# ExoMol line lists – XLIV. IR and UV line list for silicon monoxide (28Si16O)

Sergei N. Yurchenko1, Jonathan Tennyson1, Anna-Maree Syme2, Ahmad Y. Adam3,
Victoria H. J. Clark1, Bridgette Cooper1, C. Pria Dobney1, Shaun T. E.
Donnelly2, Maire N. Gorman2, Anthony E. Lynas-Gray1,5,6, Thomas Meltzer1, Alec
Owens1, Qianwei Qu1, Mikhail Semenov1,5, Wilfrid Somogyi1, Apoorva Upadhyay1,
Samuel Wright1, Juan C. Zapata Trujillo2
1 Department of Physics and Astronomy, University College London, Gower
Street, WC1E 6BT London, UK
2 School of Chemistry, University of New South Wales, 2052, Sydney, Australia
3 Physikalische und Theoretische Chemie, Fakultät für Mathematik und
Naturwissenschaften,
Bergische Universität Wuppertal, D-42097 Wuppertal, Germany
4 Department of Physics, University of Aberystwyth, Ceredigion, Wales SY23
3BZ, UK
5 Department of Physics, University of Oxford, Keble Rd., Oxford OX1 3RH, UK
6 Department of Physics and Astronomy, University of the Western Cape,
Bellville 7535, South Africa
7 Department of Science and Research, Moscow Witte University, 2nd
Kozhukhovskiy passage, Moscow, Russian Federation

###### Abstract

A new silicon monoxide (28Si16O) line list covering infrared, visible and
ultraviolet regions called SiOUVenIR is presented. This line list extends the
infrared EBJT ExoMol line list by including vibronic transitions to the
$A\,{}^{1}\Pi$ and $E\,{}^{1}\Sigma^{+}$ electronic states. Strong
perturbations to the $A\,{}^{1}\Pi$ band system are accurately modelled
through the treatment of 6 dark electronic states: $C\,{}^{1}\Sigma^{-}$,
$D\,{}^{1}\Delta$, $a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$,
$e\,{}^{3}\Sigma^{-}$ and $d\,{}^{3}\Delta$. Along with the
$X\,{}^{1}\Sigma^{+}$ ground state, these 9 electronic states were used to
build a comprehensive spectroscopic model of SiO using a combination of
empirical and ab initio curves, including the potential energy (PE), spin-
orbit (SO), electronic angular momentum (EAM) and (transition) dipole moment
curves. The ab initio PE and coupling curves, computed at the multireference
configuration interaction (MRCI) level of theory, were refined by fitting
their analytical representations to 2617 experimentally derived SiO energy
levels determined from 97 vibronic bands belonging to the $X$–$X$, $E$–$X$ and
$A$–$X$ electronic systems through the MARVEL procedure. 112 observed
forbidden transitions from the $C$–$X$, $D$–$X$, $e$–$X$, and $d$–$X$ bands
were assigned using our predictions, and these could be fed back into the
MARVEL procedure. The SiOUVenIR line list was computed using published ab
initio transition dipole moments for the $E$–$X$ and $A$–$X$ bands; the line
list is suitable for temperatures up to 10 000 K and for wavelengths longer
than 140 nm. SiOUVenIR is available from www.exomol.com and the CDS database.

###### keywords:

molecular data, opacity, astronomical data bases: miscellaneous, planets and
satellites: atmospheres, stars: low-mass.

††pagerange: ExoMol line lists – XLIV. IR and UV line list for silicon
monoxide (28Si16O)–LABEL:lastpage

##  1 Introduction

Silicon monoxide (SiO) has been observed in a wide variety of astronomical
environments since its original detection in the interstellar medium (Wilson
et al., 1971), and has been found in late-type stars (Cudaback et al., 1971)
in addition to being a key astrophysical maser (Snyder & Buhl, 1974).
Recently, there has been speculation that SiO will be present in the
atmospheres of exoplanets, notably in hot rocky planets close to their host
star where extreme temperatures cause vaporization of the silicate surface of
the planet (Schaefer et al., 2012). Significant quantities of SiO can
accumulate in a planetary atmosphere and a detection of SiO would shed light
on the interaction between the atmosphere and the crust of hot rocky
exoplanets at high temperatures (Herbort et al., 2020). Calculations have
shown that SiO absorption should dominate the infrared (IR) and ultraviolet
(UV) wavelength regions in the atmospheres of hot rocky super-Earths with
molten surfaces, with prominent features at 4, 10 and 100 $\mu$m (Ito et al.,
2015). SiO can also become a major opacity source at wavelengths shorter than
0.35 $\mu$m and dominate the transmission spectra of ultra-hot Jupiters
(Lothringer et al., 2020).

Any observation of SiO in exoplanets relies on accurate molecular opacity data
for this molecule. Currently, models use the SiO rotation-vibration-electronic
(rovibronic) line list of Kurucz (2011), which covers the
$X\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$,
$A\,{}^{1}\Pi$–$X\,{}^{1}\Sigma^{+}$ and
$E\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$ electronic bands. Computational
approaches to construct molecular line lists have undergone considerable
development since the Kurucz SiO line list (Tennyson, 2016; Tennyson &
Yurchenko, 2017). Notably, a robust computer program Duo (Yurchenko et al.,
2016) has been developed to compute the rovibronic spectra of diatomic
molecules and is able to treat multiple interacting electronic states and
their couplings, as is the case in SiO. Secondly, an efficient algorithm known
as MARVEL (Measured Active Rotational-Vibrational Energy Levels) (Furtenbacher
et al., 2007; Császár et al., 2007; Furtenbacher & Császár, 2012; Tóbiás et
al., 2019) can analyse all the available published spectroscopic data (line
positions) on a molecule and convert it into a consistent set of highly
accurate empirical-quality energy levels. These can be used to empirically
refine the spectroscopic model of a molecule, ultimately yielding line
positions that can meet the demands of high-resolution spectroscopy of
exoplanets (Birkby, 2018) and be incorporated into the line list directly by
replacing calculated values. We thus find it worthwhile to generate a new SiO
rovibronic line list that exploits these tools within the ExoMol computational
framework (Tennyson & Yurchenko, 2018). The line list has been produced for
the ExoMol Database (Tennyson & Yurchenko, 2012; Tennyson et al., 2016b,
2020), which is providing key molecular data on a range of important molecules
for the study of exoplanets and other hot atmospheres.

Experimental sources of SiO spectroscopic data selected for this work include
those of Törring (1968); Elander & Lagerqvist (1971); Field et al. (1976);
Bredohl et al. (1976); Manson et al. (1977); Lovas et al. (1981);
Mollaaghababa et al. (1991); Wallace & Livingston (1992); Tsuji et al. (1994);
Campbell et al. (1995); Wallace et al. (1996); Sanz et al. (2003) and Müller
et al. (2013). Experimentally, the $A\,{}^{1}\Pi$ – $X\,{}^{1}\Sigma^{+}$ band
of SiO has been extensively studied in the 1970s by Field et al. (1976) and
Bredohl et al. (1976). This band undergoes strong perturbations with nearby
‘dark’ electronic states $C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$,
$a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$, $e\,{}^{3}\Sigma^{-}$ and
$d\,{}^{3}\Delta$, i.e. states with transitions forbidden from the
$X\,{}^{1}\Sigma^{+}$ state. These perturbations were thoroughly analysed by
Field et al. (1976) and Bredohl et al. (1976) with a significant number of
transitions to dark states characterised. This valuable information plays an
important role in our calculations as it allows us to use experimental
constraints for these 6 dark states, which are otherwise not available.

The strong $E\,{}^{1}\Sigma^{+}$ – $X\,{}^{1}\Sigma^{+}$ system of 28Si16O was
studied by Elander & Lagerqvist (1971) who reported rovibronic transitions up
to vibrational levels with $v^{\prime}=14$. For this system, the perturbations
were not reported with the necessary level of detail and are therefore not
covered in our work. The $F$–$X$, $G$–$X$, $I$–$X$, $K$–$X$, $J$–$X$, $M$–$X$,
$L$–$X$, $N$–$X$, $P$–$X$, $O$–$X$ band systems have also been observed
experimentally (Lagerqvi et al., 1973), but not at a resolution high enough
for our analysis. These data were not included in the present study.

Lifetimes of SiO states were studied experimentally by Smith & Liszt (1972);
Elander & Smith (1973) and comparison with these data provides an important
test of the transition intensities predicted by our model.

Some of the key ab initio studies of SiO are by Langhoff & Bauschlicher
(1993); Drira et al. (1998); Chattopadhyaya et al. (2003); Guo-Liang et al.
(2008); Shi et al. (2012); Bauschlicher (2016); Feng & Zhu (2019a) and Feng &
Zhu (2019b). The most valuable is the recent work by Bauschlicher (2016) who
reported high level potential energy and transition dipole moment curves of
SiO, which we have made extensive use of in this work.

We present a new 28Si16O line list, called SiOUVenIR, built using an accurate
spectroscopic model: empirical potential energy curves (PECs), spin-orbit
curves, electronic angular momentum curves (EAMCs) and ab initio dipole moment
curves (DMCs) and transition dipole moment curves (TDMCs). As part of the
construction of the empirical curves, a MARVEL (Measured Active Rotational-
Vibrational Energy Levels) set of empirical energies for 28Si16O are produced.
The line list covers the $X\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$,
$A\,{}^{1}\Pi$–$X\,{}^{1}\Sigma^{+}$ and
$E\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$ electronic systems and includes
transitions from the ‘dark’ bands. The SiOUVenIR line list is suitable for
temperatures up to 10 000 K and spans wavelengths longer than 140 nm (72 000
cm-1) The SiOUVenIR line list builds upon the previous ExoMol study of Barton
et al. (2013), which produced the EBJT rotation-vibration line lists for the
ground electronic state $X\,^{1}\Sigma^{+}$ of the main isotopologue 28Si16O
and the four monosubstituted isotopologues 29Si16O, 30Si16O, 28Si18O, and
28Si17O.

This paper is structured as follows: In Section 2, we perform an analysis of
all the published spectroscopic literature on SiO using the MARVEL procedure.
The underlying potential energy curves (PECs), dipole moment curves (DMCs),
and angular momentum coupling curves of our SiO spectroscopic model are
discussed in Section 3. The solution of the coupled Schrödinger equations
using Duo (Yurchenko et al., 2016), fitting of the curves via deperturbation
of states and the production of the rovibronic line list are discussed in
Section 4. The line list and its applications, including lifetimes and
comparisons with lab and stellar spectra are presented in Section 5. We
conclude in Section 6.

##  2 MARVEL

All available experimental transition frequencies of SiO were extracted from
the published spectroscopic literature and analysed using the MARVEL algorithm
(Furtenbacher et al., 2007; 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-quality energy levels with the uncertainties propagated from
the input transitions. The SiO data extracted from the literature covers the
three main bands involving the $X\,^{1}\Sigma^{+}$, $A\,^{1}\Pi$, and
$E\,^{1}\Sigma^{+}$ electronic states:
$X\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$,
$A\,{}^{1}\Pi$–$X\,{}^{1}\Sigma^{+}$ and
$E\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$ as summarised in Table 1. Due to
interactions with other electronic states, 112 of the measured rovibronic
transitions belong to the four forbidden systems
$C\,{}^{1}\Sigma^{-}$–$X\,{}^{1}\Sigma^{+}$,
$D\,{}^{1}\Delta$–$X\,{}^{1}\Sigma^{+}$,
$e\,{}^{3}\Sigma^{-}$-$X\,{}^{1}\Sigma^{+}$ and
$d\,{}^{3}\Delta$-$X\,{}^{1}\Sigma^{+}$, which are usually not identified in
the experimental data but can be assigned using our variational calculations.
It is important for MARVEL that all transitions are interconnected into a
single network, which is not always the case. Here we include the empirical
energies of SiO from the accurate ExoMol EBJT line list (Barton et al., 2013)
for $v\leq{7}$ and $J\leq 100$ in order to fill gaps and connect all
experimental data into a single network. According to our experience, EBJT
calculated energies are accurate enough for this task.

###  2.1 Description of experimental sources

Figure 1: Spread of uncertainties across transition wavenumber for each
electronic band within SiO. Forbidden bands are grouped together in the bottom
right plot. Note the different y axis for the
$X\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$subplot (top left panel).  Table 1:
Breakdown of the assigned transitions by electronic bands for the sources used
in this study. A is the number of available transitions and V is the number of
validated. The mean and maximum uncertainties obtained using the MARVEL
procedure are given in cm-1.

Electronic Band |  Vibrational Bands |  $J$ Range |  Wavenumber Range cm-1 |  Mean/Max |  A/V
---|---|---|---|---|---
13BaYuTe |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (0-0), (1-0), (1-1), (2-0), (2-1), (2-2), (3-0), (3-1), (3-2), (3-3), (4-0), (4-1), (4-2), (4-3), (4-4), (5-0), (5-1), (5-2), (5-3), (5-4), (5-5), (6-0), (6-1), (6-2), (6-3), (6-4), (6-5), (7-1), (7-2), (7-3), (7-4), |  0 - 103 |  1.4 - 7216.6 |  7.37e-02/2.28e-01 |  3426/3426
68Torring |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (0-0), (1-1), (2-2), (3-3), |  0 - 1 |  1.4 - 1.4 |  3.27e-06/4.00e-06 |  4/4
71ElLa |  |  |  |  |
E ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (0-0), (0-1), (0-2), (1-0), (1-1), (1-2), (10-0), (11-0), (12-0), (13-1), (14-1), (2-0), (2-1), (2-2), (3-0), (3-1), (4-0), (5-0), (6-0), (7-0), (8-0), (9-0), |  2 - 64 |  49762.1 - 60025.8 |  1.91e-01/9.20e-01 |  2079/2076
76FiLaRe |  |  |  |  |
A ${}^{1}\Pi$-X ${}^{1}\Sigma^{+}$ |  (0-0), (1-0), (2-0), (3-0), (4-0), (5-0), (6-0), (7-0), (8-0), (9-0), |  0 - 49 |  42439.1 - 49752.3 |  5.60e-02/5.00e-01 |  1032/1032
D ${}^{1}\Delta$-X ${}^{1}\Sigma^{+}$ |  (6-0), |  35 - 36 |  42453.1 - 42558.4 |  5.00e-02/5.00e-02 |  3/3
d ${}^{3}\Delta$-X ${}^{1}\Sigma^{+}$ |  (14-0), (15-0), (9-0), |  11 - 41 |  42487.9 - 46720.6 |  5.88e-02/2.00e-01 |  34/34
e ${}^{3}\Sigma^{-}$-X ${}^{1}\Sigma^{+}$ |  (14-0), (15-0), (7-0), (9-0), |  9 - 38 |  44089.1 - 48255.8 |  5.08e-02/6.00e-02 |  24/24
C ${}^{1}\Sigma^{-}$-X ${}^{1}\Sigma^{+}$ |  (10-0), (11-0), |  13 - 42 |  44898.4 - 45912.8 |  5.00e-02/5.00e-02 |  9/9
76BrReCo |  |  |  |  |
D ${}^{1}\Delta$-X ${}^{1}\Sigma^{+}$ |  (6-1), (6-2), (6-3), |  34 - 36 |  38821.0 - 41349.4 |  3.18e-01/5.50e-01 |  18/18
A ${}^{1}\Pi$-X ${}^{1}\Sigma^{+}$ |  (0-1), (0-2), (0-3), |  28 - 39 |  38825.2 - 41372.0 |  3.82e-01/9.00e-01 |  42/42
d ${}^{3}\Delta$-X ${}^{1}\Sigma^{+}$ |  (9-1), (9-2), (9-3), |  28 - 39 |  38841.4 - 41374.5 |  3.39e-01/7.10e-01 |  24/24
77MaClDe |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (0-0), (1-1), (2-2), (3-3), (4-4), |  1 - 7 |  2.9 - 10.1 |  1.00e-05/1.00e-05 |  17/17
81LoMaOI |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (1-0), (2-1), (3-2), (4-3), (5-4), |  0 - 61 |  1213.0 - 1294.0 |  1.48e-03/3.00e-03 |  35/35
91MoGoVr |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (1-1), (10-10), (11-11), (13-13), (14-14), (15-15), (16-16), (17-17), (18-18), (19-19), (2-2), (20-20), (21-21), (22-22), (25-25), (3-3), (30-30), (31-31), (32-32), (33-33), (34-34), (36-36), (37-37), (38-38), (39-39), (4-4), (40-40), (5-5), (6-6), (7-7), (8-8), (9-9), |  5 - 9 |  7.6 - 11.4 |  3.56e-07/5.67e-07 |  62/62
92WaLi |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (2-0), (3-1), (4-2), (5-3), (6-4), |  3 - 101 |  2381.4 - 2497.3 |  1.01e-02/2.00e-02 |  507/507
94TsOhHi |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (2-0), (3-1), (4-2), (5-3), |  2 - 120 |  2420.3 - 2486.6 |  1.10e-02/2.00e-02 |  50/50
95CaKlDua |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (1-0), (10-9), (101-100), (102-101), (103-102), (104-103), (105-104), (106-105), (11-10), (12-11), (13-12), (2-1), (3-2), (4-3), (5-4), (6-5), (61-60), (62-61), (63-62), (64-63), (65-64), (66-65), (7-6), (8-7), (9-8), |  0 - 140 |  926.6 - 1241.3 |  9.52e-03/5.00e-02 |  1661/1622
95CaKlDu:lab |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (1-0), (2-1), (3-2), (4-3), (5-4), |  1 - 82 |  1080.9 - 1311.9 |  5.54e-04/6.30e-03 |  375/375
03SaMcTh |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (0-0), (1-1), (10-10), (11-11), (12-12), (13-13), (14-14), (15-15), (16-16), (17-17), (18-18), (19-19), (2-2), (20-20), (21-21), (22-22), (23-23), (24-24), (25-25), (26-26), (27-27), (28-28), (29-29), (3-3), (30-30), (31-31), (32-32), (33-33), (34-34), (35-35), (36-36), (37-37), (38-38), (39-39), (4-4), (40-40), (41-41), (42-42), (43-43), (44-44), (45-45), (5-5), (6-6), (7-7), (8-8), (9-9), |  0 - 1 |  1.0 - 1.4 |  1.02e-07/6.00e-07 |  46/46
13MuSpBi |  |  |  |  |
X ${}^{1}\Sigma^{+}$-X ${}^{1}\Sigma^{+}$ |  (0-0), (1-1), (2-2), (3-3), (4-4), (46-46), (47-47), (48-48), (49-49), (5-5), (50-50), (51-51), |  0 - 19 |  0.9 - 27.5 |  1.77e-07/5.00e-07 |  29/29

a From Müller et al. (2013).

Experimental transitions from the ground state system $X$–$X$ are the same as
used by Barton et al. (2013) to construct the EBJT line list. The other
sources, denoted by the MARVEL tag, are follows:

13MuSpBi (Müller et al., 2013): reported 29 rotational microwave transitions
of the $X$–$X$ system for a number of isotopologues of SiO, including 28Si16O,
for vibrational levels with $v$ up to 51. They have also compiled an extensive
source of all important spectroscopic data for $X$–$X$ of SiO from other
sources, some of which were sourced to the original lab or observational data,
for example their version of the sunspot IR line positions by Campbell et al.
(1995) which appears to give more decimal places.

68Torring: Four microwave transitions by Törring (1968).

77MaClDe: 17 microwave transitions by Manson et al. (1977).

81LoMaOl: 35 lines in the 1240 cm-1 region by Lovas et al. (1981).

91MoGoVr: Microwave data (62 lines) by Mollaaghababa et al. (1991) taken from
the compilation by Müller et al. (2013).

94TsOhHi: IR data (50 lines) by Tsuji et al. (1994) from the region of the
first overtone, 2400 cm-1, observed in spectra of giant M-stars.

95CaKlDu:lab and 95CaKlDu:sunspot: IR sunspot spectra of SiO by Campbell et
al. (1995), which were then recompiled by Müller et al. (2013). Here we use
the data provided by Müller et al. (2013), which appeared in two parts, as lab
and sunspot data. We retain this structure in our analysis.

92WaLi: 507 IR line positions covering the region of the 1st overtone band
(2400 cm-1) of 28Si16O were extracted from the IR Spectral Atlases of the Sun
by Wallace & Livingston (1992), see Wallace et al. (1996). There is a further
sunspot Atlas due to Wallace et al. (1994) which contains identified SiO
lines. However this Atlas does not provide line positions so it was not used.

03SaMcTh: 46 rotational (microwave) transitions of SiO by Sanz et al. (2003)
covering high vibrational excitations, up to $v=44$.

76FiLaRe,76BrReCo (Field et al., 1976; Bredohl et al., 1976): The
$A\,{}^{1}\Pi$ – $X\,{}^{1}\Sigma^{+}$ system was reported by two main
sources, Field et al. (1976), (0-0), (1-0), (2-0), (3-0), (4-0), (5-0), (6-0),
(7-0), (8-0), (9-0), 1102 lines, and Bredohl et al. (1976) (0-1), (0-2),
(0-3), 84 lines. This system is heavily perturbed by $C\,{}^{1}\Sigma^{-}$,
$D\,{}^{1}\Delta$, $a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$,
$e\,{}^{3}\Sigma^{-}$ and $d\,{}^{3}\Delta$. In this study, a significant
number of forbidden transitions to these dark states were measured, which were
all assigned as additional $A$–$X$ transitions. MARVEL requires that all
transitions are uniquely identified. To this end we initially gave a dummy
label $A^{\prime}$ to all duplicates, which were then properly resolved by
comparing to the calculated values using Duo (see below). These data are very
important as they provide (the only) access to these forbidden states, which
strongly affect the behavior of the ‘bright’ states.

71ElLa: UV data covering the $E$–$X$ system by Elander & Lagerqvist (1971),
2079 lines, $v^{\prime}_{\rm max}=14$.

13BaYuTe: 3426 pseudo-experimental values for $v=0\ldots 7$ and $J=0\ldots
100$ constructed from the EBJT energies as ‘transitions’ from the zero-point
state, $X\,{}^{1}\Sigma^{+}$, $v=0$, $J=0$ to fill gaps in the MARVEL set and
connect otherwise disjoint clusters (so called ‘orphans’ Császár &
Furtenbacher (2011)). They play the same role as MARVEL’s ‘Magic numbers’. The
uncertainties $\sigma$ of these pseudo-experimental energies were decided
based on the following scheme:

$\sigma=\left\\{\begin{array}[]{cc}0.01+0.0005J(J+1)&v\leq 6,\\\
0.2+0.0005J(J+1)&v=7\end{array}\right.$

The vibrational analysis of the higher $E$–$X$, $G$–$X$ band systems by Barrow
& Rowlinson (1954) was not included in the current study as this work does not
contain line positions.

Figure 2: The spread of uncertainties for each vibronic state.

In total, 6051 experimental and 3426 pseudo-experimental (13BaYuTe)
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 spread of transition wavenumbers and their
uncertainties is shown in Fig. 1, split by the electronic bands of the
transitions. Unsurprisingly the $X\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$
band has the lowest uncertainties, reaching an uncertainty of $10^{-7}$ cm-1
for rotational transitions from Sanz et al. (2003) and Müller et al. (2013).
The final MARVEL process for SiO resulted in one main spectroscopic network,
shown in Fig. 3, containing 2617 energy levels with rotational excitation up
to $J=103$ for molecular states below 61 881 cm-1. The energy levels are
described in Table 2, giving the $J$ and energy range for each vibronic state,
as well as the sources of data that contribute to these energy levels. The
spread of uncertainties for each vibronic state is shown in Fig. 2, clearly
showing the higher accuracy and reliability of the $X\,{}^{1}\Sigma^{+}$ state
energy levels. These energy levels were used in Duo to refine our rovibronic
spectroscopic model (PECs, SOCs and EAMCs) corresponding to all states but
$X\,{}^{1}\Sigma^{+}$, which was kept unchanged. We also did not include the
EAM coupling into the fit between $A$ and $X$ in order not to destroy the
integrity of the $X$ PEC. The MARVEL input transitions and output energy files
are given as part of the supplementary material. Figure 3 is an illustration
of the MARVEL spectroscopic network where the nodes show the ro-vibrationic
energy levels of each electronic state and the lines between them represent
the transitions. Almost all energy levels from higher electronic states
transition to low vibrational states from $X\,{}^{1}\Sigma^{+}$ which leads to
the many offshoots of higher ro-vibrational levels of the
$X\,{}^{1}\Sigma^{+}$ state.

Figure 3: Rovibrational spectroscopic networks for SiO. Each node represents a
unique rovibronic energy level with the color indicating the electronic state
and the connections between nodes are from input transition data. Table 2:
Summary of experimentally-derived MARVEL energy levels, including
uncertainties and data sources in SiO. No is the number of energy levels in
that vibronic state; Mean/Max are the mean and maximal values of
uncertainties.

$v$ |  $J$ range |  E-range (cm-1) |  No |  Mean/Max |  Sources
---|---|---|---|---|---
X ${}^{1}\Sigma^{+}$ |  |  |  |  |
0 |  0 - 103 |  0.0 - 7643.7 |  104 |  0.001/0.007 |  03SaMcTh, 13MuSpBi, 13BaYuTe, 68Torring, 77MaClDe, 71ElLa, 76FiLaRe, 81LoMaOI, 92WaLi, 94TsOhHi, 95CaKlDu:lab, 95CaKlDu:sunspot
1 |  0 - 102 |  1229.6 - 8675.5 |  102 |  0.002/0.028 |  03SaMcTh, 13MuSpBi, 13BaYuTe, 68Torring, 77MaClDe, 81LoMaOI, 91MoGoVr, 95CaKlDu:lab, 95CaKlDu:sunspot, 71ElLa, 76BrReCo, 92WaLi, 94TsOhHi
2 |  0 - 103 |  2447.3 - 9983.1 |  104 |  0.002/0.01 |  03SaMcTh, 13MuSpBi, 13BaYuTe, 68Torring, 77MaClDe, 81LoMaOI, 91MoGoVr, 92WaLi, 94TsOhHi, 95CaKlDu:lab, 95CaKlDu:sunspot, 71ElLa, 76BrReCo
3 |  0 - 103 |  3653.2 - 11135.1 |  102 |  0.003/0.01 |  03SaMcTh, 13MuSpBi, 13BaYuTe, 68Torring, 77MaClDe, 81LoMaOI, 91MoGoVr, 92WaLi, 94TsOhHi, 95CaKlDu:lab, 95CaKlDu:sunspot, 76BrReCo
4 |  0 - 103 |  4847.2 - 12275.3 |  104 |  0.004/0.02 |  03SaMcTh, 13MuSpBi, 13BaYuTe, 77MaClDe, 81LoMaOI, 91MoGoVr, 92WaLi, 94TsOhHi, 95CaKlDu:lab, 95CaKlDu:sunspot
5 |  0 - 102 |  6029.5 - 13264.1 |  103 |  0.005/0.014 |  03SaMcTh, 13MuSpBi, 13BaYuTe, 81LoMaOI, 91MoGoVr, 92WaLi, 94TsOhHi, 95CaKlDu:lab, 95CaKlDu:sunspot
6 |  0 - 101 |  7200.0 - 14244.5 |  102 |  0.005/0.01 |  03SaMcTh, 13BaYuTe, 91MoGoVr, 92WaLi, 95CaKlDu:sunspot
7 |  0 - 100 |  8358.8 - 15216.6 |  82 |  0.005/0.03 |  03SaMcTh, 13BaYuTe, 91MoGoVr, 95CaKlDu:sunspot
8 |  3 - 99 |  9514.1 - 16180.3 |  70 |  0.006/0.02 |  03SaMcTh, 91MoGoVr, 95CaKlDu:sunspot
9 |  4 - 98 |  10654.9 - 17135.6 |  60 |  0.007/0.02 |  03SaMcTh, 91MoGoVr, 95CaKlDu:sunspot
10 |  5 - 91 |  11785.4 - 17338.4 |  47 |  0.005/0.02 |  03SaMcTh, 91MoGoVr, 95CaKlDu:sunspot
11 |  5 - 83 |  12897.5 - 17493.7 |  41 |  0.006/0.02 |  03SaMcTh, 91MoGoVr, 95CaKlDu:sunspot
12 |  8 - 80 |  14026.0 - 18239.5 |  22 |  0.01/0.03 |  03SaMcTh, 95CaKlDu:sunspot
13 |  9 - 60 |  15126.7 - 17466.3 |  16 |  0.014/0.04 |  03SaMcTh, 91MoGoVr, 95CaKlDu:sunspot
A ${}^{1}\Pi$ |  |  |  |  |
0 |  2 - 47 |  42643.8 - 44047.9 |  85 |  0.054/0.275 |  76BrReCo, 76FiLaRe
1 |  2 - 45 |  43483.9 - 44759.0 |  85 |  0.052/0.141 |  76FiLaRe
2 |  1 - 49 |  44306.4 - 45803.1 |  92 |  0.043/0.05 |  76FiLaRe
3 |  1 - 47 |  45124.0 - 46485.8 |  92 |  0.044/0.05 |  76FiLaRe
4 |  1 - 49 |  45925.8 - 47388.4 |  92 |  0.043/0.05 |  76FiLaRe
5 |  1 - 43 |  46715.6 - 47834.2 |  76 |  0.044/0.078 |  76FiLaRe
6 |  1 - 42 |  47494.4 - 48551.2 |  79 |  0.045/0.057 |  76FiLaRe
7 |  3 - 34 |  48264.5 - 48949.5 |  45 |  0.046/0.05 |  76FiLaRe
8 |  3 - 31 |  49021.0 - 49583.2 |  55 |  0.045/0.064 |  76FiLaRe
9 |  4 - 21 |  49766.3 - 50012.0 |  26 |  0.05/0.05 |  76FiLaRe
C ${}^{1}\Sigma^{-}$ |  |  |  |  |
10 |  40 - 42 |  46125.4 - 46203.1 |  3 |  0.05/0.05 |  76FiLaRe
11 |  13 - 18 |  46044.6 - 46122.6 |  6 |  0.05/0.05 |  76FiLaRe
D ${}^{1}\Delta$ |  |  |  |  |
6 |  35 - 36 |  43433.3 - 43469.6 |  4 |  0.074/0.127 |  76BrReCo, 76FiLaRe
E ${}^{1}\Sigma^{+}$ |  |  |  |  |
0 |  2 - 55 |  52581.6 - 54241.7 |  54 |  0.104/0.283 |  71ElLa
1 |  3 - 64 |  53252.4 - 55463.2 |  62 |  0.089/0.212 |  71ElLa
2 |  2 - 59 |  53907.8 - 55775.0 |  58 |  0.097/0.6 |  71ElLa
3 |  3 - 61 |  54561.1 - 56531.4 |  59 |  0.105/0.2 |  71ElLa
4 |  1 - 59 |  55198.0 - 57029.0 |  59 |  0.149/0.503 |  71ElLa
5 |  4 - 58 |  55840.5 - 57583.7 |  55 |  0.144/0.283 |  71ElLa
6 |  2 - 56 |  56458.2 - 58075.2 |  55 |  0.14/0.2 |  71ElLa
7 |  5 - 55 |  57087.1 - 58619.4 |  51 |  0.137/0.2 |  71ElLa
8 |  3 - 53 |  57687.6 - 59104.2 |  51 |  0.139/0.2 |  71ElLa
9 |  1 - 53 |  58281.3 - 59689.4 |  53 |  0.137/0.212 |  71ElLa
10 |  4 - 56 |  58881.7 - 60425.5 |  53 |  0.132/0.141 |  71ElLa
11 |  1 - 51 |  59456.0 - 60734.1 |  51 |  0.142/0.2 |  71ElLa
12 |  6 - 50 |  60049.3 - 61245.7 |  45 |  0.139/0.2 |  71ElLa
13 |  10 - 47 |  60647.2 - 61660.3 |  38 |  0.139/0.2 |  71ElLa
14 |  15 - 39 |  61263.9 - 61881.1 |  25 |  0.127/0.141 |  71ElLa
d ${}^{3}\Delta$ |  |  |  |  |
9 |  29 - 39 |  43187.3 - 43615.4 |  6 |  0.11/0.136 |  76BrReCo, 76FiLaRe
14 |  37 - 41 |  46770.1 - 46957.8 |  3 |  0.05/0.05 |  76FiLaRe
15 |  12 - 24 |  46816.1 - 47073.3 |  20 |  0.044/0.05 |  76FiLaRe
e ${}^{3}\Sigma^{-}$ |  |  |  |  |
7 |  15 - 15 |  48403.6 - 48403.6 |  1 |  0.05/0.05 |  76FiLaRe
9 |  26 - 37 |  44743.3 - 45160.3 |  10 |  0.048/0.05 |  76FiLaRe
14 |  38 - 38 |  48367.7 - 48367.7 |  1 |  0.05/0.05 |  76FiLaRe
15 |  9 - 16 |  48319.4 - 48421.0 |  8 |  0.046/0.05 |  76FiLaRe

##  3 Spectroscopic Model of SiO

###  3.1 Potential energy, spin orbit and electronic angular momentum curves

The initial singlet $A\,{}^{1}\Pi$, $C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$,
$E\,{}^{1}\Sigma^{+}$ PECs of SiO were taken from Bauschlicher (2016), see
Fig. 4. The initial PECs for the triplet states (Fig. 4) and initial SOCs and
EAMCs shown in Fig. 5 were computed ab initio in this work using the MRCI/aug-
cc-pV5Z level of theory. The ab initio curves were then refined by fitting to
MARVEL energies. For the $X\,{}^{1}\Sigma^{+}$ PEC we used the empirical curve
from Barton et al. (2013).

The ab initio calculations in this work were performed using the internally
contracted multireference configuration interaction (IC-MRCI-F12c) approach
with the F12-optimised correlation consistent basis set, QZ-F12 (Peterson et
al., 2008) in the frozen core approximation. The active space and state-
averaging was chosen as in Bauschlicher (2016): occupied (8,3,3,0), closed
(5,1,1,0) with two $A_{1}$ and one $\Pi$ states.

Calculations employed the diagonal fixed amplitude ansatz 3C(FIX) (Ten-No,
2004) and a Slater geminal exponent value of $\beta=1.0$ $a_{0}^{-1}$ (Hill et
al., 2009). MOLPRO2015 (Werner et al., 2012) was used for all electronic
structure calculations. A dense grid of 130 Si–O bond lengths was used.

In Duo all PECs were represented using an Extended Morse Oscillator (EMO)
function (Lee et al., 1999) as given by

$V(r)=V_{\rm e}\;\;+\;\;(A_{\rm e}-V_{\rm
e})\left[1\;\;-\;\;\exp\left(-\sum_{k=0}^{N}B_{k}\xi_{p}^{k}(r-r_{\rm
e})\right)\right]^{2},$ (1)

where $A_{\rm e}-V_{\rm e}=D_{\rm e}$ is the dissociation energy, $A_{\rm e}$
is the corresponding asymptote, $r_{\rm e}$ is an equilibrium distance of the
PEC, and $\xi_{p}$ is the Šurkus variable (Šurkus et al., 1984) given by

$\xi_{p}=\frac{r^{p}-r^{p}_{\rm e}}{r^{p}+r^{p}_{\rm e}}$ (2)

with $V_{\rm e}=0$ for the $X\,{}^{1}\Sigma^{+}$ state. All states except
$A\,{}^{1}\Pi$ have a common dissociation limit which we fixed to the ground
state dissociation energy $D_{\rm e}$ 8.26 eV given by Huber & Herzberg
(1979); this value was used to define the $X$ PEC in (Barton et al., 2013). A
mass spectrometric dissociation energy by Hildenbrand & Murad (1969) gives
$D_{0}^{0}=$7.93 eV while Langhoff & Arnold (1979) obtained 8.1 eV computed
using an ab initio configuration interaction method.

The spin-orbit curves as the EAMC of $A$–$E$ were morphed using the expansion

$F(r)=\sum^{N}_{k=0}B_{k}\,z^{k}(1-\xi_{p})+\xi_{p}\,B_{\infty},$ (3)

where $z$ is the damped-coordinate polynomial given by (for SOCs, EAMCs
$A$–$B$, SRCs):

$z=(r-r_{\rm ref})\,e^{-\beta_{2}(r-r_{\rm ref})^{2}-\beta_{4}(r-r_{\rm
ref})^{4}},$ (4)

see also Prajapat et al. (2017) and Yurchenko et al. (2018a). Here $r_{\rm
ref}$ is a reference position chosen to be close to $r_{\rm e}$ of
$X\,{}^{1}\Sigma^{+}$ and $\beta_{2}$ and $\beta_{4}$ are damping factors.

Figure 4: Refined potential energy curves of SiO.

Figure 5: Refined electronic angular momentum and spin-orbit curves for SiO.

###  3.2 Dipole moment curves

The dipole moment curve of $X\,{}^{1}\Sigma^{+}$ was taken from Barton et al.
(2013). The (transition) DMCs $A\,{}^{1}\Pi$–$X\,{}^{1}\Sigma^{+}$,
$E\,{}^{1}\Sigma^{+}$–$X\,{}^{1}\Sigma^{+}$ (most important for this study),
$A\,{}^{1}\Pi$–$A\,{}^{1}\Pi$, $E\,{}^{1}\Sigma^{+}$–$E\,{}^{1}\Sigma^{+}$,
$E\,{}^{1}\Sigma^{+}$–$A\,{}^{1}\Pi$, $C\,{}^{1}\Sigma^{-}$–$A\,{}^{1}\Pi$
were taken from Bauschlicher (2016), and the $D\,{}^{1}\Delta$-$A\,{}^{1}\Pi$
TDMC has been computed in this work. 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. The ab initio (transitional) dipole
moment curves used in this work are shown in Fig. 6.

The ab initio DMC of $X\,{}^{1}\Sigma^{+}$ was modelled using the expansion

$\mu(r)=(1-\xi_{p})\sum^{N}_{k=0}C_{k}\,z^{k}+\xi_{p}\,C_{\infty},$ (5)

where $z$ is taken as the damped-coordinate $z$ from Eq. (4), see also
Prajapat et al. (2017) and Yurchenko et al. (2018a). This was done to reduce
the numerical errors commonly appearing in simulations of spectra of high
overtones, see Medvedev et al. (2016). Here $C_{k}$ and $C_{\rm\infty}$ are
adjustable parameters. The expansion centre $r_{\rm ref}$ is typically chosen
to be close to the equilibrium value of the ground electronic state. The $X$
state dipole moment curve from the EBJT model was transformed to the
analytical representation of Eq. (4).

The vibrational transitional dipole moment for the ground ($X$) vibrational
state $v=0$ is -3.0803 D, which compares well with the experimentally
determined values of 3.0982 ($D_{0}$) and 3.088 D ($D_{e}$) using the
molecular beam electric resonance spectra (Raymonda et al., 1970). In Table 3
we compare diagonal vibrational transition dipole moments with the
experimental values by (Raymonda et al., 1970) for a few
$v^{\prime}=v^{\prime\prime}$ vibrational bands, showing excellent agreement.

Figure 6:  Ab initio (transition) dipole moment curves used in this work are from Bauschlicher (2016) except $X$–$X$ which is taken from (Barton et al., 2013) and the $D\,{}^{1}\Delta$-$A\,{}^{1}\Pi$ TDMC which is computed in this work. Table 3: Comparison of experimental (molecular beam electric resonance spectra (Raymonda et al., 1970)) and theoretical (this work) diagonal vibrational transition dipole moments (Debye) of SiO in its ground $X$ electronic state for $v^{\prime}=v^{\prime}=0-4$.  Band | Exp. (D) | This work (D)
---|---|---
0–0 | 3.098 | 3.080
1–1 | 3.118 | 3.100
2–2 | 3.118 | 3.120
3–3 | 3.137 | 3.139
4–4 | 3.157 | 3.159

All expansion parameters or curves defining our spectroscopic model are given
as supplementary material to the paper as a Duo input file.

##  4 Deperturbation

In Duo calculations a grid of 501 Sinc DVR points ranging from 0.2 to 7 Å was
used. The vibrational basis set consisted of 40, 40, 50, 30, 30, 30, 30, 30,
30 vibrational basis functions for the $X\,{}^{1}\Sigma^{+}$, $A\,{}^{1}\Pi$,
$E\,{}^{1}\Sigma^{+}$, $C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$,
$a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$, $e\,{}^{3}\Sigma^{-}$ and
$d\,{}^{3}\Delta$ states, respectively, generated by solving the three
independent $J=0$ Schrödinger equations using the Sinc DVR method (Yurchenko
et al., 2016).

Figure 7 illustrates the main perturbations of the $A\,{}^{1}\Pi$ system
$(v=0,\ldots,9)$ caused by the nearby lying electronic states
$C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$, $a\,{}^{3}\Sigma^{+}$,
$b\,{}^{3}\Pi$, $e\,{}^{3}\Sigma^{-}$ and $d\,{}^{3}\Delta$. This figures
shows all empirical and theoretical (Duo) term values as a function of $J$,
reduced as

$\tilde{E}_{i}^{\rm red}=\tilde{E}_{i}-0.631J(J+1)$

to remove the major $J(J+1)$ contribution.

The empirical term values representing the $A$–$X$ system were obtained by
combing the experimental transition frequencies by Field et al. (1976);
Bredohl et al. (1976) and the accurate lower state $X$ of SiO as taken from
the ExoMol line list EBJT for SiO. All perturbing states are dark, i.e.
electric dipole transitions to and from the $X\,{}^{1}\Sigma^{+}$ state are
forbidden, and appear in the experiment only because of the interaction with
the ‘bright’ $A\,{}^{1}\Pi$ state, allowed for $X\,{}^{1}\Sigma^{+}$, via the
so-called intensity borrowing. These interactions are very local to the
crossing, giving rise to more transitions than would otherwise be expected for
the $A$–$X$ system. The $J$ dependencies of the energies around each crossing
break the rovibronic sequences of the $A$ term values. The shape of the
crossing is very specific to the character of the crossing state, not only to
its multiplicity, but also to the curvature exhibiting the differences between
the corresponding rotational constants: the larger the difference of $B$ from
0.631 cm-1, the steeper the $J$ curve. These shapes represent signatures of
the state in question and are used to reconstruct their relative positions on
the energy diagrams as in Fig. 7.

Field et al. (1976) reported a detailed analysis of all perturbations that
appeared in the $A$–$X$ spectra they observed, including assignments of the
crossing rovibronic energies and estimates of the corresponding spectroscopic
constants. This information was extremely valuable for our analysis. We first
used their spectroscopic constants to reconstruct rovibronic energies of the
perturbing state $C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$,
$a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$, $e\,{}^{3}\Sigma^{-}$ and
$d\,{}^{3}\Delta$ around crossing points with the program PGOPHER (Western,
2017) and then to initially refine the corresponding ab initio curves by
fitting to these empirical energies using Duo. This allowed us to correlate
the Duo energy levels to the experimental (MARVEL) values and properly assign
some of the duplicate transitions from Bredohl et al. (1976) and Field et al.
(1976) to forbidden vibronic systems $C$–$X$, $D$–$X$, $c$–$X$ and $d$–$X$,
with the help of the detailed description of interactions provided by Field et
al. (1976). This was important because the ab initio spectroscopic model (PECs
and SOCs) is not accurate enough for predicting the exact vibronic state,
especially when the corresponding vibrational quantum numbers are very high
($v=10$–20). The re-assignments of 112 transitions from Bredohl et al. (1976)
and Field et al. (1976) are listed in Tables 4 and 5.

There are no experimental line positions for the $a\,{}^{3}\Sigma^{+}$ state.
Hager et al. (1975) estimated $T_{0}$ as 33 409 cm-1, which we used to adjust
the corresponding $T_{\rm e}$ constant of this state.

Table 4:  Assignment of forbidden transitions from Bredohl et al. (1976). $\tilde{\nu}$ (cm-1) | State’ | $J^{\prime}$ | $v^{\prime}$ | e/f’ | $\Omega^{\prime}$ | $J"$ | $v^{\prime\prime}$ |  |  $\tilde{\nu}$ (cm-1) | State’ | $J^{\prime}$ | $v^{\prime}$ | e/f’ | $\Omega^{\prime}$ | $J"$ | $v^{\prime\prime}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
41349.36 | $D{}^{1}\Delta$ | 35 | 6 | e | 2 | 34 | 1 |  | 40160.70 | $d{}^{3}\Delta$ | 29 | 9 | e | 2 | 28 | 2
41247.78 | $D{}^{1}\Delta$ | 35 | 6 | e | 2 | 36 | 1 |  | 40076.77 | $d{}^{3}\Delta$ | 29 | 9 | e | 2 | 30 | 2
41299.36 | $D{}^{1}\Delta$ | 35 | 6 | f | 2 | 35 | 1 |  | 40119.35 | $d{}^{3}\Delta$ | 29 | 9 | f | 2 | 29 | 2
40137.54 | $D{}^{1}\Delta$ | 35 | 6 | e | 2 | 34 | 2 |  | 38958.90 | $d{}^{3}\Delta$ | 29 | 9 | e | 2 | 28 | 3
40036.45 | $D{}^{1}\Delta$ | 35 | 6 | e | 2 | 36 | 2 |  | 38875.30 | $d{}^{3}\Delta$ | 29 | 9 | e | 2 | 30 | 3
40087.69 | $D{}^{1}\Delta$ | 35 | 6 | f | 2 | 35 | 2 |  | 38917.80 | $d{}^{3}\Delta$ | 29 | 9 | f | 2 | 29 | 3
38937.63 | $D{}^{1}\Delta$ | 35 | 6 | e | 2 | 34 | 3 |  | 41351.85 | $d{}^{3}\Delta$ | 34 | 9 | e | 1 | 33 | 1
38837.16 | $D{}^{1}\Delta$ | 35 | 6 | e | 2 | 36 | 3 |  | 41252.76 | $d{}^{3}\Delta$ | 34 | 9 | e | 1 | 35 | 1
38888.06 | $D{}^{1}\Delta$ | 35 | 6 | f | 2 | 35 | 3 |  | 41302.91 | $d{}^{3}\Delta$ | 34 | 9 | f | 1 | 34 | 1
41335.65 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 35 | 1 |  | 40139.25 | $d{}^{3}\Delta$ | 34 | 9 | e | 1 | 33 | 2
41230.94 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 37 | 1 |  | 40040.92 | $d{}^{3}\Delta$ | 34 | 9 | e | 1 | 35 | 2
41283.97 | $D{}^{1}\Delta$ | 36 | 6 | f | 2 | 36 | 1 |  | 40090.85 | $d{}^{3}\Delta$ | 34 | 9 | f | 1 | 34 | 2
40123.76 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 35 | 2 |  | 38938.91 | $d{}^{3}\Delta$ | 34 | 9 | e | 1 | 33 | 3
40020.00 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 37 | 2 |  | 38841.40 | $d{}^{3}\Delta$ | 34 | 9 | e | 1 | 35 | 3
40072.57 | $D{}^{1}\Delta$ | 36 | 6 | f | 2 | 36 | 2 |  | 38890.89 | $d{}^{3}\Delta$ | 34 | 9 | f | 1 | 34 | 3
38924.43 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 35 | 3 |  | 41209.05 | $d{}^{3}\Delta$ | 39 | 9 | e | 0 | 40 | 1
38821.02 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 37 | 3 |  | 41266.66 | $d{}^{3}\Delta$ | 39 | 9 | f | 0 | 39 | 1
38873.36 | $D{}^{1}\Delta$ | 36 | 6 | f | 2 | 36 | 3 |  | 40111.69 | $d{}^{3}\Delta$ | 39 | 9 | e | 0 | 38 | 2
41374.54 | $d{}^{3}\Delta$ | 29 | 9 | e | 2 | 28 | 1 |  | 39999.18 | $d{}^{3}\Delta$ | 39 | 9 | e | 0 | 40 | 2
41289.93 | $d{}^{3}\Delta$ | 29 | 9 | e | 2 | 30 | 1 |  | 40056.34 | $d{}^{3}\Delta$ | 39 | 9 | f | 0 | 39 | 2
41332.80 | $d{}^{3}\Delta$ | 29 | 9 | f | 2 | 29 | 1 |  | 41322.36 | $d{}^{3}\Delta$ | 39 | 9 | e | 0 | 38 | 1
Table 5:  Assignment of forbidden transitions from Field et al. (1976). $\tilde{\nu}$ (cm-1) | State’ | $J^{\prime}$ | $v^{\prime}$ | e/f’ | $\Omega^{\prime}$ | $J"$ | $v^{\prime\prime}$ |  |  $\tilde{\nu}$ (cm-1) | State’ | $J^{\prime}$ | $v^{\prime}$ | e/f’ | $\Omega^{\prime}$ | $J"$ | $v^{\prime\prime}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
45912.81 | $C{}^{1}\Sigma^{-}$ | 13 | 11 | f | 0 | 13 | 0 |  | 46677.51 | $d{}^{3}\Delta$ | 21 | 15 | e | 1 | 20 | 0
45906.48 | $C{}^{1}\Sigma^{-}$ | 14 | 11 | f | 0 | 14 | 0 |  | 46647.05 | $d{}^{3}\Delta$ | 21 | 15 | f | 1 | 21 | 0
45899.97 | $C{}^{1}\Sigma^{-}$ | 15 | 11 | f | 0 | 15 | 0 |  | 46615.12 | $d{}^{3}\Delta$ | 21 | 15 | e | 1 | 22 | 0
45893.63 | $C{}^{1}\Sigma^{-}$ | 16 | 11 | f | 0 | 16 | 0 |  | 46673.81 | $d{}^{3}\Delta$ | 24 | 15 | e | 3 | 23 | 0
45882.76 | $C{}^{1}\Sigma^{-}$ | 17 | 11 | f | 0 | 17 | 0 |  | 46639.16 | $d{}^{3}\Delta$ | 24 | 15 | f | 3 | 24 | 0
45875.02 | $C{}^{1}\Sigma^{-}$ | 18 | 11 | f | 0 | 18 | 0 |  | 46602.92 | $d{}^{3}\Delta$ | 24 | 15 | e | 3 | 25 | 0
44940.32 | $C{}^{1}\Sigma^{-}$ | 40 | 10 | f | 0 | 40 | 0 |  | 45753.88 | $d{}^{3}\Delta$ | 37 | 14 | f | 2 | 37 | 0
44914.70 | $C{}^{1}\Sigma^{-}$ | 41 | 10 | f | 0 | 41 | 0 |  | 45698.94 | $d{}^{3}\Delta$ | 37 | 14 | e | 2 | 38 | 0
44898.36 | $C{}^{1}\Sigma^{-}$ | 42 | 10 | f | 0 | 42 | 0 |  | 42543.94 | $d{}^{3}\Delta$ | 39 | 9 | e | 0 | 38 | 0
42453.07 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 37 | 0 |  | 42487.91 | $d{}^{3}\Delta$ | 39 | 9 | f | 0 | 39 | 0
42558.45 | $D{}^{1}\Delta$ | 36 | 6 | e | 2 | 35 | 0 |  | 45713.65 | $d{}^{3}\Delta$ | 41 | 14 | f | 1 | 41 | 0
42506.71 | $D{}^{1}\Delta$ | 36 | 6 | f | 2 | 36 | 0 |  | 48254.23 | $e{}^{3}\Sigma^{-}$ | 9 | 15 | f | 1 | 9 | 0
46720.58 | $d{}^{3}\Delta$ | 12 | 15 | e | 3 | 11 | 0 |  | 48249.98 | $e{}^{3}\Sigma^{-}$ | 10 | 15 | f | 1 | 10 | 0
46703.19 | $d{}^{3}\Delta$ | 12 | 15 | f | 3 | 12 | 0 |  | 48245.56 | $e{}^{3}\Sigma^{-}$ | 11 | 15 | f | 1 | 11 | 0
46684.37 | $d{}^{3}\Delta$ | 12 | 15 | e | 3 | 13 | 0 |  | 48230.81 | $e{}^{3}\Sigma^{-}$ | 12 | 15 | f | 1 | 12 | 0
46715.29 | $d{}^{3}\Delta$ | 13 | 15 | e | 2 | 12 | 0 |  | 48225.27 | $e{}^{3}\Sigma^{-}$ | 13 | 15 | f | 1 | 13 | 0
46696.51 | $d{}^{3}\Delta$ | 13 | 15 | f | 2 | 13 | 0 |  | 48255.85 | $e{}^{3}\Sigma^{-}$ | 14 | 15 | e | 1 | 13 | 0
46676.19 | $d{}^{3}\Delta$ | 13 | 15 | e | 2 | 14 | 0 |  | 48213.84 | $e{}^{3}\Sigma^{-}$ | 14 | 15 | e | 1 | 15 | 0
46710.10 | $d{}^{3}\Delta$ | 14 | 15 | e | 2 | 13 | 0 |  | 48251.55 | $e{}^{3}\Sigma^{-}$ | 15 | 7 | e | 0 | 14 | 0
46689.90 | $d{}^{3}\Delta$ | 14 | 15 | f | 2 | 14 | 0 |  | 48206.65 | $e{}^{3}\Sigma^{-}$ | 15 | 15 | e | 0 | 16 | 0
46668.11 | $d{}^{3}\Delta$ | 14 | 15 | e | 2 | 15 | 0 |  | 48247.24 | $e{}^{3}\Sigma^{-}$ | 16 | 15 | e | 1 | 15 | 0
46700.44 | $d{}^{3}\Delta$ | 15 | 15 | e | 3 | 14 | 0 |  | 48199.56 | $e{}^{3}\Sigma^{-}$ | 16 | 15 | e | 1 | 17 | 0
46678.73 | $d{}^{3}\Delta$ | 15 | 15 | f | 3 | 15 | 0 |  | 44272.94 | $e{}^{3}\Sigma^{-}$ | 26 | 9 | e | 0 | 25 | 0
46655.61 | $d{}^{3}\Delta$ | 15 | 15 | e | 3 | 16 | 0 |  | 44264.79 | $e{}^{3}\Sigma^{-}$ | 27 | 9 | e | 0 | 26 | 0
46670.75 | $d{}^{3}\Delta$ | 16 | 15 | f | 2 | 16 | 0 |  | 44227.74 | $e{}^{3}\Sigma^{-}$ | 29 | 9 | f | 1 | 29 | 0
46646.05 | $d{}^{3}\Delta$ | 16 | 15 | e | 2 | 17 | 0 |  | 44215.43 | $e{}^{3}\Sigma^{-}$ | 30 | 9 | f | 1 | 30 | 0
46703.19 | $d{}^{3}\Delta$ | 18 | 15 | e | 3 | 17 | 0 |  | 44203.50 | $e{}^{3}\Sigma^{-}$ | 31 | 9 | f | 1 | 31 | 0
46677.20 | $d{}^{3}\Delta$ | 18 | 15 | f | 3 | 18 | 0 |  | 44183.19 | $e{}^{3}\Sigma^{-}$ | 32 | 9 | f | 1 | 32 | 0
46649.70 | $d{}^{3}\Delta$ | 18 | 15 | e | 3 | 19 | 0 |  | 44170.00 | $e{}^{3}\Sigma^{-}$ | 33 | 9 | f | 1 | 33 | 0
46696.69 | $d{}^{3}\Delta$ | 19 | 15 | e | 3 | 18 | 0 |  | 44155.74 | $e{}^{3}\Sigma^{-}$ | 34 | 9 | f | 1 | 34 | 0
46669.20 | $d{}^{3}\Delta$ | 19 | 15 | f | 3 | 19 | 0 |  | 44215.43 | $e{}^{3}\Sigma^{-}$ | 36 | 9 | e | 1 | 35 | 0
46640.24 | $d{}^{3}\Delta$ | 19 | 15 | e | 3 | 20 | 0 |  | 44110.19 | $e{}^{3}\Sigma^{-}$ | 36 | 9 | e | 1 | 37 | 0
46685.22 | $d{}^{3}\Delta$ | 20 | 15 | e | 1 | 19 | 0 |  | 44197.39 | $e{}^{3}\Sigma^{-}$ | 37 | 9 | e | 1 | 36 | 0
46656.19 | $d{}^{3}\Delta$ | 20 | 15 | f | 1 | 20 | 0 |  | 44089.08 | $e{}^{3}\Sigma^{-}$ | 37 | 9 | e | 1 | 38 | 0
46625.89 | $d{}^{3}\Delta$ | 20 | 15 | e | 1 | 21 | 0 |  | 47296.58 | $e{}^{3}\Sigma^{-}$ | 38 | 14 | f | 1 | 38 | 0

Our coupled model significantly improves the description of the perturbations.
The residual errors between the observed and calculated values (obs$-$calc)
are illustrated in Fig. 8. In fitting, we used the EMO form in Eq. (1) to
represent the PECs and the damped form in Eq. (5) to represent the morphing of
SOCs of the couplings with the perturbing states. Owing to the scarcity of
information on dark states, only a minimal number of the expansion parameters
$B_{k}$ could be obtained. The assignment of the dark states around state
crossings is illustrated in Fig. 7, where the experimental energy values (open
red circles) are overlaid with the theoretical ones. Most of the experimental
points nicely follow the $A$ state sequences, except around the crossing
points where these sequences break and follow the corresponding patterns of
the dark states, forming distinct patterns. It should be noted that the
rovibronic assignment of interacting levels exactly at their crossing is
always ambiguous owing to the mixed nature of the resonating states.

Figure 7: Perturbations of the $A\,{}^{1}\Pi$ state term values by ‘dark’
sates. Reduced energy term values $E-0.631J(J+1)$ are shown as a function of
$J$. The red open circles represent experimental levels from the $A$ system.
The filled symbols represent the calculated (Duo) levels with the red circles
indicating the $A$ states.

Figure 8 shows the residuals representing our fit. Some of the $A$ state
energies still show small perturbations due to interactions with dark
electronic states, which are not fully accounted for in our model.

Figure 8: Residuals (Obs.-Calc.) between the experimentally determined
energies of SiO ($A\,{}^{1}\Pi$ and $E\,{}^{1}\Sigma^{+}$) from our MARVEL
analysis and the Duo energies corresponding to our refined spectroscopic
model.

##  5 Line list and simulations of spectra of SiO

The SiOUVenIR line list was produced with Duo using the refined spectroscopic
model of SiO (provided as supplementary material). It contains 91 395 763
transitions and 174 250 states for $X\,{}^{1}\Sigma^{+}$ $A\,{}^{1}\Pi$,
$E\,{}^{1}\Sigma^{+}$, $C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$,
$a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$, $e\,{}^{3}\Sigma^{-}$ and
$d\,{}^{3}\Delta$, covering wavenumbers up to 72 000 cm-1 ($\lambda<140$ nm)
and $J=0\ldots 250$. 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 6 and 7, respectively; the full
files are available from www.exomol.com and CDS. The lines list contains state
uncertainties, Landé-$g$ factors (Semenov et al., 2017), lifetimes (Tennyson
et al., 2016a) and a partition function which is very similar to that of EBJT
due to the negligible contribution from the energy levels belonging to the
excited electronic states.

The calculated energies were replaced with the MARVEL values (MARVELised),
where available. We have used the labels ‘Ca’ and ‘Ma’ in the penultimate
column of the States file to indicate if the value is calculated (Duo) 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:

${\rm unc}=av+bJ(J+1),$ (6)

where $a$ and $b$ are electronic state dependent constant, defined in Table 8.

Table 6: Extract from the states file of the SiOUVenIR line list for 28Si16O.

$i$ |  Energy (cm-1) | $g_{i}$ | $J$ | unc | $\tau$ | $g$ | Parity | State | $v$ | ${\Lambda}$ | ${\Sigma}$ | $\Omega$ | Label | Calc.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1666 | 44144.709385 | 7 | 3 | 8.920000 | 9.0168E-03 | 1.1217 | + | f | d3Delta | 11 | 2 | 1 | 3 | Ca | 44144.709385
1667 | 44149.467240 | 7 | 3 | 8.920000 | 2.6619E-04 | 0.4728 | + | f | d3Delta | 11 | 2 | 0 | 2 | Ca | 44149.467240
1668 | 44154.189842 | 7 | 3 | 8.920000 | 1.0477E-04 | -0.2607 | + | f | d3Delta | 11 | 2 | -1 | 1 | Ca | 44154.189842
1669 | 44209.302522 | 7 | 3 | 8.920000 | 6.3159E-01 | 0.5301 | + | f | b3Pi | 11 | 1 | 1 | 2 | Ca | 44209.302522
1670 | 44276.546978 | 7 | 3 | 8.920000 | 1.0252E-03 | 0.0905 | + | f | b3Pi | 11 | 1 | 0 | 1 | Ca | 44276.546978
1671 | 44314.520663 | 7 | 3 | 0.050000 | 2.8958E-08 | 0.0835 | + | f | A1Pi | 2 | 1 | 0 | 1 | Ma | 44314.597828
1672 | 44342.823666 | 7 | 3 | 8.920000 | 4.8501E-01 | -0.0368 | + | f | b3Pi | 11 | 1 | -1 | 0 | Ca | 44342.823666
1673 | 44421.259298 | 7 | 3 | 7.320000 | 1.7643E-05 | 0.1667 | + | f | e3Sigma- | 9 | 0 | 1 | 1 | Ca | 44421.259298
1674 | 44463.770855 | 7 | 3 | 12.120000 | 1.8929E+00 | 0.5833 | + | f | a3Sigma+ | 15 | 0 | 1 | 1 | Ca | 44463.770855
1675 | 44472.524815 | 7 | 3 | 12.120000 | 8.5180E+00 | -0.4164 | + | f | a3Sigma+ | 15 | 0 | 0 | 0 | Ca | 44472.524815
1676 | 44644.518546 | 7 | 3 | 7.320000 | 1.8475E-02 | 0.0000 | + | f | C1Sigma- | 9 | 0 | 0 | 0 | Ca | 44644.518546
1677 | 44787.014146 | 7 | 3 | 7.320000 | 3.0209E-01 | 0.3333 | + | f | D1Delta | 9 | 2 | 0 | 2 | Ca | 44787.014146
1678 | 44814.070066 | 7 | 3 | 9.720000 | 1.1037E-02 | 1.1218 | + | f | d3Delta | 12 | 2 | 1 | 3 | Ca | 44814.070066
1679 | 44818.761972 | 7 | 3 | 9.720000 | 3.3377E-04 | 0.4729 | + | f | d3Delta | 12 | 2 | 0 | 2 | Ca | 44818.761972
1680 | 44823.424247 | 7 | 3 | 9.720000 | 1.3523E-04 | -0.2609 | + | f | d3Delta | 12 | 2 | -1 | 1 | Ca | 44823.424247
1681 | 45075.695373 | 7 | 3 | 9.720000 | 6.4266E-01 | 0.5298 | + | f | b3Pi | 12 | 1 | 1 | 2 | Ca | 45075.695373

$i$: State counting number.

$\tilde{E}$: State energy term values in cm-1, MARVEL or Calculated (Duo).

$g_{i}$: Total statistical weight, equal to ${g_{\rm ns}(2J+1)}$.

$J$: Total angular momentum.

unc: Uncertainty, cm-1.

$\tau$: Lifetime (s-1).

$g$: Landé $g$-factors.

$+/-$: Total parity.

State: Electronic state.

$v$: State vibrational quantum number.

$\Lambda$: Projection of the electronic angular momentum.

$\Sigma$: Projection of the electronic spin.

$\Omega$: Projection of the total angular momentum, $\Omega=\Lambda+\Sigma$.

Label: ‘Ma’ is for MARVEL and ‘Ca’ is for Calculated.

Table 7: Extract from the transitions file of the SiOUVenIR line list for
28Si16O.

$f$ | $i$ |  $A_{fi}$ (s-1) | $\tilde{\nu}_{fi}$
---|---|---|---
18 | 506 | 2.4093E+01 | 2097.041341
508 | 16 | 8.0507E+00 | 2099.618181
818 | 506 | 9.6724E+00 | 2100.876729
1928 | 816 | 1.0376E+01 | 2102.115330
2358 | 1926 | 1.0772E+01 | 2103.333959
3528 | 2356 | 1.1030E+01 | 2104.532592
7157 | 8325 | 1.1301E+01 | 2105.654263
3958 | 3526 | 1.1212E+01 | 2105.711204
5128 | 3956 | 1.1350E+01 | 2106.869772
6727 | 7155 | 1.1372E+01 | 2107.160357
5558 | 5126 | 1.1458E+01 | 2108.008270
5557 | 6725 | 1.1455E+01 | 2108.646782
6728 | 5556 | 1.1546E+01 | 2109.126673

$f$: Upper state counting number;
$i$: Lower state counting number;
$A_{fi}$: Einstein-$A$ coefficient in s-1;
$\tilde{\nu}_{fi}$: transition wavenumber in cm-1.

Table 8: $a$ and $b$ constants defining state dependent uncertainties via Eq. (6).  State | $a$ | $b$
---|---|---
$X\,{}^{1}\Sigma^{+}$ | 0.002 | 0.0001
$A\,{}^{1}\Pi$ | 0.5 | 0.001
$E\,{}^{1}\Sigma^{+}$ | 0.5 | 0.001
All other | 0.8 | 0.001

Figure 9 illustrates the three main bands of SiO compared against a spectrum
generated from the Kurucz (2011) line list at 2000 K. We see increased
deviation between the spectra with increased energy due to Figure 10
illustrates the evolution of the spectrum with increasing temperature.

Figure 9: Simulated SiO absorption spectra at 2000 K showing the three main
bands of the UV-IR system. A Gaussian profile of half width of half maximum
(HWHM) of 1 cm-1 was used. The spectrum based on the line list by Kurucz
(2011) is indicated for comparison.  Figure 10: Temperature dependence of the
SiO spectra using the SiOUVenIR line list. A Gaussian profile with half width
at half maximum (HWHM) of 1 cm-1 was used.

Figure 11 shows a spectrum of SiO computed using transitions between the
‘MARVELised’ (i.e. more accurate) states only. It illustrates the completeness
of the MARVEL part of the line list for high-resolution spectroscopic
applications, such as high-dispersion spectroscopy of exoplanets (Snellen,
2014).

Figure 11: A SiO absorption spectrum at $T=$ 2000 K simulated using
transitions between MARVEL states only. A Gaussian profile with HWHM of 1
cm-1was used.

As an illustration of of the quality of the SiO line list in the UV, we
compare theoretical SiO spectra with existing experimental spectroscopic
studies of SiO from this region of two prominent bands, $A$–$X$ and $E$–$X$.
Figure 12 compares absorption spectra from the $A$–$X$ bands from two
different regions, 234–236 nm and 250-290 nm to the fluorescence spectrum by
Coursimault et al. (2003) (Left) and shock-tube absorption spectrum by Park &
Arnold (1978) (Right), respectively. To reproduce the fluorescence spectrum at
the 235 nm band, a non local thermodynamic equilibrium (non-LTE) population of
$T_{\rm rot}=1000$ K and $T_{\rm vib}=5000$ K was used. Figure 13 provides an
illustration for the $E-X$ band by comparing the shock-tube absorption
spectrum by Park (1978) a SiOUVenIR synthetic spectrum computed for $T$ = 3740
K (LTE). All three cases demonstrate fairly good agreement with experiment.

Figure 12: SiO $A-X$ band. Left display: Absorption spectrum of SiO computed
using $T_{\rm rot}=1000$ K, $T_{\rm vib}=5000$ K, using the Voigt line profile
with HWHM = 0.015 nm compared to a fluorescence spectrum by Coursimault et al.
(2003); Right display: SiO absorption spectrum computed using $T$ = 3740 K and
the Voigt line profile with using HWHM = 0.03 nm compared to the shock-tube
absorption spectrum by Park & Arnold (1978).  Figure 13: SiO $E-X$ band. An
absorption spectrum of SiO computed using $T$ = 3740 K and the Voigt line
profile compared to a shock-tube absorption spectrum by Park (1978) using HWHM
= 0.04 nm. Figure 14: A comparison between cross sections generated from
SiOUVenIR and the experimental data from Hermann et al. (2001). SiOUVenIR
cross sections are generated using Lorentzian line profiles with HWHM = 4.5
cm-1, in simulated non-LTE with a rotational temperature of 3000 K and a
vibrational temperature of 4000 K.

As an astrophysical application, Fig. 15 shows a simulation of the stellar
spectrum of the star HD61913, taken from Lebzelter et al. (2012). This part of
the spectrum features the 2–0 IR band, with the band head at 4044 nm. The
small satellite bandhead belongs to the 3–1 hot band. The absorption spectrum
(air) of SiO was generated from the SiOUVenIR line list using the Lorentzian
line profile with a HWHM of 0.2 cm-1 with the Beer-Lambert law then applied
plotted next to the HD61913 spectrum. In addition to the local thermodynamic
equilibrium (LTE) spectrum calculated with the star’s effective temperature of
3530 K, a non-LTE spectrum (calculated with a rotational temperature of 1800 K
and a vibrational temperature of 3530 K) is also plotted. The non-LTE spectrum
demonstrates improved similarity to the stellar spectrum, particularly with
regard to the satellite 3–1 bandhead next to the 2–0 band head.

Lifetimes of SiO were reported in multiple theoretical (Chattopadhyaya et al.,
2003; Oddershede & Elander, 1976; Langhoff & Arnold, 1979; Drira et al., 1998;
Bauschlicher, 2016; Feng & Zhu, 2019a) and experimental (Smith & Liszt, 1972;
Elander & Smith, 1973) works. Table 9 shows a comparison of these works with
the calculated lifetimes for the $A\,{}^{1}\Pi$ and
$E\,{}^{1}\Sigma^{+}$states. In our work, the output from Duo was used as the
input for the ExoCross program (Yurchenko et al., 2018b) to calculate
lifetimes, see also Tennyson et al. (2016a). It should be noted that the
theoretical lifetimes show slow dependence on $J$ and tend to monotonically
increase with $v$ up to about 100 ns for $v=40$. This means that an observed
lifetime for the state will be temperature dependent.

For the $A\,{}^{1}\Pi$ state at $v^{\prime}=0,J^{\prime}=0$ our lifetime value
is 28.5 ns, which is in close agreement with some previous theoretical works
(Bauschlicher, 2016; Feng & Zhu, 2019a; Chattopadhyaya et al., 2003), but is
twice the calculated lifetime predicted by Langhoff & Arnold (1979) and Drira
et al. (1998), which seem to be closer to the experimental value of Smith &
Liszt (1972). While it is hard for us to reconcile these large differences, as
we use a very high accuracy ab initio model for TDMCs, we attribute the
difference with the experimental work Smith & Liszt (1972) to be due to the
lower perturbing states.

For the $E\,{}^{1}\Sigma^{+}$ state at $v^{\prime}=0,J^{\prime}=0$ our value
is 11.23 ns, which is in close agreement with some very recent calculations
(Bauschlicher, 2016; Feng & Zhu, 2019a) and experiment (Elander & Smith,
1973).

Figure 15: A comparison of the SiO spectrum (in air) using the SiOUVenIR line list with a section of the stellar spectrum of HD61913 from Lebzelter et al. (2012) showing the 2-0 (middle) and 3-1 (left) IR band region. A Lorentzian line profile with HWHM of 0.2 cm-1 was used. For the LTE spectrum, the equilibrium temperature of 3530 K was used. The non-LTE spectrum was computed for $T_{\rm rot}=1800$ K and $T_{\rm vib}=3530$ K. Table 9: Comparison of the experimental and calculated lifetimes, $\tau$ (ns), for the $A^{1}\Pi$ and $E^{1}\Sigma^{+}$ vibrational ground states. ‘This work’ values are quoted for $v^{\prime}=0,J^{\prime}=0$.  $A^{1}\Pi$ | source |  | $E^{1}\Sigma^{+}$ | source |
---|---|---|---|---|---
28.5 | This work | Cal | 11.23 | This work | Cal
9.6 | Smith & Liszt (1972) | Exp | 10.5 | Elander & Smith (1973) | Exp
28.9 | Chattopadhyaya et al. (2003) | Cal | 7.10 | Chattopadhyaya et al. (2003) | Cal
49.5 | Oddershede & Elander (1976) | Cal |  |  |
16.6 | Langhoff & Arnold (1979) | Cal | 6.80 | Langhoff & Arnold (1979) | Cal
12.5 | Drira et al. (1998) | Cal | 6.00 | Drira et al. (1998) | Cal
29.3 | Bauschlicher (2016) | Cal | 11.0 | Bauschlicher (2016) | Cal
32.34 | Feng & Zhu (2019a) | Cal | 10.58 | Feng & Zhu (2019a) | Cal

##  6 Conclusion

A new IR and UV line list for SiO is presented. The SiOUVenIR line list
supersedes the original SiO ExoMol line list from Barton et al. (2013).
SiOUVenIR is available from www.exomol.com and the CDS database, via
ftp://cdsarc.u-strasbg.fr/pub/cats/J/MNRAS/, or
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS/

As part of the line list construction, a MARVEL analysis for 28Si16O was
performed. All experimental line positions from the literature (to the best of
our knowledge) covering the IR and UV regions ($X$–$X$, $A$–$X$ and $E$–$X$
systems) were collected and processed to generate a comprehensive set of
empirical energies of 28Si16O. Forbidden experimental transitions connecting
the $X$ state with the dark electronic states of the $C$, $D$, $e$, and $d$
system were assigned using our fully coupled description of the rovibronic
system. An accurate spectroscopic model for SiO was built: ab initio TDMC and
empirically refined PECs, SOCs, EAMCs. This allowed us to make full line
assignments to transitions associated with perturbing states.

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 UV spectra show close agreement with experiment,
similarly with computed lifetimes.

This SiOUVenIR linelist was largely constructed through two online “Hackathon”
days during 2020 via the online video-conferencing Zoom platform. The team was
split into “Breakout rooms” which each had a particular task (Ab initio,
MARVEL, fitting, etc.). These tasks were largely independent for initial
iterations of this new linelist. The online Hackathon days had the advantage
of including co-authors who would not normally be able to participate in “in-
person” “Hackathons”.

## Acknowledgments

This work was supported by the STFC Projects No. ST/R000476/1, ST/S506497/1
and ST/P006736/1 and by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme through Advance Grant
number 883830. The authors acknowledge the use of the UCL Legion High
Performance Computing Facility (Legion@UCL) and associated support services in
the completion of this work, along with the Cambridge Service for Data Driven
Discovery (CSD3), part of which is operated by the University of Cambridge
Research Computing on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk).
The DiRAC component of CSD3 was funded by BEIS capital funding via STFC
capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant
ST/R00689X/1. DiRAC is part of the National e-Infrastructure. This research
was undertaken with the assistance of resources from Supercomputing Wales and
the Australian National Computational Infrastructure (NCI Australia), a NCRIS
enabled capability supported by the Australian Government. Additional
supported was provided by UK research councils EPSRC, under grant
EP/N509577/1. We would like to thank the UK Natural Environment Research
Council (NERC) for funding through grant NE/T000767/1. We also want to thank
Moscow Witte University for sponsoring the fellowship enabling this research.

##  7 Data Availability

The data underlying this article are available in the article and in its
online supplementary material.

## Supporting Information

The MARVEL input transitions file and output energy levels file, the Duo input
file, which contains all the potential energy, dipole moment and coupling
curves of SiO used in this work, and the temperature-dependent partition
function of 28Si16O up to 10000 K are given as supplementary only data to this
article. The full SiOUVenIR line list is available from www.exomol.com and the
CDS database, via ftp://cdsarc.u-strasbg.fr/pub/cats/J/MNRAS/, or
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS/.

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