Trajectory Monte Carlo

A thermodynamically stable system conserves energy. Thus, by monitoring the potential energy one can confirm that a stable (and productive) phase of the simulation has begun. Absence of systematic drift in computed averages is often used as a check on the stability of a Monte Carlo trajectory. Fluctuations in the energy... 

Quasi-Harmonic Frequencies Calculated from Monte Carlo Trajectories on the Exact Potential Surface for Trans and Gauche Butane... 

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. 

Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck... 

Figure 38. Striped domain wall in a model of uniaxially compressed N2 monolayers on graphite (a) observed in Monte Carlo simulations of 52 X 12 molecules at 10 K and a coverage of 1.026 monolayers. Dots denote the centers of the honeycomb hexagons of the graphite basal plane, and crosses mark the mean positions of the molecular centers of mass. The inset of (a) shows the herringbone order in the commensurate region at the left and right boundaries of (a). The center-of-mass distribution in the region of the domain wall of (a) sampled from the Monte Carlo trajectory is magnified in (b). (Adapted from Fig. 1 of Ref. 283.)...

A local algorithm for Metropolis Monte Carlo trajectories must be constructed carefully. Due to the finite probability of exactly repeated states in these paths, the corresponding transition probability includes a singular term [see Eq. (1.14)]. The generation algorithm for local path moves must take this singularity into account properly. Appropriate acceptance probabilities are given in [5]. (H. C. Andersen has drawn our attention to an omission in [5]. In Metropolis Monte Carlo trajectories sequences of multiple rejections can occur. Attempts to modify time slices in the interior... 

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. 

Blais NC, Tmhlar DG (1973) Monte carlo trajectories dynamics of the reaction F -F D2 on a semiempirical valencebond potential energy surface. J Chem Phys 58(3) 1090... 

What is the relationship between the molecular dynamics simulations of a continuous model and an isothermal Monte Carlo trajectory of an otherwise similar discretized (or lattice) model When only local (and small distance) moves are applied in a properly controlled random (or rather pseudorandom) scheme, the discrete models mimic the coarse-grained Brownian dynamics of the chain. The Monte Carlo trajectory could be then interpreted as the numerical solution to a stochastic equation of motion. Of course, the short-time dynamics... 

A Monte Carlo update corresponds to the discrete time step A to in the simulation process. In order to reduce correlations, typically a number of updates are performed between measurements of a quantity O. This series of updates is called a sweep and the time passed in a single sweep is At =ATAto if the sweep consists of N updates. Thus, if M sweeps are performed, the discrete time series is expressed by the vector (0(Tinit + At), 0(Tiiut + 2At),. .., 0(xaat + wAt),. .., 0(Ti it + MAt)) and represents the Monte Carlo trajectory. The period of equilibration Tjnjt sets the starting point of the measurement. For convenience, we use the abbreviation 0 = 0(Tjnjt -b wAt) and... 

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. 

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