Collision dynamics scattering cross-section

Scattering cross sections for chemical reactions may exhibit structure due to resonance or due to other dynamical effects such as interference or threshold phenomena. It is useful to have techniques that can identify resonance behavior in theoretical simulations and distinguish it from other sorts of dynamics [67]. Since resonance is associated with dynamical trapping, the concept of the collision time delay proves quite useful in this regard. Of course since collision time delay for chemical reactions is typically in the subpicosecond domain, this approach is, at present, only useful in analyzing theoretical scattering results. Nevertheless, time delay is a valuable tool for the theoretical identification of reactive resonances. 

D. G. Truhlar and J. T. Muckerman, Reactive scattering cross sections. Ill Quasiclassical and semiclassical methods, Atom-Molecule Collision Theory (R. B. Bernstein, ed.), Plenum, New York, 1979 L. M. Raff and D. L. Thompson, The classical approach to reactive scattering, Theory of Chemical Reaction Dynamics (M. Baer, ed.), CRC, Boca Raton, FL, 1985. 

Raff, LM, Thompson DL. The classical approach to reactive scattering. In Baer M, Editor. Theory of chemical reaction dynamics, Vol. 3. Boca Raton CRC Press 1985. pp. 1-121. Truhlar, DG, Muckerman JT. Reactive scattering cross section. Ill Quasiclassical and semi-classical methods. In Bemtein RB, Editor. Atom-molecule collision theory. New York Plenum Press 1979, pp. 505 6. 

The END trajectories for the simultaneous dynamics of classical nuclei and quantum electrons will yield deflection functions. For collision processes with nonspherical targets and projectiles, one obtains one deflection function per orientation, which in turn yields the semiclassical phase shift and thus the scattering amplitude and the semiclassical differential cross-section... 

Experiments have also played a critical role in the development of potential energy surfaces and reaction dynamics. In the earliest days of quantum chemistry, experimentally determined thermal rate constants were available to test and improve dynamical theories. Much more detailed information can now be obtained by experimental measurement. Today experimentalists routinely use molecular beam and laser techniques to examine how reaction cross-sections depend upon collision energies, the states of the reactants and products, and scattering angles. 

It is clear that the unmistakable resonance fingerprint provided by a narrow Lorentzian peak in the integral cross section (ICS) will be rare for reactive resonances in a collision experiment. However, a fully resolved scattering experiment provides a wealth of data concerning the reaction dynamics. We expect that the state-to-state differential cross sections (DCS) as functions of energy can be analyzed, using various methods, to reveal the presence of reactive resonances. In the following subsections, we discuss how various collision observables are influenced by existence of a complex intermediate. Many of the resonance detection schemes that have been proposed, such as the use of collision time delay, are purely theoretical in that the observations required are not currently feasible in the laboratory. Nevertheless, these ideas are also discussed since it is useful to have method available... 

The outcome of an isolated (microscopic) reactive scattering event can be specified in terms of an intrinsic fundamental quantity the reaction cross-section. The cross-section is an effective area that the reactants present to each other in the scattering process. It depends on the quantum states of the molecules as well as the relative speed of the reactants, and it can be calculated from the collision dynamics (to be described in Chapter 4). 

The aim is to establish the relation between the observable cross-sections and the collision dynamics. We denote the scattering state in the interaction region at t = 0 by x) and write the Hamiltonian in the form Hc.m. + Hre, i.e., the Hamiltonians associated with the center-of-mass motion and the relative motion. The propagator can be written in the form U(t) = exp(—iHc.mt/h)exp(—iHre t/h), and x(t)) = [/(f) x) describes the time-dependent scattering state at any time, i.e. (il x(f)) is the associated wave packet. 

A proper description of the dynamics in a collision is required to obtain stopping cross sections that compare well with experimental results over a wide range of projectile energies and that include all the channels available during the collision. Here we have shown that the END approach fulfills this requirement. The proper inclusion of non-adiabatic effects, even at the minimal level of END, provides trajectories for the projectile suitable for the calculation of scattering problems. We have... 

Resonance phenomena have been shown to play a significant role in many electron collision and photoionization problems. The long lived character of these quasi-stationary states enables them to influence other dynamic processes such as vibrational excitation, dissociative attachment and dissociative recombination. We have shown it is possible to develop ab initio techniques to calculate the resonant wavefunctions, cross sections and dipole matrix elements required to characterize these processes. Our approach, which is firmly rooted in the R-matrix concept, reduces the scattering problem to a matrix problem. By suitable inversion or diagonalization we extract the required resonance parameters. 

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