Classical trajectory Monte Carlo calculations

Fig. 4.26. Plot of the ratio aPS/aH for positron and proton collisions with helium, versus the velocity of the projectiles (following Schultz and Olson, 1988). The curve is the result of their classical trajectory Monte Carlo calculation. The triangles are based on the positron data of Fromme et al. (1986) whilst the circles are from Diana et al. (1986b).

Classical-trajectory Monte Carlo calculation for collision processes of e++(e-p) and pA + p). Phys. Rev. A 32 2640-2644. 

Classical Trajectory Monte Carlo Calculations (CTMC) have been reviewed recently by Schulz et al. [5.24]. Classical treatments of collisions retain with complete kinematic accuracy many of the features which we think of as the physics of a reaction. Their advantage is they are very flexible and can be quickly changed to describe many varied experimental systems. They are however not inexpensive in terms of computing time essentially statistical in nature, it is quite time consuming to produce meaningful cross sections for unlikely events. 

Fig. 13.6 Classical trajectory Monte Carlo (CTMC) ionization and charge transfer cross sections, with statistical standard deviations, for specific initial f plotted against reduced impact speed v/ve. The cross sections are given in units of a 2, where a = rraQ. Circles are for = 2, squares for i = 14. Also included for comparison are points from approximate CTMC calculations with the Na target core held fixed for i = 14 (triangles) and ( = 2...

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 MD/QM methodology [18] is likely the simplest approach for explicit consideration of quantum effects, and is related to the combination of classical Monte Carlo sampling with quantum mechanics used previously by Coutinho et al. [27] for the treatment of solvent effects in electronic spectra, but with the variation that the MD/QM method applies QM calculations to frames extracted from a classical MD trajectory according to their relative weights. 

While the two methods are, at face value, quite different in the ways in which full quantum dynamics is reduced to quantum-classical dynamics, there are common elements in the manner in which they are simulated. The Trotter-based scheme for QCL dynamics makes use of the adiabatic basis and is based on surface-hopping trajectories where transitions are sampled by a Monte Carlo scheme that requires reweighting. Similarly, ILDM calculations make use of the mapping hamiltonian basis and also involve a similar Monte Carlo sampling with reweighting of trajectories in the ensemble used to obtain the expectation values of quantum operators. 

The Monte Carlo method includes both a random sampling scheme and an importance sampling scheme. Both sampling schemes have been used in Section 4.1 on classical trajectory calculations. 

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]. 

These experiments are important because they are performed on a reaction for which a priori calculations of V(rAB, rBC, rCA) are likely to have their best chance of success as only three electrons are involved. Even here the accurate computation of V, frequently termed the potential-energy hypersurface, is extremely difficult. Porter and Karplus [19] have determined a semiempirical hypersurface, and Karplus, Porter, and Sharma [20] have calculated classical trajectories across it. This type of computer experiment has been mentioned before and will be described in greater detail later. The objective of Karplus et al. was to calculate aR(E) and E0. Collisions were therefore simulated at selected values of E, with other collision parameters selected by Monte Carlo procedures, and the subsequent trajectories were calculated using the classical equations of motion. Above E0, oR was found to rise to a maximum value, of the same order of magnitude as the gas-kinetic cross section, and then gradually to decrease to greater energies. 

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. ... 

Following the construction of the model is the calculation of a sequence of states (or a trajectory of the system). This step is usually referred to as the actual simulation. Simulations can be stochastic (Monte Carlo) or deterministic (Molecular Dynamics) or they can combine elements of both, like force-biased Monte Carlo, Brownian dynamics or general Langevin dynamics (see Ref. 16 for a discussion). It is usually assumed that the physical system can be adequately described by the laws of classical mechanics. This assumption will alsq be made throughout the present work. 

More recently, Karplus and Raff [39] have carried out extensive computer experiments for the reaction between K and CH3I and have found that the dimensionality of the calculation and the details of the potential are of critical importance for some considerations. They used a potential energy function similar to that employed by Blais and Bunker and calculated both two- and three-dimensional classical trajectories. As usual, the initial conditions were selected by use of a Monte Carlo procedure, and the trajectories were examined to see if reaction occurs. In both dimensions, most of the reaction energy appears as internal excitation of products. The calculated differential cross sections are markedly different for two and three dimensions and both disagree with the molecular beam experiments. The total reaction cross section obtained from the computer calculations is about 400 in both two and three dimensions, as compared to the experimental result of about 10 A. Karplus and Raff found the situation could be greatly improved by... 

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