Trajectory calculations numerical experiments

A model, classical trajectory calculation for collinear H + Hg Hg + H, using a realistic potential function, was set up as an experiment in the application of programming skill and of Runge-Kutta numerical integration methods [J. M. White, Physical Chemistry Laboratory Experiments (Englewood Cliffs, N.J. Prentice-Hall, 1975), pp. 301-326]. 

Let us briefly discuss this approach. Its idea is the comparison of the experimentally obtained distribution with the so-called the prior distribution, which would be obtained under the condition of the equiprobable energy distribution over all degrees of freedom of the products. Prior distributions, which we designate as p° ( V) and / (n), have a maximum entropy and, hence, give the minimum of information about the dynamics of the reaction. The real distribution obtained in experiment has lower entropy than that of the a priori one. As model trajectory calculations and analysis of data of numerous experiments show, the distribution functions over rotational P N) and vibrational P n) states can be expressed through the corresponding a priori distributions as follows ... 

In this chapter we elucidate the state-specific perspective of unimolec-ular decomposition of real polyatomic molecules. We will emphasize the quantum mechanical approach and the interpretation of the results of state-of-the-art experiments and calculations in terms of the quantum dynamics of the dissociating molecule. The basis of our discussion is the resonance formulation of unimolecular decay (Sect. 2). Summaries of experimental and numerical methods appropriate for investigating resonances and their decay are the subjects of Sects. 3 and 4, respectively. Sections 5 and 6 are the main parts of the chapter here, the dissociation rates for several prototype systems are contrasted. In Sect. 5 we shall discuss the mode-specific dissociation of HCO and HOCl, while Sect. 6 concentrates on statistical state-specific dissociation represented by D2CO and NO2. Vibrational and rotational product state distributions and the information they carry about the fragmentation step will be discussed in Sect. 7. Our description would be incomplete without alluding to the dissociation dynamics of larger molecules. For them, the only available dynamical method is the use of classical trajectories (Sect. 8). The conclusions and outlook are summarized in Sect. 9. 

In order to demonstrate the physical significance of asymjjtotic nonadiabatic transitions and especially the aiialj-tical theory developed an application is made to the resonant collisional excitation transfer between atoms. This presents a basic physical problem in the optical line broadening [25]. The theoretical considerations were mad( b( for< [25, 27, 28, 29, 25. 30] and their basic id( a has bec n verified experimentally [31]. These theoretical treatments assumed the impact parameter method and dealt with the time-dependent coupled differenticil equations imder the common nuclear trajectory approximation. At that time the authors could not find any analytical solutions and solved the coupled differential equations numerically. The results of calculations for the various cross sections agree well with each other and also with experiments, confirming the physical significance of the asymptotic type of transitions by the dipole-dipole interaction. 

The KAM theorem, it should be noted, has nothing to say about what happens when the strength of the perturbation increases. However, a considerable amount of experience has accumulated from detailed numerical calculations performed for many systems. One can visualise the results by studying Poincare sections if a cut is made across an invariant torus (see fig. 10.3) and a numerical calculation of trajectories is performed over a sufficiently long time, the stable orbits fill the deformed tori densely, and so result in closed curves in the two-dimensional cut, whereas the irregular or chaotic orbits yield a random speckle. 

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. 

The paper is arranged in the following way. In the next section a few aspects of the RIOSA which are relatively new are discussed and a detailed procedure is given for doing the actual calculations. The third section deals with numerical results as obtained by us in the last five years. However, in contrast to what has been published elsewhere, we concentrate only on those results which can be compared either with findings obtained by reliable methods, i.e. exact quantum mechanical (EQM), coupled states (CS) and quasiclassical trajectory (QCT) methods or with experiment. A discussion and a summary are given in the last section. 

A correct calculation of solvation thermodynamics and solution structure is conceivable only in terms of the methods of statistical physics, in particular, the computer experiment schemes, including, in the first place, the molecular dynamics (MD) and the Monte-Carlo (MC) methods [10]. By means of the MD method Newton s classic equations of motion are solved numerically with the aid of a computer assuming that the potential energy of molecular interaction is known. In this manner, the motion of molecules of the liquid may be observed , the phase trajectories found and then the values of the necessary functions are averaged over time and determined. This method permits both the equilibrium and the kinetical properties of the system to be calculated. 

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