The ExoMol Atlas of Molecular Opacities

Abstract

The ExoMol project is dedicated to providing molecular line lists for exoplanet and other hot atmospheres. The ExoMol procedure uses a mixture of ab initio calculations and available laboratory data. The actual line lists are generated using variational nuclear motion calculations. These line lists form the input for opacity models for cool stars and brown dwarfs as well as for radiative transport models involving exoplanets. This paper is a collection of molecular opacities for 52 molecules (130 isotopologues) at two reference temperatures, 300 K and 2000 K, using line lists from the ExoMol database. So far, ExoMol line lists have been generated for about 30 key molecular species. Other line lists are taken from external sources or from our work predating the ExoMol project. An overview of the line lists generated by ExoMol thus far is presented and used to evaluate further molecular data needs. Other line lists are also considered. The requirement for completeness within a line list is emphasized and needs for further line lists discussed.

1. Introduction

Molecular opacities dictate the atmospheric properties and evolution of a whole range of cool stars and all brown dwarfs. They are also important for the radiative transport properties of exoplanets. Even for a relatively simple molecule, the opacity function is generally complicated as it varies strongly with both wavelength and temperature. This is because molecules interact with light via a variety of different motions, namely rotational, vibrational and electronic, and the close-spacing of the energy levels leads to strong temperature dependence related to the thermal occupancy of the levels.

Molecular opacity functions, such as the ones provided by Sharp and Burrows [1], require very extensive input in the form of temperature-dependent spectroscopic data for atoms and molecules. At room temperature, the HITRAN database [2] provides comprehensive and validated spectroscopic parameters for molecules found in the Earth’s atmosphere and some other species. However, the complexity of molecular spectra increases rapidly with temperature and data from HITRAN and can only be used with extreme caution at higher temperatures [3]. The HITEMP database [4] is a HITRAN-style database designed to work at higher temperatures. However, HITEMP only contains spectroscopic line lists for five molecules and there are improved line lists available for each of these species, namely: H${}_2$O [5], CO${}_2$ [6,7], CO [8], NO [9], and OH [10]. The opacity of all of these species are discussed further below.

The need for extensive, high temperature spectroscopic data on molecules, many of whom do not occur in the Earth’s atmosphere, has led to a number of systematic efforts to generate the molecular line lists required [7,11,12,13,14,15,16]. In particular, a number of groups have been progressively generating line lists for key molecules. These includes the TheoReTS [16] group (Reims/Tomsk), the NASA Ames team and our own ExoMol activity based in University College London, all of which used similar theoretical procedures discussed below, and an experimental initiative led by Bernath [14]. These activities have been significantly enhanced by the discovery of exoplanets and the requirement of extensive line lists to be used in exoplanet models and characterization [17,18].

As molecular line lists are extended, complicated and have a strong temperature dependence, it can be hard to understand a priori spectral regions where they show significant absorption. This work aims to summarize the absorption by the key molecular species required for cool stars, brown dwarfs and exoplanets. Many of the line lists used are taken from those computed by the Exomol project, but these are augmented by lists taken from other sources to give a comprehensive set, or atlas, of absorbing species.

In this work, we present a collection (atlas) of molecular opacities for 52 molecules (130 isotopologues) generated using line lists from the ExoMol database. The opacities are presented in the form of cross sections for two reference temperatures, 300 K and 2000 K using the Doppler (zero-pressure) broadening. The goal of this atlas is to present in a simple, visual form an overview of spectroscopic coverage, main spectroscopic signatures as well as temperature dependence of the molecular opacities relevant for atmospheric studies of hot exoplanets and stars.

2. Methodology

The general methodology used by the ExoMol project is very similar to that used by both the TheoReTS and NASA Ames group for producing molecular line lists. It has been well-described elsewhere [19,20] and below we will discuss only general considerations.

Our technique involves the following general steps: (1) computing accurate ab initio potential energy and dipole surface moment surfaces, (2) improving the potential energy surface (PES) using available high-resolution spectroscopic data and (3) generating a comprehensive line list using the improved PES, ab initio dipole moment surface (DMS) and an appropriate variational nuclear motion program [21,22,23]. Development of these program specifically for opacity calculations has been discussed by us elsewhere [24].

Performing such calculations is always a trade-off between completeness and accuracy. A complete line list should contain information on possible transitions at all wavelengths and at all temperatures. An accurate one should reproduce line positions and transition probabilities to the high standards of laboratory high resolution spectroscopy. In practice, for most systems, it is hard if not impossible to achieve both of these goals simultaneously so compromises have to be made. Experience shows that completeness is essential [25] while accuracy is something to strive for and, of course, is essential for high resolution spectroscopic studies [26].

Starting from an ab initio PES, there are three ways of improving the accuracy of the line positions. The first method is to fit the potential to measured transitions frequencies, empirical energies or a mixture of both. This is now a standard technique that is widely employed. It is capable of giving very good results [27] if in most cases not actual experimental accuracy. The second technique is to adjust the vibrational band origins during the calculation. This technique works only if the rotation–vibration basis used is a simple product between vibrational and rotational functions. A basis in this form is used by the polyatomic nuclear motion program TROVE [23] and the vibrational basis is contracted by performing an initial rotationless ($J=0$) calculations. At this stage, the band origins can be shifted to their empirical value [28]. Finally, the ExoMol data structure [29] presents a line list as a states file with energy levels and associated quantum numbers and a highly compact transitions file of Einstein A coefficients. This structure therefore allows computed energy levels to be replaced with empirical ones after the calculation is complete; indeed, this can be done some time after the line list is computed if new empirical data becomes available [30]. This allows many transition frequencies to be generated with experimental accuracy, including ones that have not actually been measured.

To provide lists of accurate empirical energy levels, Furtenbacher et al. [31,32] developed the measured active rotation vibration energy level procedure (MARVEL).

Table 1. Summary of MARVEL analyses performed for astronomically important molecules.

Molecule	${\mathit{N}}_{\mathbf{iso}}$	${\mathit{N}}_{\mathbf{trans}}$	${\mathit{N}}_{\mathit{E}}$	Reference(s)
H${}_2$O	9	182,156	18,486	[35,36,37,38]
H${}_3^{+}$	3	1410	911	[39,40]
NH${}_3$	1	28,530	4961	[41]
C${}_2$	1	23,343	5699	[42]
TiO	1	48,590	10,564	[43]
HCCH	1	37,813	11,213	[44]
SO${}_2$	3	40,269	15,130	[45]
H${}_2$S	1	39,267	7651	[46]
ZrO	1	21,195	8329	[47]

$N_{\mathrm{iso}}$: Number of isotopologues considered; $N_{\mathrm{trans}}$: Number of transitions validated for the main isotopologue; $N_E$: Number of energy levels given for the main isotopologue.

gives a summary of molecules of astronomical importance for which the MARVEL procedure has been applied. Overviews of this methodology can be found elsewhere [33,34].

The line lists generated by the methods discussed above form the input to opacity calculations. To judge the potential influence of each molecule on the opacity, one can generate cross sections as a function of wavelength and temperature. As several of the line lists considered contain many billions of lines (see

Table 2. Datasets created by the ExoMol project and included in the ExoMol database [50].

Paper	Molecule	${\mathit{N}}_{\mathbf{iso}}$	${\mathit{T}}_{\mathbf{max}}$	${\mathit{N}}_{\mathbf{elec}}$	${\mathit{N}}_{\mathbf{lines}}$	DSName	Reference
I	BeH	1	2000	1	16,400	Yadin	[51]
I	MgH	3	2000	1	10,354	Yadin	[51]
I	CaH	1	2000	1	15,278	Yadin	[51]
II	SiO	5	9000	1	254,675	EJBT	[52]
III	HCN/HNC	1	4000	1	399,000,000	Harris	[30]
IV	CH${}_4$	1	1500	1	9,819,605,160	YT10to10	[53]
V	NaCl	2	3000	1	702,271	Barton	[54]
V	KCl	4	3000	1	1,326,765	Barton	[54]
VI	PN	2	5000	1	142,512	YYLT	[55]
VII	PH${}_3$	1	1500	1	16,803,703,395	SAlTY	[56]
VIII	H${}_2$CO	1	1500	1	10,000,000,000	AYTY	[57]
IX	AlO	4	8000	3	4,945,580	ATP	[58]
X	NaH	2	7000	2	79,898	Rivlin	[59]
XI	HNO${}_3$	1	500	1	6,722,136,109	AlJS	[60]
XII	CS	8	3000	1	548,312	JnK	[61]
XIII	CaO	1	5000	5	21,279,299	VBATHY	[62]
XIV	SO${}_2$	1	2000	1	1,300,000,000	ExoAmes	[63]
XV	H${}_2$O${}_2$	1	1250	1	20,000,000,000	APTY	[64]
XIV	H${}_2$S	1	2000	1	115,530,3730	AYT2	[65]
XV	SO${}_3$	1	800	1	21,000,000,000	UYT2	[66]
XVI	VO	1	2000	13	277,131,624	VOMYT	[67]
XIX	H${}_2$${}^{17,18}$O	2	3000	1	519,461,789	HotWat78	[68]
XX	H${}_3^{+}$	1	3000	1	11,500,000,000	MiZATeP	[69]
XXI	NO	6	5000	2	2,281,042	NOName	[9]
XXII	SiH${}_4$	1	1200	1	62,690,449,078	OY2T	[70]
XXIII	PO	1	5000	1	2,096,289	POPS	[71]
XXIII	PS	1	5000	3	30,394,544	POPS	[71]
XXIV	SiH	4	5000	3	1,724,841	SiGHTLY	[72]
XXV	SiS	12	5000	1	91,715	UCTY	[73]
XXVI	HS	6	5000	1	219,463	SNaSH	[74]
XXVI	NS	6	5000	1	3,479,067	SNaSH	[74]
XXVII	C${}_2$H${}_4$	1	700	1	49,841,085,051	MaYTY	[75]
XXVIII	AlH	3	5000	3	40,000	AlHambra	[76]
XXIX	CH${}_3$Cl	2	1200	1	166,279,593,333	OYT	[75]
XXX	H${}_2$${}^{16}$O	1	5000	1	1,500,000,000	Pokazatel	[5]
XXXI	C${}_2$	3	5000	8	6,080,920	8State	[77]
XXXII	MgO	3	5000	4	22,579,054	LiPTY	[78]

$N_{\mathrm{iso}}$: Number of isotopologues considered; $T_{\max }$: Maximum temperature for which the line list is complete; $N_{\mathrm{elec}}$: Number of electronic states considered; $N_{\mathrm{lines}}$: Number of lines, value is for the main isotopologue; DSName: Name of line list and of data set in ExoMol database [50].

), calculating cross sections can be computationally expensive. For this reason, we have developed a highly optimized program ExoCross [48], which generates wavelength-dependent cross sections as function of temperature and pressure. Here, we use ExoCross to systematically generate cross sections for the key species that are important for opacities of cool stars, brown dwarfs and exoplanets.

In generating these cross sections, we consider the main (parent) isotopologue of each species, which is taken as being in 100 % abundance. For simplicity, we use a Doppler profile on a wavenumber grid of 1 cm${}^{-1}$. The cross sections have been generated using the methodology by Hill et al. [49]. The cross sections can be also obtained at higher resolutions (up to 0.01 cm${}^{-1}$) using the cross sections App at www.exomol.com.

Cross sections were generated at two standard temperatures of 300 K and 2000 K except when the molecule is expected to be entirely in the condensed phase at 300 K. For a few, key strongly bound species, we also consider the cross section at 5000 K, which is near the upper limit of the temperature where molecules make a contribution to opacities.

3. Results

Table 2 summarises the molecules for which line lists have been provided as part of the ExoMol project. Sources for line lists of other key species are given in

Table 3. Other molecular line lists considered. These data can also be obtained from the ExoMol website.

Molecule	${\mathit{N}}_{\mathbf{iso}}$	${\mathit{T}}_{\mathbf{max}}$	${\mathit{N}}_{\mathbf{elec}}$	${\mathit{N}}_{\mathbf{lines}}$	DSName	Reference	Methodology
							Line Positions	Intensities
NH${}_3$	2	1500	1	1,138,323,351	BYTe	[79]	empirical ${}^a$	ab initio ${}^a$
LiH	1	12,000	1	18,982	CLT	[80]	ab initio	ab initio
ScH	1	5000	6	1,152,827	LYT	[81]	tuned ab initio	ab initio
NH	1		1	10,414	14BrBeWe	[82]	empirical	ab initio
CH	2	6000	4	54,086	14MaPlVa	[83]	empirical	ab initio
CO	9	9000	1	752,976	15LiGoRo	[8]	empirical	emp./ab initio ${}^b$
OH	1	6000	1	45,000	16BrBeWe	[10]	empirical	ab initio
CN	1		1	195,120	14BrRaWe	[84]	empirical	ab initio
CP	1		1	28,735	14RaBrWe	[85]	empirical	ab initio
HCl	1		1	2588	11LiGoBe	[86]	empirical	ab initio
FeH	1		2	93,040	10WEReSe	[87]	empirical	ab initio
TiH	1		3	181,080	05BuDuBa	[88]	empirical	ab initio
CO${}_2$	13	1000	1	149,587,373	Ames-2016	[7]	empirical	ab initio
TiO	1		13	45,000,000	Schwenke	[89]	tuned ab initio	ab initio
C${}_2$H${}_2$	1	1000	1	33,890,981	ASD-1000	[90]	effect. Hamilt. ${}^c$	effect. dipole ${}^c$
CrH	1		2	13,824	02BuRaBe	[91]	empirical	ab initio
CH${}_3$F	1	400	1	1,391,882,159	OYKYT	[92]	ab initio	ab initio

$N_{\mathrm{iso}}$: Number of isotopologues considered; $T_{\max }$: Maximum temperature for which the line list is complete (where stated); $N_{\mathrm{elec}}$: Number of electronic states considered; $N_{\mathrm{lines}}$: Number of lines: value is for the main isotopologue; DSName: Name of line list and of data set in ExoMol database [50]; ${}^a$ ExoMol methodology, see text; ${}^b$ A mixture of empirical and ab initio values of the dipole moment function; ${}^c$ Empirical, based on Effective Hamiltonian and Dipole moment expansions.

. Data for all these species, including cross sections, are available on the ExoMol website.

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 display temperature-dependent cross sections for each of these species. For all molecules except water, only results for the major (parent) isotopologue are given. As specified in Table 2, many ExoMol line lists also consider isotopically substituted species. In most cases, isotopic substitution leads to shifts in spectroscopic features that are observable at medium to high resolution but result in no fundamental change in structure of the spectrum. The exception is where this substitution leads to breaking of the symmetry, which can lead to new features due to changes in the way the Pauli principle applies, examples here include ${}^{12}$C${}^{12}$C to ${}^{12}$C${}^{13}$C, and changes in vibrational structure such as those encountered on moving from H${}_2$O to HDO.

For most species, absorption cross sections are plotted at two temperatures: 300 K and 2000 K. Exceptions are where the species is unlikely to have significant vapor pressure at 300 K, when this curve has been omitted, or when the species unlikely to survive at 2000 K, in which case an appropriate, lower temperature is used. The figures are grouped according to whether the line lists concerned cover the infrared only (wavenumbers below 12,000 cm${}^{-1}$), extend into the visible (wavenumbers below 20,000 cm${}^{-1}$) or cover the near ultraviolet (wavenumbers below 35,000 cm${}^{-1}$). Beyond that, the figures are grouped to contain roughly similar species. Each of the figures and species are considered in turn below.

H${}_2$O: the spectrum of water (see Figure 1) is important in a whole range of astronomical objects [14] including M-dwarfs [94] and exoplanets [95]. The ExoMol line lists for H${}_2$${}^{18}$O and H${}_2$${}^{17}$O complement the H${}_2$${}^{16}$O BT2 line list [96], which has been widely used for astronomical studies including, for instance, as the foundation of the BT-Settl cool star model atmosphere [97]. A comprehensive line list for HDO is also available [93]. In fact, we have recently updated BT2 with a new H${}_2$${}^{16}$O line list known as POKAZATEL [5]. Besides greatly improving the accuracy of the previous line list, it also extends the temperature range beyond 3000 K, which BT2 was designed for. POKAZATEL has been tested against laboratory spectra recorded in flames and was found to perform very well [98]. On the scale of the figure, BT2 and POKAZATEL are very similar for temperatures below 2000 K, although the detailed line positions given by POKAZATEL are considerably more accurate (see the laboratory study by Campargue et al. [99], for example). However, at high temperatures, POKAZATEL is much more complete. What is really noticeable is how flat the water opacity becomes at high temperatures.

Figure 2 shows cross sections for a number of other hydrides: methane, silane, phosphine, ammonia and ethylene.

CH${}_4$: Figure 2 shows the methane cross sections generated by the 10to10 line list, which contains almost ${10}^{10}$ transitions. Other line lists available for methane are notably the TheoReTS line list(s) [100,101], and a comprehensive experimental line list from Hargreaves et al. [102,103]. The 10to10 line list has been used to identify spectroscopic features in T-dwarfs [104]. Model calculations have shown that reproducing the atmospheric opacity of these methane-rich brown dwarfs requires the explicit consideration of many billions of transitions [25]. As the temperature range of 10to10 is limited, we have recently extended it with a new 34to10 line list, which contains $34\times {10}^{10}$ transitions [105]. To make use of this line list tractable in radiative transport models, the weak lines are amalgamated into so-called super-lines [16]. The temperature coverage for the various methane line lists, which broadly agree with each other, is probably now adequate for astrophysical purposes. However, there is a need for better coverage at shorter wavelengths (<$0.85$ $\mathsf{\mu }$m).

PH${}_3$: the phosphine cross sections (see Figure 2) show a very pronounced feature about 4.5 $\mathsf{\mu }$m, which displays only weak dependence on temperature. Phosphine is a well-known component of solar system gas giants [106], and it can be anticipated that this feature will at some point be used to identify PH${}_3$ in exoplanets.

NH${}_3$: the spectrum of ammonia is given in Figure 2. Ammonia spectra are well known in brown dwarfs [104] and are thought to provide the key signature for coolest class of these species known as Y-dwarfs. Plotted is the BYTe line list, which is significantly less accurate than most of the other line lists considered here and is really only complete for wavenumbers below 10,000 cm${}^{-1}$ and $T<1500$ K. A new ExoMol line list that should help resolve these issues is currently under construction. A line list for ${}^{15}$NH${}_3$ is also available [107].

C${}_2$H${}_4$: the final spectrum given in Figure 2 is of ethylene. Only one temperature is plotted since, despite containing 60 billions line, the line list is only complete up to 700 K. However, it is likely that ethylene will decompose at higher temperatures. Although not shown here, the main features of the ethylene spectrum show an unusually small sensitivity to temperature [75]. Rey et al. [108] have also computed a far-infrared ethylene line list.

Figure 3 shows cross sections for a variety of polyatomic oxides plus hydrogen cyanide.

H${}_2$O${}_2$: hydrogen peroxide cross sections are given in Figure 3. It should be noted that even at room temperature the line list for H${}_2$O${}_2$ given by the current release of HITRAN [2] is not complete, as the strong mid-infrared absorptions are missing. This is because of the difficulty of analyzing the observed spectra in this region.

HCN: the line list shown in Figure 3 is unusual for two reasons. First, it contains transition sets for two isomers, HCN and HNC, which are often considered to be distinct molecules and second it is not new; rather it is a reworking of earlier line lists [110,111] based on extensive compilations of empirical energy levels due to Mellau [112,113,114]. A similar empirically-informed line list is available for H${}^{13}$CN [115]. The updated Harris line list was used to tentatively identify HCN in super-Earth exoplanet 55 Cancri e [116]. However, the 1.54 $\mathsf{\mu }$m feature used for this detection is close to a similar acetylene feature. As discussed below, there is still no good line list for hot acetylene. The opacity of HCN is particularly important for reliable models of cool carbon stars where its inclusion leads to fundamental changes in the stellar structure [117]. The ratio between HCN and HNC can provide a potential thermometer the atmospheres of cool stars [118].

SO${}_2$: sulphur dioxide cross sections are given in Figure 3. SO${}_2$ is thought to be an important component of oxygen-rich exoplanet atmospheres [119,120] and to have been important in both early-Earth [121] and early-Mars [122].

SO${}_3$: sulphur trioxide cross sections are given in Figure 3. SO${}_3$ can be hard to detect because it usually appears in conjunction with SO${}_2$, which absorbs strongly in the same region. Interestingly, our cross sections suggest that it may be easier to distinguish SO${}_3$ from SO${}_2$ at higher temperatures as, while the spectrum SO${}_3$ retains much of it structure at higher temperatures, the spectrum of SO${}_2$ becomes increasingly flattened. The line list is only complete up to 800 K and this has been used as the upper temperature.

H${}_2$CO: Figure 3 shows temperature dependent cross sections of formaldehyde.

CO${}_2$: carbon dioxide cross sections given in Figure 3 are obtained from the Ames-2016 (4000 K) line list of Huang et al. [7,123], for the main isotopologue (${}^{12}$C${}^{16}$O${}_2$) and covers rotational excitations up to J = 220. The line list is based on an empirical potential energy surface and accurate variational nuclear motion calculations. The Ames-2016 database also contains line lists for 12 other isotopologues of CO${}_2$.

Figure 4 shows cross sections of HNO${}_3$, CH${}_3$Cl and C${}_2$H${}_2$.

HNO${}_3$: nitric acid, whose spectrum is shown in Figure 4, is not likely to be an important source of opacity. However, its spectroscopic signature can be clearly observed in the Earth’s atmosphere from space [3,124,125] and thus HNO${}_3$ could be a possible biomarker on exoplanets. Like H${}_2$O${}_2$, the HITRAN data for HNO${}_3$ is incomplete for frequencies above about 1000 cm${}^{-1}$.

CH${}_3$Cl: methyl chloride is also a potential biosignature in the search for life outside of the solar system [126]. ExoMol’s mew line list for methyl chloride covers the two major isomers, CH${}_3^{35}$Cl and CH${}_3^{37}$Cl.

C${}_2$H${}_2$: acetylene cross sections given in Figure 4 are generated using the ASD-1000 database [90]. ASD-1000 was generated using an Effective Hamiltonian and contains about 34,000,000 lines. This would appear to be too few lines for the line list to be complete at elevated temperatures as normally, hot line lists for tetratomics contain billions of transitions in order to be complete for high temperatures. A variational acetylene line list is currently under construction as part of the ExoMol project.

H${}_2$S: Figure 5 shows that hydrogen sulphide has a rather unusual spectrum. Transitions involving the vibrational fundamentals are anomalously weak [127] for this system, which shifts the peak of the vibration–rotation spectrum to shorter wavelengths than is usual: the figure shows that the strongest vibration band is an overtone lying in the 2.5 $\mathsf{\mu }$m region.

Figure 6 gives cross sections for NS as well as the alkaline earth monohydrides MgH and CaH. Together with BeH, MgH and CaH were the first set of species considered by ExoMol. These ExoMol line lists only contain pure rotation and rotation–vibration transitions within the X ${}^2{\Sigma }^{+}$ electronic ground states for each of these species.

NS: NS is one of the first ten diatomic molecules to be detected in space [128,129].

MgH: the A ${}^2\Pi$–X ${}^2{\Sigma }^{+}$ band is also important for magnesium monohydride as, in particular, its use of major importance for establishing isotopic abundances of Mg in stars [130,131,132,133,134]. Weck et al. [13] constructed a purely ab initio line list, which covers this band but is of limited accuracy. More recently, experimentally-driven studies [135,136] have sought to remedy this problem.

CaH: A ${}^2\Pi$–X ${}^2{\Sigma }^{+}$ band is also important in cool stars and brown dwarfs [137].

Figure 7 shows cross sections covering electronic spectra of three diatomic monohydrides, BeH, AlH and CH.

BeH: Darby-Lewis et al. [138] have recently developed a new model for beryllium monohydride, which both improves (marginally) on the earlier Yadin line list [51] and, more significantly, includes a treatment of the A ${}^2\Pi$–X ${}^2{\Sigma }^{+}$ band. This band has been observed in sunspot [139], but the motivation for extending the line list was actually fusion plasmas where the use of Be walls has led to the observation of A–X emissions in the plasma [140]. Darby-Lewis et al. demonstrate that their new line list gives excellent reproduction of emission spectra of both BeH and BeD, as well as predictions for the spectrum of BeT, which is important for fusion studies.

AlH: it is known to be present in sunspots through lines in its A ${}^1\Pi$–X ${}^1{\Sigma }^{+}$ electronic band which lie in the blue [141]; AlH was also recently detected around Mira-variable o Ceti by Kaminski et al. [142]. The ExoMol line list for AlH contains these electronic transitions as well as ground state lines.

CH: an extensive, empirical line list is available for CH due to Masseron et al. [83]. This line list covers transitions within the X ${}^2\Pi$ electronic ground state as well as rovibronic transitions within the A ${}^2\Delta$–X ${}^2\Pi$, B ${}^2{\Sigma }^{-}$–X ${}^2\Pi$ and C ${}^2{\Sigma }^{+}$–X ${}^2\Pi$ bands. Masseron et al. [83] also provide a line list for ${}^{13}$CH.

Figure 8 shows temperature dependent cross sections for HCl, HS, CrH, and OH.

HCl: the line list used to generate the HCl cross sections given in Figure 8 was generated alongside line lists for other hydrogen halides, namely HF, HBr and HI, using semi-empirical methods by Li. et al. [86,143].

Figure 9 shows cross sections for NaCl, KCl, PO, PN and CS.

NaCl and KCl: cross sections are shown in Figure 9. These species have large transition dipoles and hence strong transitions, and they are thought to be of importance for studies of hot super-Earths [20].

PO: phosphorus monoxide has been detected in a number of locations in space: red Supergiant Star VY Canis Majoris [144], in the wind of the oxygen-rich AGB star IK Tauri [145], and in star-forming regions [146,147]. The ExoMol line list covers transitions within the X ${}^2\Pi$ ground electronic state.

PN: it is an important molecule used to probe different regions of the interstellar medium (ISM). The ExoMol line list for PN covers just pure rotation–vibration transitions.

CS: cross sections in Figure 9 illustrate the temperature dependence of the CS spectra as generated using the comprehensive vibration–rotation ExoMol line lists for eight isotopologues of carbon monosulphide (CS) (${}^{12}$C${}^{32}$S, ${}^{12}$C${}^{33}$S, ${}^{12}$C${}^{34}$S, ${}^{12}$C${}^{36}$S, ${}^{13}$C${}^{32}$S, ${}^{13}$C${}^{33}$S, ${}^{13}$C${}^{34}$S, ${}^{13}$C${}^{36}$S) in their ground electronic states.

Figure 10 shows cross sections for the important species CO plus CN, CP and CaO.

CO: cross sections for carbon monoxide are given in Figure 10. CO is ubiquitous throughout the spectra of cool stars [148] and brown dwarfs, and can clearly be seen in exoplanets [149,150]. CO is a very strongly bound species so the molecule survives at relatively high temperatures. As can be seen from the figure, its broadband spectrum changes from comprising a series of sharp bands to becoming quasi-continuous as the temperature is raised. Unlike the other species shown in this figure, the CO cross sections through the visible are simply given by rotation–vibration transitions. We note that Li et al. [8] improved their intensities by refining their CO ab initio dipole moment curve by fitting to experiment. This is not the usual practice [151], but Li et al. found that they could not get satisfactory results using a purely ab initio approach.

CN and CP: cross sections are shown in Figure 10. These two open shell or radical molecules have the same electronic structure and their spectra are dominated by the A ${}^2\Pi$–X ${}^2{\Sigma }^{+}$ and B ${}^2{\Sigma }^{+}$–X ${}^2{\Sigma }^{+}$ bands, which for CN lie in the red and violet regions of the visible. For the heavier CP radical, these bands are shifted to longer wavelengths.

CaO: cross sections for calcium oxide are given in Figure 10. CaO is unlikely to exist in the gas phase in dense atmospheres at 300 K, this spectrum is shown for the consistency. CaO has yet to be detected in space, but it is thought of as a possible component of super-Earth exoplanet atmospheres. CaO possesses transitions with notably large cross sections, which should facilitate its detection.

Figure 11 shows cross sections for NO and PS.

NO and PS: are both radicals with X ${}^2\Pi$ electronic ground states. NO is an important species in the Earth’s atmosphere and is likely to be so in oxygen-rich exoplanets. The current ExoMol line list [9] supersedes the one provided by HITEMP [4]. For NO, the transitions considered in the figure are all within the X ${}^2\Pi$ ground state manifold, while, for PS, the B ${}^2\Pi$–X ${}^2\Pi$ electronic band is also considered. The equivalent electronic transition for NO, which is actually designated the A ${}^2\Pi$ – X ${}^2\Pi$ band, lies well into the ultra violet at about 230 nm [152].

Figure 12 shows cross sections for LiH, ScH, FeH and NH.

LiH cross sections are shown in Figure 12. They are based on ab initio calculations by Coppola et al. [80]. LiH is a four electron system so purely theoretical procedures should give reliable results. The spectrum of LiH is important for possible applications of the lithium test in brown dwarfs [153]. We note that very recently Bittner and Bernath [154] have constructed line lists for both LiF and LiCl.

ScH cross sections are shown in Figure 12. These were also based on an essentially ab initio line list, which contains an empirical adjustment for the band positions. Unlike LiH, ScH is a many-electron system with an open d-shell; such systems present considerable challenges from an ab initio perspective [155,156]; the provision of high accuracy line list for ScH will probably require further experimental input.

FeH cross sections, shown in Figure 12, are conversely based on experimental studies. FeH is an important stellar species, which has been observed in M and L dwarfs [87,157,158] as well as sunspots [159,160,161]. There is a strong need to use these laboratory studies to construct a rigorous theoretical model for the system, which can be used to generate line lists over an extended range of temperatures and wavelengths; however, ab initio calculations on the system remain a challenge [162].

NH: an empirical line list for NH from the Bernath group [82] covering rotation–vibrational transitions within the X ${}^3{\Sigma }^{-}$ ground electronic state.

Figure 13 shows cross sections for SiH, NaH and TiH.

SiH: is an accurate line list covering rotation–vibration transitions within the ground X ${}^2\Pi$ electronic state as well as transitions to the low-lying A ${}^2\Delta$ and a ${}^4{\Sigma }^{-}$ states.

NaH cross sections are shown in Figure 13. NaH remains undetected in stellar spectra but is thought to be an important opacity source in M-dwarfs [164]. However, as our cross sections show the blue A ${}^2{\Sigma }^{+}$–X ${}^2{\Sigma }^{+}$ band is extended and gives a rather flat band shape. This will make the clear detection of NaH in a stellar atmosphere difficult from its main electronic band.

TiH cross sections shown in Figure 13 represent an empirical line list for TiH from Burrows et al. [88].

Figure 13 shows cross sections for SiO, AlO, VO and TiO.

SiO cross sections are shown in Figure 14. SiO is a strongly bound molecule whose infrared spectrum is well-known in stars [165] and sunspots [166,167]. Its electronic spectrum has also been observed in sunspots [168]. At present, the ExoMol project has only provided a long-wavelength line list for SiO based on its temperature-dependent vibration–rotation and rotation spectrum. Work is in progress providing a companion line list covering its electronic bands starting from ab initio potential energy curves recently calculated by Bauschlicher Jr. et al. [169]. In order to illustrate the UV spectra of SiO, Figure 14 also shows cross sections from Kurucz’z database [170].

AlO: the spectrum of aluminium monoxide is illustrated in Figure 14. This spectrum is actually important for plasma diagnostics [171,172]. Rather remarkably, the only previous AlO line list [173], which was prepared for plasma studies, only provided relative as opposed to absolute line intensities.

VO cross sections are shown in Figure 14. The VO spectrum is heavily overlapped by that of TiO and the two molecules often occur together. VO spectral signatures are important for classifying late M dwarfs [174,175], and it is one of the dominant species in the spectra of young hot brown dwarfs [175,176,177]. VO is thought to be present in the atmosphere of hot Jupiter exoplanets [178].

TiO cross sections are shown in Figure 14. TiO is a well-known and important molecule in the atmosphere of M-dwarf stars [179]. Detections in the atmospheres of exoplanets are also beginning to be reported [180,181,182]. The figure is based on the line list of Schwenke et al. [89]; there is also a line list due to Plez [12], which has been updated in a recent release of the VALD database [183]. However, there are known problems with the available spectroscopic data on TiO (see Hoeijmakers et al. [26], for example). An ExoMol TiO line list is currently nearing completion [184]; this line list will make use of the MARVEL study on TiO [43], which leads to more accurate transition frequencies than are provided by the currently available line lists.

Figure 15 gives cross sections for the open shell diatomic species C${}_2$ and of the important molecular ion H${}_3^{+}$.

C${}_2$ has a long spectroscopic history [185,186]. In astronomical objects, its spectrum has been observed via at least six distinct electronic bands, with other band systems required to explain observed populations. An empirical line list for the Swan system was provided by Brooke et al. [187]; the figure shows cross sections generated using the newly constructed ExoMol line list [77], which considers transitions between the eight lowest electronic states in the system. Given the many perturbations between the electronic states that are hard to model and that there are astronomically observed bands not covered by this eight-state model, further work on C${}_2$ will undoubtedly be needed.

H${}_3^{+}$: the spectrum of H${}_3^{+}$ given in Figure 15 is based on ab initio calculations, which have proved themselves to be very accurate for this molecular ion [188]. H${}_3^{+}$ is an unusual species in that it has no (allowed [189]) rotational spectrum and no known electronic spectrum leaving only rotation–vibration transitions. These are well-known in the ionospheres of solar system gas giant planets [190] but have yet to be observed in hot Jupiter exoplanets [191], where they might be thought to be prominent. However, the cooling provided by H${}_3^{+}$ infrared emissions appears to be the key for the stability of atmospheres of giant extrasolar planets close to their host star [192]. This has made H${}_3^{+}$ cooling functions of great importance [193,194]. Similarly H${}_3^{+}$ has yet to observed in stellar or brown dwarf spectra, but H${}_3^{+}$ is the main source of electrons in objects such as cool white dwarfs [195]. The partition function of H${}_3^{+}$ [196] thus controls the amount of H${}^{-}$ present, which, in turn, dominates the opacity. A line list for H${}_2$D${}^{+}$ [197] is also available.

4. Conclusions

Compiling molecular opacity functions requires a large range of spectroscopic data on a large range of molecules. As discussed in this article, for many key species, there are now extensive line lists available that can be used to compute temperature-dependent opacity functions. This is a process of constant improvement and immediate improvements are indeed identified for several species discussed above. Here, we consider what species may be missing from the current compilations.

In their validation of the well-used BT–Settl model [97] atmospheres for M-dwarfs, Rajpurohit et al. [164] identified only AlH, NaH and CaOH as key species for which data was missing. In response to this, the ExoMol project has provided spectroscopic line lists for AlH [76] and NaH [59]; this leaves CaOH, which has a strong band in the visible at 557 nm, as the one key missing species.

The chemistry of oxygen-rich M-dwarf stars is somewhat simpler than that of carbon-rich stars. There are a number of carbon-containing species for which reliable line lists are not available, notably acetylene (HCCH) and C${}_3$. Acetylene spectra are well-studied experimentally [44,198,199] and recently Lyulin and Perevalov [90] produced an effective-Hamiltonian based, empirical ${}^{12}$C${}_2$H${}_2$ line list. Work on an ExoMol acetylene line list is nearing completion. Jørgensen et al. [200] produced an early, purely ab initio line list for C${}_3$. They demonstrated the importance of this species for models of cool carbon stars, but their line list cannot be considered reliable by modern standards where ab initio methods have improved considerably and tuning models to experimental data is now routine. The construction of an improved C${}_3$ line list is underway as part of the ExoMol project. Both ExoMol and TheoReTS are considering line lists for higher hydrocarbons that are needed for studies of exoplanets.

Transition metal containing diatomics provide strong sources of opacities since these open shell species often display strong electronic transitions in the near infrared and visible i.e., near the peak of the stellar flux for a cool star. A few of these species, notably TiO, VO, FeH, ScH and TiH, are considered above but there are many more possible candidates. These include CrH [201], MnH [202] and MgO [78] for which ExoMol has near complete line lists, and species such NiH, ZrO, YO, FeO. Carbides, nitrides and even sulfide species may also need to be considered in due course. Performing accurate ab initio electronic structure calculation on transition metal containing molecules remains very difficult [155,156].

Recent observations have identified a whole new class of exoplanets with masses somewhat larger than the Earth’s and orbits close to their host stars. These hot rocky super-Earths, lava planets or magma planets as they are variously known as are just beginning to be characterized [116]. Even though at this point the molecular composition of their atmosphere is essentially unknown, study of these bodies will create demands for spectroscopic data and opacities for a whole range of new molecules. We recently reviewed the data needs for hot super-Earth exoplanets [20] and refer any interested reader to that article.

Finally, line broadening must be mentioned. Molecular transitions undergo broadening due to temperature and pressure effects. While temperature-induced Doppler broadening is straightforward to model, pressure broadening is more difficult because its importance is, at least in principle, different for every transition. Broadening parameters for key collision broadeners such as H${}_2$ and He at elevated temperatures are largely missing. Approximate methods are being used to fill this void for key molecules such as water [182,203,204]. We have implemented a systematic and complete if approximate broadening “diet” [205] as part of the ExoMol data base [50]. This allows the generation of both temperature- and pressure-dependent cross sections using our program ExoCross [48]. However, there remains much work to be done in both improving temperature-dependent models for line broadening and providing the appropriate broadening parameters.