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

Abstract

We theoretically analyze the differential cross sections (DCSs) for the state-to-state reaction, H + HD(v_i = 0, j_i = 0, m_i = 0) → H₂(v_f = 0, j_f = 1,2,3, m_f = 1,..,j_f) + 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 j_f = 0,1,2,3 and m_f = 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 j_f = 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 m_f > 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.

Figure 1

Plots of d_mf, mi^J (θ_R) vs θ_R/deg for J = 10, m_i = 0 (a) m_f = 0, (b) m_f = 1, (c) m_f = 2, (d) m_f = 3. The vertical pink lines indicate the caustic angles at (a) 0°, 180°, (b) 5.7°, 174.3°, (c) 11.5°, 168.5°, (d) 17.5°, 162.5°.

Figure 2

Plots of e_mf, mi^J (θ_R) vs θ_R/deg for J = 10, m_i = 0 (a) m_f = 0, (b) m_f = 1, (c) m_f = 2, (d) m_f = 3. The vertical pink lines indicate the caustic angles at (a) 0°, 180°, (b) 5.7°, 174.3°, (c) 11.5°, 168.5°, (d) 17.5°, 162.5°.

Figure 3

Values of θ_R min^(J, m_f)/deg and θ_R max^(J, m_f)/deg (black solid circles) on a (θ_R/deg, J) plot for m_i = 0 and m_f = 1,2,3. Passing through the black solid circles are the curves, J = m_f/sin θ_R, colored red for m_f = 1, orange for m_f = 2, and green for m_f = 3.

Figure 4

Plots of | S̃_J | vs J at E_trans = 1.35 eV. The black solid circles are the numerical S matrix data, {| S̃_J |}, at integer values of J, which have been joined by straight lines. The transitions are (a) 000 → 011, (b) 000 → 021, and (c) 000 → 031.

Figure 5

Plots of | S̃_J | vs J at E_trans = 1.35 eV. The black solid circles are the numerical S matrix data, {| S̃_J |}, at integer values of J, which have been joined by straight lines. The transitions are (a) 000 → 022, (b) 000 → 032, and (c) 000 → 033.

Figure 6

Plots of arg S̃_J/rad vs J at E_trans = 1.35 eV. The black solid circles are the numerical S matrix data, {arg S̃_J/rad}, at integer values of J, which have been joined by straight lines. The transitions are (a) 000 → 011, (b) 000 → 021, and (c) 000 → 031.

Figure 7

Plots of arg S̃_J/rad vs J at E_trans = 1.35 eV. The black solid circles are the numerical S matrix data, {arg S̃_J/rad}, at integer values of J, which have been joined by straight lines. The transitions are (a) 000 → 022, (b) 000 → 032, and (c) 000 → 033.

Figure 8

Plots of four PWS N DCSs in the large-angle region from θ_R = 140° to θ_R = 180° for the transition 000 → 011 at E_trans = 1.35 eV for (a) r = 0 and (b) r = 1. Purple long-dashed curve: PWS/N/unres. Lilac dashed curve: PWS/N/res. Red solid curve: PWS/N/resΔ. Red dashed curve: Least-squares-fit to PWS/N/resΔ.

Figure 9

Plots of log σ(θ_R) vs θ_R/deg at E_trans = 1.35 eV for r = 1. Black curve: PWS. Red solid curve: PWS/N/resΔ. Blue solid curve: PWS/F/resΔ. Red dashed curves: least-squares-fits to PWS/N/resΔ in the small- and large-angle regions. Blue dashed curves: Least-squares-fits to PWS/F/resΔ in the small- and large-angle regions. The transitions are (a) 000 → 011, (b) 000 → 021, and (c) 000 → 031.

Figure 10

Plots of log σ(θ_R) vs θ_R/deg at E_trans = 1.35 eV for r = 1. Black curve: PWS. Red solid curve: PWS/N/resΔ. Blue solid curve: PWS/F/resΔ. Red dashed curves: least-squares-fits to PWS/N/resΔ in the small- and large-angle regions. Blue dashed curves: least-squares-fits to PWS/F/resΔ in the small- and large-angle regions. The transitions are (a) 000 → 022, (b) 000 → 032, and (c) 000 → 033.

Figure 11

Plots of LAM (θ_R) vs θ_R/deg at E_trans = 1.35 eV for the 000 → 011 transition, showing results for both r = 0 and r = 1. Black curve: PWS. Red solid curve: PWS/N/r = 1. Red dashed curve: PWS/N/r = 0. Blue solid curve: PWS/F/r = 1. Blue dashed curve: PWS/F/r = 0. The fainter blue solid and dashed curves show where the F LAM(θ_R) is not physically significant.

Figure 12

Plots of σ (θ_R) vs θ_R/deg at E_trans = 1.35 eV for the angular range from θ_R = 50° to θ_R = 180°. Black curve: PWS. Red curve: SOM. The transitions are (a) 000 → 011, (b) 000 → 021, and (c) 000 → 031.

Figure 13

Plots of σ (θ_R) vs θ_R/deg at E_trans = 1.35 eV for the angular range from θ_R = 50° to θ_R = 180°. Black curve: PWS. Red curve: SOM. The transitions are (a) 000 → 022, (b) 000 → 032, and (c) 000 → 033.

Figure 14

Plots of degeneracy averaged σ(θ_R) (daDCS) vs θ_R/deg at E_trans = 1.35 eV. (a) The transition 00 → 01 (red), together with experimental results and their estimated errors (blue). (b) The transition 00 → 03 (purple), together with experimental results and their estimated errors (blue). (c) Black curve: Degeneracy averaged, σ(θ_R), for the 00 → 0 transition, which is summed over j_f = 0,1,2,3. The four colored curves show the degeneracy averaged σ(θ_R) for the transitions 00 → 00 (orange), 00 → 01 (red), 00 → 02 (green), and 00 → 03 (purple).