The accuracy of the semiclassical theory of forward glory scattering for a state-to-state chemical reaction is investigated using a J-shifted Eckart parameterization for the scattering matrix element. The parameters are chosen initially to model the angular scattering of the H þ D 2 ! HD(v f ¼ 3) þ D reaction, following D. Sokolovski (Chem. Phys. Lett., 370, 805 (2003)). Then the parameters are systematically varied to generate different scattering patterns. The theory assumes that the scattering amplitude can be expanded in a Legendre partial wave series (PWS). The theoretical techniques applied to the PWS include: a nearsidefarside decomposition, and a local angular momentum (LAM) analysis; both techniques include resummations of the PWS. The semiclassical techniques used include: a uniform semiclassical approximation (USA), a primitive semiclassical approximation, a classical semiclassical approximation, an integral transitional approximation, a semiclassical transitional approximation and a semiclassical LAM. The LAM for the classical collision of two hard spheres is also employed. It is demonstrated that the USA can accurately describe glory oscillations for scattering angles on and near the forward direction (as defined by the axial caustic associated with the glory); in favourable cases the USA is accurate for sideward scattering angles and even for angles close to the backward direction.

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.

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.

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.

Yuan et al. [Nature 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...