# Isotope effect on the formation of CN by radiative association reactions

L. Giani, M. Šimsová Née Zámečníková, G. Di Genova, C. Ceccarelli, N.

Balucani, G. Nyman

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L. Giani, M. Šimsová Née Zámečníková, G. Di Genova, C. Ceccarelli, N. Balucani, et al. Iso-tope effect on the formation of CN by radiative association reactions. AIP ADVANCES, 2024, 14, 10.1063/5.0192682. insu-04836895

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# Isotope effect on the formation of CN by radiative association reactions

L. Giani 🌐 ; M. Šimsová née Zámečníková 🌐 ; G. Di Genova 🌐 ; C. Ceccarelli 🌐 ; N. Balucani 🌐 ;

G. Nyman 🌐

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# Isotope effect on the formation of CN by radiative association reactions

Cite as: AIP Advances 14, 035109 (2024); doi: 10.1063/5.0192682

Submitted: 20 December 2023 • Accepted: 8 February 2024 •

Published Online: 6 March 2024

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L. Giani,<fnref n="1" />,<fnref n="2" />,ᵃ ID M. Šimsová née Zámečníková,<fnref n="3" />,⁴ᵃ ID G. Di Genova,<fnref n="2" /> ID C. Ceccarelli,<fnref n="1" /> ID N. Balucani,<fnref n="1" />,<fnref n="2" /> ID and G. Nyman<fnref n="3" />,⁴ᵃ ID

## AFFILIATIONS

## ABSTRACT

The formation of CN and its isotopologue C¹⁵N through radiative association has been investigated. We considered two processes for the collisions of ground state C(³P) and N(⁴S) atoms (A<fnref n="2" />Π → X<fnref n="2" />Σ⁺ and b⁴Π → a⁴Σ⁺), as well as, for the first time, we investigated the radiative association of ground state C(³P) and excited N(²D) atoms (B<fnref n="2" />Σ⁺ → X<fnref n="2" />Σ⁺). The cross-sections for the three processes have been calculated using semi-classical, quantum mechanical, and Breit-Wigner theories. The rate constants, derived from the combination of semi-classical and Breit-Wigner results, have been fitted to the Kooij formula to be used in astrochemical modeling. The rate constant for the B<fnref n="2" />Σ⁺ → X<fnref n="2" />Σ⁺ process dominates in the investigated temperature range (10–10 000 K), when its own asymptotic energy is used as a reference level. Moreover, the B<fnref n="2" />Σ⁺ → X<fnref n="2" />Σ⁺ process presents the most pronounced isotope effect on the rate constant. We suggest considering these newly investigated radiative association processes for the formation of CN and C¹⁵N in the interstellar medium.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0192682

## I. INTRODUCTION

The CN (cyano) radical plays a crucial role in the synthesis of nitrogen-bearing molecules in the Interstellar Medium (ISM), and understanding its formation is pivotal in exploring the initial phases of nitrogen-based chemistry.<fnref n="1" /> A viable method to form CN radicals in the ISM is via radiative association (RA) reactions—processes where two species collide and form a molecule by releasing some of the excess energy by emitting a photon. It is a very unlikely process under the typical conditions of Earth, but it can be relevant in astrophysical contexts where the other competitive processes (stabilization through collisions with a third body) cannot occur.<fnref n="2" /> Radiative association reactions can be studied via laboratory experiments using selected-ion flow tube (SIFT) or ion-trapping techniques, such as the ion cyclotron resonance (ICR).<fnref n="3" />,⁴ However, owing to several limitations of the experimental techniques, the direct measurement of radiative association rate constants has been limited to a few small

ionic species.⁵⁻⁸ Thus, several theoretical methods have been developed, often based on perturbation theory (PT), semi-classical (SC), and classical methods, to obtain the estimates of the rate constants of such processes.<fnref n="2" />,⁹⁻¹¹ These methods and a Sturmian approach¹² have been fruitfully used for diatomic systems by many research groups.<fnref n="2" />,¹³⁻²⁴

A number of theoretical studies of the formation of CN through the radiative association of ground-state carbon C(³P) and nitrogen N(⁴S) atoms have been carried out.²⁵⁻²⁸ They considered all the combinations of the X<fnref n="2" />Σ⁺ and A<fnref n="2" />Π molecular electronic states (i.e., A<fnref n="2" />Π → X<fnref n="2" />Σ⁺, A<fnref n="2" />Π → A<fnref n="2" />Π, X<fnref n="2" />Σ⁺ → A<fnref n="2" />Π, and X<fnref n="2" />Σ⁺ → X<fnref n="2" />Σ⁺) and found the A<fnref n="2" />Π → X<fnref n="2" />Σ⁺ process to be the dominant one. More recently, eight new radiative association processes for the collision of ground and excited carbon and nitrogen atoms have been studied.²⁹ None of the processes involving collisions of excited C(¹D) and N(²D) atoms is found to be relevant at low temperatures (10–100 K). Instead, the b⁴Π → a⁴Σ⁺ process is found to con-

<footnote n="1">¹ Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France</footnote>

<footnote n="2">² Dipartimento di Chimica, Biologia e Biotecnologie, Universita degli Studi di Perugia, Perugia 06123, Italy</footnote>

<footnote n="3">³ Department of Chemistry and Molecular Biology, University of Gothenburg, Box 462, 40530 Göteborg, Sweden</footnote>

<footnote>ᵃ Authors to whom correspondence should be addressed: lisa.giani@univ-grenoble-alpes.fr; martina.simsova@proton.me; and nyman@chem.gu.se</footnote>
tribute significantly to the total rate constant of the reaction at low temperatures, together with the already well-known A²Π → X²Σ⁺ process.

Atomic nitrogen can be produced from N₂ in various processes involving energetic particles and photons, including extreme ultraviolet (EUV) photodissociation, electron impact-induced dissociation, dissociative recombination of N₂⁺, and galactic cosmic ray absorption.³⁰⁻³² These processes can produce atomic nitrogen in its ground state ⁴S as well as in its first electronically excited state ²D. Many studies have demonstrated the importance of N(²D) atoms in the formation of prebiotic nitrogen-containing molecules such as nitriles and imines in Titan's atmosphere.³³⁻⁴⁰ However, the metastable N(²D) has also been observed in several interstellar environments and comets.⁴¹⁻⁴⁶ Given the relatively long radiative lifetime of the ²D state (6.1 × 10⁴ and 1.4 × 10⁵ s for the ²D₃/₂ and ²D₅/₂ fine structure states, respectively⁴⁷), a possible contribution of N(²D) to the chemistry of denser regions of the ISM, such as protoplanetary disks or cometary comae, cannot be excluded.

Radiative association reactions can exhibit a significant isotope effect, leading to a difference of more than one order of magnitude in the rate constant.⁴⁸ C¹⁴N and C¹⁵N have been detected in several star-forming regions,⁴⁹⁻⁵⁴ the places where Sun-like stars and planets are formed, as well as in a comet (see Fig. 1).⁵⁵ Nevertheless, the effect of the isotope substitution of the nitrogen atom on the rate of formation of CN through radiative association reactions has never been explored. In astrochemistry, radiative association reactions are important in the formation of simple diatomic molecules, which, in turn, serve as building blocks for more complex molecules. Any difference in the rate constant of formation of C¹⁴N and C¹⁵N could potentially influence the fractionation of other nitrogen-containing species. Therefore, investigating the isotope effect on the formation rates of CN can provide insights into the observed fractionation of CN parent molecules and help us better understand the mechanisms responsible for nitrogen fractionation.

<figure><img src="image_13.png" /><figcaption>FIG. 1. The C¹⁴N/C¹⁵N ratio observed in several astronomical sources. The gray lines show the isotope ratios in the solar system and the local ISM.</figcaption></figure>

In this work, we calculated the cross-sections and rate constants of the formation of CN and its isotopologue C¹⁵N by radiative association of ground-state carbon and nitrogen atoms,

$$C(^{3}\text{P}_{\text{g}}) + \text{N}(^{4}\text{S}_{\text{u}}) \rightarrow \text{CN}(\text{A}^{2}\Pi) \rightarrow \text{CN}(\text{X}^{2}\Sigma^{+}) + \text{h}\nu, \quad (1)$$

$$C(^{3}\text{P}_{\text{g}}) + \text{N}(^{4}\text{S}_{\text{u}}) \rightarrow \text{CN}(\text{b}^{4}\Pi) \rightarrow \text{CN}(\text{a}^{4}\Sigma^{+}) + \text{h}\nu. \quad (2)$$

Moreover, we investigated the formation of CN and C¹⁵N through a radiative association process for the collision of ground-state carbon and excited nitrogen atoms,

$$C(^{3}\text{P}_{\text{g}}) + \text{N}(^{2}\text{D}_{\text{u}}) \rightarrow \text{CN}(\text{B}^{2}\Sigma^{+}) \rightarrow \text{CN}(\text{X}^{2}\Sigma^{+}) + \text{h}\nu. \quad (3)$$

Section II contains a brief review of the radiative association (RA) theory used in the calculations. Section III describes the potential energy curves (PECs) and transition dipole-moment functions used for the calculations. Cross-sections and rate constants are presented in Sec. IV and discussed in Sec. V.

## II. THEORETICAL METHODS

The thermal rate constant for a collisional process, such as radiative association, is given by

$$k(T) = \left(\frac{8}{\pi\mu}\right)^{\frac{1}{2}} \left(\frac{1}{k_{\text{B}}T}\right)^{\frac{3}{2}} \int_{0}^{\infty} E\sigma(E)e^{-E/k_{\text{B}}T} dE, \quad (4)$$

where E is the initial relative translational kinetic energy, μ is the reduced mass, and σ(E) is the cross-section. The radiative association cross-section can be determined using a semi-classical (SC) or quantum-mechanical (QM) approach (see Secs. II A and II B). A detailed derivation of the two methods can be found in the studies by Nyman et al.,² Babb and Kirby,⁵⁶ and Zygelman and Dalgarno,⁹,¹⁰ while only the most important formulas are reported in this paper.

### A. Quantum-mechanical approach

The quantum mechanical (QM) method is based on the perturbation theory (PT). The rate for spontaneous emission (Einstein A coefficient for spontaneous emission) from an initial continuum state ΛE to all final bound states Λ'ν'J' is

$$A_{\Lambda, \Lambda'}(E, J) = \frac{64}{3} \frac{\pi^4}{(4\pi\epsilon_0)h} \sum_{\nu, J'} \left(\frac{\nu_{E\Lambda'\nu'J'}}{c}\right)^3 \frac{S_{\Lambda, J \rightarrow \Lambda' J'}}{2J' + 1} \\ \times |\langle\Psi_{JE}^A(R)|D(R)|\Psi_{J'\nu'}^{\Lambda'}(R)\rangle|^2, \quad (5)$$

where ν_{EΛ'ν'J'} is the frequency of the emitted photon, c is the speed of light, S_{Λ, J → Λ'J'} are the Hönl–London factors of the corresponding transition. |⟨Ψ_{JE}^A(R)|D(R)|Ψ_{J'ν'}^{Λ'}(R)⟩|^2 is a term coming from the transition dipole moment (TDM) matrix between initial and final states, in which Ψ_{JE}^A(R) and Ψ_{J'ν'}^{Λ'}(R) are the normalized continuum-state and bound-state wave functions, respectively.
The spontaneous emission rate $A_{\Lambda,\Lambda'}(E, J)$ can be turned into a cross-section, giving

$$ \sigma_{\Lambda \to \Lambda'}(E) = \frac{h^3}{8\pi\mu E} p \sum_J (2J + 1) A_{\Lambda, \Lambda'}(E, J). \quad (6) $$

Here, $p$ is the statistical weight factor that accounts for the probability of approach in a specific electronic state, which is given by

$$ p = \frac{(2S + 1)(2 - \delta_{0,\Lambda})}{(2L_{\Lambda} + 1)(2S_{\Lambda} + 1)(2L_{B} + 1)(2S_{B} + 1)}, \quad (7) $$

where $S$ is the total electronic spin quantum number of the diatom, $\Lambda$ is the quantum number for the projection of the electronic angular momentum of the diatom on the internuclear axis, $L_\Lambda$ and $S_\Lambda$ are the electronic orbital and spin angular momentum quantum numbers, respectively, for atom A, and $L_B$ and $S_B$ are the corresponding quantities for atom B.

## B. Semi-classical approach

The semi-classical (SC) method is based on the optical potential method. The rate of transition from an electronic state characterized by $\Lambda$ to another electronic state characterized by $\Lambda'$ can be expressed as

$$ A_{\Lambda, \Lambda'}(R) = \frac{64}{3} \frac{\pi^4}{(4\pi\epsilon_0)h} \left(\frac{2 - \delta_{0,\Lambda+\Lambda'}}{2 - \delta_{0,\Lambda}}\right) \frac{|D(R)|^2 v_{\Lambda \to \Lambda'}(R)^3}{c^3}, \quad (8) $$

where $D(R)$ is the transition dipole-moment function at the transition frequency $v(R)$, which is defined as

$$ v_{\Lambda \to \Lambda'}(R) = \frac{V_{\Lambda}(R) - V_{\Lambda'}(R)}{h}. \quad (9) $$

When different dissociation limits are considered, the energy difference between the asymptotes has to be taken into account in the photon energy. The cross-section is calculated by integrating the transition rate $A_{\Lambda,\Lambda'}(R)$ over all the classical trajectories, giving

$$ \sigma_{\Lambda \to \Lambda'}(E) = 4\pi \sqrt{\frac{\mu}{2E}} p \int_0^\infty b db \int_{R_c}^\infty \frac{A_{\Lambda, \Lambda'}(R)}{\sqrt{1 - \frac{V_{\Lambda}(R)}{E} - \frac{b^2}{R^2}}} dR, \quad (10) $$

where $b$ is the impact parameter and $R_c$ is the classical outer turning point.

## C. Resonances

The presence of quasi-bound states inside the centrifugal barrier and the tunneling through that barrier cause shape resonances to be present in the cross-section. This indirect contribution is accounted for by the QM method, but the presence of narrow resonances makes the integral in Eq. (4) hard to evaluate numerically. Moreover, the QM method tends to overestimate the heights of long-lived resonances for which the tunneling widths are smaller or comparable to the radiative widths.²⁴,⁵⁷ We have solved this by using Breit-Wigner (BW) theory, derived for narrow resonances,⁵⁸,⁵⁹ to evaluate the resonance contribution, and then add it to the semi-classical cross-section. The BW theory for narrow resonances can

also be used in combination with the quantum mechanical perturbation theory (QMPT) method.¹⁸ Another solution is to use the Sturmian approach, see Refs. 12 and 24 for details.

A state can decay either by tunneling back through the barrier or by radiating (emitting a photon). Both processes have a corresponding finite lifetime $\tau$ and an energy width $\Gamma$ due to the uncertainty principle. The inverse of the lifetime gives the energy width, and the total width of a resonance is given by the sum of the radiative and tunneling widths,

$$ \Gamma_{\nu J \Lambda \to \Lambda'}^{\text{total}} = \Gamma_{\nu J \Lambda \to \Lambda'}^{\text{rad}} + \Gamma_{\nu J \Lambda}^{\text{tun}} = \hbar \left( \frac{1}{\tau_{\nu J \Lambda \to \Lambda'}} + \frac{1}{\tau_{\nu J \Lambda}^{\text{tun}}} \right). \quad (11) $$

Then, the contribution to the cross-section due to a single resonance is given by

$$ \sigma_{\Lambda \to \Lambda'}^{\text{res}}(E, J) = \frac{\pi \hbar^2}{2\mu E} p (2J + 1) \frac{\Gamma_{\nu J \Lambda \to \Lambda'}^{\text{rad}} \Gamma_{\nu J \Lambda}^{\text{tun}}}{(E - E_{\nu J \Lambda})^2 + \frac{(\Gamma_{\nu J \Lambda \to \Lambda'}^{\text{rad}} + \Gamma_{\nu J \Lambda}^{\text{tun}})^2}{4}}, \quad (12) $$

where $E_{\nu J \Lambda}$ is the energy of the quasi-bound level and $\Gamma^{\text{tun}}$ and $\Gamma^{\text{rad}}$ are the tunneling and radiative widths, respectively, of the corresponding resonance. The radiative width $\Gamma^{\text{rad}}$ can be obtained as the sum of the widths due to radiative decay from the initial state to all final bound states,

$$ \Gamma_{\nu J \Lambda \to \Lambda'}^{\text{rad}} = \hbar \sum_{v'v'} A_{\nu J \Lambda \to v'v' \Lambda'}. \quad (13) $$

The total resonance contribution to the rate constant is given by

$$ k_{\Lambda \to \Lambda'}^{\text{res}}(T) = \hbar^2 p \left(\frac{2\pi}{\mu k_B T}\right)^{3/2} \sum_{\nu J} \frac{2J + 1}{1/\Gamma_{\nu J \Lambda \to \Lambda'}^{\text{rad}} + 1/\Gamma_{\nu J \Lambda}^{\text{tun}}} e^{-E_{\nu J \Lambda}/k_B T}. \quad (14) $$

## III. MOLECULAR DATA AND COMPUTATIONAL DETAILS

The potential energy curves (PECs) for the $X^2\Sigma^+$, $A^2\Pi$, $a^4\Sigma^+$, $b^4\Pi$, and $B^2\Sigma^+$, as well as the transition dipole moments (TDMs) for the $A^2\Pi \to X^2\Sigma^+$ and $b^4\Pi \to a^4\Sigma^+$ transitions, are taken from Zhang et al.;²⁹ The TDM for the $B^2\Sigma^+ \to X^2\Sigma^+$ is taken from Brooke et al.⁶⁰ The PECs and TDMs are shown in Fig. 2. Note that while the ab initio data of both PECs and TDMs for the $A^2\Pi \to X^2\Sigma^+$ and $B^2\Sigma^+ \to X^2\Sigma^+$ processes start at 0.8 Å, the data for the $b^4\Pi \to a^4\Sigma^+$ transition start at 0.95 Å.

For internuclear distances $R$ smaller than 4.6, 4.8, 4.2, 3.8, and 4.5 Å for the $X^2\Sigma^+$, $A^2\Pi$, $a^4\Sigma^+$, $b^4\Pi$, and $B^2\Sigma^+$ potentials, respectively, the ab initio points were interpolated using the natural cubic spline technique. The long-range asymptotic behavior of the potentials has been obtained by extrapolation using an analytical function that accounts for the dispersion interactions,

$$ V(R) = -\frac{C_6}{R^6} + V(R \to \infty), \quad (15) $$

where the $C_6$ coefficient was determined for each potential by fitting the last ab initio points to Eq. (15) and $V(R \to \infty)$ is the dissociation limit. The obtained $C_6$ values are 47.56, 50.03, 41.17, 52.52,
<figure><img src="image_14.png" /><figcaption>**FIG. 2.** Top: Potential energy curves of CN electronic states corresponding to the C(³P<sub>g</sub>) + N(⁴S<sub>u</sub>) and C(³P<sub>g</sub>) + N(²D<sub>u</sub>) dissociation limits. The difference between the dissociation limits (2.38 eV) is retrieved from the NIST database.<sup>47</sup> Bottom: Transition dipole moments for the A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup>, b<sup>4</sup>Π → a<sup>4</sup>Σ<sup>+</sup>, and B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup> processes of the CN radical. The ab initio data are taken from Zhang *et al*.<sup>29</sup> and Brooke *et al*.<sup>50</sup></figcaption></figure>

and 59.86 $E_h \cdot a_0^6$ for the X<sup>2</sup>Σ<sup>+</sup>, A<sup>2</sup>Π, a<sup>4</sup>Σ<sup>+</sup>, b<sup>4</sup>Π, and B<sup>2</sup>Σ<sup>+</sup> potentials, respectively. The four potentials X<sup>2</sup>Σ<sup>+</sup>, A<sup>2</sup>Π, a<sup>4</sup>Σ<sup>+</sup>, and b<sup>4</sup>Π correlate with the ground states of C and N atoms, while the B<sup>2</sup>Σ<sup>+</sup> potential correlates with the excited N(²D<sub>u</sub>), which is 2.38 eV higher than the ground state N(⁴S<sub>u</sub>).<sup>47</sup> Then, if V(R → ∞) = 0 for all states except for B<sup>2</sup>Σ<sup>+</sup>, then V(R → ∞) = 2.38 eV for this state.

The transition dipole-moment functions were assumed to have the asymptotic form,

$$D(R) = Ae^{-bR} + C, \quad (16)$$

where both A and b were calculated by fitting the last ab initio points of the TDMs.

Due to some difficulties with our SC calculation for the B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup> transition (the method is very sensitive to kinks in the potentials), we used a particular treatment of the short-range behavior of the X<sup>2</sup>Σ<sup>+</sup> and B<sup>2</sup>Σ<sup>+</sup> potentials: we first inverted the first two ab initio points of the two curves in order to remove a non-physical crossing of the curves, and then we extrapolated the potentials at R < 1.1 Å using Eq. (16).

The bound-state wave functions are found using a discrete variable representation (DVR) method,<sup>61</sup> while the scattering-state wave functions are obtained using the Numerov method.<sup>62</sup> The resonances are computed using the LEVEL 8.0 code,<sup>63</sup> which returns the tunneling widths Γ<sup>tun</sup>, the transition rates A<sub>vJΛ→v'J'A'</sub> [used to compute the radiative widths Γ<sup>rad</sup> through Eq. (13)], the energy of the resonances E<sub>vJΛ</sub>, and bound-state levels E<sub>v'JΛ'</sub>. All the parameters of the final rovibrational bound states that have been used

<table><caption>**TABLE I.** Process characteristics for radiative association of C(³P<sub>g</sub>) with N(⁴S<sub>u</sub>) and N(²D<sub>u</sub>). n<sub>tot</sub> is the total number of resonances supported by the initial electronic states, D<sub>0</sub> is the dissociation energy of the final electronic state from the ground state (v' = 0, J' = 0), n<sub>v</sub> is the number of vibrational (J' = 0) states of the final electronic state, J<sub>max</sub> is the maximum value of the rotational quantum number of the final electronic state, and n<sub>rv</sub> is the total number of bound rovibrational states of the final electronic state.</caption><thead><tr><th>System</th><th>Process</th><th>n<sub>tot</sub></th><th>D<sub>0</sub> (cm<sup>-1</sup>)</th><th>n<sub>v</sub></th><th>J<sub>max</sub></th><th>n<sub>rv</sub></th></tr></thead><tbody><tr><td rowspan="3">C<sup>14</sup>N</td><td>A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup></td><td>1531</td><td>-61 302.9</td><td>51</td><td>263</td><td>5606</td></tr><tr><td>b<sup>4</sup>Π → a<sup>4</sup>Σ<sup>+</sup></td><td>645</td><td>-22 538.1</td><td>32</td><td>189</td><td>2267</td></tr><tr><td>B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup></td><td>1589</td><td>-61 302.9</td><td>51</td><td>263</td><td>5606</td></tr><tr><td rowspan="3">C<sup>15</sup>N</td><td>A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup></td><td>1569</td><td>-61 318.7</td><td>51</td><td>267</td><td>5708</td></tr><tr><td>b<sup>4</sup>Π → a<sup>4</sup>Σ<sup>+</sup></td><td>666</td><td>-22 548.0</td><td>33</td><td>192</td><td>2336</td></tr><tr><td>B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup></td><td>1641</td><td>-61 318.7</td><td>51</td><td>267</td><td>5708</td></tr></tbody></table>

in the QM calculations are reported In Table I. The LEVEL 8.0 code uses a different function to extrapolate potentials at their long range. It considers a sum of C<sub>6</sub>/R<sup>6</sup>, C<sub>7</sub>/R<sup>7</sup>, C<sub>8</sub>/R<sup>8</sup>, and C<sub>9</sub>/R<sup>9</sup> terms. Therefore, to obtain an accurate description of the potentials, the long-range behavior of potentials has been extrapolated starting at larger internuclear distances R, specifically at 6.5, 6.5, 5.0, 5.5, and 5.5 Å for the X<sup>2</sup>Σ<sup>+</sup>, A<sup>2</sup>Π, a<sup>4</sup>Σ<sup>+</sup>, b<sup>4</sup>Π, and B<sup>2</sup>Σ<sup>+</sup> potentials, respectively. The statistical weight factors are p = 4/36, p = 8/36, and p = 2/90 for reaction A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup>, b<sup>4</sup>Π → a<sup>4</sup>Σ<sup>+</sup>, and B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup>, respectively. The Hönl–London factors for doublets and quartets were approximated by using Hönl–London factors valid for the corresponding singlet transitions. The validity of this approximation was shown by Antipov *et al.*,<sup>26</sup> Gustafsson *et al.*,<sup>64</sup> and Golubev *et al.*<sup>65</sup>

For the B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup> process, some modifications to the SC and PT codes have been done to account for the different dissociation limits of the potentials involved. The spin-orbit and non-adiabatic couplings are neglected.

## IV. RESULTS

In this section, we provide cross-sections and thermal rate constants for the three transitions that we have studied, namely, A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup>, b<sup>4</sup>Π → a<sup>4</sup>Σ<sup>+</sup>, and B<sup>2</sup>Σ<sup>+</sup> → X<sup>2</sup>Σ<sup>+</sup>.

### A. The A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup> transition

The radiative association cross-section for the A<sup>2</sup>Π → X<sup>2</sup>Σ<sup>+</sup> process for the formation of C<sup>14</sup>N and C<sup>15</sup>N has been calculated using the SC, PT, and SC + BW methods.

The SC and PT results, shown in Fig. 3, are in agreement in the whole energy range shown here—0.002–4 eV for both isotopologues. The SC cross-section takes into account only the direct contribution to the cross-section and, thus, is the baseline of the PT cross-section, while the PT cross-section presents a resonance structure that is due to the tunneling through the centrifugal barrier in the effective potential.

The PT method is used to test whether the LEVEL code provides a sufficient amount of resonances. The BW theory reproduces the number of resonances well, as well as their positions obtained with the PT method; however, some discrepancies are also observed
<figure><img src="image_15.png" /><figcaption>FIG. 3. Calculated SC (green), PT (magenta), and SC + BW (blue) cross-sections of the A$^2\Pi \rightarrow X^2\Sigma^+$, b$^4\Pi \rightarrow a^4\Sigma^+$, and B$^2\Sigma^+ \rightarrow X^2\Sigma^+$ processes for the formation of C$^{14}$N (top) and C$^{15}$N (bottom). For B$^2\Sigma^+ \rightarrow X^2\Sigma^+$ process, the collision energy is referenced to the C($^3P_g$) + N($^2D_u$) dissociation limit.</figcaption></figure>

(see Fig. 3). At energies below 0.001 eV, the PT resonances are shifted to slightly lower energies with respect to the BW ones. This is because of a different function used by the LEVEL code to extrapolate the long-range behavior of the potentials with respect to the PT code.

The results obtained for C$^{15}$N do not differ significantly from the results obtained for C$^{14}$N. The small difference in the SC cross-section at low temperatures (the C$^{15}$N cross-section is 1.5% larger than that of C$^{14}$N at 10 K) is due to the different reduced masses of the systems. The resonance structures also look similar since the number of bound states and resonances is not significantly different between the two isotopologues (see Table I).

The computed rate constants are shown in Fig. 4. The total rate constants are dominated by the contribution of the direct processes. The SC rate constant does not depend on the reduced mass of the system. The small difference between the SC rate constants of the formation of C$^{14}$N and C$^{15}$N is due to the small numerical uncertainties of the calculations. The SC rate constant is lower than the SC + BW rate constant since it does not consider the resonance contribution. The resonance contribution to the rate constant is larger at low temperatures, where quantum effects become more important.

We used the SC + BW combination of results to derive the rate constant of the process, and we fitted the results to the following Arrhenius-Kooij formula:

$$k(T) = \alpha \left(\frac{T}{300K}\right)^{\beta} \exp\left(-\frac{\gamma}{T}\right). \quad (17)$$

The fitting parameters are reported in Table II. At 10 K, the SC + BW rate constant for C$^{14}$N is 5% larger than the one for C$^{15}$N, while at larger temperatures (>100 K), they become almost identical.

## B. The b$^4\Pi \rightarrow a^4\Sigma^+$ transition

For the b$^4\Pi \rightarrow a^4\Sigma^+$ process, the SC, PT, and SC + BW cross-sections of both C$^{14}$N and C$^{15}$N are shown in Fig. 3. The SC and PT results are in good agreement in the whole energy range. The same features observed in the cross-section of the A$^2\Pi \rightarrow X^2\Sigma^+$ process are also observed for the b$^4\Pi \rightarrow a^4\Sigma^+$ process. The numbers of resonances and bound states are lesser than those for the A$^2\Pi \rightarrow X^2\Sigma^+$ process due to a smaller well of a$^4\Sigma^+$ than X$^2\Sigma^+$ (see Table I). Furthermore, for these systems, we used the PT method to validate the BW results, and a combination of SC and BW cross-sections is used to evaluate the rate constant of the process. The positions of the resonances obtained with the BW and QM methods are in good agreement, even if some of the discrepancies observed for the A$^2\Pi \rightarrow X^2\Sigma^+$ process as well can be observed for the b$^4\Pi \rightarrow a^4\Sigma^+$ process. The rate constants' results (SC and SC + BW) are presented in Fig. 4. As for the A$^2\Pi \rightarrow X^2\Sigma^+$ process, the rate constants of the formation of C$^{14}$N and C$^{15}$N are very similar since the major contribution to the rate constant is due to direct RA. The parameters obtained from the fit of the rate constants through Eq. (17) are reported in Table II.
<figure><img src="image_16.png" /><figcaption>FIG. 4. Calculated SC (dashed lines) and SC + BW (solid lines) rate constants of the A$^2\Pi \rightarrow X^2\Sigma^+$, b$^4\Pi \rightarrow a^4\Sigma^+$ and B$^2\Sigma^+ \rightarrow X^2\Sigma^+$ processes for C$^{14}$N (blue) and C$^{15}$N (red).</figcaption></figure>

### C. The B$^2\Sigma^+ \rightarrow X^2\Sigma^+$ transition

The SC, PT, and SC + BW cross-sections for the B$^2\Sigma^+ \rightarrow X^2\Sigma^+$ process of both C$^{14}$N and C$^{15}$N are shown in Fig. 3. The SC cross-section agrees well with the PT results in the whole energy range, except for energies larger than 0.5 eV, for which the QM cross-section is lower than the SC one. The same behavior has been previously observed by Svensson et al.,⁶⁶ Kathir et al.,⁶⁷ and Jones et al.⁶⁸ The SC cross-section falls to zero at a collision energy of 0.98 eV, which relates to the energy difference between the two electronic states. For larger collision energies, the radiative association conditions are no longer satisfied [E + ΔE < $h\nu(R$), where ΔE = 2.38 eV is the asymptotic energy difference between the B and X states] and the cross-section becomes zero. The BW method reproduces the resonances found by the PT theory well. The calculated SC and SC + BW rate constants for C$^{14}$N and C$^{15}$N are presented in Fig. 4. There is almost no difference between the SC rate constants of C$^{14}$N and C$^{15}$N. The SC + BW rate constants are slightly different, due to a larger resonance contribution in the case of C$^{14}$N. The fitting parameters of the rate constants are reported in Table II.

### V. DISCUSSION

The rate constants for the formation of C$^{14}$N and C$^{15}$N via the collision of ground-state C($^3$P) with N($^4$S), and via ground-state

<table><caption>TABLE II. Fitting parameters for the formation of C$^{14}$N and C$^{15}$N following the collision of C($^3$P) atoms with N($^4$S) (sum of A$^2\Pi \rightarrow X^2\Sigma^+$ and b$^4\Pi \rightarrow a^4\Sigma^+$ processes) or N($^2$D) atoms (B$^2\Sigma^+ \rightarrow X^2\Sigma^+$ process). Note that the y parameter is just a fitting parameter that makes the fit better and it should not be interpreted as an activation barrier.</caption><thead><tr><th>T (K)</th><th>α (cm<sup>3</sup> s<sup>-1</sup>)</th><th>β</th><th>γ (K)</th></tr></thead><tbody><tr><td colspan="4">C(<sup>3</sup>P) + <sup>14</sup>N(<sup>4</sup>S)</td></tr><tr><td>10-100</td><td>9.3926 × 10<sup>-19</sup></td><td>8.5576(-2)</td><td>-0.9218</td></tr><tr><td>100-500</td><td>9.0307 × 10<sup>-19</sup></td><td>8.0355(-2)</td><td>-4.5858</td></tr><tr><td>500-2500</td><td>10.3405 × 10<sup>-19</sup></td><td>1.3734(-2)</td><td>46.649</td></tr><tr><td>2500-5000</td><td>22.8939 × 10<sup>-19</sup></td><td>-0.2422</td><td>683.25</td></tr><tr><td>5000-10 000</td><td>148.2720 × 10<sup>-19</sup></td><td>-0.7289</td><td>3201.53</td></tr><tr><td colspan="4">C(<sup>3</sup>P) + <sup>14</sup>N(<sup>2</sup>D)</td></tr><tr><td>10-30</td><td>1.9199 × 10<sup>-18</sup></td><td>-9.5850(-2)</td><td>2.1565</td></tr><tr><td>30-100</td><td>2.7582 × 10<sup>-18</sup></td><td>0.1664</td><td>-5.0608</td></tr><tr><td>100-500</td><td>2.5809 × 10<sup>-18</sup></td><td>0.1318</td><td>-8.1857</td></tr><tr><td>500-2000</td><td>2.8205 × 10<sup>-18</sup></td><td>0.1047</td><td>30.2161</td></tr><tr><td>2000-10 000</td><td>52.5919 × 10<sup>-18</sup></td><td>-0.8531</td><td>2311.77</td></tr><tr><td colspan="4">C(<sup>3</sup>P) + <sup>15</sup>N(<sup>4</sup>S)</td></tr><tr><td>10-100</td><td>9.6325 × 10<sup>-19</sup></td><td>9.9260(-2)</td><td>-0.9312</td></tr><tr><td>100-500</td><td>9.0074 × 10<sup>-19</sup></td><td>8.6169(-2)</td><td>-6.2155</td></tr><tr><td>500-2500</td><td>10.3651 × 10<sup>-19</sup></td><td>1.2507(-2)</td><td>46.3639</td></tr><tr><td>2500-5000</td><td>23.3453 × 10<sup>-19</sup></td><td>-0.2247</td><td>696.533</td></tr><tr><td>5000-10 000</td><td>149.5710 × 10<sup>-19</sup></td><td>-0.7332</td><td>3197.23</td></tr><tr><td colspan="4">C(<sup>3</sup>P) + <sup>15</sup>N(<sup>2</sup>D)</td></tr><tr><td>10-40</td><td>1.6115 × 10<sup>-18</sup></td><td>6.0559(-2)</td><td>1.7528</td></tr><tr><td>40-200</td><td>2.0037 × 10<sup>-18</sup></td><td>9.4364(-2)</td><td>-2.0673</td></tr><tr><td>200-1000</td><td>1.8752 × 10<sup>-18</sup></td><td>0.2181</td><td>-25.7237</td></tr><tr><td>1000-2500</td><td>3.7976 × 10<sup>-18</sup></td><td>-7.8989(-2)</td><td>325.862</td></tr><tr><td>2500-10 000</td><td>76.5674 × 10<sup>-18</sup></td><td>-1.0176</td><td>2905.41</td></tr></tbody></table>

C($^3$P) colliding with excited N($^2$D), are shown in Fig. 5. We approximate the total rate constant for the formation of CN via collisions of the atoms in their ground electronic states by summing over A → X and b → a processes. Since N($^2$D) atoms can be produced directly from N$_2$ through several processes stated in Sec. I, we report the thermal rate constant for C($^3$P) + N($^2$D) → CN + $h\nu$ relative to the corresponding asymptotic energy level. If a thermal population is assumed for the population of the atomic electronic levels, the B → X process would be irrelevant at low temperatures with respect to processes (1) and (2). The B → X thermal rate constant would become comparable to those of processes (1) and (2) starting from ~6000 K, i.e., when the thermal population of the upper electronic asymptote becomes significant. Calculated relative to its own asymptote, the B → X cross-sections dominate over the sum of A → X and b → a cross-sections at all collision energies up to at least 1 eV.

A comparison between our rate constant results and the rate constants calculated by Zhang et al.²⁹ and Gustafsson and Forrey²⁸
<figure><img src="image_17.png" /><figcaption>FIG. 5. Calculated rate constant for collisions of ground-state C and N (A$^2\Pi \rightarrow X^2\Sigma^+$ + b$^4\Pi \rightarrow a^4\Sigma^+$) in magenta and collisions of ground-state C and excited N (B$^2\Sigma^+ \rightarrow X^2\Sigma^+$) in green, with $^{14}$N (solid) and $^{15}$N (dashed) using BW + SC approaches.</figcaption></figure>

is shown in Fig. 6. While our results for the A$^2\Pi \rightarrow X^2\Sigma^+$ are in good agreement with the results reported by Gustafsson and Forrey,²⁸ the agreement is not found with Zhang *et al.*²⁹ The difference at low energies is due to a different function used for the long-range extrapolation of the potentials ($-C_6/R^6$ in our case and $-C_5/R^5 - C_6/R^6$ in the case of Zhang *et al.*²⁹). For collisions between C($^3$P) and N($^4$S) or N($^2$D) atoms, the leading term in the description of the interaction is the $-C_6/R^6$ term (dispersion interaction). Since the C$_5$ term (quadrupole-quadrupole interaction) does not have a physical meaning for the system of interest, we decided not to use it for the description of the interaction. The differences in the rate constants are also due to different evaluations of the resonance contributions: we used a combination of SC and BW cross-sections to obtain the rate constants, while in Zhang *et al.*,²⁹ QM cross-sections were used to obtain the rate constants.

<figure><img src="image_18.png" /><figcaption>FIG. 6. Comparison of our calculated BW + SC rate constants (solid lines) and rate constants reported in Zhang *et al.*²⁹ (dashed lines) for the A$^2\Pi \rightarrow X^2\Sigma^+$ (orange) and b$^4\Pi \rightarrow a^4\Sigma^+$ (blue) processes of C$^{14}$N. The rate constant for the A$^2\Pi \rightarrow X^2\Sigma^+$ process reported by Gustafsson and Forrey²⁸ is also shown (green dotted line).</figcaption></figure>

In the case of the b$^4\Pi \rightarrow a^4\Sigma^+$ process, our cross-section differs from previous results in the low energy range ($E < 0.01$ eV). In this low-energy region, the cross-section depends on the long-range part of the potential and the cross-section slope should reflect the coefficient used to model the potential. However, while the slope of our cross-section perfectly reflects the use of a $-C_6/R^6$ long-range potential, we could not reproduce the slope of the cross-section reported by Zhang *et al.*²⁹ even if the $-C_5$ coefficient was used. While in our case, the slope in the low energy range is -1/3, corresponding to the use of $-C_6/R^6$, in the case of Zhang *et al.*,²⁹ the slope does not seem to correspond to their reported use of $-C_5/R^5 - C_6/R^6$ long-range potential.

While, for the A → X and B → X processes, the SC + BW rate constants of formation of the lighter isotope are larger than the rate constant of formation of the heavier isotope (broader resonances for the lighter isotope resulting in larger resonance contribution to the rate constant), for the b → a process, the trend is opposite, which results from the details of the resonances in the two cases. The A → X and b → a processes do not show a significant isotope effect (≤5%) since the major contribution to the rate constant is due to direct processes.²⁶ For the B → X process, the isotope effect is more evident, with a rate of formation of C$^{14}$N, which is 29% larger than the rate of formation of C$^{15}$N at 10 K, due to a larger contribution from quantum resonances. The rate constant for the B → X process, calculated using its own asymptotic energy as a reference level, dominates over the sum of A → X and b → a processes. Even though the electronically excited state of atomic nitrogen N($^2$D) is expected to be much less abundant than the ground-state N($^4$S) in most environments, the larger rate constant found for C($^3$P) + N($^2$D) collisions can make this process competitive in regions where N($^2$D) is particularly abundant. The inclusion of these results in astrochemical models will allow us to quantify the contribution of excited N($^2$D) atoms in producing CN in specific regions of the ISM.

## VI. CONCLUSIONS

In this work, we studied the formation of CN and its isotopologue C$^{15}$N by radiative association, following the collisions of ground-state C($^3$P) and N($^4$S) atoms and ground-state C($^3$P) and excited N($^2$D) atoms. We calculated the cross-sections for processes (1)–(3) using semi-classical, quantum-mechanical, and a combination of semi-classical with the Breit–Wigner theory approaches, respectively. The rate constants from 10 to 10 000 K have been obtained from the last method. The evaluation of process (1) shows good agreement between our results and the rate constants reported by Gustafsson and Forrey.²⁸ We do not obtain good agreement with the rate constants reported by Zhang *et al.*²⁹ for processes (1) and (2), which may be due to the different long-range potentials and methods used. The newly studied process (3) yields the largest cross-sections over most of the energy range studied for the formation of CN by radiative association. The isotopic effect is more pronounced for process (3) than for processes (1) and (2), with a rate constant of formation of C$^{14}$N that is 1.5 times larger than the rate constant of formation of C$^{15}$N. The radiative association processes studied here are possible routes for the formation of CN in dense regions of the interstellar medium.
## SUPPLEMENTARY MATERIAL

Supplementary material is available online including the rate constants, derived using the SC + BW cross sections, of the A²Π → X²Σ⁺, b⁴Π → a⁴Σ⁺, B²Σ⁺ → X²Σ⁺ and A²Π → X²Σ⁺ + b⁴Π → a⁴Σ⁺ radiative association processes for C¹⁴N and C¹⁵N.

## ACKNOWLEDGMENTS

This work has been supported by a grant from Labex OSUG (Investissements d'avenir—ANR10 LABX56). This project has received funding within the European Union's Horizon 2020 research and innovation program from the European Research Council (ERC) for the project "The Dawn of Organic Chemistry" (DOC) (Grant No. 741002) and from the Marie Skłodowska-Curie for the project "Astro-Chemical Origins" (ACO) (Grant No. 811312). M.Š.n.Z. and G.N. acknowledge the support of Knut and Alice Wallenberg Foundation (Project No. KAW 2020.0081). G.N. also acknowledges the support of the Swedish Research Council (Grant No. 2020-05293).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**L. Giani:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing—original draft (equal); Writing—review & editing (equal). **M. Šimsová née Zámečníková:** Conceptualization (equal); Formal analysis (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing—review & editing (equal).

**G. Di Genova:** Conceptualization (equal); Methodology (equal); Validation (equal). **C. Ceccarelli:** Conceptualization (supporting); Project administration (supporting); Supervision (supporting); Validation (supporting); Writing—review & editing (supporting).

**N. Balucani:** Conceptualization (supporting); Project administration (supporting); Supervision (supporting); Validation (supporting); Writing—review & editing (supporting). **G. Nyman:** Conceptualization (lead); Methodology (lead); Project administration (lead); Supervision (lead); Validation (equal); Writing—review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

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