# Hyperfine excitation of CN by He

F. Lique, J. Klos

► To cite this version:

F. Lique, J. Klos. Hyperfine excitation of CN by He. Monthly Notices of the Royal Astronomical Society: Letters, 2011, 413 (1), pp.L20-L23. 10.1111/j.1745-3933.2011.01023.x. hal-03047426

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# Hyperfine excitation of CN by He

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

Accepted 2011 January 31. Received 2011 January 28; in original form 2011 January 7

## ABSTRACT

Modelling of molecular emission spectra from interstellar clouds requires the calculation of rates for excitation by collisions with the most-abundant species. We calculate hyperfine structure resolved excitation rate coefficients of the cyano radical (CN) by He. State-to-state rate coefficients between the 37 lowest hyperfine levels of CN were calculated for low temperatures in the range 5–30 K. The new 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

The existence of accurate collisional rate coefficients is crucial to the interpretation of molecular line observations. Without these rates, only approximate estimates of the molecular column density are possible assuming local thermodynamic equilibrium (LTE), which at the typical densities of dense interstellar clouds is generally not a good approximation (Lique et al. 2009). For molecules for which rates are known, non-LTE calculations may be performed that enable us not only to complete accurate column density estimates, but also to complete estimates of kinetic temperature and volume density from spectral lines. Therefore, astrophysical models improve significantly in sophistication with the knowledge of molecular collision rates.

Of more than 150 molecules known to exist in space, rate coefficients exist for fewer than 30 and, despite recent efforts by the molecular astrophysics community, rate coefficients are still missing for a large number of interstellar molecules. An example of a key chemical species for which collisional excitation coefficients have been unavailable until now is the cyano radical (CN). The CN molecule is one of the most widely distributed and best-known species in the interstellar medium (ISM). CN was the second interstellar molecule to be identified, going back to the 1940s (McKellar 1940; Adams 1941). A microwave detection in the Orion nebula and W51 by Jefferts, Penzias & Wilson (1970) followed. Since then, CN has been extensively observed (Fuente, Martin-Pintado & Gaume 1995; Falgarone et al. 2008; Hily-Blant et al. 2008; Milam, Woolf & Ziurys 2009; Pérez-Beaupuits et al. 2009) and most recently its anion (CN⁻) has also been detected (Agúndez et al. 2010). In diffuse molecular gas, CN not only is a probe of relatively dense regions, but also plays a role as a tracer of high-density gas. The CN thermal

lines also probe dense regions in molecular clouds and CN Zeeman observations are therefore a unique tool for measuring magnetic field strengths in star formation regions (Falgarone et al. 2008).

The determination of the rate coefficients for collisions of the CN molecule with most-abundant interstellar species has been one of important goals in past decades to improve modelling of the ISM. Hence, CN rate coefficients have been tentatively estimated from a scaling of the CO or CS rate coefficients (Truong-Bach et al. 1987; Fuente et al. 1995), but the scaling from CO or CS fails because CN is an open-shell molecule and has a $^2\Sigma^+$ ground state. Taking into account the sensitivity of the astrophysical modelling to the rate coefficients (Daniel, Cernicharo & Dubernet 2006; Lique, Cernicharo & Cox 2006a; Lique, Spielfiedel & Cernicharo 2006b; Sarrasin et al. 2010), it is very important to provide accurate rate coefficients for the CN molecule directly from quantum calculations.

This Letter aims to advance the above discussion by providing the necessary data to obtain accurate estimates of CN abundances from observational data. The calculations of rate coefficients for the rotational excitation of the CN molecule is a true challenge. Indeed, the CN molecule is an open-shell species (i.e. with a fine structure) and the presence of the nitrogen atom with a non-zero nuclear spin also leads to a hyperfine structure. The fine and the hyperfine structures have to be taken into account if one wants to provide accurate rate coefficients. As the first step, we concentrate on the collisional excitation of CN by a He atom (as a substitute for H₂). Recently, some of us have presented the calculation of rate coefficients between the fine structure levels of CN by He (Lique et al. 2010, hereinafter Paper I). Hence, we present, in this Letter, an extension of the previous calculations to the hyperfine levels of CN at low temperatures.

This Letter is organized as follows. After an outline of the computational details in Section 2, the collisional rate coefficients are presented in Section 3. Conclusions of this work are drawn in Section 4.

<footnote n="1">¹LOMC - FRE 3102, CNRS-Université du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre, France</footnote>

<footnote n="2">²Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742, USA</footnote>

<footnote>*E-mail: francois.lique@univ-lehavre.fr</footnote>
## 2 COMPUTATIONAL METHODOLOGY

### 2.1 Potential energy surface

For low-energy rotational excitation, we determined an accurate potential energy surface (PES) by means of partially spin-restricted coupled cluster calculations with single, double and perturbative triple excitations [RCCSD(T)] (Knowles, Hampel & Werner 1993, 2000). We kept the CN bond distance (r) frozen at its experimental equilibrium geometry ($r_c = 2.21a_0$; Huber & Herzberg 1979). We used the aug-cc-pVQZ basis set of Woon & Dunning (1994) for the three atoms, augmented by the bond functions optimized by Cybulski & Toczyłowski (1999) placed at a mid-distance between the CN's centre of mass and He. Details of computations and plots of the PES are given in Paper I. For the weakly bound CN-He system, the global minimum of the interaction energy is $-21.28 \text{ cm}^{-1}$ at $R = 6.8a_0$ and $\theta = 118^\circ$.

Rate coefficients obtained from this PES were found to be in excellent agreement with the experimental rotational (with unresolved fine and hyperfine structure) rate coefficients of Fei et al. (1994) (see Paper I). This agreement demonstrates the accuracy of the PES and suggests that the rate coefficients obtained from this potential will be accurate enough to allow for improved astrophysical modelling of the ISM.

### 2.2 Scattering calculations

The main goal of this work is the use of the 2D CN-He PES to determine hyperfine state-resolved excitation and de-excitation integral cross-sections and rate coefficients of CN molecules by He.

For CN in its ground electronic $^2\Sigma^+$ state, the molecular energy levels can be described in the Hund's case (b) limit. Here, the fine structure levels are labelled by $N_j$, where N is the rotational angular momentum and j is the total molecular angular momentum quantum number with $j = N + S$, where S is the electronic spin. For molecules in a $^2\Sigma^+$ state, $S = 1/2$. Hence, two kinds of levels exist, the levels with $j = N + 1/2$ and those with $j = N - 1/2$. The nitrogen atom also possesses a non-zero nuclear spin ($I = 1$). The coupling between I and j results in a splitting of each level into three hyperfine levels (except for the $j = 1/2$ level which is split into only two levels). Each hyperfine level is designated by a quantum number $F(F = I + j)$ varying between $|I - j|$ and $I + j$.

We used Alexander's description (Alexander 1982) of the inelastic scattering between an atom and a diatomic molecule in a $^2\Sigma^+$ electronic state and a fully-quantum close-coupling calculation in order to obtain the $S'(jl; j'l')$ diffusion matrix between fine structure levels of CN. J and l denote the total angular momentum ($J = j + l$) and the orbital angular momentum quantum numbers, respectively. The integral cross-sections corresponding to transitions between hyperfine levels of the CN molecule can be obtained from a scattering S-matrix between fine structure levels using the re-coupling method of Alexander & Dagdigian (1985). Inelastic cross-sections associated with a transition from an initial hyperfine level $N_jF$ to a hyperfine level $N' j' F'$ were thus obtained as follows:

$$ \sigma_{N_j F \rightarrow N' j' F'} = \frac{\pi}{k_{N_j F}^2} (2F' + 1) \sum_K \times \begin{Bmatrix} j & j' & K \\ F' & F & I \end{Bmatrix} P^K (j \rightarrow j'). \quad (1) $$

The $P^K(j \rightarrow j')$ are the tensor opacities defined by

$$ P^K(j \rightarrow j') = \frac{1}{2K + 1} \sum_{ll'} |T^K jl; j'l'|^2. \quad (2) $$

The reduced T-matrix elements (where $T = 1 - S$) are defined by Alexander & Davis (1983):

$$ T^K jl; j'l' = (-1)^{j-l'} (2K + 1) \sum_J (-1)^J (2J + 1) \times \begin{Bmatrix} l' & j' & J \\ j & l & K \end{Bmatrix} T^J jl; j'l'. \quad (3) $$

All the scattering calculations were performed with the HIBRIDON package<fnref n="1" /> modified to take into account the hyperfine structure. The calculation parameters and the energy grid were taken from Paper I.

From the rotationally inelastic cross-sections $\sigma_{N_j F \rightarrow N' j' F'}(E_c)$, one can obtain the corresponding thermal rate coefficients at temperature T by an average over the collision energy ($E_c$) (Smith 1980):

$$ k_{N_j F \rightarrow N' j' F'}(T) = \left( \frac{8}{\pi \mu k_B^3 T^3} \right)^{1/2} \times \int_0^\infty \sigma_{N_j F \rightarrow N' j' F'} E_c e^{-\frac{E_c}{k_B T}} dE_c, \quad (4) $$

where $k_B$ is Boltzmann's constant and $\mu$ is the reduced mass of the CN-He complex.

## 3 RESULTS

Using the computational scheme described above, we have obtained inelastic cross-sections up to the total energy of $250 \text{ cm}^{-1}$ for transitions between the first 37 hyperfine levels with $N, N' \le 6$. Calculations up to the total energy of $250 \text{ cm}^{-1}$ allow us to determine the corresponding rate coefficients up to 30 K. The complete set of (de-)excitation rate coefficients with $N, N' \le 6$ is available online at the LAMDA<fnref n="2" /> website.

Figs 1 and 2 present the temperature variation of the CN-He rate coefficients for selected $N = 1, j, F \rightarrow N' = 0, j', F'$ and $N = 2, j, F \rightarrow N' = 1, j', F'$ transitions, respectively. Table 1 shows the hyperfine resolved CN-He rate coefficients for $N = 4, j, F \rightarrow N' = 3, j', F'$ transitions at 20 K.

First of all, one can see, from Fig. 2 and Table 1, a propensity rule for $\Delta j = \Delta N$ transitions for whichever hyperfine level involved. This propensity rule is more pronounced when N increases. This behaviour, predicted theoretically (Alexander, Smedley & Corey 1986), is a general feature of collisions of molecules in a $^{2S+1}\Sigma$ electronic state, as shown previously for $O_2$-He (Corey, Alexander & Schaefer 1986; Lique 2010) or SO-He collisions (Lique et al. 2005).

Next, we have to distinguish between $\Delta j = \Delta N$ and $\Delta j \neq \Delta N$ transitions in order to discuss the hyperfine selection rules. For $\Delta j = \Delta N$ transitions, we have a strong propensity rule in favour of $\Delta j = \Delta F$ transitions; the propensity rule is more pronounced when

<footnote n="1">¹ The HIBRIDON package was written by M. H. Alexander, D. E. Manolopoulos, H.-J. Werner and B. Follmeg, with contributions from P. F. Vohralik, D. Lemoine, G. Corey, R. Gordon, B. Johnson, T. Orlikowski, A. Berning, A. Degli-Esposti, C. Rist, P. Dagdigian, B. Pouilly, G. van der Sanden, M. Yang, F. de Weerd, S. Gregurick, J. Kłos and F. Lique, http://www2.chem.umd.edu/groups/alexander/</footnote>

<footnote n="2">² http://www.strw.leidenuniv.nl/~moldata</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Temperature variation of the hyperfine resolved CN–He rate coefficients for $N = 1, j, F \rightarrow N' = 0, j', F'$ transitions.</figcaption></figure>

$N$ increases. This trend is the usual trend for open-shell molecules (Alexander & Dagdigian 1985; Corey & Alexander 1988).

For $\Delta j \neq \Delta N$ transitions, it is very difficult to find a clear propensity rule. The final distribution seems to be governed by two rules: the rate coefficients show propensity in favour of $\Delta j = \Delta F$ transitions, but are also proportional to the degeneracy ($2F' + 1$) of the final hyperfine level. For example, for $N = 4, j = 4.5, F \rightarrow N' = 3, j' = 2.5, F'$ transitions at 20 K, when $F = 3.5$, all the final hyperfine levels from $N' = 3, j' = 2.5$ seem to be reasonably equally populated (at least for $F = 1.5$ and 2.5), whereas, when $F = 5.5$, a propensity in favour of $F' = 3.5$ is observed [$\Delta j = \Delta F$ transitions and the highest degeneracy ($2F' + 1$) for this state].

Fig. 3 presents the temperature variation in CN–He rate coefficients for the ‘quasi-elastic’ $N = 2, j, F \rightarrow N' = 2, j', F'$ transitions.

One can note that the order of magnitude of these ‘quasi-elastic’ rate coefficients is about the same as that of the pure inelastic rate coefficients. For transitions inside a rotational level $N$, we have a clear propensity rule in favour of $\Delta j = 0$ transitions and $\Delta F = \pm 1$ that are about five times larger than the other ones.

## 4 SUMMARY AND CONCLUSIONS

We have used quantum scattering calculations to investigate the rotational energy transfer in collisions of CN($X^2\Sigma^+$) with He atoms. The calculations are based on a new, highly correlated 2D PES calculated at the RCCSD(T) level using a large atomic basis set.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Temperature variation of the hyperfine resolved CN–He rate coefficients for $N = 2, j, F \rightarrow N' = 1, j', F'$ transitions.</figcaption></figure>

Hyperfine state-resolved rate coefficients for transitions involving the lowest 37 energy levels of the CN molecule were determined for temperatures ranging from 5 to 30 K.

The collisions exhibit strong propensity rules in favour of $\Delta j = \Delta F$ transitions for $\Delta j = \Delta N$ transitions. There is no notable propensity for the other transitions. It is important to specify that the rate coefficients presented here are not independent of the initial hyperfine level and not proportional to the degeneracy ($2F' + 1$) of the final hyperfine level. These important results show that the $M_j$ randomizing limit method of Alexander & Dagdigian (1985) cannot be used for such a system.

The CN–He hyperfine rate coefficients presented in this work are the first one for this molecule. Taking into account the importance of the CN molecule in the interstellar chemistry, we hope that astronomers will analyse the impact of the new rate coefficients on the excitation of CN by modelling the pumping of its rotational energy levels through large velocity gradient and non-local radiative transfer models.

Finally, the significant abundance of $H_2$ in the ISM implies that this molecule is the primary collision partner for any other species. At low temperature, $H_2(j = 0)$ is the most-abundant collision partner. It is generally assumed (Schöier et al. 2005) that rate coefficients with He can provide a good estimate of rate coefficients for collision with para-$H_2(j = 0)$. Results on the rotational excitation of CO (Wernli et al. 2006), SiS (Lique et al. 2008; Kłos & Lique 2008) and HNC (Dumouchel et al. 2010, 2011) show that rate
<table><caption>**Table 1.** Hyperfine resolved CN–He rate coefficients (×10⁻¹² cm³ molecule⁻¹ s⁻¹) for N = 4, j, F → N' = 3, j', F' transitions at 20 K.</caption><thead><tr><th></th><th></th><th colspan="2">j' = 2.5</th><th colspan="3">j' = 3.5</th></tr><tr><th></th><th></th><th>F' = 1.5</th><th>F' = 2.5</th><th>F' = 3.5</th><th>F' = 2.5</th><th>F' = 4.5</th></tr></thead><tbody><tr><td rowspan="3">j = 3.5</td><td>F = 2.5</td><td>6.37</td><td>1.14</td><td>0.09</td><td>0.36</td><td>0.12</td><td>0.02</td></tr><tr><td>F = 3.5</td><td>0.26</td><td>6.46</td><td>0.89</td><td>0.09</td><td>0.31</td><td>0.10</td></tr><tr><td>F = 4.5</td><td>0.02</td><td>0.23</td><td>7.35</td><td>0.01</td><td>0.08</td><td>0.41</td></tr><tr><td rowspan="3">j = 4.5</td><td>F = 3.5</td><td>0.09</td><td>0.06</td><td>0.01</td><td>7.20</td><td>0.72</td><td>0.04</td></tr><tr><td>F = 4.5</td><td>0.03</td><td>0.08</td><td>0.05</td><td>0.20</td><td>7.17</td><td>0.59</td></tr><tr><td>F = 5.5</td><td>0.00</td><td>0.03</td><td>0.13</td><td>0.01</td><td>0.18</td><td>7.78</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Temperature variation of the hyperfine resolved CN–He rate coefficients for N = 2, j, F → N' = 2, j', F' transitions.</figcaption></figure>

coefficients for collisions with para-H₂(j = 0) can be up to a factor of 3 higher or lower than those for collisions with He, depending on the selected transition, but that the He rate coefficients scaled by a factor of 1.4 provide the correct order of magnitude of the H₂(j = 0) rate coefficients. Therefore, the present results provide a reasonable first estimate of collisional rate coefficients for collisions of CN with H₂. On the other hand, it may be unadvisable to use the present He rate coefficients as an estimate of the H₂(j > 0) rate coefficients, since the He and H₂(j > 0) rate coefficients usually differ significantly. Specific calculations with para- and ortho-H₂ must be performed in order to provide accurate CN rate coefficients for higher temperature.

## ACKNOWLEDGMENTS

We would like to thank Millard Alexander for his advice and help in the implementation of the hyperfine structure code in the HIBRIDON v4.3 package. This research was supported by the CNRS national program 'Physique et Chimie du Milieu Interstellaire'. The computational component of this work was done using the SIOLINO cluster at the Observatory of Paris. JK is grateful for financial support from the US National Science Foundation Grant CHE-0848110 to M. H. Alexander.

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