Electron-impact excitation of diatomic hydride cations II: OH$^+$ and SH$^+$

R-matrix calculations combined with the adiabatic-nuclei-rotation and Coulomb-Born approximations are used to compute electron-impact rotational rate coefficients for two open-shell diatomic cations of astrophysical interest: the hydoxyl and sulphanyl ions, OH$^+$ and SH$^+$. Hyperfine resolved rate coefficients are deduced using the infinite-order-sudden approximation. The propensity rule $\Delta F=\Delta j=\Delta N=\pm 1$ is observed, as is expected for cations with a large dipole moment. A model for OH$^+$ excitation in the Orion Bar photon-dominated region (PDR) is presented which nicely reproduces Herschel observations for an electron fraction $x_e=10^{-4}$ and an OH$^+$ column density of $3\times 10^{13}$~cm$^{-2}$. Electron impact electronic excitation cross sections and rate coefficients for the ions are also presented.


INTRODUCTION
Cross sections for electron collisions with molecular ions can be very large (>1000Å 2 ). If the ion in question contains a permanent dipole moment, the electron-impact rotational excitation rate coefficients far exceed those of H and H2 meaning that in comparatively electron-rich regions, electron collisions can become the dominant excitation process. Rotational rate coefficients have already been used to quantify interstellar electron densities (Jimenez-Serra et al. 2006;Harrison et al. 2013;Hamilton et al. 2016), but the rate coefficients for many key species remain unknown. In this paper we consider (de)excitation of the hydoxyl and sulphanyl ions: OH + and SH + , respectively. The species both have electronic ground states of 3 Σ − symmetry which adds an extra complication as the rotational levels display fine structure due to the electron spin of the two unpaired electrons and hyperfine structure due to the nuclear spin of the hydrogen atom.
Both OH + and SH + were only detected in the interstellar medium within the last decade; OH + being first observed by Wyrowski et al. (2010) and SH + by Benz & et al (2010) and Menten et al. (2011). However, the ions are now known to be widespread (Gerin et al. 2016). In particular ⋆ E-mail: james.hamilton@ucl.ac.uk † E-mail: alexandre.faure@univ-grenoble-alpes.fr ‡ E-mail: j.tennyson@ucl.ac.uk OH + has now been found in a variety of locations including translucent interstellar clouds (Kre lowski et al. 2010;Gupta & et al. 2010) and both OH + and SH + have been recently observed in absorption across the z = 0.89 molecular absorber towards PKS 1830-211 (Muller et al. 2016(Muller et al. , 2017. They have been also detected in emission in dense photon-dominated regions where electron collision processes are thought to be important (van der Tak et al. 2013a;Nagy et al. 2013). A number of these observations resolve the fine (and sometimes hyperfine) structure in the transitions (Benz & et al 2010;Gerin & et al. 2010;Godard et al. 2012;Nagy et al. 2013).
To date there is only one laboratory measurement of electron-impact rotational rate coefficients for a molecular ion was by Shafir et al. (2009) for HD + ; this experiment actually measured de-excitation and only gave enough information to show agreement with the theoretical predictions. This means that thus far astronomically important electronimpact rotational rate coefficients for molecular ions have all been computed (Faure & Tennyson 2001, 2003. In a recent paper (Hamilton et al. 2016), we used improved theory to compute rotational rate coefficients for three closed shell hydride cations, ArH + , CH + and HeH + ; these hydrides were chosen due to their significant role in the interstellar medium (ISM), see Faure et al. (2017) for example. In this work, electron-impact rate coefficients are calculated for the open-shell ions OH + and SH + . R-matrix calculations are combined with the adiabatic-nuclei-rotation (ANR) approx-imation to produce rotational cross sections at electron energies below 5 eV. We also present electron impact electronic excitation cross sections for the two ions considered. While these are unlikely to be important for models of interstellar medium, OH + can be found in planetary ionospheres (Fox et al. 2015), and cometary coma (Nordholt et al. 2003;Haider & Bhardwaj 2005;Rubin et al. 2009), as well as around Enceladus (Gupta & et al. 2010). In these environments electron impact electronic excitation may well be important.
Section 2 describes the R-matrix calculations and the procedure used to derive the cross-sections and rate coefficients is briefly introduced. In Section 3, we present and discuss the calculated rate coefficients. A model for the excitation of OH + in the Orion bar photon-dominated region (PDR) is also presented in Section 4. Conclusions are summarized in Section 5.

R-MATRIX CALCULATIONS
Inelastic electron collision calculations with molecular ions OH + and SH + were performed using the R-matrix method (Tennyson 2010) within the Quantemol-N (Tennyson et al. 2007) expert system to run the UK molecular Rmatrix codes (UKRMol) (Carr et al. 2012). Details follow closely the calculations performed in our previous paper (Hamilton et al. 2016), denoted I below, and are not repeated here. The calculations produce T-matrices which are processed by electron-impact rotational excitation code ROTIONS (Rabadán & Tennyson 1998), which employs the Coulomb-Born approximation to include the effects of high partial waves (Norcross & Padial 1982). In particular, ∆N = 1 transitions (N is the molecular ion rotational angular momentum) are strongly influenced by the long-range dipole moment and ROTIONS uses the Coulomb-Born approximation to include the contributions of partial waves with ℓ > 4. These long-range effects are unimportant for transitions with ∆N > 1 (Faure & Tennyson 2001). Experimental values of the dipole moments were used in these calculations where available.

OH +
The OH + target was represented using an augmented augcc-pVTZ GTO basis set. The use of augmented basis sets improves the treatment of the more diffuse orbitals for the excited states in the calculation. The ground state of OH + is X 3 Σ − which has the configuration [1 σ 2 σ 3 σ] 6 [1 π] 2 . The target was represented using CAS-CI treatment freezing the lowest energy 1 σ 2 orbital and placing the highest 6 electrons in orbitals [2-8 σ, 1-3 π] 6 . This target was constructed in an R-matrix sphere of radius 13 a0. Nine electronically excited states were used in the close-coupling expansion.
The vertical excitation energies (VEEs) of the excited states of OH + calculated using this model at an equilibrium bondlength of 1.0289Å are given in Table 1, where the VEEs are compared to measured values. The VEEs calculated in this work compare well to the measured adiabatic excitation energies (AEEs). VEEs naturally exceed AEEs and in this particular case the A 3 Π has a much larger equilibrium bondlength (1.134Å) than the b 1 Σ + state (1.032Å), which  Bekooy et al. (1985) c Ultraviolet spectroscopy Merer et al. (1975) results in a different order of the states at R = 1.029Å. The excited states a 1 ∆, b 1 Σ + and A 3 Π are within the electron energy range of interest in this investigation. Calculated equilibrium geometry dipole moment and rotational constant of OH + are compared to the best available values in Table 2. Isotopic substitution shifts the centre-of-mass and hence, for ionic system, alters the permanent dipole moment. Oxygen exists in three isotopes giving 16 OH + , 17 OH + and 18 OH + . While 16 O is the most abundant isotope, the abundance of 18 O is not negligible with an isotopic ratio 16 O/ 18 O=498.7±0.1 for the Solar System (Vienna Standard Mean Ocean Water value) (Asplund et al. 2009;Meija et al. 2016). The abundance of 17 O is much lower with an isotopic ratio 16 O/ 17 O=2632±7 (Asplund et al. 2009;Meija et al. 2016). To our knowledge, only the main isotopologue 16 OH + has been detected in the interstellar medium so far. For this reason the discussion and results presented in the main paper will be concerned with only 16 OH + (henceforth referred to as OH + ) but data for the other isotopologues are also included in the supplementary data to this article.

SH +
The SH + target was represented using a non augmented Dunning cc-pVTZ GTO basis set. Unlike OH + , an augmented basis set could not be used as it gave linear dependence problems and did not produce smooth results. The ground state of SH + has the configuration [1σ 2σ 3σ 1π 4σ 5σ] 14 [2π] 2 . The target was represented us- Table 3. Vertical excitation energies for the lowest 7 excited states of SH + compared with measured values.

State
This Work (eV) Previous (eV) Dunlavey et al. (1979) b Calculated, Bruna et al. (1983) c Observed, Rostas et al. (1984) d Observed, Horani et al. (1967) Senekowitsch et al. (1985) b Empirical, Müller et al. (2014) ing CAS-CI treatment freezing electrons of the lowest energy 1-3 σ and 1π orbitals and placing the highest 6 electrons in orbitals [4-8 σ, 2-4 π, 1 δ] 6 . This target was constructed in an R-matrix sphere of radius 10 a0. The VEEs of the excited states of SH + calculated from this model at the equilibrium bondlength of 1.3744Å are given in Table 3 and compared to published values. The VEEs calculated in this work compare well to the measured VEEs. The calculated equilibrium geometry dipole moment and rotational constant of SH + are compared to the best available values in Table 4. Sulphur exists as four isotopes giving 32 SH + , 33 SH + , 34 SH + and 36 SH + . While 32 S is the most abundant isotope, the abundance of 34 S is significant with an isotopic ratio 32 S/ 34 S∼22 for the Solar System (Asplund et al. 2009;Meija et al. 2016). The abundances of the other isotopes are much lower with isotopic ratios 32 S/ 33 S∼125 and 32 S/ 36 S 5000 (Asplund et al. 2009;Meija et al. 2016). Both isotopologues 32 SH + and 34 SH + have been detected in the (extragalactic for 34 SH + ) interstellar medium (Muller et al. 2017). The discussion and results presented in the main paper will be concerned with only 32 SH + (henceforth referred to as SH + ) but data for the other isotopologues are also included in the supplementary data to this article.

Cross-sections and rate coefficients
Working in C2v symmetry, each of the above calculations produces eight fixed-nuclei T-matrices for each molecule: the four symmetries A1, A2, B1, B2 for both doublet and quartet states of the N + 1 electron systems. These T-matrices are used to calculate the electronic excitation cross sections using standard equations (Tennyson 2010) and, once converted to the C∞v point group, rotational excitation cross sections using the program ROTIONS (Rabadán & Tennyson 1998) using the rotational constants and isotope specific dipole moments given in Tables 2 and 4. ROTIONS computes the rotational excitation cross sections for each doublet and quartet state independently. The total rotational cross sections are thus obtained as the (weighted) sum of the doublet and quartet cross sections.

Electronic transitions
Electronic excitation cross sections were computed for collision energies E coll in the range 0.01-5 eV. We consider electronic transitions from the ground state of each cation to all states with electronic thresholds below the 5 eV upper limit. The electronic thresholds are calculated using the fixed nuclei approximation. Assuming that the electron velocity distribution is Maxwellian, rate coefficients for excitation transitions were obtained for temperatures in the range 1 -5000 K.

Rotational transitions
As in I, we use a combination of the adiabatic nuclear rotation (ANR) method (Chang & Temkin 1970) with Coulomb-Born completion (for dipolar transitions only). To allow for threshold effect we used an empirical correction: below the excitation threshold cross sections are set to zero, see Faure et al. (2006) for details. The validity of this approach was confirmed recently for HeH + where the fullrovibrational multichannel quantum defect theory (MQDT) calculations byČurík & Greene (2017) were found in good agreement with the ANR/Coulomb-Born calculations of I.
Rotational transitions between levels with N 11 were considered. However, transitions were restricted to ∆N 8 owing to the finite number of partial waves in the T-matrices (ℓ 4). Rotational excitation cross sections were computed for collision energies E coll in the range 0.01-5 eV. For transitions with a rotational threshold below 0.01 eV, cross sections were extrapolated down to the threshold using a 1/E coll (Wigner's) law, as recommended by Faure et al. (2006). Rate coefficients for excitation transitions were obtained for temperatures in the range 1 -3000 K assuming a thermal electron energy distribution. The principle of detailed balance was used to compute de-excitation rate coefficients.

Hyperfine transitions
As discussed in the introduction, the fine and hyperfine structures of the OH + and SH + ions are resolved in astronomical observations. It is therefore necessary to provide hyperfine-resolved rate coefficients for these two ions. In the Hund's case (b) coupling scheme, the fine structure levels are labelled by (N, j) where j = N + S is the total angular momentum quantum number and S = 1 is the electronic spin. The hyperfine structure levels are labelled by (N, j, F ) where F = j + I is the hyperfine quantum number and I = 1/2 is the nuclear spin of the hydrogen atom. Each rotational level is thus split into 3 fine-structure levels (j = N − 1, j = N, j = N + 1) (except N = 0) and each fine-structure level is in turn split into 2 hyperfine levels (F = j ± 1/2) (except (N, j) = (1, 0)). The fine and hyperfine splittings are ∼ 1 cm −1 and ∼ 0.001 cm −1 , respectively, i.e. they are much lower than the rotational and collisional energies. Thus, assuming that the electronic and nuclear spins play a spectator role during electron-molecule collisions, hyperfine-resolved rate coefficients can be computed using the simple infinite-order-sudden (IOS) approximation. Within this approximation, which is similar in spirit to the ANR approximation, the pure rotational rate coefficients obey the following equation (Corey & McCourt 1983): where [N ′ ] represents (2N ′ + 1) and ( ) is a Wigner "3-j" symbol. In practice, the rate coefficients k N→N ′ (T ) computed with ROTIONS do not strictly follow Equ. (1) due to the Coulomb-Born completion and the threshold correction applied to the cross sections. Equ. (1) is however satisfied to within 25%, down to 10 K. Within the IOS approximation, the fine-structure rate coefficients can be obtained as follows (Corey & McCourt 1983;Lique et al. 2016): where { } is a "6-j" Wigner symbol and k0→L(T ) are the rotational rate coefficients computed with ROTIONS. Similarly, the hyperfine-resolved rate coefficients can be obtained as (Daniel et al. 2005;Lique et al. 2016): In practice, however, the hyperfine rate coefficients for transitions with N = N ′ were computed as (Neufeld & Green 1994;Faure & Lique 2012): This scaling procedure guarantees the following equality: thus ensuring that the summmed hyperfine rate coefficients are identical to the ANR/Coulomb-Born pure rotational rate coefficients. In addition, in order to improve the results at low temperatures, the fundamental excitation rate coefficients k0→L(T ) were replaced by the de-excitation fundamental rate coefficients using the detailed balance relation (within the IOS approximation) k0→L(T ) = [L]kL→0(T ), as in Faure & Lique (2012).

RESULTS
There are no previous studies on these systems against which we can compare. We start by considering results for electronimpact excitation of OH + . Fig. 1 shows the rate coefficients for the electronic excitation of OH + (X 3 Σ − ) after electron impact. This figure shows that the excitation of OH + (X 3 Σ − ) to OH + (a 1 ∆) has a lower temperature threshold than the subsequent transitions and has a greater magnitude over the investigated temperature range. This is to be expected due to the electron energy threshold of this transition, as shown in Table 1. This figure also shows that while the rate coefficients for excitation to OH + (b 1 Σ + ) and OH + (A 3 Π) have a similar temperature threshold, the rate coefficient for excitation to OH + (A 3 Π) dominates at higher temperatures and in fact is converging towards the rate coefficient for excitation to OH + (a 1 ∆). This is a consequence of the fact that the OH + (X 3 Σ − ) to OH + (A 3 Π) transition is dipole allowed so this excitation tends to dominate at high impact energies.. State-to-state Einstein coefficients for the 3 Σ − − 3 Π band can be found in Gómez-Carrasco et al. (2014). Fig. 2 presents rate coefficients for electron-impact rotational excitation of OH + from its rotational ground state. The processes are dominated by the ∆N = 1 transition due to the long-range effect of the dipole moment discussed above. As ∆N increases the temperature threshold of the  Table 5. Hyperfine de-excitation rate coefficients in cm 3 s −1 for OH + in initial levels (N, J, F ) = (1, 2, 5/2) and (1, 2, 3/2). Powers of ten are given in parentheses. process increases and the magnitude of the rate coefficients decreases. Table 5 presents rate coefficients for electron-impact hyperfine de-excitation of OH + from the initial levels (N, J, F ) = (1, 2, 5/2) and (1, 2, 3/2). These two levels are the upper states of the observed transition of OH + at 972 GHz that will be discussed in the next section. It can be noticed that transitions with ∆F = ∆j = ∆N = ±1 are collisionally favored, as observed previously for other 3 Σ − targets colliding with neutrals (see Lique et al. 2016, and references therein). We note that radiatively the selection rules ∆F = 0, ±1 holds strictly and transitions with ∆F = ∆j = ∆N are the strongest ones. We also observe that de-excitation rate coefficients decrease significantly with temperature, typically by a factor of 10 between 10 and 1000 K. +   Fig. 3 shows the rate coefficients for the electronic excitation of SH + (X 3 Σ − ) after electron impact. This figure shows that the temperature thresholds of the three transitions con- sidered in this work are fairly similar. The rate coefficient for the transition to SH + (a 1 ∆) dominates from relatively low temperatures whereas the rate coefficients for transitions to SH + (b 1 Σ + ) and SH + (A 3 Π) remain very similar up to around 2000 K. At higher temperatures, the rate coefficient for the transition to SH + (b 1 Σ + ) exceeds that for the transition to SH + (A 3 Π). This latter does however tend to converge towards the former as the temperature increases still further. Fig. 4 presents rate coeffcients for electron-impact rotational excitation of SH + from its rotational ground state. The processes are again dominated by the ∆N = 1 transition, particularly at low temperatures. As ∆N increases the temperature threshold of the process increases and the magnitude of the rate coefficient decreases with the exception of the rate coefficient for the ∆N = 4 transition which comes to exceed that of the ∆N = 3 transition above ∼90 K. Table 6 presents rate coefficients for electron-impact hyperfine de-excitation of SH + from the initial levels (N, J, F ) = (1, 2, 5/2) and (1, 2, 3/2). These two levels are the upper states of the transition of SH + at 526 GHz first detected with Herschel (Benz & et al 2010). Again we can notice that transitions with ∆F = ∆j = ∆N = ±1 are favoured and that de-excitation rate coefficients decrease by a factor of ∼10 between 10 and 1000 K.

SH
The supplementary data associated with this paper include: • Electronic excitation cross sections and rate coefficients for 16 OH + and 32 SH + . Data include all electronic states with thresholds below 5 eV.
• Rotation excitation cross sections and rate coefficients for the three isotopes of OH + and the four isotopes of SH + . Rotational excitation datasets are published for transitions with starting values of N = 0 to N = 11.

OH + EXCITATION IN THE ORION BAR
The first detection of OH + in emission in a Galactic source was reported by van der Tak et al. (2013a) using the Herschel Space Observatory. These authors presented line profiles and maps of OH + line emission toward the Orion Bar PDR. The Orion Bar PDR is the archetypal edge-on molecular cloud surface illuminated by far-ultraviolet radiation from nearby massive stars. The analysis of the chemistry and excitation of OH + by van der Tak et al. (2013a) suggests an origin of the emission at visual extinctions AV ∼ 0.1 − 1 where most of the electrons are provided by the ionized carbon atoms and hydrogen is predominantly in atomic form. This is also the region where CH + and SH + emissions originate (Nagy et al. 2013). In such an environment, the dominant formation pathway for OH + is O + + H2 and the main destruction route is OH + +H2 (van der Tak et al. 2013a). The reaction of OH + with H is endothermic. Chemical pumping may thus play a role in the excitation of OH + only if the molecular fraction f (H2) = 2N (H2)/(2N (H2)+N (H)) is large enough. Given that f (H2) is expected to be low (< 10%) in the PDR layers where OH + ions form, the impact of chemical pumping should be small, as found by Gómez-Carrasco et al. (2014). This is in contrast with CH + which reacts rapidly with H to form C + + H2 .
We have thus assumed that the excitation of OH + is entirely driven by inelastic collisions with electrons and hydrogen atoms. The hyperfine collisional data presented above for OH + + e − and those of Lique et al. (2016) for OH + + H were combined with spectroscopic data from CDMS and implemented in a non-LTE radiative transfer model. We have employed the public version of the RADEX code 1 which uses the escape probability formulation assuming an isothermal and homogeneous medium. The cosmic microwave background (CMB) is the only background radiation field with a temperature of 2.73 K. Radiative pumping by local dust and starlight is neglected in order to focus on collisional excitation effects. We assume that OH + probes a homogeneous region corresponding to the "hot gas at average density" described by Nagy et al. (2017) for the Orion Bar: the atomic hydrogen density is taken as n(H) = 2×10 5 cm −2 and the kinetic temperature as T k = 500 K, that is a thermal pressure of 10 8 K.cm −3 which is typical of dense PDR. We adopted a typical electron fraction x(e) = n(e − )/n(H) = 10 −4 , as expected if carbon is fully ionized. The line width was fixed at 4 km.s −1 , as observed by van der Tak et al. (2013b). Assuming a unit filling factor, the OH + column density is the single free parameter adjusted to best reproduce the integrated intensities measured by van der Tak et al. (2013b). We have employed the three transitions observed by these authors at 909.159, 971.804 and 1033.119 GHz, corresponding to the transitions (N, j, F ) = (1, 0, 1/2) → (0, 1, 3/2), (1, 2, 5/2) → (0, 1, 3/2) and (1, 1, 3/2) → (0, 1, 3/2), respectively, which are the strongest hyperfine components in each fine-structure line. It must be noted that the transition (N, j, F ) = (1, 2, 5/2) → (0, 1, 3/2) is actually blended with the transition (1, 2, 3/2) → (0, 1, 1/2) at 971.805 GHz. Since RADEX does not treat the overlap of lines, it was necessary to extract the excitation temperature and line center opacity of the blended transitions. Assuming Gaussian shapes, the opacities were summed to simulate a composite line whose intensity was integrated over velocity range from -10 to +10 km.s −1 . Overlap effects should be properly included in the radiative transfer treatment but given the low opacity of the lines (τ < 2) their impact is expected to be moderate here.
Very good agreement is observed in Fig. 5 between our model and the observations for a OH + column density of 3 × 10 13 cm −2 . Indeed, the calculations agree, essentially within error bars, with Herschel data at 971.804 and 1033.119 GHz. They are also consistent with the upper limit at 909.159 GHz. Our column density is a factor of ∼ 3 lower than the value derived by van der Tak et al. (2013b). These authors have employed similar physical conditions but different collisional data and they included chemical terms, which explains the difference. On the other hand, we note that our result is in good agreement with the column density derived by van der Tak et al. (2013b) using the abundance predicted by the Meudon PDR code (1.6 × 10 13 cm −2 ). Finally, the contribution of electron collisions was found to be moderate, of the order of 10-20%, at an electron fraction xe = 10 −4 . The excitation of OH + in the Orion Bar is therefore dominated by hydrogen collisions. The impact of electron-impact excitation would be much larger in environments with high ionisation fractions such as supernova remnants (Barlow et al. 2013;Hamilton et al. 2016) or planetary nebulae (Aleman et al. 2014).

CONCLUSIONS
Electronic and rotational excitation cross sections and rate coefficients have been produced and made available for a range of rotational transitions of the open-shell hydrides OH + and SH + and their isotopologues. The electronic structure calculations were validated where possible against published data. The calculated excitation thresholds, calculated dipole transition moments and rotational constants of both hydrides were validated against measured values or values recommended by the CDMS (Müller et al. 2005) and these comparisons are very good.
The R-matrix method was used to calculate T-matrices from which electronically and rotationally inelastic cross sections were calculated. No published data were available to validate these inelastic cross sections but the reliability of the ANR/Coulomb-Born approach was previously confirmed both experimentally and theoretically. Rate coefficients were calculated by integration of the cross sections using Maxwell-Boltzman distribution of electron velocities. Hyperfine de-excitation rate coefficients were deduced from the rotational data using the IOS approximation. As with the closed shell hydrides (Hamilton et al. 2016), the rotational excitation rate coefficients of the ∆N = 1 transitions were found to be strongly influenced by the long-range effect of the dipole moment and have the largest magnitudes. This result was found to translate in the hyperfine propensity rule ∆F = ∆j = ∆N = ±1.
The electron-impact excitation data were combined with the results of Lique et al. (2016) for OH + +H collisions in order to model the rotational/hyperfine excitation of OH + in the Orion Bar PDR. Very good agreement with the observations of van der Tak et al. (2013b) was obtained for a OH + column density of 3 × 10 13 cm −2 , which is similar to the prediction of the Meudon PDR model. We recommend using the present data in any model of OH + excitation in regions where the electron fraction is larger than 10 −4 .
Finally, electron collisions can seed processes besides rotational excitation and electronic excitation. For molecular ions both dissociative recombination (DR) and vibrational excitation can be astrophysically important processes. The mechanisms for these differ somewhat from that considered above as their cross sections are dominated by the contribution of resonances. They thus require rather more extensive theoretical procedures, see for example Little et al. (2014). We note that electron-impact vibrational excitation and DR rate coefficients have very recently been computed by Stroe & Fifirig (2018).