Issue 
A&A
Volume 638, June 2020



Article Number  A72  
Number of page(s)  5  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/202038041  
Published online  12 June 2020 
Hyperfine excitation of SH^{+} by H
^{1}
LOMC – UMR 6294, CNRSUniversité du Havre, 25 rue Philippe Lebon, BP 1123, 76 063 Le Havre cedex, France
email: francois.lique@univlehavre.fr
^{2}
Instituto de Física Fundamental, CSIC, C/Serrano, 123, 28006 Madrid, Spain
^{3}
Departamento de Quimica Fisica, Facultad de Ciencias Quimicas, Universidad de Salamanca, 37008 Salamanca, Spain
^{4}
Firat University, Department of Physics, 23169 Elazi, Turkey
Received:
27
March
2020
Accepted:
24
April
2020
Context. SH^{+} is a surprisingly widespread molecular ion in diffuse interstellar clouds. There, it plays an important role by triggering the sulfur chemistry. In addition, SH^{+} emission lines have been detected at the UVilluminated edges of dense molecular clouds, socalled photodissociation regions (PDRs), and toward highmass protostars. An accurate determination of the SH^{+} abundance and of the physical conditions prevailing in these energetic environments relies on knowing the rate coefficients of inelastic collisions between SH^{+} molecules and hydrogen atoms, hydrogen molecules, and electrons.
Aims. We derive SH^{+}–H fine and hyperfineresolved rate coefficients from recent quantum calculations for the SH^{+}–H collisions, including inelastic, exchange, and reactive processes.
Methods. The method we used is based on the infiniteorder sudden approach.
Results. Statetostate rate coefficients between the first 31 fine levels and 61 hyperfine levels of SH^{+} were obtained for temperatures ranging from 10 to 1000 K. Finestructure resolved rate coefficients present a strong propensity rule in favor of Δj = ΔN transitions. The Δj = ΔF propensity rule is observed for the hyperfine transitions.
Conclusions. The new rate coefficients will help significantly in the interpretation of SH^{+} spectra from PDRs and UVirradiated shocks where the abundance of hydrogen atoms with respect to hydrogen molecules can be significant.
Key words: molecular data / molecular processes / radiative transfer / methods: laboratory: molecular
© F. Lique et al. 2020
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Submillimeter absorption lines from the groundrotational state of SH^{+} were first detected toward the W3 IRS5 highmass starforming region with Herschel/HIFI (Benz et al. 2010). In parallel, and using the APEX telescope, Menten et al. (2011) detected rotational absorption lines produced by SH^{+} in the lowdensity (n_{H} ≲ 100 cm^{−3}) diffuse clouds in the line of sight toward the strong continuum source SgrB2(M) in the Galactic center. Despite the very endothermic formation route of this hydride ion (for a review, see Gerin et al. 2016), subsequent absorption measurements of multiple lines of sight with Herschel demonstrated the ubiquitous presence of SH^{+} in diffuse interstellar clouds (Godard et al. 2012).
SH^{+} rotational lines have also been detected in emission toward the Orion Bar photodissociation region (PDR; Nagy et al. 2013), a strongly UVirradiated surface of the Orion molecular cloud (Goicoechea et al. 2016). In warm and dense PDRs (n_{H} ≳ 10^{5} cm^{−3}) such as the Orion Bar, SH^{+} forms by exothermic reactions of S^{+} with vibrationally excited H_{2}(with v ≥ 2, see details in Agúndez et al. 2010, Zanchet et al. 2013, 2019). High angular resolution images taken with ALMA show that SH^{+} arises from a narrow layer at the edge of the PDR, the photodissociation front that separates atomic from molecular gas (Goicoechea et al. 2017). In these PDR layers, the abundance of hydrogen atoms is comparable to that of hydrogen molecules, which are continuously being photodissociated. Both H and H_{2}, together with electrons (arising from the ionization of carbon atoms, see e.g. Cuadrado et al. 2019), drive the collisional excitation of molecular rotational levels and atomic finestructure levels.
In addition to PDRs, the SH^{+} line emission observed toward massive protostars likely arises from the UVirradiated cavities of their molecular outflows (Benz et al. 2010, 2016). In these UVirradiated shocks, the density of hydrogen atoms can be high as well. All in all, the molecular abundances and physical conditions in these regions, where atomic and molecular hydrogen can have comparable abundances, are not well understood.
In the interstellar medium (ISM), molecular abundances are derived from excitation line modeling. Assuming local thermodynamic equilibrium (LTE) conditions in the interstellar media with low densities is generally not a good approximation, as discussed by Roueff & Lique (2013). Hence, the population of molecular levels is driven by the competition between collisional and radiative processes. It is then essential to determine accurate collisional data between the involved molecules and the most abundant interstellar species, which are usually electrons and atomic and molecular hydrogen, in order to obtain reliably modeled spectra.
The computation of collisional data for the SH^{+} started recently. First, Rmatrix calculations combined with the adiabaticnucleus rotation and CoulombBorn approximations was used to compute electronimpact rotational rate coefficients, and hyperfineresolved rate coefficients were deduced using the infiniteorder sudden approximation (Hamilton et al. 2018). Then, timeindependent closecoupling quantumscattering calculations were employed by Dagdigian (2019) to compute hyperfineresolved rate coefficients for (de)excitation of SH^{+} in collisions with both para and orthoH_{2}.
Collisional data with atomic hydrogen are much more challenging to compute because of the possible reactive nature of the SH^{+}–H collisional system. However, we have recently overcome this difficult problem and presented quantum mechanical calculations of cross sections and rate coefficients for the rotational excitation of SH^{+} by H, including the reactive channels (Zanchet et al. 2019), using new accurate potential energy surfaces.
Unfortunately, it was not possible to include the fine and hyperfine structure of the SH^{+}(^{3}Σ^{−}) molecule in the quantum dynamical calculations, but they are resolved in astronomical observations. This made the new set of data difficult to use in astrophysical applications.
The aim of this work is to use the quantum statetostate rate coefficients for the SH^{+}(^{3}Σ^{−})–H inelastic collisions to generate a new set of fine and hyperfine resolved data that can be used in radiative transfer models. The paper is organized as follows: Sect. 2 provides a brief description of the theoretical approach. In Sect. 3 we present the results. Concluding remarks are presented in Sect. 4.
2. Computational method
2.1. Potential energy surfaces
The collisions between SH^{+}(X^{3}Σ^{−}) and H(^{2}S) can take place on two different potential energy surfaces (PESs), the ground quartet (^{4}A″) and doublet (^{2}A″) electronic states of the H_{2}S^{+} system. In this work, we use the H_{2}S^{+} quartet and doublet PESs that were previously generated by Zanchet et al. (2019).
Briefly, the stateaverage complete active space (SACASSCF) method (Werner & Knowles 1985) was employed to calculate the first ^{4}A″ together with the two first ^{2}A′ and the three first ^{2}A″ electronic states. The obtained stateaverage orbitals and multireference configurations were then used to calculate both the lowest ^{4}A″ and ^{2}A″ states energies with the internally contracted multireference configuration interaction method (icMRCI; Werner & Knowles 1988) and the Davidson correction (Davidson 1975). For sulfur and hydrogen atoms, the augmented correlationconsistent quintuple zeta (augccpV5Z) basis sets were used, and all calculations were made using the MOLPRO suite of programs (Werner et al. 2012). These energies were then fit using the GFIT3C procedure (Aguado & Paniagua 1992; Aguado et al. 1998).
The two PESs exhibit completely different topographies. The ^{4}A″ electronic state does not present any minimum out of the van der Waals wells in the asymptotic channels and does not present any barrier to the SH^{+} + H → H_{2} + S^{+} reaction. This reaction is exothermic on this surface, and reactive collisions are likely to occur in competition with the inelastic collisions.
On the other hand, the ^{2}A″ state presents a deep insertion HSH well and does not present any barrier either. For this state, in contrast with the previous case, the SH^{+} + H → H_{2} + S^{+} reaction is endothermic, and only inelastic collisions can occur (pure or involving H exchange).
2.2. Timeindependent and wave packet calculations
During a collision between SH^{+} and H, three processes compete: the inelastic (1), reactive (2), and exchange (3) processes:
where v and N designate the vibrational and rotational levels, respectively, of the molecule (SH^{+} or H_{2} when the reaction occurs). Only collisions with SH^{+} molecules in their groundvibrational state v = 0 are considered in this work. Therefore, the vibrational quantum number v is omitted hereafter.
The spinorbit couplings between the different H_{2}S^{+} were ignored and the collisions on the ground quartet and doublet electronic states were studied separately. Because of their different topography, the dynamical calculations were treated differently on the two PESs.
The reaction dynamics on the ^{4}A″ state was studied with a timeindependent treatment based on hyperspherical coordinates. On this PES, the SH^{+} + H → H_{2} + S^{+} collision is a barrierless and exothermic reaction for which it has been shown (Zanchet et al. 2019) that the reactivity is strong (k > 10^{−10} cm^{3} s^{−1}), even at low temperatures. The competition between all the three processes (inelastic, exchange, and reactivity) is therefore rigorously taken into account. We used the ABC reactive code of Skouteris et al. (2000) to carry out closecoupling calculations of the reactive, inelastic, and exchange cross sections. The cross sections were obtained following the approach described in Tao & Alexander (2007), which was recently used to study the rotational excitation of HeH^{+} (Desrousseaux & Lique 2020) by H.
We computed the cross sections for the first 13 rotational levels of the SH^{+} molecule (0 < N < 12) for a collisional energy ranging from 0 to 5000 cm^{−1} and for all values of the total angular momentum J leading to a nonzero contribution in the cross sections. More details about the scattering calculations can be found in Zanchet et al. (2019).
The ground doublet (^{2}A″) electronic state exhibits a deep well. Therefore timeindependent treatment cannot be used, and the dynamics was studied from a quantum wavepacket method using the MADWAVE3 program of Zanchet et al. (2009). In this electronic state, the reactive channels are largely endothermic and are not open at the collisional energies considered in this work.
The inelastic and exchange cross sections in the ^{2}A″ state were calculated with the usual partial wave expansion as
where J is the total angular momentum quantum number, and Ω, Ω′ are the projections of the total angular momentum on the reactant and product bodyfixed zaxis, respectively. α = I, α′=I, or E denotes the arrangement channels, inelastic or exchange. k^{2} = 2μ_{r}E_{k}/ℏ^{2} is the square of the wave vector for a collision energy E_{k}, and are the transition probabilities, that is, the square of the corresponding Smatrix elements. We computed the cross sections from the N = 0 rotational states to the first 13 rotational levels of the SH^{+} molecule (0 < N′< 12) for a collisional energy ranging from 0 to 5000 cm^{−1}. Because of the high computational cost of these simulations, they were only performed for J = 0, 10, 15, 20, 25, 30, 40, 50, …, 110, while for intermediate J values, they were interpolated using a uniform Jshifting approximation as recently used for the OH^{+}–H and CH^{+}–H collisional systems (Werfelli et al. 2015; Bulut et al. 2015). The convergence analysis and the parameters used in the propagation for each of the two PESs are described in detail in Zanchet et al. (2019).
For the two sets of calculations the corresponding cross sections are summed for the doublet and quartet states independently because the two rearrangement channels, inelastic and exchange, yield the same products. Finally, the cross sections for each of the two electronic states of quartet (with an electronic spin S = 3/2) and doublet (with an electronic spin S = 1/2) spin multiplicity were summed with the proper degeneracy factor to give the total collision cross sections as
As shown in Zanchet et al. (2019), the magnitude of the excitation cross sections obtained on the doublet states is stronger than that on the quartet states because of the nonreactive character of the collision and of the deep well, which both favor inelastic collisions.
From the total collision cross sections σ_{N → N′}(E_{k}), we can obtain the corresponding thermal rate coefficients at temperature T by an average over the collision energy (E_{k}),
where k_{B} is Boltzmann’s constant and μ is the reduced mass. The crosssection calculations carried out up to a kinetic energy of 5000 cm^{−1} allowed computing rate coefficients for temperatures ranging from 10 to 1000 K.
In all these calculations, the spinrotation couplings of SH^{+} were not included, and therefore the present set of rate coefficients cannot be directly used to model interstellar SH^{+} spectra where the fine and hyperfine structure is resolved.
2.3. Infiniteorder sudden calculations
In this section, we describe how the statetostate fine and hyperfine rate coefficients for the SH^{+}–H collisional system were computed using infiniteorder sudden (IOS) methods (Goldflam et al. 1977; Faure & Lique 2012) using the above k_{0 → N′}(T) rate coefficients as “fundamental” rate coefficients (those out of the lowest level)
For SH^{+} in its ground electronic ^{3}Σ^{−} state, the molecular energy levels can be described in the Hund case (b) limit^{1}. The finestructure levels are labeled by Nj, where j is the total molecular angular momentum quantum number defined by j = N + S. S is the electronic spin. For molecules in a ^{3}Σ^{−} state, S = 1. This means that three kinds of levels (j = N − 1, j = N and j = N + 1) exist, except for the N = 0 rotational level, which is a single level.
The hydrogen atom also possesses a nonzero nuclear spin (I = 1/2). The coupling between I and j results in a splitting of each level into two hyperfine levels (except for the N = 1, j = 0 level, which remains only one level). Each hyperfine level is designated by a quantum number F (F = I + j) that varies between I − j and I + j.
Using the IOS approximation, we obtain the rate coefficients among the fine structure levels from the k_{0 → L}(T) fundamental rate coefficients with the following formula [e.g., ][]corey83:
where the parentheses and braces are the 3j and 6j symbols. In the usual IOS approach, k_{0 → L}(T) is calculated for each collision angle. Here, however, we used the k_{0 → L}(T) rate coefficients of Eq. (6), which are obtained with a more accurate quantum method.
The hyperfine resolved rate coefficients can also be obtained from the fundamental rate coefficients following Faure & Lique (2012),
In addition, we note that the fundamental excitation rates k_{0 → L}(T) were in practice replaced by the deexcitation fundamental rates using the detailed balance relation,
where
ε_{L} is the energies of the rotational levels L. This procedure was indeed found to significantly improve the results at low temperatures because of the strong threshold effects.
The fine and hyperfine splittings of the rotational states are of a few cm^{−1} and of a few 0.001 cm^{−1}, respectively, and can be neglected compared to the collision energy at T > 30 − 50 K so that our approach is expected to be reasonably accurate for the entire temperature range considered in this work. Lique et al. (2016) have investigated the accuracy of the IOS approach in the case of OH^{+}–H collisions. It was shown to be reasonably accurate (within a factor of 2), even at low temperature, so that we can anticipate a similar accuracy for our collisional system. In addition, we note that with the present approach, some fine and hyperfine rate coefficients are strictly zero. This selection rule is explained by the 3j and 6j Wigner symbols that vanish for some transitions. In a more accurate approach, these rate coefficients will not be strictly zero, but will generally be weaker than the other rate coefficients.
3. Results
Using the computational method described above, we generated fine and hyperfine resolved rate coefficients for the SH^{+}–H collisional system using the doublet and quartet pure rotational rate coefficients in order to provide the astrophysical community with the first set of data for the SH^{+}–H collisional system. In all the calculations, we considered all the SH^{+} energy levels with N, N′≤10, and we included in the calculations all fundamental rate coefficients with L ≤ 12. The complete set of (de)excitation rate coefficients is available online from the LAMDA (Schöier et al. 2005) and BASECOL (Dubernet et al. 2013) websites.
3.1. Fine and hyperfinestructure excitation
The thermal dependence of the finestructureresolved statetostate SH^{+}–H rate coefficients is illustrated in Fig. 1 for selected N = 2, j → N′ = 1, j′ transitions.
Fig. 1. Temperature variation of the fine structure resolved deexcitation rate coefficients for the SH^{+} molecule in collision with H for selected N = 2, j → N′ = 1, j′ transitions. 
The temperature variation of the deexcitation rate coefficients is relatively smooth, except at low temperature (T < 50 K), where they increase rapidly. The weak temperature dependence of the rate coefficients (except at low temperature) could have been anticipated on the basis of the Langevin theory for ion–neutral interactions.
A strong propensity rule exists for Δj = ΔN transitions. This Δj = ΔN propensity rule was predicted theoretically (Alexander & Dagdigian 1983) and is general for molecules in the ^{3}Σ^{−} electronic state. It was also observed previously for the O_{2}(X^{3}Σ^{−})He (Lique 2010), NH(X^{3}Σ^{−})–He (Toboła et al. 2011) or OH^{+}–H (Lique et al. 2016) collisions.
Figure 2 presents the temperature variation of the hyperfinestructureresolved statetostate SH^{+}–H rate coefficients for selected N = 2, j = 3, F = 3.5 → N′ = 1, j′,F′ transitions.
Fig. 2. Temperature variation of the hyperfine structure resolved deexcitation rate coefficients for the SH^{+} molecule in collision with H for the N = 2, j = 3, F = 3.5 → N′ = 1, j′,F′ transitions. 
For Δj = ΔN transitions, we have a strong propensity rule in favor of Δj = ΔF hyperfine transitions. This trend is the usual trend for openshell molecules (Alexander 1985; Dumouchel et al. 2012; Kalugina et al. 2012; Lique et al. 2016). For Δj ≠ ΔN transitions, it is much more difficult to find a clear propensity rule. The final distribution seems to be governed by two rules: the rate coefficients show propensity in favor of Δj = ΔF transitions, but are also proportional to the degeneracy (2F′+1) of the final hyperfine level, as was found for the CN–paraH_{2} system (Kalugina et al. 2012).
3.2. Comparison with SH^{+}–H_{2} rate coefficients
We then compared the new SH^{+}–H rate coefficients with those reported recently for the hperfine excitation of SH^{+} by H_{2} (Dagdigian 2019). The SH^{+} molecule has been observed in media where atomic and molecular hydrogen are significant colliding partners, and this comparison should allow evaluating the effect of the different collisional partners. In Fig. 3 we compare the SH^{+}–H and SH^{+}–H_{2} (both para and orthoH_{2}) rate coefficients for a selected number of transitions.
Fig. 3. Comparison between SH^{+}–H and SH^{+}–H_{2} (both para and orthoH_{2}) rate coefficients for a selected number of hyperfine (N = 2, j = 3, F = 3.5 → N′ = 1, j′,F′) transitions. 
In astrophysical applications, when collisional data are not available, it is very common to derive collisional data from collisional rate coefficients calculated for the same molecule in collision with another colliding partner. This approach (Lique et al. 2008) consists of assuming that the excitation crosssections are similar for the two colliding systems and that the rate coefficients differ only by a scaling factor due to the reduced mass that appears in Eq. (6). Hence, the following scaling relationship can be used:
At low temperatures, the rate coefficients for collisions with H do not have the highest magnitude, as expected from the scaling relationships. They can even be one order of magnitude weaker. We also note that the differences between the H and H_{2} rate coefficients depend on the transitions and on the temperature, which makes it impossible to extrapolate accurate H collisional data from H_{2} collisional data, or vice versa. This therefore confirms that it is unrealistic to estimate unknown collisional rate coefficients by simply applying a scaling factor to existing rate coefficients. This result has previously been observed for water (Daniel et al. 2015) and ammonia (Bouhafs et al. 2017). However, when the temperature increases, the agreement improves, and scaling techniques would lead to a reasonable estimate of the H or H_{2} rate coefficients for temperatures above 500 K.
4. Summary and conclusion
The fine and hyperfine excitation of SH^{+} by H were investigated. We obtained fine and hyperfine resolved rate coefficients for transitions involving the lowest levels of SH^{+} for temperatures ranging from 10 to 1000 K. Finestructureresolved rate coefficients present strong propensity rules in favor of the Δj = ΔN transition. The Δj = ΔF propensity rule is observed for the hyperfine transitions.
As a molecule that can be observed from groundbased observatories (Müller et al. 2014) in the Milky Way and beyond (Muller et al. 2017), we expect that these new data will significantly help in the accurate interpretation of SH^{+} rotational emission spectra from dense PDRs and massive protostars, enable this molecular ion to act as tracer of the energetics of these regions, and of the first steps of sulfur chemistry.
For ^{3}Σ^{−} electronic groundstate molecules, the energy levels are usually described in the intermediate coupling scheme (Gordy & Cook 1984; Lique et al. 2005). However, the use of IOS scattering approach implies that the Hund case (b) limit should be used.
Acknowledgments
We acknowledge the FrenchSpanish collaborative project PICS (Ref. PIC2017FR7). F. L. acknowledges financial support from the European Research Council (Consolidator Grant COLLEXISM, Grant agreement 811363), the Institut Universitaire de France and the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP cofunded by CEA and CNES. The research leading to these results has received funding from MICIU under grants No. FIS201783473C2 and AYA201785111P. N.B. acknowledges the computing facilities by TUBITAKTRUBA. This work was performed using HPC resources from GENCICINES (Grant A0070411036).
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All Figures
Fig. 1. Temperature variation of the fine structure resolved deexcitation rate coefficients for the SH^{+} molecule in collision with H for selected N = 2, j → N′ = 1, j′ transitions. 

In the text 
Fig. 2. Temperature variation of the hyperfine structure resolved deexcitation rate coefficients for the SH^{+} molecule in collision with H for the N = 2, j = 3, F = 3.5 → N′ = 1, j′,F′ transitions. 

In the text 
Fig. 3. Comparison between SH^{+}–H and SH^{+}–H_{2} (both para and orthoH_{2}) rate coefficients for a selected number of hyperfine (N = 2, j = 3, F = 3.5 → N′ = 1, j′,F′) transitions. 

In the text 
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