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 XiahouC.; ConnorJ. N. L.Phys. Chem. Chem. Phys.2021, 23, 13349–1336934096934. Motivation comes from the state-of-the-art experiments and simulations of Yuan et al.Nature Chem.2018, 10, 653–65829686377 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.


INTRODUCTION
The H + H 2 → H 2 + H reaction and its isotopic variants are important benchmarks in the theory of chemical reaction dynamics. In particular, measurements and calculations of state-to-state differential cross sections (DCSs) can provide detailed information on the dynamics and mechanism of this class of reactions.
Recently, an important experimental advance has been reported by Yuan et al. 1 for the H + HD → H 2 + D reaction. They have measured, for the first time, fast oscillations in the small-angle region of the degeneracy-averaged DCSs (abbreviated as daDCSs). They reported daDCSs for the following two transitions. The purpose of the present paper is to analyze and quantitatively understand the daDCSs of Yuan et al. 1 To do this, we start with the helicity (or body-fixed) representation for the scattering amplitude. We then have to consider the following 16 state-to-state DCSs where m i and m f are the helicity quantum numbers for the initial and final states, respectively. In our earlier paper 6 (denoted XC1), which is a companion to this one, we studied theoretically the DCSs for the four state-to-state transitions in reaction (R2) with m f = 0, namely In particular, we analyzed the dynamics of the angular scattering for reaction (R3) in order to understand the physical content of the structure in the four helicity-resolved DCSs. 6 We discovered: glory scattering at small angles, broad or "hidden" nearside rainbows, Nearside-Farside (NF) interference effects (sometimes called Fraunhofer diffraction/ oscillations), a "CoroGlo" test to distinguish corona and forward glory scattering, and a "distorted mirror image" mechanism present at intermediate and large angles. 6 In this paper, we focus on the DCSs for state-to-state transitions with nonzero helicities. This reduces the number of DCSs in reaction (R2) to 12. A further reduction is possible because the DCSs for m f = −1, −2, −3 are equal to those for m f = +1, +2, +3, respectively. This leaves the following six DCSs to be analyzed. We will often write 000 → 011, 000 → 021, 000 → 031, 000 → 022, 000 → 032, and 000 → 033 for the six transitions or, more simply, 011, 021, 031, 022, 032, and 033. There is a fundamental difference between DCSs with m f = 0 and those with m f > 0 for reactions of the type (R3) and (R4). All DCSs with m f > 0 are identically equal to zero in the forward (θ R = 0°) and backward (θ R = 180°) directions in the center-of-mass reference frame, which is a consequence of the conservation of angular momentum. Furthermore, the partial wave series (PWS) for the scattering amplitude for m f = 0 uses a basis set of Legendre polynomials, whereas for m f > 0 the basis set consists of reduced rotation matrix elements (also called Wigner or little d f unctions), which simplify to associated Legendre functions when m i = 0. This means the theoretical analysis is more complicated and difficult for the m f > 0 case compared to m f = 0. Now there has been one previous NF analysis of DCSs for chemical reactions with m f > 0, which was made more than 20 years ago. 7 In this work, Dobbyn et al. 7 made the following important observation (on page 1117): "...although the PWS becomes more complicated for more general types of collisions, this has little impact on the physical insight provided by a NF analysis".
Thus, in this paper, (two variants of) NF theory will be used to provide physical insight into the reaction dynamics. Note that the NF theory was used extensively in XC1. 6 In particular, an NF analysis has the advantage that the semiclassical (asymptotic) picture is still evident, even though semiclassical techniques, such as the stationary phase or saddle point methods, are not applied. Note that Yuan et al. 1 have conjectured on the role an NF analysis plays in explaining oscillatory structures in their DCSs. The two NF theories we use are (1) For a PWS with a basis set of Wigner functions, we use three NF decompositions: unrestricted ( unres NF), 8 restricted ( res NF), 9,10 and restrictedΔ ( resΔ NF). 7 The unres NF decomposition is a straightforward generalization 8 of the NF decomposition for a Legendre PWS. 8,11 The unres NF DCSs incorrectly diverge as θ R → 0°, 180°. In contrast, the res NF and resΔ NF DCSs correctly go to zero as θ R → 0°, 180°. 7,9,10 The properties of the caustics of Wigner functions as well as those for reduced rotation matrix elements of the second kind play an important role in the definitions of res NF and resΔ NF. 7,9,10 We also perform a resummation for a PWS of Wigner functions, 12 since it is well-known that a resummation can improve the physical effectiveness of an NF decomposition. 13−19 In fact, the present paper is the first time that resummation theory has been combined with the res NF and resΔ NF decompositions.
The above remarks apply, in particular, to NF analyses of the full DCSs for the six transitions. We also report the results (including resummations) of the unres NF decomposition for the Local Angular Momentum (LAM), since this provides important additional insights into the reaction dynamics. 13 This paper is organized as follows: Section 2 summarizes the partial wave theory and explains our conventions and definitions for the DCSs and LAMs. This section also includes a discussion of the caustic properties that we need and summarizes the unres NF, res NF, and resΔ NF decompositions. Section 3 outlines the resummation for a PWS of Wigner functions. The properties of the input scattering matrices for the six transitions are presented in Section 4; we use the same accurate scattering matrix elements employed by Yuan et al. 1 in a simulation of their experiments. In Section 5, we discuss in detail the behavior of the unres NF, res NF, and resΔ NF DCSs at small and large angles, as this has not been done before. Our results for the full and NF DCSs and LAMs, including resummations, are presented and discussed in Sections 6 and 7, respectively. The SOM DCSs at intermediate and large angles are presented and discussed in Section 8. We report daDCSs in Section 9, where we make comparisons with the experimental data. Our conclusions are in Section 10. Most of our results are presented graphically.
Appendix A proves that the state-to-state m i = 0 DCSs for m f = −1, −2, −3 and m f = +1, +2, +3 are equal, respectively. In applications of the NF theory, it is essential to use unambiguous and consistent definitions for the special functions (of the first and second kinds) employed in the various NF decompositions. In Appendix B, we gather together the precise mathematical definitions that we use, since there is often more than one definition in the literature.
We also emphasize the following: This paper complements and extends XC1, 6 where additional discussions and references can be found. These two papers illustrate the potency of the NF theory for divers applications.

PARTIAL WAVE THEORY
2.1. Partial Wave Series. We start with the helicity (or body-fixed) PWS representation of the scattering amplitude for reaction (R4) at a fixed translational (or total) energy 22,23 which is called the unrestricted NF decomposition. The adjective "unrestricted" is added because eq 7 can be used for all θ R ∈ (0,π) with no restriction on the sum over J. We also sometimes write unres NF. The corresponding N and F DCSs are given by With the help of eqs 2 and (8), we obtain Equation 9 is the Fundamental Identity for Full and NF DCSs and is exact. 25 Similarly, we can define (unrestricted) N and F LAMs There is an exact Fundamental Identity for Full and NF LAMs analogous to eq 9, although more complicated in form. 25 As before, the args in eq 10 are not necessarily principal values in order that the derivatives be well-defined.
However, there is a problem with the unrestricted decomposition for m f > 0. Although the unres NF decomposition • We see that the little e function diverges as θ R → 0°, 180°, which means that the N,F components of eq 6 also diverge. Then we have the unfortunate situation in the interesting forward and backward regions that σ (N,F) (θ R )→∞, whereas σ(θ R ) → 0; i.e., although the NF decomposition (5)−(7) is mathematically exact, it is not physically meaningful at small and large angles. The Journal of Physical Chemistry A pubs.acs.org/JPCA Article classically allowed regions) separated from two nonoscillatory regions (classically forbidden regions) when θ R is close to 0°, 180°. We can distinguish between these regions using the notion of caustics, which are discussed next.
This in turn implies that the NF decomposition (7) should work best when θ R satisfies the inequality (12).
An inspection of Figure 3 shows, for given values of θ R , J, and m f , that there is a minimum value of J, denoted J min (m f ) (θ R ), such that θ R satisfies the inequality (12). For m f > 0, we have where int(x) ≡ integer part of x. Sometimes, +1 is added to the right-hand-side of eq 13 to exclude the case where J = m f . In practice, it makes little difference whether +1 is added or not. 9, 10 We confirmed this is the case in our calculations for all six transitions. The physical reason is that the PWS (1) receives its main numerical contribution from partial waves with J ≫ m f , helped by the (2J + 1) factor.
The comments just given lead us to introduce the restricted nearside-farside decomposition, denoted res NF, which we discuss next.
2.4. Restricted Nearside-Farside Decomposition ( res NF). The decomposition in which partial waves with J < J min (m f ) (θ R ) are omitted from eqs 1 and (7) defines res NF. 9,10 The restricted N,F subamplitudes are given by And the corresponding res NF DCSs are Note that res NF is an approximate decomposition because it omits partial waves from classically forbidden regions of θ R ; that is, it neglects the following terms in the PWS (1) The contribution of each partial wave in eq 16 is nonoscillatory and small in magnitude. Notice that eqs 13 and (14) are, in general, discontinuous functions of θ R . As a result, the corresponding DCSs also exhibit discontinuities, although, as we shall see in Section 5, they are usually small and confined to small and large angles. In addition, there is no global LAM for res NF because of the discontinuities. Having identified the Δf(θ R ) term of eq 16, we can include it in the res NF decompositionthis gives rise to the restrictedΔ nearsidefarside decomposition, denoted resΔ NF, which we discuss next.
2.5. RestrictedΔ Nearside-Farside Decomposition ( resΔ NF). The resΔ NF decomposition is obtained when we combine eq 16 with eq 14 to obtain an improved version of res NF. We have for the subamplitudes 7 And the corresponding resΔ NF DCSs are Notice that resΔ NF is an exact NF decomposition, unlike eq 14, which is approximate. Similar to res NF, eq 17 is also a discontinuous function of θ R , although the discontinuities are usually small in the corresponding DCSs and confined to small and large anglesexamples are provided in Section 5. In addition, there is again no global LAM for resΔ NF because of these discontinuities. Note: If, for a given θ R , we have that J min (m f ) (θ R ) is equal to m f , then unres NF, res NF, and resΔ NF become equivalent.
Practical Remark. It can often happen that J min (m f ) (θ R ), for particular values of θ R and m f , can exceed J max , that is, J min (m f ) (θ R ) > J max . For example, J min (m f = 3) (θ R = 0.1°) = 1718, but J max = 40. Then many computer programs applied directly to eqs 14− (18) will crash as they attempt to use values of SJ that are undefined for J > J max , when the upper limit of J = ∞ has been replaced by J = J max . This problem can be avoided by adding sufficient SJ ≡ 0 to the PWS for J > J max .

RESUMMATION OF THE PARTIAL WAVE SERIES
It is well-established that a resummation of a Legendre PWS can significantly improve the physical effectiveness of an NF decomposition. 13−19 Totenhofer et al. 19 have provided an extensive discussion of the Legendre case. This same improvement in NF physical effectiveness occurs for a basis set of little d functions, although this has only been studied for a single example, namely, Ar + HF rotationally inelastic scattering. 12 It has been found previously that the biggest effect for cleaning the N,F DCSs and N,F LAMs of unphysical oscillations occurs on going from resummation order, r = 0 (no resummation, i.e., eq 1) to resummation order, r = 1. There is usually a smaller cleaning effect for further resummations, r = 1 to r = 2 and r = 2 to r = 3.
Whiteley et al. 12 have resummed the PWS (1), which we now write as f r=0 (θ R ), from r = 0 to r = 1. We do not repeat the derivation here, which exploits the recurrence relation obeyed by the little d functions; rather, we simply write down the final result for the resummed representation for f r=1 (θ R ). From eq (3.9) of ref 12 with m i = 0, we have Equation 19 also assumes that (1 + β cos θ R ) ≠ 0. Notice that eq 20 is valid for J = m f , as proven in the Appendix of ref 12.
For this case, we see from eq 22 that g m f m f = 0.
An NF decomposition of eq 19 can now be made with the corresponding N,F subamplitudes given by (θ R ≠ 0,π) Notice that the full amplitudes, f r=0 (θ R ) and f r=1 (θ R ), are independent of β and numerically the same for a given value of θ R . This is also true for the full LAMs, LAM r=0 (θ R ) and LAM r=1 (θ R ).
In our applications, we need to choose a value for the resummation parameter β. We extend the prescription used by Anni et al. 13 Figure 4 shows graphs of |SJ| versus J for the three transitions 000 → 011, 000 → 021, 000 → 031, while Figure 5 shows plots for the remaining three transitions 000 → 022, 000 → 032, 000 → 033. Figures 6 and 7    The Journal of Physical Chemistry A pubs.acs.org/JPCA Article exception that the global maxima of the |SJ| curves are always at J = 0 when m f = 0 (see Figure 1 of XC1 (i.e., ref 6)). • Figures 6 and 7 show that the plots of arg SJ/rad versus J are roughly quadratic in shape. The kinks in some of the curves are seen to correspond to near-zeros in |SJ|, where the phase of SJ varies more rapidly with J. The curves in Figures 6 and 7 have similar properties to the arg SJ/rad plots for m f = 0 (see Figure 2 of XC1 (i.e., ref 6)). • Note that, in the NF analysis, only the values of S̃J at J = 0,1,2,... are used. To help guide the eye, the points (black solid circles) in Figures 4−7 have been joined by straight lines. This was also done in Figures 1 and 2 of XC1. 6 When we want a smooth continuation of the {S̃J} to real values of J, for example, for use in an asymptotic (semiclassical) analysis, we would typically use a cubic B-spline interpolation. 6 Notice also that the kinks do not affect the NF analysis nor the asymptotic analysis, as explained in XC1. 6 We next consider in more detail the properties of the unres NF, res NF, and resΔ NF decompositions.

PROPERTIES OF THE UNRESTRICTED, RESTRICTED
AND RESTRICTEDΔ NEARSIDE-FARSIDE DECOMPOSITIONS INCLUDING RESUMMATIONS In Sections 2.2, 2.4, and 2.5, we developed the theory for the unres NF, res NF, and resΔ NF decompositions, respectively, for r = 0; the extension of the theory to r = 1 was given in Section 3. In the present section, we investigate in detail how these three decompositions (including resummations) influence the corresponding N,F DCSs at small and large angles, as this has not been investigated before. In Figure 8, we plot four N DCSs for the 000 → 011 transition at large angles, namely, for θ R = 140 • −180 • . The upper panel shows DCSs for r = 0, and the lower panel shows DCSs for r = 1.
We begin our discussion with the res N DCS (lilac dashed curve) and the resΔ N DCS (red solid curve) in Figure 8. We note the following: • By construction, the res N and resΔ N DCSs tend to zero as θ R →180 • . Their discontinuities are clearly visible on the scale of the drawings. The density of jumps increases as θ R →180 • ; this is expected from Figure 3. • The resΔ N DCS is usually smaller than the res N DCS for both r = 0 and r = 1. Now both the res N subamplitude and the Δf(θ R )/2 term in eq 17 are complex-valued quantities, which means that destructive interference can occur, resulting in the resΔ N DCS being smaller than the res N DCS. • The first discontinuity for increasing θ R occurs at θ R ≈ 150°for r = 0 but at θ R ≈ 161 • for r = 1. This behavior can be understood because J min (m f =1) (θ R ), which is equal to int(m f /sin θ R ) by eq 13, jumps from J = 1 at θ R ≈ 149.9°t o J = 2 at θ R ≈ 150.0°, causing a discontinuity in the PWS (14) and in the resulting resΔ N and res N DCSs.
In contrast, for r = 1, the J = m f = 1 term is put equal to zero by the choice of β in eq 27, resulting in the PWS (25) starting at J = m f + 1 = 2. Then the first jump occurs for J = 2 at θ R ≈ 160.5°to J = 3 at θ R ≈ 160.6°.
• Because of congestion in the graphs, it is difficult for the eye to follow the jumps in the resΔ N and res N DCSs in Figure 8, especially as θ R → 180°. However, it is the general trend in these DCSs that is of interest. We The Journal of Physical Chemistry A pubs.acs.org/JPCA Article therefore made least-squares-fits to the resΔ N DCSs. These are shown as red dashed curves for r = 0 and r = 1 in Figure 8a,b, respectively.
We also plotted the unres N DCSs (purple long-dashed curves) for r = 0 and r = 1 in Figure 8. As expected, they tend to infinity as θ R → 180°. The beneficial effect of cleaning can be seen because the unres N r = 0 DCS starts to diverge at θ R ≈ 150°, whereas for r = 1, the unres N DCS diverges at a larger angle, namely, θ R ≈ 170°.
The results discussed above for the 000 → 011 transition have all been for N DCSs at large angles. We also did a similar analysis for the N DCSs at small angles and obtained analogous results (not shown). In addition, we also calculated unres F, res F, and resΔ F DCSs at large and small angles and obtained comparable results (also not shown). Finally, we performed unres NF, res NF, and resΔ NF analyses for the remaining five transitions at large and small angles, again finding similar results (not shown) to the 011 case.
The least-squares-fits to the resΔ NF DCSs are used in the next section, where we report NF analyses of the full DCSs for all the transitions. Figure 9 shows logarithmic plots of the full and resΔ N, resΔ F r = 1 DCSs versus θ R for the 000 → 011, 000 → 021, and 000 → 031 transitions. The corresponding DCSs for the 000 → 022, 000 → 032, and 000 → 033 transitions are displayed in Figure  10. For clarity of viewing, notice that, at large and small angles, least-squares-fits to the resΔ N and resΔ F DCSs are plotted, as explained in Section 5. We use the following color conventions for the DCSs in Figures 9 and 10 as well as in some other figures.
• The DCS = 0 a 0 2 sr −1 at θ R = 0°followed by the next observation listed here.
• Fast oscillations in an angular range extending up to θ R ≈ 50°, accompanied by a decreasing DCS. This behavior merges into the next observation listed here. • An increasing DCS with slow oscillations, which extend into the large-angle region. • The DCS = 0 a 0 2 sr −1 at θ R = 180°.
The full DCSs for the remaining five transitions exhibit similar properties to the 011 case and are not discussed separately. We can also compare with the four full DCSs for the m f = 0 case shown in Figure 3 of XC1. 6 We see that the m f = 0 and m f > 0 DCSs are alike, the main difference being (a) the m f = 0 DCSs are nonzero at θ R = 0°,180°unlike the m f > 0 DCSs, (b) the angular regions separating the fast and slow oscillations are slowly varying for m f = 0, whereas there are pronounced minima when m f > 0.
Next, we examine the resΔ N, resΔ F r = 1 DCSs in Figures 9  and 10, making use of the exact Fundamental Identity for Full and N,F DCSs given by eq 9, which is also valid for the r = 1 case. 25 In angular regions where there are fast oscillations, we see that the resΔ N and resΔ F r = 1 DCSs are varying relatively slowly with θ R , which tells us that the fast oscillations in the full DCSs arise from NF interference. Another name for the fast oscillations is Fraunhofer diffraction/oscillations. In contrast, the slow oscillations are seen to be resΔ N-dominated. Thus, we have the important result from the NF analysis that the fast and slow oscillations arise from different physical mechanisms. This is also the case for the m f = 0 DCSs. 6 We can extract useful information from the periods Δθ R of the fast oscillations. A simple NF model shows that these oscillations are analogous to the interference pattern from the  The Journal of Physical Chemistry A pubs.acs.org/JPCA Article well-known "Young's double-slit experiment", as explained in a molecular scattering context in Appendix A of ref 28. This analogy was also used in XC1, 6 and it yields the simple relation where J eff is an effective total angular momentum variable characteristic of the NF oscillations. For example, for the m f = 0 DCSs, we have J eff = J g + 1/2, where J g is the glory angular momentum variable, defined as the position of a local maximum in a plot of arg S(J)/rad versus J (see Figure 2 of XC1 6 ). Figures 9 and 10 show that Δθ R usually lies in the range of Δθ R = 6°−7°, which is similar to the m f = 0 DCSs. Then eq 28 gives J eff = 30.0−25.7. An inspection of Figures 6  and 7 shows that these values for J eff are also close to a local maximum in the arg S(J)/rad plots.

FULL AND NEARSIDE-FARSIDE LAMs INCLUDING RESUMMATIONS
A full and N,F LAM analysis provides information on the value of the total angular momentum variable that contributes to the scattering at an angle θ R , under semiclassical conditions. An important tool 16,25 for interpreting a LAM plot is the exact Fundamental Identity for Full and N,F LAMs, which is also valid for r = 1 and is analogous to the identity for DCSs given by eq 9. Figure 11 shows a full and N,F LAM plot for the 000 → 011 transition using the unres NF decomposition for r = 0 and r = 1. We first make the following observations on the f ull LAM.
• The full LAM shows oscillations at small angles. At intermediate and larger angles it becomes monotonic and increases except for θ R around 140°.
• The full LAM changes from F dominance to N dominance as θ R increases in the small-angle region. This is the same behavior shown by the F and N r = 1 DCSs in Figure 9a. • The spike at θ R ≈ 46.2°corresponds to the minimum in the full DCSsee Figure 9a. Thus, the full LAM plot provides a clear indication of a change in the mechanism for the reaction as θ R increases. Next, we examine the N and F r = 0,1 LAMs in Figure 11 and note the following.
• The effect of cleaning the N,F r = 0 LAMs is very striking, with nonphysical oscillations for θ R ≲ 50°being replaced by slower variations in the N,F r = 1 LAMs. • It is known that N and F LAMs that are nearly zero, or oscillate about zero, are nonphysical. 13,15,16 It can be seen in Figure 11 that this occurs for both the r = 0 and r = 1 F LAMs when θ R ≳ 50°. The corresponding curves are drawn in a fainter blue compared to the F LAMs for θ R ≲ 50°. • An averaging of the N and F r = 1 LAMs for 10°≤ θ R ≤ 45°gives −28.1 and 28.2, respectively. The value J ≈ 28 is consistent with the information obtained from the periodicity of the fast oscillations in Section 6. An inspection of Figure 6a shows that J ≈ 28 is close to a local maximum in the arg S(J)/rad plot. • The N r = 1 LAM for θ R ≳ 50°is close to the full N LAM. And both of them monotonically increase (except for θ R ≈ 140°) and are similar to the LAM for a hardsphere collision. 13 This implies that the SOM model, which is an approximate N theory incorporating hardsphere dynamics, should be approximately valid at intermediate and large scattering angles. This point is confirmed in Section 8. The properties of the full and N,F r = 0,1 LAMs for the other transitions are similar to those for the 011 case and are not shown separately. Overall, we can say that the information given by the LAM analysis is consistent and complementary to that in the DCS plots of Figures 9 and 10.

SEMICLASSICAL OPTICAL MODEL (SOM) DCSs AT INTERMEDIATE AND LARGE ANGLES
The SOM is a simple procedure, introduced by Herschbach, 20,21 for calculating the DCSs of state-to-state reactions. In XC1, 6 we applied the SOM to the four m f = 0 transitions and showed that the SOM provided valuable insights into structures in the DCSs at intermediate and large angles. In particular, we found that the SOM and PWS DCSs are distorted mirror images of the corresponding P J ≡ |SJ| 2 versus J plots, with J = 0,1,2,.... The theory for the SOM has been given in XC1, 6 and below we just state the working equations when m f > 0.
The SOM DCS is given by 29) with P J ≡ P(J) and where J = m f , m f + 1,.... In eqs 29 and (30), d is the sum of the radii of two hard spheres representing the reactants and is the only adjustable parameter in the theory. The above equations assume that J ≤ kd; otherwise, σ SOM (θ R ) ≡ 0. Notice that the SOM only depends on the value of the modulus |SJ| and is   Figures 12 and  13 for the six transitions in the range of θ R = 50°−180°. Now, for the m f = 0 case, we obtained values of d by fitting the SOM DCS to the PWS DCS at, or close to, θ R = 180°. This does not work for m f > 0 because the PWS DCSs are equal to 0 a 0 2 sr −1 at θ R = 180°. Instead, we obtained d by fitting the SOM DCS at, or close to, the PWS peak nearest to θ R = 180°. An exception is the 000 → 031 transition in Figure 12c, for which the second nearest peak was used (note, this DCS exhibits the most detailed structure out of the six transitions). The values we used for d are given in the figures.
It can be seen in Figures 12 and 13 that the SOM reproduces the main features in the PWS DCSs, with larger deviations as the PWS DCSs become more structured. This is encouraging, considering the simplicity of the SOM, and, like the m f = 0 case, it tells us that the SOM and PWS DCSs are distorted mirror images of the corresponding P J versus J plots. As expected, the SOM does not reproduce the NF interference (or Fraunhofer) oscillations in the PWS DCSs for θ R ≲ 50°( not shown). Finally, we note that the values for d lie in the range of 1.44−1.97a 0 , which are much less than the sum of the radii at the saddle point for the BKMP2 potential energy surface, which is d ‡ = r HH ‡ + r HD ‡ = 3.514 a 0 . This tells us, as was also found for m f = 0, that the scattering at intermediate and large angles arises mainly from small values of J or, equivalently, from small-impact parameters.

DEGENERACY AVERAGED DIFFERENTIAL CROSS SECTIONS (daDCSs)
In this section, we calculate degeneracy averaged DCSs (daDCSs) and compare with the experimental daDCSs for the two transitions v i = 0, j i = 0 → v f = 0, j f = 1 and v i = 0, j i = 0 → v f = 0, j f = 3. The usual definition of a daDCS is In our applications, we have a single initial state, namely, v i = 0, j i = 0, m i = 0, so eq 31 simplifies to A further simplification is possible when m i = 0 because, as shown in Appendix A, DCSs for m f = −1,−2,−3 are equal to those for m f = +1,+2,+3, respectively. We can write eq 32 in the form where the sum is zero if j f = 0. Equation 33 can also be written in a more compact way, namely where the prime on the Σ sign means "multiply the first term in the sum by 1/2". We can also substitute eqs 1 and (2) into eq 34, obtaining  Figure 14a compares the calculated daDCS using eq 35 for the transition 00 → 01 with the experimental data. The corresponding results for the 00 → 03 transition are in Figure  14b. A single scaling factor has been applied to the experimental data to compare with the calculations. 1 The results in Figure 14a,b are an extension of the corresponding figures of Yuan et al. 1 because we included estimated experimental uncertainties. These are a 10% error in the measurements and an angular uncertainty of 1.5°. 1 It can be seen that the agreement between the calculated and experimental daDCSs is very good, in particular, for the NF interference (Fraunhofer) oscillations at θ R ≲ 40°; these are shown in more detail in the insets. In the experiments, also note that ∼97% of the HD molecules in the molecular beam are in their ground state, and the translational energy uncertainty is ∼1.2%. 1 If an experiment cannot resolve individual j f states, then it is necessary to sum over these states. We have (36) Figure 14c shows a plot of σ 00→0 (θ R ) versus θ R over the whole angular range. It can be seen that the structure in σ 00→0 (θ R ) (black curve) is largely washed out, even though the individual σ 00→0j f (θ R ) in eq 36 (colored curves) possess distinct fast and slow oscillations, although less pronounced than the helicityresolved DCSs in Figures 9 and 10. We can also relate our results and notations to the figures in the Supporting Information (SI) and main text of ref 1, with the following clarifications noted.

CONCLUSIONS
We have theoretically analyzed structures in the DCSs of the ground-state reaction H + HD → H 2 + D for the product states 011, 021, 031, 022, 032, and 033. The calculations extend and complement our previous analyses in XC1 6 for the cases 000, 010, 020, and 030, making 10 DCSs in all. The motivation comes from the experiments and simulations of Yuan et al., 1 who have measured for the first time fast oscillations in the small-angle region of the daDCSs for j f = 1 and 3 as well as slow oscillations in the large-angle region. Our main theoretical tools were two variants of Nearside-Farside theory: (1) We applied unrestricted, restricted, and restrictedΔ NF decompositions, including resummations, to the helicity PWS, which is expanded in a basis set of little d functions. We analyzed in detail the properties of restricted and restrictedΔ NF DCSs and showed that they 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 calculated LAMs to obtain further insights into the reaction dynamics. Properties of little e functions played an important role in the NF analysis, as do the caustics associated with the little d and little e functions. (2) We applied an approximate N theory at intermediate and large angles, namely, the Semiclassical Optical Model.
We showed that the fast oscillations at small angles (sometimes called Fraunhofer diffraction or oscillations) arise from an NF interference effect. In contrast, the slow oscillations at large angles are an N effect and arise in the DCS as a distorted mirror image of the corresponding P J versus J plot. We also compared with the experimental daDCSs, obtaining very good agreement.
Our analyses confirm the earlier insight of Dobbyn et al. 7 that as the PWS increases in complexity, this has little impact on the physical insight provided by an NF analysis.