Some Remarks on How to Select Initial Conditions

To start the mathematical integration of the equations of motion for one particular trajectory, a set of initial values of coordinates and either velocities or momenta must be specified. These, however, are dependent on the experimental conditions which need be reproduced, such as collision energy, intramolecular vibrational energies etc... In addition, some other variables, for instance intramolecular instantaneous elongations, molecular orientations, impact parameter, etc. are necessarily specified in classical mechanics but are not observable microscopically because of the Uncertainty Principle. The ensemble of these result in a set of trajectories associated with a given set of observable initial conditions. 

Many trajectories are necessary to describe all the different events that are summed up to form a unique wave describing the global chemical reaction under observable conditions in quantum mechanics. In this respect, a set of classical trajectories which spread around a mean trajectory in classical mechanics corresponds roughly to the quantum mechanical spreading (through space or time) of the density probability function around its center. 

Since the number of trajectories is necessarily limited (in particular when computing the forces requires a great amount of computer time), good criteria for selecting the initial conditions are of prime interest. The problem is so important that it was discussed at length several times before In this section we just consider what can be done in the case of constrained systems. The approximation of the constraints is so rough as to eliminate the need for any of the refined corrections that allowed to get very accurate results for small free molecular systems (cf. Chapter A). 

Several domains of application need to be distinguished. In the thermal chemistry of unimolecular reactions, a complex process, which is not known to us, leads to the 

In the thermal chemistry of bimolecular reactions studied within the framework of trajectories calculations, it is normally possible to obtain the rate constant of the reaction as a function of temperature(cf. Chapter A). However, this can require rather tedious calculations, because of the necessity of averaging the results over a large distribution of individual trajectories. We do not discuss further the problem of bimolecular reactions. 

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