Introduction

The recent observation of scattering resonances in low-energy molecular collisions arguably has been one of the most exciting breakthroughs in modern atomic and molecular physics. Scattering resonances can only be understood from the framework of quantum mechanics, and they are testimony to the wavelike quantum nature of matter. In a simplified picture, these resonances may be regarded as the orbiting of the two collision partners around each other that only occurs at low energies, typically in the 10−3 to 10 K range, as here the de Broglie wavelength is sufficiently large for quantum effects to dominate. Even though classically forbidden, they can occur either by tunneling through a potential barrier (an orbiting or shape resonance), or by the transient excitation of the molecule to a state of higher energy (a Feshbach resonance).

Scattering resonances manifest themselves by a dramatic increase in the integral cross sections (ICSs) at the resonance energies, accompanied by rapidly changing differential cross sections (DCSs). They are extremely sensitive to the details of the potential energy surface (PES) describing the interaction between the colliding molecules; a change in PES of typically less than a percent can already induce major differences in the resonance structures. Not surprisingly, there has been a long-term quest to observe (and fully resolve!) these scattering resonances experimentally, ideally in collision experiments retrieving both state-to-state integral and DCSs1. Using a crossed-beam approach, resonances were first observed in 1972 for H atoms scattering with Hg atoms2 and later other collision partners3. Total ICSs were recorded by scanning the collision energy with mechanical velocity selectors while monitoring the hydrogen beam depletion. Over the next few decades, resonance states were mostly probed through IR-spectroscopy of weakly bound dimers4. In 1993, evidence of a resonance in reactive scattering was observed for F + H2 → HF + H collisions5. By applying merged-beam approaches, resonances in ICSs were observed since 2012 in Penning ionization reactions, at collision energies down to 0.01 cm−1 6,7,8,9,10,11,12.

Resonances in state-to-state inelastic ICSs were first recorded in 2012 for CO-H2 at collision energies down to 4 cm−1 13,14, using crossed molecular beams with a small and variable intersection angle in combination with state-selective Resonance Enhanced Multi Photon Ionization (REMPI) detection. This approach has since been used to study scattering resonances in state-to-state ICSs for O2-H2, CO-He, C-He/H2/D2 and D2O-H2 collisions15,16,17,18,19,20,21, although pronounced resonance structures could not always be resolved in these studies. The highest experimental resolution thus far is obtained using the Stark deceleration and velocity map imaging (VMI) techniques, that in addition to measurements of ICSs enabled the probing of the energy dependence of DCSs in the resonance region for the NO-He and NO-H2 systems at collision energies down to 0.2 cm−1 22,23,24,25,26. The unprecedented high resolution obtained in these experiments required quantum chemistry calculations beyond the CCSD(T) gold standard level of theory. For the benchmark NO-He system, only theory with corrections at the CCSDT(Q) level could reproduce the experimental obervations, epitomizing the extreme sensitivity of resonance features to details of the PES26. VMI has recently also been applied to record ICSs and DCSs for elastic scattering of He* with D2 down to 0.7 cm−1 27, as well as to probe resonance effects in inelastic collisions between Zeeman decelerated C atoms and H2 molecules at energies down to 0.5 cm−1 28.

Despite these breakthroughs, many open questions still remain. Can we, for instance, extend the unprecedented experimental precision and exquisit agreement with state-of-the-art quantum theory to more complex systems beyond benchmark systems like NO-He? Stepping up the complexity ladder is essential to test new multi-electron quantum chemistry methods and validate the approximations inevitably needed to describe larger systems, and would help bridge the gap between ab initio methods typically used for small weakly-interacting systems and density-functional or semi-empirical methods that are more relevant for heavier and strongly interacting systems. It is also essential to test quantum scattering methods, since for larger molecules, there is an increasing number of effects that need to be taken into account to properly describe how the system evolves over the PES. A change of system may also facilitate the manipulation of resonance structures using external electric or magnetic fields. The idea is that at energies below  ~1 Kelvin, the interaction energy of a polar molecule with external electric and magnetic fields is on the order of the collision energy itself, offering the distinctive opportunity to engineer interaction Hamiltonians and control the collision outcome.

The systems used thus far to probe scattering resonances are often predominantly chosen for reasons of experimental feasibility, but they are unfortunately not very favorable to further break new ground. The NO radical, for instance, is particularly easy to produce and detect, but its modest dipole moment of 0.16 D makes scattering resonances involving NO rather immune to electric fields. This low dipole moment also prohibits reaching lower energies by beam merging, which requires a sufficiently strong electric field-induced force to bend the beam’s trajectory. In these respects, molecules like OH and ND3 are much more appealing, and have been prime candidates in cold molecular research ever since the field started in the 1990’s29. However, they are either difficult to produce in large quantities required for controlled scattering experiments or lack sensitive detection schemes that allow for high-resolution VMI detection. For the latter, one of the strongest bottlenecks is the requirement for a state-selective REMPI detection scheme that ionizes the molecules near threshold, thus imparting negligible recoil energy to the detected ions. Such schemes are generally lacking for most molecules of interest or are impedingly insensitive. Being a tour-de-force experimentally, measurements of state-to-state resolved ICSs and DCSs of quantum scattering resonances involving molecules thus remain restricted to systems involving the NO radical, and it is until now unclear how and if the major hurdles can be overcome to extend these studies to other systems.

Cold collision studies involving ND3 are particularly relevant, as ND3 has been the system of choice in many seminal experiments on the manipulation of neutral polar molecules. The molecule was used in the first demonstration of electrostatic trapping30,31, AC trapping32,33, a buncher34, mirror35, storage ring36, synchrotron37,38, beamsplitter39,40, fountain41, cryofuge42, co-trapping with laser-cooled atoms43, as well as in the first demonstration of the increased spectral resolution by the elongated interaction time afforded by decelerated molecules44. Since the discovery of NH3 in the interstellar medium in 196845, and of ND3 in 200246, rotationally inelastic collisions involving ammonia have attracted considerable interest47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64. Since then, observed inversion transitions are used to probe the temperature of molecular clouds65,66.

Here, we report measurements of scattering resonances in the ICS and DCS for inelastic inversion-deexcitation collisions between ND3 (\({1}_{1}^{-}\to {1}_{1}^{+}\)) and H2 or HD. The collision energy is varied between 0.5 and 25 cm−1, scanning over several groups of resonances for both systems. DCSs are probed with high resolution using VMI in combination with a near-recoil free \(1+{1}^{{\prime} }\) REMPI scheme using Vacuum Ultra Violet (VUV) light67, solving a long-standing bottleneck in using ND3 in high-resolution imaging experiments. We find that a new PES at the CCSD(T)/AVTZ+MB level of theory, with a correction based on CCSDT(Q)/AVDZ calculations, is required to find quantitative agreement between the predicted and measured resonance positions.

Results

The experiments were performed by crossing a state-selected and velocity-controlled ND3 packet emerging from a Stark decelerator with a beam of para-H2 or HD at an angle of 5.2°. The H2 (HD) beam traveled at a fixed velocity of v2 = 865 m/s (815 m/s) after expanding from a cryogenically cooled pulsed valve at 35 K (40 K). Using the Stark decelerator, the velocity of the ND3 packets was tuned between 350 and 980 m/s, resulting in collision energies Ecol between 0.5 and 25 cm−1. The collision energy resolution ranged from 0.1 cm−1 at the lowest energies to 2.5 cm−1 at the highest energies. The jk = 11 rotational ground state of para-ammonia is split into two inversion components with opposite parity, of which the Stark decelerator only transmits molecules in the \({1}_{1}^{-}\) upper inversion component.

We first measured ICSs for \({1}_{1}^{-}\to {1}_{1}^{+}\) inelastic inversion-deexcitation collisions by state-selectively ionizing scattered ND3 (\({1}_{1}^{+}\)) molecules using a convenient 2+1 REMPI scheme at 321 nm, see Fig. 1. For ND3-H2, three distinct resonance features were observed at energies ~1, 6, and 13 cm−1. For ND3-HD, the resonances are less pronounced but three resonance features causing a series of inflection points could clearly be discerned.

Fig. 1: Integral cross section (ICSs) as a function of the collision energy, Ecol.
Fig. 1: Integral cross section (ICSs) as a function of the collision energy, Ecol.
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ICSs for ND3 (\({1}_{1}^{-}\to {1}_{1}^{+}\)) collisions with (a) H2 or (b) HD, both with j2 = 0. Each panel shows a comparison between experimental (black) and predicted cross sections based on the CCSD(T)+ET(Q) PES (blue). Horizontal error bars reflect the energy calibration uncertainty, computed by propagating the uncertainty in v2 (see SI). Vertical error bars represent statistical uncertainties, calculated as the standard deviation of the mean over hundreds of samples (see Methods section). All error bars represent a 95% confidence interval. The smoothing of the predicted cross sections due to the experimental resolution was taken into account based on simulations. Source data are provided as a Source Data file.

We compared the experimentally observed ICSs with the calculated ICSs based on the available PES computed by Maret et al.66,68. This PES was calculated at the CCSD(T)/AVDZ level of theory, after which the correlation part of the interaction energy was scaled using CCSD(T)/AVTZ calculations by performing a two-point CBS extrapolation (see SI). In the scattering calculations, a rotational basis up to j = 6 for ND3, j2 = 2 for para-H2 and j2 = 4 for HD was used. The resulting ICSs, convoluted with the experimental resolution, gave unsatisfactory agreement with the experimentally observed ICSs, as the resonance structures appeared shifted to higher collision energies (see SI).

We therefore attempted to calculate PESs at a higher level of theory. We found that CBS extrapolation to AVTZ and AVQZ gave incorrect results. Unfortunately, a complete basis set extrapolation, as performed for NO-He (using AVnZ for n = 4, 5, 624), is currently unrealistic. Compared to NO-He, ND3-H2 is much more computationally demanding even though it has fewer electrons (12 instead of 17) and no open-shell character. The increase in computational cost is caused by the increased number of degrees of freedom (5 instead of 2) in the rigid rotor approximation. Where the NO-He potential could be defined by a grid of 912 geometries, our ND3-H2 potential required 29,569 points. We computed this PES at the CCSD(T)/{AVTZ+MB} level of theory and applied an additional correction ET(Q) that depended on the radial coordinate only (see SI). This correction function was determined by comparing calculations at the CCSD(T)/AVDZ and CCSDT(Q)/AVDZ level of theory, similar to recent work for NO-He24. Instead of evaluating ET(Q) at every point of the PES as was done previously for NO-He, we averaged the radial dependence found for a few fixed angular coordinates. For the N-D bond distance a value of 1.946 a0 was used, which is the vibrational average of NH369. The inclusion of midbond functions was found to yield more accurate energies at a lower computational cost compared to using the AVQZ basis set, based on a handful of test geometries for which calculations up to CCSD(T)/AV6Z were performed (see SI). We found that the resonances at energies ~1 cm−1 and below responded extremely sensitively to the level of theory we used (see SI).

Our new CCSD(T)+ET(Q) PES was found to be deeper than the Maret PES by ~2%, which caused the resonances to shift to lower energies in better agreement with the experiments, although an intensity mismatch across the sampled collision energies remained, see Fig. 1. We tested several further modifications to the potential at less computationally expensive levels of theory. Most notably, we explicitly included the umbrella coordinate of ND3, as vibrational motion may impact the low-energy scattering behavior70,71. Although full dimensional calculations are currently not feasible for ND3-H2, we evaluated every point of the 5D PES at ten different umbrella angles to yield a new 6D PES beyond the rigid rotor approximation. This modification was found to have a too small effect to explain the discrepancy between experiment and theory, however, consistent with previous results for collisions of NH3 with rare gas atoms61,72. Furthermore, we changed the N-D bond length and used a global scaling factor, but these efforts did not result in a better agreement between experiment and theory (see SI).

Characterization of the resonances was achieved by performing a full partial wave analysis and by calculating the scattering wavefunctions at the resonance energies. While scattering, the total parity \({{{\mathcal{P}}}}\) as well as the total angular momentum with quantum number \({{{\mathcal{J}}}}\) are conserved (see SI). We found that both parities contribute near-equally to the scattering cross sections, such that a detailed analysis for one value of \({{{\mathcal{P}}}}\) sufficed. The total angular momentum is obtained by coupling the partial wave with quantum number ℓ with the rotational angular momenta j and j2 of the ND3 and H2 molecules, respectively. We calculated for each value of \({{{\mathcal{J}}}}\) the individual contribution to the scattering cross section (see Fig. 2), and found that groups of overlapping resonances cause the observed resonance features in the ICSs. The resonance observed at a collision energy ~1.2 cm−1 in ND3-H2 appeared relatively pure, with a dominant contribution of \({{{\mathcal{J}}}}=3\).

Fig. 2: Calculated cross sections based on the CCSD(T)+ET(Q) PES as a function of the collision energy.
Fig. 2: Calculated cross sections based on the CCSD(T)+ET(Q) PES as a function of the collision energy.
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Cross sections for ND3 (\({1}_{1}^{-}\to {1}_{1}^{+}\)) collisions with H2 (a, c) or HD (b, d). a, b Calculated ICS (black), with the contribution of angular momentum states \({{{\mathcal{J}}}}\) (see legend). Black triangles mark the energies at which the DCS was probed experimentally. c, d 2D image plot of the calculated Differential Cross Section (DCS) as function of the collision energy and scattering angle. The DCS is normalized for each energy separately to emphasize the angular structure. Source data are provided as a Source Data file.

For each resonance, we could derive the values of ℓin and ℓout that represent the relevant partial wave of the entrance and exit channels, respectively, as well as the resonant partial wave \({\ell }_{{{{\rm{res}}}}}\) that characterizes the quasi-bound state from which the resonance originates. Since we exclusively studied \({1}_{1}^{-}\to {1}_{1}^{+}\) inversion changing collisions, the value for ℓ can only change from ℓin even to ℓout odd or vice versa. The partial waves are further constrained by the conservation of \({{{\mathcal{J}}}}\), which for the \({1}_{1}^{-}\to {1}_{1}^{+}\) transition implies that \({\ell }_{{{{\rm{in/out}}}}}=\left\{{{{\mathcal{J}}}}-1,{{{\mathcal{J}}}},{{{\mathcal{J}}}}+1\right\}\). Together with the calculated scattering wavefunction, we could infer that during the collision the partial waves evolve from \({\ell }_{{{{\rm{in}}}}}=\left\{2,4\right\}\) via a resonance state with \({\ell }_{{{{\rm{res}}}}}=4\) to ℓout = 3 (see SI). The resonance state could be associated with the \({2}_{1}^{-}\) rotational level of ND3, and could hence be characterized as a Feshbach resonance (the \({2}_{1}^{-}\) state is asymptotically closed at a collision energy of 1.2 cm−1). The resonance appears as a relatively broad feature in the ICS, indicating that the corresponding quasi-bound state is short-lived.

Using similar reasoning, we could fully characterize the ten most prominent resonances for ND3-H2. We found that nearly all resonances are of Feshbach character, except for the two resonances at Ecol = 7.87 and 7.97 cm−1 that could be characterized as shape resonances, and the resonance at Ecol = 14.47 cm−1 that could be best described as a combined Feshbach-shape resonance (see SI).

We further investigated the resonances by calculating the DCSs as a function of collision energy, that directly reflect the partial wave composition of a resonance. The theoretical DCSs computed from the CCSD(T)+ET(Q) PES, see Fig. 2, showed a pattern of diffraction oscillations whose spacing scales with \(1/\sqrt{{E}_{{{{\rm{col}}}}}}\) 73,74,75. The diffraction pattern was interrupted at energies that coincided with a resonance, reflecting the vastly different scattering behavior in which only selected partial waves dominated when a resonance was accessed. The angular distributions changed rapidly as the collision energy was tuned over the resonances, with the appearance and disappearance of pronounced scattering flux in particular angular regions, such as the strong backward scattering in energy windows between adjacent groups of resonances. Such rapid variation was also observed in previous work on NO-He24, and is the result of the rapidly changing partial wave composition underlying each resonance. The interference between individual partial waves can strongly enhance or reduce the flux in specific angles, and the observation of such fast evolution of DCSs is testimony of the quantum nature of scattering resonances in low-energy collisions.

We probed the DCSs experimentally using VMI in combination with a new recoil-free \(1+{1}^{{\prime} }\) REMPI scheme involving VUV67. The 2+1 REMPI scheme used for ICS measurements was not suitable for this purpose, as this scheme imparts 17 m/s recoil velocity to the ND3 ions, blurring the images. The lack of a suitable REMPI scheme hampered measurements of DCSs in low-energy collisions before, but was essential in our experiment to record angular distributions with sufficient resolution to infer the energy dependence of DCSs in the resonance region. We recorded a total of 20 high-resolution scattering images at different collision energies accross the resonance regions, see Figs. 3 and 4. We observed clear diffraction oscillations with additional structures featuring strong backscattering at selected energies, consistent with the rapidly changing DCSs around the resonances we found theoretically. We quantitatively compared the experimental images with simulated images based on the theoretically predicted DCSs and the kinematics of the experiment. Angular scattering distributions were extracted from all images, and in general excellent agreement was found between the experimental and simulated distributions. For ND3-H2, a deviation was found in the backscattered region at Ecol = 3.4 and 4.1 cm−1, which we attributed to the small shift between the experimentally observed and theoretically predicted resonance position in the ICS, in combination with the sudden appearance of backscattering close to these collision energies. We found that only the CCSD(T)+ET(Q) PES gave good agreement between experiment and theory; as resonant features in the DCS are shifted along with those in the ICS, we found that simulated images based on the PES by Maret et al. did not capture the observed energy dependence of the DCSs well.

Fig. 3: Experimental (exp) and simulated (sim) velocity mapped images for ND3(\({1}_{1}^{-}\to {1}_{1}^{+}\))-H2 scattering at several collision energies.
Fig. 3: Experimental (exp) and simulated (sim) velocity mapped images for ND3( $${1}_{1}^{-}\to {1}_{1}^{+}$$ 1 1 − → 1 1 + )-H2 scattering at several collision energies.
Full size image

The simulated images were generated using the calculated DCSs based on the CCSD(T)+ET(Q) PES, as shown in Fig. 2b. They are oriented such that the forward direction faces right. Each image pair was normalized by their integrated intensity. The angular distributions extracted from the images are shown for every collision energy to the right of every image pair. Source data are provided as a Source Data file.

Fig. 4: Experimental (exp) and simulated (sim) velocity mapped images for ND3(\({1}_{1}^{-}\to {1}_{1}^{+}\))-HD scattering at several collision energies.
Fig. 4: Experimental (exp) and simulated (sim) velocity mapped images for ND3( $${1}_{1}^{-}\to {1}_{1}^{+}$$ 1 1 − → 1 1 + )-HD scattering at several collision energies.
Full size image

The simulated images were generated using the calculated DCSs based on the CCSD(T)+ET(Q) PES, as shown in Fig. 2d. They are oriented such that the forward direction faces right. Each image pair was normalized by its integrated intensity. The angular distributions extracted from the images are shown for every collision energy to the right of every image pair. Source data are provided as a Source Data file.

Discussion

Our joint experimental and theoretical study of partial wave scattering resonances in ND3-H2 and ND3-HD at collision energies down to 0.5 cm−1 underlines the level of detail that can now be obtained in scattering experiments involving polyatomic systems. As only the second molecular system for which we were able to probe resonances in both state-to-state ICS and DCS with high resolution, the success attained here unlocks possiblities to probe fully controlled quantum dynamics studies beyond the benchmark NO-He system. Since ND3 has a strong, near-linear Stark effect, collision systems involving ND3 appear the most attractive forum to study the effects of external electric fields on resonance structures and partial wave dynamics. Unlike NO, Stark energies of  ~1 cm−1 are readily obtained for ND3 in experimentally attainable electric fields of  ~75 kV/cm. The distinct resonance features as observed here at collision energies ~1 cm−1 are expected to sensitively respond to external fields, paving the way to modify the collision dynamics and control the scattering outcome. Moreover, in the initial \({1}_{1}^{-}\) and final \({1}_{1}^{+}\) state, the ND3 molecules possesses two distinctive angular momentum projection states belonging to mj = 0 and ∣mj∣ = 1. Molecules in either of these states can be prepared before the collision using the decelerator76, whereas transitions to either of the two final projection states would appear as distinctive rings in the images separared by the field induced Stark shift. The recoil-free detection scheme for ND3 as demonstrated here enables sufficient resolution to fully separate these rings, offering the unprecendented opportunity to steer and control low-energy resonances, and to simultaneously probe how stereodynamics influences partial wave dynamics.

Methods

The experiments were performed using a crossed molecular beam setup described before24. A supersonic beam of ND3 was created by expanding 2% ND3 seeded in a carrier gas into a vacuum chamber through a Nijmegen Pulsed Valve77. The mean velocity of the molecular beam could be coarsely tuned between 450 m/s and 900 m/s by using different carrier gas mixtures. For a given carrier gas, precise tuning of the velocity was achieved by passing the beam through a 2.6 m long Stark decelerator. The velocity-controlled packets of ND3 emerging from the decelerator exclusively resided in the \({j}_{k}^{p}={1}_{1}^{-}\) upper inversion component of the rotational ground state of E-symmetry ND3, often called para-ammonia by analogy to NH331,44. Population in the \({1}_{1}^{+}\) lower inversion component was effectively eliminated from the beam. Molecules initially in the \({1}_{1}^{+},| {m}_{j}|=1\) component are high-field seeking and deflected from the beam axis inside the decelerator, whereas molecules initially in the \({1}_{1}^{+},{m}_{j}=0\) component are immune to electric fields and travel through the decelerator in free flight greatly reducing particle densities.

The ND3 packets traveled in free flight for 529.5 mm towards the interaction region, where they were intercepted by a cryogenic beam of H2 or HD at an angle of 5.2°. These beams were produced by supersonically expanding neat beams of H2 or HD through a temperature-stabilized Even-Lavie valve that was mounted on the second stage of a cold head. A pure sample of para-H2 (j2 = 0) could be produced by first condensing normal H2 over a NiSO4 catalyst. The beams were collimated 47.5 mm before the beam crossing point by 3 mm diameter pinholes.

Inelastic \({1}_{1}^{-}\to {1}_{1}^{+}\) inversion-deexcitation collisions were probed by state-selectively ionizing scattered ND3 (\({1}_{1}^{+}\)) molecules. As this is the only open channel at the lowest collision energies, studying cold inelastic collisions involving other rotational levels requires preparing the system in a rotationally excited state26,76. The ions were then mapped onto a microchannel plate using an advanced high-resolution velocity map imaging spectrometer78. The extraction field was 20 V/cm, too low to cause a significant Stark shift or associated change in the collision dynamics. For ICS measurements, a 2+1 REMPI scheme was employed at 320 nm by inducing the B ← X transition in ND3 using a single dye laser. For DCS measurements, we employed a \(1+{1}^{{\prime} }\) REMPI scheme that ionized ND3 at threshold, thereby minimizing blurring effects in the images due to ion recoil. In this scheme, the B ← X transition was induced using a single photon ~160 nm, after which a photon of 448 nm excited the ND3 molecule to a Rydberg state above the ionization potential that autoionizes with near-zero recoil67. The 160 nm photons were generated using difference frequency mixing in xenon gas using two dye lasers, while a third dye laser was used to generate the ionizing photons at 448 nm.

To cover a range of collision energies between 0.5 and 25 cm−1, seven seed-gas mixtures were used to prepare the ND3 beam. For a given seed gas, ICSs were recorded by switching to a different velocity (energy) every four seconds using the decelerator, continuously cycling back and forth over the collision energy range allowed by the seed gas. Each data point, except the outer edges, was covered by at least two different seed gas mixtures, such that ICSs measured during different experimental runs could be stitched together. Every four seconds, a background measurement was performed by detuning the secondary beam in time such that no collisions were probed. Every two hours, the measurement was interrupted to probe the density of the incoming \({1}_{1}^{-}\) beam, confirming that long-term drifts in the experiment did not deteriorate the beam intensity. Each velocity range was repeated during at least two days, with a typical amount of  ~200 recorded cycles per day. The normalized signal was then computed as the difference between the accumulated scattering and background measurement, divided by the initial ND3 beam density. Calibration of the collision energy was performed using three independent methods by recording ICSs and scattering images while scanning the ND3 velocity accross the region of minimal collision energy (see SI). The experimentally obtained normalized signals were corrected for flux-to-density effects using extensive Monte Carlo simulations of the experiment (see SI). Scattering images were recorded one energy at a time, over a period ranging from several hours to a few days. For these measurements, the background signal was sampled by toggling the overlap of the secondary beam on/off every 30 seconds. For each image, the laser powers were reduced to yield only a few events per shot, such that event counting and centroiding could be applied to accumulate a high-resolution image.

The scattering calculations were performed by means of the close-coupling method in the body-fixed frame, as already described elsewhere for collisions of NH3 and ND3 with H2 and D251,58,68.

For this purpose, various PESs were employed. The five-dimensional PES of Maret et al.66, in which ammonia and hydrogen are considered as rigid rotors, was first used. Additional five-dimensional PESs were generated at various levels of theory (CCSD(T), CCSD(T)-F12a, CCSDT, CCSDT(Q)) using the MOLPRO quantum chemistry package79 and various basis sets (see SI) on a grid of 29,568 unique geometries. The PES was expanded in angular functions, and the radial coefficients were interpolated with the reproducing kernel Hilbert space method. The long-range part of the PES was constructed by developing the radial coefficients in inverse powers of the distance. A six-dimensional PES with non-rigid NH3 was also constructed by repeating this process for several values of the umbrella angle that describes the inversion motion of ammonia. The impact of isotopic substitution (in the case of ND3-H2 and ND3-HD) was taken into account by performing a coordinate transformation on the fitted PESs to reflect the change in the position of the centers of mass, after which a new angular expansion of the PES was carried out.

Integral and differential cross sections were computed by solving the coupled channel equations in the body-fixed frame over a grid of 400 values of the kinetic energy in the range 0.01–25 cm−1. The rotational basis included all levels with j ≤ 6 (for ND3) and j2 ≤ 4 (for H2/HD).

The inversion of ND3 was treated using two different models. For PESs with rigid ND3, a two-state model was adopted in which the inversion-tunneling states are taken as linear combinations of the two rigid equilibrium structures. When the PES included an explicit dependence on the inversion angle, the umbrella motion of ammonia was treated using a Hamiltonian that comprises a specific kinetic and potential term dependent on the inversion angle. This follows the approach tested for low-energy collisions of ammonia with rare gas atoms (see e.g., refs. 59,72).

To examine the character of scattering resonances, a partial wave analysis was performed for all values of the total angular momentum \({{{\mathcal{J}}}}\) and for each parity \({{{\mathcal{P}}}}\) by computing the scattering wavefunctions.