Stereodynamics of rotationally inelastic scattering in cold He + HD collisions

Molecular hydrogen is the most abundant molecule in the universe, and it is also the simplest neutral molecule. Collisions of H₂ and its isotopomers with He and H₂ have long served as prototypes for inelastic rotational and vibrational transitions in molecules, while H + H₂ and its isotopic counterparts have served as benchmarks for reactive scattering.^1–7 These systems continue to attract considerable interest as they are amenable to precise calculations of their interaction potentials and highly resolved experimental studies. Indeed, the HD molecule has been the topic of a number of recent studies focusing on the geometric phase effect in chemical reactions^5–7 and stereodynamics of cold molecular collisions.^8–14 The latter topic has attracted considerable attention as recent developments in intra-beam technologies^15,16 allow molecular collisions to be explored near 1 K. In this regime, collisions of light molecules such as HD occur through a few low angular momentum partial waves. Stereodynamics of such collisions is often controlled by isolated resonances and allows theoretical predictions to be benchmarked against experiments.^12–14,17 Experiments on cold and ultracold molecules^18–22 have largely focused on dimers of alkali metal atoms prepared using photoassociation or Feshbach resonance methods, and HD presents an excellent test case for benchmarking theory and experiment for a prototype non-alkali metal dimer system.

Recently, the stereodynamics of rotationally inelastic scattering of HD with He was examined by Perreault et al.^10,11 in which the Stark induced Adiabatic Raman Passage (SARP) technique was used to prepare HD initially in the v = 1, j = 2 ro-vibrational state with control of its alignment and orientation. By analyzing the angular distribution of the inelastically scattered HD in the v′ = 1, j′ = 0 state, they concluded that only even partial waves contribute, and in particular, an isolated incoming d-wave (l = 2) shape resonance near a collision energy of 1 cm⁻¹–3 cm⁻¹ accounts for the observed angular distribution. However, the experiment does not provide energy resolved measurements, although 90% of the collisions were attributed to energies <5 K, drastically limiting the number of partial waves.

In this Communication, we provide a theoretical analysis of stereodynamics of rotationally inelastic scattering of HD (v = 1, j = 2 → v′ = 1, j′ = 0) by collisions with He based on rigorous quantum scattering calculations on a highly accurate ab initio potential energy surface (PES) for the HeH₂ complex. It is found that the integral cross section (ICS) for j = 2 → j′ = 0 transition exhibits a stereodynamic preference with a universal trend below <0.8 cm⁻¹ that also features a pronounced peak due to an incoming p-wave (l = 1) shape resonance. In the energy range of 1 cm⁻¹–3 cm⁻¹, the ICS displays a weak shoulder feature arising from a l = 2 shape resonance. The stereodynamic effect in this regime and at higher energies is significantly influenced by the interference between different partial waves. We show that dominant features of the experimental angular distribution measured by Perreault et al.^10,11 include signatures of both l = 1 (odd) and l = 2 (even) partial wave shape resonances.

Rotationally inelastic scattering of HD by He has been extensively reported in the literature primarily for astrophysical applications.^23–25 The HD molecule breaks the homonuclear symmetry of H₂ and angular dependence of the interaction potential for He + HD features both even and odd terms compared to only even terms for He + H₂. In this study, we adopt a modified version of the PES for the HeH₂ system developed by Bakr, Smith, and Patkowski²⁶ referred to as the BSP3 PES.²⁷ Its uncertainty in the van der Walls well region is estimated to be less than 0.04 K, well below the mean collision energies involved in the experiments of Perreault et al.¹⁰ The quantum scattering calculations are carried out based on the close-coupling (CC) scheme using the MOLSCAT code²⁸ (see also the supplementary material).

In the SARP experiment,¹⁰ the HD molecule is prepared in a rotational state j with a specific projection component m as |j, m⟩ or a coherent superposition of rotational states with different m,^8,13 where the z-axis for the projection of j is taken parallel to the initial relative velocity for the collision. For j = 2, in the experimental H-SARP preparation (the HD bond is preferentially aligned parallel to the initial relative velocity), the rotational state is described as |j, m⟩ = |2, 0⟩, while for the V-SARP preparation (the HD bond is preferentially aligned perpendicular to the initial relative velocity), the rotational state is described as $3/8(|2,−2+|2,2)−(1/2)|2,0$⁠. The HD is also prepared in a state $1/2(|2,−1+|2,1)$ using X-SARP in the experiment.¹⁰ 

For collisions between a structureless atom (¹S) and a closed-shell molecule (¹Σ), the scattering amplitude for a rotational transition between oriented (m-specified) rotational states is given by^29–32 

where θ and ϕ are the scattering polar angles in the center of mass frame in which the z-axis is parallel to the initial relative velocity for the collision, E is the total energy, J is the quantum number of the total angular momentum of the collision complex defined as J = j + l with l denoting the orbital angular momentum between collision partners, T(E) is the T-matrix obtained from the S-matrix as T(E) = 1 − S(E), and Y_l′m−m′ denotes a spherical harmonic.

To obtain differential and integral cross sections, DCS and ICS, it is necessary to evaluate the square of modulus of the scattering amplitude. The DCS is given as

where k is the magnitude of the initial wave vector. Since the ϕ-dependence is not measured in the experiment, our interest is the θ-dependence obtained after averaging over ϕ,

where f_{jm→j′m′}(θ, E) is the θ-dependent scattering amplitude defined by the relation f_{jm→j′m′}(θ, ϕ, E) = f_{jm→j′m′}(θ, E)exp{−i(m − m′)ϕ}. This yields |f_{jm→j′m′}(θ, ϕ, E)|² = |f_{jm→j′m′}(θ, E)|², and thus, the integral over ϕ in Eq. (3) results in a constant factor of 2π due to the absence of ϕ-dependence in the integrand. In other words, as long as we focus on a transition process from a specific |j, m⟩ state, the DCS is independent of ϕ. The ICS (σ_{jm→j′m′}) is obtained by taking an integral of the DCS over θ from 0 to π.

Our goal is to examine the quenching of HD from j = 2 → j′ = 0 (m′ = 0) prepared in various initial bond-axis alignments and initial rotational state orientations specified with m. Thus, the m-dependence (m = 0, ±1, and ±2) of f_{j=2,m→j′=0,m′=0} and associated cross sections are fundamental quantities because the cross section for any initial orientation or alignment is given by the sum of these fundamental cross sections. For example, the ICSs with the V-SARP and X-SARP preparations are given using the square of modulus of the coherent expansion coefficients for the initial rotational states as σ^V = (3/8)(σ_2,−2→0,0 + σ_2,2→0,0) + (1/4)σ_2,0→0,0 and σ^X = (1/2) (σ_2,−1→0,0 + σ_2,1→0,0), respectively. Note that σ^X = σ_2,−1→0,0 = σ_2,1→0,0.

Figure 1 (a) shows the ICS for j = 2 → j′ = 0 as a function of the collision energy for non-polarized (isotropic) collisions and different initial orientations of the rotational state as well as the initial bond-axis alignments with the SARP preparations. Two features are notable: a prominent peak near 0.2 cm⁻¹ from an l = 1 partial wave and a shoulder at 1 cm⁻¹–3 cm⁻¹ from an l = 2 partial wave. For j = 2, the partial wave l = 0 occurs for J = 2, the l = 1 contribution arises from J = 1 and 3, and l = 2 from J = 0, 2 and 4. For both l = 1 and 2, the component associated with the highest J and outgoing partial wave l′ given as l + 2 has the largest contribution (see Fig. S1 of the supplementary material). We note that J = l′ due to j′ = 0 and the conservation of J.

Previous calculations by Zhou and Chen³³ for (j = 2 → j′ = 0 in v = 0) using the BSP PES of Bakr et al.²⁶ have yielded similar ICS as our isotropic result, including the pronounced (l = 1) and the weak shoulder (l = 2) resonances. Additional tests by scaling the BSP3 PES by ±1% did not alter the resonance structures. We tested three other available HeH₂ PESs^26,34,35 and all yielded very similar results as the BSP3 PES employed in this study. For these tests, we used the modified version of the PES by Boothroyd et al.³⁴ and Muchnick and Russek³⁵ reported by Bakr et al.²⁶ A comparison of the ICSs from these PESs indicates that the resonance features are robust to possible uncertainties in the PES (see Fig. S2 of the supplementary material). Moreover, recent line shape parameters of H₂ and HD immersed in He evaluated using BSP3 PES achieved excellent (subpercent) agreement with ultra-accurate experimental results,^36,37 validating the spectroscopic-level accuracy of the PES used in this study.

Around the primary peak (l = 1), the ICS clearly shows a stereodynamic preference (m-dependence). The ICS with m = 0 is the largest in the whole energy range, while |m| = 2 is the lowest except around round 2 cm⁻¹ (in the vicinity of the l = 2 resonance) where the order of |m| = 1 and 2 is reversed. Below, we will discuss the origin of the observed m-dependence. As 75% of the V-SARP ICS comes from m = ±2, the H-SARP ICS (m = 0) should be larger in the entire energy range, consistent with experimental results. In the lowest energy region, the stereodynamic preference disappears as s-wave scattering dominates, and the Wigner threshold behavior of inelastic cross section emerges (see also Fig. S1 in the supplementary material).

Now, we analyze the origin of the stereodynamic preference for the j = 2, m → j′ = 0, m′ = 0 transition. In the low energy regime, the scattering amplitude [Eq. (1)] is dominated by the s-wave (l = 0, J = l′ = 2) in the incident channel. The initial orientation (m) dependence of the scattering amplitude comes from the two 3-j symbols. Note that m-dependence in the spherical harmonic Y_l′m will not remain in the ICS because the integral of |Y_l′m|² over θ yields unity in evaluating ICS. The product of the two 3-j symbols results in the m-independent factor of 1/5, while each 3-j symbol has a m-dependent phase factor³⁸ (see also the supplementary material). A striking feature is that the initial orientation independence comes from purely algebraic character of the scattering amplitude, and thus, the m-independence of the ICS (j = 2 → j′ = 0) is universal for atom (¹S) + molecule (¹Σ) systems in the low energy limit. Also, this m-independence for l = 0 holds at any collision energy. We note that the m-independence for l = 0 applies to systems including additional angular momenta as demonstrated for the F + H₂ → HF + H reaction by Aldegunde et al.³⁹ Threshold laws for m-changing collisions have been discussed by Krems and Dalgarno.⁴⁰ 

From the above analysis, it is clear that the origin of the stereodynamic preference observed at low energies (below 0.8 cm⁻¹) in Fig. 1(a) is due to the l = 1 partial wave. For j = 2, there are two terms associated with l = 1 in the scattering amplitude: J = l′ = 1 and J = l′ = 3. Again m-dependence for these terms comes from the two 3-j symbols. For J = l′ = 1, the product of the two 3-j symbols is m-dependent as $−(4−m2)/90$ which yields zero for |m| = 2.^38,41 As shown in Eq. (3), the square of the modulus of the scattering amplitude determines the magnitude of the cross section, and thus, the ratio of the ICS is 4:3 for m = 0 vs |m| = 1. Similarly, for J = l′ = 3, the ratio of the ICSs becomes 9:8:5 for m = 0, |m| = 1, and |m| = 2, respectively⁴¹ (see also the supplementary material). Importantly, the stereodynamic preference for m = 0 and |m| = 1 is similar for both J = 1 and 3. We note that, for the ICS, there is no interference effect from the cross terms arising from these two terms and that with l = 0 (J = 2).

Thus, a universal trend exists for the stereodynamic preference for the ICS in low energy rotational quenching (j = 2 → j′ = 0). We observe no m-dependence in the s-wave regime followed by a gradual universal trend in m-dependence with increasing collision energy due to the contribution from an l = 1 partial wave [m = 0 (H-SARP) > |m| = 1 (X-SARP) > (V-SARP) > |m| = 2]. This universal trend is regardless of the presence of an l = 1 shape resonance, but in the present case, the shape resonance accentuates this feature. In Fig. 1(b), we show initial HD orientation (m) dependence of ICS ignoring interference between various l waves in the scattering amplitude. In comparison with (a), we observe that it is possible to describe the ICS accurately without the interference terms (cross terms) of the scattering amplitude until (including) the region of the l = 1 shape resonance. At higher energies (above 0.8 cm⁻¹), the interference effect between higher-order partial waves plays a crucial role in the stereodynamics.

As stated above, the experimental study¹⁰ concluded that only even partial waves, in particular, a resonantly enhanced l = 2 partial wave, contribute to the inelastically scattered HD angular distribution. In contrast, the calculated ICS reveals that contribution from the l = 1 resonance is significant in the relevant collision energy regime and its background contribution is non-negligible even in the region of the l = 2 resonance, ∼1 cm⁻¹–5 cm⁻¹ (see Fig. S1 of the supplementary material). However, if the intensities of the l = 1 and l = 2 resonances are reversed or if collisions below 1 cm⁻¹ are ineffective in causing rotational relaxation under the experimental conditions, then it is conceivable that experimental results will show primarily signatures of the l = 2 resonance. As discussed earlier, resonance features are very stable and robust across all available HeH₂ PESs.

Next, we consider angular distribution of inelastically scattered HD and contributions from even and odd partial waves. To simulate the experimental angular distribution, it is necessary to calculate the differential rate constant¹² averaged over the relative velocity distribution of He and HD in the experiment. The angular distribution is given by^10,12

We note that the relative velocity distribution P(v_r) reported by Perreault et al.¹⁰ is not symmetric about v_r = 0 (see also Fig. S3 of the supplementary material). The DCS dσ(θ)/dθ depends on the relative velocity v_r and the collision energy as given in Eq. (3). In particular, near a resonance, the DCS shows significant energy dependence (see also Figs. S4–S7 of the supplementary material).

In Fig. 2, we present the differential rate constants for even and odd partial wave contributions derived from Eq. (4). As discussed by Perreault et al.,¹⁰ by choosing only even or odd partial waves, the symmetric angular distribution about θ = 90° is obtained regardless of the symmetry of the relative velocity distribution about v_r = 0. The results exhibit distinctly different features for even and odd partial waves. The experimental results¹⁰ (see Fig. 4) for H-SARP and V-SARP exhibit two prominent peaks centered around 25° and 165° with a shallow peak near 90°. The shallow peak is more prominent for the V-SARP preparation. While the experimental results were attributed to signatures of an l = 2 resonance,¹⁰ the computed H-SARP and X-SARP results for odd partial waves (primarily l = 1) appear to be in better agreement with experiment. The computed V-SARP results show markedly different angular dependence compared to the experimental data and exhibit distinctly different features for even and odd l values. In particular, the strong central peak seen for even l is absent for odd l. The m-resolved differential rate constants that include contributions from both even and odd partial waves are displayed in Fig. 3 as a function of the scattering angle. The dominant contribution from l = 1 is clearly reflected in the overall rate constant.

In Fig. 4, we compare our results that include contributions from both even and odd partial waves with the experimental angular distributions.¹⁰ For comparison, we omitted contributions from DCS below 0.01 cm⁻¹ (|v_r| ∼ 12 m/s) in evaluating the integrals in Eq. (4) to limit contributions from purely s-wave scattering. However, this truncation has only a very small effect as higher or lower values of energy truncations yield very similar results (see Fig. S8 of the supplementary material). Our theoretical results for H-SARP and X-SARP are still in close agreement with experiment even if contributions from both even and odd partial waves are included. The main difference is that the symmetry about θ = 90° is now absent in the theoretical results. In the analysis of the experimental results,¹⁰ it was assumed that the relative velocity dependence of the DCS is negligible, leading to an additional assumption that the DCS is symmetric about θ = π/2 to explain the observed symmetry of the angular distribution. However, this assumption is not fully valid as DCS exhibits strong energy dependence, particularly in the vicinity of resonances, and the experimental relative velocity distribution is not sufficiently narrow to capture just the peak of a single (partial wave) resonance.

In summary, we have carried out explicit quantum calculations of integral and differential cross sections for rotational quenching of state prepared HD (v = 1, j = 2 → v′ = 1, j′ = 0) in collisions with He at relative collision energies below 10 K. Our results show a prominent l = 1 shape resonance near 0.2 cm⁻¹ and a weak shoulder feature from an l = 2 partial wave between 1 cm⁻¹ and 3 cm⁻¹. These two resonances control the stereodynamics of the j = 2 → j′ = 0 rotational transition in HD. We have also identified a universal trend for the stereodynamic preference in the low energy regime for the integral cross section and its features in the vicinity of an l = 1 resonance. It is found that the interference effect due to cross terms in the square of modulus of the scattering amplitude plays a critical role in determining the stereodynamic preference at higher energies where higher-order partial wave contributes. This interference effect alters the stereodynamic preference near an l = 2 resonance. The angular distribution of the scattered HD molecule is found to be strongly influenced by the l = 1 resonance. The calculated angular distributions for the H-SARP and X-SARP preparations are found to be in overall good agreement with experiment but do not depict a symmetric profile around θ = 90° due to the strong energy dependence of the scattering amplitude and the interference between even and odd partial wave contributions. The V-SARP results also show broad agreement with experiment except that the central peak reflects contributions from both even and odd partial waves leading to a double peak structure. We believe that measurements that include the energy dependence of the angular distribution will allow a more accurate comparison with experiment.

See the supplementary material associated with this article for computational details of the scattering calculation, partial wave contributions in the integral cross section, sensitivity to the potential energy surface, universal trend in the stereodynamic preference at low energies, relative velocity distribution of He and HD, energy dependence of the differential cross section, and low energy cutoff effect on the differential rate constants.