Trajectory simulation

Born-Oppenheimer Direct Dynamics Classical Trajectory Simulations 

Department of Chemistry and Department of Computer Science, Wayne State University, Detroit, Michigan 48202 

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  

Reviews in Computational Chemistry, Volume 19 edited by Kenny B. Lipkowitz, Raima Latter, and Thomas R. Cundari ISBN 0-471-23585-7 Copyright 2003 Wiley-VCH, John Wiley Sons, Inc. 

For the most general case (see section on integrating classical equations of motion), T depends on both the momenta p and coordinates q. The index i in the equations above is the number of coordinates or conjugate momenta for the Hamiltonian. If Cartesian coordinates are used, this number is 3N, where N is the number of atoms. 

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For some systems qiiasiperiodic (or nearly qiiasiperiodic) motion exists above the unimoleciilar tlireshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low uninioleciilar tliresholds, widely separated frequencies and/or disparate masses [12,, ]. Thus, classical trajectory simulations perfomied for realistic Hamiltonians predict that, for some molecules, the uninioleciilar rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. 

Apparent non-RRKM behaviour occurs when the molecule is excited non-randomly and there is an initial non-RRKM decomposition before IVR fomis a microcanonical ensemble (see section A3.12.2). Reaction patliways, which have non-competitive RRKM rates, may be promoted in this way. Classical trajectory simulations were used in early studies of apparent non-RRKM dynamics [113.114]. 

Lenzer T, Luther K, Troe J, Gilbert R G and Urn K F 1995 Trajectory simulations of collisional energy transfer in highly excited benzene and hexafluorobenzene J. Chem. Phys. 103 626-41... 

Summarizing, from a mathematical point of view, both forward and backward analysis lead to the insight that long term trajectory simulation should be avoided even with symplectic discretizations. Rather, in the spirit of multiple as opposed to single shooting (cf. Bulirsch [4, 18]), only short term trajectories should be used to obtain reliable information. 

Here, we give here a brief outline of the methods as introduced in Refs. 43, 44, and 47. Suppose that the initial state of the system is tpoili, ,<]n) From ipo, the Wigner phase-space distribution D(qi,..., qn pi,. ..,Pn] is computed. This distribution is used to sample initial positions and momenta, . .., for a classical trajectory simulation of the process of... 

The first energy derivative is called the gradient g and is the negative of the force F (with components along the a center denoted Fa) experienced by the atomic centers F = -g. These forces, as discussed in Chapter 16, can be used to carry out classical trajectory simulations of molecular collisions or other motions of large organic and biological molecules for which a quantum treatment of the nuclear motion is prohibitive. 

Monte Carlo electron trajectory simulations provide a pictorial view of the complei electron—specimen interaction. As shown in Figure 2a, which depicts the interac-... 

Macromolecular fluctuations are characterized by correlation times which are closely related to the underlined relaxation processes. Such relaxation times can be evaluated by classical trajectory simulations using... 

Since the forward peak is clearly from high J collisions, it is clearly produced via a rapidly rotating intermediate exhibiting an enhanced time delay. Further insight into the associated dynamics is provided by a classical trajectory simulation by Skodje. The forward peak results from the sideway collisions of the H atom on the HD-diatom (see Fig. 37). At the point where the transition state region is first reached, the collision complex is already oriented about 70° relative to the center-of-mass collision axis. The intermediate then rotates rapidly with an angular frequency of u> J/I, where / is the moment of inertia of the intermediate. If the intermediate with a time delay of the order of the lifetime r, the intermediate can rotate... 

A dynamical model for SN2 nucleophilic substitution that emerges from the trajectory simulations is depicted in Figure 9. The complex formed by a collision between the reactants is an intermolecular complex CinterR. To cross the central barrier, this complex has to undergo a unimolecular transition in which energy is... 

Hu, X. and Martens, C. C. Classical-trajectory simulation of the cluster-atom association reaction... 

Lipeng Sun and William L. Hase, Born-Oppenheimer Direct Dynamics Classical Trajectory Simulations. 

More informative are the stochastic trajectory simulations run by Muhl-hausen et al. (M WT), on empirical interaction potential surfaces for scattering and desorption Although the major thrust was to understand the direct beam scattering results of NO/Ag(l 11), extension of these calculations allows for comparison to the desorption of NO from Pt(lll) Important insights derived from the NO/Ag(lll) calculations were ... 

Fig. 4.36. Projection of a 3D trajectory simulation of a stable ion onto the x- and y-coordinate. Reproduced from Ref. [110] with permission. Elsevier Science, 1998.

Ion trajectory simulations allow for the visualization of the ion motions while travelling through a quadrupole mass analyzer (Fig. 4.36). Furthermore, the optimum number of oscillations to achieve a certain level of performance can be determined. It turns out that best performance is obtained when ions of about 10 eV kinetic energy undergo a hundred oscillations. [110]... 

D. Shemesh and R. B. Gerber. Classical trajectory simulations of photoionization dynamics of tryptophan intramolecular energy flow, hydrogen-transfer processes and conformational transitions, J. Phys. Chem. A, 110 8401-8408 (2006). 

The classical trajectory simulations of Rydberg molecular states carried out by Levine ( Separation of Time Scales in the Dynamics of High Molecular Rydberg States, this volume) remind me of the related question asked yesterday by Prof. Woste (see Berry et a]., Size-Dependent Ultrafast Relaxation Phenomena in Metal Clusters, this volume). Here I wish to add that similar classical trajectory studies of ionic model clusters of the type A B have been carried out by... 

Instead, classical trajectory simulations are performed to determine the fraction of trajectories crossing the dividing surface that actually contribute to the formation of product molecules [5,6]. If the surface is the optimal one corresponding to a minimum value for the rate constant, all trajectories crossing the dividing surface from the reactant side to the product side will lead to the formation of products. If not, a certain fraction of the trajectories crossing the dividing surface will turn around, recross the surface, and therefore not make a contribution to the formation of products. 

Besides the chamber diameter, the width of the apertures is an important structural parameter. It affects the density (pressure) of the sample gas and the overall gas flow into the system. Ion trajectory simulations identified an optimum ion yield for entrance and exit orifices of 40 pm and a chamber diameter of 320 pm [19], as experimentally validated in the real systems. 

Figure 7 shows the structures of the ion optics with the ionization chamber on the left and the entrance of the mass-analyzer on the right, which are described in the next section. The arrangement and shape of these structures were also computed via ion trajectory simulations. A result is shown in Fig. 8. At an extraction voltage (IE) of -25 V and separator input voltage (WFP) of 0 V an ion focus voltage (IFO) of 67 V generates the highest ion yield for the separator.

Fig. 10 Trajectory simulations of the ions inside the energy filter with a focal point in the center of the outlet aperture (a) and with a focal point before the outlet aperture (b)...

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