Exact Quantum-Mechanical Calculation of a Collinear Collision of a Particle with a Harmonic Oscillator

103

It has been observed in the past that the usual Airy uniform approximation gives probabilities greater than one, especially for near elastic collisions. By mapping the phase onto -t cosy + ky +A rather than (1!3)y 3 -tY + A one obtains a uniform appr~ximation involving Bessel functions of the first kind, which approaches unity for the elastic collision. This Bessel uniform approximation is no more complicated than the Airy and also gives good agreement with exact quantum results, even if probabilities are large.

Section: Application To a Nonreactive Systemmentioning

confidence: 99%

Semiclassical transition probabilities by an asymptotic evaluation of the S matrix for elastic and inelastic collisions. Bessel uniform approximation

Stine

Marcus

1973

103

“…[15][16][17][18][19] However, it has also been established for some time that perturbation theory can fail for highfrequency oscillators in gas-phase collisions. 20 More recent studies have found that the conventional perturbation theory approach 14 gives lifetimes that differ significantly from experimental measurements in both clusters 21 and liquids. 22,23 This has been attributed to the potential energy surfaces 23 and, more significantly, to the need to modify the conventional perturbation theory approach using quantum correction factors, [23][24][25][26] i.e., by choosing an appropriate q(T).…”

Section: ͑11͒mentioning

confidence: 99%

A general method for implementing vibrationally adiabatic mixed quantum-classical simulations

Thompson

2003

An approach for carrying out vibrationally adiabatic mixed quantum-classical molecular dynamics simulations is presented. An appropriate integration scheme is described for the vibrationally adiabatic equations of motion of a diatomic solute in a monatomic solvent and an approach for calculating the adiabatic energy levels is presented. Specifically, an iterative Lanczos algorithm with full reorthogonalization is used to solve for the lowest few vibrational eigenvalues and eigenfunctions. The eigenfunctions at one time step in a mixed quantum-classical trajectory are used to initiate the Lanczos calculation at the next time step. The basis set size is reduced by using a potential-optimized discrete variable representation. As a demonstration the problem of a homonuclear diatomic molecule in a rare gas fluid (N 2 in Ar͒ has been treated. The approach is shown to be efficient and accurate. An important advantage of this approach is that it can be straightforwardly applied to polyatomic solutes that have multiple vibrational degrees-of-freedom that must be quantized.

“…See Appendix A for details. Using the propagation method, we obtained approximately three-place accuracy in any probability (squared amplitude of a solution vector element) greater than w- 6 • We tested the accuracy of our solution vectors in two ways. First, a vector's probabilities should sum to 1; our sum values were always one to four decimal places.…”

Section: Methodsmentioning

confidence: 99%

“…The Hamiltonian JC for the collision of A striking B 2 is 6 : (4) where M=mA/(mA+2mB). of energy and length are fiw and one-half the classical ground state vibrational amplitude, respectively.…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanical calculations of rotational-vibrational scattering in homonuclear diatom-atom collisions

Wagner

McKoy

1973

Most calculations of the vibrational scattering of diatom-atom collisions use the breathing sphere approximation (BSA) of orientation averaging the intermolecular potential. The resulting angularly symmetric potential can not cause rotational scattering. We determine the error introduced by the BSA into observables of the vibrational scattering of low-energy homonuclear diatom-atom collisions by comparing two quantum mechanical calculations, one with the BSA and the other with the full angularly asymmetric intermolecular potential. For ·reasons of economy the rotational scattering of the second calculation is restricted by the use of special incomplete channel sets in the expansion of the scattering wavefunction. Three representative collision systems are studied: H2-Ar, 0 2-He, and I2-He. From our calculations, we reach two conclusions. First, the BSA can be used to analyze accurately experimental measurements of vibrational scattering. Second, measurements most sensitive to the symmetric part of the intermolecular potential are, in order, elastic cross sections, inelastic cross sections, and inelastic differential cross sections. Elastic differential cross sections are sensitive to the potential only if the collision is" sticky," with scattering over a wide range of angles; 12-He is such a collision. Otherwise the potential sensitivity of elastic differential cross sections is concentrated in the experimentally difficult region of very small angle scattering.