# First rate coefficients for an interstellar anion: application to the CN⁻—H₂ collisional system

J. Klos, F. Lique

► To cite this version:

J. Klos, F. Lique. First rate coefficients for an interstellar anion: application to the CN⁻—H₂ collisional system. Monthly Notices of the Royal Astronomical Society, 2011, 418 (1), pp.271-275. 10.1111/j.1365-2966.2011.19481.x. hal-03047417

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# First rate coefficients for an interstellar anion: application to the CN⁻–H₂ collisional system

J. Kłos<fnref n="1" />* and F. Lique<fnref n="2" />*

Accepted 2011 July 20. Received 2011 July 13; in original form 2011 June 16

## ABSTRACT

Recently, several molecular anions have been detected in the interstellar medium. Accurate modelling of their microwave emission spectra requires the calculation of rates for rotational excitations by collisions with the most abundant species. We report the first collisional data for an interstellar anion, the rate coefficients for the rotational excitation of CN⁻ by H₂. State-to-state rate coefficients between the 11 lowest rotational levels of CN⁻ were calculated for temperatures ranging from 5 to 100 K. Collisions with para- and ortho-H₂ are taken into account. The CN⁻ rate coefficients are compared with those of the neutral CO molecule, and, as expected, it is found that the rate coefficients for the anion are significantly larger than those of the neutral species. The new set of rate coefficients will help significantly in the interpretation of the CN⁻ emission lines observed with current and future telescopes, and enable this molecule to become a powerful astrophysical tool.

**Key words:** molecular data – molecular processes – scattering.

## 1 INTRODUCTION

During the last 5 years, several anions have been detected in the interstellar and circumstellar media. The first anion detected in space was C₆H⁻ (McCarthy et al. 2006). Then, C₄H⁻ (Cernicharo et al. 2007), C₈H⁻ (Brünken et al. 2007), C₃N⁻ (Thaddeus et al. 2008) and C₅N⁻ (Cernicharo et al. 2008) have also been detected. These anions are all linear multiply-bonded carbon chains with closed-shell electronic ground-state configurations. The detections of anions have led to interesting research on chemistry of negative molecular ions. Now, one believes that they are formed by radiative attachment of an electron to the neutral molecule with the same chemical formula (Millar et al. 2007). The abundance of these anions relative to the neutral counterparts increases with both size and the electron affinity of the neutral molecule, as expected for formation by radiative electron attachment (Herbst & Osamura 2008). However, this process cannot explain the abundance of small interstellar anions. Accurate rate coefficients enable accurate determination of the molecular abundance in the interstellar medium (ISM; Lique et al. 2009; Sarrasin et al. 2010). Then, one needs accurate collisional rate coefficients for the interpretation of anions molecular line observations and for solving the problem of the abundance of small anions.

Very recently, the CN⁻ anion has been discovered in the C star envelope IRC +10216 (Agúndez et al. 2010). The identification

of CN⁻ has been supported by the calculation of rate coefficients for CN⁻–para-H₂ (*j* = 0) collisional system. The rate coefficients provided in this paper were obtained from preliminary calculations based on an CN⁻–H₂ potential energy surface (PES) averaged over H₂ orientations. We believe that it is important to extend these preliminary calculations and take into account the rotation of H₂ both in the PES and in the scattering calculations in order to accurately evaluate rate coefficients for this interstellar anion.

To the best of our knowledge, there are no calculations for interstellar anions in collision with He or H₂. The present calculations will provide the first accurate estimate of the order of magnitude of the rate coefficients of interstellar anions. One has to mention that rate coefficients for anions with He will not be relevant for molecular clouds’ modelling. Indeed, the interaction of charged molecules with He is expected to be generally different from the one with H₂ (i.e. possible presence of reactive channels Ramazani et al. 2009), and one could anticipate that at the opposite of what has been found for neutral species (Wernli et al. 2006; Lique et al. 2008; Dumouchel, Faure & Lique 2010; Dumouchel, Kłos & Lique 2011), rate coefficients of anions with He may not be an estimate of the one with H₂.

In this paper, we present calculations of a completely new five-dimensional (5D) PES for the ground electronic state of the CN⁻–H₂ collisional system calculated at the coupled clusters’ level. Using this accurate PES averaged over ground vibrational state of CN⁻, rotational excitation of CN⁻ with both para- and ortho-H₂ were studied. Collisional cross-sections for the first 11

<footnote n="1">¹Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742, USA</footnote>

<footnote n="2">²LOMC - FRE 3102, CNRS-Université du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre, France</footnote>

<footnote>*E-mail: jklos@umd.edu (JK); francois.lique@univ-lehavre.fr (FL)</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1. Body-fixed Jacobi coordinates of CN⁻–H₂.</figcaption></figure>

rotational levels of CN⁻ in collision with H₂($i_2 = 0, 1$) are reported for kinetic energies up to at least 800 cm⁻¹, yielding converged rate coefficients up to 100 K. The paper is organized as follows. Section 2 describes *ab initio* calculations and modelling of the PES, and Section 3 contains a concise description of the scattering calculations. In Section 4, we present and discuss our results.

## 2 POTENTIAL ENERGY SURFACE

### 2.1 Coordinates

We define the body-fixed coordinate system as shown in Fig. 1. The geometry of the CN⁻–H₂ system with H₂ treated as rigid rotor is then characterized by three angles $\theta, \theta'$ and $\phi$, and the distance R between the centres of masses of H₂ and CN⁻. The polar angles of the CN⁻ and H₂ molecules with respect to R are denoted, respectively, $\theta$ and $\theta'$, while $\phi$ denotes the dihedral angle, which is the relative polar angle between the anion and H₂ bonds. As shown previously (Faure et al. 2005; Jankowski & Szalewicz 2005), a better description of the intermolecular potential is obtained by fixing the molecular distance at its average value in the ground vibrational level rather than at the equilibrium distance. Accordingly, we used a H₂ bond distance $r_{H-\text{H}} = 1.448\,76a_0$. During the *ab initio* calculations of the potential, we vary the interatomic distance r of the CN⁻ molecule. The final $V(r, R, \theta, \theta', \phi)$ potential thus is 5D.

### 2.2 Calculation of the interaction energy

The PES was calculated within the supermolecular approach using single- and double-excitation coupled cluster method with perturbative contributions from connected triple excitations [CCSD(T)] (Knowles, Hampel & Werner 1993, 2000). The electronic structure calculations were performed using the MOLPRO package (Werner et al. 2010). In all calculations, to determine the interaction potential $V(r, R, \theta, \theta', \phi)$, the basis set superposition error was corrected at all geometries with the Boys and Bernardi counterpoise procedure (Boys & Bernardi 1970):

$$V(r, R, \theta, \theta', \phi) = E_{\text{CN}^-\text{H}_2}(r, R, \theta, \theta', \phi) - E_{\text{CN}^-}(r, R, \theta, \theta', \phi) - E_{\text{H}_2}(r, R, \theta, \theta', \phi), \quad (1)$$

where the energies of the CN⁻ and H₂ subsystems are computed with the full (four atoms plus bond functions) basis set.

The four atoms were described by the augmented, correlation-consistent triple zeta basis set (aug-cc-pVTZ) of Woon & Dunning (1994) augmented by the (3s, 2p, 1d) bond functions defined by Williams et al. (1995), placed at mid-distance between the CN⁻ and H₂ centres of mass. The radial scattering coordinate R was assigned 45 values from 50.0a₀ to 3.5a₀, the $\theta$ grid ranged from 0° to 180° in steps of 20° plus geometry with $\theta = 90°$, whereas $\theta'$ and $\phi$ ranged from 0° to 180° and 0° to 90°, respectively in 22.5 increment. The geometry with $\theta = 0°$ and $\theta' = 0°$ corresponds to C–N⁻···H–H collinear arrangement.

For a van der Waals system, where the ground state is described well by a predominant configuration at all computed geometries, the CCSD(T) level of theory and the use of aug-cc-pVTZ basis set augmented by mid-bond functions is expected to yield reliable results (Lique & Kłos 2008).

The scattering calculations will be performed using 4D potential with the CN⁻ treated as a rigid rotor. For this purpose, we average the 5D PES with a vibrational ground-state wavefunction of the CN⁻ diatomic. The CN⁻ ground-state vibrational wavefunction was determined by solving diatomic nuclear Schrödinger equation with the sinc discrete variable representation (DVR) method of Colbert & Miller (1992) using the CN⁻ potential of Špirko & Polák (2008). The averaging procedure results in 4D $V_{v=0}(R, \theta, \theta', \phi)$ PES which will be used in subsequent scattering calculation using MOLSCAT program (Hutson & Green 1994).

The global minimum of the $V_{v=0}(R, \theta, \theta', \phi)$ PES is located at $R = 6.38a_0$ for the collinear C–N⁻···H–H geometry ($\theta = 0°, \theta' = 0°$) with a well depth of 875.4 cm⁻¹. The depth of this well comes from the charge interaction with quadrupole and higher moments of the H₂ molecule and other induction or dispersion terms. The collinear geometry with the C atom facing the H atom of the H₂ is located at $R = 7.00a_0$ with a well depth of 751.2 cm⁻¹. Fig. 2 represents the cut of the PES for $\theta' = 0°$ and $\phi = 0°$.

There is a significant barrier for the H₂ rotation (~820 cm⁻¹), the interaction energy for the T-shape geometry, where H₂ is an upper bar of the T letter ($\theta = 0°, \theta' = 90°$), is only -54.5 cm⁻¹. This means that the H₂ in this complex is rather hindered rotor.

<figure><img src="image_3.png" /><figcaption>Figure 2. Contour plot of the cut of the CN⁻–H₂ PES for the H₂ orientation corresponding to the minimum energy.</figcaption></figure>
## 2.3 Analytical representation

The set of angular *ab initio* points for each *R*, *r* distance is fitted to the expansion in bispherical harmonics identical to those used in *MOLSCAT* program Hutson & Green (1994):

$$V(r, R, \theta, \theta', \phi) = \sum_{l_1, l_2, l} v_{l_1, l_2, l}(r, R) A_{l_1, l_2, l}(\theta, \theta', \phi), \quad (2)$$

where $A_{l_1, l_2, l}(\theta, \theta', \phi)$ is formed from coupled spherical harmonics which are associated with the rotational angular momenta of CN⁻ and H₂. In expression (2), the index $l_2$ is even because H₂ is homonuclear.

To construct the above expansion, at each point of the radial grid *R*, we have fitted the *ab initio* points using a linear least-squares method to expression (2) with $l_1^{\max} = 8$ and $l_2^{\max} = 4$. The choice of such expansion parameters limits results to a total of 68 radial expansion coefficients $v_{l_1, l_2, l}$. The quality of the angular expansion fits was very good, the rms error was of the order of $10^{-4}$ to $10^{-5}$ cm⁻¹.

To obtain analytical representation of the radial coefficients $v_{l_1, l_2, l}$, we have used the reproducing Kernel Hilbert space method of Ho & Rabitz (1996) that interpolates between points and extrapolates in the long-range as $R^{-3}$ (charge-quadrupole term).

## 3 SCATTERING CALCULATIONS

We consider collisions of CN⁻ with para- and ortho-H₂ such as

$$\mathrm{CN}^{-}(j_1) + \mathrm{o-}; \mathrm{p-H}_{2}(j_2) \rightarrow \mathrm{CN}^{-}(j'_1) + \mathrm{o-}; \mathrm{p-H}_{2}(j'_2), \quad (3)$$

where $j_1$ and $j_2$ designate the rotational levels of CN⁻ and H₂, respectively. The calculations are restricted to low/moderate collisional energies ($E_c < 1000 \text{ cm}^{-1}$). The possible rotational excitation of H₂ ($j_2 = 0 \rightarrow j_2 = 2$ or $j_2 = 1 \rightarrow j_2 = 3$) has been taken into account in the calculations, but we do not show these results since the corresponding rate coefficients are at least three orders of magnitude lower than those without H₂ excitation. The rotational energy levels of the CN⁻ and H₂ molecules were computed using the experimental spectroscopic constants of Huber & Herzberg (1979).

Because the full close-coupling (CC) approach of Green (1975) is computationally demanding for molecules with small rotational constants, we have used the coupled-states (CS) approximation (McGuire & Kouri 1974) for the determination of collision-energy-dependent integral cross-sections. The validity of the CS approach compared to a full CC approach is tested in Table 1. It is found that the CS approach can lead to inaccuracies of 20–50 per cent for very low total energy cross-sections (<50 cm⁻¹), but agreement improves between CC and CS cross-sections when the total energy is greater than 50 cm⁻¹, and we expect that the corresponding rate coefficients will be accurate since they are not as sensitive to the accuracy of description of the resonances. The relative accuracy of the CS approximation compared to full CC approach can

<table><caption>**Table 1.** Comparison of CS and CC integral cross-sections at three values of total energy. Cross-sections in units of Å².</caption><thead><tr><th rowspan="2">Transition<br>(j<sub>1</sub>, j<sub>2</sub>) → (j'<sub>1</sub>, j'<sub>2</sub>)</th><th colspan="2">50 cm<sup>-1</sup></th><th colspan="2">100 cm<sup>-1</sup></th><th colspan="2">250 cm<sup>-1</sup></th></tr><tr><th>CC</th><th>CS</th><th>CC</th><th>CS</th><th>CC</th><th>CS</th></tr></thead><tbody><tr><td>(1,0) → (0,0)</td><td>40.57</td><td>41.33</td><td>26.39</td><td>25.08</td><td>12.66</td><td>13.87</td></tr><tr><td>(2,0) → (0,0)</td><td>7.06</td><td>9.63</td><td>5.51</td><td>8.45</td><td>4.91</td><td>4.47</td></tr><tr><td>(2,0) → (1,0)</td><td>65.30</td><td>67.03</td><td>40.25</td><td>38.90</td><td>19.93</td><td>22.53</td></tr><tr><td>(3,0) → (1,0)</td><td>19.75</td><td>11.58</td><td>8.14</td><td>12.29</td><td>7.82</td><td>9.70</td></tr></tbody></table>

be explained by the relatively small energy spacing between CN⁻ rotational levels.

The standard time-independent coupled scattering equations were solved using the *MOLSCAT* code developed by Hutson & Green (1994). The calculations were performed using the propagator of Manolopoulos (1986), with the reduced mass of the system being 1.871 amu. Calculations were carried out at values of the total energy ranging from 3.0 to 1000 cm⁻¹ for collision of CN⁻ with para-H₂ and from 122.4 to 1200 cm⁻¹ for collision of CN⁻ with ortho-H₂. The integration parameters were chosen to ensure convergence of the cross-sections over this range. In all of the calculations, the integration range extended from 4.0 to 80a₀, due to the large well depth of the PES. The *STEPS* parameter was increased at the lowest energies to constrain the step length of the integrator below 0.1–0.2a₀. We have extended the CN⁻ rotational basis sufficiently to ensure convergence of the inelastic cross-sections. At the largest total energy considered, the CN⁻ rotational basis was extended to $j_1 = 20$ ($E \approx 1100$ K) to ensure convergence of the first 11 (up to $j_1 = 10$) rotational levels of CN⁻. The rotational basis of H₂ was extended to $j_2 = 2$ and 3 for calculations with para- and ortho-H₂, respectively. The inclusion of the $j_2 = 3$ channel improves the convergence of the ortho-H₂ cross-sections. This channel is needed due to the fact that the well depth of the potential is quite substantial and because of large anisotropy. The maximum value of the total angular momentum *J* used in the calculations was set automatically to be large enough that the inelastic cross-sections were converged to within 0.05 Å². All 68 terms contained in the Legendre polynomial expansion of the potential defined by equation (2) were included in the calculation. Calculations with negative ions are much harder than with neutral species since the strong interaction in the PES imply to add many closed channel in the scattering calculations and to perform calculations for large total angular momentum *J* in order to converge the inelastic cross-sections.

From the rotationally inelastic cross-sections $\sigma_{j_1 j_2 \rightarrow j'_1 j'_2}(E_k)$, one can obtain the corresponding thermal rate coefficients at temperature *T* by an average over the collision energy ($E_k$) (Smith 1980):

$$k_{j_1 j_2 \rightarrow j'_1 j'_2}(T) = \left( \frac{8}{\pi \mu k_B^3 T^3} \right)^{1/2} \times \int_0^\infty \sigma_{j_1 j_2 \rightarrow j'_1 j'_2}(E_k) E_k e^{(-E_k/k_B T)} dE_k, \quad (4)$$

where $k_B$ is Boltzmann's constant and $\mu$ the reduced mass. To obtain precise values of the rate coefficients, the energy grid was chosen to be sufficiently fine to include scattering resonances which are observed in the cross-sections at low energies (not shown here). Calculations up to kinetic energies of 800 cm⁻¹ allow us to determine rate coefficients up to 100 K.

## 4 RESULTS

Fig. 3 illustrates the variation with temperature of the de-excitation rate coefficients of CN⁻ in collision with para-H₂($j_2 = 0$) and ortho-H₂($j_2 = 1$). One can see that there are no significant differences between the collisions with the two species of H₂. The rate coefficients for collisions with H₂($j_2 = 0$) and H₂($j_2 = 1$) are about the same order of magnitude. This trend is opposite to the trend observed for several interstellar neutral species like CO (Wernli et al. 2006), SiS (Klos & Lique 2008), HNC (Dumouchel et al. 2011) or H₂O (Dubernet et al. 2006).

Then, we were interested in the propensity rules of pure CN⁻ rotational transitions. Fig. 4 presents downward rotational rate
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Typical variation with temperature of collisional de-excitation rate coefficients of CN⁻ by para-H₂(j₂ = 0) (panel a) and ortho-H₂(j₂ = 1) (panel b).</figcaption></figure>

coefficients out of CN⁻(j₁ = 5) levels at 50 K for collisions with para- and ortho-H₂. It is interesting to note that for collisions with both para- and ortho-H₂, inelastic rate coefficients with Δj₁ = 1 are larger than those with Δj₁ = 2. This plot also confirmed that

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Propensity rules for transitions out of the initial j₁ = 5 state of the CN⁻ molecule with para- and ortho-H₂ collisions for T = 50 K.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Temperature variation of the CN⁻ (solid lines) and CO (dashed lines) de-excitation rate coefficients for collisions with para-H₂(j₂ = 0).</figcaption></figure>

no significant differences exist between the collisions with the two species of H₂.

Finally, the number of interstellar anions detected in space became important, and there were, up to this work, no estimates of the collisional rate coefficients for these negatively charged species. One way to estimate the anionic rate coefficients could be by using the rate coefficients of the corresponding neutral species or rate coefficients for an isoelectronic molecule. In the present case, the CN⁻ rate coefficients could be estimated from the CN (Lique et al. 2010; Lique & Kłos 2011) or CO rate coefficients. The CO molecule is isoelectronic with CN⁻ and has the same ground electronic state ¹Σ⁺. Hence, we compare in Fig. 5, on a small data sample, the temperature variation of the de-excitation rate coefficients of CN⁻ in collisions with para-H₂(j₂ = 0) with those obtained recently for the neutral CO molecule in collision with para-H₂(j₂ = 0) (Wernli et al. 2006). One has to mention that other CO-H₂ rate coefficients were computed recently by Yang et al. (2010), and they are in good agreement with those of Wernli et al. (2006) for small Δj transitions.

On can clearly see on Fig. 5 that the CN⁻ rate coefficients are significantly larger than those of CO. Indeed, the CN⁻-H₂(j₂ = 0) rate coefficients are up to a factor of 10 larger than those of CO-H₂(j₂ = 0). The large difference can be explained by the shape of the PES. The well depth of the CO-H₂ PES of Jankowski & Szalewicz (2005) is 93 cm⁻¹, whereas that of CN⁻-H₂ PES is 875 cm⁻¹, which is about nine times larger. This comparison shows that the impact of collision will be more important for interstellar anions than for neutral species, but also that neutral rate coefficients cannot be used for modelling of interstellar line emissions of anions.

## 5 SUMMARY AND CONCLUSIONS

We have applied 4D quantum CC scattering calculations to investigate rotational energy transfer in collisions of the CN⁻ molecule with para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1) molecules. The calculations are based on a new reliable 5D *ab initio* PES for the CN⁻-H₂ complex and expanded in a bispherical angular basis.

Integral cross-sections for transitions involving the lowest 11 levels of the CN⁻ molecule and j₂ = 0, 1 levels of H₂ molecule were determined for kinetic energies up to a least 800 cm⁻¹. Corresponding rate coefficients were given for temperature ranging from 5 to 100 K. We have found that the rate coefficients are rather weakly dependent on the rotational level of H₂.
We have also compared the present set of CN⁻-H₂ rate coefficients with those recently computed for CO-H₂. Significant differences were found showing that the rate coefficients for anions cannot be simply obtained from neutral rate coefficients. This means that rate coefficients for species like C₄H⁻ or C₃N⁻ could not be derived from those corresponding to neutral C₄H, C₃N or HC₃N (Wernli et al. 2007) molecules.

Finally, the CN⁻-H₂ rate coefficients presented in this work are the first ones not only for this molecule, but also for an interstellar anions. Taking into account the importance of the anions in the interstellar chemistry, we hope that astronomers will analyse the impact of the new rate coefficients on the rotational excitation of CN⁻, and we also expect that these results will help in the interpretation of the CN⁻ emission lines observed with current and future telescopes.

## ACKNOWLEDGMENTS

This research was supported by the CNRS national programme 'Physique et Chimie du Milieu Interstellaire'. JK is grateful to M. H. Alexander for financial support from the US National Science Foundation, Grant CHE-0848110.

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