ExoMol molecular line lists - XXVI: spectra of SH and NS

Line lists for the sulphur-containing molecules SH (the mercapto radical) and NS are computed as part of the ExoMol project. These line lists consider transitions within the $X$ ${}^2\Pi$ ground state for $^{32}$SH, $^{33}$SH, $^{34}$SH and $^{\text{32}}$SD, and $^{14}$N$^{32}$S, $^{14}$N$^{33}$S, $^{14}$N$^{34}$S, $^{14}$N$^{36}$S and $^{15}$N$^{32}$S. Ab initio potential energy (PEC) and spin-orbit coupling (SOC) curves are computed and then improved by fitting to experimentally observed transitions. Fully ab initio dipole moment curves (DMCs) computed at high level of theory are used to produce the final line lists. For SH, our fit gives a root-mean-square (rms) error of 0.03 cm$^{-1}$ between the observed ($v_{\rm max}=4$, $J_{\rm max} = 34.5$) and calculated transitions wavenumbers; this is extrapolated such that all $X$ $^2\Pi$ rotational-vibrational-electronic (rovibronic) bound states are considered. For $^{\text{32}}$SH the resulting line list contains about 81000 transitions and 2300 rovibronic states, considering levels up to $v_{\rm max} = 14$ and $J_{\rm max} = 60.5$. For NS the refinement used a combination of experimentally determined frequencies and energy levels and led to an rms fitting error of 0.002 cm$^{-1}$. Each NS calculated line list includes around 2.8 million transitions and 31000 rovibronic states with a vibrational range up to $v=53$ and rotational range to $J=235.5$, which covers up to 23000 cm$^{-1}$. Both line lists should be complete for temperatures up to 5000 K. Example spectra simulated using this line list are shown and comparisons made to the existing data in the CDMS database. The line lists are available from the CDS (http://cdsarc.u-strasbg.fr) and ExoMol (www.exomol.com) data bases.


INTRODUCTION
Sulphur chemistry is important in a variety of astronomical environments including the interstellar medium (ISM) (Oppenheimer & Dalgarno 1974;Duley et al. 1980;Vidal et al. 2017), hot cores (Charnley 1997;Woods et al. 2015), comets (Canaves et al. 2002;Rodgers & Charnley 2006;Canaves et al. 2007), starburst and other galaxies Martín 2005), exoplanets (Visscher et al. 2006;Zahnle et al. 2009), and brown dwarfs and low-Mass dwarf stars (Visscher et al. 2006). The ExoMol project aims at providing comprehensive molecular line lists for exoplanet and other atmospheres. ExoMol has provided line lists for several sulphur-baring molecules: CS (Paulose et al. 2015), SO2 (Underwood et al. 2016a), H2S (Azzam et al. 2016), SO3 (Underwood et al. 2016b), and PS (Prajapat et al. 2017); a line list for SiS has also just been completed (Upadhyay et al. 2018). In this work we extend this coverage by providing line lists for the major isotopologues of SH and NS. For both species we only consider transitions within the ground electronic state: both SH and NS have an X 2 Π ground state. The excited electronic states lie above 30,000 cm −1 and 23,000 cm −1 for SH and NS, respectively, and thus the line lists presented here will be accurate for the visible, infrared and radio spectral regions. Both species are well-know astronomically from transitions within the ground state.
The diatomic mercapto radical SH has long been of interest to astronomers, but proved challenging to detect. It was definitively detected in the interstellar medium (ISM) (Neufeld et al. 2012), in asymptotic-giant-branch (AGB) stars (Yamamura et al. 2000) and the Sun's atmosphere (Berdyugina & Livingston 2002), tentatively detected in comets (Krishna Swamy & Wallis 1987) and predicted to occur in brown dwarfs (Visscher et al. 2006) and hot Jupiter exoplanets (Visscher et al. 2006;Zahnle et al. 2009) as one of the major sulphur-bearing gases after H2S. The ISM detection was difficult due to the location of the key rotational transition which was inaccessible both from the ground and the Herschel telescope; after a number of failed searches in the ISM (Meeks et al. 1969;Heiles & Turner 1971), Neufeld et al. (2012) finally detected SH in the terahertz region using SOFIA (Stratospheric Observatory For Infrared Astronomy) by its 1383 GHz 2 Π 3 2 J = 5 2 − 3 2 transition. The A 2 Σ + -X 2 Π band, a UV absorption band not considered in this paper, has also been used to detect SH in translucent interstellar clouds (Zhao et al. 2015) and in the Sun's atmosphere (Berdyugina & Livingston 2002). Zahnle et al. (2009) generated an A-X line list for SH.
In our own atmosphere, SH is known to react with NO2, O2 and O3: SH is produced in the troposphere by oxidation of H2S by the OH radical (Ravichandran et al. 1994).
Experimentally, SH spectra have been studied since 1939 (Glockler & Horwitz 1939;Lewis & White 1939) with over 100 experimental publications to date. This work is based on the measured transitions from experimental studies presented in Table 1.
Transitions for NS are much more astronomically accessible and it was one of the first ten diatomic molecules to be detected in space (Somerville 1977;Lovas et al. 1979), with the first positive detection by Gottlieb et al. (1975) using the J = 5/2 − 3/2 transition of the 2 Π 1/2 state at 115.6 GHz towards Sagittarius B2. Radio astronomy has also been used to detect NS in giant molecular clouds (McGonagle et al. 1992;Leurini et al. 2006;Belloche et al. 2013), cold dark clouds (McGonagle et al. 1994), comets (Irvine et al. 1999;Biver 2005), extragalactically (Martín et al. 2003) and the NGC 253 starburst region Meier et al. (2015).
Experimentally, there have been significant experimental work focusing on the excited electronic states; however, this is not of relevance to this work. Numerous experimental studies on NS have been made on the spectra of the ground state. Laser magnetic resonance (LMR) studies include those of Carrington et al. (1968), Uehara & Morino (1969), Anacona (1994) and Anacona (1995). Experimental measurements of rovibrational transitions within the ground state are reported in a series of papers (Narasimham & Balasubramanian 1971;Matsumura et al. 1980;Lovas & Suenram 1982;Anacona et al. 1986;Sinha et al. 1988;Lee et al. 1995;Amano et al. 1969). The experimental frequencies used in this work are summarised in Table 2.
Early electronic structure calculation on NS were made by Bialski & Grein (1976), Salahub & Messmer (1976) and Karpfen et al. (1978). Subsequently, Lie et al. (1985) and Karna & Grein (1986) performed configuration interaction studies on the low-lying and Rydberg states of NS. CCSD(T) calculation of equilibrium geometries of NS for plasma applications were made by Czernek &Živný (2004). The most recent theoretical study on NS is that of Gao et al. (2013) who undertook calculations at the MRCI+Q+DK/AV5Z level of theory for the PECs of low-lying electronic states. However, while there are some computed dipole moment (Lie et al. 1985;Gao et al. 2013) and spin-orbit coupling (Shi et al. 2012) curves, we perform new ab initio calculations to ensure the uniform quality of our model.

METHOD AND SPECTROSCOPIC MODELS
Our general method is to start from high quality ab initio potential energy curves (PECs), associated coupling curves and dipole moment curves (DMCs). Since ab initio transition frequencies are not accurate enough, the PECs and couplings are refined using empirical energy levels and transition wavenumbers from laboratory spectra. Ab initio DMCs are found to give the best results. The nuclear motion problem is solved using the program Duo ) which allows for full couplings between the curves, see Tennyson et al. (2016b) for a full discussion of the theory. Duo has been successfully used to produce line lists for a number of diatomic molecules AlO, PS, PN, ScH, VO, NO, CaO, SiH (Patrascu et al. 2015;Lodi et al. 2015;McKemmish et al. 2016;Yurchenko et al. 2016b;Wong et al. 2017;Prajapat et al. 2017;Yurchenko et al. 2018). The refined PECs, coupling curves and DMCs together form a spectroscopic model for the diatomic system, which can be useful beyond the immediate line list application considered here.
Both SH and NS have X 2 Π ground states. In this case the spin-orbit (SO) coupling splits the PEC in two curves, which are often denoted 2 Π 1 2 and 2 Π 3 2 . The SO splitting coupling presents a significant contributions to the energies of these molecules: 360 cm −1 and 220 cm −1 for the v = 0 states of SH and NS, respectively. Another important coupling for spectroscopy of the 2 Π systems is the due to the presence of electronic angular momentum (EAM), which causes the Λ-doubling effect.
We use the extended Morse oscillator (EMO) functions (Lee et al. 1999) to represent the PECs, both ab initio and refined.
In this case the PEC is given by where Ae − Ve is the dissociation energy, re is an equilibrium distance of the PEC, and ξp is theŠurkus variable given by: The corresponding expansion parameters are obtained by fitting to the experimental data (energies and frequencies) of the molecule in question, as detailed below.
To model the SO coupling we use ab initio curves computed using high levels of theory with the program MOL-PRO (Werner et al. 2012). These curves are then refined by fitting to the experimental data using the morphing approach (Meuwly & Hutson 1999;Skokov et al. 1999). In this approach, the ab initio curves represented on a grid of bond lengths are 'morphed' using the following expansion: where z is either taken as theŠurkus variable z = ξp or the damped-coordinate given by: see also Prajapat et al. (2017) and Yurchenko et al. (2018). Here r ref is a reference position equal to re by default and β2 and β4 are damping factors. When used for morphing, the parameter B∞ is usually fixed to 1. The Λ-doubling effects in Duo can be modelled directly using an effective Λ-doubling function, in case of 2 Π we use the (p + 2q) effective coupling (Brown & Merer 1979) given by: HLD leads to a linearĴ-dependence, which is justified for the heavy molecule like NS. In this case for α LD p2q (r) we use a simple, one-parameter function: For the SH molecule, which is affected by a stronger centrifugal distortion, this is not appropriate. Here we follow the approach recently used for solving another hydrogen-containing 2 Π system, SiH , where the Λ-doubling is modelled via an EAM interaction with a closely lying 2 Σ-state (Brown & Merer 1979). In the case of X 2 Π of SH, the closest Σ state is A 2 Σ − . The latter is introduced with a dummy potential curved in the EMO representation, while the EAM-curve is given by the 1st order ξp-type expansion in Eq. (3) (see also below). The dipole moment curves (DMC) of SH and NS are computed using a high level ab initio theory on a grid of bond length values ranging from about 0.8 to 8Å. In order to reduce the numerical noise in the intensity calculations of high overtones (see recent recommendations by Medvedev et al. (2016) the DMCs are represented analytically. The expansion with a damped z coordinate in Eq. (3) is employed (Prajapat et al. 2017;Yurchenko et al. 2018).
All these functional forms are included in Duo (functions.f90). The corresponding expansion parameters as well as their grid representations can be found in the Duo input files provided as supplementary data.

SH
The Duo model for SH consists of two PECs, X 2 Π and A 2 Σ + , represented by EMO forms in Eq. (1), diagonal (X-X) and non-diagonal (X-A) SO coupling curves (ab initio) morphed by functions using Eq. (3), an EAM coupling curve (X-A) also represented by Eq. (3), an by a diagonal X-X DMC. The A state PEC is only used to support the Λ-doubling effect in the Xstate energies and is not included in SH the line list. We use the ab initio SO coupling curves obtained at the MRCI+DKH4+Q level of theory, where DKH4 is the fourth-order Douglas-Kroll-Hess representation of the relativistic Hamiltonian and Q denotes a Davidson correction. An AWC5Z Gaussian Type basis set was used (Dunning Jr. 1989;Woon & Dunning 1993;Peterson & Dunning Jr. 2002;Szalay et al. 2012). The ab initio PE, SO, EAM and DM curves of SH used in this work are shown in Figs 1-4.
The PEC, SO and EAM expansion parameters were obtained by fitting to the experimental frequencies from the sources listed in Table 1 with a root-mean-square (rms) error of 0.03 cm −1 . The empirical vibrational information is limited to only ∆v = 0 and ∆v = 1 transitions with vmax = 4, which complicates obtaining a globally accurate model from the fitting. The refined curves are shown in Figs 1-3. The quality of the fit is illustrated in Fig. 5, where the Obs.-Calc. residuals for all experimental data are shown and in Tables 3 and 4. Most of the ∆v = 0 (vmax = 4) and ∆v = 1 (vmax = 1) frequencies are reproduced within 0.005 cm −1 , except for the Π 3/2 band, which is found to diverge at J = 25 by about 0.15 cm −1 . Our final Table 1. List of experimental data used in refinement of the SH X 2 Π potential energy curves.
(1, 0) 1,149 -1,251 Lee et al. (1995) WS 0.5 -7.5 (0,0) 2.3 -11.6 value for De (4.46 eV), corresponding to the best fit, is higher than the experimental X 2 Π dissociation energy of 3.62±0.03 eV (D0) by Continetti et al. (1991), as well as with the ab initio values recommended by Csaszar et al. (2003) De = 3.791 eV and D0 = 3.625 eV. Therefore we limit our extrapolations to high vibrational excitations to those that do not to exceed this D0 value. The final spectroscopic model (provided as a Duo input file in the supplementary material) was then used to generate line lists for the following isotopologues: 32 SH, 33 H, 34 H, 36 SH (Jmax = 64.5) and 32 SD (Jmax = 89.5). In the Duo calculations we used a sinc DVR method based on the grid of 501 points equally distributed from 0.85 to 5.0Å. The equilibrium value of the ab initio (MRCI/AWC5Z) dipole moment of SH (X 2 Π) is µe = 0.801 D (at r = 1.3565Å). The vibrationally averaged dipole moments, given by µv = v|µ|v where |v is the vibrational eigenfunction of X 2 Π at the limit of J = 0, are µ0 = 0.794 D and µ1 = -0.017 D.

NS
The electronic structure of the lowest seven states of NS was intensely studied by Gao et al. (2013). In this work we only concentrate on the ground electronic state spectrum of NS. The program MOLPRO (Werner et al. 2012) was used to compute ab initio PEC, SOC and DMC for the NS X 2 Π ground state along with the spin-orbit coupling curve for this state on a grid of 143 geometries distributed between 0.8Å and 2.7Å using the MRCI method and Douglas-Kroll Hamiltonian (dkroll=2) with an aug-cc-pVQZ-DK basis set and the Davidson correction included. The (2s,2p)/N and (2s,2p)/O complete active space (CAS) is defined by 8330/4110 in the C2v symmetry employed by MOLPRO.   The ab initio PEC, SO and DMC are shown in Figs. 6-8. The ab initio DMC of NS was modelled using the dampedvariable expansion in Eq. (3). The equilibrium dipole value µe is 1.834 D (at re = 1.494Å), while the vibrationally averaged µ0 is 1.825 D, which is in a good agreement with the experimental (Stark) value of 1.81 D due to Amano et al. (1969).
In order to fit the PEC and SO curves of NS to the experimental data for the X 2 Π state, several steps were taken. Firstly, the program PGOPHER (Western 2017) was used to construct a list of rovibronic energies using the molecular parameters published by Sinha et al. (1988). The MARVEL procedure (Furtenbacher et al. 2007) was then used to transform the list of measured experimental transitions summarised in Table 2 into several 'networks' of derived energies: these were used to check the rovibronic energies determined using PGOPHER. This list of experimentally derived energies was then used to perform an initial fit of the data, which was then improved by using the actual experimentally measured frequencies.
The Duo calculations were based on the sinc DVR method comprising 701 points evenly distributed between 0.9Å and 3.3Å. The ab initio X 2 Π PEC of NS was represented using the EMO form in Eq. (1) and refined by fitting to 358 experimental frequencies covering the rotational excitations up to J = 32.5 and vibrational states up to v = 5; however, only v = 1 − 0 transitions are for an the vibration band, the remaining are microwave transitions for which ∆v = 0.
In the fits, we also included the 161 PGOPHER term values (J 23.5) generated from the constants by Sinha et al. (1988). Using experimentally-derived energies together with the measured frequencies helps to constrain the fitted value to the absolute energies, not only to the separation between them. This tends to make fits more stable and prevent drifts between states (see also Patrascu et al. (2015)). In the refinement, the effects of the spin-orbit coupling and Λ-doubling were taken into account: the ab initio SOC was morphed and the Λ-doubling curves was refined by using the expression in Eq. (3). The refined PEC and SOC of NS are shown in Figs. 6 and 7. The fitted parameters are presented in the supplementary material.
Our final model reproduces the experimental frequencies with an rms error of 0.002 cm −1 . The experimentally derived energies are reproduced with an rms error of 0.03 cm −1 . Fig.9 shows the difference between the transition frequencies (cm −1 ) calculated using the refined curves (Calc.) and those experimentally measured (Obs.). The error for most of the data is within 0.002 cm −1 , which is comparable to that obtained by the effective rotational methods. Table 5 shows a sample of the Obs.-Calc. residuals for the rotational energies (v = 0) as a function of J, while Table 6 compares some residuals for v = 0 and v = 1.

SH
Line lists for the five most important isotopologues of SH were computed using Duo. Table 7 summarises the statistics for these line lists. Those for SH contain almost 200,000 lines while the heavier D atom means that the 32 SD line list contains more than double this number of transitions. In order to further improve the numerical stability of intensity calculations for high overtones, we follow the procedure used by Wong et al. (2017)   data. Extracts of the states and transitions files are given in Tables 8 and 9 respectively. Apart from the energy term values, statistical weights and quantum numbers, the states file also contains the lifetimes ) and the Landé g-factors (Semenov et al. 2017).

NS
For NS, five line lists were computed for the isotopologues 14 N 32 S, 14 N 33 S, 14 N 34 S, 14 N 36 S and 15 N 32 S (see Table 7). The line lists are based in the lowest vmax = 60 vibrational eigenfunctions with the rotational quantum number J ranging from 0.5 to 200.5 and the maximum energy term value Emax was set from 0 cm −1 to 38 964.6 cm −1 . The frequency window was set to 23 000 cm −1 which is just below the next electronic state, a 4 Π (Gao et al. 2013). The values of vmax and Emax correspond   to the dissociation limit of 4.83 eV determined by (Czernek &Živný 2004). A dipole moment cutoff of 10 −8 D was also used. Again , Tables 10 and 11 show extracts from the corresponding states and transition files.

Partition Functions
Partition functions were computed up to 5000 K for every species considered at 1 K intervals. These can be found in the supplementary material. We have also fitted the partition function to the function form of Vidler & Tennyson (2000): an(log 10 T ) n .
Table 12 gives the expansion coefficients for the parent isotopologues, fits for other species can be found in the supplementary material, which reproduce the ExoMol partition function within 2-3 %. In general our partition functions are in excellent agreement with those available from other sources, namely from Sauval & Tatum (1984), Barklem & Collet (2016) and the CDMS database (Müller et al. 2005), once allowance is made for the nuclear spin conventions employed: ExoMol uses the HITRAN convention which leads to a factor of 2 for 32 SH and 3 for 14 N 32 S compared to the 'astronomers' convention employed by Sauval & Tatum (1984) and Barklem & Collet (2016). The only significant disagreement is for SH below 1000 K, where the results of Sauval & Tatum (1984) follow the wrong trend; Sauval & Tatum (1984) only aimed to be accurate above 1000 K.
Our partition functions should be complete up to 5000 K. For SH, the completeness is within 0.3 %, which corresponds to the number of states in our line list above the experimental dissociation energy (Continetti et al. 1991) at T = 5000 K. For NS, we mainly miss the contributions from the a 4 Π rovibronic states not considered here (Te = 24,524 cm −1 , Gao et al. (2013)). This should not exceed 1 % judging by the contribution to Q(T ) from the X 2 Π energies of NS above 24,524 cm −1 .   (Barklem & Collet) x 3 (Sauval & Tatum) x 3 This work Figure 10. Temperature dependence of the partition functions of SH (left) and NS (right) computed using our line lists and compared to those by Sauval & Tatum (1984) and Barklem & Collet (2016).  Figure 11 gives an overview of the spectrum of SH at different temperatures. Note how intensities drop exponentially across the entire frequency range shown: no unphysical, plateau-like structures at higher frequencies is present (Medvedev et al. 2016). This illustrates that our measures to prevent this spurious effect were successful. Figure 12 compares a simulated emission spectrum at T = 2000 K (HWHW = 0.01 cm −1 ) to the experimental spectrum of Winkel & Davis (1984). The agreement is good, especially considering the complex coupling and the limited amount of the experimental data. Figure 13 shows a comparison of an SH spectrum at T = 296 K computed using the ExoMol line list with that from the CDMS database (Müller et al. 2005). The CDMS spectrum was obtained using the equilibrium dipole moment of 0.7580 D from Meerts & Dymanus (1974), while our value is 0.794 D. Figure 14 shows a comparison of spectra for the main isotopologue of NS as a function of temperature. Figure 15 shows the effects on the spectra of NS when the main isotopes of N and S are substituted. It can be seen that effect of substitution of the N atom leads to redshifts of up to 28 cm −1 and that of substituting S is up to 10 cm −1 .

NS
To illustrate the accuracy of our line lists, spectra have been simulated and compared to the existing CDMS database (Müller et al. 2005) for the rotational and fundamental bands of NS, see Fig.16   splitting was averaged in the CDMS transitions of NS. Our intensities are in good agreement with those from CDMS. Some difference between the intensities from the fundamental band is due to the different ab initio dipole moments used: our transition dipole µ1 for the fundamental band is 0.049 D, while CDMS used µ1 = 0.045 D from unpublished work by H. Müller. [H]

Lifetimes
Lifetimes for states within the X 2 Π ground state can be computed in straightforward manner from our line lists ). These are included in states file, see Tables 8 and 10 above. Figure 17 presents lifetimes states associated with the main isotopologues of SH and NS. As can be seen for NS, lifetimes for each vibrational state decrease by approximately up to one order of magnitude as energy is increased.

CONCLUSIONS
New line lists for the electronic ground states of major isotopologues of both SH and NS are generated using a high level ab initio theory refined to available experimental data. For SH, each line list contains approximately 81 000 transitions and 2300 states, with a range up to the dissociation limit of 29 234 cm −1 , vibrational coverage up to 14 and rotational coverage to J = 60.5. For NS, the line lists contain 2.7 -2.9 million transitions up to 23 000 cm −1 and 31 000 -33 000 states with a range up to the dissociation limit of 38 964.6 cm −1 , with vibrational coverage up to ν=54 and rotational coverage to J = 235.5. These are the only available hot line lists for these molecules. The line lists, which are named SNaSH, are available from the CDS (http://cdsarc.u-strasbg.fr) and ExoMol (www.exomol.com) data bases. Our model is affected by the limitations of the experimental data as well as by the ab initio accuracy, especially for the dipole moment calculations. The latter is critical for accurate retrievals of molecular opacities in different astronomical bodies. It is typical for hot diatomics that experimental data on the dipole moments are either absent or extremely inaccurate, which emphasises the importance of the ab initio calculations of this property.
Considering the importance of the SH molecule for star spectroscopy, it would be important to extend the present study with the inclusion of the A 2 Σ + electronic state, which would allow accurate modelling and prediction in the high-energy visible and UV spectral regions.
Our new line lists should enable detection of and inclusion in models of SH and NS for exoplanet temperatures which are largely not covered by experimental results. The ExoMol project has already provided line lists for for several sulphurcontaining molecules, namely CS (Paulose et al. 2015), SiS (Upadhyay et al. 2018), PS (Prajapat et al. 2017), H2S (Azzam et al. 2016), SO2 (Underwood et al. 2016a) and SO3 (Underwood et al. 2016b). SO is probably the only major sulphur-baring species which is important at elevated temperatures which is missing from this list.

ACKNOWLEDGEMENTS
This work was supported by the UK Science and Technology Research Council (STFC) No. ST/M001334/1 and the COST action MOLIM No. CM1405. This work made extensive use of UCL's Legion high performance computing facility.