A single resonance Regge pole dominates the forward-angle scattering of the state-to-state F + H2 → FH + H reaction at Etrans = 62.09 meV

The aim of the present paper is to bring clarity, through simplicity, to the important and long-standing problem: does a resonance contribute to the forward-angle scattering of the F + H2 reaction? We reduce the problem to its essentials and present a well-defined, yet rigorous and unambiguous, investigation of structure in the differential cross sections (DCSs) of the following three state-to-state reactions at a translational energy of 62.09 meV: F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 0, 1, 2, mf = 0) + H, where vi, ji, mi and vf, jf, mf are the initial and final vibrational, rotational and helicity quantum numbers respectively. Firstly, we carry out quantum-scattering calculations for the Fu-Xu-Zhang potential energy surface, obtaining accurate numerical scattering matrix elements for indistinguishable H2. The calculations use a time-independent method, with hyperspherical coordinates and an enhanced Numerov method. Secondly, the following theoretical techniques are employed to analyse structures in the DCSs: (a) full and Nearside-Farside (NF) partial wave series (PWS) and local angular momentum theory, including resummations of the full PWS up to second order. (b) The recently introduced "CoroGlo" test, which lets us distinguish between glory and corona scattering at forward angles for a Legendre PWS. (c) Six asymptotic (semiclassical) forward-angle glory theories and three asymptotic farside rainbow theories, valid for rainbows at sideward-scattering angles. (d) Complex angular momentum (CAM) theories of forward and backward scattering, with the Regge pole positions and residues computed by Thiele rational interpolation. Thirdly, our conclusions for the three PWS DCSs are: (a) the forward-angle peaks arise from glory scattering. (b) A broad (hidden) farside rainbow is present at sideward angles. (c) A single Regge pole contributes to the DCS across the whole angular range, being most prominent at forward angles. This proves that a resonance contributes to the DCSs for the three transitions. (d) The diffraction oscillations in the DCSs arise from NF interference, in particular, interference between the Regge pole and direct subamplitudes.

Glories, hidden rainbows and nearside-farside interference effects in the angular scattering of the state-to-state H + HD → H2 + D reaction

Yuan et al. [Nat. Chem., 2018, 10, 653] have reported state-of-the-art measurements of differential cross sections (DCSs) for the H + HD → H2 + D reaction, measuring for the first time fast oscillations in the small-angle forward region of the DCSs. We theoretically analyse the angular scattering dynamics in order to quantitatively understand the physical content of structure in the DCSs. We study the H + HD(vi = 0, ji = 0, mi = 0) → H2(vf = 0, jf = 0,1,2,3, mf = 0) + D reaction for the whole range of scattering angles from θR = 0° to θR = 180°, where v, j, m are the vibrational, rotational and helicity quantum numbers respectively for the initial and final states. The restriction to mf = 0 arises because states with mf ≠ 0 have DCSs that are identically zero in the forward (θR = 0°) and backward (θR = 180°) directions. We use accurate quantum scattering matrix elements computed by Yuan et al. at a translational energy of 1.35 eV for the BKMP2 potential energy surface. The following theoretical techniques are employed to analyse the DCSs: (a) full and nearside-farside (NF) partial wave series (PWS) and local angular momentum theory, including resummations of the full PWS up to third order. We also use window representations of the scattering matrix, which give rise to truncated PWS, (b) six asymptotic (semiclassical) small-angle glory theories and four N rainbow theories, (c) we introduce "CoroGlo" tests, which let us distinguish between glory and corona scattering at small angles for Legendre PWS, (d) the semiclassical optical model (SOM) of Herschbach is employed to understand structure in the DCSs at intermediate and large angles. Our conclusions are: (a) the small-angle peaks in the DCSs arise mainly from glory scattering. For the 000 → 020 transition, there is also a contribution from a broad, or hidden, N rainbow, (b) at larger angles, the fast oscillations in the DCSs arise from NF interference, (c) the N scattering in the fast oscillation region contains a hidden rainbow for the 000, 020, 030 cases. For the 000 → 020 transition, the rainbow extends up to θR ≈ 60°; for the 000 and 030 cases, the angular ranges containing a N rainbow are smaller, (d) at intermediate and backward angles, the slowly varying DCSs, which merge into slow oscillations, are explained by the SOM. Physically it shows this structure in a DCS arises from direct scattering and is a distorted mirror image of the corresponding probability versus total angular momentum quantum number plot.

Application of the Partial Wave QP Decomposition to the Angular Scattering of the State-to-State F + H2 Reaction at Etrans = 0.04088 eV

We analyze the physical content of structures present in the product differential cross sections (DCSs) of the benchmark F + H₂(v_i, j_i, m_i) → FH(v_f, j_f, m_f) + H reaction, where v, j, and m are the vibrational, rotational, and helicity quantum numbers, respectively, for the initial and final states. We analyze three state-to-state transitions: 000 → 300, 000 → 310, and 000 → 320. Accurate quantum S matrix elements are employed at a translational energy of 0.04088 eV for the Fu-Xu-Zhang potential energy surface. Our analysis of the DCSs uses a new technique called the QP decomposition; it makes an exact decomposition of the scattering (S) matrix into a Q part and a P part. The P part consists of a partial wave (PW) sum of Regge poles (involving both positions and residues) together with a rapidly oscillating quadratic phase. The Q part of the decomposition is then constructed exactly by subtracting the rapidly oscillating phase and the PW Regge pole sum from the input PW S matrix. In practice, it is convenient to make a small modification, which we call the QmodPmod decomposition. All our calculations use only integer values of the total angular momentum quantum number, namely, J = 0, 1, 2,... We find that the QmodPmod decomposition is successful and physically meaningful, in that the properties of Qmod matrix are simpler than those of the input S matrix. We then carry out a QmodPmod analysis of the DCSs, which provides novel insights into interference structures present in the angular scattering. In particular, we find for all three reactions that Regge resonances contribute across the whole angular range of the DCSs, being particularly pronounced at small angles. The techniques of nearside-farside decomposition and local angular momentum analysis for resummed Legendre PW series are also employed to provide additional insights into the angular scattering.

Deriving uniform semiclassical approximations for one-dimensional fermionic systems

A complete derivation is provided of the uniform semiclassical approximations to the particle and kinetic energy densities of N noninteracting bounded fermions in one dimension. The employed methodology allows the inclusion of non-perturbative quantum effects, including tunneling and quantum oscillations, via an infinite resummation of the Poisson summation formula. We explore the analytic behavior, physical meaning, and the relationship between the semiclassical uniform approximations for the fermionic kinetic energy and particle densities.

Rainbows, supernumerary rainbows and interference effects in the angular scattering of chemical reactions: an investigation using Heisenberg's S matrix programme

Rainbow and supernumerary rainbow in the angular scattering of a state-to-state chemical reaction.

State-to-State F + H2 Reaction at Etrans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering

We report two new contributions for understanding the quantum dynamics of the benchmark state-to-state reaction, F + H2(vi, ji, mi) → FH(vf, jf, mf) + H, where (vi, ji, mi) and (vf, jf, mf) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. We analyze product differential cross sections (DCSs) for the transitions, 000 → 300, 000 → 310, and 000 → 320, at a translational energy of 0.04088 eV using the potential energy surface of Fu-Xu-Zhang. The two new contributions are as follows: (1) We exploit the recently introduced QP decomposition of J. N. L. Connor [ J. Chem. Phys . 2013 , 138 , 124310 ] to transform numerical partial-wave scattering (S) matrix elements for the three transitions into parametrized (analytic) formulas, in which all terms in the three parametrized S matrices have a direct physical interpretation. In particular, they contain the positions and residues of Regge poles in the first quadrant of the complex angular momentum (CAM) plane. We obtain very close agreement between the values of the parametrized and numerical S matrix elements. (2) We then apply a uniform asymptotic Watson/CAM theory, which allows a Regge pole to be close to a saddle point. It uses the parametrized S matrices and is applied to the partial wave series (PWS) representation for the scattering amplitude to understand structure in a DCS in terms of three contributing subamplitudes. We prove using this powerful CAM theory that resonance Regge poles contribute to the small-angle scattering in the DCSs for all three transitions, with the oscillations at larger angles arising from nearside-farside interference. We obtain very good agreement between the uniform asymptotic Watson/CAM DCSs and the corresponding PWS DCSs, except for angles close to the forward and backward directions, where (as expected) the Watson/CAM formulas become nonuniform.

Angular Scattering Dynamics of the CH4 + Cl → CH3 + HCl Reaction Using Nearside-Farside, Local Angular Momentum, and Resummation Theories

The differential cross section (DCS) for the CH4 + Cl → CH3 + HCl reaction is studied at six total energies where all of the species are in their ground states. The scattering (S) matrix elements have been calculated by the rotating line umbrella method for a dual-level ab initio analytic potential energy surface. We make the first application to this reaction of nearside-farside (NF) and local angular momentum (LAM) techniques, including resummation orders (r) of 0, 1, 2, and 3 for the partial-wave series representation of the full scattering amplitude. We find that resummation usually cleans the NF r = 0 DCSs of unphysical oscillations, especially at small angles. This cleaning effect is typically most pronounced when changing from no resummation (r = 0) to r = 1; further resummations from r = 1 to r = 2 and from r = 2 to r = 3 have smaller effects. The NF DCS analyses show that the reaction is N-dominated at sideward and large angles, whereas at small angles there are oscillations caused by NF interference. The NF LAM analysis provides consistent and complementary information, in particular for the total angular momenta that contribute to the reaction at different scattering angles. The NF analyses also provide justification for simpler N-dominant dynamical theories such as the semiclassical optical model, which provides an explanation for the distorted mirror image effect for the moduli of the S matrix elements and the DCSs, as well as the use of a hard-sphere DCS over limited angular ranges.