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.

Nearside-Farside Analysis of the Angular Scattering for the State-to-State H + HD → H2 + D Reaction: Nonzero Helicities.

We theoretically analyze the differential cross sections (DCSs) for the state-to-state reaction, H + HD(vi = 0, ji = 0, mi = 0) → H2(vf = 0, jf = 1,2,3, mf = 1,..,jf) + D, over the whole range of scattering angles, where v, j, and m are the vibrational, rotational, and helicity quantum numbers for the initial and final states. The analysis extends and complements previous calculations for the same state-to-state reaction, which had jf = 0,1,2,3 and mf = 0, as reported by Xiahou, C.; Connor, J. N. L. Phys. Chem. Chem. Phys. 2021, 23, 13349-13369. Motivation comes from the state-of-the-art experiments and simulations of Yuan et al. Nature Chem. 2018, 10, 653-658 who have measured, for the first time, fast oscillations in the small-angle region of the degeneracy-averaged DCSs for jf = 1 and 3 as well as slow oscillations in the large-angle region. We start with the partial wave series (PWS) for the scattering amplitude expanded in a basis set of reduced rotation matrix elements. Then our main theoretical tools are two variants of Nearside-Farside (NF) theory applied to six transitions: (1) We apply unrestricted, restricted, and restrictedΔ NF decompositions to the PWS including resummations. The restricted and restrictedΔ NF DCSs correctly go to zero in the forward and backward directions when mf > 0, unlike the unrestricted NF DCSs, which incorrectly go to infinity. We also exploit the Local Angular Momentum theory to provide additional insights into the reaction dynamics. Properties of reduced rotation matrix elements of the second kind play an important role in the NF analysis, together with their caustics. (2) We apply an approximate N theory at intermediate and large angles, namely, the Semiclassical Optical Model of Herschbach. We show there are two different reaction mechanisms. The fast oscillations at small angles (sometimes called Fraunhofer diffraction/oscillations) are an NF interference effect. In contrast, the slow oscillations at intermediate and large angles are an N effect, which arise from a direct scattering, and are a "distorted mirror image" mechanism. We also compare these results with the experimental data.

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(vi, ji, mi) → FH(vf, jf, mf) + 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.

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

In earlier research, we have demonstrated that broad "hidden" rainbows can occur in the product differential cross sections (DCSs) of state-to-state chemical reactions. Here we ask the question: can pronounced and localized rainbows, rather than broad hidden ones, occur in reactive DCSs? Further motivation comes from recent measurements by H. Pan and K. Liu, J. Phys. Chem. A, 2016, 120, 6712, of a "bulge" in a reactive DCS, which they conjecture is a rainbow. Our theoretical approach uses a "weak" version of Heisenberg's scattering matrix program (wHSMP) introduced by X. Shan and J. N. L. Connor, Phys. Chem. Chem. Phys., 2011, 13, 8392. This wHSMP uses four general physical principles for chemical reactions to suggest simple parameterized forms for the S matrix; it does not employ a potential energy surface. We use a parameterization in which the modulus of the S matrix is a smooth-step function of the total angular momentum quantum number, J, and (importantly) its phase is a cubic polynomial in J. We demonstrate for a Legendre partial wave series (PWS) the existence of pronounced rainbows, supernumerary rainbows, and other interference effects, in reactive DCSs. We find that reactive rainbows can be more complicated in their structure than the familiar rainbows of elastic scattering. We also analyse the angular scattering using Nearside-Farside (NF) PWS theory and NF PWS Local Angular Momentum (LAM) theory, including resummations of the PWS. In addition, we apply full and NF asymptotic (semiclassical) rainbow theories to the PWS - in particular, the uniform Airy and transitional Airy approximations for the farside scattering. This lets us prove that structure in the DCSs are indeed rainbows, supernumerary rainbows as well as other interference effects.

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.

This paper considers the asymptotic (semiclassical) analysis of a forward glory and a rainbow in the differential cross section (DCS) of a state-to-state chemical reaction, whose scattering amplitude is given by a Legendre partial wave series (PWS). A recent paper by C. Xiahou, J. N. L. Connor and D. H. Zhang [Phys. Chem. Chem. Phys., 2011, 13, 12981] stated without proof a new asymptotic formula for the scattering amplitude, which is uniform for a glory and a rainbow in the DCS. The new formula was designated "6Hankel" because it involves six Hankel functions. This paper makes three contributions: (1) we provide a detailed derivation of the 6Hankel approximation. This is done by first generalizing a method described by G. F. Carrier [J. Fluid Mech., 1966, 24, 641] for the uniform asymptotic evaluation of an oscillating integral with two real coalescing stationary phase points, which results in the "2Hankel" approximation (it contains two Hankel functions). Application of the 2Hankel approximation to the PWS results in the 6Hankel approximation for the scattering amplitude. We also test the accuracy of the 2Hankel approximation when it is used to evaluate three oscillating integrals of the cuspoid type. (2) We investigate the properties of the 6Hankel approximation. In particular, it is shown that for angles close to the forward direction, the 6Hankel approximation reduces to the "semiclassical transitional approximation" for glory scattering derived earlier. For scattering close to the rainbow angle, the 6Hankel approximation reduces to the "transitional Airy approximation", also derived earlier. (3) Using a J-shifted Eckart parameterization for the scattering matrix, we investigate the accuracy of the 6Hankel approximation for a DCS. We also compare with angular scattering results from the "uniform Bessel", "uniform Airy" and other semiclassical approximations.

Rainbows and glories in the angular scattering of the state-to-state F + H2 reaction at E(trans)=0.04088 eV.

State-of-the-art differential cross sections (DCSs) have been reported by Wang et al. [Proc. Nat. Acad. Sci. (U.S.), 2008, 105, 6227] for the state-to-state F + H(2)→ FH + H reaction using fully quantum-state-selected crossed molecular beams. We theoretically analyze the angular scattering of this reaction, in order to quantitatively understand the physical content of structure in the DCSs. Three transitions are studied, v(i)=0, j(i)=0, m(i)=0 → v(f)=3, j(f)=0, 1, 2, m(f)=0 at a translational energy of 0.04088 eV, where v, j, m are the vibrational, rotational and helicity quantum numbers respectively for the initial and final states. The input to our analyses consists of accurate quantum scattering (S) matrix elements computed for the Fu-Xu-Zhang potential energy surface, as used by Wang et al. in a computational simulation of their experimental DCSs. We prove that the pronounced peak at forward angles observed in the experimental and simulated DCSs for all three transitions is a glory. At larger angles, it is demonstrated that the 000 → 300 and 000 → 310 DCSs both possess a broad farside rainbow, which is accompanied by diffraction oscillations. We confirm the conjecture of Wang et al. that these diffraction oscillations arise from nearside-farside (NF) interference. We find that the reaction is N dominant for all three transitions. The theoretical techniques used to analyze the angular scattering include uniform semiclassical theories of glory and of rainbow scattering. We also make the first application of a semiclassical formula that is uniform for both glory + rainbow scattering. In addition, structure in the DCSs is analyzed using NF theory and local angular momentum theory, in both cases with three resummations of the partial wave series for the scattering amplitude. We make the first explicit application of the Thiele rational interpolation formula to extract the position and residue of the leading Regge pole from a set of S matrix elements, thereby making contact with complex angular momentum theories of DCSs, which interpret the angular scattering in terms of Regge resonances. Our calculations complement the exit-valley vibrationally-adiabatic analysis of Wang et al.

A new rainbow: angular scattering of the F + H2(v(i) = 0, j(i) = 0) --> FH(v(f) = 3, j(f) = 3) + H reaction.

The angular scattering of a state-to-state chemical reaction contains fundamental information on its dynamics. Often the angular distributions are highly structured and the physical interpretation of this structure is an important and difficult problem. Here, we report a surprising finding for the benchmark F + H(2) --> FH + H reaction, when the product molecule FH is in a vibrational state with quantum number = 3 and a rotational state with quantum number = 3. We demonstrate that the differential cross section (DCS) is an example of (attractive) rainbow scattering, being characterized by an Airy function and its derivative. The rainbow reveals its presence in the DCS by interference with the repulsive (or nearside) scattering producing characteristic diffraction oscillations. The rainbow is broad, which explains why it has not been recognized in the many earlier theoretical and experimental investigations of this reaction. There is an angular region in the DCS where the rainbow dominates, but with the unusual property that the DCS is less intense than in adjoining angular regions. The reaction investigated is F + H(2)(v(i) = 0, j(i) = 0, m(i) = 0) --> FH(v(f) = 3, j(f) = 3, m(f) = 0) + H, where v(i), j(i), m(i) and v(f), j(f), m(f) are initial and final vibrational, rotational and helicity quantum numbers, respectively. The relative translational energy is 0.119 eV. We use rigorous semiclassical (asymptotic) techniques that provide physical insight as well as a mathematically sound and numerically accurate description of the angular scattering. The semiclassical DCS agrees very closely with the exact quantum DCS. The semiclassical scattering amplitude is used to assess the physical effectiveness of the Fuller nearside-farside decomposition for the partial wave series of the F + H(2) reaction, including the effect of one resummation. We also compare the semiclassical and exact quantum nearside, farside, and full local angular momenta and find good agreement. Although our new rainbow has unusual and unexpected properties, similar rainbows are predicted to occur in the DCSs of many state-to-state chemical reactions, since the semiclassical analysis is generic and not specific to the present F + H(2) example.

Local angular momentum-local impact parameter analysis: derivation and properties of the fundamental identity, with applications to the F+H2, H+D2, and Cl+HCl chemical reactions.

The technique of local angular momentum-local impact parameter (LAM-LIP) analysis has recently been shown to provide valuable dynamical information on the angular scattering of chemical reactions under semiclassical conditions. The LAM-LIP technique exploits a nearside-farside (NF) decomposition of the scattering amplitude, which is assumed to be a Legendre partial wave series. In this paper, we derive the "fundamental NF LAM identity," which relates the full LAM to the NF LAMs (there is a similar identity for the LIP case). Two derivations are presented. The first uses complex variable techniques, while the second exploits an analogy between the motion of the scattering amplitude in the Argand plane with changing angle and the classical mechanical motion of a particle in a plane with changing time. Alternative forms of the fundamental LAM-LIP identity are described, one of which gives rise to a CLAM-CLIP plot, where CLAM denotes (Cross section) x LAM and CLIP denotes (Cross section) x LIP. Applications of the NF LAM theory, together with CLAM plots, are reported for state-to-state transitions of the benchmark reactions F+H2-->FH+H, H+D2-->HD+D, and Cl+HCl-->ClH+Cl, using as input both numerical and parametrized scattering matrix elements. We use the fundamental LAM identity to explain the important empirical observation that a NF cross section analysis and a NF LAM analysis provide consistent (and complementary) information on the dynamics of chemical reactions.