# Collisional excitation of HCN by H2O

Michal Zoltowski, F Lique, P. Zuchowski, J. Klos, J. Loreau, D. Kedziera

► To cite this version:

Michal Zoltowski, F Lique, P. Zuchowski, J. Klos, J. Loreau, et al. Collisional excitation of HCN by H2O. Monthly Notices of the Royal Astronomical Society, 2025, 540 (1), pp.626-632. 10.1093/mn-ras/staf477. hal-05105513

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# Collisional excitation of HCN by H₂O

M. Żółtowski<sup>1,1,2*</sup> F. Lique,<sup>1</sup> P. Żuchowski,<sup>3</sup> J. Kłos,<sup>4,5</sup> J. Loreau<sup>6</sup> and D. Kędzierza<sup>2</sup>

Accepted 2025 March 17. Received 2025 March 14; in original form 2025 February 20

## ABSTRACT

Hydrogen cyanide (HCN) is one of the commonly observed molecules in the atmosphere of comets. As it may serve as a crucial building block in the reaction chain leading to the formation of amino acids, measuring its abundance, and comparing it to that observed in the interstellar medium can provide clues about star-forming processes and the emergence of life. However, to model HCN observations accurately, a non-local thermodynamic equilibrium analysis is necessary. This requires information about the radiative properties of HCN and its collisional excitation due to collisions with water (H₂O) since water is the most abundant projectile in comae. Here, we present a new set of rate coefficients for the rotational excitation of HCN colliding with both *para*- and *ortho*-H₂O. The calculations were performed using the statistical adiabatic channel model method and a recently published HCN–H₂O interaction potential energy surface. We provide rate coefficients for transitions between the rotational energy levels of HCN up to *j* = 10 for temperatures up to 100 K. We present a comparison of the excitation of HCN due to collisions with *para*- and *ortho*-H₂O and discuss its potential impact on astrophysical modelling. Finally, we compared our new data with those available in the literature and found significant differences, especially at low temperatures.

**Key words:** astrochemistry – molecular data – comets – scattering.

## 1 INTRODUCTION

Comets play a crucial role in understanding the formation of our Solar system. These rocky objects covered by ice contain frozen molecules that were formed during the early stages of our planetary system's formation. As a comet approaches the Sun, the icy mantle covering its nucleus begins to melt, releasing the icy molecules into a gas-phase envelope called a coma. By detecting these molecules, we can retrieve valuable information about the chemical composition and physical conditions that prevailed during the Solar system's formation (Bodewits et al. 2022).

One of these molecules is hydrogen cyanide (HCN). It is one of the most commonly observed molecules in comets. Its deposition on Earth by comets or other small cosmic bodies is likely, and it may have served as a crucial building block in the reaction chains that led to the formation of the first amino acids and proteins, ultimately contributing to the emergence of life on our planet (Matthews & Minard 2008). Additionally, by comparing the abundance of HCN observed in comets to that observed in the interstellar medium, we can gather insight into the chemical evolution during star formation processes.

To properly model observations, it is essential to know the energy level populations of the observed molecules. Radiative transfer the-

ory is then crucial in this context. In a cometary coma, local thermodynamic equilibrium (LTE) cannot be easily maintained (Bockelée-Morvan et al. 2004). Consequently, one must consider both the collisional and radiative properties of the observed molecules to model observations. The radiative properties of molecules are described by Einstein coefficients. For HCN the Einstein coefficients are well documented and readily available in the literature (Endres et al. 2016). In contrast, collisional properties, described by rate coefficients, present a significant challenge, as their determination requires a two-step computational process that is both memory-intensive and time-consuming. Indeed, to derive rate coefficients, the potential energy surface (PES) of the molecule interacting with the most abundant projectile must first be calculated. In the second step, scattering calculations are performed on the computed PES. For modelling HCN observations in comets, it is necessary to consider the HCN–H₂O collision system, as water is usually the most abundant species in comets.

To date, the only available data in the literature for this system are those provided by Dubernet & Quintas-Sánchez (2019), who employed the coupled-states (CS) approximation method (McGuire & Kouri 1974) to perform the scattering calculations. This is an approximation to the exact state-of-the-art quantum close-coupling (CC) scattering method of Arthurs & Dalgarno (1960). Using such a method, they approximated the rotation of the whole collisional system, by omitting Coriolis coupling. Despite using this approximation, the computational cost of the scattering calculations was so

© 2025 The Author(s).

Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

<footnote n="1"><sup>1</sup>IPR (Institut de Physique de Rennes), University Rennes 1, UMR 6251, CNRS, F-35000 Rennes, France</footnote>

<footnote n="2"><sup>2</sup>Faculty of Chemistry, Nicolaus Copernicus University in Toruń, 7 Gagarina Street, PL-87-100 Toruń, Poland</footnote>

<footnote n="3"><sup>3</sup>Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń, 5 Grudziadzka Street, PL-87-100 Toruń, Poland</footnote>

<footnote n="4"><sup>4</sup>Joint Quantum Institute, College Park, MD 20742, USA</footnote>

<footnote n="5"><sup>5</sup>Department of Physics, University of Maryland, College Park, MD 20742, USA</footnote>

<footnote n="6"><sup>6</sup>KU Leuven, Department of chemistry B-3001 Leuven, Belgium</footnote>

<footnote>*E-mail: mzoltowski@umk.pl</footnote>
high that the authors were unable to fully converge their calculations. In addition, they restricted their study to the para nuclear symmetry of H₂O. They provided thermalized rate coefficients for transitions between the rotational levels of HCN up to *j* = 7 and for temperatures up to 150 K. The objective of this work is to improve upon the existing data. Hence, we present new calculations of the HCN–H₂O collisional system using the quantum statistical adiabatic channel model (SACM) for rotational transitions between the energy levels of HCN up to *j* = 10, considering both nuclear spin symmetries of H₂O for temperatures up to 100 K.

This paper is organized as follows. In Section 2, we describe the PES used in this work and provide details of the scattering calculations. In Section 3, we present our new collisional data, discuss the differences arising from excitations by para- and ortho-H₂O, and compare our results with those of Dubernet & Quintas-Sánchez (2019). Finally, in Section 4, we summarize our findings and discuss future perspectives.

## 2 METHODOLOGY

Providing collisional rate coefficients requires two steps of calculation. First, under the assumption of the Born–Oppenheimer approximation, the electronic Schrödinger equation is solved using quantum chemistry methods to obtain a PES. Then, using the PES, the excitation cross-sections can be computed by solving the nuclear part of the Schrödinger equation with the help of quantum, statistical, and classical scattering theory, or a combination of these approaches, such as the mixed quantum–classical method (MQCT). It is worth highlighting the MQCT here, as it has recently been successfully used to treat collisional excitation in complex systems such as H₂O–H₂O (Mandal & Babikov 2023) and CO–CO (Bostan et al. 2024).

Calculated cross-sections are averaged over the collisional energy assuming a Maxwell–Boltzmann distribution of the collisional energy to provide rate coefficients. In this section, we present the methodology employed for the calculations of collisional data for the HCN–H₂O system.

### 2.1 Potential energy surface

In this work, we used the PES of the HCN–H₂O system recently published by some of us (Vindel-Zandbergen et al. 2023). Both monomers were treated as rigid rotors, resulting in a five-dimensional interaction potential. The PES was calculated using the AUTOPES code (Metz & Szalewicz 2020), within symmetry-adapted perturbed theory and with the description of monomers based on the density functional theory [SAPT(DFT); Misquitta et al. 2005]. The 2833 ab initio points were calculated using an augmented correlation-consistent polarized valence quadruple-zeta (aug-cc-pVQZ) basis set. Two minima were identified on the PES: the first, a global minimum, is found at a configuration where the hydrogen atom of HCN approaches the oxygen atom of the water molecule. This minimum occurs at a distance of 7.09 *a*₀ between the centres of mass of the two monomers (H₂O and HCN), with an energy of –1852.34 cm⁻¹. The second, a local minimum, is found at a configuration where the nitrogen atom of HCN approaches one of the hydrogen atoms of the water molecule. This minimum is located at a distance of 6.91 *a*₀, with an energy of –1338.40 cm⁻¹. This PES was used for rovibrational state calculations, and the results obtained from those calculations were in excellent agreement with the experimental measurements performed by Fillery-Travis, Legon & Willoughby (1983). For more detailed information about calculations of the

new HCN–H₂O PES, we refer the reader to the paper of Vindel-Zandbergen et al. (2023). To make this PES suitable for scattering calculations, we expanded it into a set of radial functions. Details regarding this procedure are presented in Appendix A.

### 2.2 Scattering calculations – statistical adiabatic channel model

From the features of the PES (very anisotropic with a very deep well) and information from the previous study by Dubernet & Quintas-Sánchez (2019), we can anticipate that state-of-the-art calculations using the CC method (Arthurs & Dalgarno 1960) will not be feasible to perform. We have decided to use the modified SACM, introduced to inelastic scattering studies by Loreau, Faure & Lique (2018a) and based on the original work by Quack & Troe (1974, 1975). We will briefly introduce this method below. For detailed explanations, we refer the reader to recent references where the method is benchmarked against fully quantum–mechanical results for several collisional systems (Loreau et al. 2018a; Loreau, Lique & Faure 2018b; Desrousseaux et al. 2021; Konings et al. 2021).

The starting point of this method is the coupled radial equations, as presented in the CC formalism (Phillips, Maluendes & Green 1995). We perform a diagonalization of the Hamiltonian, excluding the kinetic energy term. This diagonalization must be performed for each total angular momentum *J* and for each parity. The total angular momentum *J* represents the coupling between the orbital angular momentum *l* of the system and the angular momenta *j*₁ and *j*₂ of the colliding partners (in our case, *j*₁ and *j*₂ represent the angular momenta of H₂O and HCN, respectively). From each diagonalization at a given *J* and parity, we obtain a set of adiabatic curves as a function of distance. Each adiabatic curve has a centrifugal barrier that depends on the orbital quantum number *l* and corresponds asymptotically to a specific internal state of the colliding molecules (the sum of energy of the rotational energy levels of H₂O and HCN). The SACM method assumes that a complex can form and an inelastic collision can occur if the kinetic energy exceeds the centrifugal barrier in the entrance/exit channel. The probability is determined solely by counting all possible exit channels. This implies that all exit channels are equally weighted, and the *S* matrix is constructed as shown in equation (1) for open channels (where *N*(*E*, *J*) is the number of open channels at a given collisional energy *E* and total angular momentum *J*) and is zero for closed channels:

$$|S_{\text{if}}(E, J)|^2 = \begin{cases} \frac{1}{N(E, J)} & \text{for open channels,} \\ 0 & \text{for closed channels,} \end{cases} \qquad (1)$$

where *i* and *f* are initial and final states, respectively. The cross-section is calculated using

$$\sigma_{\text{if}}(E) = \frac{\pi}{(2j_1 + 1)(2j_2 + 1)k^2} \sum_{J=0}^{\infty} (2J + 1)|S_{\text{if}}(E, J)|^2, \qquad (2)$$

where *k* is the wave vector:

$$k^2 = \left(\frac{2\mu}{\hbar^2}\right) (E_{\text{tot}} - E_1 - E_2), \qquad (3)$$

$E_{\text{tot}}$ represents the total energy, $E_1$ and $E_2$ are the internal energies of the H₂O and HCN monomers, respectively. The difference between the total energy and the sum of the monomers' internal energies corresponds to the collisional energy *E*.

The SACM method offers the advantage of being significantly less computationally expensive than the CC method. However, the SACM method is based on adiabatic curves obtained from the diagonalization of the CC Hamiltonian, excluding the kinetic
term, and still includes the Coriolis effect, in contrast to the CS approximation. Additionally, the adiabatic curves are independent of energy, meaning that they need to be computed only once for a given total angular momentum. Furthermore, calculations converge much faster with the size of the rotational basis, as it is not necessary to include a large number of closed channels in the calculations. Moreover, it has been demonstrated (Loreau et al. 2018a, 2022; Desrousseaux et al. 2021) that the SACM method can achieve accuracy within a factor of 1.5 compared to the CC method at low temperatures, which is within the range of acceptable inaccuracy for modelling astrophysical observations.

## 2.3 Convergence criteria

To obtain accurate results with the SACM method, we must ensure that our calculations converge with respect to the size of the rotational basis and that we include a sufficient number of *J* values in the summation. The rotational basis is constructed by combining the rotational levels of the molecules in the collisional system and the angular momentum of the collisional systems.

In this work, we use the following notation: for H₂O, (j₁,kₐ,kc), where j₁ represents the rotational quantum number with projections kₐ and kc along the principal axes of inertia; and for HCN, j₂, where j₂ denotes the rotational quantum number. In addition, the difference kₐ − kc allows us to distinguish between *para*- and *ortho*-nuclear spin isomers of H₂O. Specifically, for the *para* isomer, kₐ − kc is an even number, whereas for the *ortho* isomer, kₐ − kc is an odd number. Fortunately, the *ortho*- and *para*-levels of H₂O do not interconvert during inelastic collisions, allowing us to treat them separately and saving a significant amount of computing time and memory.

Our objective is to obtain collisional data for H₂O levels up to *j*₁ = 2 and HCN levels up to *j*₂ = 10, for temperatures up to 100 K. Hence, the internal energy of the highest channel considered (*j*₁, *j*₂ = 2, 10) is having an energy of 297.96 cm⁻¹ for *ortho*-H₂O and of 298.72 cm⁻¹ for *para*-H₂O. [Rotational energy levels were calculated using following rotational constants: *B*₀ = 1.4782 cm⁻¹ (Maki 1974) for HCN and *A* = 27.8770 cm⁻¹, *B* = 14.5120 cm⁻¹, *C* = 9.2850 cm⁻¹ (Herzberg 1966) for H₂O.] To ensure convergence of rate coefficients for temperatures up to 100 K, we need to compute cross-sections for collisional energies up to approximately 500 cm⁻¹. This requirement implies that our rotational basis must include all energy levels with energies up to approximately 800 cm⁻¹. To achieve this, we include in the basis all rotational levels up to *j*₁ = 8 for H₂O and *j*₂ = 22 for HCN. However, with such a large basis, we encounter numerical limits e.g. at *J* = 3, there are already more than 20 000 channels, making calculations unfeasible. Results obtained with this extensive basis serve as benchmark data. We explored various combinations of *j*₁, *j*₂ = max to find a balance between computational efficiency and accuracy. Ultimately we selected a basis that includes all levels with *j*₁ up to 5 and *j*₂ up to 15. A systematic comparison of cross-sections and rate coefficients between results obtained with the basis *j*₁ = 5, *j*₂ = 15 and those obtained with *j*₁ = 8, *j*₂ = 22 are presented in Figs 1 and 2. At kinetic energies of 100 and 300 cm⁻¹, we observe an almost perfect agreement between the results obtained from both bases. However, at a kinetic energy of 500 cm⁻¹, deviations start to appear. As could have been anticipated the results obtained with the basis *j*₁ = 5, *j*₂ = 15 start to overestimate the cross-sections compared to those from the basis *j*₁ = 8, *j*₂ = 22 because of the lack of open channels in the basis.

Nevertheless, upon examining the plots of rate coefficients in Fig. 2, we find that these deviations do not affect the rate coefficients even at a temperature of 100 K, which is our goal. Hence, using

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Systematic comparison of cross-sections for all inelastic transitions between energy levels of HCN (*j*₂ up to 10) and H₂O (*j*₁ up to 2), calculated with the basis used in this work, as a function of the 'fully' converged basis (see text for details). Comparisons were performed at kinetic energies of 100 cm⁻¹ (upper panel), 300 cm⁻¹ (middle panel), and 500 cm⁻¹ (lower panel) for total angular momenta *J* up to 3. The solid line represents exact results, dashed lines indicate a factor of 2 difference, and the dashed-dotted lines indicates a factor of 3 difference.</figcaption></figure>

the basis $j_1 = 5$ and $j_2 = 15$, we calculated adiabatic curves for $J$ up to 150. This number of $J$ was required to obtain convergence of cross-sections with respect to the total number of angular momenta included to better than 5 per cent.

## 3 RESULTS AND DISCUSSION

We obtained state-to-state cross-sections and rate coefficients for transitions between rotational levels of HCN up to $j_2 = 10$ and *para/ortho*-H₂O up to $j_1 = 2$.

It is important to note that for using rate coefficients in radiative transfer modelling, they must fulfil detailed balance rule (Faure, Lique & Loreau 2020; Żółtowski et al. 2023). Thus, we have two options for presenting our data in a manner that upholds the detailed balance rule.

The first option consists of choosing one state of the projectile and neglecting the possibility that both the target and the projectile can be simultaneously excited/deexcited. Typically, we choose the ground state of the projectile in such cases. The second option, which is more commonly practiced, consists of providing 'thermalized' rate coefficients. These 'thermalized' rate coefficients are obtained by summing over all possible final states of the projectile and averaging over the initial states of the projectile, assuming a thermal population
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Systematic comparison of rate coefficients for all inelastic transitions between energy levels of HCN ($j_2$ up to 10) and H₂O ($j_1$ up to 2), calculated with the basis used in this work, as a function of the 'fully' converged basis (see text for details). Comparisons were performed at temperatures of 10 K (upper panel), 50 K (middle panel), and 100 K (lower panel) for total angular momenta *J* up to 3. The solid line represents exact results, dashed lines indicate a factor of 2 difference, and the dashed-dotted line indicates a factor of 3 difference.</figcaption></figure>

distribution:

$$k_{j_2 \to j'_2}(T) = \sum_{j_1 k_a k_c \ j'_1 k'_a k'_c} P_{j_1 k_a k_c}(T) k_{j_1 k_a k_c, j_2 \to j'_1 k'_a k'_c, j'_2}(T), \quad (4)$$

where

$$P_{j_1 k_a k_c}(T) = \frac{g_{j_1} e^{(-E_{j_1 k_a k_c} / k_b T)}}{\sum_{j_1 k_a k_c} g_{j_1} e^{(-E_{j_1 k_a k_c} / k_b T)}}, \quad (5)$$

where $g_{j_1}$ is the degeneracy factor of the rotational level, and $k_b$ is the Boltzmann constant.

Hence, we prepared four sets of rate coefficients. The first two sets 'p-H₂O – ground' and 'o-H₂O – ground', consider H₂O in its ground rotational states. In these two sets of data, we use state-to-state data where the rotational state of H₂O remains in its ground level before and after the collision (for para-H₂O: $j_{1,k_a,k_c} = 0,0$; for ortho-H₂O: $j_{1,k_a,k_c} = 1,0$). The third and fourth sets, the 'thermalized set' (p-H₂O – thermalized, o-H₂O – thermalized), were generated using equation (4). A comparison of all sets of rate coefficients is presented in Fig. 3.

We analysed deexcitation transitions of HCN from the level $j_2 = 10$ to all lower lying rotational levels at three temperatures: 10, 50, and 100 K. At 10 K, for low $\Delta j_2$ transitions, we observe a difference of about a factor of 2 between the 'ground' and the 'thermalized' sets. For both para and ortho symmetry, the differences between the 'ground' and the 'thermalized' sets decrease with increasing $\Delta j_2$. Additionally, the differences between rate

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Rate coefficients for the rotational deexcitation of HCN from the rotational energy level of $j_2 = 10$ to the next 10 lower levels at temperatures of 10 K (upper panel), 50 K (middle panel), and 100 K (lower panel). The terms 'o-H₂O – thermalized' and 'p-H₂O – thermalized' represent the thermalized rate coefficients for ortho and para symmetry, respectively. The 'o-H₂O – ground' and 'p-H₂O – ground' refer to rate coefficients where only the ground level of ortho and para symmetry is considered. See text for more details.</figcaption></figure>

coefficients from the 'ground' set and the 'thermalized' set increase with increasing temperature. The differences between these two sets of rate coefficients can be directly attributed to the fact that, in the case of the 'ground' set, only a state-to-state rate coefficients with fixed rotational level of water is considered (i.e. a transition where the rotational state of water before and after collision remains in ground level). In contrast, the 'thermalized' set involves summing the values of the rate coefficients over all possible final states of water for a given initial state. Even when accounting for the population of the given initial state, the inclusion of multiple possible transitions often leads to larger rate coefficients compared to the single transition considered in the 'ground' set. From this analysis, we conclude that using rate coefficients with a fixed state of water at the ground level is a crude approximation as already discussed in Faure et al. (2020) and at the opposite of what is found and regularly used when considering interstellar collisions induced by H₂ (Roueff & Lique 2013). Even at very low temperatures, where the ground state is the most populated, we observe differences of up to 50 per cent between the 'ground' and 'thermalized' sets of rate coefficients.

Focusing now on the results obtained with the 'thermalized' set, we do observe a slight deviation between the excitation rate coefficients induced by para-H₂O and ortho-H₂O. At lower temperatures (10 and 50 K), the rate coefficients for ortho-H₂O are larger up to $\Delta j_2 = 3$ transitions. Beyond $\Delta j_2 = 4$ transitions, the para-H₂O rate coefficients become dominant. However, at 100 K, the ortho-H₂O rate coefficients are consistently larger than those of para-H₂O. To further investigate the differences between the excitation induced by para- and ortho-H₂O projectiles, we performed a systematic comparison of rate coefficients. This comparison was conducted at temperatures of 10, 50, and 100 K using the thermalized rate coefficients. The results are presented in Fig. 4. In the low-temperature regime, the rate coefficients for ortho-H₂O and para-H₂O differ by up to a factor of ~2. However, as the temperature increases, these
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Systematic comparison of the thermalized rate coefficients for *para*-H₂O and *ortho*-H₂O at temperatures of 10, 50, and 100 K for all rotational transitions of HCN calculated in this work. The solid line corresponds to equal results, dashed lines represent a factor of 2 difference, and dashed-dotted lines represent a factor of 3 difference.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Comparison of the thermalized rate coefficients for selected deexcitation transitions. The coefficients calculated in this work using the SACM method are represented by solid lines, while the dashed lines denote the CS approximation rate coefficients provided by Dubernet & Quintas-Sánchez (2019).</figcaption></figure>

differences decrease, becoming less than 20 per cent at 100 K. These differences at low temperatures can be explained by threshold effects that are especially strong in this temperature regime. They are due the different rotational energy level structure of *ortho*- and *para*-H₂O. Hence, limiting collisional data to only one symmetry of water in radiative transfer modelling may result in errors in the determination of the physical and chemical conditions at low temperatures.

We compared our results for HCN-*para*-H₂O with those obtained by Dubernet & Quintas-Sánchez (2019). We examined how the rate coefficients vary with increasing Δj₂ transitions. In Fig. 5, we present rate coefficients for deexcitation transitions of HCN

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Systematic comparison of the SACM rate coefficients calculated in this work and the CS rate coefficients calculated by Dubernet & Quintas-Sánchez (2019) at temperatures of 10, 50, and 100 K for transitions of HCN between rotational levels up to j = 7. The solid line represents exact results, while the dashed lines indicate a factor of 2 difference, and the dashed-dotted lines represent a factor of 3 difference.</figcaption></figure>

starting from j₂ = 4 to the next four lower rotational levels. We compare the thermalized rate coefficients from this work to the thermalized rate coefficients of Dubernet & Quintas-Sánchez (2019). At low temperatures, our rate coefficients tend to be higher than those reported by Dubernet & Quintas-Sánchez (2019), with differences reaching up to 40 per cent for some transitions. As the temperature increases, the differences systematically decrease, and are 5–10 per cent at 100 K. Except for the Δj = 1 transitions, we observe that the differences between our results and those of Dubernet & Quintas-Sánchez (2019) decrease with increasing Δj and temperature. To fully compare our results for HCN-*para*-H₂O with those of Dubernet & Quintas-Sánchez (2019), we examined thermalized rate coefficients for all possible deexcitation transitions of HCN up to j₂ = 7, as it was the highest rotational level for which Dubernet & Quintas-Sánchez (2019) provided data. The comparison was performed at temperatures of 10, 50, and 100 K and the results are presented in Fig. 6. For most of the transitions, the differences between the two data sets remain within a factor of 2, although several transitions exhibit discrepancies as large as a factor of 3 or more at low temperatures. Overall, our rate coefficients tend to be larger at lower temperatures, with differences decreasing with increasing temperatures.

The differences between our results and those of Dubernet & Quintas-Sánchez (2019) can be attributed to the use of different PESs, different scattering methods, and different sizes of the rotational basis. Starting with the different PES, Dubernet & Quintas-Sánchez (2019) employed a PES developed in a previous study (Quintas-Sánchez & Dubernet 2017), while we used a recently published PES of Vindel-Zandbergen et al. (2023). The two PESs differ by about 2–3 per cent at global and local minima. These differences are primarily due to the calculation methods used: SAPT(DFT) in Vindel-Zandbergen et al. (2023) and coupled cluster in Quintas-Sánchez & Dubernet (2017). Detailed explanations of the differences between these PESs can be found in Vindel-Zandbergen et al. (2023). However, we do not expect that the differences between the two PES could significantly contribute to the differences seen
between the two sets of collisional data. As for different scattering methods, our calculations are based on the SACM method, whereas Dubernet & Quintas-Sánchez (2019) used the CS method. Previous studies (Loreau et al. 2018a, b) have demonstrated that the SACM method is a reliable alternative to the exact CC method at low temperatures but becomes less accurate as temperature increases. In contrast, the CS approximation struggles to reproduce the CC results at low temperatures, although the differences diminish as the temperature increases (e.g. Żóltowski et al. 2021).

Finally, the size of the rotational basis sets used in the calculations is a crucial factor. In the case of the quantum method, to obtain converged results, we need to include not only open channels but also closed ones. Particularly for the HCN-H₂O system, Dubernet & Quintas-Sánchez (2019) noted that their results were highly sensitive to the size of the basis set included. Due to the high computational cost of the CS method, Dubernet & Quintas-Sánchez (2019) had to use a reduced rotational basis set (j₁ = 4, j₂ = 15) and could not take into account all necessary rotational energy levels, making full convergence impossible to achieve.

The SACM method requires a smaller basis set compared to pure quantum methods. We need to consider only open channels, or even less than that to achieve convergence of the cross-sections as demonstrated in this work. In our calculations we used a slightly larger basis than Dubernet & Quintas-Sánchez (2019) (j₁ = 5, j₂ = 15), and that allowed us to obtain fully converged results.

A more detailed discussion regarding the accuracy of our data and data of Dubernet & Quintas-Sánchez (2019) is currently not feasible. Indeed, such an analysis would require comparisons with state-of-the-art calculations, which are presently unavailable, or with experimental measurements, which, to the best of our knowledge, do not exist.

## 4 CONCLUSIONS

We presented a new scattering study of the excitation of HCN by collisions with *para*- and *ortho*-H₂O. Calculations were performed using the PES provided by Vindel-Zandbergen et al. (2023), and the SACM approach. We obtained rate coefficients for transitions between rotational levels of HCN up to *j*₂ = 10 and H₂O up to *j*₁ = 2 for temperatures up to 100 K.

We compared the efficiency of HCN excitation due to collisions with *para*- and *ortho*-H₂O. The results show that at low temperatures, the differences in the rate coefficients due to collisions with *para*- and *ortho*-H₂O can be as high as a factor of 2. However, as the temperature increased, these differences decreased, being about 20 per cent at 100 K. Due to these differences at low temperatures, it is crucial to include collisional data of HCN with both symmetries of water in the modelling of observations of low-temperature comae. Our data for the excitation of HCN induced by *para*-H₂O were compared to the data obtained by Dubernet & Quintas-Sánchez (2019) using the CS approximation. The differences varied between a factor of 2 and 3 for most of the rate coefficients. These discrepancies, while reasonable considering the two very different numerical approaches, can be attributed to differences in the PES, the scattering methods employed, and the size of rotational basis sets used in the calculations. However, it is not currently possible to evaluate the absolute accuracy of these results, as no exact theoretical or experimental data are available for direct comparison. We believe that our new data will be valuable for analysing cometary observations and will enable the determination of the physical and chemical conditions of observed comets with greater precision. Therefore, we strongly encourage

astronomers to incorporate our new HCN-H₂O data into their models when interpreting observations of comets.

## ACKNOWLEDGEMENTS

MŻ is grateful for the support of the National Science Centre (NCN) through Grant 'Sonatin 8' No. 2024/52/C/ST9/00110. FL acknowledges Rennes Metropole for financial support. This research was supported by the CNRS national program 'Physique et Chimie du Milieu Interstellaire'. JL acknowledges support from Internal Funds KU Leuven through Grant No. C14/22/082. We wish to acknowledge the support from the CEA/GENCI for awarding us access to the TGCC/IRENE supercomputer within the A0110413001 project and from the Wroclaw Centre for Networking and Supercomputing (Project 218). JK is grateful to the U.S. National Science Foundation for support through Grant No. AST-2009253 (Subaward No. 362627Sub1). PŻ and DK are grateful for the support of the National Science Centre (NCN) through Grant 'Sonata Bis 9' No. 2019/34/E/ST4/00407.

## DATA AVAILABILITY

Sets of state-to-state cross-sections, rate coefficients, and thermalized rate coefficients of HCN-*para*/ortho-H₂O will be submitted to the BASECOL (Dubernet et al. 2024) and EMAA data bases (https://emaa.osug.fr).

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## APPENDIX A: EXPANSION OF THE PES IN ANGULAR FUNCTIONS

To perform our scattering calculations using the SACM method, we expanded the five-dimensional PES of the HCN-H₂O system into angular functions. These functions describe the relative rotations of the HCN molecule around its centre of mass and the rotations of the R distance vector around the H₂O molecule. Our approach follows the methodologies outlined in Phillips et al. (1994), Valiron et al. (2008), and Dagdigian (2020), which detail the expansion into angular functions:

$$V(R, \theta, \phi, \theta', \phi') = \sum_{l_1 m_1 l_2 l} v_{l_1 m_1 l_2 l}(R) \tilde{t}_{l_1 m_1 l_2 l}(\theta, \phi, \theta', \phi'), \quad (A1)$$

where $\tilde{t}_{l_1 m_1 l_2 l}(\theta, \phi, \theta', \phi')$ represents normalized (Valiron et al. 2008) spherical tensor shown below:

$$\tilde{t}_{l_1 m_1 l_2 l}(\theta, \phi, \theta', \phi') = \alpha_{l_1 m_1 l_2 l} (1 + \delta_{m_1 0})^{-1} \\ \times \sum_{r_1 r_2} l_1 l_2 l r_1 r_2 r Y_{l_2 r_2}(\theta', \phi') Y_{lr}(\theta, \phi) \\ \times [\delta_{m_1 r_1} + (-1)^{l_1+m_1+l_2+l} \delta_{-m_1 r_1}]. \quad (A2)$$

The normalization of the tensor is provided by the $\alpha_{l_1 m_1 l_2 l}$ factor:

$$\alpha_{l_1 m_1 l_2 l} = [2(1 + \delta_{m_1 0})^{-1} (2l_1 + 1)^{-1}]^{-1/2}. \quad (A3)$$

The radial R and angular variables are defined in the body-fixed frame, where the C₂ symmetry axis of the water molecule aligns

with the z-axis, and the water molecule lies in the x-z plane. The

<figure><img src="image_8.png" /><figcaption>**Figure A1.** Selected radial terms expansion coefficients as a function of distance. The isotropic $v_{0000}$ term, dipole-dipole (D-D) $v_{1012}$ term, dipole-quadrupole (D-Q) $v_{1023}$ term, quadrupole-dipole (Q-D) $v_{2013}$ term, and quadrupole-quadrupole (Q-Q) $v_{2024}$ term.</figcaption></figure>

origin of the coordinate system is placed at the centre of mass of the water molecule, and the R distance vector points toward the centre of mass of the HCN molecule, similar to the set-up used by Quintas-Sánchez & Dubernet (2017) in their H₂O-HCN study. The spherical coordinates $(\theta, \phi)$ describe the rotations of the R vector, while the spherical coordinates $(\theta', \phi')$ describe the rotations of the HCN molecule around its centre of mass.

We determined the $v_{l_1 m_1 l_2 l}(R)$ coefficients by performing a least-squares fit of the PES from Vindel-Zandbergen et al. (2023) to a finite expansion with terms defined by maximum values of $l_1 = 10$, $m_1 = 10$, and $l_2 = 10$. This expansion yields 1956 radial $v_{l_1 m_1 l_2 l}(R)$ coefficients for R distances ranging from 3.5 bohr to 10 000 $a_0$. In Fig. A1, we present selected radial expansion coefficients. The isotropic $v_{0000}$ coefficient exhibits a minimum of 1632.84 cm⁻¹ at R = 7.4 $a_0$. The most attractive coefficient is the $v_{1012}$ term, which corresponds to the electrostatic interaction of two molecular dipoles in the long range. Additionally, we show the dipole-quadrupole and quadrupole-quadrupole terms.

This paper has been typeset from a TeX/LA TeX file prepared by the author.