Classical trajectory simulations initial conditions

Classical Trajectory Simulations Final Conditions Classical Trajectory Simulations Initial Conditions Trajectory Simulations of Molecular Collisions Classical Treatment. 

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. 

In classical molecular dynamics simulations, reaction probabilities in general are determined by averaging over the results of many trajectories whose initial conditions are usually picked at random. The statistical uncertainty of the calculated reaction probabilities is then given by 1 /V N, where N is the number of calculated trajectories. This also means that it is computationally very demanding to determine small reaction probabilities since any calculated probability below 1 / JN is statistically not significant. 

By choosing the initial conditions for an ensemble of trajectories to represent a quantum mechanical state, trajectories may be used to investigate state-specific dynamics and some of the early studies actually probed the possibility of state specificity in unimolecular decay [330]. However, an initial condition studied by many classical trajectory simulations, but not realized in any experiment is that of a micro-canonical ensemble [331] which assumes each state of the energized reactant is populated statistically with an equal probability. The classical dynamics of this ensemble is of fundamental interest, because RRKM unimolecular rate theory assumes this ensemble is maintained for the reactant [6,332] as it decomposes. As a result, RRKM theory rules-out the possibility of state-specific unimolecular decomposition. The relationship between the classical dynamics of a micro-canonical ensemble and RRKM theory is the first topic considered here. 

Figure 6. Classical trajectory simulation of quantum evolution of the Q function for the same initial conditions and interaction times as in Fig. 4. In our simulation 10,000 trajectories were calculated.

The early classical trajectory simulations of cluster collisions were done around 1980. The focus of those studies was to investigate the dynamics of the initial stages of nucle-ation. More recently the focus has been on chemical reactions in clusters. The motivation for this work is to study solvation effects under conditions where the systems can be fully explored because of their relatively small sizes and simplicity. By varying the number of atoms and molecules in the clusters, it is possible to trace the behavior of chemical reactions fi-om the gas phase to the condensed phase, thus studying the evolution of changes in the dynamics as the effects of the medium go from none to liquid or crystalline. 

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 ... 

The classical trajectory simulations were carried out with VENUS interfaced with the semiempirical electronic structure theory computer program MOPAC. To simulate experimental conditions for (gly-H) -I-diamond collisions, the center of a beam of (gly-H)+ ion projectiles is aimed at the center of the surface, with fixed incident angle 0, and fixed initial translational energy, E,. The radius of the beam was chosen so that the beam overlapped a unit area on the surface. For each trajectory, the projectile was randomly placed in the cross section of this beam and then randomly rotated about its center of mass so that it had an initial random orientation with respect to the surface. The azimuthal angle, %, between the beam and a fixed plane perpendicular to the surface, was sampled randomly between 0 and 2n. Such a random sampling of X simulates collisions with different domains of growth on the diamond surface. 

In what is called BO MD, the nuclear wavepacket is simulated by a swarm of trajectories. We emphasize here that this does not necessarily mean that the nuclei are being treated classically. The difference is in the chosen initial conditions. A fully classical treatment takes the initial positions and momenta from a classical ensemble. The use of quantum mechanical distributions instead leads to a seraiclassical simulation. The important topic of choosing initial conditions is the subject of Section II.C. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

The master equation evolves the classical degrees of freedom on single adiabatic surfaces with instantaneous hops between them. Each single (fictitious) trajectory represents an ensemble of trajectories corresponding to different environment initial conditions. This choice of different environment coordinates for a given initial subsystem coordinate will result in different trajectories on the mean surface the average over this collection of classical evolution segments results in decoherence. Consequently, this master equation in full phase space provides a description in terms of fictitious trajectories, each of which accounts for decoherence. When the approximations that lead to the master equation are valid, this provides a useful simulation tool since no oscillatory phase factors appear in the trajectory evolution. 

The major problem in implementing classical trajectories is that the typical experiment samples a wide distribution of initial conditions. Since each trajectory is run for a specified set of initial coordinates and momenta, a large number of trajectories must be run with a wide range of initial conditions in order to simulate a particular experimental situation. An efficient choice of initial conditions is essential to getting useful information from trajectory calculations. 

As has been mentioned above, a new method for the treatment of the dynamics of mixed classical quantum system has been recently suggested by Jung-wirth and Gerber [50,51]. The method uses the classically based separable potential (CSP) approximation, in which classically molecular dynamics simulations are used to determine an effective time-dependent separable potential for each mode, then followed by quantum wave packet calculations using these potentials. The CSP scheme starts with "sampling" the initial quantum state of the system by a set of classical coordinates and momenta which serve as initial values for MD simulations. For each set j (j=l,2,...,n) of initial conditions a classical trajectory [q (t), q 2(t),..., q N(t)] is generated, and a separable time-dependent effective potential V (qj, t) is then constructed for each mode i (i=l,2,...,N) in the following way ... 

In the classical trajectory approach, if a potential energy surface is available, one prescribes initial conditions for a particular trajectory. The initial variables are selected at random from distributions that are representative of the collisions process. The initial conditions and the potential energy function define a classical trajectory which can be obtained by numerical integration of the classical equations of motion. Then another set of initial variables is chosen and the procedure is repeated until a large number of trajectories simulating real collision events have been obtained. The reaction parameters can be obtained from the final conditions of the trajectories. Details of this technique are given by Bunker.29... 

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