Collisional excitation of N⁺(³P) in interstellar clouds Free

ABSTRACT

The detection of the N⁺ ion in ionized interstellar regions, as well as its use as a tracer for atomic nitrogen (N) in interstellar media, makes it an interesting ion from the astrophysical point of view. Accurate determination of the N⁺ abundance in the interstellar clouds requires detailed modelling of its excitation conditions. Hence, collisional excitation induced by the dominant interstellar species has to be studied. In this paper, we study the collisional excitation of N⁺ by He and H₂. We provide new potential energy surfaces describing the interactions between N⁺ and He and between N⁺ and H₂ considered as a structureless collider. Quantum calculations of the excitation cross-sections for collisions of N⁺ with He and H₂ are provided, as well as the first excitation rate coefficients for N⁺ induced by these two species. The results are compared to collisional data for the excitation induced by electrons. It is shown that the excitation of N⁺ by a neutral collider can play a significant role in weakly ionized regions as well as in warm molecular clouds.

1 INTRODUCTION

N and N₂, the dominant nitrogen-bearing elements (Maret, Bergin & Lada 2006; Le Gal et al. 2014) in the interstellar medium (ISM) are respectively impossible and difficult to detect spectroscopically through infrared or millimeter-wavelength transitions. This is due to the fact that N and N₂ do not have fine-structure levels and dipole moment, respectively. Proxy is thus used to infer their abundance in interstellar clouds (Le Gal et al. 2014). The N⁺ ion is detected in the warm and ionized ISM (Goldsmith et al. 2015; Langer et al. 2017; Doherty et al. 2020) and could qualify as such a proxy. Indeed, N⁺ is considered as an important tracer for N in (fully) ionized regions. It can also be used for discriminating between fully and weakly ionized regions. Despite not (yet) detected in cold molecular clouds, N⁺ is also expected to be important constituent of such region that initiates the nitrogen chemistry and particularly the formation of ubiquitous ammonia (Dislaire et al. 2012).

The spin–orbit interaction in ionized nitrogen N⁺ splits its ground electronic state into a triplet of fine-structure levels ³P_j with j = 0, 1, 2. The ³P₁ and ³P₂ levels are at 48.738 cm⁻¹ and 130.8 cm⁻¹ above ³P₀ (Shen et al. 2022), respectively. Transitions with Δj = 1 are allowed and thus make the detection of N⁺ possible.

Collisional data are needed for interpreting N⁺ observations and estimating physical conditions. Indeed, in environments that do not reach local thermodynamic equilibrium (LTE), the population of N⁺ levels has to be modelled through the competition between radiative and collisional processes (Roueff & Lique 2013). Collisional data for N⁺, although known for the excitation induced by electrons (Tayal 2011), are still missing for the excitation induced by heavier species such as He or H₂. Without them, astrophysical observations must be interpreted on the basis of the LTE assumption, which does not allow to make the most of the highly resolved molecular spectra that recently developed facilities do offer. Due to the lack of collisional data available for N⁺ with the main neutral colliders (He, H, and H₂¹), Persson et al. (2014) had to model spectral observations based on scaled C–H rate coefficients. Such approximation may lead to significant errors in the observations, interpretation, and exploitation.

Calculation of these data requires first to compute an interaction potential between N⁺ and the projectile (H₂ or He in the present study). He and H₂ are the dominant colliders in cold molecular clouds where N⁺ is expected to play a significant role in nitrogen chemistry. In warm (ionized) clouds, where N⁺ has been detected, H is the dominant neutral projectile. Calculations with H projectile are difficult to perform and we estimate that calculations with He and H₂ projectiles can serve as template for the modelling of the impact of neutral colliders (Roueff & Lique 2013). Two studies (Soldán & Hutson 2002; Aïssaoui, Knowles & Bouledroua 2020) have provided potential energy curves (henceforth PECs) for N⁺–He in the last two decades. However, their PECs are based on coupled cluster methods with an incomplete basis. Nowadays, computing facilities should allow to improve the description of the N⁺–He interaction. At the opposite, the determination of global N⁺–H₂ potential energy surfaces (henceforth PESs) involved in the collisional process as well as in the non-adiabatic and spin–orbit couplings between them is a very challenging task because of the reactive nature of the system. In order to overcome this difficulty, Gómez-Carrasco et al. (2022) have recently provided PESs for the reactant and product channels separately without considering the intermediate region and have simulated the dynamics of the N⁺ + H₂ → NH⁺ + H reaction using statistical theories.

Full-dimensional quantum time independent close-coupling (CC) calculations are the method of choice to obtain accurate collisional rate coefficients. Such an approach can be easily used for non-reactive collisional systems such as N⁺–He. However, in the case of reactive systems such as N⁺–H₂ (Gómez-Carrasco et al. 2022), this approach is impractical due to its memory and CPU requirements and approximate treatments have to be considered.

In this paper, we provide new highly accurate PECs for studying the collisional excitation of N⁺ by He. We also provide PECs averaged over H₂ rotations for N⁺–H₂ interaction. This later potential relies on important approximations, as it treats this system as a non-reactive system. Then, we report inelastic cross-sections of the collisional systems from quantum CC calculations, and we derive collisional data for both systems. This paper is organized as follows: in Section 2, we present the new N⁺–He PECs, as well as the spherically averaged PECs for the N⁺–H₂ interaction. Section 3 presents the inelastic cross-sections and the rate coefficients obtained from our new PECs. Finally, in Section 4, we discuss our results and their astrophysical implications.

2 N⁺–He AND N⁺–H₂ INTERACTION POTENTIALS

2.1 N⁺–He potential energy curves

The interaction of a closed-shell rare-gas atom with a ³P-state atom gives rise to two electronic states: a ³Σ state and a doubly degenerate ³Π state. Ab initio calculations were performed for internuclear distances in the range 1.8 < R < 100 a₀ for the ³Σ electronic state and 2.5 < R <100 a₀ for the ³Π electronic state, using the spin restricted coupled cluster method with single and double excitation and a perturbative treatment of triple excitation [RCCSD(T)] (Knowles, Hampel & Werner 1993, 2000) and the molpro package (Werner et al. 2020).

As is usually the case in standard ab initio calculations, the spin–orbit coupling was neglected, but taken into account in the scattering calculations. Calculations were performed with the augmented correlation consistent [aug-cc-pVXZ (X = T, Q, 5)] basis set and extrapolated to the complete basis set (CBS) using the procedure of Peterson, Woon & Dunning Jr (1994). Standard corrections were applied to account for the basis set superposition error, using Boys & Bernardi (1970)’s procedure.

In order to obtain an analytical representation, suitable for scattering calculations, we have used the following procedure. For short-range (SR) R-distances, we have used the Born–Mayer formula that traditionally describes the ion–atom curves such as:

with α and β being constants parameters (Pauly 1979). Three coefficients (C₄, C₆, and C₈) adjusted from the ab initio energy values at 15, 20, and 25 a₀ were used to describe the PES at long-range (LR) R-distances using the following analytical formula:

Table 1 shows the value calculated for the SR and LR parameters.

C₄, C₆, and C₈ correspond to the leading terms in the LR expansion of the N⁺–He interaction: dipole–dipole induction coefficient, dispersion, and induction coefficients, respectively, (Côté & Dalgarno 2000; Bouchelaghem & Bouledroua 2014).

Finally, we used a cubic spline for interpolating smoothly our ab initio data points.

Fig. 1 presents the N⁺–He PECs obtained. The PECs are characterized by a minimum at 2.9 a₀ of −2037 cm⁻¹ for the ³Σ state and at 5.0 a₀ of –204.05 cm⁻¹ for the ³Π state. The well is thus much deeper for ³Σ than for ³Π, which can easily be explained since the ³Σ state corresponds to approaches of He along the unoccupied p orbital. In that case, He can get much closer to the ion and the well will thus be much deeper. Note that our results, although improving upon former PECs published by Soldán & Hutson (2002) and Aïssaoui et al. (2020) as relying on an extrapolation to the CBS, are in good agreement with them, with a difference of less than 0.2 per cent and 4.5 per cent for the ³Σ and ³Π minima, respectively. Table 2 shows and compares the main characteristics of the PECs obtained by Soldán & Hutson (2002) (SH02), Aïssaoui et al. (2020) (A20), and in this work.

2.2 N⁺–H₂ spherically averaged potential energy curves

As mentioned in the introduction, obtaining global PESs for the H₂N⁺ system is highly challenging because of the reactive nature of this system, as exemplified by the work done by Gómez-Carrasco et al. (2022). Indeed, these authors only computed a PES for the entrance (N⁺ + H₂) and exit (NH⁺ + H) channel, neglecting the intermediate (⁠|$\mathrm{NH}_2 ^+$|⁠) region. Here, we used a different strategy and we selected a reduced dimensional approach, where the H₂ molecule is described as a structureless sphere. This approach requires to keep the intermolecular distance between the two H atoms fixed, which amounts to treating the complex as non-reactive. Then, we averaged the PECs thus obtained for limited H₂ orientations which means that we treat H₂ as a pseudo-atom rather than as a molecule.

Hence, as shown in Fig. 2, the N⁺–H₂ PECs are described by the two Jacobi coordinates R, the distance from the centre of mass of H₂ molecules to the N⁺ atom, and θ, the angle between R and the H₂ bond axis r. We considered that the H₂ molecule only approaches the ion linearly (θ = 0°) or perpendicularly (θ = 90°).

Ab initio calculations were performed using the RCCSD(T) approach with the augmented correlation consistent triple zeta (aug-cc-pVTZ) basis set. We chose to stay at this level of precision given the level of approximations involved in the spherically averaged approach that we adopted. Calculations were performed in the range 1.3 < R < 100 a₀ for all electronic states considered here. Interactions between the ion N⁺ and H₂ along the linear approach give rise to three electronic states: a ³Σ state and two degenerate ³Π states. Regarding the perpendicular approach, the degeneracy disappears and the ³Σ and ³Π state split into ³A₂ (for ³Σ), ³B₁ and ³B₂ (for ³Π) electronic states in the C_2v symmetry group, respectively.

Thus, we calculated a total of five PECs, two for the linear approach (³Σ and ³Π) and three for the perpendicular approach (³A₂, ³B₁, and ³B₂). Fig. 3 presents the five PECs obtained.

N⁺–H₂ spherically averaged PECs (⁠|$V^{i}_{av}(R)$|⁠) are obtained by averaging over the linear and perpendicular approaches using the following formula, with Vⁱ the interaction potential at a given distance R:

Note that we tested three different ways of proceeding to this average to get the N⁺–H₂ PECs for ³Σ and ³Π pseudo-states. First, the average was done over the linear potential and the perpendicular potential using only the ³B₁ state and the ³A₂ state for the latter; only the ³B₂ state and the ³A₂ state; and finally an average over the ³B₁, ³B₂ state, and the ³A₂ state. Given that we did not observe significant differences between the final PECs thus obtained, we adopted the averaged curve over ³B₁ and ³B₂ for the scattering calculations, as displayed in Fig. 4, albeit physically meaningless. Such a weak influence of the use of ³B₁ or ³B₂ states for the generation of the N⁺–H₂ spherically averaged PECs can be explained by the fact that the two ³B₁ or ³B₂ states are almost degenerate up to R-distances of 2.4 a₀.

Similarly to what has been done for the analytical representation of the N⁺–He PECs, the extrapolation to SR distances is based on the Born–Mayer formula, with α = 1223.76 and β = 3.05 au for the ³Σ state and α = 9328.61 and β = 8.42 au for ³Π state. The extrapolation for LR distances relies on multipolar expansion (R⁻ⁿ terms) through the C₃, C₄, and C₅ coefficients, whose value is determined from LR ab initio points and correspond, respectively, to 0.0514, −2.7823, and 1.0847 au for the Σ state; and 0.0619, −3.2309, and 3.2563 au for the Π state. Finally, ab initio points were smoothly connected on the basis of a cubic spline interpolation.

Fig. 4 presents the PECs obtained for the ³Σ and the ³Π states through this spherically averaged approach. Compared to the PECs of N⁺–He, we observe an inversion of the states’ hierarchy: the potential well is deeper for the ³Π state than for ³Σ one, and the minima are much deeper than for He (see Table 3 for PECs characteristics).

3 QUANTUM SCATTERING CALCULATIONS

3.1 State-to-state cross-sections

Scattering calculations for N⁺–He and N⁺–H₂ were performed with the CC approach (Arthurs & Dalgarno 1960), following the method described by Alexander, Orlikowski & Straub (1983); Orlikowski & Alexander (1984) for ³P-atom – structureless projectile collisions and implemented in the hibridon code (Alexander et al. 2021). The spin–orbit couplings dependence on R was neglected. As is usually done, we use the asymptotic experimental spin–orbit splittings of N⁺(³P), Δ_{j = 1}  = 48.738 cm⁻¹, and Δ_{j = 2}  = 130.8 cm⁻¹, respectively, to determine spin–orbit corrected PECs. The hybrid log-derivative/airy propagator implemented in the hibridon code has been used, starting from R_min = 1.5 a₀ to R_mid = 25 a₀ for the log-derivative method, and from R_mid = 25 a₀ to R_max = 250 a₀ for the Airy propagator for N⁺–He collisions and from R_min = 1.2 a₀ to R_mid = 40 a₀ for the log-derivative method, and from R_mid = 40 a₀ to R_max = 200 a₀ for the Airy propagator for N⁺–H₂ collisions. All these parameters were adequate to ensure a convergence of cross-sections to better than 1 per cent. These calculations have been performed for total energies up to 10 000 cm⁻¹. For the energy grid, we have used a variable step. For energies smaller than 100 cm⁻¹, the step is 0.2 cm⁻¹. Then, between 100 and 200 cm⁻¹, the step is increased to 0.5 cm⁻¹, between 200 and 400 cm⁻¹ to 2, between 400 and 600 cm⁻¹ to 10. Finally, between 600 and 1000 cm⁻¹, the step is increased to 20 cm⁻¹ and to 100 cm⁻¹ above that limit. The reduced mass of the complexes were μ = 3.11 u for N⁺–He and μ = 1.76 u for N⁺–H₂. For the former, the maximum total angular momentum considered was J = 20 for energies lower than 300 cm⁻¹, J = 30 for energies between 300 and 1000, J = 50 for energies between 1000 and 2500, and J = 80 for energies greater than 2500 cm⁻¹. For the latter, the maximum total angular momentum considered was J = 30 for energies lower than 300 cm⁻¹, J = 50 for energies between 300 and 1000 and J = 80 for energies superior to 1000 cm⁻¹.

Fig. 5 shows the state-to-state cross-sections for the de-excitation transitions of N⁺ induced by collisions with He. We can see that the transition from ³P₂ to ³P₁ dominates the de-excitation process and that the ³P₁ to ³P₀ transition is weak, which is not surprising as this transition is classically forbidden (Monteiro & Flower 1987). This behaviour has been noticed for others atoms with a ³P electronic ground state in collision with He (Alexander et al. 1983; Monteiro & Flower 1987; Lique et al. 2018).

Fig. 6 presents the state-to-state cross-sections of N⁺ with H₂. Cross-sections at low collision energy (E_c) exhibit strong Feshbach and shape resonances due to the attractive potential wells (Costes & Naulin 2016). These resonances disappear when E_c is large compared to the well depth. The transition from ³P₂ to ³P₁ is the most efficient; and the transition from ³P₁ to ³P₀ dominates over the ³P₂ to ³P₀ at low energy, thanks to a coupling through ³P₂ that is stronger at low energy, as noted in Alexander et al. (1983).

3.2 Rate coefficients

From the inelastic cross-sections σ(E_c), one can derive the corresponding rate coefficients at temperature T by an averaging over the collision energy (E_c):

where k_B is the Boltzmann constant, and i and j denote the initial and final levels of the target, respectively. Calculations of the cross-sections at total energies up to 10 000 cm⁻¹ allow the determination of rates up to 2000 K without loss of accuracy.

Unsurprisingly, the rate coefficients derived from the N⁺–He cross-sections reflect the hierarchy of transitions observed for the latter, as shown in Fig. 7. The transition from the ³P₂ state to the ³P₁ one largely dominates the other transitions and the ³P₁ to ³P₀ remains negligible. Likewise, in the case of N⁺–H₂, the same behaviour is observed for the transitions’ hierarchy with respect to rate coefficients than for the cross-sections, with the transition from 3P₁ to 3P₀ dominating over the 3P₂ to 3P₀ up to ∼500 K.

3.3 Discussion and comparison with electronic excitation rate coefficients

As mentioned in the introduction, H₂ is the most abundant partner in interstellar molecular clouds and thus the dominant collisional partner in such media. Despite significant update for many interstellar atoms and molecules (Roueff & Lique 2013; van der Tak et al. 2020), collisional data are still difficult to produce for collisions with H₂, especially for reactive molecules and ions. When data for collisions with He are available, it has often been proposed to derive H₂-rate coefficients from He-rate coefficients calculated for the same molecule. Such an approach (Schöier et al. 2005; Lique et al. 2008) consists in assuming that the excitation cross-sections are similar for both colliding systems and that the rate coefficients would only differ by a scaling factor, that corresponds to the square root of the ratio between the reduced masses appearing in the conversion of the cross-sections into rate coefficients (equation 4). Hence, the following scaling relationship could be used in our case:

Note that this approximation is only supposed to be used for H₂ at low temperatures (e.g. when most H₂ is in its ground state), but has already been challenged for many species; for a general criticism of this assumption, see Roueff & Lique (2013).

Here, we discuss the validity of estimating H₂-rate coefficients from He ones when it comes to our system, i.e, to determine whether He can be a good template model for understanding and quantifying the collisional excitation of N⁺ by H₂. One caveat that must be addressed in order to answer this question is the following: we do not have state-of-the-art results for H₂, but only quantum scattering calculations performed on the basis of interaction potentials containing important approximations. N⁺–H₂ was treated as a non-reactive system given that the intermolecular distance between the two H was fixed, and H₂ was treated as a pseudo-atom given that we averaged the PESs over H₂ rotations. There is no way to determine how these approximations impact the collisional rate coefficients obtained for N⁺–H₂. We have reasons however to believe that, albeit these approximations, our calculations may provide the right order of magnitude of the fine structure resolved rate coefficients. Thus, we may be allowed to use them at least to interpret the exact results obtained for He and to determine on this basis whether the 1.4 scaling factor assumption is valid for this system.

We see in Figs 5 and 6 that, as expected, given the deeper potential well in the N⁺–H₂ interaction as compared to N⁺–He one, H₂ cross-sections are much larger for the He ones. A difference of one order of magnitude can be observed between the corresponding cross-sections. The order of magnitude of the cross-sections is not the only difference between the two collisional complexes. The behaviour of the state-to-state cross-sections observed for the N⁺–H₂ system is very different from the one observed for N⁺–He one. Whereas the 3P₁ to 3P₀ transition induced by He collisions was negligible, for N⁺–H₂ collisions, it is not only much more important proportionally, and it actually dominates over the ³P₂ to ³P₀ transition thanks to an efficient coupling of the ³P₀ and ³P₁ states with the ³P₂ one. Our results show that He is not a correct model to describe the excitation of N⁺ induced by H₂.

In astrophysical media where N⁺ is found, electronic collisions are also supposed to play a significant role for the excitation of N⁺. Thus, it is important to compare the efficiency of electron and neutral collisional partners (He and H₂) for the excitation of N⁺. In Fig. 7, we compare rate coefficients for the de-excitation of N⁺ induced by electron impact (Tayal 2011) to the de-excitation rate coefficients induced by He and H₂ collisions computed in this work. Rate coefficients for the (de-)excitation of N⁺ by electrons are ∼2–3 orders of magnitude higher than that of the collisional (de-)excitation induced by H₂. Such a difference suggests that in astrophysical environments where the electronic fraction is low (n_e− ≤ 10⁻⁴), collisions with H₂ will compete or even dominate the excitation of N⁺, and indicates that N⁺–H₂ collisional data should be used in order to accurately model N⁺ excitation in interstellar clouds. At the opposite, in highly ionized media, the electronic excitation will dominate.

4 The excitation of N⁺ in interstellar media

Calculations of critical densities can help to identify more precisely when such N⁺ rate coefficients would need to be considered in radiative transfer models. The concept of critical density refers to the ratio between the Einstein coefficients, expressed in s⁻¹ and the collisional de-excitation rate coefficients for a given transition (in cm⁻³ s⁻¹). Put differently, the ratio of the Einstein coefficients A_{i → j} to the collisional de-excitation rate coefficients k_{i → j} defines the critical density n* of a given line, i.e. the density at which photon de-excitation and collisional de-excitation are equal.

Above such density, LTE conditions can generally be safely assumed. Critical densities for electrons had already been calculated by Goldsmith et al. (2015), on the basis of Tayal’s collisional rates. Fig. 8 displays the critical densities for He, H₂, and electrons for the two allowed radiative transitions as a function of temperature.

It can be seen that for ionized warm media where N⁺ has been detected (with densities lower than 10² cm⁻³), LTE conditions are not met and that rate coefficients are required to model the N⁺ emissions, although most often ignored to this day.

Then, we decided to test the impact of our rate coefficients on the excitation of N⁺ in astrophysical media, by performing radiative transfer calculations for physical conditions corresponding to media where N⁺ is detected. Non-LTE calculations were performed with the radex code (Van der Tak et al. 2007). Both collisional and radiative processes are taken into account. We focus on the calculation of the excitation temperature for N⁺ transitions, with a background temperature of 2.73 K and a N⁺ column density of 10¹⁴ cm⁻². H₂ is used as the only representative of neutral colliders.²

Fig. 9 shows the excitation temperature of ³P₁ to ³P₀ and ³P₂ to ³P₁ transitions as a function of the density of neutral species, with the electronic density kept fixed at n_e− = 10⁻² cm⁻³ and n_e− = 1 cm⁻³, respectively. A kinetic temperature of 2000 K³ was adopted. One observes that at low densities of neutral species, even with a neutral to electron abundance ratio greater than 10², the excitation temperature remains weakly sensitive to the density of neutral species. It is only above a critical threshold of n(H₂) ≥ 10³–10⁴ cm⁻³ that the excitation temperature increases and that the media start to thermalize very fast as the density increases, with an excitation temperature equal to the kinetic temperature around n(H₂) ≥ 10⁵ cm⁻³. Below this threshold, collisions are inefficient and do not contribute significantly to the excitation of the N⁺ ion, regardless of the ratio neutral species to electrons and of the density of electrons. Note that this neutral density value is much higher than the one typically observed in warm ionized clouds, i.e. 0.1 to 100 cm⁻³. Thus, environments where N⁺ is detected do not satisfy the LET, and collisional rate coefficients are needed to appropriately model observations.

As a second application of our rate coefficients, we decided to verify assumptions made about N⁺ in modelling the nitrogen chemistry in cold environments such as dark clouds. As emphasized in Dislaire et al. (2012), N⁺ + H₂ → NH⁺ + H is identified as a key reaction for the formation of nitrogen hydrides in dark clouds (Agúndez & Wakelam 2013; Hily-Blant et al. 2013; Le Gal et al. 2014). Whether the reaction N⁺ + H₂ → NH⁺ + H is efficient enough to reproduce the observed amount of ammonia in dark clouds has actually been debated since the original claim made by Le Bourlot (1991), a claim later challenged by updated calculations of the rate coefficients for N⁺ + H₂ → NH⁺ + H (Dislaire et al. 2012), which felt below the critical value that had been inferred to explain the observed abundance of ammonia. However, due to the fact that in such cold environments, N⁺ cannot be detected, assumptions have to be made about the distribution of its fine structure population. It is thus interesting to check the assumption according to which its population follows an LTE distribution in cold molecular clouds.

In such environments, the exact population of N⁺ fine structure levels is not known and can influence the reactivity of N⁺. LTE conditions are usually assumed but not checked because of the lack of collisional data (Gómez-Carrasco et al. 2022). Thus, we decided to perform radiative transfer calculations to determine the population levels of the fine structure levels of N⁺ as a function of the density of neutral species (n(H₂)), where ‘neutral species’ refer to 90 per cent of H₂ and 10 per cent of He, at 10, 20, and 50 K, respectively, (see Fig. 10). At 10 K, as could have been anticipated, almost 100 per cent of N⁺ is in its ground state (not shown here), even at high densities according to the LTE distribution. But the level of population changes as the temperature increases: at 20 K already, one observes 2–5  per cent of N⁺ in level ³P₁ at densities of 10⁴–10⁵ cm⁻³. At 50 K, 20 per cent to 30 per cent of the whole N⁺ is in the ³P₁ level for a density of neutral species of 10⁴–10⁵ cm⁻³. Such values are below the LTE conditions that predict ∼10 per cent and ∼40 per cent of N⁺ is in ³P₁ level, respectively. Given that the N⁺ + H₂ → NH⁺ + H rate coefficients depend on the N⁺ energy level population, this means that the reaction will not happen as predicted by the model. This could possibly have an important impact on the determination of the age of the clouds when the abundance of NH₃ is taken as the main tracer of it.

5 CONCLUSION

We have calculated new PECs for N⁺–He at the RCCSD(T) level of theory with an extrapolation to the CBS. The calculations were found to be in good agreement with former theoretical studies. Using these new PECs, we have performed CC calculations in order to determine collisional excitation cross-sections for N⁺–He and derive collisional rate coefficients.

N⁺–H₂ has also been studied. These data have been computed also using RCCSD(T) approach on the basis of a spherically averaged approach to the PECs for N⁺–H₂, relying on important approximations. The comparison of the N⁺–H₂ rate coefficients with the N⁺–He one shows that significant differences exist. The differences are of such nature, e.g. differences in the hierarchy of different de-excitation transitions, that even scaled by the square root of the collision reduced mass, the He rate coefficients cannot be reasonable approximation to the N⁺–H₂ rate coefficients.

Then, we considered the astrophysical consequences of our calculations. First, we have shown that media where N⁺ is typically observed are not at LTE, and that LTE conditions are reached only for densities much higher than the typical conditions of warm ionized clouds. This illustrates the importance of taking into account the collisional rates coefficients. Furthermore, we have tested some important assumptions that are traditionally part of the modelling of nitrogen and ammonia. Our results show that the assumption of LTE distribution of the N⁺ energy levels at temperatures below 50 K is not verified above 10 K. This could have an important impact on the modelling of ammonia, especially on the determination of the age of the clouds where N⁺ is detected. Finally, we emphasize that the present study should be completed by the theoretical study of the collisional excitation of N⁺ by H and H⁺, the dominant heavy projectiles in warm (ionized) clouds in order to fully interpret and exploit the N⁺ observations recently performed.

ACKNOWLEDGEMENTS

This project received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement no. 811363) and from the European Union Horizon 2020 research and innovation programme under the Marie Slodowska-Curie grant agreement no. 101026214.

DATA AVAILABILITY

The data underlying this article will be made available through the EMAA,⁴ LAMDA (Schöier et al. 2005; van der Tak et al. 2020), and BASECOL (Dubernet, M.-L. et al. 2013) data bases. The PECs computed in this work are available upon request from the authors.

Footnotes

References