Trajectory Simulations Initial Conditions

An average rotational energy is computed by averaging over the longest vibrational period /vib of the product  

The standard deviation for (fp) may also be calculated. Inserting ( [j) into equation (27) gives the average vibrational energy. 

Levine and R. B. Bern.stein, Molecular Reaction Dynam-ic.s and Chemical Reactivity , Oxford University Pre.s.s, New York, 1987, pp. 36-44. 

Gutzwiller. Chaos in Cla.ssical and Quantum Mechanics , Springer-Verlag, New York, 1990, pp. 207-227. 

Potential Energy Surfaces. Molecular Structure and Reaction Dynamics , Taylor Franci.s, London, 1985, 

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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. 

As is well known in calculations of rare events [6, 16, 96], it is notoriously inefficient to start the simulation of trajectories with initial conditions sampled in one of the metastable states. Instead, initial conditions should be sampled at the transition state, which ensures that all trajectories cross the transition state at least once and thereby drastically improves the sampling of reactive events. In the model system (48), such an ensemble is given by... 

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. 

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. 

It is appropriate to consider first the question of what kind of accuracy is expected from a simulation. In molecular dynamics (MD) very small perturbations to initial conditions grow exponentially in time until they completely overwhelm the trajectory itself. Hence, it is inappropriate to expect that accurate trajectories be computed for more than a short time interval. Rather it is expected only that the trajectories have the correct statistical properties, which is sensible if, for example, the initial velocities are randomly generated from a Maxwell distribution. 

If the constant temperature algorithm is used in a trajectory analysis, then the initial conditions are constantly being modified according to the simulation of the constant temperature bath and the relaxation of the molecular system to that bath temperature. The effect of such a bath on a trajectory analysis is less studied than for the simulation of equilibrium behavior. 

Another approach is to run multiple MD simulations with different initial conditions. The recent work by Karplus and coworkers [19] observes that multiple trajectories with different initial conditions improve conformational sampling of proteins. They suggest that multiple trajectories should be used rather than a single long trajectory. 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

Reactive trajectories can be identified a priori, without any numerical simulation, if the moving separatrices are used instead of the standard dividing surfaces. In relative coordinates, the portion of the barrier ensemble that is forward reactive can immediately be identified from Fig. 3 those trajectories are reactive whose initial velocity is so large that it lies above both the stable and unstable manifolds. Because the initial conditions in the ensemble (49) lie at Ax(0) = —xj, this reactivity criterion reads explicitly... 

There are two important consequences of this equality for computer simulations of many-body systems. First, it means that statistically averaged properties of these systems are accessible from simulations that are aimed at generating trajectories -e.g., molecular dynamics, or ensemble averages such as Monte Carlo. Furthermore, for sufficiently long trajectories, the time-averaged properties become independent of the initial conditions. Stated differently, it means that for almost all values of qo, Po, the system will pass arbitrarily close to any point x, p, in phase space at some later time. 

US studies can produce informative free energy landscapes but assume that degrees of freedom orthogonal to the surface equilibrate quickly. The MD time needed for significant chain or backbone movement could exceed the length of typical US simulations (which are each typically on the nanosecond timescale). However, in spite of this caveat, US approaches have been very successful. One explanation for this success lies in the choice of initial conditions US simulations employ initial coordinates provided by high-temperature unfolding trajectories, which themselves have been found to yield predictive information about the nature of the relevant conformational space. 

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)... 

Consequently, we can ensure that the simulation process is correct. Note that if there is at least one positive Lyapunov exponent, trajectories obtained from two very close initial conditions diverge, and when all Lyapunov exponents are negative the same trajectories converge. A practical procedure to numerically determine the Lyapunov exponents is given in the Appendix. 

When potential surfaces are available, quasiclassical trajectory calculations (first introduced by Karplus, et al.496) become possible. Such calculations are the theorist s analogue of experiments and have been quite successful in simulating molecular reactive collisions.497 Opacity functions, excitation functions, and thermally averaged rate coefficients may be computed using such treatments. Since initial conditions may be varied in these calculations, state-to-state cross sections can be obtained, and problems such as vibrational specificity in the energy release of an exoergic reaction and vibrational selectivity in the energy requirement of an endo-... 

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. 

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