Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

Abstract

Three new contributions to the complex angular momentum (CAM) theory of differential cross sections (DCSs) for chemical reactions are reported. They exploit recent advances in the Padé reconstruction of a scattering (S) matrix in a region surrounding the ReJ axis, where J is the total angular momentum quantum variable, starting from the discrete values, J = 0, 1, 2, .... In particular, use is made of Padé continuations obtained by Sokolovski, Castillo, and Tully [Chem. Phys. Lett. 313, 225 (1999)] for the S matrix of the benchmark F + H2(v(i) = 0, j(i) = 0, m(i) = 0) → FH(v(f) = 3, j(f) = 3, m(f) = 0) + H reaction. Here v(i), j(i), m(i) and v(f), j(f), m(f) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. The three contributions are: (1) A new exact decomposition of the partial wave (PW) S matrix is introduced, which is called the QP decomposition. The P part contains information on the Regge poles. The Q part is then constructed exactly by subtracting a rapidly oscillating phase and the PW P matrix from the input PW S matrix. After a simple modification, it is found that the corresponding scattering subamplitudes provide insight into the angular-scattering dynamics using simple partial wave series (PWS) computations. It is shown that the leading n = 0 Regge pole contributes to the small-angle scattering in the centre-of-mass frame. (2) The Q matrix part of the QP decomposition has simpler properties than the input S matrix. This fact is exploited to deduce a parametrized (analytic) formula for the PW S matrix in which all terms have a direct physical interpretation. This is a long sort-after goal in reaction dynamics, and in particular for the state-to-state F + H2 reaction. (3) The first definitive test is reported for the accuracy of a uniform semiclassical (asymptotic) CAM theory for a DCS based on the Watson transformation. The parametrized S matrix obtained in contribution (2) is used in both the PW and semiclassical parts of the calculation. Powerful uniform asymptotic approximations are employed for the background integral; they allow for the proximity of a Regge pole and a saddle point. The CAM DCS agrees well with the PWS DCS, across the whole angular range, except close to the forward and backward directions, where, as expected, the CAM theory becomes non-uniform. At small angles, θ(R) ≲ 40°, the PWS DCS can be reproduced using a nearside semiclassical subamplitude, which allows for a pole being close to a saddle point, plus the farside surface wave of the n = 0 pole sub-subamplitude, with the oscillations in the DCS arising from nearside-farside interference. This proves that the n = 0 Regge resonance pole contributes to the small-angle scattering.

The 6Hankel asymptotic approximation for the uniform description of rainbows and glories in the angular scattering of state-to-state chemical reactions: derivation, properties and applications

“…We again use the relations (14) and (15) connecting Ai() and the Hankel functions of the first and second kinds. This lets write eqn (27) as the sum of two terms…”

“…If g (x) = 1 and f (a;x) is exactly a cubic polynomial in x, eqn (4), (12), (27) and the 2Hankel formula (34) are exact results for I(a) because the 3-jets are then exact representations of f (a;x). This is useful for the checking of computer programs.…”

Polarization of molecular angular momentum in the chemical reactions Li + HF and F + HD

“…80 The differential cross sections of the HF channel exhibit a completely different behavior. The products are mainly forward scattered with remarkable nearside-farside oscillations 86,87 in the full range of angles, suggesting again resonance contribution 88 especially at the higher collision energy. As will be shown in Ref.…”

Computer Software for Understanding Resonances and Resonance-Related Phenomena in Chemical Reactions

“…Future work Angular distributions, also known as differential cross sections (DCS), contain other detailed information of the reaction mechanism, and are more difficult to interpret. Although there has been much work on the analysis of DCSs in terms of trajectories and resonance poles 12,13,16,14,18,19,27,26, no computer codes have yet been produced for public use. This Section is, therefore, more a statement of intent than a description of existing software.…”