State-to-state rate coefficients for HCS⁺ in rotationally inelastic collisions with H₂ at low temperatures

ABSTRACT

HCS⁺ ions have been detected in several regions of the interstellar medium (ISM), but an accurate determination of the chemical-physical conditions in the molecular clouds where this molecule is observed requires detailed knowledge of the collisional rate coefficients with the most common colliders in those environments. In this work, we study the dynamics of rotationally inelastic collisions of HCS⁺ + H₂ at low temperature, and report, for the first time, a set of rate coefficients for this system. We used a recently developed potential energy surface for the HCS⁺–H₂ van der Waals complex and computed state-to-state rotational rate coefficients for the lower rotational states of HCS⁺ in collision with both para- and ortho-H₂, analysing the influence of the computed rate coefficients on the determination of critical densities. Additionally, the computed rate coefficients are compared with those obtained by scaling the ones from HCS⁺ in collision with He (an approximation that is sometimes used when data is lacking), and large differences are found. Furthermore, the approximation of using the rates for the HCO⁺ + H₂ collision as a rough approximation for those of the HCS⁺ + H₂ system is also evaluated. Finally, the complete set of de-excitation rate coefficients for the lowest 30 rotational states of HCS⁺ by collision with H₂ is reported from 5 to 100 K.

astrochemistry, molecular data, molecular processes, scattering, ISM: molecules

1 INTRODUCTION

The HCS⁺ molecule has been widely observed in the interstellar medium (ISM) (Turner 1996; Rodríguez-Baras et al. 2021), being one of the species whose quantities are essential for obtaining elemental abundances in molecular clouds (Fuente et al. 2019). After its first detection in the Orion Nebulae (Thaddeus, Guélin & Linke 1981), HCS⁺ molecules have been observed in a variety of ISM environments (Irvine, Good & Schloerb 1983; Turner 1996; Houde et al. 2000; Lucas & Liszt 2002; Nomura & Millar 2004; Leurini et al. 2006; Fuente et al. 2016; Potapov et al. 2016). This ion is formed in the ISM fundamentally by the reactions of CS⁺ with H₂, C⁺ with H₂S, and CS with |${\rm H}_{3}^{+}$| and HCO⁺ (McAllister 1978; Montaigne et al. 2005); while the main mechanism for its destruction is by reaction with electrons (Millar 1983). HCS⁺ has shown to be non-reactive with most of the interstellar molecules at relevant collision energies, including H₂ (Millar et al. 1985).

In typical molecular clouds, temperatures range from five to a few tens of kelvins, and the densities range from 10⁴ to 10⁶cm⁻³ (Denis-Alpizar et al. 2020). Therefore, collisions are rare, and a Boltzmann distribution cannot describe the rotational population of the interstellar molecules in such environments. In such cases, analysis of the chemical-physical conditions of ISM regions where a molecule is observed should be performed using non-local thermal-equilibrium (non-LTE) models. But these models require knowledge of the Einstein coefficients, as well as the rotational rate coefficients of the observed molecules with the most common colliders in the ISM (i.e., H₂, He, H, and electrons). Moreover, since sulfur-containing molecules have not been observed in as much abundance as expected in the ISM (Rivière-Marichalar et al. 2019), accurate molecular data allowing a better estimation of the conditions where molecules such as HCS⁺ are observed, are indeed valuable.

The collision of HCS⁺ ions with He has been studied in detail by Monteiro (1984) and Dubernet, Quintas-Sánchez & Tuckey (2015). However, no set of rate coefficients has been reported so far for its collisions with H₂. When accurate molecular data is lacking, rate coefficients for the collision of a neutral molecule with H₂ are usually estimated by scaling from those obtained for collisions with He (Schöier et al. 2005). However, this is often not a good approximation, particularly in the case of ions, due to the different behaviour in the long range of the potential energy surface (PES). Also, in the LAMDA data base (Schöier et al. 2005; Van der Tak et al. 2020), the rates for HCS⁺ + H₂ collisions are assumed to be the same as the ones obtained for HCO⁺ (isovalent with HCS⁺) by collisions with H₂; but the accuracy of this approximation deserves to be investigated.

Very recently, we reported (Quintas-Sánchez, Dawes & Denis-Alpizar 2021) the first PES for the HCS⁺–H₂ van der Waals system. The global minimum of this surface is five times deeper than in the case of HCS⁺–He, but is only half that found for HCO⁺–H₂. Such large differences make it even more important to evaluate the approximations used so far for estimating the state-to-state rate coefficients for this system. The main goal of this work is to study the rotational relaxation of HCS⁺ by H₂ at low temperatures and report the first accurate set of rate coefficients for this collision system. The paper is organized as follows: in the next section, the methods employed are presented, while the results are discussed in Section 3. Finally, the conclusions are discussed and summarized in Section 4.

2 METHODS

In our previous work (Quintas-Sánchez et al. 2021), we presented the first theoretical spectroscopy and scattering study of the HCS⁺–H₂ system. Using a new intermolecular PES, calculated for the ground electronic state of the complex, we computed the rovibrational bound states, rotational constants, as well as (de)-excitation cross-sections for the lowest levels in the collision of HCS⁺ with both para- and ortho-H₂. Both the HCS⁺ and H₂ molecules were considered as rigid rotors. In the case of linear triatomic molecules, close-coupling calculations including the bending motions have shown (Stoecklin et al. 2013; Denis-Alpizar, Stoecklin & Halvick 2014, 2015; Stoecklin, Denis-Alpizar & Halvick 2015) that the rigid rotor approximation is valid for collisional energies lower than the bending frequency. The HCS⁺ molecule has a relatively large bending frequency of 784 cm⁻¹ (Puzzarini 2005); therefore, close-coupling calculations for collisional energies lower – or slightly higher – than this value are expected to provide accurate results. The employed coordinates are shown in Fig. 1. R connects the centres of mass of the H₂ and HCS⁺ molecules, while θ₁, θ₂, and φ describe the angular orientations of both systems.

Figure 1.

Internal coordinates used to describe the HCS⁺ + H₂ system.

Using the same highly accurate PES and state-of-the-art methods, in this paper we study the dynamics of HCS⁺ rotational relaxation via collisions with both para- and ortho-H₂, focusing on the state-to-state rate coefficients at low temperatures. Details of the recently developed four-dimensional PES can be found in Quintas-Sánchez et al. (2021). Briefly, the analytical representation of the PES was constructed using an automated interpolating moving least-squares (IMLS) methodology, freely available as a software package under the name AUTOSURF (Quintas-Sánchez & Dawes 2019). The fitting basis and other aspects of the procedure were the same as for previous similar systems and have been described in detail elsewhere (Majumder, Ndengue & Dawes 2016; Dawes & Quintas-Sánchez 2018; Quintas-Sánchez & Dawes 2019, 2021). The surface was constructed from 3023 ab initio energies, computed with explicitly correlated coupled-cluster theory, CCSD(T)-F12b, (Knizia, Adler & Werner 2009) with two sizes of basis (VTZ-F12/VQZ-F12) (Peterson, Adler & Werner 2008) extrapolated to the complete basis set (CBS) limit. The final estimated root-mean-squared error was 0.19 cm⁻¹. The PES was found to have two minima defining the isomers of the system: a global minimum of −752.6 cm⁻¹ in a T-shaped configuration, with the HCS⁺ molecule acting as the stem of the T (with the H atom pointing to the H₂ molecule) and a centre-of-mass separation of 4.148 Å; and a secondary minimum of −509.1 cm⁻¹ in a non-planar crossed structure, at a centre-of-mass separation of 3.341 Å. As mentioned in the previous section, the global minimum of this surface (−752.6 cm⁻¹) is five times deeper than in the case of HCS⁺ + He [−138.5 cm⁻¹, Dubernet et al. (2015)], but is only half that found for HCO⁺ + H₂ [−1509.4 cm⁻¹, Denis-Alpizar et al. (2020)]. Additionally – to facilitate comparison with the HCS⁺ + He PES – the PESs of the HCS⁺ + H₂ and HCO⁺ + H₂ systems were averaged over the para-H₂ (j = 0) rotational wavefunction. Fig. 2 shows the two-dimensional (2D) plots of these averaged surfaces, as well as the surface of the HCS⁺ + He complex. Here, not only the differences in the depth of the wells are observed, but also in the anisotropy of the PESs.

Figure 2.

Comparison of the PESs for the HCS⁺–He (upper panel), HCS⁺–H₂ (middle panel), and HCO⁺–H₂ (lower panel) systems. The angle describes the orientation of the HCS⁺ or HCO⁺ fragment. Contours are separated by 40 cm⁻¹. The HCS⁺–H₂ and HCO⁺–H₂ PESs were averaged over the para-H₂ wavefunction, see text for details.

Because our quantum scattering calculations require an expansion of the angular dependence of the PES in products of spherical harmonics, the 4D IMLS potential was recast in that representation. First, the IMLS-PES was used to generate a grid of potential energies for |$50\, 700$| configurations. Then, the grid of energies were fitted following the same procedure recently used for studying the HCO⁺–H₂ complex (Denis-Alpizar et al. 2020). Let us recall only the main features here . Briefly, the analytical function employed has the form:

$$\begin{eqnarray} V(R,\theta _1,\theta _2,\varphi) = \sum _{l_1=0}^{4} \sum _{l_2=0}^{10} \sum _{m=0}^{\min (l_1,l_2)} f^{l_1,l_2}_m(R) \\ \times \bar{P}_{l_1}^m(\cos \theta _1) \bar{P}_{l_2}^m(\cos \theta _2) \cos (m\varphi) \end{eqnarray}$$

(1)

where the angular part is represented by the multiplication of normalized associated Legendre polynomials and cosine functions. H₂ has a centre of symmetry; thus, l₁ is restricted to even values (Nasri et al. 2015). For each value of R, the energies were fitted using a least-squares method, with the weight function: w(E) = min (V₀/E; 10), where |$V_0=5\, 000$| cm⁻¹. Each coefficient |$f_m^{l_1,l_2}(R)$| was then fitted using the Reproducing Kernel Hilbert Space (RKHS) procedure, so that (Ho & Rabitz 1996):

$$\begin{eqnarray} f^{l_1,l_2}_m(R)=\sum _{k=1}^N \alpha _k^{l_1,l_2,m} q^{2,2}(R,R_k)~, \end{eqnarray}$$

(2)

where N is the number of radial points of the grid {R_k}, and q^{2, 2}(R, R_k) are 1D kernels defined as:

$$\begin{eqnarray} q^{2,2}\left(R_{\rm k}, R\right)=\frac{1}{3 R_{\gt }^{3}}-\frac{ R_{\lt }}{5 R_{\gt }^{4}} , \end{eqnarray}$$

(3)

where R_> and R_< are the greater and smaller values between R_k and R. The |$\alpha _{\rm k}^{l_1,l_2,m}$| coefficients are obtained by solving the system of linear equations |$\mathbf {q}{(\mathbf {R}_i,\mathbf {R}_j})\mathbf {\alpha } =\mathbf {f}^{l_1,l_2}_m(\mathbf {R}_i)$|⁠, where i and j label the different radial geometrical configurations of the grid. The large number of energies in the grid, together with the use of the least-square and kernel procedure, allows a fit of high quality.

The dynamics of the collision of HCS⁺ with para-H₂ and ortho-H₂ was studied using the Didimat code (Guillon et al. 2008). This code solves the close-coupling equations in the space-fixed frame. The log-derivative propagator (Manolopoulos 1988) was employed, and the maximum propagation distance was set to 100 a₀. The rotational constants of HCS⁺ and H₂ were fixed to their experimental values: |$B_{\rm H_2}=60.853$| cm⁻¹ (Huber & Herzberg 1979) and |$B_{\rm HCS^+}=0.71173$| cm⁻¹ (Tang & Saito 1995).

The cross-sections from a given initial rotational state j of HCS⁺ were computed for collisional energies in the [10⁻², 10³] cm⁻¹ interval. Up to j = 20, the basis included 35 rotational states of HCS⁺, while for initial rotational states of HCS⁺ from j = 21 up to j = 30, the basis was increased to include 40 states. Only one rotational state of H₂ was included in the basis for the close-coupling calculations. The small influence – in the scattering calculation – of the number of rotational states of H₂ included in the basis was discussed in our previous paper (Quintas-Sánchez et al. 2021). The convergence of the quenching cross-sections is checked automatically by the code as a function of the total angular momentum (J) for each collision energy. At the highest collisional energy, 1000 cm⁻¹, the maximum value of J needed to reach convergence was J = 99.

Finally, the state-to-state de-excitation rate coefficients (⁠|$k_{j_i \rightarrow j_f}$|⁠) are computed by the average of the rotational cross-sections (⁠|$\sigma _{j_i \rightarrow j_f}$|⁠) over a Maxwell–Boltzmann distribution at a given temperature T, as

$$\begin{eqnarray} k_{j_i \rightarrow j_f}(T)=\sqrt{\frac{8}{\pi \mu k_{\mathrm{B}}^{3} T^{3}}} \int _{0}^{\infty } E_{\mathrm{c}}\, \sigma _{j_i \rightarrow j_f}(E_{\mathrm{c}})\, e^{-\frac{E_{\mathrm{c}}}{k_{\mathrm{B}} T}}\, \mathrm{d} E_{\mathrm{c}}~, \end{eqnarray}$$

(4)

where j_i and j_f are the initial and final rotational states of HCS⁺, E_c is the collisional energy, and k_B is the Boltzmann constant.

3 RESULTS AND DISCUSSION

Rotational de-excitation cross-sections of HCS⁺ in collision with para- (solid lines) and ortho-H₂ (dashed lines) are shown in Fig. 3 from the initial rotational states j = 7 (upper panel), j = 6 (middle panel), and j = 5 (lower panel). Note that in the figure a logarithmic scale is used for both the x- and y-axes. The curves show typical shape- and Feshbach-resonances for van der Waals systems, which are associated with the formation of quasi-bound states due to the effective potential and the coupling between open and closed channels. As expected, all cross-sections decrease with increasing collision energy and decrease for larger increments of Δj. It should be noted that the calculations were carried out up to 1000 cm⁻¹ because, at collisional energies higher than the opening of the first excited vibrational level of the triatomic molecule, the bending–rotation coupling could affect the magnitude of the cross-sections, as was observed for the C₃ +H₂ collision system (Stoecklin et al. 2015). Future work in this direction would be valuable.

Figure 3.

Rotational de-excitation cross-sections of HCS⁺ in collision with para- (solid lines) and ortho-H₂ (dashed lines) from j_i = 7 (upper panel), j_i = 6 (middle panel), and j_i = 5 (lower panel). Rotational transitions of HCS⁺ are labelled as j_i → j_f.

As can be seen in Fig. 3, the similarities in the de-excitation cross-sections by collision with para-H₂ and ortho-H₂ observed for j < 4 in our preliminary study (Quintas-Sánchez et al. 2021) remain for larger values of j. Comparing the para-H₂ and ortho-H₂ cross-sections, the results converge to be quite similar at higher collision energies (especially above 100 cm⁻¹), while pronounced differences are observed at lower energies. It is also worth noticing that for most transitions, the cross-sections for collision with ortho-H₂ are appreciably larger – noticeable even on the log-scale plot – for energies below 50 cm⁻¹. At 10 K, the ratio between the rates for para- and ortho-H₂, including all de-excitation transitions from the initial rotational states j = 1 to j = 7, varies from 1.1 to 1.6 (with an average value of 1.2), while at 100 K, this value ranges from 1.0 to 1.2 (with an average value of 1.09). Such general behaviour has been also found in several studies of other ions in collisions with H₂; e.g., HCO⁺ (Denis-Alpizar et al. 2020), DCO⁺ (Denis-Alpizar et al. 2020), N₂H⁺ (Balança et al. 2020), SH⁺ (Dagdigian 2019), CF⁺ (Desrousseaux et al. 2019), C₃N⁻ (Lara-Moreno, Stoecklin & Halvick 2019), C₆H⁻ (Walker et al. 2017), and CN⁻ (Kłos & Lique 2011). This was discussed in detail by Lara-Moreno et al. (2019) for the C₃N⁻  +  H₂ system, and the similarities were attributed to the features of the short-range interaction. This interaction appears to be dominated by the attractive isotropic term of the PES, A|$_{00}^0$|⁠, which gives non-zero contributions in the coupling matrix element [see e.g. Eq. 9 from Green (1975)] for both para-H₂ (j = 0) and ortho-H₂ (j = 1). Henceforth, we will focus only on the rate coefficients of HCS⁺ in collisions with para-H₂.

Here, we address two assumptions often used as approximations to estimate rate coefficients for collisions with H₂ when no other – more accurate – theoretical data is available, namely: (i) the use of a mass scaling factor applied to data obtained for collisions with He; and (ii) to substitute the species of interest (in this case, HCS⁺) by a ‘similar’ molecule for which such data is previously known (in this particular case, its isovalent ion HCO⁺).

Table 1 shows the rate coefficients computed in this work – at several temperatures, for different j → j′ transitions – and those reported for the collision with He (Dubernet et al. 2015). The use of a mass scaling factor for estimating the rate coefficients for the collision with para-H₂ from those with He is a very common approximation in the absence of rates with H₂. However, as shown in the table, for this system, the ratio of accurate data |$k_{j\rightarrow j^{\prime }}^{{\rm HCS}^{+}+{\rm H}_{2}}/k_{j\rightarrow j^{\prime }}^{{\rm HCS}^{+}+{\rm He}}$| varies from 1.9 to 6.9, while only a factor of 1.4 would be applied using this approximation (Schöier et al. 2005). The maximum differences are observed at very low temperatures; but even at 60 K, the average ratio is 2.6. Therefore, using the mass scaling factor is not a good approximation – even on average – for HCS⁺ + H₂ collisions. This is not too surprising as the long-range behaviour of the PES for the collision of an ion with He is dominated by the induction term (which has only an R⁻⁴ dependence), while for the case of collisions with H₂, the long-range interaction is dominated by the electrostatic charge–quadrupole term (which has an R⁻³ dependence). A full set of rate coefficients for the collision of HCS⁺ with H₂ can be found in the supplementary materials.

Table 1.

Rotational de-excitation rate coefficients (in cm³molecule⁻¹s⁻¹) of HCS⁺ in collision with H₂ at several temperatures for different j → j′ transitions. The rate coefficients for HCS⁺ + He collisions reported by (Dubernet et al. 2015), and the corresponding ratio (⁠|$k^{{\rm HCS}^{+}+{\rm H}_{2}}\, /\, k^{{\rm HCS}^{+}+{\rm He}}$|⁠, comparable to the approximate factor of 1.4) are also included. A full set of rotational rate coefficients of HCS⁺ by collisions with H₂ can be found in the supplementary materials.

		T = 10 K			T = 20 K			T = 60 K
j	j′	|$k^{HCS^+-H_2}$|	|$k^{HCS^+-He}$|	Ratio	|$k^{HCS^+-H_2}$|	|$k^{HCS^+-He}$|	Ratio	|$k^{HCS^+-H_2}$|	|$k^{HCS^+-He}$|	Ratio
1	0	1.07 × 10−10	4.89 × 10−11	2.2	1.02 × 10−10	5.12 × 10−11	2.0	1.10 × 10−10	5.71 × 10−11	1.9
2	0	5.43 × 10−11	2.22 × 10−11	2.4	5.38 × 10−11	2.45 × 10−11	2.2	5.95 × 10−11	2.94 × 10−11	2.0
2	1	2.18 × 10−10	8.15 × 10−11	2.7	1.93 × 10−10	8.15 × 10−11	2.4	1.82 × 10−10	8.70 × 10−11	2.1
3	0	3.27 × 10−11	8.19 × 10−12	4.0	2.83 × 10−11	9.00 × 10−12	3.1	2.53 × 10−11	1.05 × 10−11	2.4
3	1	1.14 × 10−10	3.99 × 10−11	2.9	1.10 × 10−10	4.28 × 10−11	2.6	1.14 × 10−10	4.89 × 10−11	2.3
3	2	2.14 × 10−10	7.96 × 10−11	2.7	2.05 × 10−10	8.27 × 10−11	2.5	2.02 × 10−10	9.08 × 10−11	2.2
4	0	2.34 × 10−11	3.38 × 10−12	6.9	2.18 × 10−11	3.85 × 10−12	5.7	2.01 × 10−11	4.92 × 10−12	4.1
4	1	7.64 × 10−11	1.59 × 10−11	4.8	6.52 × 10−11	1.75 × 10−11	3.7	5.53 × 10−11	1.97 × 10−11	2.8
4	2	1.54 × 10−10	5.46 × 10−11	2.8	1.47 × 10−10	5.69 × 10−11	2.6	1.47 × 10−10	6.01 × 10−11	2.4
4	3	2.13 × 10−10	8.04 × 10−11	2.6	2.06 × 10−10	8.45 × 10−11	2.4	2.06 × 10−10	9.22 × 10−11	2.2
5	0	1.64 × 10−11	2.78 × 10−12	5.9	1.39 × 10−11	2.96 × 10−12	4.7	1.10 × 10−11	3.33 × 10−12	3.3
5	1	5.45 × 10−11	8.12 × 10−12	6.7	5.20 × 10−11	9.31 × 10−12	5.6	4.64 × 10−11	1.06 × 10−11	4.4
5	2	8.44 × 10−11	1.89 × 10−11	4.5	7.39 × 10−11	2.10 × 10−11	3.5	6.52 × 10−11	2.28 × 10−11	2.9
5	3	1.68 × 10−10	6.43 × 10−11	2.6	1.66 × 10−10	6.63 × 10−11	2.5	1.62 × 10−10	6.81 × 10−11	2.4
5	4	2.17 × 10−10	8.59 × 10−11	2.5	2.14 × 10−10	9.03 × 10−11	2.4	2.12 × 10−10	9.51 × 10−11	2.2

The LAMDA data base (Schöier et al. 2005), given the lack of data for HCS⁺, takes the rates for HCS⁺ + H₂ to be the same as for HCO⁺ + H₂ collisions. However, this is not an accurate approximation, as Fig. 4 clearly illustrates. Furthermore, Fig. 4 shows that the rates decrease with increasing |Δj| and a typical |Δj| = 1 propensity for collision with non-homonuclear molecules such as HCS⁺ is also observed. This propensity was also observed in collisions with He, cf. Table 1.

Figure 4.

Rotational rate coefficients of HCS⁺ in collision with H₂ at 10 and 100 K from j = 10. The rates for HCO⁺+ H₂ are also included.

The astrophysical implications of the new set of rate coefficients obtained here can be explored by determining the critical densities (⁠|$\eta _{\rm c}^{(j)}$|⁠), i.e. the density of H₂ for which radiative and collisional de-excitation processes are equally important. For densities much larger than these critical values, the LTE model can be accurately applied. At a given temperature, the critical density can be determined as (Yazidi, Ben Abdallah & Lique 2014):

$$\begin{eqnarray} \eta_{\rm c}^{(j)}(T)=\frac{A_{j\rightarrow j-1}}{{\sum \limits _{j^{\prime }\lt j}k_{j\rightarrow j^{\prime }}}(T)}~, \end{eqnarray}$$

(5)

where |$k_{j\rightarrow j^{\prime }}$| is the rotational rate coefficient for the j → j′ transition, and A_{j → j − 1} is the Einstein coefficient for the de-excitation between the states j and j − 1.

Table 2 shows |$\eta _{\rm c}^{(j)}$| at 10 and 60 K for Δj = −1 transitions from the lower rotational states of HCS⁺. The critical density has a weak dependence on T (at least for low temperatures), as was previously noticed for the rates, cf. Fig. 4. The values of |$\eta _{\rm c}^{(j)}$| obtained using the rates for HCO⁺ + H₂, and those scaled from HCS⁺ + He collisions, are also included in Table 2. Both approximations are shown to be inaccurate. If the rates for HCO⁺ + H₂ are used, |$\eta _{\rm c}^{(j)}$| is underestimated by up to a factor of 2.3. If the rates employed are those scaled from HCS⁺ + He, |$\eta _{\rm c}^{(j)}$| is overestimated up to a factor of 2.2. This is because the scaled rates are too small and appear in the denominator of equation (5). Therefore, it is clearly preferable to employ accurate system-specific rates for non-LTE modelling of this system instead of those scaled from HCS⁺ + He collisions or calculated for the HCO⁺ + H₂ system.

Table 2.

Critical densities (in 10⁴ cm⁻³) for Δj = −1 transitions of the HCS⁺ molecule obtained using the rate coefficients computed in this work, at 10 and 60 K, are compared with those scaled from HCS⁺ + He collisions, and with those obtained for the HCO⁺ + H₂ system.

ji	jf	HCS++H2	HCS++He	HCO++H2
			(scaled)
		T = 10 K
1	0	1.08	1.69	0.47
2	1	4.08	7.64	1.98
3	2	11.12	22.45	4.79
4	3	21.15	45.66	9.70
5	4	36.47	78.17	17.88
6	5	58.85	83.25	28.99
		T = 60 K
1	0	1.05	1.45	0.59
2	1	4.60	6.81	2.54
3	2	11.76	19.08	6.41
4	3	23.05	39.82	12.20
5	4	39.66	70.38	21.32
6	5	62.44	113.86	34.05

4 SUMMARY AND CONCLUSIONS

Using a recently developed PES for the HCS⁺–H₂ van der Waals system, a quantum scattering study of the rotational relaxation of HCS⁺ ions by collisions with H₂ was performed at the rigorous close-coupling level. This work expands on our previous study of this system, reporting extensive results at low temperature. Similarities between the rotational de-excitation rate coefficients of HCS⁺ by para- and ortho-H₂ were found for higher collision energies (especially above 100 cm⁻¹), while pronounced differences were observed at very low energies. The astrophysical implications of the computed set of rate coefficients were explored by determining the critical densities. Furthermore, two common approximations to estimate the rate coefficients when specific data is lacking were evaluated: scaling the ones obtained from HCS⁺ in collision with He, and the use of the rates for HCO⁺ + H₂ as a proxy for HCS⁺ + H₂. Both approximations proved to be inaccurate for this case. Finally, a complete set of rate coefficients for the lowest 30 rotational states of HCS⁺ in collisions with H₂ is reported between 5 and 100 K. These data will help inform models and data bases relevant to ISM chemistry.

SUPPORTING INFORMATION

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

ACKNOWLEDGEMENTS

Support from the project CONICYT/FONDECYT/REGULAR/N^o1200732 is gratefully acknowledged. RD and EQ-S are supported by the U.S. Department of Energy (Award DE-SC0019740). Computing resources were supported by the National Science Foundation under Grant No. OAC-1919789.

DATA AVAILABILITY

The data underlying this article are available in the article and in its online supplementary material.