Classical trajectory simulations

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]. 

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. 

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... 

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. 

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. 

In conclusion, these gas-phase measurements provide new elues to the role of solvation in ion-moleeule reaetions. For the first time, it is possible to study intrinsie reactivities and the extent to which the properties of gas-phase ion-moleeule reaetions relate to those of the eorresponding reactions in solution. It is clear, however, that gas-phase solvated-ion/moleeule reaetions in which solvent moleeules are transferred into the intermediate elusters by the nucleophile cannot be exaet duplieates of solvated-ion/ molecule reactions in solution in which solvated reactants exchange solvent molecules with the surrounding bulk solvent [743]. For a selection of more recent experimental [772] and theoretical studies of Sn2 reactions in gas phase and solution by classical trajectory simulations [773], molecular dynamics simulations [774, 775], ab initio molecular orbital calculations [776, 777], and density functional theory calculations [778, 779], see the references given. For studies of reactions other than Sn2 ion-molecule processes in the gas phase and in solution, see reviews [780, 781]. 

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

Initiated by the pioneering work of Bunker [323,324] classical trajectory simulations have been extensively used to study the decomposition of energized molecules. In a unimolecular classical trajectory study, the motions of atoms for an ensemble of molecules are simulated by solving their classical equations of motion, usually in the form of Hamilton s equations, i.e.,... 

As discussed in the previous sections, to have the most complete comparison with experiment it is important to use an accurate potential energy function U(q), usually obtained from high-level electronic structure theory calculations. In the past, this potential has always been represented by an analytical function. However, with the extraordinary increase in computer speed and enhancement in computer algorithms, it is now possible to calculate trajectories with the derivatives of the potential, dV/dqi in Eq. (60), obtained directly from an electronic structure theory, without the need for an analytical functional fit. Such on-the-fly calculations are called direct dynamics classical trajectory simulations [325]. They are particularly important for molecules with many degrees of freedom, for which the construction of analytical PES s is impossible. 

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 9. Snapshots of the phase space distribution (PSD) obtained from classical trajectory simulations based on the fewest-switches surface-hopping algorithm of a 50 K initial canonical ensemble [46], Na atoms are indicated by black circles, and F atoms are indicated by gray crosses. Dynamics on the hrst excited state starting at the Cj structure (t = 0 fs) over the structure with broken Na-Na bond t = 90 fs) and subsequently over broken ionic Na-F bond (t = 220 fs) toward the conical intersection region (t = 400 fs), Dynamics on the ground state after branching of the PSD from the hrst excited state leads to strong spatial delocalization (t = 600 fs). The C2v isomer can be identihed at 800 fs in the center-of-mass distribution. See color insert.

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.

Fig. 15.5. Poincare surface of section for a two-dimensional model of HOCl the HO distance is frozen in the classical trajectory simulations. The symbol y is the Jacobi angle and py is the corresponding momentum. Different symbols in square brackets denote different types of regular trajectories. Adapted from Ref. [66].

Direct dynamics simulations, in which the methodology of classical trajectory simulations is coupled to electronic structure, have had and will continue to have an enormous impact on the use of computational chemistry to develop [111,112] the theory of unimolecular kinetics. In these simulations the derivatives of the potential, required for numerically integrating the classical trajectory, are obtained directly from electronic stmcture theory without the need for an analytic PES. Direct dynamics is particularly important for studying the unimolecular dynamics of molecules with many degrees of freedom, for which it is difficult to construct an accurate analytic PES. 

W. H. Miller, W. L. Hase, and L. Darling, A. simple model for correcting the zero point energy problem in classical trajectory simulations of polyatomic molecules, J. Chem. Phys. 91 2863 (1989). 

MONTE CARLO SAMPLING FOR CLASSICAL TRAJECTORY SIMULATIONS... 

There are several components to a classical trajectory simulation [1-4]. A potential-energy function F(q) must be formulated. In the past F(q) has been represented by an empirical function with adjustable parameters or an analytic fit to electronic structure theory calculations. In recent work [6] the potential energy and its derivatives dV/dqt have been obtained directly from an electronic structure theory, without an intermediate analytic fit. Hamilton s equations of motion [Eq. (1.1)] are solved numerically and numerous algorithms have been developed and tested for doing this is an efficient and accurate manner [1-4]. When the trajectory is completed, the final values for the momenta and coordinates are transformed into properties that may be compared with experiment, such as product vibrational, rotational, and relative translational energies. 

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