ExoMol line lists -- {XLVI}: Empirical rovibronic spectra of silicon mononitrate (SiN) covering the 6 lowest electronic states and 4 isotopologues

Silicon mononitride ($^{28}$Si$^{14}$N, $^{29}$Si$^{14}$N, $^{30}$Si$^{14}$N, $^{28}$Si$^{15}$N) line lists covering infrared, visible and ultraviolet regions are presented. The \name\ line lists produced by ExoMol include rovibronic transitions between six electronic states: \XS, \AS, \BS, \DS, \asi, \bsi. The \ai\ potential energy and coupling curves, computed at the multireference configuration interaction (MRCI/aug-cc-pVQZ) level of theory, are refined for the observed states by fitting their analytical representations to 1052 experimentally derived SiN energy levels determined from rovibronic bands belonging to the $X$--$X$, $A$--$X$ and $B$--$X$ electronic systems through the MARVEL procedure. The SiNful line lists are compared to previously observed spectra, recorded and calculated lifetimes, and previously calculated partition functions. SiNful is available via the \url{www.exomol.com} database.


INTRODUCTION
Silicon is considered to be the second most abundant element in Earths crust (CRC Handbook 2016) and the seventh most abundant element on the planet according to McDonough & Sun (1995); low silicon abundance within a host is considered to be a better indicator of a potential planet detection than planet-metallicity correlation according to Brugamyer et al. (2011). So far there have been multiple observations of SiN in different media in space: interstellar medium (Schilke et al. 2003;Turner & Dalgarno 1977;Feldman et al. 1983), red giant stars (Davis 1947;Gratton 1952), and envelopes of carbon stars (Turner 1992). Apart from being an important astrophysical species, SiN has multiple different applications: quantum dot production (Xie et al. 2017), quantum optomechanics (Serra et al. 2018), protective coating for biological tissues and dental implants (Pettersson et al. 2013;Raza et al. 2020), membranes used in filtering and biosensor systems (Vlassiouk et al. 2009;Lee et al. 2014).
A large number of theoretical investigations of SiN have been performed since the discovery of the molecule (Xing et al. 2018(Xing et al. , 2013Oyedepo et al. 2011;Li et al. 2009;Kerkines & Mavridis 2005;Jungnickel et al. 2000;Singh et al. 1999;Cai et al. 1998;Borin 1996;Chong 1994;Chen et al. 1993;Mclean et al. 1992;Curtiss et al. 1991;Melius & Ho 1991;Muller-Plathe & Laaksonen 1989;Ziurys et al. 1984;Bruna et al. 1984;Preuss et al. 1978;Gohel & Shah 1975). The most recent ab initio work at the time of writing was carried out by Xing et al. (2018) who conducted a thorough analysis of the transition dipole moment curves and potential energy curves for the lowest 8 doublets (X 2 Σ + , A 2 Π, B 2 Σ + , C 2 Π, D 2 ∆, E 2 ∆,F 2 Π, G 2 ∆), of SiN, additionally reporting on Frank-Condon factors and lifetimes for the first 15 vibrational levels of each electronic state, for which they used program LEVEL due to Le Roy (1998). The states were studied at the Internally Contracted Multirefrence Configuration Interaction with Davidson correction (icMRCI+Q) level with additional extrapolation procedure employed as described by Oyeyemi et al. (2014). A lot of work in the 2018 paper was built on the foundation of their earlier paper (Xing et al. 2013), where they reported PECs for the lowest 8 doublets (X 2 Σ + , A 2 Π, B 2 Σ + , C 2 Π, D 2 ∆, E 2 ∆, F 2 Π, G 2 ∆) and 4 quartets (a 4 Σ + , b 4 Π, c 4 ∆, d 4 Σ − ) and the lowest sextet state 1 6 Σ + . Apart form PECs this paper also reported spectroscopic constants and discussed the effects of the spin-orbit couplings.
The work described below will present original ab initio potential energy curves (PECs), spin-orbit curves (SOCs), electronic angular momentum curves (EAMCs) and (transition) dipole moment curves (DMCs) calculated using a high level of theory with MOLPRO (Werner et al. 2020) and then empirically fitted using data obtained from a MARVEL (measured active rotation vibration energy levels) ) procedure and generated from effective Hamiltonians using PGOPHER (Western 2017). The refined spectroscopic model is then used to compute molecular line lists for four isotopologues of SiN. Both the refinement and line list calculations are performed using the rovibronic program Duo (Yurchenko et al. 2016). We then use our line lists to give high accuracy comparison to previously reported experimental spectra, calculated and experimental lifetimes, and partition function.

Ab initio MODEL
In this work six lowest electronic states, namely X 2 Σ + , A 2 Π, B 2 Σ + , D 2 ∆, a 4 Σ + , b 4 Π, were studied using the state-averaged complete active space self consistent field (CASSCF) and multireference configuration interaction (MRCI) methods and augcc-pVQZ basis sets. The calculations were performed using C2v point group symmetry. Potential energy curves, illustrated in Fig. 1, spin-orbit coupling and electronic angular moment (Fig. 2), dipole moment curves and transition dipole moment curves (Fig. 3) were computed using MOLPRO2020 (Werner et al. 2020). If the MOLPRO calculations at some geometries did not converge, they were interpolated or extrapolated from the neighbouring points as part of the rovibronic calculations (see below). We used an adaptive ab initio grid consisting of 121 bond lengths ranging from 1.1 to 3.08Å with more points around the equilibrium region of the X 2 Σ + state centred at 1.585Å, see Figs. 1-3, where the density of the ab initio grid is shown. The grid points with the corresponding ab initio values (if converged) are included in the supplementary material.
For ab initio calculation we have considered the states of SiN correlating with two dissociation asymptotes, Si( 3 P) + N( 4 S) and Si( 3 P) + N( 2 D), which generate 2 Σ + (three states), 2 Π (one state), 4 Σ + (one states) and 4 Π (one state) which show six electronic states, X 2 Σ + , A 2 Π, B 2 Σ + , D 2 ∆, a 4 Σ + and b 4 Π which are mentioned above in ab initio model section. Under the C2v point group symmetry, A 1 corresponds to Σ + /∆. Thus, the 2 ∆ state arose from the calculation of 2 Σ + . Molecular orbitals for the subsequent CI calculations were obtained for each spin and symmetry species from state averaged CASSCF calculations  where the state averaging was achieved over all six electronic states considered in this work.
Within the C2v point group symmetry 14 (8σ, 3πx, 3πy) orbitals which contained 6 closed (4σ, 1πx, 1πy) orbitals were used for all ab initio calculations. Thus, the active space represents 8 active (4σ, 2πx, 2πy) orbitals with 9 active electrons and spans the valence orbitals 5σ-8σ,2π, 3π. All CI calculations were carried out with the internally contracted CI method ). The CI calculations used the same set of reference configurations used in the CASSCF calculation.
The EAMCs represent the Lx and Ly matrix elements, which are important for the accurate description of the lambda doubling effects in Π states originating from the interactions with the Σ and ∆ states. In Fig. 2 (right) the Lx components are shown, with Ly related to them by symmetry.
The definition of SOCs, which help form a complete and self-consistent description of a spectroscopic model, requires knowledge of the magnetic quantum numbers MS (i.e. the projection of the electronic spin Σ), which are specified in Table 1.
All of the ab initio curves are later mapped to a different, denser grid used in the solution of the rovibronic problem via interpolation and extrapolation as described in the Spectroscopic Model section below.
and are shown in Fig. 3. The phases of these non-diagonal TDMCs were selected to be consistent with the phases of the ab initio curves produced in our MOLPRO calculations (Patrascu et al. 2014). There is a discontinuity in X 2 Σ + |µx|B 2 Σ + at 2.5Å that we attribute to the interaction with the higher electronic states, which are not included in this model. Due to the large displacement from the equilibrium corresponding to very high energies this discontinuity does not provide any material effect on our results, as can be seen from the spectra reproduced below.

MARVEL
All available experimental transition frequencies of SiN were extracted from the published spectroscopic literature and analysed using the MARVEL procedure Császár et al. 2007;Furtenbacher & Császár 2012;Tóbiás et al. 2019). This procedure takes a set of assigned transition frequencies with measurement uncertainties and converts it into a consistent set of empirical energy levels with the uncertainties propagated from the input transitions. The transition data extracted as part of this work from the literature covers the three main bands of SiN involving the X 2 Σ + , A 2 Π, and B 2 Σ + electronic states: X 2 Σ + -X 2 Σ + , A 2 Π-X 2 Σ + and B 2 Σ + -X 2 Σ + , as summarised in Table 2.

Description of experimental sources
All the available sources of experimental transitions considered in this work are listed below: 92ElHaGu by Elhanine et al. (1992): an infrared (IR) study of the A 2 Π-X 2 Σ + system through 724 transitions. Unfortunately, the original line data with assignments were lost/unavailable and only the derived spectroscopic constants for A 2 Π and X 2 Σ + states remain. Using their X 2 Σ + constants we produced four X 2 Σ + -X 2 Σ + pseudo-experimental lines (MARVEL Magic numbers) to help connecting spectroscopic networks in the MARVEL analysis. We also used their extended set of the spectroscopic constants in PGOPHER (Western 2017) to generate pseudo-experimental energies for the first five vibrational states of A 2 Π and the first three vibrational states of X 2 Σ + states (up to J = 49.5) for the refinement of our spectroscopic model (see below). Elhanine et al. (1992) is the only existing information on the vibrationally excited A 2 Π states of SiN and is therefore crucial for providing MARVEL energies for states of A 2 Π with v > 1. The PGOPHER file used to generate the energies for fitting is provided as part of the supplementary information.
75Linton by Linton (1975): This UV study reported 460 lines of the C 2 Π-A 2 Π system. Sadly due to the low quality of this data and the lack of information provided in the paper to reconstruct the full set of quantum numbers required for MARVEL, Table 2. Breakdown of the assigned transitions by electronic bands for the sources used in this MARVEL study. A and V are the numbers of the available and validated transitions, respectively. The mean and maximum uncertainties (Unc.) obtained using the MARVEL procedure are given in cm −1 .
In total, 1987 experimental and 9 pseudo-experimental transitions were processed via the online MARVEL app (available through a user-friendly web interface at http://kkrk.chem.elte.hu/marvelonline) using the Cholesky (analytic) approach with a 0.5 cm −1 threshold on the uncertainty of the "very bad" lines. The final MARVEL process for 28 Si 14 N resulted in one main spectroscopic network, containing 1054 energy levels and 1456 validated transitions, with the rotational excitation up to J = 44.5 and covering energies up to 30 308 cm −1 . These energy levels in conjunction with energies generated from PGOPHER were used to refine our ab initio rovibronic spectroscopic model (PECs, SOCs and EAMCs) presented above. The MARVEL input transitions and output energy files are given as part of the supplementary material.

ROVIBRONIC CALCULATIONS
To obtain rovibronic energies and wavefunctions for the six electronic states in question, we solve a set of fully coupled Schrödinger equation for the motion of nuclei using the Duo program (Yurchenko et al. 2016). Duo uses the Hunds case a basis set with the vibrational basis functions obtained by solving uncoupled vibrational Schrödinger equations for each electronic state in question with using a sinc DVR basis set (Guardiola & Ros 1982). Atomic masses are used to represent the kinetic energy operator. In Duo calculations, an equidistant grid of 501 radial points ranging from 1.1 to 5.0Å. The ab initio curves were cubic-spline interpolated to map the ab initio curves onto the denser Duo grid. For the extension outside the ab initio bond lengths (r > 3.080Å), the ab initio curves were extrapolated using the functional forms given by (Yurchenko et al. 2016) f short other (r) = A + Br, for short range and f long other (r) = A/r 2 + B/r 3 for long range, where A and B are stitching parameters. A more detailed description of the Duo methodology was previously given by Yurchenko et al. (2016), see also Tennyson et al. (2016b).

Refining the spectroscopic model
For our spectroscopic model of SiN we initially used ab initio PECs for the doublets X 2 Σ + , A 2 Π, B 2 Σ + , D 2 ∆ and quartets b 4 Π, a 4 Σ + , as well as all appropriate SOCs and EAMCs of SiN. The X 2 Σ + , A 2 Π and B 2 Σ + PECs and the associated couplings were then refined by fitting to our empirical set of MARVEL and PGOPHER term values of 28 Si 14 N as described above.
For the refinements, the PECs for the X 2 Σ + , A 2 Π, and B 2 Σ + states were parameterised using the Extended Morse Oscillator (EMO) function (Lee et al. 1999) as given by where Ae − Ve = De is the dissociation energy, Ae is the corresponding asymptote, re is an equilibrium distance, and ξp is thě Surkus variable (Šurkus et al. 1984) given by with Ve = 0 for the X 2 Σ + state. The X 2 Σ + and A 2 Π states have a common asymptote which was fixed for the analytical form of the potential to the ground state dissociation energy De 4.75 eV based on the experiment by Naulin et al. (1993) (see also Kerkines & Mavridis (2005)). Similarly the B 2 Σ + state was adjusted to be 19233 cm −1 above the X 2 Σ + dissociation using the atomic excitation energy of N (Kramida et al. 2021). The processed and refined PECs used in Duo can be seen in Fig. 4. For the parameterisation of SOCs and the EAMCs, the ab initio curves were morphed using a polynomial decay expansion as given by: where z is the damped-coordinate polynomial given by: The refined curves f (r) are the represented as with B∞ = 1 in order for F (r) → 1 at r → ∞. Morphing allows one to retain the original shape of the property with a minimum  Figure 6. The residuals (Obs.-Calc.) between the experimentally determined energies of SiN (X 2 Σ + , A 2 Π and B 2 Σ + ) from our MARVEL analysis (open), pseudo-experimental PGOPHER (filled) generated energies and Duo energies corresponding to our refined spectroscopic model. Only MARVEL energies are available for B 2 Σ + , hence the vibrational level labels are not differentiated by source.
number of varied parameters, see e.g. Prajapat et al. (2017) and Yurchenko et al. (2018). In Eq. (6), r ref is a reference position chosen to be close to re of X 2 Σ + and β2 and β4 are damping factors, typically chosen to be 8 × 10 −1 and 2 × 10 −2 . The morphed and extended by Duo SOCs and EAMCs are shown in Fig. 5. The 1062 MARVEL and 854 PGOPHER energy levels values were used to refine the PECs, SOCs, EAMCs into analytical form as described above. Figure 6 shows the residuals representing how well our model compares to the MARVEL and PGOPHER energies. Unfortunately the most significant work from the number of transitions provided is Jenkins & de Laszlo (1928), where the accuracy of the transitions was 0.1 cm −1 at best. This is the main reason for the spread of uncertainties in the X 2 Σ + and B 2 Σ + states. Similarly most data for the A 2 Π state had to be supplemented using the PGOPHER calculations due to the difficulty of obtaining experimental data for the state.

Dipole moment curves
Most of the (transition) dipole moment curves from Fig. 3 were left unchanged and used as grid points in Duo calculations apart from the X 2 Σ + |µz|X 2 Σ + , which was fitted to analytical form of the polynomial decay given in Eq. (5). Having an analytical form helps to reduce the numerical noise arising due to interpolation of ab initio curves present in high overtone transitions, see Medvedev et al. (2016). The comparison between ab initio X 2 Σ + -X 2 Σ + DMC and its analytical form is shown in Fig. 7, with the deviations at very large radial displacement not affecting the intensities for the energy excitations selected for this study.
All expansion parameters or curves defining our spectroscopic model are given as supplementary material to the paper as a Duo input file.  Figure 7. Comparison of the ab initio and fitted DMC for the X 2 Σ + -X 2 Σ + system.

LINE LIST AND SIMULATIONS OF SPECTRA OF SIN
The SiNful line list was produced with Duo using the empirically refined and ab initio curves as described above. For the main isotopologue 28 Si 14 N, it contains 43 646 806 transitions and 131 935 states for X 2 Σ + , A 2 Π, B 2 Σ + , D 2 ∆, a 4 Σ + and b 4 Π, covering wavenumbers up to 58 000 cm −1 v = 0 . . . 30 and J = 0 . . . 245.5. For the isotopologue line lists only the atomic masses were adjusted in the Duo input files. Further details on the line list statistics covering the isotpologues can be seen in Table 3. The line list is provided in State and Transition files, as is customary for the ExoMol format (Tennyson et al. 2020). Extracts from the States and Trans files are shown in Tables 4 and 5, respectively; the full files are available from www.exomol.com. The States file contains energy term values, state uncertainties, Landé-g factors (Semenov et al. 2017), lifetimes (Tennyson et al. 2016a) and quantum numbers. The Transition file contains Einstein A coefficients. The partition functions are also included as part the standard line list compilation.
The calculated energies were replaced with the MARVEL values (MARVELised), where available. We have used the labels 'Ca','EH' and 'Ma' in the penultimate column of the States file to indicate if the energy value is calculated using Duo, derived using PGOPHER or MARVELised, respectively.
The uncertainty values in the States file correspond to two cases: the MARVEL uncertainties are used for MARVELised energies, while for the calculated values the following approximate expression is used: where a and b are electronic state dependent constant, given in Table 6. For the X 2 Σ + , A 2 Π and B 2 Σ + states uncertainties were estimated based on the progression of residuals from Fig. 6 as average increases of obs.-calc. in v and J for each state shown.

Overall Spectra
To demonstrate the accuracy of the SiNful line list, several spectra were calculated, analysed and compared to available laboratory measurements. Figure 8 illustrates the main bands of SiN at 2000 K. The dominance of the X 2 Σ + -A 2 Π in the 0-5000 cm −1 region confirms why it is hard to detect the comparatively weak X 2 Σ + -X 2 Σ + transitions. Additionally, while the X 2 Σ + -B 2 Σ + band system appears to be overall stronger than A 2 Π-B 2 Σ + , the latter should still be detectable in the 18000 -24000 cm −1 region according to our model, in line with observations of these vibronic bands by Ojha & Gopal (2006) and Foster et al. (1985). Additionally, in the region above 28000 cm −1 the A 2 Π -D 2 ∆ band becomes dominant, which agrees    with previous vibronic observations in this region by Bredohl et al. (1976). Figure 9 shows how the spectrum of 28 Si 14 N changes with increasing temperature.
5.2 B 2 Σ + -X 2 Σ + band B 2 Σ + -X 2 Σ + is the strongest electronic band in the system, with the largest number of experimental observations due to it being the easiest to detect. In Figure 10 we simulate the B 2 Σ + -X 2 Σ + (5,4) and (4,3) band at rotational temperature of 392 K and 412 K respectively to provide direct comparison with the experiment. The simulated spectra is also adjusted to be in air rather than in vacuum to align with the experimental spectra. This is achieved by using the IAU standard of conversion adopted by Morton (1991): where λair and λvac are wavelengths in air and vacuum respectively. The overall agreement on the rotational structure is within the uncertainty provided.

5.3
The A 2 Π-X 2 Σ + band system was first detected by Jevons (1913), however because of the vicinity of the A 2 Π and X 2 Σ + states it has proved difficult to study experimentally. Figure 11 shows the comparison of our simulated spectra of the A 2 Π-X 2 Σ + (1,0) band with the experiment of Foster et al. (1985). The spectrum is simulated at the effective temperature of 740 K. The position of the bands agrees to 0.05 cm −1 which is within our calculated uncertainties. Figure 12 shows a comparison of the simulated spectra (shown as sticks) for different silicon isotopes of the SiN molecule with the experimental spectra of Yamada et al. (1988). The overall agreement for 28 Si 14 Nis ∼0.016 cm −1 , for 29 Si 14 N is ∼0.035 cm −1 and for 30 Si 14 N is ∼0.055 cm −1 . The intensities were adjusted based on the natural abundance for each silicon isotope. These are defined as 92.2%, 4.68% and 3.09% for 28 Si 14 N , 29 Si 14 N, 30 Si 14 N respectively. Additionally, in the same range a Q22(0.5) line of A 2 Π-X 2 Σ + (0,0) for 29 Si 14 N should be present, but it was probably too weak to be observed during the experiment. This line is indicated with the an asterisk.  Figure 10. Simulated absorption spectra of SiN showing the B 2 Σ + -X 2 Σ + (5,4) and (4,3) bands at temperature of 392 K and 412 K respectively. A comparison to the experimental spectra from Schofield & Broida (1965) is provided. A Gaussian profile HWHM of 0.2 cm −1 was used.

Lifetimes
There are few precise experimental measurements of the lifetimes of different electronic states of SiN in the literature. Walkup et al. (1984) report the B 2 Σ + v = 1 vibrational state to have a lifetime of 200 ns ±10 ns (Trot = 500 K). Our calculated value for this vibronic state is lower: 130 ns (J = 0.5), slowly increasing to 150 ns for J = 100 ns and even 200 ns for J = 160.5 (v = 1, B 2 Σ + ). Additionally there have been several theoretical works (Xing et al. 2018;Kaur & Baluja 2015) which showed that for the A 2 Π state the lifetimes decrease with the vibrational levels as lower levels are much less energetically accessible due to the proximity of the X 2 Σ + state. This can be seen in more detail in  Figure 11. Simulated absorption spectrum of SiN at effective temperature of 740 K showing the A 2 Π-X 2 Σ + (1,0) band. A Gaussian profile of HWHM of 0.009 cm −1 was used. A comparison with the experimental spectra from Foster et al. (1985) is provided, with calculated spectra in red and experimental in black.  Walkup et al. (1984) for v ′ = 1 of B 2 Σ + . 0  1460  1660  124  129  1  610  679  130  136  2  385  429  137  143  3  281  312  143  150  4  221  245  150  157  5  182  202  156  164  6  155  172  163  170  7  135  149  169  176  8  121  132  175  181  9  109  119  181  187  10 98 .

Partition function
For the partition function of 28 Si 14 N, we follow the ExoMol and HITRAN (Gamache et al. 2017) convention and include the full nuclear spin. This means that our convention is different to that of recently reported by Barklem & Collet (2016) with whom we compare our results in Fig. 13. In order for comparison to be in equivalent convention, their partition function was multiplied by the factor of three as 14 N has a nuclear spin degeneracy of 3 and 28 S has nuclear spin degeneracy of 0. The differences at higher temperatures can be attributed to incompleteness of the model used by Barklem & Collet (2016), which is demonstrated by comparing partition functions by using only X 2 Σ + state, X 2 Σ + and A 2 Π states and all states: the partition  (Yamada et al. 1988).
function of Barklem & Collet (2016) appears to be based on the X 2 Σ + state only. As evident from Fig. 13, the contribution from the A 2 Π state cannot be neglected due to its low excitation energy. Partition functions in for T = 1 − 3000 K in steps of 1 K are available the 4 isotopologue, 28 Si 14 N, 29 Si 14 N, 30 Si 14 N and 28 Si 15 N, are available via the ExoMol website.

CONCLUSION
New IR and UV line lists called SiNful for isotopologues of SiN are presented. SiNful is available from www.exomol.com (Tennyson et al. 2020) and from www.zenodo.org (European Organization For Nuclear Research & OpenAIRE 2013). As part of the line list construction, a MARVEL analysis for 28 Si 14 N was performed. All experimental line positions from the literature (to the best of our knowledge) currently available for the X-X, A-X and B-X systems were processed to generate a MARVEL set of empirical energies of SiN. An accurate spectroscopic model for SiN was built: ab initio (T)DMC and empirically refined PECs, SOCs, EAMCs, using both previously derived MARVEL energies as well as PGOPHER generated energies.
The line list was MARVELised, where the theoretical energies are replaced with the MARVEL values (where available). The line list provides uncertainties of the rovibronic states in order to help in high-resolution applications. Comparisons of simulated spectra with experiment for both A-X and B-X system show close agreement. Additionally lifetimes, Landé-g factors and partition functions are provided. The importance of inclusion of the energies of the excited state A 2 Π when computing the partition function of SiN is demonstrated.  Figure 13. Comparison of the partition functions for 28 Si 14 N: values from this work and the work of Barklem & Collet (2016).

DATA AVAILABILITY
The data underlying this article are available in the article and in its online supplementary material, including (1) the Duo input files for each isotopologue containing all the potential energy, (transition) dipole moment and coupling curves of SiN used in this work, (2) the MARVEL data set, (3) digitalised line data from Linton (1975), (4) extracted ab initio curves as part of the excel, (5) PGOPHER file used to generate energy levels for two low lying states, (6) the temperature-dependent partition function of 28 Si 14 N up to 10000 K. The SiNful line lists for 28 Si 14 N, 29 Si 14 N, 30 Si 14 N and 28 Si 15 N are available from www.exomol.com.