Classical trajectories Monte Carlo sampling

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. 

Peslherbe, G. H. Wang, H. Hase, W. L. Monte Carlo sampling for classical trajectory simulations, Adv. Chem. Phys. 1999,105,171-201. 

MONTE CARLO SAMPLING FOR CLASSICAL TRAJECTORY SIMULATIONS... 

MONTE CARLO SAMPLING TOR CLASSICAL TRAJECTORY SIMULATIONS 195... 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix formulation yields a consistent treatment of electronic populations and coherences, and the momentum changes associated with an electronic transition can be directly derived from the formalism without the need of ad hoc assumptions. Employing a Monte-Carlo sampling scheme of local classical trajectories, however, we have to face two major complications, that is, the representation of nonlocal phase-space operators and the sampling problem caused by rapidly varying phases. At the present time, the... 

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),... 

A chemical reaction cannot be described by only two particles interacting. A minimum of three particles is necessary. Unfortunately, for such three-bodied systems it is not possible to obtain analytical solutions and it becomes necessary to use numerical methods. Among these, the Monte Carlo method assumes particular relevance for the calculation of classical trajectories using random sampling. 

Since the early 1960s classical trajectory simulations, with Monte Carlo sampling of initial conditions, have been widely used to study the uni-molecular and intramolecular dynamics of molecules and clusters reactive and nonreactive collisions between atoms, molecules, and clusters and the collisions of these species with surfaces. " For a classical trajectory study of a system, the motions of the atoms for the system under study are determined by numerically integrating the system s classical equations of motion. These equations are usually expressed in either Hamilton s form ... 

Procedures for selecting initial values of coordinates and momenta for an ensemble of trajectories has been described in detail in recent chapters entitled Monte Carlo Sampling for Classical Trajectory Simulations and Classical Trajectory Simulations Initial Conditions. In this section a brief review is given of methods for selecting initial conditions for trajectory simulations of unimolecular and bimolecular reactions and gas-surface collisions. 

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]. 

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. ... 

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