Collisional trajectories

The equations of motion for granular flows have been derived by adopting the kinetic theory of dense gases. This approach involves a statistical-mechanical treatment of transport phenomena rather than the kinematic treatment more commonly employed to derive these relationships for fluids. The motivation for going to the formal approach (i.e., dense gas theory) is that the stress field consists of static, translational, and collisional components and the net effect of these can be better handled by statistical mechanics because of its capability for keeping track of collisional trajectories. However, when the static and collisional contributions are removed, the equations of motion derived from dense gas theory should (and do) reduce to the same form as the continuity and momentum equations derived using the traditional continuum fluid dynamics approach. In fact, the difference between the derivation of the granular flow equations by the kinetic approach described above and the conventional approach via the Navier Stokes equations is that, in the latter, the material properties, such as viscosity, are determined by experiment while in the former the fluid properties are mathematically deduced by statistical mechanics of interparticle collision. 

As indicated above, we have assumed that the molecules of B are stationary to obtain the above expression. In practice, Figure 5.1 shows that, for each pair of molecules, A and B, involved in a collisional trajectory, we can define a relative velocity v g, which is related to their velocities, and Vg, according to... 

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... 

Grigoleit U, Lenzer T and Luther K 2000 Temperature dependence of collisional energy transfer in highly excited aromatics studied by classical trajectory calculations Z. Phys. Chem., A/F214 1065-85... 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

Reactants AB+ + CD are considered to associate to form a weakly bonded intermediate complex, AB+ CD, the ground vibrational state of which has a barrier to the formation of the more strongly bound form, ABCD+. The reactants, of course, have access to both of these isomeric forms, although the presence of the barrier will affect the rate of unimolecular isomerization between them. Note that the minimum energy barrier may not be accessed in a particular interaction of AB+ with CD since the dynamics, i.e. initial trajectories and the detailed nature of the potential surface, control the reaction coordinate followed. Even in the absence (left hand dashed line in Figure 1) of a formal barrier (i.e. of a local potential maximum), the intermediate will resonate between the conformations having AB+ CD or ABCD+ character. These complexes only have the possibilities of unimolecular decomposition back to AB+ + CD or collisional stabilization. In the stabilization process,... 

Figure 2. Simplified picture of atom-atom collisional ionization with crossing distance r. Heavy solid lines represent trajectories of neutral systems. At the first crossing (r= rj some fraction (1 - PJ of trajectories make adiabatic transitions and are represented by dashed lines (ion pairs). Those making diabatic transitions remain neutral and continue their flight relatively unaffected. Each of these trajectories then encounters r = r<- again, and again each trajectory can make an adiabatic or diabatic transition, resulting in ion pairs or neutrals depending on the trajectory. The ultimate production of ions requires one transition to be diabatic and one to be adiabatic, in either order. The inner circle represents the repulsive core.

Hase s trajectory value for the association rate constant, /cp of 1.04 cm- s maybe used in conjunction with the above Langevin value of the collisional stabilization rate constant to yield a unimolecular dissociation rate constant of 3.75 x 10 ° s and a lifetime of 27 ps. In each case, these values are in excellent agreement with the order of magnitude of lifetimes predicted by Hase s calculations for cr/CHjCl collisions at relative translational energies of 1 kcal mor , rotational temperatures of 300 K, and vibrational energies equal to the zero-point energy of the system. 

The reaction of trimethylene biradical was successfully treated by means of dynamics simulations by two groups with different PESs as described above.11 15 The success led one of the groups to extend the study to analyze the collisional and frictional effects in the trimethylene decomposition in an argon bath.16 A mixed QM/MM direct dynamics trajectory method was used with argon as buffer medium. Trimethylene intramolecular potential was treated by AM1-SRP fitted to CASSCF as before, and intermolecular forces were determined from Lennard-Jones 12-6 potential energy functions. 

Kaukonen et al. (1991) reported classical trajectory calculations for [Na4Cl3]+Ar + Cl with n = 12 and 32 and for [Na14Cl12] + 2Ar30 + Cl-. Their results showed that it is possible to tune the relative probabilities for different product isomers by varying the initial vibrational temperature of the reactants and relative translations energy between collisional partners and/or the number of embedding Ar atoms. 

Most of our present understanding of the dynamics and of the collisional mechanisms of elementary chemical reactions comes from classical approaches, from simple classical models, and from quasiclassical trajectory studies. More recently, quantum mechanical results on the dynamics of directmode reactions have become available. 

In this article, particular attention has been paid to the results of laboratory experiments that have provided information about the molecular dynamics of electronically adiabatic atom-exchange reactions. For a reaction like A + BC - AB + C, where the possibility of crossing between different electronic states can safely be ignored, a theoretical study comprises three distinct stages [20] (a) the determination of the potential-energy hypersurface, (b) the solution of the equations that describe the motion of A, B, and C under the influence of V, and (c) the averaging of the results of these trajectory calculations so that they reflect the distribution of collisional... 

In order to obtain properly averaged results, either collision parameters for each trajectory must be selected by Monte Carlo methods or, when starting values are systematically chosen, the final results must be integrated over complete statistical distributions. The purpose of a Monte Carlo selection technique is to ensure that the distributions of each parameter within a sample of trajectories approach the true statistical distributions as the size of the sample grows. Some examples of how this can be done for different types of distribution function will be described below. Before starting the integration, it is generally necessary to transform the selected values of the collisional... 

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