long, depending upon the system studied. 

Goursaud, S., Sizun, M., and Fiquet-Fayard, F. (1976). Energy partitioning and isotope effects in the fragmentation of triatomic negative ions Monte Carlo Scheme for a classical trajectory study, J. Chem. Phys. 65, 5453-5461. 

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. 

The amount of Monte Carlo selection that has been employed in different studies has varied. For example, Blais and Bunker [48, 305] used a complete Monte Carlo procedure in their studies of the K + CH3I reaction, although the distribution of one or more parameters could be suppressed, allowing them to observe how particular results depended on different features of the collisions. On the other hand, Karplus et al. [20] adopted a rather different approach in their investigation of the H + Ha system. A batch of trajectories was calculated with particular values of v, b and vibrational and rotational energies of H2. The remainder of the variables were chosen by Monte Carlo methods. The vibrational and rotational energies corresponded to individual rotational states in the zero-point vibrational level. By averaging the results... 

Molecular dynamics uses classical mechanics to study the evolution of a system in time. At each point in time the classical equations of motion are solved for a system of particles (atoms), interacting via a set of predefined potential functions (force field), after which the solution obtained is applied to predict positions and velocities of the particles for a (short) step in time. This step-by-step process moves the system along a trajectory in phase space. Assuming that the trajectory has sampled a sufficiently large part of phase space and the ergodicity principle is obeyed, all properties of interest can then be computed by averaging along the trajectory. In contrast to the Monte Carlo method (see below), the MD method allows one to calculate both the structural and time-dependent characteristics of the system. An interested reader can find a comprehensive description of the MD method in the books by Allen and Tildesley or Frenkel and Smit. ... 

In Bunker s study, representative three-atom molecules were selected using a Monte Carlo method, after which the computer program followed the internal motions of the molecules by solving Newtonian equations of motion and determined the time it took for the molecules to break apart. A large number of molecules had to be considered because very few randomly chosen molecules came apart in a length of time that was practical." Over the next two years, 200 hours of computer time produced distributions of lifetimes for various model molecules. Out of more than 300,000 trajectories studied,... 

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