Semiclassical Theory of the Three-Turning-Point Problem

Abstract

A new theory is presented for phase shifts resulting from scattering by atoms with attractive potentials, i.e., the three-turning-point problem. A simple formula, valid in and below the three-turning-point region is derived η=η0−12Tan−1{cosα sinβ/[(−1)nsinα−cosβ]}+θ[(−1)nsinα−cosβ]12π,where the angle α is related to the Stokes constant for the two outer turning points; β is proportional to twice the action integral minus π times the number of WBKJ wavefunction nodes in the inner region, (n−1); and τ is a unit step function which is zero for negative arguments. The theory predicts virtual energy states for incident energies in the three-turning-point range whose location compares well with numerical quantum calculations. The resonance portion of the phase shifts shows only semiquantitative comparison with the quantum results.

Resonance Theory of Termolecular Recombination Kinetics: H+H+M→H2M

“…It is possible, however, to study the tunneling phenomenon and quasibound states via semiclassical methods. 16 , [32][33][34][35] Next, possibilities for the reaction intermediates X 2 i are considered, and it is shown under what circumstances quasibound states account for the most significant contributions. It is assumed that the relevant interaction potentials of a pair of ground-state X atoms are known and that the collisions proceed adiabatically along them.…”

“…In general for an effective potential with three classical turning points 33 ,34 there are three "types" of collisions: (1) "normal" collisions with energies E greater than the barrier height V max; (2) "nonresonance" collisions with E< V max; these have a very small probability of tunneling and hence becoming trapped in the classically inaccessible region inside the rotational barrier; and (3) "resonance" collisions, in which tunneling leads to the subsequent "trapping" of a pair of X atoms in the classically inaccessible region. 12 -17 , [33][34][35][36][37] Quasibound states are distinguished by lifetimes long compared to a vibrational period for the complex or compared to a typical "collision time." Hence they can be treated as virtual bound states in the final rate determining step (3c).…”

Diatomic predissociation linewidths

“…Others have pointed to the spectroscopically important peak in the relative amplitude of the wavefunction inside and outside the barrier region [14,[18][19][20]; however the intimate connection [12] between this characteristic and the behaviour of the phase shift has received relatively little emphasis in the molecular literature. Again in the semi-classical field attention has concentrated on the behaviour of the phase shift [24][25][26][27], or equivalently (see below) on the location of complex energy poles in the scattering matrix [28][29][30], but the behaviour of the amplitude ratio has been largely ignored.…”