ExoMol XXVI: Line list for SH

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

Sergei N. Yurchenko, Wesley Bond, Maire N. Gorman, Lorenzo Lodi,Laura K. McKemmish, William Nunn, Rohan Shah and Jonathan Tennyson
Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
Department of Physics, Aberystwyth University, Penglais, Aberystwyth, Ceredigion, UK, SY23 3BZ
Email: j.tennyson@ucl.ac.uk
July 31, 2019Accepted XXXX. Received XXXX; in original form XXXX
July 31, 2019Accepted XXXX. Received XXXX; in original form XXXX
Abstract

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   ground state for SH, SH, SH and SD, and NS, NS, NS, NS and NS. 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 between the observed (, ) and calculated transitions wavenumbers; this is extrapolated such that all   rotational-vibrational-electronic (rovibronic) bound states are considered. For SH the resulting line list contains about 81 000 transitions and 2 300 rovibronic states, considering levels up to and . 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. Each NS calculated line list includes around 2.8 million transitions and 31 000 rovibronic states with a vibrational range up to and rotational range to , which covers up to 23 000 cm. 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.

keywords:
molecular data; opacity; astronomical data bases: miscellaneous; planets and satellites: atmospheres; stars: low-mass
pubyear: 2018pagerange: 1References

1 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 et al., 2005; 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), SO (Underwood et al., 2016a), HS (Azzam et al., 2016), SO (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   ground state. The excited electronic states lie above 30,000 cm and 23,000 cm 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, 1988) 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 HS. 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 transition. The    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 line list for SH.

In our own atmosphere, SH is known to react with NO, O and O: SH is produced in the troposphere by oxidation of HS 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.

A number of multi-reference configuration interaction (MRCI) level theoretical calculations have been performed on SH (Raimondi et al., 1975; Hirst & Guest, 1982; Bruna & Hirsch, 1987), with the most recent study been those of Kashinski et al. (2017) and Vamhindi & Nsangou (2016). Spin-orbit splitting of the ground state potential energy curve (PEC) was calculated by Baeck & Lee (1990) and Qui-Xia et al. (2008). Lifetimes for SH have previously been calculated by McCoy (1998) and Brites et al. (2008) with Resende & Ornellas (2001).

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 transition of the 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.

2 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 (Yurchenko et al., 2016a) 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   ground states. In this case the spin-orbit (SO) coupling splits the PEC in two curves, which are often denoted and . The SO splitting coupling presents a significant contributions to the energies of these molecules: 360 cm and 220 cm for the states of SH and NS, respectively. Another important coupling for spectroscopy of the systems is the due to the presence of electronic angular momentum (EAM), which causes the -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

(1)

where is the dissociation energy, is an equilibrium distance of the PEC, and is the Šurkus variable given by:

(2)

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 MOLPRO (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:

(3)

where is either taken as the Šurkus variable or the damped-coordinate given by:

(4)

see also Prajapat et al. (2017) and Yurchenko et al. (2018). Here is a reference position equal to by default and and are damping factors. When used for morphing, the parameter is usually fixed to 1.

The -doubling effects in Duo can be modelled directly using an effective -doubling function, in case of we use the effective coupling (Brown & Merer, 1979) given by:

(5)

leads to a linear -dependence, which is justified for the heavy molecule like NS. In this case for we use a simple, one-parameter function:

(6)

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 system, SiH (Yurchenko et al., 2018), where the -doubling is modelled via an EAM interaction with a closely lying -state (Brown & Merer, 1979). In the case of   of SH, the closest state is  . The latter is introduced with a dummy potential curved in the EMO representation, while the EAM-curve is given by the 1st order -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 8 Å. 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 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.

2.1 Sh

The Duo model for SH consists of two PECs,   and  , represented by EMO forms in Eq. (1), diagonal () and non-diagonal () SO coupling curves (ab initio) morphed by functions using Eq. (3), an EAM coupling curve () also represented by Eq. (3), an by a diagonal DMC. The state PEC is only used to support the -doubling effect in the -state 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 14.

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. The empirical vibrational information is limited to only and transitions with , which complicates obtaining a globally accurate model from the fitting. The refined curves are shown in Figs 13. 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 () and () frequencies are reproduced within 0.005 cm, except for the band, which is found to diverge at by about 0.15 cm. Our final value for (4.46 eV), corresponding to the best fit, is higher than the experimental dissociation energy of 3.62 eV () by Continetti et al. (1991), as well as with the ab initio values recommended by Csaszar et al. (2003) = 3.791 eV and = 3.625 eV. Therefore we limit our extrapolations to high vibrational excitations to those that do not to exceed this value.

Source No. of transitions Vibrational bands
Bernath et al. (1983) 50 (1-0) 11.5
Winkel & Davis (1984) 285 (1-0), (2-1), (3-2) 34.5
Ram et al. (1995) 175 (1-0), (2-1), (3-2), (4-3) 16.5
Yamamura et al. (2000) 30 (1-0), (2-1), (3-2) 25.5
Eliet et al. (2011) 6 (0-0) 4.5
Martin-Drumel et al. (2012) 70 (0-0), (1-1) 16.5
Table 1: List of experimental data used in refinement of the SH   potential energy curves.
Study Method J Range (cm)
Anacona et al. (1986) WS 2.5 – 6.5 , 2.3 – 10.1
Sinha et al. (1988) FTS 0.5 – 35.5 (1, 0) 1,149 – 1,251
Lee et al. (1995) WS 0.5 – 7.5 (0,0) 2.3 – 11.6
Table 2: Experimental sources of NS spectroscopic data used in the refinement of the PEC. FTS=Fourier Transform Spectrometry, WS=millimetre and sub-millimetre Wave spectrometry.

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: SH, H,H, SH () and SD (). In the Duo calculations we used a sinc DVR method based on the grid of 501 points equally distributed from 0.85 to 5.0 Å. The equilibrium value of the ab initio (MRCI/AWC5Z) dipole moment of SH ( ) is  D (at  Å). The vibrationally averaged dipole moments, given by where is the vibrational eigenfunction of   at the limit of , are  D and = -0.017 D.

The experimental value of the SH () dipole moment of 0.7580(1) D was obtained by Meerts & Dymanus (1974) in a Stark experiment. Benidar et al. (1991) reported anomalously weak intensities of the fundamental band of SH and obtained a very rough estimate for a relative dipole moment value of of (0.011 D)/0.63 D = 0.017, which compares favourably to our value .

Obs. Calc. Obs.-Calc.
1.5 - 1.5 46.1289 46.1293 -0.0004
2.5 + 1.5 64.5513 64.5519 -0.0006
3.5 - 0.5 86.8367 86.8418 -0.0051
3.5 + 1.5 82.9811 82.9808 0.0003
4.5 - 0.5 106.2538 106.2508 0.0030
5.5 - 1.5 119.6169 119.6179 -0.0010
6.5 + 1.5 137.8762 137.8771 -0.0009
7.5 + 1.5 156.1850 156.1846 0.0004
8.5 - 0.5 181.9686 181.9684 0.0002
9.5 + 0.5 200.5907 200.5909 -0.0002
10.5 - 0.5 219.0671 219.0677 -0.0006
11.5 + 1.5 228.2230 228.2231 -0.0001
12.5 - 0.5 255.5412 255.5415 -0.0003
13.5 - 0.5 273.5067 273.5068 -0.0001
14.5 - 1.5 281.0630 281.0639 -0.0009
15.5 + 1.5 298.3891 298.3901 -0.0010
Table 3: Example of Observed Calculated residuals for SH frequencies as a function of for the (0, 0) band where .
Band Obs. Calc. Obs.-Calc.
1.5 + 0.5 (0, 0) 48.5370 48.5318 0.0052
1.5 - 1.5 (1, 0) 2642.8296 2642.8262 0.0034
1.5 + 0.5 (1, 0) 2644.8974 2644.8939 0.0035
3.5 - 1.5 (1, 1) 80.5572 80.5590 -0.0018
3.5 + 1.5 (1, 1) 80.5901 80.5907 -0.0006
1.5 - 0.5 (2, 1) 2546.2628 2546.2735 -0.0107
1.5 + 1.5 (2, 1) 2544.5747 2544.5867 -0.0120
2.5 - 0.5 (3, 2) 2464.0678 2464.0150 0.0528
3.5 - 0.5 (3, 2) 2479.3627 2479.3160 0.0467
3.5 - 1.5 (4, 3) 2376.6704 2376.6516 0.0188
3.5 + 1.5 (4, 3) 2376.6944 2376.6727 0.0217
Table 4: Example of Observed Calculate residuals for SH frequencies for various vibrational bands as a function of where .
Figure 1: Potential energy curves of SH: fitted (solid) and ab initio (dashed).
Figure 2: Spin-orbit curves of SH: , fitted (solid) and ab initio (dashed) and , fitted only (solid).
Figure 3: An empirical EAM curve () of SH.
Figure 4: The diagonal   ab initio dipole moment curve of SH.
Figure 5: Observed Calculated residuals for SH.

2.2 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   ground state along with the spin-orbit coupling curve for this state on a grid of 143 geometries distributed between 0.8 Å and 2.7 Å 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 C symmetry employed by MOLPRO.

The ab initio PEC, SO and DMC are shown in Figs. 68. The ab initio DMC of NS was modelled using the damped-variable expansion in Eq. (3). The equilibrium dipole value is 1.834 D (at  Å), while the vibrationally averaged 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).

Figure 6: Ab initio PEC and refined PEC for the   state of NS, see Eq. (1).
Figure 7: Ab initio and refined SO curves of NS.
Figure 8: ab initio Dipole moment Curve of NS.

In order to fit the PEC and SO curves of NS to the experimental data for the   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 0.9 Å and 3.3 Å. The ab initio   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 and vibrational states up to ; however, only transitions are for an the vibration band, the remaining are microwave transitions for which .

In the fits, we also included the 161 PGOPHER term values () 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 -doubling were taken into account: the ab initio SOC was morphed and the -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. The experimentally derived energies are reproduced with an rms error of 0.03 cm. Fig.9 shows the difference between the transition frequencies (cm) calculated using the refined curves (Calc.) and those experimentally measured (Obs.). The error for most of the data is within 0.002 cm, 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 () as a function of , while Table 6 compares some residuals for and .

Figure 9: Observed Calculated residuals for NS.
Obs. Calc. Obs.-Calc.
0.5 + 0.5 2.3026 2.3024 0.0002
1.5 + 1.5 3.8760 3.8757 0.0003
2.5 + 0.5 5.3808 5.3806 0.0002
3.5 - 1.5 6.9758 6.9760 -0.0002
3.5 - 0.5 6.9198 6.9196 0.0002
4.5 + 1.5 8.5257 8.5259 -0.0002
5.5 + 0.5 10.0095 10.0098 -0.0003
6.5 + 0.5 11.5359 11.5356 0.0003
Table 5: Example of Observed minus calculated R-branch frequencies as a function of for the NS (0, 0) band.
Band Obs. Calc. Obs.-Calc.
0.5 + 0.5 (1, 0) 1206.5519 1206.5507 0.0012
1.5 - 1.5 (1, 0) 1207.9205 1207.9196 0.0008
2.5 + 0.5 (1, 0) 1209.5552 1209.5541 0.0011
3.5 - 0.5 (1, 0) 1211.0378 1211.0369 0.0009
4.5 + 0.5 (1, 0) 1212.5078 1212.5071 0.0007
5.5 - 0.5 (1, 0) 1213.9651 1213.9647 0.0005
6.5 + 0.5 (1, 0) 1215.4100 1215.4096 0.0004
7.5 - 1.5 (1, 0) 1216.7633 1216.7624 0.0009
8.5 + 0.5 (1, 0) 1218.2612 1218.2612 0.0000
9.5 + 0.5 (1, 0) 1219.6812 1219.6802 0.0010
10.5 + 1.5 (1, 0) 1221.0099 1221.0094 0.0005
11.5 + 1.5 (1, 0) 1222.3989 1222.3991 -0.0002
12.5 - 0.5 (1, 0) 1223.8239 1223.8231 0.0008
13.5 + 1.5 (1, 0) 1225.1387 1225.1391 -0.0004
14.5 - 1.5 (1, 0) 1226.4888 1226.4894 -0.0005
15.5 + 1.5 (1, 0) 1227.8262 1227.8265 -0.0003
17.5 - 0.5 (1, 0) 1230.4566 1230.4575 -0.0008
18.5 + 0.5 (1, 0) 1231.7462 1231.7477 -0.0015
20.5 + 1.5 (1, 0) 1234.3120 1234.3126 -0.0005
22.5 - 0.5 (1, 0) 1236.7884 1236.7888 -0.0004
24.5 + 1.5 (1, 0) 1239.2594 1239.2603 -0.0009
25.5 + 1.5 (1, 0) 1240.4629 1240.4637 -0.0008
28.5 - 0.5 (1, 0) 1243.9352 1243.9340 0.0012
30.5 - 1.5 (1, 0) 1246.2757 1246.2745 0.0012
32.5 + 0.5 (1, 0) 1248.4186 1248.4170 0.0015
34.5 - 0.5 (1, 0) 1250.5983 1250.5935 0.0048
2.5 - 0.5 (1, 1) 5.3493 5.3497 -0.0003
3.5 - 0.5 (1, 1) 6.8637 6.8634 0.0003
4.5 - 0.5 (1, 1) 8.4028 8.4025 0.0003
5.5 + 0.5 (1, 1) 9.9289 9.9287 0.0002
2.5 + 1.5 (2, 2) 5.3371 5.3368 0.0003
3.5 + 0.5 (2, 2) 6.8201 6.8194 0.0007
4.5 - 0.5 (2, 2) 8.3339 8.3331 0.0008
2.5 - 0.5 (3, 3) 5.2622 5.2609 0.0013
3.5 - 1.5 (3, 3) 6.8043 6.8032 0.0011
2.5 + 0.5 (4, 4) 5.2046 5.2031 0.0015
3.5 - 0.5 (4, 4) 6.6935 6.6914 0.0021
Table 6: Example of Observed Calculated residuals for NS frequencies for various vibrational bands ( branch).

3 Results and Discussion

3.1 Line lists

3.1.1 Sh

SH SH SH SH SD NS NS NS NS NS
60.5 60.5 60.5 60.5 84.5 235.5 236.5 237.5 239.5 240.5
number of energies 2326 2326 2328 2334 4532 31502 31802 32089 32620 33051
number of lines 81,348 81,274 81,319 81,664 219,463 2,755,796 2,795,487 2,831,482 2,901,113 2,957,016
Table 7: Statistics for the SH and NS line lists.

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 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) and apply a cutoff of  D to all matrix elements of the dipole moment . The full line lists are given in the ExoMol format (Tennyson et al., 2016c) as supplementary 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 (Tennyson et al., 2016a) and the Landé -factors (Semenov et al., 2017).

Energy (cm) -factor Parity e/f State
1 360.537424 4 0.5 inf -0.000697 + e X2Pi 0 1 -0.5 0.5
2 2959.255407 4 0.5 0.708340 -0.000699 + e X2Pi 1 1 -0.5 0.5
3 5461.024835 4 0.5 0.247650 -0.000699 + e X2Pi 2 1 -0.5 0.5
4 7865.670867 4 0.5 0.130790 -0.000700 + e X2Pi 3 1 -0.5 0.5
5 10172.812283 4 0.5 0.083811 -0.000699 + e X2Pi 4 1 -0.5 0.5
6 12381.905056 4 0.5 0.060462 -0.000699 + e X2Pi 5 1 -0.5 0.5
7 14492.276445 4 0.5 0.047367 -0.000697 + e X2Pi 6 1 -0.5 0.5
8 16503.167217 4 0.5 0.039468 -0.000694 + e X2Pi 7 1 -0.5 0.5
9 18413.775027 4 0.5 0.034505 -0.000687 + e X2Pi 8 1 -0.5 0.5
10 20223.296263 4 0.5 0.031348 -0.000679 + e X2Pi 9 1 -0.5 0.5
11 21930.957245 4 0.5 0.029391 -0.000669 + e X2Pi 10 1 -0.5 0.5
12 23535.998338 4 0.5 0.028279 -0.000659 + e X2Pi 11 1 -0.5 0.5
13 25037.635167 4 0.5 0.027810 -0.000646 + e X2Pi 12 1 -0.5 0.5
14 26435.111738 4 0.5 0.027863 -0.000627 + e X2Pi 13 1 -0.5 0.5
15 27727.807976 4 0.5 0.028375 -0.000599 + e X2Pi 14 1 -0.5 0.5


: State counting number.

: State energy in cm.

: Total statistical weight, equal to .

: Total angular momentum.

: Lifetime (s).

: Landé -factors.

: Total parity.

: Rotationless parity.

State: Electronic state.

: State vibrational quantum number.

: Projection of the electronic angular momentum.

: Projection of the electronic spin.

: Projection of the total angular momentum, .

Table 8: Extract from the states file of the SH line list.
(s)
1037 1051 5.28E-006 12167.591629
501 399 4.09E-006 12167.733027
1625 1633 1.56E-005 12168.620635
355 372 9.54E-006 12169.526762
868 828 7.14E-003 12170.134555
896 800 6.94E-003 12170.552236
1064 1024 4.92E-006 12170.716947
385 342 1.16E-005 12170.931908
175 252 1.27E-005 12171.792693
804 821 5.70E-007 12172.527689
205 222 1.29E-005 12172.803180
1528 1584 3.80E-004 12172.908871
1510 1520 1.61E-005 12173.738888
1552 1517 7.02E-006 12174.190774
620 520 7.76E-006 12174.773993
1551 1562 3.86E-004 12174.802908
112 69 1.58E-008 12174.960464
818 832 2.67E-005 12175.374485

: Upper state counting number;
: Lower state counting number;
: Einstein-A coefficient in s;
: transition wavenumber in cm.

Table 9: Extract from the transitions file of the SH line list.

3.1.2 Ns

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

Energy (cm) Parity e/f State
1 0.000000 6 0.5 inf -0.000767 + e X2Pi 0 1 -0.5 0.5
2 1204.267014 6 0.5 7.5810E-001 -0.000767 + e X2Pi 1 1 -0.5 0.5
3 2391.811399 6 0.5 3.7574E-001 -0.000767 + e X2Pi 2 1 -0.5 0.5
4 3562.543092 6 0.5 2.4892E-001 -0.000767 + e X2Pi 3 1 -0.5 0.5
5 4716.365128 6 0.5 1.8592E-001 -0.000767 + e X2Pi 4 1 -0.5 0.5
6 5853.187808 6 0.5 1.4840E-001 -0.000767 + e X2Pi 5 1 -0.5 0.5
7 6972.935476 6 0.5 1.2360E-001 -0.000767 + e X2Pi 6 1 -0.5 0.5
8 8075.513299 6 0.5 1.0604E-001 -0.000767 + e X2Pi 7 1 -0.5 0.5
9 9160.809357 6 0.5 9.3014E-002 -0.000767 + e X2Pi 8 1 -0.5 0.5
10 10228.711651 6 0.5 8.2993E-002 -0.000767 + e X2Pi 9 1 -0.5 0.5
11 11279.122350 6 0.5 7.5073E-002 -0.000767 + e X2Pi 10 1 -0.5 0.5
12 12311.955993 6 0.5 6.8677E-002 -0.000767 + e X2Pi 11 1 -0.5 0.5
13 13327.125579 6 0.5 6.3426E-002 -0.000767 + e X2Pi 12 1 -0.5 0.5
14 14324.536677 6 0.5 5.9056E-002 -0.000767 + e X2Pi 13 1 -0.5 0.5
15 15304.087622 6 0.5 5.5383E-002 -0.000767 + e X2Pi 14 1 -0.5 0.5
16 16265.666432 6 0.5 5.2272E-002 -0.000767 + e X2Pi 15 1 -0.5 0.5
17 17209.152585 6 0.5 4.9619E-002 -0.000767 + e X2Pi 16 1 -0.5 0.5
18 18134.433115 6 0.5 4.7350E-002 -0.000767 + e X2Pi 17 1 -0.5 0.5
19 19041.403694 6 0.5 4.5404E-002 -0.000767 + e X2Pi 18 1 -0.5 0.5
20 19929.953675 6 0.5 4.3734E-002 -0.000767 + e X2Pi 19 1 -0.5 0.5


: State counting number.

: State energy in cm.

: Total statistical weight, equal to .

: Total angular momentum.

: Lifetime (s).

: Landé -factors.

: Total parity.

: Rotationless parity.

State: Electronic state.

: State vibrational quantum number.

: Projection of the electronic angular momentum.

: Projection of the electronic spin.

: , projection of the total angular momentum.

Table 10: Extract from the states file for the line list of NS.
(s)
7738 7381 1.0816E-12 21129.956080
5205 5270 1.5211E-12 21129.964779
5098 5376 1.5212E-12 21129.965742
11335 11591 3.1996E-13 21129.967612
13834 13865 1.5996E-13 21129.972884
6281 5916 1.5599E-14 21129.973205
7633 7486 1.0819E-12 21129.974657
17595 17799 1.3308E-13 21130.010597
18641 18492 2.9298E-12 21130.032427
14941 14800 1.0700E-15 21130.038476
15071 14910 3.7284E-14 21130.042388
22202 21929 5.7979E-15 21130.049792
7889 7931 7.6269E-17 21130.090860
12655 12499 4.4455E-18 21130.105215
5327 5386 7.8126E-18 21130.120789
21381 21404 7.3821E-16 21130.145157
13229 12887 5.8001E-15 21130.148827
25770 25777 3.0568E-15 21130.159305
18726 18407 2.9242E-12 21130.204387
12449 12495 2.5432E-13 21130.208111
15334 15370 2.3191E-13 21130.223604


: Upper state counting number;
: Lower state counting number;
: Einstein-A coefficient in s;
: transition wavenumber in cm.

Table 11: Extract from the transition file for the line list of NS.

3.2 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):

(7)

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 SH and 3 for NS 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  K. For NS, we mainly miss the contributions from the   rovibronic states not considered here ( 24,524 cm, Gao et al. (2013)). This should not exceed 1 % judging by the contribution to from the   energies of NS above 24,524 cm.

SH NS
1.20403 1.10562
-0.47064 0.19180
2.88449 1.01142
-6.62867 -1.40930
7.73660 1.96534
-5.21848 -1.82956
2.16261 1.00323
-0.54158 -0.31160
0.07488 0.05081
-0.00437 -0.00338
Table 12: Expansion coefficients for the partition function given by Eq. (7). Parameters for other isotopologues can be found in the supplementary material.
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).

3.3 Spectra

3.3.1 Sh

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  K (HWHW = 0.01 cm) 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  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 11: Temperature dependence of our simulated SH absorption spectrum. A Gaussian line profile with HWHM=1 cm is used. The curves become flatter with increasing temperature.
Figure 12: Comparison of simulated spectra at 2000 K (HWHM=0.01 cm) using the new ExoMol line list for SH and experimental spectra by Winkel & Davis (1984).
Figure 13: Comparison of simulated spectra using the new ExoMol line list for SH and the CDMS database for the band.

3.3.2 Ns

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 and that of substituting S is up to 10 cm.

Figure 14: Temperature dependence of our simulated NS absorption spectrum. A Gaussian line profile with HWHM=1 cm is used. The curves become flatter with increasing temperature.
Figure 15: Comparison of spectra (298 K) for the fundamental band of NS for NS (left) and NS (right) against 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. In order to make this comparison, hyperfine 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 for the fundamental band is 0.049 D, while CDMS used  D from unpublished work by H. Müller.

Figure 16: Detailed Comparison our results with those given by CDMS (Müller et al., 2005) for pure rotational transitions (left) and the fundamental band (right).

3.4 Lifetimes

Lifetimes for states within the   ground state can be computed in straightforward manner from our line lists (Tennyson et al., 2016a). 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.

Figure 17: Lifetimes calculated for levels of SH and NS in their the   electronic states. For SH the lifetimes increase with vibrational excitation; for NS the longest lived states are for and the lifetimes decrease with vibrational excitation.

4 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, vibrational coverage up to 14 and rotational coverage to . For NS, the line lists contain 2.7 – 2.9 million transitions up to 23 000 cm and 31 000 – 33 000 states with a range up to the dissociation limit of 38 964.6 cm, with vibrational coverage up to =54 and rotational coverage to . 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   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 sulphur-containing molecules, namely CS (Paulose et al., 2015), SiS (Upadhyay et al., 2018), PS (Prajapat et al., 2017), HS (Azzam et al., 2016), SO (Underwood et al., 2016a) and SO (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.

5 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.

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