Rotational excitation of the interstellar NH₂ radical by H₂

Neutral nitrogen hydrides (NH, NH₂, and NH₃) are highly abundant species in a variety of astrophysical regions. Among them, the NH₂ radical, even if not generally the most abundant one, is of key importance for the chemistry of these media. It was detected for the first time in the interstellar medium (ISM) by van Dishoeck et al.¹ who pointed out that this simple hydride is crucial for testing the production pathways of nitrogen-bearing molecules. Indeed, the NH₂ chemistry is directly related to that of the abundant and ubiquitous ammonia, NH₃. In addition, NH₂ exhibit spin symmetry states whose relative abundances are sensitive to the H₂ ortho-to-para ratio (OPR) in the gas phase. NH₂ observations² may be able to put constraints on the H₂ OPR in dense gas.

The Infrared Space Observatory was later used to observe infrared absorption lines of NH₂^3,4 and recently, the Herschel Space Observatory⁵ with the Heterodyne Instrument for the Far-Infrared (HIFI)⁶ allowed the observation of several low-lying rotational transitions of NH₂ at a very high spectroscopic resolution.⁷ These (absorption) lines were employed to derive NH:NH₂:NH₃ abundance ratios of ∼2:1:1 in lukewarm diffuse clouds⁸ and ∼3:1:20 in the colder envelope of low-mass protostars.⁹ Emission lines were also detected more recently in denser and hot ISM regions.² In addition, anomalous (non statistical) ortho-to-para ratios of NH₂ and NH₃ were derived in the diffuse gas.^2,10 These ratios were successfully reproduced by gas-phase models including a rigorous nuclear-spin chemistry,^9,11 suggesting that NH₂, just like NH₃, is mainly formed in the gas-phase via a series of successive hydrogen abstraction reactions $NHn+$ + H₂ (n = 0–3) followed by electronic dissociative recombination. A recent study has also emphasized the importance of the H-exchange reaction NH₂ + H in the ortho-para conversion of NH₂.¹² 

However, the analysis of the NH₂ rotational spectra, especially those in emission, was hampered by the lack of collisional rate coefficients. Without these data, only approximate estimates of the molecular column density are possible assuming local thermodynamic equilibrium (LTE), which is generally not a good approximation. Persson et al.² estimated collisional rate coefficients assuming quenching rate coefficient of 5 × 10⁻¹¹ cm³ s⁻¹ and state-specific downward rates for radiatively allowed transitions that scale in proportion to radiative line strengths. Such estimates are very approximate and the accurate determination of the NH₂ abundance would greatly benefit from accurate collisional data. The main collider in the dense ISM is generally molecular hydrogen, H₂.

Scattering studies implying nonlinear polyatomic molecules and H₂ are still sparse. To date, calculations of rate coefficients for the collisional excitation by para- and ortho-H₂ (hereafter p-H₂ and o-H₂, respectively) have been performed only for the four interstellar molecules H₂CO,¹³ NH₃,¹⁴ CH₃OH,¹⁵ H₂O,¹⁶ and SO₂.¹⁷ 

To the best of our knowledge, there are no collisional data available for the NH₂–H₂ system. The only relevant study for astrophysics implying the NH₂ molecules is the measurement of integral cross sections for NH₂–He rotational transitions performed by Dagdigian.¹⁸ However, no rate coefficients were given in this work. NH₂–H collisions were also studied¹⁹ but the NH₂ molecule was in its first excited electronic state that is negligibly populated in astrophysical media.

NH₂ is an asymmetrical rotor with two forms caused by the different relative orientations of the hydrogen nuclear spins. In collision with H₂, the two forms behave like two distinct species: ortho-NH₂ and para-NH₂ (hereafter denoted as o-NH₂ and p-NH₂, respectively). In addition, NH₂ has a complex rotational structure resulting from the open-shell character of the NH₂ ground electronic state $B12$⁠. Hence, each rotational level is split by spin-rotation interaction in a fine structure of two sublevels identified by the total angular momentum j₁ with $j1→=N1→+S→$ (where N₁ is the rotational angular momentum and S is the electronic spin). Moreover, fine structure levels are further split into three components through hyperfine interactions, due to the coupling between the nitrogen nuclear spin and the total angular momentum j₁. Finally, a second hyperfine structure resulting from the coupling between the nuclear spins of the hydrogen nuclei splits all o-NH₂ sublevels into three new sublevels.

The calculation of collisional cross sections taking into account this complex structure is an extremely challenging task. Calculations of collisional data for open-shell molecules in collision with H₂ have been achieved only recently²⁰ for linear molecules and cannot be easily extended to a polyatomic top. This is why, in the present work, we neglect the fine and hyperfine structure of the NH₂ target and we provide data for transitions between rotational levels. Collisional data including these specific structures may be deduced from the present calculations using decoupling approximations.^21,22

In this paper, we used a new accurate full-dimensional NH₄ potential energy surface (PES)²³ to compute the cross sections for the collisional excitation of the first 15 rotational levels of both o- and p-NH₂ by H₂. The paper is organized as follows. The PES and the scattering calculations are presented in Section II. In Section III, we report state-to-state resolved cross sections for the rotational excitation of NH₂ by H₂. Concluding remarks are drawn in Section IV.

The NH₂ radical is known to react with H₂ to form NH₃ through an exothermic pathway by a direct hydrogen abstraction mechanism. This reactive channel NH₂ + H₂ → NH₃ + H was found to have a barrier of 3340 cm⁻¹.²³ The reaction is thus quite slow with rate coefficients lower than 10⁻¹⁵ cm³ s⁻¹ for temperatures lower than 500 K.^23,24 Under such circumstances, neglecting the reactive pathway in the treatment of the NH₂–H₂ collision should have only a small influence on the description of the inelastic processes,^25,26 as long as low to moderate collision energies are considered. We also note that the hydrogen exchange process is also negligibly small under such conditions.

An accurate rigid-rotor five-dimensional PES was then constructed for the NH₂–H₂ system in its electronic and vibrational ground state, suitable for low-energy inelastic rotational calculations, from the recently computed nine-dimensional global PES of NH₄.²³ This PES was determined at the UCCSD(T)-F12a/aug-cc-pVTZ level of theory and the ab initio points were fitted using the permutation-invariant polynomial neutral network (PIP-NN) method²⁷ with a root mean squared error (RMSE) of 27 cm⁻¹. The RMSE of 27 cm⁻¹ is for the full nine-dimensional PES (including the reactive path). The fitting RMSE for the non-reactive NH₂–H₂ region relevant for this work is much smaller, of the order of few cm⁻¹.

In this work, the intermolecular potential is described as a function of five coordinates, namely, the intermolecular distance R from the NH₂ center of mass to the H₂ center of mass, and four relative angles (θ, φ) and (⁠$θ′$⁠, $φ′$⁠) which describe, respectively, the collision direction and the H₂ orientation relative to the NH₂ body-fixed system. The body-fixed Jacobi coordinate system used in our calculations is presented in Fig. 1.

As the original routine from Li and Guo²³ employs internuclear coordinates, the following transformation was employed to determine the Cartesian positions of the two hydrogen atoms (H_a with coordinates x_a, y_a, z_a and H_b with coordinates x_b, y_b, z_b) of the H₂ molecule in the NH₂ body-fixed coordinate system:

where $rH2$ is the bond length of H₂ fixed at its vibrationally averaged distance $⟨rH2⟩0=1.449a0$⁠. The NH₂ molecule, which lies in the (xoz) plane (see Fig. 1), was also kept rigid with an averaged geometry taken from the experimental work of Davies et al.:^28,$⟨rNH⟩0=1.936a0$ and $⟨HNH^⟩0=103.33°$⁠. We note that employing state-averaged geometries is a reliable approximation for including zero-point vibrational effects within a rigid-rotor PES, as discussed in previous studies on H₂O–H₂.^29,30 In addition, it was shown recently for the CO–H₂ system that state-averaged geometries also give scattering results very close to full-dimensional calculations.³¹ Here, the lowest vibrational bending frequency of NH₂ is ≃ 1500 cm⁻¹.³² Hence, vibrational excitation is closed at the investigated energies and can be safely neglected in the scattering calculations.

The original fit of Li and Guo²³ was employed to generate interaction energies on a very dense grid of 81 000 geometries in $R,θ,φ,θ′$⁠, and $φ′$⁠. An asymptotic potential of 2062.47 cm⁻¹ (corresponding to the above monomer averaged geometries) was subtracted from these interaction energies. 27 values in R were selected in the range [3.75–20.00]a₀ and this radial grid was combined with 3000 random angular geometries $θ,φ,θ′,φ′$⁠. The PES $V(R,θ,φ,θ′,φ′)$ was expanded over angular functions for all R-distances using the following expression:³³ 

where

where $(3j)$ is a “3-j” symbol, Y_pq is a spherical harmonic, $δij$ is a Kronecker delta, equal to one if i = j and to zero otherwise, and the sum is over r₁, r₂, r. The indices p₁, p₂, and p refer to the tensor ranks of the angle dependence of the NH₂ orientation, the H₂ orientation, and the collision vector orientation, respectively. In Eq. (8), the index of the C_2υ symmetry of NH₂ requires that q₁ be even and the homonuclear symmetry of H₂ similarly constrains p₂ to be even. The expansion coefficients υ_p1q1p2p(R) were obtained through a least-squares fit on the random grid of 3000 orientations at each intermolecular separation. We initially included all anisotropies up to p₁ = 10, p₂ = 6, and p = 16, resulting in 810 basis functions. We then selected only significant terms using a Monte Carlo error estimator (defined in the work of Rist and Faure³⁴), resulting in a final set of 146 expansion functions with anisotropies up to p₁ = 10, p₂ = 6, and p = 13. The RMSE was found to be lower than 1 cm⁻¹ for $R>4.75$a₀. A cubic spline interpolation of the coefficients (υ_p1q1p2p)(R) was finally performed over the whole R range and it was smoothly connected using a switching function to standard extrapolations (exponential and power laws at the short and long-range, respectively) in order to provide continuous radial expansion coefficients for the scattering calculations. Figure 2 shows a sample of expansion coefficients (υ_p1q1p2p) of the NH₂–H₂ PES, as a function of intermolecular distance R.

Two-dimensional plots of the NH₂–H₂ PES are presented in Figs. 3–5.

The global minimum of the 5D fitted PES is located at $θ=θ′=0°$ with a depth of −213.00 cm⁻¹ and at an intermolecular distance $R=6.05a0$⁠. This minimum corresponds to the most stable configuration of the NH₂–H₂ complex so that H₂ is approaching the N atom of the NH₂ molecule along the C₂ axis.

In Fig. 3, we show a contour plot of the interaction energy for fixed $θ′=φ=φ′=0°$⁠. This plot shows the anisotropy of the interaction with respect to the NH₂ rotation.

Figure 4 shows the interaction energies for $φ=φ′=0°$ and R = 6.05 a₀. We found a relatively strong anisotropy of the PES with respect to the NH₂ and H₂ rotation. It means that the rotational state of H₂ will probably influence the magnitude of the NH₂ excitation cross sections.

A secondary minimum is found at $θ=113°$⁠, with an energy deduced from our fit of −122.29 cm⁻¹. It occurs when H₂ is approaching along the direction of the NH bond and H₂ is perpendicular to the NH₂ plane (⁠$θ′=φ′=90°$⁠). The contour plot is displayed in Fig. 5.

Finally, it is interesting to compare the NH₂–H₂ with the H₂O–H₂ PES of Valiron et al.²⁹ The shape of the two PESs is quite similar. Indeed, Valiron et al.²⁹ reported that the H₂O–H₂ global minimum is at the same configuration than that of NH₂–H₂, the well depth of H₂O–H₂ and NH₂–H₂ PESs being similar (−235.14 cm⁻¹ for the H₂O–H₂ complex vs. −213 cm⁻¹ for the NH₂–H₂ complex).

Furthermore, for both NH₂–H₂ and H₂O–H₂ PESs, a secondary minimum is found when H₂ is approaching along the direction of the NH (OH) bond with the H₂ molecule perpendicular to the NH₂ (H₂O plane). The H₂O–H₂ secondary minimum found at $θ=119°$ and $θ′=90°$ with a well depth equal to −199.40 cm⁻¹ is, however, deeper than that of NH₂–H₂ (−122.29 cm⁻¹) found at $θ=113°$ and $θ′=90°$⁠.

We used the fitted NH₂–H₂ PES to study the rotational excitation of NH₂ by H₂.

We focus only on the collisional excitation of the rotational states of NH₂, since the computation of fine/hyperfine structure resolved cross sections is a true challenge. The rotational energy levels of NH₂ are labelled by three numbers: the angular momentum N₁ and the pseudo-quantum numbers k_a and k_c which correspond to the projection of N₁ along the axis of the least and greatest moments of inertia, respectively. The para states correspond to k_a + k_c odd and the ortho states to k_a + k_c even. Extension to the fine/hyperfine structure using approximate treatment will be considered in a future study.

The rotational levels of NH₂ were obtained using the rotational constants from Müller et al.³⁵ Figure 6 shows the rotational energy levels of both o- and p-NH₂. For the H₂ molecule, the rotational states are denoted by j₂ throughout this paper. The para states of H₂ have even rotational states $j2=0,2,…$ and the ortho states have odd rotational states, $j2=1,3,…$⁠.

As inelastic (nonreactive) collisions cannot interconvert the ortho- and para-forms, the calculations were done separately for the four spin combinations, namely, p-NH₂–p-H₂, p-NH₂–o-H₂, o-NH₂–p-H₂, and o-NH₂–o-H₂. NH₂ molecule is treated as a rigid rotor. We used the quantum close-coupling (CC) approach to obtain the inelastic cross sections as described in Phillips et al.³⁶ All scattering calculations have been performed with the version 14 of the MOLSCAT code.³⁷ The coupled equations were solved using the modified log-derivative airy propagator of Alexander and Manolopoulos.³⁸ The reduced mass of the system is 1.790 367 amu.

Inelastic cross sections were obtained between levels with a rotational energy $Erot≤422$ cm⁻¹, that is, up to $N1kakc=440$ for o-NH₂ and $N1kakc=441$ for p-NH₂.

The collisions were studied for the total energy ranging from 32 to 1500 cm⁻¹. The integration parameters were chosen to ensure convergence of the cross sections in this range. We carefully spanned the energy range to guarantee a good description of the resonances. For collision with p-H₂, the energy steps are 0.2 cm⁻¹ below 100 cm⁻¹, 0.5 cm⁻¹ from 100 to 200 cm⁻¹, 1.0 cm⁻¹ from 200 to 500 cm⁻¹, 2.0 cm⁻¹ from 500 to 700 cm⁻¹, 5.0 cm⁻¹ from 700 to 1000 cm⁻¹, and 20 cm⁻¹ from 1000 to 1500 cm⁻¹. For collision with o-H₂, the energy steps are 0.2 cm⁻¹ below 300 cm⁻¹, 1.0 cm⁻¹ from 200 to 500 cm⁻¹, 2.0 cm⁻¹ from 500 to 700 cm⁻¹, 5.0 cm⁻¹ from 700 to 1000 cm⁻¹, and 20 cm⁻¹ from 1000 to 1500 cm⁻¹. At each collision energy, the total angular momentum J_tot was set large enough to converge cross sections, the value of J_tot varies from 19 at low energies to 64 for energies larger than 1000 cm⁻¹.

Since a given N₁ rotational number includes a large number of NH₂ sub-rotational energy, rotational levels with internal energies above E_max = 650 cm⁻¹ were eliminated for total energies $Etot≤200$ cm⁻¹. This E_max parameter was progressively increased up to E_max = 2200 cm⁻¹ for E_tot = 1500 cm⁻¹. These large values of E_max are needed to converge cross sections. A similar effect was previously observed for methyl-formate (HCOOCH₃) colliding with helium³⁹ and for formaldehyde (H₂CO) colliding with H₂.⁴⁰ 

It was also crucial to optimize the rotational basis of H₂ in order to keep calculations feasible in terms of both central processing unit (CPU) time and memory. Tests of the p-H₂ and o-H₂ basis were performed at different values of total energy. For collisions with p-H₂$(j2=0)$⁠, the inclusion of the H₂$(j2=2)$ level was necessary to obtain cross sections converged to better than 5%, even when these channels were energetically closed. For o-H₂$(j2=1)$⁠, it was found that inclusion of the H₂$(j2=3)$ level in the basis does not have a noticeable influence on the magnitude of the cross sections. Hence, for the determination of rotational excitation cross section of NH₂ in collision with o-H₂, only the H₂$(j2=1)$ basis was retained.

Figure 7 shows the collisional energy dependence of the de-excitation integral cross sections of o- and p-NH₂ in collision with o-H₂$(j2=1)$ and p-H₂$(j2=0)$⁠. One can first observe resonances that appear for energies lower than 250 cm⁻¹. These resonances in the de-excitation cross sections are related to the presence of the attractive potential well with a depth of −213 cm⁻¹ that allows the H₂ molecule to be temporarily trapped and hence quasi-bound states to be formed before the complex dissociates. The cross sections for collisions with p-H₂$(j2=0)$ seem to display a richer resonance structure than the cross sections for collisions with o-H₂$(j2=1)$ that appear to have a smoother energy dependence. Actually, there are many more, and hence overlapping, resonances for the cross sections for collisions with o-H₂$(j2=1)$ because of a larger number of (quasi-)bound states due to the contribution of an additional coupling term N₁ + j₂ = j₁₂ absent in collisions with p-H₂$(j2=0)$⁠. Second, regarding the magnitude of the cross sections, we observe a global decrease of their intensity with increasing $ΔN1$⁠. However it is interesting to note that for collisions with p-H₂$(j2=0)$⁠, the magnitude of cross sections with $ΔN1=2$ can be larger than those with $ΔN1=1$⁠, while the trend is a rather monotonic decrease for o-H₂$(j2=1)$⁠.

Also, when the collisional energy increases, the magnitude of the cross sections for $ΔN1=1$ and $ΔN1>1$ tends to be closer whether for collisions with p- or o-H₂. This behavior is expected and observed for many systems like HNC–H₂,⁴¹ HCl–H₂,⁴² and O₂–H₂.⁴³ 

In order to have an overview of the differences that exist between the excitation cross sections with p- and o-H₂ colliders, we show, in Fig. 8, a comparison between the two sets of cross sections for all the de-excitation transitions from all initial levels up to $N1kakc=440$ for o-NH₂ and up to $N1kakc=441$ for p-NH₂ at a fixed collision energy of 100 cm⁻¹. The horizontal axis presents the cross sections for collisions with p-H₂$(j2=0)$ whereas the vertical axis presents the cross sections for collisions with o-H₂$(j2=1)$⁠.

Examination of the plots for collisions of o- and p-NH₂ with both o- and p-H₂ shows that the two sets of data agree generally within a factor of 3. The largest cross sections are those for collision with o-H₂, with only a few exceptions. This trend was already observed for several interstellar species like SiS,⁴⁴ HCl,⁴² SO₂,⁴⁵ or H₂O.⁴⁶ 

This behavior can also be explained by looking at the radial coefficients $υp1q1p2p$ of the expansion Equation (7), as plotted in Fig. 2. The radial coefficients contributing to cross sections with $j2→j2′$ transitions are those with p₂ in the range $j2−j2′<p2<j2+j2′$⁠. Then, for collisions with p-H₂$(j2=0)$ only the terms with p₂ = 0 contribute whereas for collisions with o-H₂$(j2=1)$ the p₂ = 0, 2 terms contribute. The radial coefficients with p₂ = 2 are not negligible compared to the one with p₂ = 0 explaining why the cross sections for collisions with o-H₂$(j2=1)$ are larger than the ones for collisions with p-H₂$(j2=0)$⁠. The non-negligible contribution of the radial coefficients with p₂ = 2 can be explained notably by the dipole-quadrupole interaction that exists for collisions with o-H₂ and that vanishes for collisions with p-H₂.

Then, we were interested in the propensity rules in the NH₂–H₂ collisional system. Figures 9 and 10 show, at 3 different collisional energies (50, 500, and 1000 cm⁻¹), the rotational de-excitation cross sections from p-NH₂(N_1kakc = 4₂₃) and o-NH₂(N_1kakc = 4₁₃) for both p- and o-H₂ collisions.

Despite the fact that no strong propensity rules are present, we actually observe a slight propensity rule in favor of transitions with odd $Δka$ and $Δkc$ which is seen for p-NH₂ in collision with both p-H₂ and o-H₂, while for collisions of o-NH₂ with both p-H₂ and o-H₂ we notice a more marked propensity in favor of even $Δka$ and $Δkc$⁠. Furthermore, a propensity rule in favor of transitions with $Δkc=0$ exists for collisions of both o- and p-NH₂ with both o- and p-H₂. This propensity is pointed out in Figs. 9 and 10 where the transitions 4₁₃-3₁₃ of o-NH₂ and 4₂₃-3₀₃ p-NH₂ are indeed found to be favored. This trend may be explained by the difficulty of reorienting the angular momentum vector with respect to the axis of the greatest moment of inertia.

Because of the similarity of molecular geometries of H₂O and NH₂, it is interesting to compare the propensity properties of the two systems. From the propensity rules found for the H₂O–H₂ system⁴⁷ (⁠$ΔN1$ = 0, ±1; $Δka$ = 0, ±1; $Δkc$ = 0, ±1), it can be seen that transitions with conservation of k_c or small variation of k_c are also favored for this system, which might be general to C_2υ molecules.

Finally, as collisions with helium are often used to model collisions with p-H₂$(j2=0)$⁠, it is interesting to compare our theoretical results with the experimental state resolved cross sections for rotationally inelastic collisions of NH₂(⁠$B12$⁠) with He measured by Dagdigian.¹⁸ Dagdigian used a crossed beam experiment and resolved fluorescence spectroscopy to study rotational energy transfer within the NH₂ electronic state. Table I presents close-coupling calculation for NH₂ with p-H₂$(j2=0)$ collisions at the kinetic energy 467 cm⁻¹ used in experiment. By comparing our results with the experimental relative cross sections out of the NH₂$(N1kakc=000$⁠) state induced by collision with He,¹⁸ we found that the p-H₂ collisional partner is much more efficient than He for inducing $ΔN1>1$ transitions. This probably mainly reflects the difference in the interaction potential where the NH₂–H₂ well depth is expected to be much larger than the NH₂–He one. Then we confirm again that for hydrides molecules, He cannot be used as a model for H₂, as already found for HCl⁴² or H₂O.⁴⁶ 

We have presented in this paper a study of the interaction between NH₂ and hydrogen molecules. Cross sections for the rotational (de)excitation of o- and p-NH₂ colliding with both p-H₂$(j2=0)$ and o-H₂$(j2=1)$ have been computed. All transitions among both the 15 lowest levels of o-NH₂ and p-NH₂ were considered.

The cross sections for collisions with para- and ortho-H₂ differ, the magnitude of the ortho-H₂ ones being dominant. Propensity rules are discussed and it is found that no rigorous selection rules are defined although transitions with $Δkc=0$ seem to be slightly favored.

The present results should therefore be adopted in any radiative transfer model of NH₂ in environments with T ≤ 150 K. In particular, Persson et al.² used a non-LTE radiative transfer model using rate coefficients estimated from an assumed quenching rate coefficient of $5×10−11$ cm³ s⁻¹ and radiative selection rules, i.e., $ΔN1$ = 0, ±1; $Δka$ = ±1, ±3, etc.; $Δkc$ = ±1, ±3, etc. Our results indicate that inelastic cross sections corresponding to radiatively forbidden transitions (e.g., $Δkc=0$⁠) can be as probable as (or even stronger than) those corresponding to radiatively allowed transitions. Hence, previously published NH₂ abundance should be revised accordingly, using our new data.

This work has been supported by the Agence Nationale de la Recherche (ANR-HYDRIDES) Contract No. ANR-12-BS05-0011-01 and by the program “Physique et Chimie du Milieu Interstellaire” (PCMI) funded by CNRS and CNES. We thank the CPER Haute-Normandie/CNRT/s, Electronique, Matériaux. H.G. acknowledges the financial support of the United States Department of Energy (Grant No. DE-SC0015997) and J.L. acknowledges the National Natural Science Foundation of China (Grant No. 21573027).