the temperature and pressure coverage of absorption cross-sections is regularly expanded for improved accuracy of retrievals, for example, using Fourier transform measurements of the molecule contained in a cold cell (Harrison 2018).

Other planets of the Solar System (and their moons) contain many trace gases, which are typically retrieved using experimental absorption cross-sections. For example, Titan, with its rich hydrocarbon atmosphere, requires cross-section measurements of nitrogen broadened species at low temperatures (e.g. Sung et al. 2013, 2018; Vuitton et al. 2019). Similarly, H₂ and He broadened measurements are required for the Jovian planets (e.g. Sung, Toon & Crawford 2016b; Dodangodage et al. 2020). It is expected that some of these species will be present at elevated temperatures in warm Gas Giant exoplanetary atmospheres (Fleury et al. 2023b), therefore recent studies have also provided experimental absorption cross-sections at elevated temperatures (e.g. Alrefae et al. 2014; Hargreaves et al. 2015; Beale et al. 2016; Bernath & Fernando 2021; Adil et al. 2023).

The limited number of absorption cross-section measurements for specific temperatures and pressures makes them less flexible than line lists. However, techniques have been developed to interpolate between measurements with polynomial coefficients (Buehler et al. 2022) as well as producing pseudo-line lists (Steffens et al. 2023). Each technique does not identify quantum mechanical transitions, as is the case for spectroscopic line lists (see earlier in Section 2), but instead empirical parameters are determined that can reproduce the absorption cross-section intensity over the range of measurements. However, care should be taken when extrapolating these techniques to conditions beyond those used from the initial studies.

Ariel will cover the ~2500–5000 cm⁻¹ (~2–4 µm) region at a resolving power of 100 (see Table 1), which spans the spectral range where key molecular functional groups (e.g. C-H stretch at ~3000–3300 cm⁻¹) exhibit strong absorption. The majority of the absorption cross-sections described above and included in HITRAN have spectral features within this region. Some of these molecules have already been proposed as biosignatures (Zhan et al. 2021) or technosignatures (Haqq-Misra et al. 2022), due to their distinct spectral features and limited pathways for synthesis. Indeed, it was in this spectral range that the recent tentative detection of dimethyl sulphide (DMS or (CH₃)₂S) was made from transit spectra of K2-18b recorded using JWST (Madhusudhan et al. 2023) and absorption cross-sections in HITRAN.

Polycyclic aromatic hydrocarbons (PAHs) and tholins are known to contribute significantly to the optical properties of Solar System planetary spectra (Abdel-Shafy & Mansour 2016; Zhao et al. 2018). They have an optical slope similar to cloud/hazes, potentially causing a degeneracy in the interpretation of spectral data caused by aerosols (see Section 6, and, for example, fig. 1 of Ecolano et al. (2022)). Ecolano et al. (2022) explore the detectability of PAHs observed with Ariel (Tinetti et al. 2018) and Twinkle (Edwards et al. 2019a), finding there is a possibility of detection. Dubey et al. (2023) investigate the formation processes of various PAHs on the thermalized atmospheres of hot Jupiter exoplanets, and found that planets with an effective temperature of around 1300 K are the most promising targets for investigating PAHs, particularly for atmospheres with a high carbon-to-oxygen ratio (C/O) and high metallicity. A follow-up study by Grübel et al. (in preparation) focuses on the intercomparison of different cloud/haze, PAHs and tholin models while retrieving the best case scenario for a particular planet, Wasp-6 b. The cross-sections presented by Ecolano et al. (2022), based on calculations of Draine & Li (2007), represent a combined effect observed from a collection of crystalline PAHs. These cross-sections are consistent with observations of the

interstellar medium (ISM) and thus presumably originate from a diverse PAH mixture. Given the sheer amount of different molecules, they currently represent the most convenient way to implement PAHs in atmospheric retrievals. The NASA Ames PAH IR Spectral Data base is a source of cross-section data for PAHs (Boersma et al. 2014; Bauschlicher et al. 2018; Mattioda et al. 2020),<fnref n="13" /> offering extensive data on thousands of PAHs, although primarily for the identification within interstellar emission spectra rather than for their detection in exoplanetary atmospheres.

**Vibrational frequency calculations:** The computation of a full ro-vibrational line list for one species can be very time consuming, and often takes years to produce. There has thus been work on computing the vibrational spectra for a large number of species, and then adding rotational substructure (for example using software such as PGOPHER; Western 2017), in order to indicate where prominent spectral features are expected. Root mean square errors of such computations are typically on the order of 15–25 cm⁻¹ (Zapata Trujillo & McKemmish 2022a), with higher order computational chemistry or laboratory-derived data required to be incorporated for higher accuracies. There are some notable absorption features in the observed spectra of some exoplanet atmospheres, for example in WASP-39b's atmosphere at 4.56 µm as observed by JWST (Alderson et al. 2023), which have yet to be identified among the species with available ro-vibrational spectra. There is benefit in identifying potential species or types of species which could fit such features, which could then give incentive for more detailed work into the full ro-vibrational spectra. The computation of such vibrational frequencies for large numbers of molecules is facilitated by studies into the best practice for appropriate model chemistry choices for harmonic frequency calculations, such as Zapata Trujillo & McKemmish (2022a, 2023). VIBFREQ1295 (Zapata Trujillo & McKemmish 2022b) is a data base of vibrational frequency calculations. RASCALL (Rapid Approximate Spectral Calculations for ALL) (Sousa-Silva et al. 2019) is an approach which can simulate spectra consisting of approximate band centres and qualitative intensities for a huge number of species. The approximate spectra for a large number of phosphorus-bearing species was computed by Zapata Trujillo et al. (2021), building upon the RASCALL data base by using quantum chemistry to improve on the accuracy of the approximate spectra for these species.

In addition, computations are available which focus on the vibrational spectra of nanoclusters, such as (TiO₂)ₙ, which can be important for exoplanet atmosphere models or observations in the IR (Sindel et al. 2023). Vibrational frequencies of other nanoclusters have been computed, such as (VO)ₙ up to n = 10 (Lecoq-Molinos et al. 2024), (VO₂)ₙ up to n = 10 (Lecoq-Molinos et al. 2024), (V₂O₅)ₙ up to n = 4 (Lecoq-Molinos et al. 2024), TiₓOᵧ (Jeong et al. 2000), and (MgO)ₓ (Chen, Felmy & Dixon 2014).

### 2.1.4 How high-resolution laboratory spectroscopy can help improve the accuracy of theoretically computed line lists

There are several ways in which laboratory spectroscopy can contribute to the data needs of exoplanetary atmospheric studies, some of which are already in active use and some require more investment. The most common and efficient methodology of the production of spectroscopic data is semi-empirical, in which molecular spectra are first measured at given laboratory conditions (temperature, pressures, wavelength range etc), analysed, and then inter- or extrapolated to conditions dictated by the retrieval frameworks. A typical example

<footnote n="13">¹³https://www.astrochem.org/pahdb/</footnote>
C₂H₆. MARVEL studies for N₂O isotopologues (see TENNYSON et al. (2024) for ¹⁴N¹⁶O), CO₂ (see IBRAHIM et al. (2024); ALATOOM et al. (2024), CO, HCN/HNC, HCO⁺, BH and ¹⁵NH₃ are being performed as well as updates to ¹⁴NH₃, HCCH, and inevitably, water.

(ii) **TheoReTS:** line lists for several species including CH₂, CH₃, NH₃, H₂CCN, H₂NCN, CH₃Cl, C₃H₄, H₂O₂, CCl₄, CH₃CN or C₂H₆ are being worked on, and will be gradually uploaded to the TheoReTS database (see Section 2.1.1.5).

(iii) **CaSDa:** the CaSDa database is working on producing new data for the existing species listed in Table 5, including new bands and new isotopologues (e.g. for CH₃D). They also plan to develop new line lists for species such as SiH₄ and C₃H₆O₃ (see Section 2.1.1.1).

(iv) **NASA Ames:** line lists for sulphur-bearing species OCS and CS₂ (see Section 2.1.1.4).

(v) **HITRAN and HITEMP:** Work toward the next edition of the HITRAN database is underway, which will include works highlighted in the HITRAN2020 paper (Gordon et al. 2022) and additional recent literature. Examples include expanding the 3800-4800 cm⁻¹ region of NH₃, the addition of new molecules

(vi) **Non-LTE conditions:** although most studies of exoplanet atmospheres assume species to be in local thermodynamic equilibrium (LTE), effects arising from non-LTE conditions are known to be important in Earth's upper atmosphere, as well as in the atmospheres of other Solar System planets. In the most accurate treatment, non-LTE radiative transfer calculations determine the populations of all the atomic and molecular states involved by considering the relevant species' rates with respect to radiative decay and collisional excitation, de-excitation as well as potentially ionization and other processes (Wright et al. 2022). However, complete data sets are rarely available for this purpose; even for the Non-LTE of H atoms, some of the collisional data are not so well known. In particular, data for collisions of H atoms with heavy particles (atoms/molecules) are lacking (Muñoz & Schneider 2019). Studies such as Wright et al. (2022) are working towards improving data and models for non-LTE effects in exoplanet atmospheres. An upcoming paper (García Muñoz et al., Icarus, 2024, in review) also considers non-LTE effects and emphasises the importance of H₂O-H₂O excitation collisions, especially for the vibrational modes of the molecule.

(vii) **Vibrational modes of molecular clusters:** vibrational spectra from DFT calculations are in progress for species such as (TiO)ₙ clusters up to n = 10 and (SiO)ₙ clusters up to n = 20 (Lecoq-Molinos, in prep.).

(viii) **Quantum computing approaches:** *ab initio* line list parameters are currently calculated using modern high-performance computers, particularly crucial for line lists containing millions to billions of transitions. However, the algorithms used can become computationally intensive and time consuming, scaling with the complexity of the molecule and temperature coverage. Quantum computers could be an alternative solution, even though fault-tolerant quantum computers are unlikely to be available in the immediate future. Variational Quantum Algorithms (VQAs) and, in particular, Variational Quantum Eigensolvers (QVEs) are promising class of algorithms to estimate the eigenstates and the corresponding eigenvalues of a given Hamiltonian (Cerezo et al. 2020). This kind of technology demonstrates the potential,

from a theoretical point of view, to describe the electronic structure of complex molecules faster. In the near future, a quantum speedup would be ideal to describe precise line lists (Innan et al. 2023).

## 2.4 Data needs: What's missing and urgent?

(i) **Species occurring in JWST studies:** the recent potential detection of CS₂ in the exoplanetary atmosphere of TOI-270d (Holmberg & Madhusudhan 2024; Benneke et al. 2024) is an example of molecules requiring improvement of the spectral coverage of associated molecular data. SO₂ was recently observed in the atmosphere of hot gas giant exoplanet WASP-39 b (Tsai et al. 2023; Powell et al. 2024), with theoretical studies such as Janssen et al. (2023) predicting the importance of sulphur species in hot rocky exoplanet atmospheres also. These studies highlight the need for line lists of sulphur species such as CS₂ up to high temperatures.

(ii) **Ti- and V-bearing species:** Hoeijmakers et al. (2020) predict significant concentrations of gas-phase TiO₂, VO₂, and TiS in the atmosphere of hot Jupiter exoplanet WASP-121b. These could be important absorbers which currently do not have available accurate ro-vibrational line lists.

(iii) **Species relevant to lava planets:** recent studies suggested that modelling the atmospheric properties of hot-Super Earth (lava) exoplanets require different types species to be considered, including KOH, NaOH, SiO₂, CaOH, FeO, MgO, MgOH (Schaefer et al. 2012; Tennyson & Yurchenko 2017; Mahapatra et al. 2017). There have therefore been calls for line lists for these species spanning the IR and visible spectral region. We note these calls have been somewhat addressed and there are now available line lists from ExoMol for KOH and NaOH (Owens et al. 2021), SiO₂ (Owens et al. 2020), CaOH (Owens et al. 2022b), MgO (Li et al. 2019). Theoretically, FeO is particularly challenging; there are more than fifty low-lying electronic states (Sakellaris et al. 2011; TENNYSON & YURCHENKO 2017).

(iv) **Carbon and hydrocarbon species:** for example, C₃ which can be important for high-temperature carbon atmospheres (Aringer et al. 2009) and for which an ExoMol line list is nearing completion (Lynas-Gray et al. 2024), or C₄H₈ which is considered to be important in the upper atmosphere of low-metallicity objects (Bilger et al. 2013). Martin-Drumel et al. (2023) report recent laboratory work on the infrared spectra of C₃, with a combined fit of available infrared and optical laboratory data performed using the PGOPHER (Western 2017) package. Allene and propyne, CH₃C₂H, are isomers of one another which have both been detected on Titan (Lombardo et al. 2019; Nixon et al. 2013). Propene has also been observed on Earth (Blake et al. 1996), and is expected on Saturn (Moses et al. 2000). There are some known cross-sections for propene (Bernath et al. 2023), including some hosted by the HITRAN database (Sung et al. 2018), but not a line list.

(v) **PAHs:** can be important for Earth and Titan-like atmospheres (López-Puertas et al. 2013). As previously mentioned, one of the main sources of PAH cross-sections is the NASA Ames PAH database (Boersma et al. 2014; Bauschlicher et al. 2018; Mattioda et al. 2020)<fnref n="15" />, which is tailored for characterising species in the cold temperatures of the ISM. To be able to fully comprehend the abundance and role

<footnote n="15">¹⁵ https://www.astrochem.org/pahdb/</footnote>
illustrating the need for such extrapolations is that most of the existing experimental molecular spectra have been recorded at, or close to, the ambient atmospheric conditions on Earth, i.e. room temperature and pressure, and for specific spectroscopic ranges dictated by the experimental setup. A typical retrieval procedure, on the other hand, requires a broad range of temperatures, pressures, and wavelengths, on dense grids of the corresponding values (for example, pressures between at least $1 \times 10^{-5}$–100 bar, and temperatures ~100–3000 K). This is achieved by representing the experimental data in some functional forms, the so-called spectroscopic models, of different complexity and extrapolation power.

This is a powerful, well established procedure, which is however very hungry for experimental data. Some molecules have almost no spectra measured, especially those corresponding to extreme conditions such as suggested for the atmospheres of the so-called Lava planets, including SiO₂, KOH, NaOH, and MgOH (Schaefer, Lodders & Fegley 2012; Tennyson & Yurchenko 2017). Apart from the mainstream atmospheric molecules, such as those described in Section 2.1.2, with relatively accessible spectra, there is a serious lack of analysed laboratory spectra for a large number of molecules at higher temperatures and lower wavelengths.

The so-called effective Hamiltonians (such as used by CaSDa) based on Taylor-type expansions with respect to the angular momenta with a large set of empirical spectroscopic constants provide very accurate description of the experimental spectra (usually recorded at room temperature) but with rather limited extrapolation power (e.g. to higher temperatures and especially to higher energies). These methods usually benefit from an extensive rotational coverage of individual vibrational bands in a given spectroscopic regions and thus are ideally suitable for the analysis of the FTS-like broad-band techniques, as mentioned above. The so-called empirically adjusted ab initio techniques (employed by ExoMol, TheoReTS, and NASA Ames) are based on the empirical refinement of the underlying inter-nuclear ab initio spectroscopic properties, such e.g. potential energy surfaces (PES). These methods hugely benefit from analysed laboratory data (line positions and energies), ideally with an extensive coverage of the vibrational excitations at the lowest rotational excitations to minimize the coupling between the rotational and vibrational degrees of freedom. Put simply, experimental information on vibrations is especially important to increase the quality of the extrapolation of the empirical model, while the rotational information is more useful for interpolation and benchmarking of the models. These type of empirical techniques are less accurate than effective Hamiltonians when it comes to matching the experiment, but are very powerful when extrapolating to higher temperatures and even higher energies.

The MARVEL (Measured Active Rotation–Vibration Energy Levels) (Furtenbacher et al. 2007) process involves analysing high-accuracy laboratory measured spectra to get out experimentally determined energy levels. These energy levels are then used in theoretically computed line lists to improve the accuracy of all transitions which involve those levels. In the case of the ExoMol data base, the MARVEL process has been applied to molecular line lists for species such as C₂H₂ (Chubb et al. 2018a), TiO (McKemmish et al. 2017), H₂CS (Mellor et al. 2022), C₂ (Furtenbacher et al. 2016; McKemmish et al. 2020), CaOH (Wang et al. 2020b), H₂S (Chubb et al. 2018b), NH₃ (Al-Derzi et al. 2015; Furtenbacher et al. 2020), AlO (Bowesman et al. 2021), H₂CO (Al-Derzi et al. 2021; Germann et al. 2024), H₃⁺ (Bowesman et al. 2022), and OCS (Xu & Tennyson 2024) as well as ongoing efforts to refine the line list of PH₃. Due to new laboratory data becoming available, the MARVEL process for improving the accuracy of theoretical line lists need to be assessed

periodically. The opacities computed from line lists therefore also ideally need recomputing periodically, as the accuracy and coverage of line lists improve (see Section 4).

Regardless of the method used to improve the accuracy of theoretically computed line lists, their key common feature is the strong dependence on experimental laboratory data. Even the best modern ab initio methods, except perhaps of very limited systems, do not have the required quality to be used in atmospheric applications and need to be either completely constructed from scratch, such as for the effective Hamiltonians, by fitting the corresponding expansion, the so-called spectroscopic, parameters, or refined, such as for the ab initio methods, by fitting the corresponding analytic descriptions of the spectroscopic models to the experimental data.

Our general appeal to the experimental spectroscopists who would like to engage with exoplanetary atmospheric studies is to talk to the atmospheric modellers about their direct needs in terms of the species, temperature, and wavelength coverage and also talk to the line list providers about their needs in terms of the vibrational and rotational coverage.

## 2.2 Data used by retrieval codes

Owing to the importance of the line lists for atmospheric retrievals, it would not be an exaggeration to say that all, or close to all, the line lists discussed throughout this section, have already been or are in the process of being included in the main (exo-)planetary retrieval frameworks. It is the usual practice of exoplanetary atmospheric modellers to implement new line lists or important updates into their retrieval programs periodically; this usually involves building molecular opacities in the form of cross-sections or k-tables. Details on opacities used in various retrieval codes can be found in Section 4.

## 2.3 What's being worked on?

(i) **ExoMol**: at the time of writing, the following line lists are in progress by the ExoMol project (see Section 2.1.1 'ExoMol'), either as new species or improvements/updates to existing line lists; diatomics: OH, PN, CO, CH, CS, OH⁺, NO⁺, NiH, CrH, AlF, BH, ZrS; triatomics in IR: HCN, HOD; triatomics (electronic): HCN; larger polyatomic (IR): C₂H₂, C₂H₆. MARVEL studies for N₂O isotopologues [see Tennyson et al. (2024a) for ¹⁴N¹⁶O], CO₂ (see Ibrahim et al. 2024; Alatoom et al. 2024), CO, HCN/HNC, HCO⁺, BH, CS₂, C₂H₄, and ¹⁵NH₃ are being performed as well as updates to ¹⁴NH₃, HCCH, and inevitably, water.

(ii) **TheoReTS**: line lists for several species including CH₂, CH₃, NH₃, H₂CCN, H₂NCN, CH₃Cl, C₃H₄, H₂O₂, CCl₄, CH₃CN, or C₂H₆ are being worked on, and will be gradually uploaded to the TheoReTS data base (see Section 2.1.1 'TheoReTS').

(iii) **CaSDa**: the CaSDa data base is working on producing new IR data for the existing species listed in Table 5, including new bands and new isotopologues (e.g. for CH₃D). They also plan to develop new line lists for species such as SiH₄ and C₃H₆O₃ (see Section 2.1.1 'CaSDa').

(iv) **NASA Ames**: line lists for sulphur-bearing species OCS and CS₂ (see Section 2.1.1 'NASA Ames data base').

(v) **HITRAN and HITEMP**: Work toward the next edition of the HITRAN data base is underway, which will include works highlighted in the HITRAN2020 paper (Gordon et al. 2022) and additional recent literature. Examples include expanding the 3800–4800 cm⁻¹ region of NH₃, the addition of new molecules such as S₂ (Gomez, Hargreaves & Gordon 2024).
of PAHs on exoplanets, there is a need for new PAH opacities to be calculated or measured over a broad parameter space of environmental conditions and easily implementable for retrieval applications. This necessity extends beyond the infrared spectrum, as the availability of temperature-dependent UV cross-sections is quite limited (see Section 8 for the availability of UV PAH photionisation and photoabsorption cross-sections).

(vi) **Species relevant to Archean atmospheres:** another special type of exoplanetary atmosphere with special data requirements are archean atmospheres, necessitating line lists for species including HC₃N (Rimmer et al. 2019, 2021).

(vii) **Molecular ions:** Bourgalais et al. (2020) performed laboratory experiments to help identify ions observable by Ariel in the upper atmospheres of sub-Neptunes. Their results pointed towards H₃O⁺ and H₃⁺ being the most promising. Partly as a consequence of this study ExoMol provides line lists for H₃O⁺ (Yurchenko et al. 2020) and H₃⁺ (Mizus et al. 2017; Bowesman et al. 2023). The ExoMol database also provides line lists for a number of other ions including CH⁺ (Pearce et al. 2024), OH⁺ Hodges & Bernath (2017); Hodges et al. (2018) and HeH⁺ (Amaral et al. 2019). Other potentially important ions for which there are no suitable line lists at present include HCO⁺, N₂H⁺ and H₂O⁺, as well as NO⁺ which is the dominant molecular ion in the Earth's ionosphere.

(viii) **Radicals:** such as OH, HO₂, C₂H, CH₂, C₃H₃, NH₂, N₂H₃ are known to be important on Earth (Seinfeld & Pandis 2016), Mars (McElroy & Donahue 1972), Titan (Wilson & Atreya 2004), and other planetary atmospheres (Catling & Kasting 2017).

(ix) **Line lists extending into the UV:** for all major species such as, H₂O, CO₂, CO, OH, CH₄, HCN, N₂, NH₃, C₂H₂, C₂H₄, C₂H₆, H₃⁺, H₂⁺, H₃O⁺, OH⁺, SO, SH, H₂S, SO₂, OCS, S₂, CS. See Section 8 for further discussion on UV photoabsorption and photodissociation data.

(x) **Isotopologues:** Very recently Lew et al. (2024) used JWST to detect isotopologues of CO. Similar studies have been performed on cool stars from the ground (Pavlenko et al. 2020b,a); such isotope-resolved studies have the potential to provide a lot of extra information about atmospheres. Some atoms have more than one isotope with relatively large terrestrial abundances, e.g. Cl is 75% ³⁵Cl and 25% ³⁷Cl. In other cases, isotopic abundances can be very different from Solar abundances, e.g. Mg and C. The importance of H means that even if deuterium (D) is a minor fraction, D containing molecules such HD, HOD and CH₃D can be highly abundant. We note that all these species have different symmetry to the parent (unsubstituted) molecule which means that the spectra of the two species differ significantly. The ExoMol data base provides line lists for HOD (Voronin et al. 2010) and the weak IR spectrum of HD (Amaral et al. 2019), while CH₃D has been considered by Theorets (Rey et al. 2014b). Lines for isotopically substituted species which do not break their symmetry are straight forward to compute given a spectroscopic model for the parent molecule, and this done routinely by ExoMol; methods to improve the accuracy of these line lists have been developed (Polyansky et al. 2017; McKemish et al. 2024). A list of natural (terrestrial) abundances of various species included in the HITRAN database can be

found online<fnref n="16" />, as well as elemental isotopic compositions from NIST<fnref n="17" />.

(xi) **Vibrational modes of molecular clusters:** Köhn et al. (2021) point out that the onset of cloud formation may be observable through vibrational bands of (TiO₂)₄, (TiO₂)₅, and (TiO₂)₆, potentially via the Mid-Infrared Instrument on JWST or the Extremely Large Telescope's mid-IR imager, but more complete line-list data are required first. We note that the full quantum chemical computation of line lists for such large clusters is challenging, and so may require computations of vibrational frequencies only, at least as a starting point. There is some discussion on this in Section 2.1.3.2. We note some vibrational data for TiO₂ nanoclusters have recently been computed (Sindel et al. 2023).

# 3 LINE SHAPES: MOLECULAR AND ATOMIC

## 3.1 Theoretical approaches to line-shape parameters: Molecules

**The Voigt profile.** In order to simulate a spectrum of a single molecule at a given pressure and temperature, three main components are required; line position, line intensity and line profile. In this section, the latter is reviewed. The most popular line profile used in the majority of atmospheric applications (and beyond) is the Voigt profile. It refers to the so-called “intermediate” pressure regime and accounts for collision (pressure) and Doppler (thermal motion) broadenings considered as statistically independent processes and has been the standard for high-resolution line-by-line modelling of infrared molecular absorption (Armstrong 1967). The Voigt profile represents a convolution of the associated Lorentzian and Gaussian profiles, representing the collisional and Doppler effects, and tends to these two shapes in the limiting cases of high and low gas densities, respectively. The Voigt profile has no closed analytical form and is often related to the complex probability function. Several special numerical algorithms exist in the literature for efficient evaluation of the Voigt profile (Humblicek 1982; Kuntz 1997), see the recent computational and numerical comparison by Zhang et al. (2024); some numerical approaches allow one to take advantage of the convolution of the entire spectrum at once (van den Bekerom & Pannier 2021). While for the Doppler contribution, the corresponding (half) line width γD is a simple function of the molecular mass, gas temperature and the line positions (frequency):

$$ \gamma_D = \sqrt{\frac{2N_A k_B T \ln 2}{M} \frac{\tilde{v} f_i}{c}}, \quad (1) $$

there is no simple analytic description of the line (half) width γL, defining the Lorentzian contribution, at least in the general case. The molecular parameters γL strongly depend on the system (the molecule and the perturbing partner), as well as on the gas conditions, i.e. pressure and temperature. Together with the line positions v̄, transition probabilities (e.g. oscillator strength gf or Einstein A coefficients), the Lorentzian width (usually half-width-at-maximum, or HWHM) γL are an integral part of spectroscopic databases. They are typically obtained empirically either through direct measurements with different level of parametric modelling or

<footnote n="16">¹⁶ https://hitran.org/docs/iso-meta/</footnote>

<footnote n="17">¹⁷ http://physics.nist.gov/Comp</footnote>
(vi) **Non-LTE conditions:** although most studies of exoplanet atmospheres assume species to be in local thermodynamic equilibrium (LTE), effects arising from non-LTE conditions are known to be important in Earth's upper atmosphere, as well as in the atmospheres of other Solar System planets. In the most accurate treatment, non-LTE radiative transfer calculations determine the populations of all the atomic and molecular states involved by considering the relevant species' rates with respect to radiative decay and collisional excitation, de-excitation as well as potentially ionization and other processes (Wright, Waldmann & Yurchenko 2022). However, complete data sets are rarely available for this purpose; even for the Non-LTE of H atoms, some of the collisional data are not so well known. In particular, data for collisions of H atoms with heavy particles (atoms/molecules) are lacking (García Muñoz & Schneider 2019). Studies such as Wright et al. (2022) are working towards improving data and models for non-LTE effects in exoplanet atmospheres. In a recent work, García Muñoz, Asensio Ramos & Faure (2024) also considered non-LTE effects and emphasized the importance of H₂O–H₂O excitation collisions, especially for the vibrational modes of the molecule.

(vii) **Vibrational modes of molecular clusters:** vibrational spectra from DFT calculations are in progress for species such as (TiO)ₙ clusters up to *n* = 10 and (SiO)ₙ clusters up to *n* = 20 (Lecoq-Molinos, in preparation).

(viii) **Quantum computing approaches:** ab initio line list parameters are currently calculated using modern high-performance computers, particularly crucial for line lists containing millions to billions of transitions. However, the algorithms used can become computationally intensive and time consuming, scaling with the complexity of the molecule and temperature coverage. Quantum computers could be an alternative solution, even though fault-tolerant quantum computers are unlikely to be available in the immediate future. Variational quantum algorithms (VQAs) and, in particular, variational quantum eigensolvers (QVEs) are promising class of algorithms to estimate the eigenstates and the corresponding eigenvalues of a given Hamiltonian (Cerezo et al. 2021; Tilly et al. 2022). This kind of technology demonstrates the potential, from a theoretical point of view, to describe the electronic structure of complex molecules faster. In the near future, a quantum speedup would be ideal to describe precise line lists (Innan, Al-Zafar Khan & Bennai 2024).

## 2.4 Data needs: what's missing and urgent?

The main molecules expected to be observed by *Ariel* and *JWST* are the usual atmospheric suspects, H₂O, CO₂, and CH₄, as well as the recently detected SO₂. The line lists for these species are generally in a good shape both in terms of temperature completeness and in terms of accuracy, at least as far as space-based observations in the IR are concerned. This applies to other potentially detectable, common atmospheric absorbers such as NH₃, HCN, N₂O, NO, C₂H₂ etc. While the existing data for these species should be at a high level of readiness for the majority of low-to-medium resolution studies including *JWST* or *Ariel*, not all line lists extend to low enough wavelengths to cover the entire range observed by these telescopes – see for example, the wavelength ranges in tables 7–15 of Chubb et al. (2020b) for a number of high-temperature line lists and should be considered when interpreting observed spectra which includes a wavelength region not covered by a given line list. Not taking this into account could cause some biases in analyses which extend beyond the wavelength range of current line lists. Extending current line lists into the optical is therefore very important. Less mainstream or more

exotic molecules are not at such a good readiness level, some of which we review in the following. The line lists for high resolution (HR) applications require even more work. With some exceptions, such as H₂O or CO₂ in the mid-IR, HR applications demand further improvement of the quality and quantity of the laboratory data at high temperatures for most of the molecules. The data needs discussed below are mostly for the IR and optical regions as directly imposed by the observational and associated retrieval coverage of *Ariel* and *JWST*. However, the data needs in shorter wavelengths, UV, and even far-UV, are dictated by the atmospheric chemistry models (see Section 7).

(i) **Species occurring in JWST studies:** The recent potential detection of CS₂ in the exoplanetary atmosphere of TOI-270d (Benneke et al. 2024; Holmberg & Madhusudhan 2024) is an example of a molecule requiring improvement of the spectral coverage of associated molecular data, highlighting the need for line lists of sulphur species such as CS₂ up to high temperatures. The recent observation of SO₂ in the atmosphere of hot gas giant exoplanet WASP-39 b (Tsai et al. 2023; Powell et al. 2024) and in the atmosphere of warm Neptune WASP-107 b (Dyrek et al. 2024) prompted an increase in interest in sulphur species, with theoretical studies such as Janssen et al. (2023) predicting the importance of them also in hot rocky exoplanet atmospheres. The SO₂ line list of Underwood et al. (2016a) is considered complete up to high temperatures of ~2000 K in the wavelength range it covers, but only covers wavelengths down to 1.25 µm. This will lead to an underestimation of opacity in the wavelength region below 1.25 µm for model atmospheres including SO₂. The main reason for the lack of wavelength coverage of the line list is that the quantum chemistry computations used for computing a high-temperature line list can become extremely computationally extensive with increasing energy (decreasing wavelength), especially for molecules composed of heavier atoms. Considering the recent SO₂ detections in exoplanet atmospheres, having a high temperature line list extending to lower wavelengths is a high priority.

(ii) **Ti- and V-bearing species:** Hoeijmakers et al. (2020) predict significant concentrations of gas-phase TiO₂, VO₂, and TiS in the atmosphere of hot Jupiter exoplanet WASP-121b. These could be important absorbers, potentially accessible both by the high and low resolution observational methods, which currently do not have any available ro-vibrational (IR) or rovibronic (Vis and UV) line lists.

(iii) **Species relevant to lava planets:** recent studies suggested that modelling the atmospheric properties of hot-Super Earth (lava) exoplanets require different types species to be considered, including KOH, NaOH, SiO₂, CaOH, FeO, MgO, MgOH (Schaefer et al. 2012; Mahapatra, Helling & Miguel 2017; Tennyson & Yurchenko 2017). There have therefore been calls for line lists for these species spanning the IR and visible spectral region. We note these calls have been somewhat addressed and there are now available line lists from ExoMol for KOH and NaOH (IR) (Owens et al. 2021), SiO₂ (IR) (Owens et al. 2020), CaOH (IR, Vis) (Owens et al. 2022b), MgO (IR, UV) (Li et al. 2019). Theoretically, FeO is particularly challenging; there are more than fifty low-lying electronic states (Sakellaris, Miliordos & Mavridis 2011; Tennyson & Yurchenko 2017).

(iv) **Carbon and hydrocarbon species:** for example, C₃ which can be important for high-temperature carbon atmospheres (Aringer et al. 2009). An ExoMol IR line list for C₃ has just been completed (Lynas-Gray et al. 2024). Martin-Drumel et al. (2023) report recent laboratory work on the infrared spectra of C₃, with a combined fit of available infrared and optical laboratory data performed using
semi-empirically, where a more sophisticated theory tuned to available experimental data is involved such as, e.g., the semi-empirical approaches (Bykov et al. 2004) representing a simplification of the Robert-Bonamy formulae to Anderson-type expressions with fitted correction-factor parameters or modified complex Robert-Bonamy calculations (Ma et al. 2007) with adjusted atom-atom interaction parameters. Application of fully first-principles methods are currently limited to simple system or low temperatures (Wcisło et al. 2021).

The pressure-broadening of spectral lines corresponding to pure rotational (microwave, MW) and rovibrational (infrared, IR) transitions is mainly governed by inelastic collisions (De Lucia & Green 1988) and exhibit (except for light perturbers such as hydrogen and helium) strongly pronounced dependence on the rotational quantum number *J*.

Apart from the line broadening effect, collisions also affect the line positions by introducing the so-called line shift δ. These are also provided by the spectroscopic databases, especially those focusing on high resolution applications, such as terrestrial, planetary (Sung & Varanasi 2005), Solar and in some cases stellar, but so far have not featured very much in the low-mid resolution spectroscopy of exoplanets, as would be applicable to the Ariel mission. It is also noteworthy that ro-vibrational line shifts are typically smaller than the associated line widths by some orders of magnitude.

In the context of the Ariel mission, the temperature and pressure dependencies of line widths are crucial. For Voigt line widths γ (in cm⁻¹atm⁻¹), the following power law is almost always assumed:

$$ \gamma(T) = \gamma_0(T_{\text{ref}}, P_{\text{ref}}) \left(\frac{T}{T_{\text{ref}}}\right)^n \frac{P}{P_{\text{ref}}}, \quad (2) $$

where *T*<sub>ref</sub> and *P*<sub>ref</sub> are some reference temperature and pressure (e.g., 1 atm and 296 K, respectively). Then the parameters γ₀ (reference value of HWHM) and *n* (temperature exponents) are tabulated by the databases for each perturber. This model works quite well for many molecular systems but for a given set of parameters it can deviate when a description of a large range of temperatures is required.

The Lorentzian (and therefore Voigt) profile is known to fail in the far wings leading to significant accumulated overestimation of the opacity at high pressure. As a work-around, a cut-off the far wings at some distance from the centre of the line is applied. The common standards are to use a 25 cm⁻¹ cut-off, employed by HITRAN, ExoMol and now also by MAESTRO (Molecules and Atoms in Exoplanet Science: Tools and Resources for Opacities) (Gharib-Nezhad et al. 2024). A detailed analysis of line-wing profiles and cutoffs in the context of exoplanet atmospheres, including the Voigt profile, can be found in Gharib-Nezhad et al. (2024).

**Beyond the Voigt line profile.** Recent experimental studies powered by the development of experimental techniques in the last few decades have demonstrated the importance of a departure of the line shape description from the simple Voigt profile model, at least for high resolution applications. This motivated the spectroscopic databases also to provide parameters for more accurate profile models. One of such departures is due to the effect of collisional (Dicke) line narrowing.

The line narrowing can be described either as a reduction of the Doppler component (velocity-changing collisions resulting in molecular confinement) or the dependence of the collisional relaxation rates on the relative molecular speed. In the first case, velocity-changing collisions are often considered as strong (no memory effects, Maxwell-Boltzmann re-distribution of velocities after each collision) and the most frequently used associated profile is that of Rautian & Sobel'man (1966, 1967), although the soft-collision model (light perturbers, many collisions required to modify significantly the active-molecule velocity) developed by Galatry (1961) is used by some researchers too. In the second case, the speed-dependence (SD) of the collisional relaxation rates dominates the line narrowing and the corresponding models forming the so-called Speed-Dependent Voigt (SDV) profiles. The most accurate description of the dependence on the absolute absorber's speed (Berman 1972; Pickett 1980) is by the hypergeometric confluent function (hgSDV model), which however, leads to high computational costs. As a more practical approach, a modification of hgSDV based on the mean quadratic speed (Rohart et al. 1994; Köhler et al. 1995; Rohart et al. 2007) is used (qSDV model, often referred to as simply SDV), for which an efficient numerical-calculation algorithm is available (Boone et al. 2007).

Both mechanisms, with the velocity-changing collisions and with the speed dependence viewed as independent, are accounted for in “combined” line-shape models, such as Speed-Dependent Rautian (SDR) or Speed-Dependent Galaxy (SDG) profiles. The partial correlation of these effects is considered by the partially Correlated quadratic-Speed-Dependent Hard-Collision profile (pCqSDHCP), developed by Tran et al. (2013) (called the Hartmann-Tran profile or HTP profile) and recommended for high resolution spectroscopy in a IUPAC Technical Report (Tennyson et al. 2014). Parameters for this advanced profile have been incorporated into the HITRAN database (Gordon et al. 2022) since 2016 for the hydrogen molecule as well as some transitions of water vapor and molecular oxygen. The line narrowing effects, being routinely used in high-resolution terrestrial and planetary applications, have not been considered in the mainstream exoplanetary spectroscopic studies yet, and likely first requires studies to demonstrate the impact of such line profiles on different atmospheric models.

Another important deficiency of the common line profile methodology used in spectral modelling is due to the breakdown of the isolated lines approximation. This phenomenon, also caused by molecular collisions, is associated with the so-called line-mixing or line-interference effect. While the line narrowing effects are mostly noticeable at lower pressure (well below 1 atm for most gases), line-mixing usually becomes relevant at higher pressure when individual lines start to broaden and overlap. Line-mixing effects can be especially significant in the line wings regions where intensity is never truly zero, and there is always some degree of overlapping with others lines. Because of this effect, the intensity in the wings of a band falls off faster than predicted by the sum of Lorentzians (see figures in Hartmann et al. 2010). To a lesser extent, the troughs between individual lines or branches can also be affected. In practice, as stated above, cross-sections calculations employ sums of Voigt profiles with certain wing cut-offs, say ±25 cm⁻¹ from line centres. Such treatment bypasses a proper account of line-mixing and prevents overestimation of absorption away from band centres which would come from addition of billions of line wings (recall that the wings of the Voigt profile fall off as Lorentzians ∝ ω⁻² and not exponentially). However, such cut-offs may lead to another non-physical situation, where the calculated cross-sections between strong bands end up being exactly zero. In
the PGOPHER (Western 2017) package; these and other data were used as input for a MARVEL study (Tennyson 2024). Molecular line lists are required for the following hydrocarbon species, in IR for direct retrievals and at shorter wavelengths for atmospheric chemistry modellings (see more in Section 7). C₄H is considered to be important in the upper atmosphere of low-metallicity objects (Bilger, Rimmer & Helling 2013). Allene (CH₂CHCH) and propyne (CH₃CCH) are isomers of one another which have both been detected on Titan (Lombardo et al. 2019; Nixon et al. 2013). Propene has also been observed on Earth (Blake et al. 1996), and is expected on Saturn (Moses et al. 2000). There are some known cross-sections for propene (Bernath et al. 2023), including some hosted by the HITRAN database (Sung et al. 2018), but not a line list.

(v) **PAHs:** can be important for Earth and Titan-like atmospheres (López-Puertas et al. 2013). As previously mentioned, one of the main sources of PAH cross-sections is the NASA Ames PAH data base (Boersma et al. 2014; Bauschlicher et al. 2018; Mattioda et al. 2020),<fnref n="14" /> which is tailored for characterizing species in the cold temperatures of the ISM. To be able to fully comprehend the abundance and role of PAHs on exoplanets, there is a need for new PAH opacities to be calculated or measured over a broad parameter space of environmental conditions and easily implementable for retrieval applications. This necessity extends beyond the infrared spectrum, as the availability of temperature-dependent UV cross-sections is quite limited (see Section 8 for the availability of UV PAH photoionization and photoabsorption cross-sections).

(vi) **Species relevant to Archean atmospheres:** another special type of exoplanetary atmosphere with special data requirements are archean atmospheres, necessitating line lists for species including HC₃N (Rimmer et al. 2019, 2021).

(vii) **Molecular ions:** Bourgalais et al. (2020) performed laboratory experiments to help identify ions observable by *Ariel* in the upper atmospheres of sub-Neptunes. Their results pointed towards H₃O⁺ and H₃⁺ being the most promising. Partly as a consequence of this study ExoMol provides IR line lists for H₃O⁺ (Yurchenko et al. 2020b) and H₃⁺ (Mizus et al. 2017; Bowesman et al. 2023). The ExoMol data base also provides line lists for a number of other ions including CH⁺ (Pearce et al. 2024), OH⁺ Hodges & Bernath (2017), Hodges et al. (2018) and HeH⁺ (Amaral et al. 2019) covering a very broad wavelength range. Other potentially important ions for which there are no suitable line lists at present include HCO⁺, N₂H⁺, and H₂O⁺, as well as NO⁺ which is the dominant molecular ion in the Earth's ionosphere.

(viii) **Radicals:** such as OH, HO₂, C₂H, CH₂, C₃H₃, NH₂, N₂H₃ are known to be important on Earth (Seinfeld & Pandis 2016), Mars (McElroy & Donahue 1972), Titan (Wilson & Atreya 2004), and other planetary atmospheres (Catling & Kasting 2017). Out of these species, the required molecular data are only in a good shape for OH. There are high quality (accurate and complete) rovibronic data for OH applicable for IR and UV studies (Bernath 2020; Mitev, Yurchenko & Tennyson 2024). Construction of an IR line list for CH₂ is in progress as part of TheoReTs.

(ix) **Line lists are needed** extending into the UV and optical. Extension down to the UV is needed for all major species such as, H₂O, CO₂, CO, OH, CH₄, HCN, N₂, NH₃, C₂H₂, C₂H₄, C₂H₆, H₃⁺, H₃O⁺, OH⁺, SO, SH, H₂S, SO₂, OCS, S₂, CS. See Section 8 for further discussion on UV photoabsorption and photodissociation data. In some of these cases (such as HCN, NH₃, C₂H₂, C₂H₄, C₂H₆, H₃O⁺, H₂S, SO₂, OCS), data in the optical as well as the

UV is also lacking or only partially covered for line lists which are appropriate up to high temperatures. Extending even just down to the optical region is even more pressing for direct use of these line lists in interpreting and modelling *Ariel* and *JWST* observations. We note that spectroscopic measurements of excited states of some of these species in the UV region are available in the literature, for example for NH₃ from Langford et al. (1998). See Section 2.1.4 for a discussion on how measured data such as these can be used to compute an accurate and complete line list up to high temperatures.

(x) **Isotopologues:** Very recently Lew et al. (2024) used *JWST* to detect isotopologues of CO. Similar studies have been performed on cool stars from the ground (Pavlenko, Yurchenko & Tennyson 2020b; Pavlenko et al. 2020a); such isotope-resolved studies have the potential to provide a lot of extra information about atmospheres. Some atoms have more than one isotope with relatively large terrestrial abundances, e.g. Cl is 75 per cent ³⁵Cl and 25 per cent ³⁷Cl. In other cases, isotopic abundances can be very different from Solar abundances, e.g. Mg and C. The importance of H means that even if deuterium (D) is a minor fraction, D containing molecules such HD, HOD, and CH₃D can be highly abundant. We note that all these species have different symmetry to the parent (unsubstituted) molecule which means that the spectra of the two species differ significantly. The ExoMol data base provides line lists for HOD (Voronin et al. 2010) and the weak IR spectrum of HD (Amaral et al. 2019), while CH₃D has been considered by TheoReTs (Rey, Nikitin & Tyuterev 2014a). Lines for isotopically substituted species which do not break their symmetry are straightforward to compute given a spectroscopic model for the parent molecule, and this done routinely by ExoMol; methods to improve the accuracy of these line lists have been developed (Polyansky et al. 2017; McKemmish et al. 2024). A list of natural (terrestrial) abundances of various species included in the HITRAN data base can be found online,<fnref n="15" /> as well as elemental isotopic compositions from NIST.<fnref n="16" />

(xi) **Vibrational modes of molecular clusters:** Köhn et al. (2021) point out that the onset of cloud formation may be observable through vibrational bands of (TiO₂)₄, (TiO₂)₅, and (TiO₂)₆, potentially via the mid-infrared instrument on *JWST* or the extremely large telescope's mid-IR imager, but more complete line-list data are required first. We note that the full quantum chemical computation of line lists for such large clusters is challenging, and so may require computations of vibrational frequencies only, at least as a starting point. There is some discussion on this in Section 2.1.3 'Vibrational frequency calculations'. We note some vibrational data for TiO₂ nanoclusters have recently been computed (Sindel et al. 2023).

# 3 LINE SHAPES: MOLECULAR AND ATOMIC

## 3.1 Theoretical approaches to line-shape parameters: molecules

**The Voigt profile.** In order to simulate a spectrum of a single molecule at given pressure and temperature, three main components are required; line position, line intensity, and line profile. In this section, the latter is reviewed. The most popular line profile used in the majority of atmospheric applications (and beyond) is the Voigt profile. It refers to the so-called 'intermediate' pressure regime and accounts for collision (pressure) and Doppler (thermal motion) broadenings considered as statistically independent processes and

<footnote n="14">¹⁴https://www.astrochem.org/pahdb/</footnote>

<footnote n="15">¹⁵https://hitran.org/docs/iso-meta/</footnote>

<footnote n="16">¹⁶http://physics.nist.gov/Comp</footnote>
reality, line-mixing would cause a sub-Lorentzian shape that far enough from strong absorption bands would contribute to continuum absorption (see Section 5 for a discussion on continuum absorption). An in-depth discussion of wing cut-offs is presented in Gharib-Nezhad et al. (2024). An empirical method of far-wing corrections involving $\chi$-factors is discussed in the following sections.

At higher pressure where even line cores significantly overlap, line-mixing can lead to a noticeable intensity redistribution within a branch or an entire band. This is known to happen for closely spaced lines in Q-branches and band heads even at sub-atmospheric pressures, and these features may appear stronger and more narrow than one would expect from the sum of isolated lines. This presents potential interest for more reliable confirmation of molecular species' detection, where features like these are often the most noticeable. The case of very high pressure has been considered before in order to describe the deep venusian atmosphere (Filippov et al. 2013; Buldyreva et al. 2013). In such conditions, the rotational structure is entirely collapsed and the band shape differs significantly from the sum of individual spectral lines. This is relevant to modelling of radiative transfer in hot dense atmospheres.

Proper analysis of line-mixing is complicated, and usually a perturbative approach is taken on a band-by-band basis for the most intense bands. At present, line-mixing is not typically implemented in suites for producing opacities for exoplanetary atmospheric retrievals, although these effects may become more important with newer instruments that can probe high-pressure regions of around 1 - 10 bar.

**Calculations of line widths and shifts.** The needs of MW/IR pressure-broadened line widths and pressure-induced line shifts for atmospheric applications motivated an extensive development of theoretical methods. The most developed and commonly accepted hypothesis of the impact approximation completed by various choices for the translational dynamics gave birth to semi-classical (Anderson 1949; Tsao & Curnutte 1962), classical (Gordon 1966), and quantum-mechanical (Baranger 1958a; Shafer & Gordon 1973) approaches. Besides, there exist statistical approaches, e.g. Margenau (1935), whose two-particle static limit is the opposite case to the impact approximation. Static limit works well for low molecular velocities (negligible rate of change of intermolecular interactions) and is applicable at large frequency detunings, i.e. in the far spectral wings which are hardly observable at IR wavelengths. Therefore, approaches based on the impact approximation remain major tools for predicting collisional line-shape parameters. High computational cost and absence of refined potential-energy surfaces (PES) set limits to the use of quantum-mechanical methods as soon as polyatomic/"exotic" molecular pairs and/or high temperatures are concerned.

**Quantum-mechanical approaches.** The most rigorous method, close coupling (CC), is usually applied to predict parameters of purely rotational transitions at room temperature, and only some studies address hot environments. Such data, provided the vibrational contribution to line-broadening is small, can technically be reused for other lines in the infrared. Certain simplifications of this formalism, such as the coupled-states (CS) approximation (used, for example, in Thibault et al. 2011), the infinite-order sudden (IOS) approximation (Goldflam et al. 1977), and the nearest neighbour Coriolis coupling (NNCC) (Yang et al. 2018; Selim et al.

2022) can make calculations more manageable while preserving the accuracy in the high-temperature limit for most lines in the band. The main issue of quantum-mechanical approaches at higher temperatures is that the number of rotational functions needed to construct a complete basis grows too fast, making calculations of the scattering matrix impractical. A useful feature of quantum-mechanical approaches is that they allow estimating parameters of beyond-Voigt profiles mentioned above. Semiclassical approaches can provide the Lorentzian component of the profile, and with some extra work, the speed-dependence parameters. If all the parameters of the extended Hartmann-Tran profile are required, one could employ the generalised Hess approach along with CC for the scattering matrix computation. This has been done recently for H₂ and HD with helium as the perturber (Wcisło et al. 2021; Stankiewicz et al. 2021) up to 1000 K.

**Semi-classical approaches.** The need for accurate potential energy surfaces (PES) and long-time statistics accumulation also disadvantage purely classical computations. As a result, advanced semi-classical approaches are typically employed, such as the formalism of Robert and Bonamy (Robert & Bonamy 1979) and its fully complex implementation (Lynch et al. 1996) further supplied by exact trajectories (Buldyreva et al. 1999), a corrected average over the perturber's states (Antony et al. 2006; Ma et al. 2007) and an account for line coupling (Ma et al. 2013). In cases where large volumes of line-shape data are to be produced with a minimal CPU cost, the semi-empirical method (Bykov et al. 2004) is used which represents a simplification of the Robert-Bonamy expressions to the Anderson-Tsao-Curnutte forms with a correction factor with empirically fitted parameters. Although some experimental values are necessary, the adjusted model parameters are practically temperature-independent and can be used for calculations at other required temperatures (not too low to keep the validity of the trajectory notion).

**Machine Learning.** Given the high volume of line broadening data that is required to model the effects of various different exoplanet compositions, machine learning appears to provide a practical approach to the problem. Recently Guest et al. (2024) have complete an initial machine learning study of air-broadening based on data in the HITRAN database. Studies which look at the effect of changing the broadener are underway.

**UV and visible frequency ranges.** Spectra which involve transitions between electronic states (i.e. rotation-vibration-electronic or rovibronic transitions), typically between the ground and an excited electronic state, occur, with some exceptions, in the UV and/or visible region. In these cases, the line broadening due to predissociative effects can dominate the collisional (pressure) broadening by some orders of magnitude. These broadening are caused by the associated lifetimes of *quasi-bound* transitions decreased. A new data format has recently been proposed to capture (Tennyson et al. 2023) and use (Yurchenko et al. 2023) the predissociative line broadening in spectroscopic databases.

In the case of rovibronic transitions between *bound* low-lying vibrational levels of electronic states, available measurements show *no* well pronounced *J*-dependencies in the collisional broadening of rovibronic transitions. The absence of rotational dependence of the collisional (pressure) broadening means that rotational energy transfer is of minor importance and the broadening parameters are mainly influenced by elastic collisions. For such transitions line widths and line
has been the standard for high-resolution line-by-line modelling of infrared molecular absorption (Armstrong 1967). The Voigt profile represents a convolution of the associated Lorentzian and Gaussian profiles, representing the collisional and Doppler effects, and tends to these two shapes in the limiting cases of high and low gas densities, respectively. The Voigt profile has no closed analytical form and is often related to the complex probability function. Several special numerical algorithms exist in the literature for efficient evaluation of the Voigt profile (Humblicek 1982; Kuntz 1997), see the recent computational and numerical comparison by Zhang et al. (2024); some numerical approaches allow one to take advantage of the convolution of the entire spectrum at once (van den Bekerom & Pannier 2021). While for the Doppler contribution, the corresponding (half) line width $γ_D$ is a simple function of the molecular mass, gas temperature, and the line positions (frequency):

$$γ_D = \sqrt{\frac{2N_A k_B T \ln 2}{M} \frac{\tilde{v}_{fi}}{c}}, \qquad (1)$$

there is no simple analytic description of the line (half) width $γ_L$, defining the Lorentzian contribution, at least in the general case. The molecular parameters $γ_L$ strongly depend on the system (the molecule and the perturbing partner), as well as on the gas conditions, i.e. pressure and temperature. Together with the line positions $\tilde{v}$, transition probabilities (e.g. oscillator strength $gf$ or Einstein A coefficients), the Lorentzian width (usually half-width-at-maximum, or HWHM) $γ_L$ are an integral part of spectroscopic data bases. They are typically obtained empirically either through direct measurements with different level of parametric modelling or semi-empirically, where a more sophisticated theory tuned to available experimental data is involved such as, e.g. the semi-empirical approaches (Bykov, Lavrentieva & Sinitsa 2004) representing a simplification of the Robert-Bonamy formulae to Anderson-type expressions with fitted correction-factor parameters or modified complex Robert-Bonamy calculations (Ma, Tipping & Boulet 2007) with adjusted atom-atom interaction parameters. Application of fully first-principles methods are currently limited to simple system or low temperatures (Wcisło et al. 2021).

The pressure-broadening of spectral lines corresponding to pure rotational (microwave, MW) and rovibrational (infrared, IR) transitions is mainly governed by inelastic collisions (De Lucia & Green 1988) and exhibit (except for light perturbers such as hydrogen and helium) strongly pronounced dependence on the rotational quantum number J.

Apart from the line broadening effect, collisions also affect the line positions by introducing the so-called line shift $δ$. These are also provided by the spectroscopic data bases, especially those focusing on high resolution applications, such as terrestrial, planetary (Sung & Varanasi 2005), Solar, and in some cases stellar, but so far have not featured very much in the low-mid resolution spectroscopy of exoplanets, as would be applicable to the *Ariel* mission. It is also noteworthy that ro-vibrational line shifts are typically smaller than the associated line widths by some orders of magnitude.

In the context of the *Ariel* mission, the temperature and pressure dependencies of line widths are crucial. For Voigt line widths $γ$ (in cm$^{-1}$atm$^{-1}$), the following power law is almost always assumed:

$$γ(T) = γ_0(T_{\text{ref}}, P_{\text{ref}}) \left(\frac{T_{\text{ref}}}{T}\right)^n \frac{P}{P_{\text{ref}}}, \qquad (2)$$

where $T_{\text{ref}}$ and $P_{\text{ref}}$ are some reference temperature and pressure (e.g. 1 atm and 296 K, respectively). Then the parameters $γ_0$ (reference value of HWHM) and $n$ (temperature exponents) are tabulated by the data bases for each perturber. This model works quite well for many

molecular systems but for a given set of parameters it can deviate when a description of a large range of temperatures is required.

The Lorentzian (and therefore Voigt) as an isolated line profile model is known to fail in the far wings, always leading to an overestimation of the opacity at high pressure, especially for the systems with closely spaced lines (Chapter IV Hartmann, Boulet & Robert 2008), e.g. in heavier molecules. As a work-around, a cut-off the far wings at some distance from the centre of the line is applied. The common standards are to use a 25 cm$^{-1}$ cut-off, employed by HITRAN, ExoMol, and now also by MAESTRO (Molecules and Atoms in Exoplanet Science: Tools and Resources for Opacities) (Gharib-Nezhad et al. 2024). A detailed analysis of line-wing profiles and cutoffs in the context of exoplanet atmospheres, including the Voigt profile, can be found in Gharib-Nezhad et al. (2024).

**Beyond the Voigt line profile.** Recent experimental studies powered by the development of experimental techniques in the last few decades have demonstrated the importance of the departure of the line shape description from the simple Voigt profile model, at least for high resolution applications. This motivated the spectroscopic data bases also to provide parameters for more accurate profile models. One of such departures is due to the effect of collisional (Dicke) line narrowing.

The line narrowing can be described either as a reduction of the Doppler component (velocity-changing collisions resulting in molecular confinement) or the dependence of the collisional relaxation rates on the relative molecular speed. In the first case, velocity-changing collisions are often considered as strong (no memory effects, Maxwell-Boltzmann re-distribution of velocities after each collision) and the most frequently used associated profile is that of Rautian & Sobel'man (1966, 1967), although the soft-collision model (light perturbers, many collisions required to modify significantly the active-molecule velocity) developed by Galatry (1961) is used by some researchers too. It is now understood that both mechanisms can be important across any pressure regimes, from very low to very high (D'Eu, Lemoine & Rohart 2002).

In the second case, the speed-dependence (SD) of the collisional relaxation rates dominates the line narrowing and the corresponding models forming the so-called speed-dependent Voigt (SDV) profiles. The most accurate description of the dependence on the absolute absorber's speed (Berman 1972; Pickett 1980) is by the hypergeometric confluent function (hgSDV model), which however, leads to high computational costs. As a more practical approach, a modification of hgSDV based on the mean quadratic speed (Rohart, Mäder & Nicolaisen 1994; Köhler, Mäder & Nicolaisen 1995; Rohart et al. 2007) is used (qSDV model, often referred to as simply SDV), for which an efficient numerical-calculation algorithm is available (Boone, Walker & Bernath 2007).

Both mechanisms, with the velocity-changing collisions and with the speed dependence viewed as independent, are accounted for in 'combined' line-shape models, such as speed-dependent Rautian (SDR) or speed-dependent Galatry (SDG) profiles. The partial correlation of these effects is considered by the partially correlated quadratic-speed-dependent hard-collision profile (pCqSDHCP), developed by Tran, Ngo & Hartmann (2013) (called the Hartmann-Tran profile or HTP profile) and recommended for high resolution spectroscopy in a IUPAC technical report (Tennyson et al. 2014). Parameters for this advanced profile have been incorporated into the HITRAN data base (Gordon et al. 2022) since 2016 for the hydrogen molecule as well as some transitions of water vapour and molecular oxygen. The line narrowing effects, being routinely used in high-resolution terrestrial and planetary applications, have not been
shifts have the same orders of magnitude. As mentioned by Margenau (1936), lines of rovibronic (bound-bound) transitions are broadened and shifted similarly to atomic lines of a similar type and a similar energy jump. Indeed, the first theoretical approaches to pressure-induced line widths/shifts used the model of a classical oscillating radiator whose phase is modified during collision by a random amount, leading thus, via Fourier-integral analysis, to a “dispersion” (Lorentz) profile. For example, Lindholm (1945) and Foley (1946) successfully developed such theories. Foley (1946), in particular, argued that non-adiabatic effects are small for atoms due to large gaps between their electronic levels but can occur for molecular IR transitions because of similar energy differences between the rotational levels of the radiator and perturber. Detailed analyses of some representative one-term interactions (dipole-dipole, quadrupole-quadrupole and dispersive) in the framework of phase-shift theory were made by Mizushima (1951) who also considered velocity averaging and derived line-width and line-shift expressions (intended for microwave transitions) with explicit temperature dependencies. These expressions served as a basis for the power law of Eq. (2). Later, Hindmarsh et al. (1967) accounted additionally for short-range repulsive forces via a Lennard-Jones 12-6 potential model but for simplicity conducted their derivation with the mean-thermal-velocity approximation. Temperature dependence of collisional broadening and shifts of atomic and molecular rovibronic lines has been also addressed by Jablonski et al. (2013). Quantum-mechanical adiabatic approaches were initiated by Jablonski (1937, 1945) who assumed the gas to be a very large “molecule” and calculated the stationary states of internal motion. A unified model relying on the Franck-Condon principle and containing both impact and quasistatic limits was proposed by Szudy & Baylis (1975). The authors retrieved the expressions of Lindholm (1945) and Foley (1946) in the classical limit and suggested also a first-order correcting term leading to line shapes with an asymmetry effect.

When there is an interaction with one or more dissociative states, i.e., predissociations, the broadening can vary significantly due to a variation in the lifetimes of the excited states. The broadening in this case is a manifestation of such interaction with the continuum. The extent of the line broadening (or lifetime of the state in question) derives from the strength of the predissociation, i.e., the strength of the coupling between the bound excited states and the dissociative states. Typically non-radiative lifetimes of electronically excited states are ca. $10^{-8}$s which result in widths of $5 \times 10^{-4}$ cm$^{-1}$. For lifetimes of $10^{-13}$ s the linewidth will be $50$ cm$^{-1}$ which will result in the rotational structure disappearing. For nonradiative lifetimes shorter than $10^{-15}$ s, the vibrational structure disappears and will resemble a continuum. These types of predissociation-induced line broadenings have been well documented experimentally for over half a century (see chapter 7.4.2 in Léfeuvre-Brion & Field (2004)).

## 3.2 Theoretical approaches to line-shape parameters: Atoms

The broadening of alkali lines by rare gases is a problem which has been extensively investigated in experimental and theoretical work (Allard & Kielkopf 1982; Burrows & Volobuyev 2003; Allard et al. 2016, 2019; Peach et al. 2020; Allard et al. 2023, 2024). A theory of spectral line broadening

has been developed to calculate neutral atom spectra given the interaction and transition moments for relevant states of the radiating atom with other atoms in its environment. Within this framework it is possible to compute the complete spectrum with a unified approach. Unlike impact theories of line broadening which predict a Lorentzian line or the approximation methods of Szudy & Baylis (1975, 1996) the unified theory of Allard et al. (1999) provides an accurate spectrum from the line centre to the extreme wing. Complete details and the derivation of the theory are given by Allard et al. (1999). The approach is based on quantum theory of spectral line shapes by Baranger (1958b,c) with an adiabatic representation to include the degeneracy of atomic levels.

The spectrum $I(\Delta\omega)$ can be written as the Fourier transform (FT) of the dipole autocorrelation function $\Phi(s)$,

$$I(\Delta\omega) = \frac{1}{\pi} Re \int_{0}^{+\infty} \Phi(s) e^{-i\Delta\omega s} ds, \quad (3)$$

where $\Delta\omega$ is the angular frequency difference from the unperturbed centre of the spectral line. The autocorrelation function $\Phi(s)$ is calculated with the assumptions that the radiator is stationary in space, the perturbers are mutually independent, and in the adiabatic approach the interaction potentials give contributions that are scalar additive. This last simplifying assumption allows the total profile $I(\Delta\omega)$ to be calculated when all the perturbers interact, as the FT of the $N^{th}$ power of the autocorrelation function $\phi(s)$ of a unique atom-perturber pair. Therefore,

$$\Phi(s) = (\phi(s))^N, \quad (4)$$

that is to say, the interperturber correlations are neglected. For a perturber density $n_p$,

$$\Phi(s) = e^{-n_p g(s)}, \quad (5)$$

where decay of the autocorrelation function with time leads to atomic line broadening. When $n_p$ is high, the spectrum is evaluated by computing the FT of Eq. (5). The real part of $n_p g(s)$ damps $\Phi(s)$ for large $s$ but this calculation is not feasible when extended wings have to be computed at low density because of the very slow decrease of the autocorrelation function. An alternative is to use the expansion of the spectrum $I(\Delta\omega)$ in powers of the density described in Royer (1971). For the implementation of alkali lines perturbed by helium and molecular hydrogen in atmosphere codes, the line opacity is calculated by splitting the profile into a core component described with a Lorentzian profile, and the line wings computed using an expansion of the autocorrelation function in powers of density. The impact approximation determines the asymptotic behavior of the unified line shape correlation function. In this way the results are applicable to a more general line profile and opacity evaluation for the same perturbers at any given layer in the photosphere or planetary atmosphere.

In Allard et al. (2019) the use of a density expansion in the opacity tables is reviewed. When the expansion is stopped at the first order it is equivalent to the one-perturber approximation. Previous opacity tables were constructed to third order allowing us to obtain line profiles up to $N_{He}/N_{H2}=10^{19}$ cm$^{-3}$. The new tables are constructed to a higher order allowing line profiles to $N_{He}/N_{H2}=10^{21}$ cm$^{-3}$.
considered in the mainstream exoplanetary spectroscopic studies yet, and likely first requires studies to demonstrate the impact of such line profiles on different atmospheric models.

Another important deficiency of the common line profile methodology used in spectral modelling is due to the breakdown of the isolated lines approximation. This phenomenon, also caused by molecular collisions, is associated with the so-called line-mixing or line-interference effect. While the line narrowing effects are mostly noticeable at lower pressure (well below 1 atm for most gases), line-mixing usually becomes relevant at higher pressure when individual lines start to broaden and overlap. Line-mixing effects can be especially significant in the line wings regions where intensity is never truly zero, and there is always some degree of overlapping with others lines. Because of this effect, the intensity in the wings of a band falls off faster than predicted by the sum of Lorentzians (see figures in Hartmann et al. 2010). To a lesser extent, the troughs between individual lines or branches can also be affected. In practice, as stated above, cross-sections calculations employ sums of Voigt profiles with certain wing cut-offs, say $\pm 25 \text{ cm}^{-1}$ from line centres. Such treatment bypasses a proper account of line-mixing and prevents overestimation of absorption away from band centres which would come from addition of billions of line wings (recall that the wings of the Voigt profile fall off as Lorentzians $\propto \omega^{-2}$ and not exponentially). However, such cut-offs may lead to another non-physical situation, where the calculated cross-sections between strong bands end up being exactly zero. In reality, line-mixing would cause a sub-Lorentzian shape that far enough from strong absorption bands would contribute to continuum absorption (see Section 5 for a discussion on continuum absorption). An in-depth discussion of wing cut-offs is presented in Gharib-Nezhad et al. (2024). An empirical method of far-wing corrections involving $\chi$-factors is discussed in the following sections.

At higher pressure where even line cores significantly overlap, line-mixing can lead to a noticeable intensity redistribution within a branch or an entire band. This is known to happen for closely spaced lines in Q-branches and band heads even at sub-atmospheric pressures, and these features may appear stronger and more narrow than one would expect from the sum of isolated lines. This presents potential interest for more reliable confirmation of molecular species' detection, where features like these are often the most noticeable. The case of very high pressure has been considered before in order to describe the deep venusian atmosphere (Buldyreva, Gennadiev & Filippov 2013; Filippov et al. 2013). In such conditions, the rotational structure is entirely collapsed and the band shape differs significantly from the sum of individual spectral lines. This is relevant to modelling of radiative transfer in hot dense atmospheres.

Proper analysis of line-mixing is complicated, and usually a perturbative approach is taken on a band-by-band basis for the most intense bands. At present, line-mixing is not typically implemented in suites for producing opacities for exoplanetary atmospheric retrievals, although these effects may become more important with newer instruments that can probe high-pressure regions of around 1–10 bar.

**Calculations of line widths and shifts.** The needs of MW/IR pressure-broadened line widths and pressure-induced line shifts for atmospheric applications motivated an extensive development of theoretical methods. The most developed and commonly accepted hypothesis of the impact approximation completed by various choices for the translational dynamics gave birth to semi-classical (Anderson 1949; Tsao & Curnutte 1962), classical (Gordon 1966), and quantum-mechanical (Baranger 1958c; Shafer & Gordon 1973) approaches. Besides, there are statistical approaches, e.g. Margenau

(1935), whose two-particle static limit is the opposite case to the impact approximation. Static limit works well for low molecular velocities (negligible rate of change of intermolecular interactions) and is applicable at large frequency detunings, i.e. in the far spectral wings which are hardly observable at IR wavelengths. Therefore, approaches based on the impact approximation remain major tools for predicting collisional line-shape parameters. High computational cost and absence of refined potential-energy surfaces (PES) set limits to the use of quantum-mechanical methods as soon as polyatomic/'exotic' molecular pairs and/or high temperatures are concerned.

**Quantum-mechanical approaches.** The most rigorous method, close coupling (CC), is usually applied to predict parameters of purely rotational transitions at room temperature, and only some studies address hot environments. Such data, provided the vibrational contribution to line-broadening is small, can technically be reused for other lines in the infrared. Certain simplifications of this formalism, such as the coupled-states (CS) approximation (used, for example, in Thibault et al. 2011), the infinite-order sudden (IOS) approximation (Goldflam, Green & Kouri 1977), and the nearest neighbour Coriolis coupling (NNCC) (Yang et al. 2018; Selim, van der Avoird & Groenenboom 2022) can make calculations more manageable while preserving the accuracy in the high-temperature limit for most lines in the band. The main issue of quantum-mechanical approaches at higher temperatures is that the number of rotational functions needed to construct a complete basis grows too fast, making calculations of the scattering matrix impractical. A useful feature of quantum-mechanical approaches is that they allow estimating parameters of beyond-Voigt profiles mentioned above. Semiclassical approaches can provide the Lorentzian component of the profile, and with some extra work, the speed-dependence parameters. If all the parameters of the extended Hartmann-Tran profile are required, one could employ the generalized Hess approach along with CC for the scattering matrix computation. This has been done recently for H₂ and HD with helium as the perturber (Stankiewicz et al. 2021; Wcisło et al. 2021) up to 1000 K.

**Semiclassical approaches.** The need for accurate potential energy surfaces (PES) and long-time statistics accumulation also disadvantage purely classical computations. As a result, advanced semiclassical approaches are typically employed, such as the formalism of Robert and Bonamy (Robert & Bonamy 1979) and its fully complex implementation (Lynch, Gamache & Neshya 1996) further supplied by exact trajectories (Buldyreva, Bonamy & Robert 1999), a corrected average over the perturber's states (Antony et al. 2006; Ma et al. 2007) and an account for line coupling (Ma, Boulet & Tipping 2013). In cases where large volumes of line-shape data are to be produced with a minimal CPU cost, the semi-empirical method (Bykov et al. 2004) is used which represents a simplification of the Robert-Bonamy expressions to the Anderson-Tsao-Curnutte forms with a correction factor with empirically fitted parameters. Although some experimental values are necessary, the adjusted model parameters are practically temperature-independent and can be used for calculations at other required temperatures (not too low to keep the validity of the trajectory notion).

**Machine learning.** Given the high volume of line broadening data that is required to model the effects of various different exoplanet compositions, machine learning appears to provide a practical approach to the problem. Recently Guest, Tennyson & Yurchenko (2024) have completed an initial machine learning study of air-broadening based on data in the HITRAN data base. Studies which look at the effect of changing the broadener are underway.
## 3.3 State of the art - Data availability

### 3.3.1 Molecular line profiles

**Voigt parameters:** it is now recognised in the (exo-)atmospheric community that Earth's conditions with the oxygen and nitrogen as the main collisional broadeners are not default ones, especially for hot Jupiter exoplanets, where hydrogen and helium dominate molecular collisions. Although, as noted below, broadening parameters are becoming available for a small number of systems which deviate from the Voigt profile, the vast majority of available data are for Voigt broadening parameters. Even so, collisional broadening parameters which are required for computing Voigt profiles are missing or incomplete for a larger number of the relevant molecules (see Section 2) and are therefore in practice substituted by crude “guestimates” when generating molecular opacities for the subsequent use in the radiative transfer calculations. Indeed, computing opacities without line profiles is meaningless, i.e. “something” is always better than “nothing”, but the errors involved are usually difficult to quantify. The impact of using incorrect broadening parameters was shown by Niraula et al. (2022, 2023) to impact retrieved exoplanet properties.

HITRAN now includes different broadening species applicable to planetary atmospheres, as shown in Table 6. The addition of these parameters to the database is carried out by searching the literature for accurate experimental measurements, which can be fit to a rotationally-dependent function (see Wilzewski et al. 2016; Tan et al. 2019, 2022). This enables the functions to be applied to the high-temperature lines lists of databases such as ExoMol and HITEMP. These broadeners are representative of the typical species in the (exo-)atmosphere under study: a mix of H₂ and He for gas giants, CO₂ for rocky planets and H₂O for tropical atmospheric scenarios (or water-dominated atmospheres). The temperature dependence of line broadening parameters has also recently been under investigation. Indeed line shapes are described by different parameters such as the pressure broadening and pressure shift coefficients and their respective dependence on temperature; this applies to both molecules and atoms. The species and broadeners for which Voigt broadening parameters are currently available in the ExoMol database can be found online<fnref n="18" />. A complete set of air-broadening parameters have been generated using machine learning (Guest et al. 2024) and a large number of broadening parameters based on the theory of Buldyreva et al. (2022) have also been added recently.

**Empirical χ-factor corrections:** Solar System applications have shown that the Voigt profile, tending to a Lorentz profile with increasing pressure, is not applicable for high-pressure conditions where a sub-Lorentzian profile needs to be used (Bézard et al. 2011). The pressure is so high in the lower layers of the atmosphere that the line-interference (intensity-transfer, also called line mixing) effects, in particular line (and, therefore, band) wings, need to be accurately modeled. For now, the band shapes obtained in the impact-approximation (Markov) limit of collisional-broadening theory (which is invalid for large detunings from the band centre) are empirically corrected via so-called χ-factors, as, e.g., Tonkov et al. (1996); Bézard et al. (2011) for CO₂

on Venus and the recent confirmation by laboratory measurements (Mondelain et al. 2022, 2023). Empirical χ-factor corrections were also used to model far-wing absorption for CO₂ at high pressures and temperatures (Tran et al. 2011; Cole et al. 2023), CH₄ (Tran et al. 2011) and H₂O (Tran et al. 2011). The artificial χ-factors as well as the forced re-normalization procedure in the impact-approximation models become unnecessary if a non-Markovian approach operating with a frequency-dependent relaxation matrix and a symmetric metric in the line space is employed (Buldyreva & Daneshvar 2013; Filippov et al. 2013), but the collision-induced absorption (while relevant) should be correctly modeled; see Section 5. Even for correctly modelling methane absorption in the cold ice giant atmospheres, for example, having more precise line profiles can be important.

**Double-power law:** recently, Gamache & Vispoel (2018) suggested to correct Eq. (2) by adding a supplementary power term (so-called double-power law), with an analogous two-term expression for the shifts, in order to model line widths and shifts over large temperature intervals with the same parameters' sets. So far this has only been applied to the H₂O-CO₂ collisional system, with adequate parameters published in Régalia et al. (2019), and for the He-broadening of H₂ and HD (Wcisło et al. 2021; Stankiewicz et al. 2021).

**The importance of laboratory measured broadening parameters:** as mentioned earlier in this section, accurate laboratory broadening measurements are critical for the production of rotationally-dependent broadening parameters, many of which have been compiled in the HITRAN database (see Table 6). A few examples of laboratory measurements of broadening coefficients to study Earth, Mars, Jupiter and other planetary compositions for key absorbers include Smith et al. (2014); Sung et al. (2020); Devi et al. (2016); Lavrentieva & Dudaryonok (2020). Available measurements that are relevant to high-temperature atmospheres, are not all necessarily relevant to high-metallicity atmospheres (e.g., Gharib-Nezhad et al. 2019; Yousefi et al. 2021; Baldacchini et al. 2001; Goldenstein et al. 2013), or are only limited to a small spectral region. Further details can be found in Hartmann et al. (2018). Nevertheless, such measurements in the laboratory are an important component of compiling broadening parameters for a large range of species, spectral regions, broadening species, temperatures, and pressures.

### 3.3.2 Atomic line profiles

**Na and K:** the resonance line doublets of atoms such as Na and K can have a large impact on the spectra of exoplanets and Brown Dwarfs, as demonstrated by, for example, Allard et al. (2019); Whiteford et al. (2023). Such strong lines as these require detailed computations to model the line profile and wings far from the line centre. Typically works such as Allard et al. (2016) for K perturbed by H₂<fnref n="19" />, Allard et al. (2024) for K perturbed by He<fnref n="20" />, Allard et al. (2019) for Na perturbed by H₂<fnref n="21" />, Allard et al. (2023) for Na perturbed by

<footnote n="18">¹⁸ https://exomol.com/data/data-types/broadeningCoefficients/</footnote>

<footnote n="19">¹⁹ https://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/589/A21</footnote>

<footnote n="20">²⁰ https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/683/A188</footnote>

<footnote n="21">²¹ http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/628/A120</footnote>
**UV and visible frequency ranges.** Spectra which involve transitions between electronic states (i.e. rotation-vibration-electronic or rovibronic transitions), typically between the ground and an excited electronic state, occur, with some exceptions, in the UV and/or visible region. In these cases, the line broadening due to pre-dissociative effects can dominate the collisional (pressure) broadening by some orders of magnitude. These broadening are caused by the associated lifetimes of quasi-bound transitions decreased. A new data format has recently been proposed to capture (Tennyson et al. 2023) and use (Yurchenko et al. 2024) the pre-dissociative line broadening in spectroscopic data bases.

In the case of rovibronic transitions between bound low-lying vibrational levels of electronic states, available measurements show no well pronounced J-dependencies in the collisional broadening of rovibronic transitions. The absence of rotational dependence of the collisional (pressure) broadening means that rotational energy transfer is of minor importance and the broadening parameters are mainly influenced by elastic collisions. For such transitions line widths and line shifts have the same orders of magnitude. As mentioned by Margenau (1936), lines of rovibronic (bound-bound) transitions are broadened and shifted similarly to atomic lines of a similar type and a similar energy jump. Indeed, the first theoretical approaches to pressure-induced line widths/shifts used the model of a classical oscillating radiator whose phase is modified during collision by a random amount, leading thus, via Fourier-integral analysis, to a 'dispersion' (Lorentz) profile. For example, Lindholm (1945) and Foley (1946) successfully developed such theories. Foley (1946), in particular, argued that non-adiabatic effects are small for atoms due to large gaps between their electronic levels but can occur for molecular IR transitions because of similar energy differences between the rotational levels of the radiator and perturber. Detailed analyses of some representative one-term interactions (dipole-dipole, quadrupole-quadrupole, and dispersive) in the framework of phase-shift theory were made by Mizushima (1951) who also considered velocity averaging and derived line-width and line-shift expressions (intended for microwave transitions) with explicit temperature dependencies. These expressions served as a basis for the power law of equation (2). Later, Hindmarsh, Petford & Smith (1967) accounted additionally for short-range repulsive forces via a Lennard-Jones 12-6 potential model but for simplicity conducted their derivation with the mean-thermal-velocity approximation. Temperature dependence of collisional broadening and shifts of atomic and molecular rovibronic lines has been also addressed by Cybulski et al. (2013). Quantum-mechanical adiabatic approaches were initiated by Jablonski (1937, 1945) who assumed the gas to be a very large 'molecule' and calculated the stationary states of internal motion. A unified model relying on the Franck-Condon principle and containing both impact and quasi-static limits was proposed by Szudy & Baylis (1975). The authors retrieved the expressions of Lindholm (1945) and Foley (1946) in the classical limit and suggested also a first-order correcting term leading to line shapes with an asymmetry effect.

When there is an interaction with one or more dissociative states, i.e. pre-dissociations, the broadening can vary significantly due to a variation in the lifetimes of the excited states. The broadening in this case is a manifestation of such interaction with the continuum. The extent of the line broadening (or lifetime of the state in question) derives from the strength of the pre-dissociation, i.e. the strength of the coupling between the bound excited states and the dissociative states. Typically non-radiative lifetimes of electronically excited states are ca. $10^{-8}$ s which result in widths of $5 \times 10^{-4}$ cm$^{-1}$. For lifetimes of $10^{-13}$ s the linewidth will be $50$ cm$^{-1}$ which will result

in the rotational structure disappearing. For non-radiative lifetimes shorter than $10^{-15}$ s, the vibrational structure disappears and will resemble a continuum. These types of pre-dissociation-induced line broadenings have been well documented experimentally for over half a century (see chapter 7.4.2 in Léfeuvre-Brion & Field 2004).

## 3.2 Theoretical approaches to line-shape parameters: atoms

The broadening of alkali lines by rare gases is a problem which has been extensively investigated in experimental and theoretical work (Allard & Kielkopf 1982; Burrows & Volobuyev 2003; Allard, Spiegelman & Kielkopf 2016; Allard et al. 2019, 2023, 2024; Peach et al. 2020). A theory of spectral line broadening has been developed to calculate neutral atom spectra given the interaction and transition moments for relevant states of the radiating atom with other atoms in its environment. Within this framework it is possible to compute the complete spectrum with a unified approach. Unlike impact theories of line broadening which predict a Lorentzian line or the approximation methods of Szudy & Baylis (1975, 1996) the unified theory of Allard et al. (1999) provides an accurate spectrum from the line centre to the extreme wing. Complete details and the derivation of the theory are given by Allard et al. (1999). The approach is based on quantum theory of spectral line shapes by Baranger (1958a, b) with an adiabatic representation to include the degeneracy of atomic levels.

The spectrum $I(\Delta\omega)$ can be written as the Fourier transform (FT) of the dipole autocorrelation function $\Phi(s)$,

$$I(\Delta\omega) = \frac{1}{\pi} \mathrm{Re} \int_{0}^{+\infty} \Phi(s) e^{-i\Delta\omega s} \mathrm{d}s, \quad (3)$$

where $\Delta\omega$ is the angular frequency difference from the unperturbed centre of the spectral line. The autocorrelation function $\Phi(s)$ is calculated with the assumptions that the radiator is stationary in space, the perturbers are mutually independent, and in the adiabatic approach the interaction potentials give contributions that are scalar additive. This last simplifying assumption allows the total profile $I(\Delta\omega)$ to be calculated when all the perturbers interact, as the FT of the $N^{th}$ power of the autocorrelation function $\phi(s)$ of a unique atom-perturber pair. Therefore,

$$\Phi(s) = (\phi(s))^N, \quad (4)$$

that is to say, the interperturber correlations are neglected. For a perturber density $n_p$,

$$\Phi(s) = e^{-n_p g(s)}, \quad (5)$$

where decay of the autocorrelation function with time leads to atomic line broadening. When $n_p$ is high, the spectrum is evaluated by computing the FT of equation (5). The real part of $n_p g(s)$ damps $\Phi(s)$ for large $s$ but this calculation is not feasible when extended wings have to be computed at low density because of the very slow decrease of the autocorrelation function. An alternative is to use the expansion of the spectrum $I(\Delta\omega)$ in powers of the density described in Royer (1971). For the implementation of alkali lines perturbed by helium and molecular hydrogen in atmosphere codes, the line opacity is calculated by splitting the profile into a core component described with a Lorentzian profile, and the line wings computed using an expansion of the autocorrelation function in powers of density. The impact approximation determines the asymptotic behavior of the unified line shape correlation function. In this way the results are applicable to a more general line profile and opacity evaluation for the same perturbers at any given layer in the photosphere or planetary atmosphere.
He<fnref n="22" />, or Burrows & Volobuyev (2003) for Na and K perturbed by a mixture of H₂ and He, are used in the computation of opacities for exoplanet and Brown Dwarf atmospheres. See also Sharp & Burrows (2007) for empirical estimates used for other atomic (and molecular) line broadening parameters.

For the strongest atomic lines (K, Na, He) individual line absorption can dominate the entire observed spectroscopic region and very accurate shapes are mandatory, as described above. For others (Fe, Ca, Mg etc.), the atomic features are less intense and can be very dense, comparable to the molecular lines. In this case, the spectra are less sensitive to the individual lines. We do not go in detail here on these other species, and instead focus only on Na and K which are most relevant to high-temperature exoplanet atmospheres due to their extensive line wings.

### 3.4 Data used by retrieval codes

As previously highlighted in this section, the vast majority of atmospheric modelling and retrieval codes use the Voigt profile as the current standard for temperature- and pressure-broadened opacities.

The broadening parameters used for computing opacities (either on-the-fly or pre-computed; see Section 4) will have an impact on the resulting radiative transfer models (e.g. Anisman et al. 2022a; Gharib-Nezhad & Line 2019). It is important to compute (or choose, if pre-computed) opacities which are relevant for the atmosphere which is being modelled, both in terms of temperature and pressure, and also what the main broadening species are (e.g. H₂/He-dominated, H₂O-dominated, CO₂-dominated atmospheres). Often broadening data are supplied alongside line lists, which may then be typically used in the computation of opacities. One of the main differences between opacity computations is the extent out to which the line wings are calculated from the line centre (see, for example, Gharib-Nezhad et al. 2024). The ExoMolOP database (Chubb et al. 2021) employs the line profiles of Allard et al. (2016) and Allard et al. (2019) in computing the resonance line opacities of K and Na, respectively, broadened by H₂. For the molecular opacity Voigt line profiles in the ExoMolOP database, averaged-J broadening parameters for H₂ and He broadening were used for those species with available data at the time, as outlined in Table 2 of Chubb et al. (2021). The Voigt line wings were computed up to 500 Voigt widths from the line centres, up to a maximum cutoff of 25 cm⁻¹. The EXOPLINES database (Gharib-Nezhad et al. 2021) uses a Voigt profile with a variable line-wing cutoff depending on the pressure (30 cm⁻¹ for P ≤ 200 bar and 150 cm⁻¹ for P > 200 bar), but note the lack of available broadening coefficients for the species they compute opacities for (including a number of metal hydrides). They therefore estimated parameters to use in a similar way to Chubb et al. (2021), based on the similarity between molecular symmetry or dipole moments that these absorbers have with the available Lorentz coefficients. Some works, such as Gharib-Nezhad & Line (2019) and Anisman et al. (2022a) use opacities which were computed assuming H₂O-broadening parameters, extracted from the HITRAN2020 database (Gordon et al. 2022). The opacities computed for the Helios-r2 retrieval code (Kitzmann et al. 2020; Grimm et al. 2021) are computed using

a Voigt profile. The impact of the spectral line wing cut-off has been recently reviewed by Gharib-Nezhad et al. (2024), with the recommendation for the following line wing cut-offs to be used for current computations using the Voigt profile: 25 cm⁻¹ for P ≤ 200 bar and 100 cm⁻¹ for P > 200 bar.

### 3.5 What's being worked on?

**Machine learning approaches:** so far, the theoretical methods used for line broadening are very limited in what they can realistically provide, in terms of the molecular species, spectral ranges and temperature coverage. Machine learning (ML) represents an attractive alternative to fill the existing data gaps in broadening data due to its relative simplicity, which has been recently explored to produce air broadening data in the ExoMol data base (Guest et al. 2024). Considering the success of ML in a wide range of applications and the fast progress in the development of ML techniques, the work in this direction will continue.

**(Vib)rotational line-shape parameters:** the extreme conditions experienced by most observable exoplanets (high temperatures and huge levels of ionisation) lead to the formation of “exotic” species whose spectroscopic parameters are often impossible to measure in standard laboratory experiments. A significant number of required perturbers and non-existing empirical PESs to practice “standard” theoretical approaches described in Section 3.1. Therefore, for a rough J-independent theoretical estimate for IR/MW line width (in cm⁻¹atm⁻¹), the following expression for the Lorentzian HWHM has been recently suggested (Buldyreva et al. 2022):

$$ \gamma_L = 1.7796 \cdot 10^{-5} \frac{m}{(m-2)} \frac{1}{\sqrt{T}} \sqrt{\frac{m_a + m_p}{m_a m_p}} d^2, \quad (6) $$

where only the index m = 2(q − 1) of the leading long-range interaction V ~ R⁻q, the active- and perturbing-molecule masses m_a, m_p and the kinetic diameter of the molecular pair d = (d_a + d_p)/2 are required at the input temperature T. Besides the neutral active molecules, Eq. (6) holds also for ionic absorbers. It has been employed to produce line-width data (Buldyreva et al. 2024b) for more than fifty active species (and their isotopologues) self-perturbed and perturbed by He, Ar, O₂, H₂, N₂, CO, CO₂, NO, H₂O, NH₃ and CH₄ at the reference temperature of 296 K. These new data are progressively integrated into ExoMol as new files of ExoMol Diet (Barton et al. 2017) and into the definition .def files for all line lists of a considered molecule.

**Rovibronic line-shape parameters:** the same goal of providing at least approximate values of line-shape parameters for electronic transitions in hot-temperature active molecules concerned by Ariel has initiated recently a study (Buldyreva et al. 2024a) revisiting classical phase-shift theory. Besides a general analysis for arbitrary molecular pairs (including the points on the temperature dependence and Maxwell-Boltzmann averaging over relative velocities), improvements of the potential and trajectory descriptions were attempted. Detailed calculations, based on specially computed state-of-the-art PESs, were done for some representative molecular pairs (NO and OH perturbed by N₂ and Ar, selected because of available experimental data). As intended, for the considered molecular partners the calculated line-shape parameters were found to be consistent with measurements within 30–40% in wide temperature ranges (300–3000 K). Although this

<footnote n="22">²² https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/674/A171</footnote>
In Allard et al. (2019) the use of a density expansion in the opacity tables is reviewed. When the expansion is stopped at the first order it is equivalent to the one-perturber approximation. Previous opacity tables were constructed to third order allowing us to obtain line profiles up to $N_{\text{He}}/N_{\text{H}_2}=10^{19} \text{ cm}^{-3}$. The new tables are constructed to a higher order allowing line profiles to $N_{\text{He}}/N_{\text{H}_2}=10^{21} \text{ cm}^{-3}$.

## 3.3 State-of-the-art – data availability

### 3.3.1 Molecular line profiles

**Voigt parameters:** it is now recognized in the (exo-)atmospheric community that Earth's conditions with the oxygen and nitrogen as the main collisional broadeners are not default ones, especially for hot Jupiter exoplanets, where hydrogen and helium dominate molecular collisions. Although, as noted below, broadening parameters are becoming available for a small number of systems which deviate from the Voigt profile, the vast majority of available data are for Voigt broadening parameters. Even so, collisional broadening parameters which are required for computing Voigt profiles are missing or incomplete for a larger number of the relevant molecules (see Section 2) and are therefore in practice substituted by crude 'guestimates' when generating molecular opacities for the subsequent use in the radiative transfer calculations. Indeed, computing opacities without line profiles is meaningless, i.e. 'something' is always better than 'nothing', but the errors involved are usually difficult to quantify. The impact of using incorrect broadening parameters was shown by Niraula et al. (2022, 2023) to impact retrieved exoplanet properties.

HITRAN now includes different broadening species applicable to planetary atmospheres, as shown in Table 7. The addition of these parameters to the data base is carried out by searching the literature for accurate experimental measurements, which can be fit to a rotationally dependent function (see Wilzewski et al. 2016; Tan et al. 2019, 2022). This enables the functions to be applied to the high-temperature line lists of data bases such as ExoMol and HITEMP. These broadeners are representative of the typical species in the (exo-)atmosphere under study: a mix of H₂ and He for gas giants, CO₂ for rocky planets, and H₂O for tropical atmospheric scenarios (or water-dominated atmospheres). The temperature dependence of line broadening parameters has also recently been under investigation. Indeed line shapes are described by different parameters such as the pressure broadening and pressure shift coefficients and their respective dependence on temperature; this applies to both molecules and atoms. The species and broadeners for which Voigt broadening parameters are currently available in the ExoMol data base can be found online.<fnref n="17" /> A complete set of air-broadening parameters have been generated using machine learning (Guest et al. 2024) and a large number of broadening parameters based on the theory of Buldyreva, Yurchenko & Tennyson (2022) have also been added recently.

**Empirical χ-factor corrections:** Solar System applications have shown that the Voigt profile, tending to a Lorentz profile with increasing pressure, is not applicable for high-pressure conditions where a sub-Lorentzian profile needs to be used (Bézard et al. 2011). The pressure is so high in the lower layers of the atmosphere that the line-interference (intensity-transfer, also called line mixing) effects, in particular line (and, therefore, band) wings, need to be accurately modelled. For now, the band shapes obtained in the impact-approximation (Markov) limit of collisional-broadening theory (which is invalid for large detunings from the band centre)

are empirically corrected via so-called χ-factors, as, e.g. Tonkov et al. (1996) and Bézard et al. (2011) for CO₂ on Venus and the recent confirmation by laboratory measurements (Mondelain et al. 2022, 2023). Empirical χ-factor corrections were also used to model far-wing absorption for CO₂ at high pressures and temperatures (Tran et al. 2011; Cole et al. 2023), CH₄ (Tran et al. 2011), and H₂O (Tran et al. 2011). The artificial χ-factors as well as the forced re-normalization procedure in the impact-approximation models become unnecessary if a non-Markovian approach operating with a frequency-dependent relaxation matrix and a symmetric metric in the line space is employed (Buldyreva & Daneshvar 2013; Filippov et al. 2013), but the collision-induced absorption (while relevant) should be correctly modelled; see Section 5. Even for correctly modelling methane absorption in the cold ice giant atmospheres, for example, having more precise line profiles can be important.

**Double-power law:** Gamache & Vispoel (2018) suggested to correct equation (2) by adding a supplementary power term (the so-called double-power law), with an analogous two-term expression for the shifts, in order to model line widths and shifts over large temperature intervals with the same parameters' sets. So far, this has been applied to the H₂O-CO₂ collisional system, with respective parameters published in Régalia et al. (2019), to CO₂-H₂O reported in Vispoel & Gamache (2024), and to He-broadening of H₂ and HD (Wcisło et al. 2021; Stankiewicz et al. 2021).

**The importance of laboratory measured broadening parameters:** as mentioned earlier in this section, accurate laboratory broadening measurements are critical for the production of rotationally dependent broadening parameters, many of which have been compiled in the HITRAN data base (see Table 7). A few examples of laboratory measurements of broadening coefficients to study Earth, Mars, Jupiter, and other planetary compositions for key absorbers include Smith et al. (2014), Sung et al. (2020), Devi et al. (2016), and Lavrentieva & Dudaryonok (2020). Available measurements that are relevant to high-temperature atmospheres, are not all necessarily relevant to high-metallicity atmospheres (e.g. Baldacchini et al. 2001; Goldenstein, Jeffries & Hanson 2013; Gharib-Nezhad et al. 2019; Yousefi et al. 2021), or are only limited to a small spectral region. Further details can be found in Hartmann et al. (2018b). Nevertheless, such measurements in the laboratory are an important component of compiling broadening parameters for a large range of species, spectral regions, broadening species, temperatures, and pressures.

### 3.3.2 Atomic line profiles

**Na and K:** the resonance line doublets of atoms such as Na and K can have a large impact on the spectra of exoplanets and Brown Dwarfs, as demonstrated by, for example, Allard et al. (2019) and Whiteford et al. (2023). Such strong lines as these require detailed computations to model the line profile and wings far from the line centre. Typically works such as Allard et al. (2016) for K perturbed by H₂,<fnref n="18" /> Allard et al. (2024) for K perturbed by He,<fnref n="19" /> Allard et al. (2019) for Na perturbed by H₂,<fnref n="20" /> Allard et al. (2023) for Na perturbed by He,<fnref n="21" /> or Burrows & Volobuyev (2003) for Na and K perturbed by a mixture of H₂ and He, are used in the computation of opacities for exoplanet and Brown Dwarf atmospheres. See also Sharp & Burrows (2007)

<footnote n="17">¹⁷https://exomol.com/data/data-types/broadeningCoefficients/</footnote>

<footnote n="18">¹⁸https://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/589/A21</footnote>

<footnote n="19">¹⁹https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/683/A188</footnote>

<footnote n="20">²⁰http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/628/A120</footnote>

<footnote n="21">²¹https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/674/A171</footnote>
phase-theory approach requires knowledge of the isotropic interaction PESs associated with the excited and ground radiator's states, it has a much lower computational cost with respect to quantum-mechanical methods which, in addition, become unpractical at high temperatures.

**3.6 Data needs: What's missing and urgent?**

In general, we would advocate for a consistent set of parameters which cover an adequate spectral range for each species of importance. This includes species which require new or improved linelists in Section 2.4. In the frame of the Ariel mission, this is a challenging request as Ariel aims to observe about 1000 exoplanets ranging from Jupiters and Neptunes down to super-Earth size in the visible and the infrared with its meter-class telescope (Edwards et al. 2019a). This implies very different broadeners and pressure and temperature conditions. Line shapes accurate at high temperature, up to 2500 K, are high up on the list. This includes temperature exponents of the line parameters as temperature dependent line mixing effects for the main molecular species at least. The most important broadeners to be considered remain the usual suspects; H₂, He, CO₂ and H₂O.

(i) **Species occuring in JWST studies:** see the first point in Section 2.4, with line broadening parameters also required up to high temperatures for the listed species.

(ii) **Species relevant to lava planets:** the broadening for molecules important in lava planet environments require very different types of line-broadening to other commonly studied planets, which are generally not currently available (see, for example, Tennyson & Yurchenko 2017). This includes species such as KOH, NaOH, SiO₂, CaOH, FeO, MgO, MgOH (Schaefer et al. 2012; Tennyson & Yurchenko 2017; Mahapatra et al. 2017).

(iii) **H₂ and He broadeners at high temperatures:** there has been significant progress recently in the provision of line broadening parameters for H₂/He dominated atmospheres such as of hot Jupiters or hot Neptunes. Even so, it should be acknowledged that while many species in HITRAN have broadening parameters for terrestrial-relevant molecules, H₂ and He broadening parameters are not available for some important species such as CH₄.

(iv) **N₂ broadening:** it is known from studies of the Solar System that broadening by N₂ is required for the remote sensing of Titan (He et al. 2022b). Currently, N₂ is not separated from “air” in HITRAN, but is included as a broadening gas for many absorption cross-sections included in HITRAN that have been observed in Titan’s atmosphere.

(v) **Species relevant to heavy atmospheres:** the availability of broadening data with species such as CO₂ and H₂O as broadeners is also relatively limited (see e.g. Table 6).

(vi) **Rotational- and vibrational-dependence:** while the rotational (J)-dependence is often not known for many species (see, e.g., Gharib-Nezhad et al. 2021), the vibrational dependence is even less studied.

(vii) **Beyond Voigt profiles:** introducing a new line profile, such as Hartmann-Tran (see Section 3.1), implies introducing new line parameters. This complicates the release of parameters and recipes to the community. Efforts are made, and for instance, HAPI<fnref n="23" /> (Kochanov et al. 2016) enables any

user to choose a set of adequate line parameters depending on the atmospheric conditions of their spectra. Nevertheless, issues arise when the available parameters do not entirely cover the required spectral range and different datasets need to be combined. This results in a lack of traceability and poor reliability on the retrievals' results. We would therefore advocate for a consistent set of parameters which cover an adequate spectral range.

(viii) **Line mixing:** A recent overview of line mixing, which causes potentially large deviations from a Voigt profile at high pressures, can be found in Gharib-Nezhad et al. (2024). We are not aware of any exoplanetary models which currently allow for the effects of line mixing, but these effects may become more important with newer instruments that can probe high-pressure regions of around 1 - 10 bar.

(ix) **Deuterated species:** deuterated molecular species are targeted in the (exo-)planetary community to infer planetary evolution. Typical targets for these kinds of investigation are hydrocarbons and water: C₂HD, CH₃D, and HDO (Villanueva et al. 2022). Broadening parameters for these species are therefore important.

(x) **Atomic species:** line parameters for atomic species are generally not readily available (with the previously noted exceptions of the strong Na and K doublets broadened by H₂ and He), which can lead to uncertainties in the calculated cross-sections. As shown in Grimm et al. (2021), the use of a Voigt profile for atomic lines is not always suitable. For instance the wings of autoionization lines are more appropriately described by a so-called Fano profile (Fano 1961), which decreases much faster than a Voigt profile. Using a Voigt profile for these lines will result in an overestimate of the absorption, creating an artificial continuum. Other problematic lines for atomic species are resonance lines, where using a normal Voigt profile can also lead to over- or underestimated absorption.

(xi) **Limitations of experimental studies on some species:** as noted in Section 2.1.2.2, experimental measurements can be particularly challenging for reactive (or toxic) species, especially at high temperatures, which can limit the availability of parameters. It is therefore not uncommon for some spectroscopic parameters, such as pressure-broadening (Tan et al. 2022), to be based only on limited laboratory measurements for some molecules. For example, inorganic salts are very difficult to sublimate and keep in the gas phase. These molecules, where possible, could be targets of theoretical studies instead.

**4 COMPUTED CROSS-SECTIONS AND OPACITIES**

Line lists (see Section 2) give all the information required (such as upper and lower energy levels, transitions, and typically either Einstein-A coefficients or intensities at a reference temperature) to generate a cross-section at a given temperature and pressure. Line lists will have a maximum temperature up to which they are considered complete; cross-sections computed above these temperatures will have opacity missing. This is because hot band transitions between energy levels of higher energy than are populated for the maximum line list temperature will not be taken into account. When computing cross-sections various choices need to be made, such as what line list, line profile, line-wing cutoff, and broadening parameters to use (see Sections 2 and 3). These will partly

<footnote n="23">²³ https://hitran.org/hapi/</footnote>
for empirical estimates used for other atomic (and molecular) line broadening parameters.

For the strongest atomic lines (K, Na, He) individual line absorption can dominate the entire observed spectroscopic region and very accurate shapes are mandatory, as described above. For others (Fe, Ca, Mg etc.), the atomic features are less intense and can be very dense, comparable to the molecular lines. In this case, the spectra are less sensitive to the individual lines. We do not go in detail here on these other species, and instead focus only on Na and K which are most relevant to high-temperature exoplanet atmospheres due to their extensive line wings.

## 3.4 Data used by retrieval codes

As previously highlighted in this section, the vast majority of atmospheric modelling and retrieval codes use the Voigt profile as the current standard for temperature- and pressure-broadened opacities.

The broadening parameters used for computing opacities (either on-the-fly or pre-computed; see Section 4) will have an impact on the resulting radiative transfer models (e.g. Gharib-Nezhad & Line 2019; Anisman et al. 2022b). It is important to compute (or choose, if pre-computed) opacities which are relevant for the atmosphere which is being modelled, both in terms of temperature and pressure, and also what the main broadening species are (e.g. H₂/He-dominated, H₂O-dominated, and CO₂-dominated atmospheres). Often broadening data are supplied alongside line lists, which may then be typically used in the computation of opacities. One of the main differences between opacity computations is the extent out to which the line wings are calculated from the line centre (see for example, Gharib-Nezhad et al. 2024). The ExoMolOP data base (Chubb et al. 2020b) employs the line profiles of Allard et al. (2016, 2019) in computing the resonance line opacities of K and Na, respectively, broadened by H₂. For the molecular opacity Voigt line profiles in the ExoMolOP data base, averaged-J broadening parameters for H₂, and He broadening were used for those species with available data at the time, as outlined in table 2 of Chubb et al. (2020b). The Voigt line wings were computed up to 500 Voigt widths from the line centres, up to a maximum cutoff of 25 cm⁻¹. The EXOPLINES data base (Gharib-Nezhad et al. 2021) uses a Voigt profile with a variable line-wing cutoff depending on the pressure (30 cm⁻¹ for P ≤ 200 bar and 150 cm⁻¹ for P > 200 bar), but note the lack of available broadening coefficients for the species they compute opacities for (including a number of metal hydrides). They therefore estimated parameters to use in a similar way to Chubb et al. (2020b), based on the similarity between molecular symmetry or dipole moments that these absorbers have with the available Lorentz coefficients. Some works, such as Gharib-Nezhad & Line (2019) and Anisman et al. (2022b) use opacities which were computed assuming H₂O-broadening parameters, extracted from the HITRAN2020 data base (Gordon et al. 2022). The opacities computed for the Helios-r2 retrieval code (Kitzmann et al. 2020; Grimm et al. 2021) are computed using a Voigt profile. The impact of the spectral line wing cut-off has been recently reviewed by Gharib-Nezhad et al. (2024), with the recommendation for the following line wing cut-offs to be used for current computations using the Voigt profile: 25 cm⁻¹ for P ≤ 200 bar and 100 cm⁻¹ for P > 200 bar.

## 3.5 What's being worked on?

**Machine learning approaches:** so far, the theoretical methods used for line broadening are very limited in what they can realistically

provide, in terms of the molecular species, spectral ranges and temperature coverage. Machine learning (ML) represents an attractive alternative to fill the existing data gaps in broadening data due to its relative simplicity, which has been recently explored to produce air broadening data in the ExoMol data base (Guest et al. 2024). Considering the success of ML in a wide range of applications and the fast progress in the development of ML techniques, the work in this direction will continue.

**(Vib)rotational line-shape parameters:** the extreme conditions experienced by most observable exoplanets (high temperatures and huge levels of ionization) lead to the formation of 'exotic' species whose spectroscopic parameters are often impossible to measure in standard laboratory experiments. A significant number of required perturbers and non-existing empirical PESs to practice 'standard' theoretical approaches described in Section 3.1. Therefore, for a rough J-independent theoretical estimate for IR/MW line width (in cm⁻¹atm⁻¹), the following expression for the Lorentzian HWHM has been recently suggested (Buldyreva et al. 2022):

$$ \gamma_L = 1.7796 \times 10^{-5} \frac{m}{(m-2)} \frac{1}{\sqrt{T}} \sqrt{\frac{m_a + m_p}{m_a m_p}} d^2, \quad (6) $$

where only the index m = 2(q - 1) of the leading long-range interaction V ~ R⁻⁴, the active- and perturbing-molecule masses m_a, m_p, and the kinetic diameter of the molecular pair d = (d_a + d_p)/2 are required at the input temperature T. Besides the neutral active molecules, equation (6) holds also for ionic absorbers. It has been employed to produce line-width data (Buldyreva et al., in preparation) for more than fifty active species (and their isotopologues) self-perturbed and perturbed by He, Ar, O₂, H₂, N₂, CO, CO₂, NO, H₂O, NH₃, and CH₄ at the reference temperature of 296 K. These new data are progressively integrated into ExoMol as new files of ExoMol Diet (Barton et al. 2017) and into the definition.def files for all line lists of a considered molecule.

**Rovibronic line-shape parameters:** the same goal of providing at least approximate values of line-shape parameters for electronic transitions in hot-temperature active molecules concerned by Ariel has initiated recently a study (Buldyreva et al. 2024) revisiting classical phase-shift theory. Besides a general analysis for arbitrary molecular pairs (including the points on the temperature dependence and Maxwell-Boltzmann averaging over relative velocities), improvements of the potential and trajectory descriptions were attempted. Detailed calculations, based on specially computed state-of-the-art PESs, were done for some representative molecular pairs (NO and OH perturbed by N₂ and Ar, selected because of available experimental data). As intended, for the considered molecular partners the calculated line-shape parameters were found to be consistent with measurements within 30–40 per cent in wide temperature ranges (300–3000 K). Although this phase-theory approach requires knowledge of the isotropic interaction PESs associated with the excited and ground radiator's states, it has a much lower computational cost with respect to quantum-mechanical methods which, in addition, become unpractical at high temperatures.

## 3.6 Data needs: what's missing and urgent?

In general, we would advocate for a consistent set of parameters which cover an adequate spectral range for each species of importance. This includes species which require new or improved line lists in Section 2.4. The need for the accurate treatment of line broadening effects in atmospheric retrievals is known to play an important role in atmospheric models, and more so with emerging observational
be determined by the requirements of the cross-sections be-
ing computed: will they be used for modelling Earth-like
planets, in which case a line list such as HITRAN might be
the most appropriate, or hotter exoplanets, in which case a
line list computed or measured up to high temperatures is
more appropriate, such as those provided by databases such
as HITEMP (Rothman et al. 2010), ExoMol (Tennyson et al.
2020), or TheoReTs (Rey et al. 2016b). If the cross-sections
are for modelling or retrieving atmospheric properties of a
hot gas giant exoplanet, then the atmospheric composition
will be typically dominated by H₂ and He. The choice of
broadening parameters in Section 3 will thus be H₂ and He
for such atmospheres, whereas for a planet with a heavier
atmosphere, heavier broadening species such as H₂O or N₂
will be more appropriate. Partition functions are also re-
quired for computing cross-sections; these are typically in-
cluded as part of a line list (see Section 2), with tools such
as TIPS (Gamache et al. 2021) available for computing par-
tition functions for the HITRAN2020 database. Studies such
as Niraula et al. (2022) and Niraula et al. (2023) highlight
how different choices, including line lists, broadening coeffi-
cients, and far-wing behaviours, can impact interpretations of
observed spectra. Niraula et al. (2023) explore the impact of
opacity choices on the atmospheric properties of the JWST
NIRSpec/G395H transmission spectra of WASP-39 b (Alder-
son et al. 2023). Villanueva et al. (2024) present comparison
tests designed for the exoplanetary community, in order to
identify the effects of different input data choices, such as
opacity format and line list source.

Opacity files are typically provided as either: 1) high-resolution cross-sections, 2) sampled cross-sections, or 3) correlated-k coefficients, also known as k-tables. The general process is to compute high-resolution cross-sections (typically with a grid spacing of around 0.1 cm⁻¹ or 0.01 cm⁻¹, or R = λ/Δλ ~ 1,000,000) for a grid of temperatures and pressures (typically spanning pressures of at least ~1×10⁻⁵ - 100 bar, and temperatures ~100 K - 3000 K), and then, usually for reasons of computational feasibility, to either sample these cross-sections at a lower resolution (~λ/Δλ ~ 10,000 - 20,000), or compute k-tables at ~λ/Δλ ~ 300 - 3000, if the application is for lower-resolution applications such as characterising Ariel or lower-resolution JWST observations. These resolutions are approximately equivalent to one another for achieving the same accuracy when used in an atmospheric modelling or retrieval code. The general principle of k-tables is to order spectral lines within a given spectral bin, producing a smooth cumulative distribution function to represent opacity, which can then be more efficiently sampled than cross-sections being sampled without any ordering of the spectral lines, with less loss of opacity for the same resolving power (see, for example, Min 2017; Chubb et al. 2021; Leconte 2021, for further details).

**Super-lines:** very large line lists can contain billions of
lines between millions of energy level states, for example the
latest ExoMol CH₄ line list consists of over 50 billion transi-
tions between 9 million states (Yurchenko et al. 2024a). The
computation of opacities from such large line lists can benefit
from techniques such as the super-lines method (Rey et al.
2016b; Yurchenko et al. 2017; Tennyson et al. 2020), which
yields a vast improvement in computational speed when com-
puting high-resolution cross-sections. Either line intensities
can be computed at a given temperature, and then the in-
tensities of all lines within a specified spectral bin summed

together to create a so-called super-line. If a high enough
resolving power, such as R = $\frac{\lambda}{\Delta\lambda}$ ~ 1,000,000, is used for
the creation of the super-lines then the overall opacity will
be conserved (Yurchenko et al. 2017; Tennyson et al. 2020;
Chubb et al. 2021). Alternatively, some line lists, such as the
CH₄ line lists of ExoMol (Yurchenko et al. 2017) and Theo-
ReTs (Rey et al. 2014a, 2016b), are already partitioned into
a large set of relatively weak lines and a small set of impor-
tant, stronger lines. When computing opacities, the weaker
lines can be combined into a set of super-lines (or effective
lines as done for CH₄ in HITEMP (Hargreaves et al. 2020)),
with broadening of the stronger lines treated individually.
Irwin et al. (2019) employed a similar “pseudo continuum”
approach when employing the NH₃ line list of Coles et al.
(2019) to analyse the visible spectrum of Jupiter, finding a
great increase in efficiency with very little loss of accuracy.

## 4.1 State of the art - Data availability

There are some programs which have been developed by
line list producers to generate cross-section data from line
lists, such as ExoCross<sup>24</sup> (Yurchenko et al. 2018) and pyEx-
oCross<sup>25</sup> (Zhang et al. 2024) (ExoMol), and HAPI (HI-
TRAN)<sup>26</sup> (Kochanov et al. 2016). Other tools have been
developed by line list users, such as HELIOS-K<sup>27</sup> (Grimm
& Heng 2015), Exo-k<sup>28</sup> (Leconte 2021), RADIS<sup>29</sup> (Pannier
& Laux 2019; van den Bekerom & Pannier 2021) and oth-
ers (Min 2017). Various databases are available which have
made use of tools such as these to compute opacities. Each
will have their own set of assumptions and choices used
when computing the opacities, depending on the intended
applications: some databases provide high resolution cross-
ections and some sampled cross-sections or k-tables (see the
introduction to this section), often formatted for direct use in
a particular atmospheric retrieval or modelling code. Some
codes, which employ a so-called line-by-line method, work
directly from line lists, and so do not require these opacity
files. In such an approach assumptions about line broaden-
ing are made by the user at the time of running the code
and so it can be considered to be more flexible, however it
can become computationally unfeasible for atmospheric re-
trievals which cover a large range of pressures, temperatures,
and wavelengths. Table 9 gives a non-exhaustive list of opac-
ity databases employed by typical codes used to compute
models or run retrievals relevant for the Ariel space mission,
along with other applications. Some additional notes on these
databases are given in Section 4.2. It is important, although
not always the case, for modelling and retrieval studies to
report on the opacities they use and either cite or explain the
assumptions that went into computing those opacities. This
allows for traceability and for analyses to be reproducible.

<footnote n="24">24 https://github.com/Trovemaster/exocross</footnote>

<footnote n="25">25 https://github.com/ExoMol/PyExoCross</footnote>

<footnote n="26">26 https://hitran.org/hapi/</footnote>

<footnote n="27">27 https://github.com/exoclime/HELIOS-K</footnote>

<footnote n="28">28 https://perso.astrophy.u-bordeaux.fr/~jleconte/exo-k-doc/index.html</footnote>

<footnote n="29">29 https://github.com/radis/</footnote>
facilities, regardless of if they are space-based (lower resolution) such as JWST and Ariel or ground-based (high resolution) such as the VLT or ELT (Hedges & Madhusudhan 2016). Studies such as Gharib-Nezhad & Line (2019), Niraula et al. (2022), and Anisman et al. (2022b) highlight the impact of using different broadening species for computing spectral line shapes on model atmospheric spectra, indicating that incorrect line shapes will likely lead to biases when analysing observed spectra. Gharib-Nezhad & Line (2019) find that emission spectra are particularly affected by incorrect broadening parameters, with the differences observed highly dependent on the pressure-temperature structure of the atmosphere. There will be an impact on the radiative balance and thermal structure of the atmosphere. In the frame of the Ariel mission, the requirement of a consistent set of parameters which cover an adequate spectral range for each species of importance is a challenging request as Ariel aims to observe about 1000 exoplanets ranging from Jupiters and Neptunes down to super-Earth size in the visible and the infrared with its meter-class telescope (Edwards et al. 2019b). This implies very different broadeners and pressure and temperature conditions. Line shapes accurate at high temperature, up to 2500 K, are high up on the list. This includes temperature exponents of the line parameters as temperature-dependent line mixing effects for the main molecular species at least. The most important broadeners to be considered remain the usual suspects; H₂, He, CO₂, and H₂O.

(i) **Species occurring in JWST studies:** see the first point in Section 2.4, with line broadening parameters also required up to high temperatures for the listed species.

(ii) **Species relevant to lava planets:** the broadening for molecules important in lava planet environments require very different types of line-broadening to other commonly studied planets, which are generally not currently available (see for example, Tennyson & Yurchenko 2017). This includes species such as KOH, NaOH, SiO₂, CaOH, FeO, MgO, MgOH (Schaefer et al. 2012; Mahapatra et al. 2017; Tennyson & Yurchenko 2017), whose spectra will be broadened by exotic molecules.

(iii) **H₂ and He broadeners at high temperatures:** there has been significant progress recently in the provision of line broadening parameters for H₂/He dominated atmospheres such as of hot Jupiters or hot Neptunes. Even so, it should be acknowledged that while many species in HITRAN have broadening parameters for terrestrial-relevant molecules, H₂ and He broadening parameters are not available for some important species such as CH₄.

(iv) **N₂ broadening:** it is known from studies of the Solar System that broadening by N₂ is required for the remote sensing of Titan (He et al. 2022b). Currently, N₂ is not separated from 'air' in HITRAN, but is included as a broadening gas for many absorption cross-sections included in HITRAN that have been observed in Titan's atmosphere.

(v) **Species relevant to heavy atmospheres:** the availability of broadening data with species such as CO₂ and H₂O as broadeners is also relatively limited (see e.g. Table 7).

(vi) **Rotational- and vibrational-dependence:** while the rotational (J)-dependence is often not known for many species (see e.g. Gharib-Nezhad et al. 2021), the vibrational dependence is even less studied but generally considered to be less important, see Guest et al. (2024) for example.

(vii) **Beyond Voigt profiles:** introducing a new line profile, such as Hartmann-Tran (see Section 3.1), implies introducing new line parameters. This complicates the release of parameters and recipes to the community. Efforts are made, and for instance,

HAPI<fnref n="22" /> (Kochanov et al. 2016) enables any user to choose a set of adequate line parameters depending on the atmospheric conditions of their spectra. Nevertheless, issues arise when the available parameters do not entirely cover the required spectral range and different data sets need to be combined. This results in a lack of traceability and poor reliability on the retrievals' results. We would therefore advocate for a consistent set of parameters which cover an adequate spectral range. Note that the various representations of the profiles are not orthogonal fits so the parameters cannot be used reliably in other functional forms.

(viii) **Line mixing:** A recent overview of line mixing, which causes potentially large deviations from a Voigt profile at high pressures, can be found in Gharib-Nezhad et al. (2024). We are not aware of any exoplanetary models which currently allow for the effects of line mixing, but these effects may become more important with newer instruments that can probe high-pressure regions of around 1–10 bar.

(ix) **Deuterated species:** deuterated molecular species are targeted in the (exo-)planetary community to infer planetary evolution. Typical targets for these kinds of investigation are hydrocarbons and water: C₂HD, CH₃D, and HDO (Villanueva et al. 2022) all have the property that they have different symmetry to their parent isotopologue. Broadening parameters for these species are important but the change of symmetry means the usual practice of transferring broadening parameters between isotopologues will not work for these cases.

(x) **Atomic species:** line parameters for atomic species are generally not readily available (with the previously noted exceptions of the strong Na and K doublets broadened by H₂ and He), which can lead to uncertainties in the calculated cross-sections. As shown in Grimm et al. (2021), the use of a Voigt profile for atomic lines is not always suitable. For instance the wings of autoionization lines are more appropriately described by a so-called Fano profile (Fano 1961), which decreases much faster than a Voigt profile. Using a Voigt profile for these lines will result in an overestimate of the absorption, creating an artificial continuum. Other problematic lines for atomic species are resonance lines, where using a normal Voigt profile can also lead to over- or underestimated absorption.

(xi) **Limitations of experimental studies on some species:** as noted in Section 2.1.2 (high-temperature experimental measurements) experimental measurements can be particularly challenging for reactive (or toxic) species, especially at high temperatures, which can limit the availability of parameters. It is therefore not uncommon for some spectroscopic parameters, such as pressure-broadening (Tan et al. 2022), to be based only on limited laboratory measurements for some molecules. For example, inorganic salts are very difficult to sublimate and keep in the gas phase. These molecules, where possible, could be targets of theoretical studies instead. The lack of the line shape data also means the lack of understanding of their importance on the atmospheric characterization. More work, both experimental and theoretical is required.

## 4 COMPUTED CROSS-SECTIONS AND OPACITIES

Line lists (see Section 2) give all the information required (such as upper and lower energy levels, transitions, and typically either Einstein-A coefficients or intensities at a reference temperature) to generate a cross-section at a given temperature and pressure. Line lists will

<footnote n="22">²²https://hitran.org/hapi/</footnote>
## 4.2 Data used by retrieval codes

Molecular opacities (cross-sections and k-tables) are now included in many radiative transfer codes and associated libraries, such as those listed in Table 2. The opacity databases listed in Table 9 are available online and although some are formatted with specific retrieval or modelling codes in mind, they can generally be adapted for more general use or for different codes. ExoMolOP (Chubb et al. 2021) has k-tables and cross-sections for currently around 80 species in the ExoMol database formatted for direct input into TauREx3 (Al-Refaie et al. 2021), ARCiS (Ormel & Min 2019; Min et al. 2020), NEMESIS (Irwin et al. 2008), petitRADTRANS (Mollière et al. 2019) (see Table 2). The EXOPLINES (Gharib-Nezhad et al. 2021), Gandhi2020 (Gandhi et al. 2020), MAESTRO (Batalha et al. 2022) and DACE (Grimm et al. 2021) databases/compilations all contain high-resolution cross-sections for various pressures and temperatures. These can be sampled down to a smaller desired resolution for practical use, or converted to k-tables. Various tools currently exist for this, including Exo-k<fnref n="30" /> (Leconte 2021), with more tools expected to become openly available in the near future. We note there are various file formats used for storing opacity data which are often different for different retrieval codes; popular choices include HDF5 (Mollière et al. 2019; Al-Refaie et al. 2021), pickle (Al-Refaie et al. 2021), or FITS (Min et al. 2020).

The need to interpret observational spectra covering a wide range of atmospheric conditions robustly and benchmark them across various studies inspired the creation of the MAESTRO (Molecules and Atoms in Exoplanet Science: Tools and Resources for Opacities) database. MAESTRO is a NASA-supported, open opacity service accessible to the community through a web interface<fnref n="31" />. Primary aims of the MAESTRO project are to incorporate the most up-to-date line lists, establish community standards for computing opacity data (see, e.g., Gharib-Nezhad et al. 2024), and offer insights and guidance on assumptions and limitations regarding data relevancy and completeness, particularly concerning different atmospheric compositions. The overall goal is to create one uniform and standard database for opacities. MAESTRO currently hosts high-resolution data from various sources such as EXOPLINES (Gharib-Nezhad et al. 2021) and ExoMolOP (Chubb et al. 2021), which span a large range of pressures, temperatures, and wavelengths. The ExoMolOP data hosted by the MAESTRO database are the high-resolution cross-sections which were used to compute the k-table or sampled cross-section files mentioned above. There is an option for opacity-producers to submit data to MAESTRO via GitHub<fnref n="32" />.

We note the importance of checking opacity files for their original line list source to check their accuracy, spectral coverage, and see what temperature they are suitable to be used up to; see Section 2 for a discussion of completeness of the line lists for different species, which depends on the temperature up to which a line list is considered complete.

## 4.3 What's being worked on?

Opacities need to be updated periodically as new line lists become available or are updated; work is in progress to keep these databases up-to-date, including adding important isotopologues to ExoMolOP. The MAESTRO database is also expanding, with new species and isotopologues to be added in due course. With existing instruments such as HST/UVIS and potential future instruments such as the Habitable Worlds Observatory (HWO), comes the requirement for opacity data which spans into the UV region, starting around 0.1 - 0.2µm. Work is ongoing to compute opacities down to these wavelengths for the species which already have some UV data available, and for future line lists which are in progress. Such opacities will be incorporated into databases such as ExoMolOP and MAESTRO.

## 4.4 Data needs: What's missing and urgent?

(i) **External updates and advances:** as mentioned at the start of this section, there are many variables and assumptions that are made when computing opacities. As new and improved data become available (line lists, broadening parameters, theoretical knowledge), then opacities will ideally need to be recomputed to keep to date with the latest advances. The requirements of new instruments will also have an impact here; opacities for use in characterising atmospheres observed with JWST, for example, need to be at higher resolution than those for use in characterising observations of the same planets using HST or Spitzer. Theoretical advances in line shape theory could also impact new opacities. As mentioned above, opacities which extend into the UV are also needed.

(ii) **Heavy and other atmosphere applications:** a focus on heavier atmospheres will require opacities tailored differently (i.e. different broadening parameters) to those for lighter gas giant atmospheres. In addition, stellar atmospheres also have their own requirements (e.g. Cristallo et al. 2007).

(iii) **Isotopologues:** although there are exceptions and updates are being made, opacities and line lists can be focused on the main isotopologues of a species only. For some species, however, isotopologues are more important than in others; for example natural elemental abundances of ⁷⁹Br/⁸¹Br and ³⁵Cl/³⁷Cl require molecules containing these elements to be formed from cross-sections combined from a combination of the two main isotopologues. There is also some interest in observing different isotopologues in exoplanet (and brown dwarf) atmospheres (Mollière & Snellen 2019; Barrado et al. 2023), which require accurate opacities to be computed for a variety of individual isotopologues for species of interest.

# 5 COLLISION INDUCED ABSORPTION (CIA) AND CONTINUUM DATA

## 5.1 Collision induced absorption (CIA)

CIA arises as broad features in atmospheric spectra due to additional spectral absorption occurring during collisions between pairs of species, with the interactions inducing a dipole moment. CIA is a distinct process from the collisional (pressure) broadening of absorption lines. Even species with no permanent dipole moment, such as H₂, N₂, and O₂ can have an observable effect if they are abundant enough (Karman

<footnote n="30">³⁰ https://perso.astrophy.u-bordeaux.fr/~jleconte/exo_k-doc/index.html</footnote>

<footnote n="31">³¹ https://science.data.nasa.gov/opacities/</footnote>

<footnote n="32">³² https://github.com/maestro-opacities/submit-data</footnote>
have a maximum temperature up to which they are considered complete; cross-sections computed above these temperatures will have opacity missing. This is because hot band transitions between energy levels of higher energy than are populated for the maximum line list temperature will not be taken into account. When computing cross-sections various choices need to be made, such as what line list, line profile, line-wing cutoff, and broadening parameters to use (see Sections 2 and 3). These will partly be determined by the requirements of the cross-sections being computed: will they be used for modelling Earth-like planets, in which case a line list such as HITRAN might be the most appropriate, or hotter exoplanets, in which case a line list computed or measured up to high temperatures is more appropriate, such as those provided by data bases such as HITEMP (Rothman et al. 2010), ExoMol (Tennyson et al. 2020), or TheoReTs (Rey et al. 2016a). If the cross-sections are for modelling or retrieving atmospheric properties of a hot gas giant exoplanet, then the atmospheric composition will be typically dominated by H₂ and He. The choice of broadening parameters in Section 3 will thus be H₂ and He for such atmospheres, whereas for a planet with a heavier atmosphere, heavier broadening species such as H₂O or N₂ will be more appropriate. Partition functions are also required for computing cross-sections; these are typically included as part of a line list (see Section 2), with tools such as TIPS (Gamache et al. 2021) available for computing partition functions for the HITRAN2020 data base. Studies such as Niraula et al. (2022, 2023) highlight how different choices, including line lists, broadening coefficients, and far-wing behaviours, can impact interpretations of observed spectra. Niraula et al. (2023) explore the impact of opacity choices on the atmospheric properties of the JWST NIRSpec/G395H transmission spectra of WASP-39 b (Alderson et al. 2023). Both Niraula et al. (2022) and Niraula et al. (2023) conclude that the incompleteness and inaccuracy of line lists is the biggest issue in the accurate retrieval of atmospheric properties from observations. Niraula et al. (2022) find strong biases due to incorrect line broadening assumptions, with the effects due to incomplete knowledge of far line wings greater for super-Earths than warm-Jupiters. They find the strongest biases, however, are due to using a line list which is incomplete at the atmospheric temperature being probed (i.e. using HITRAN instead of HITEMP or ExoMol for a hot Jupiter exoplanet atmosphere). Villanueva et al. (2024) present comparison tests designed for the exoplanetary community, in order to identify the effects of different input data choices, such as opacity format and line list source.

Opacity files are typically provided as either: (1) high-resolution cross-sections, (2) sampled cross-sections, or (3) correlated-k coefficients, also known as k-tables. The general process is to compute high-resolution cross-sections (typically with a grid spacing of around 0.1 cm⁻¹ or 0.01 cm⁻¹, or $R = \frac{\lambda}{\Delta\lambda} \sim 1000000$) for a grid of temperatures and pressures (typically spanning pressures of at least $\sim 1 \times 10^{-5}-100$ bar, and temperatures $\sim 100$ K–3000 K), and then, usually for reasons of computational feasibility, to either sample these cross-sections at a lower resolution ($\frac{\lambda}{\Delta\lambda} \sim 10000-20000$), or compute k-tables at $\frac{\lambda}{\Delta\lambda} \sim 300-5000$, if the application is for lower resolution applications such as characterizing Ariel or lower-resolution JWST observations. These resolutions are approximately equivalent to one another for achieving the same accuracy when used in an atmospheric modelling or retrieval code. The general principle of k-tables is to order spectral lines within a given spectral bin, producing a smooth cumulative distribution function to represent opacity, which can then be more efficiently sampled than cross-sections being sampled without any ordering of the spectral lines, with less loss of opacity for the same resolving power (see for

example, Min 2017; Chubb et al. 2020b; Leconte 2021, for further details).

**Super-lines:** very large line lists can contain billions of lines between millions of energy level states, for example the latest ExoMol CH₄ line list consists of over 50 billion transitions between 9 million states (Yurchenko et al. 2024d). The computation of opacities from such large line lists can benefit from techniques such as the super-lines method (Rey et al. 2016a; Yurchenko et al. 2017; Tennyson et al. 2020), which yields a vast improvement in computational speed when computing high-resolution cross-sections. Either line intensities can be computed at a given temperature, and then the intensities of all lines within a specified spectral bin summed together to create a so-called super-line. If a high enough resolving power, such as $R = \frac{\lambda}{\Delta\lambda} \sim 1000000$, is used for the creation of the super-lines then the overall opacity will be conserved (Yurchenko et al. 2017; Chubb et al. 2020b; Tennyson et al. 2020). Alternatively, some line lists, such as the CH₄ line lists of ExoMol (Yurchenko et al. 2017) and TheoReTs (Rey et al. 2014b, 2016a), are already partitioned into a large set of relatively weak lines and a small set of important, stronger lines. When computing opacities, the weaker lines can be combined into a set of super-lines (or effective lines as done for CH₄ in HITEMP; Hargreaves et al. 2020), with broadening of the stronger lines treated individually. Irwin et al. (2019) employed a similar 'pseudo continuum' approach when employing the NH₃ line list of Coles et al. (2019b) to analyse the visible spectrum of Jupiter, finding a great increase in efficiency with very little loss of accuracy.

## 4.1 State-of-the-art – data availability

There are some programs which have been developed by line list producers to generate cross-section data from line lists, such as ExoCross<fnref n="23" /> (Yurchenko et al. 2018d) and pyExoCross<fnref n="24" /> (Zhang et al. 2024) (ExoMol), and HAPI (HITRAN)<fnref n="25" /> (Kochanov et al. 2016). Other tools have been developed by line list users, such as HELIOS-K<fnref n="26" /> (Grimm & Heng 2015), Exo-k<fnref n="27" /> (Leconte 2021), RADIS<fnref n="28" /> (Pannier & Laux 2019; van den Bekerom & Pannier 2021), and others (Min 2017). Various data bases are available which have made use of tools such as these to compute opacities. Each will have their own set of assumptions and choices used when computing the opacities, depending on the intended applications: some data bases provide high resolution cross-sections and some sampled cross-sections or k-tables (see the introduction to this section), often formatted for direct use in a particular atmospheric retrieval or modelling code. Some codes, which employ a so-called line-by-line method, work directly from line lists, and so do not require these opacity files. In such an approach assumptions about line broadening are made by the user at the time of running the code and so it can be considered to be more flexible, however it can become computationally unfeasible for atmospheric retrievals which cover a large range of pressures, temperatures, and wavelengths. Table 10 gives a non-exhaustive list of opacity data bases employed by typical codes used to compute models or run retrievals relevant for the *Ariel* space mission, along with other applications. Some additional notes

<footnote n="23">²³https://github.com/Trovemaster/exocross</footnote>

<footnote n="24">²⁴https://github.com/ExoMol/PyExoCross</footnote>

<footnote n="25">²⁵https://hitran.org/hapi/</footnote>

<footnote n="26">²⁶https://github.com/exoclime/HELIOS-K</footnote>

<footnote n="27">²⁷https://perso.astrophy.u-bordeaux.fr/~jleconte/exo-k-doc/index.html</footnote>

<footnote n="28">²⁸https://github.com/radis/</footnote>
<table><caption>**Table 9.** An overview of some typical opacity databases designed for the (exo)planetary atmosphere community.</caption><thead><tr><th>Database</th><th>Types of data</th><th>Reference</th><th>Link</th></tr></thead><tbody><tr><td>DACE</td><td>Many molecular and atomic cross-sections, 0.01 cm<sup>-1</sup> spacing</td><td>Grimm et al. (2021)</td><td>https://dace.unige.ch/opacityDatabase/</td></tr><tr><td>ExoMoLOP</td><td>Sampled cross-sections and k-tables for many molecules</td><td>Chubb et al. (2021)</td><td>https://exomol.com/data/data-types/opacity/</td></tr><tr><td>EXOPLINES</td><td>High-resolution cross-sections for selected species</td><td>Gharib-Nezhad et al. (2021)</td><td>https://doi.org/10.5281/zenodo.4458189</td></tr><tr><td>Gandhi2020</td><td>High-resolution cross-sections for 6 molecules, 0.01 cm<sup>-1</sup> spacing</td><td>Gandhi et al. (2020)</td><td>https://osf.io/mgnw5/?view_only=5d58b814328e4600862ccfae4720acc3</td></tr><tr><td>MAESTRO</td><td>High-resolution cross-sections from various sources</td><td>Batalha et al. (2022)</td><td>https://science.data.nasa.gov/opacities/</td></tr></tbody></table>

et al. 2019). CIA data is labelled by the relevant pairs of species involved in these collisions; for hot Jupiter atmospheres H₂-H₂ and H₂-He are the most important pairs. Sufficient wavenumber coverage and appropriate temperature ranges for the data are necessary for the effect to be included.

## 5.2 Continuum

Continuum absorption is a process, and therefore dataset, which is typically presented as distinct from CIA. There are two main contributions which make up continuum absorption data: 1) excess opacity due to far line wings (see Section 3), and 2) dimer absorption, i.e. the absorption due to molecules which collide and become loosely bonded; to date this is most commonly associated with water (Ptashnik et al. 2011b; Shine et al. 2016).

There is potentially some community consensus lacking on how the continuum absorption and CIA processes are distinguished in experiment and calculations, and how they are treated numerically in radiative transfer codes. In some cases (see Section 5.3.2) continuum data contains some CIA absorption also, so care needs to be taken to not double count these contributions. We note the temperature coverage of continuum data is limited.

Probably the most well used source of continuum data, particularly for terrestrial applications and Earth-like planets, is the water vapour continuum. The continuum is a source of opacity which should be included in addition to line-by-line molecular absorption of H₂O (and other species, where necessary and available). It is typically tailored to be used alongside a particular opacity source; e.g. the MT_CKD continuum data (Mlawer et al. 2012; Mlawer et al. 2023) is designed to work alongside opacities computed from the HITRAN line list with a 25 cm⁻¹ Voigt line wing cut-off. This is due to the continuum data making up for the excess opacity due to far line wings.

## 5.3 State of the art - Data availability

### 5.3.1 Collision induced absorption (CIA)

The primary source for CIA data is the HITRANonline website<fnref n="33" />, where coverage for the number of pairs of species is greatest. For hot Jupiters, with a H₂- and He-dominated atmosphere, H₂-H₂, H₂-He, and He-He are the most important sources of CIA to be included. However for Earth-like or heavier atmospheres, a wider range of species will need to be considered. Another compilation of CIA data can be found via the Laboratoire de Météorologie Dynamique

Generic GCM (LMDG) online repository<fnref n="34" /> (see, e.g., Charney et al. 2021), which can be found via the Exo-k website<fnref n="35" />.

Table 10 summarises the current status of CIA data available in the HITRAN database. An overview of the transitions and pairs of species included at the time of the 2019 update of the HITRAN CIA parameters can be found in Karman et al. (2019), with updates for the 2020 release of the HITRAN database outlined in Gordon et al. (2022). Table 10 shows the coverage and limitations of the current state of CIA data for the species where data exists and also the wavenumber regions and temperatures for which this data is applicable.

The accurate spectroscopic parameters obtained from the comb-referenced spectroscopy measurements of O₂ by Fleurybaey et al. (2021) in Table 8, combined with other CRDS measurements (Mondelain et al. 2019; Kassi et al. 2021), were recently utilised (Adkins et al. 2023) for fitting a parameterised representation of the theoretical calculations by Karman et al. (2018b) for O₂-O₂ and O₂-N₂ collision-induced absorption.

### 5.3.2 Continuum

There are various databases and literature which provide continuum data, such as the MT_CKD (Mlawer-Tobin-Clough-Kneizys-Davies) database<fnref n="36" /> (Mlawer et al. 2012; Mlawer et al. 2019). Both self (due to the interaction of the species with itself) and foreign (due to interaction with the species with another species in air) continuum data are available. The MT_CKD water vapor continuum was recently made available through HITRANonline (Mlawer et al. 2023). H₂O continuum data is also available from the CAVIAR lab (Ptashnik et al. 2011a; Shine et al. 2016), and was used in recent analyses of the impact of continuum data in exoplanet atmospheres by Anisman et al. (2022b). Original sources of continuum data measured in the laboratory include works such as Baranov et al. (2008); Odintsova et al. (2020). As well as H₂O, the MT_CKD continuum model also includes carbon dioxide, ozone, nitrogen and oxygen (Mlawer et al. 2012). It includes a number of collision-induced continuum absorption bands, so care needs to be taken not to double count when including both continuum and collision induced absorption (CIA) data (Mlawer et al. 2012).

Although we do not go into detail here, we note that other continuum opacities, such as H⁻, can be important for characterising hot gaseous exoplanet atmospheres. For example, opacity data was computed and implemented into retrieval codes used in the observational study of Lewis et al. (2020)

<footnote n="33">³³ https://hitran.org/cia</footnote>

<footnote n="34">³⁴ https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/continuum_data</footnote>

<footnote n="35">³⁵ https://perso.astrophy.u-bordeaux.fr/~jleconte/exo-k-doc/where_to_find_data.html</footnote>

<footnote n="36">³⁶ https://github.com/AER-RC/MT_CKD</footnote>
<table><caption>**Table 10.** An overview of some typical opacity data bases designed for the (exo)planetary atmosphere community.</caption><thead><tr><th>Data base</th><th>Types of data</th><th>Reference</th><th>Link</th></tr></thead><tbody><tr><td>DACE</td><td>Many molecular and atomic cross-sections, 0.01 cm<sup>-1</sup> spacing</td><td>Grimm et al. (2021)</td><td>https://dace.unige.ch/opacityDatabase/</td></tr><tr><td>ExoMoLOP</td><td>Sampled cross-sections and k-tables for many molecules</td><td>Chubb et al. (2020b)</td><td>https://exomol.com/data/data-types/opacity/</td></tr><tr><td>EXOPLINES</td><td>High-resolution cross-sections for selected species</td><td>Gharib-Nezhad et al. (2021)</td><td>https://doi.org/10.5281/zenodo.4458189</td></tr><tr><td>Gandhi2020</td><td>High-resolution cross-sections for six molecules, 0.01 cm<sup>-1</sup> spacing</td><td>Gandhi et al. (2020)</td><td>https://osf.io/mgnw5/?view_only=5d58b814328e4600862cfae4720acc3</td></tr><tr><td>MAESTRO</td><td>High-resolution cross-sections from various sources</td><td>Batalha et al. (2022)</td><td>https://science.data.nasa.gov/opacities/</td></tr></tbody></table>

on these data bases are given in Section 4.2. It is important, although not always the case, for modelling and retrieval studies to report on the opacities they use and either cite or explain the assumptions that went into computing those opacities. This allows for traceability and for analyses to be reproducible.

## 4.2 Data used by retrieval codes

Molecular opacities (cross-sections and k-tables) are now included in many radiative transfer codes and associated libraries, such as those listed in Table 2. The opacity data bases listed in Table 10 are available online and although some are formatted with specific retrieval or modelling codes in mind, they can generally be adapted for more general use or for different codes. ExoMoLOP (Chubb et al. 2020b) has k-tables and cross-sections for currently around 80 species in the ExoMoL data base formatted for direct input into TauREx3 (Al-Refaie et al. 2021), ARCiS (Ormel & Min 2019; Min et al. 2020), NEMESIS (Irwin et al. 2008), petitRADTRANS (Mollière et al. 2019) (see Table 2). The EXOPLINES (Gharib-Nezhad et al. 2021), Gandhi2020 (Gandhi et al. 2020), MAESTRO (Batalha et al. 2022), and DACE (Grimm et al. 2021) data bases/compilations all contain high-resolution cross-sections for various pressures and temperatures. These can be sampled down to a smaller desired resolution for practical use, or converted to k-tables. Various tools currently exist for this, including Exo-k<fnref n="29" /> (Leconte 2021), with more tools expected to become openly available in the near future. We note there are various file formats used for storing opacity data which are often different for different retrieval codes; popular choices include HDF5 (Mollière et al. 2019; Al-Refaie et al. 2021), pickle (Al-Refaie et al. 2021), or FITS (Min et al. 2020).

The need to interpret observational spectra covering a wide range of atmospheric conditions robustly and benchmark them across various studies inspired the creation of the MAESTRO (Molecules and Atoms in Exoplanet Science: Tools and Resources for Opacities) data base. MAESTRO is a NASA-supported, open opacity service accessible to the community through a web interface.<fnref n="30" /> Primary aims of the MAESTRO project are to incorporate the most up-to-date line lists, establish community standards for computing opacity data (see e.g. Gharib-Nezhad et al. 2024), and offer insights and guidance on assumptions and limitations regarding data relevancy and completeness, particularly concerning different atmospheric compositions. The overall goal is to create one uniform and standard data base for opacities. MAESTRO currently hosts high-resolution data from various sources such as EXOPLINES (Gharib-Nezhad et al. 2021) and ExoMoLOP (Chubb et al. 2020b), which span a large range of pressures, temperatures, and wavelengths. The ExoMoLOP data hosted by the MAESTRO data base are the high-resolution cross-sections which were used to compute the k-table or sampled cross-section files mentioned above. There is an option for opacity-producers to submit data to MAESTRO via GitHub.<fnref n="31" />

We note the importance of checking opacity files for their original line list source to check their accuracy, spectral coverage, and see what temperature they are suitable to be used up to; see Section 2 for a discussion of completeness of the line lists for different species, which depends on the temperature up to which a line list is considered complete.

<footnote n="29">²⁹https://perso.astrophy.u-bordeaux.fr/~jleconte/exo_k-doc/index.html</footnote>

<footnote n="30">³⁰https://science.data.nasa.gov/opacities/</footnote>

<footnote n="31">³¹https://github.com/maestro-opacities/submit-data</footnote>
<table><caption>**Table 10.** Collision induced absorption (CIA) data sets available from HITRANonline with the wavenumber and temperature ranges for which they provide coverage, and the citations for the original sources.</caption><thead><tr><th>System</th><th>ν<sub>range</sub>/cm<sup>-1</sup></th><th>λ<sub>range</sub>/µm</th><th>T<sub>min</sub>/K</th><th>T<sub>max</sub>/K</th><th>Reference</th></tr></thead><tbody><tr><td>H<sub>2</sub>-CH<sub>4</sub> (equilibrium)</td><td>0 - 1,946</td><td>> 5.14</td><td>40</td><td>400</td><td>Borysow &amp; Frommhold (1986a)</td></tr><tr><td>H<sub>2</sub>-CH<sub>4</sub> (normal)</td><td>0 - 1,946</td><td>> 5.14</td><td>40</td><td>400</td><td>Borysow &amp; Frommhold (1986a)</td></tr><tr><td>H<sub>2</sub>-H<sub>2</sub></td><td>20 - 10,000</td><td>1 - 500</td><td>200</td><td>3,000</td><td>Abel et al. (2011); Fletcher et al. (2018)</td></tr><tr><td>H<sub>2</sub>-H</td><td>100 - 11,000</td><td>0.9 - 100</td><td>1,000</td><td>2,500</td><td>Gustafsson &amp; Frommhold (2003)</td></tr><tr><td>H<sub>2</sub>-He</td><td>20 - 20,000</td><td>0.5 - 500</td><td>200</td><td>9,900</td><td>Abel et al. (2012)</td></tr><tr><td>He-H</td><td>50 - 11,000</td><td>0.9 - 200</td><td>1,500</td><td>10,000</td><td>Gustafsson &amp; Frommhold (2001)</td></tr><tr><td>N<sub>2</sub>-H<sub>2</sub></td><td>0 - 1,886</td><td>> 5.3</td><td>40</td><td>400</td><td>Borysow &amp; Frommhold (1986b)</td></tr><tr><td>N<sub>2</sub>-He</td><td>1 - 1,000</td><td>10 - 10,000</td><td>300</td><td>300</td><td>Bar-Ziv &amp; Weiss (1972)</td></tr><tr><td>N<sub>2</sub>-N<sub>2</sub></td><td>0 - 5,000</td><td>> 2</td><td>70</td><td>330</td><td>Karman et al. (2015); Baranov et al. (2005)<br>Lafferty et al. (1996); Hartmann et al. (2017)<br>Sung et al. (2016b); Chistikov et al. (2019)</td></tr><tr><td>N<sub>2</sub>-air</td><td>1,850 - 5,000</td><td>2 - 5.4</td><td>228</td><td>330</td><td>Baranov et al. (2005); Lafferty et al. (1996)<br>Hartmann et al. (2017); Menoux et al. (1993)</td></tr><tr><td>N<sub>2</sub>-H<sub>2</sub>O</td><td>1,930 - 2,830</td><td>3.5 - 5.2</td><td>250</td><td>350</td><td>Hartmann et al. (2018)</td></tr><tr><td>O<sub>2</sub>-CO<sub>2</sub></td><td>12,600 - 13,839</td><td>0.7 - 0.8</td><td>296</td><td>296</td><td>Vangvichith et al. (2009)</td></tr><tr><td>O<sub>2</sub>-N<sub>2</sub></td><td>1,300 - 13,840</td><td>0.7 - 7.7</td><td>193</td><td>296</td><td>Baranov et al. (2005); Lafferty et al. (1996)<br>Thibault et al. (1997); Orlando et al. (1991)<br>Menoux et al. (1993); Maté et al. (1999)<br>Tran et al. (2006); Karman et al. (2018a)</td></tr><tr><td>O<sub>2</sub>-O<sub>2</sub></td><td>1,150 - 29,800</td><td>0.3 - 8.7</td><td>193</td><td>293</td><td>Baranov et al. (2004); Maté et al. (1999)<br>Thibault et al. (1997); Orlando et al. (1991)<br>Tran et al. (2006); Karman et al. (2018a)<br>Tran et al. (2006); Karman et al. (2018a)<br>Spiering &amp; van der Zande (2012); Spiering et al. (2011)<br>Thalman &amp; Volkamer (2013); Kassi et al. (2021)<br>Adkins et al. (2023)</td></tr><tr><td>O<sub>2</sub>-air</td><td>1,300 - 13,839</td><td>0.7 - 7.7</td><td>193</td><td>300</td><td>Maté et al. (1999); Spiering &amp; van der Zande (2012)<br>Thibault et al. (1997); Orlando et al. (1991)<br>Tran et al. (2006); Karman et al. (2018a)<br>Drouin et al. (2017)</td></tr><tr><td>CO<sub>2</sub>-CO<sub>2</sub></td><td>1 - 3,250</td><td>3.1 - 10,000</td><td>200</td><td>298</td><td>Gruszka &amp; Borysow (1997); Baranov &amp; Vigasin (1999)<br>Baranov (2018)</td></tr><tr><td>CO<sub>2</sub>-H<sub>2</sub></td><td>0 - 2,000</td><td>> 5</td><td>200</td><td>350</td><td>Wordsworth et al. (2017)</td></tr><tr><td>CO<sub>2</sub>-He</td><td>0 - 1,000</td><td>> 10</td><td>300</td><td>300</td><td>Bar-Ziv &amp; Weiss (1972)</td></tr><tr><td>CO<sub>2</sub>-CH<sub>4</sub></td><td>1 - 2,000</td><td>5 - 10,000</td><td>200</td><td>350</td><td>Wordsworth et al. (2017)</td></tr><tr><td>CO<sub>2</sub>-Ar</td><td>0 - 300</td><td>> 33</td><td>200</td><td>400</td><td>Turbet et al. (2020)<br>Odintsova et al. (2021)</td></tr><tr><td>CH<sub>4</sub>-He</td><td>1 - 1,000</td><td>> 10</td><td>40</td><td>350</td><td>Taylor et al. (1988)</td></tr><tr><td>CH<sub>4</sub>-Ar</td><td>1 - 697</td><td>14.3 - 10,000</td><td>70</td><td>296</td><td>Samuelson et al. (1997)</td></tr><tr><td>CH<sub>4</sub>-N<sub>2</sub></td><td>0 - 1379</td><td>> 7.3</td><td>40</td><td>400</td><td>Finenko et al. (2021, 2022)</td></tr><tr><td>CH<sub>4</sub>-CH<sub>4</sub></td><td>0 - 990</td><td>> 10.1</td><td>200</td><td>800</td><td>Borysow &amp; Frommhold (1987)</td></tr></tbody></table>

using data from John (1988). Retrieval analysis found evidence for significant amounts of H⁻ in the atmosphere of hot Jupiter HAT-P-41b, thought to be produced through a combination of photochemical and collisional processes (Lewis et al. 2020). Evidence for H⁻ was also found by Mikal-Evans et al. (2022) in the atmosphere of ultrahot gas giant exoplanet WASP-121b and by Arcangeli et al. (2018) for very hot gas giant exoplanet WASP-18b.

## 5.4 Data used by retrieval codes

### 5.4.1 Collision induced absorption (CIA)

Collision induced absorption data is readily usable in retrieval codes such as ARCiS, TauREx3, petitRADTRANS, and others mentioned in Table 2. The HITRAN formatted data (.cia) provides a basis for including CIA data<fnref n="37" /> (Karman et al.

2019). As noted above, CIA including H₂ and He is commonly included in the atmospheric modelling of hydrogen and helium rich atmospheres and the corresponding data is distributed with the default data sets for retrieval codes such as TauREx3 and ARCiS. Additional species can be included in the modelling and retrieval codes as they become available or are updated, following the HITRAN format to maximise interoperability. Though data format does not pose a major limitation, restricted wavelength range coverage for the CIA data poses constraints on modelling where CIA between additional species is considered.

The HITRAN CIA format supports different sets of absorption data for the same pair of species but for varying temperatures. While this is provided in some data sets, it relies on implementation in the retrieval code or downstream processing library in order to query based on temperature. This is not always supported.

<footnote n="37">³⁷ https://hitran.org/data/CIA/CIA_Readme.pdf</footnote>
## 4.3 What's being worked on?

Opacities need to be updated periodically as new line lists become available or are updated; work is in progress to keep these data bases up-to-date, including adding important isotopologues to ExoMolOP. The MAESTRO data base is also expanding, with new species and isotopologues to be added in due course. With existing instruments such as HST/UVIS and potential future instruments such as the Habit-able Worlds Observatory (HWO), comes the requirement for opacity data which spans into the UV region, starting around 0.1–0.2 µm. Work is ongoing to compute opacities down to these wavelengths for the species which already have some UV data available, and for future line lists which are in progress. Such opacities will be incorporated into data bases such as ExoMolOP and MAESTRO.

## 4.4 Data needs: what's missing and urgent?

(i) **External updates and advances:** as mentioned at the start of this section, there are many variables and assumptions that are made when computing opacities. As new and improved data become available (line lists, broadening parameters, theoretical knowledge), then opacities will ideally need to be recomputed to keep to date with the latest advances. The requirements of new instruments will also have an impact here; opacities for use in characterizing atmospheres observed with *JWST*, for example, need to be at higher resolution than those for use in characterizing observations of the same planets using *HST* or *Spitzer*. Theoretical advances in line shape theory could also impact new opacities. As mentioned above, opacities which extend into the UV are also needed.

(ii) **Heavy and other atmosphere applications:** a focus on heavier atmospheres will require opacities tailored differently (i.e. different broadening parameters) to those for lighter gas giant atmospheres. In addition, stellar atmospheres also have their own requirements (e.g. Cristallo et al. 2007).

(iii) **Isotopologues:** although there are exceptions and updates are being made, opacities and line lists can be focused on the main isotopologues of a species only. For some species, however, isotopologues are more important than in others; for example natural elemental abundances of $^{79}$Br/$^{81}$Br and $^{35}$Cl/$^{37}$Cl require molecules containing these elements to be formed from cross-sections combined from a combination of the two main isotopologues. There is also some interest in observing different isotopologues in exoplanet (and brown dwarf) atmospheres (Mollière & Snellen 2019; Barrado et al. 2023), which require accurate opacities to be computed for a variety of individual isotopologues for species of interest.

# 5 COLLISION INDUCED ABSORPTION (CIA) AND CONTINUUM DATA

## 5.1 Collision induced absorption (CIA)

Collision-induced absorption (CIA) arises as broad features in atmospheric spectra due to additional spectral absorption occurring during collisions between pairs of species, with the interactions inducing a dipole moment. Symmetric molecules, such as $O_2$, $N_2$, $H_2$ do not possess any permanent or vibrationally induced electric dipole and are thus microwave and IR inactive, although even if extremely weak, magnetic dipole and electric quadrupole moments can contribute to the absorption. Nevertheless, significant absorption is induced through the creation of a transient dipole via intermolecular interactions during their collisions. This process enables one to reveal the presence of non-polar species in the atmosphere by

detecting collision induced supramolecules signatures. CIA in dense gases was discovered in 1949 by Crawford and Welsh in the infrared range (Crawford, Welsh & Locke 1949). Almost immediately after this discovery, it was realized that CIA of electromagnetic radiation should be important in just about any cool and dense environment, including atmospheres of solar and extrasolar planets (Trafton 1964; Mayor & Queloz 1995), cool white dwarfs (Shipman 1977; Mould & Liebert 1978; Bergeron, Saumon & Wesemael 1995; Saumon & Jacobson 1998), brown dwarfs (Burgasser, Burrows & Kirkpatrick 2006; Geballe et al. 2009), cool main sequence stars (Linsky 1969; Hansen & Sterl Phinney 1998), and the so-called first stars (Rees 1976; Bromm & Larson 2004; Ripamonti & Abel 2004; Bromm et al. 2009; Loeb 2010).

The term 'collision-induced absorption' is usually used to refer to any bimolecular absorption that is absent from the isolated molecules but reveals itself as a result of intermolecular interaction (Vigasin & Mokhov 2017). To understand this process theoretically, a detailed statistical analysis of the wide range of thermally exciting interacting molecules that make up molecular pairs is needed. Three categories of molecular pairs, all related to a range of the potential energy surface of the associated van der Waals complex, can be distinguished: true bound, metastable, and free pairs. The simultaneous presence of bound-bound (dimer), bound-free, free-bound, and free-free (collisional) transitions in collision-induced processes makes the spectral analysis and the theoretical treatment very complex.

For decades, CIA has been known to contribute significantly to the transmission of radiation in the Earth's atmosphere, primarily via $N_2-N_2$, $N_2-O_2$, and $O_2-O_2$ (Farmer & Houghton 1966; Rinsland et al. 1981, 1982), as well as to play a crucial role in the radiation budget of other planets (Tipping & Poll 1985). Different molecular pairs have been identified as major contributors to the opacity of planetary atmospheres, such as $H_2-H_2$, $H_2-He$ in gas giant planets (Richard et al. 2012), $O_2-CO_2$ and $CO_2-CO_2$ in Venus' atmosphere (Gruszka & Borysow 1997) and $H_2-N_2$, $N_2-CH_4$ in Titan's (Borysow & Tang 1993). In the Solar System, the $H_2$ dimer has been detected on Jupiter and Saturn from the comparison between satellite measurements and laboratory absorption spectra recorded by McKellar (1988). Several ionic complexes were also identified for example in the E-ring of Saturn, where the successive solvation of sodium cations by water was proposed to assign the observed mass spectrum (Postberg et al. 2009) $H_2-H_2$ and $H_2-He$ CIA are considered as a thermometer to measure the thermal structure of the upper tropospheres of the gas giants (Conrath, Gierasch & Ustinov 1998). The opacity of Jupiter in the far infrared is dominated by the CIA of $H_2-H_2$ and $H_2-He$ (Trafton 1964; Borysow, Moraldi & Frommhold 1984; Richard et al. 2012), making them vital components of not only the solar system gas giant planets but also Jupiter-like exoplanetary atmospheres. The importance of CIA warming in exoplanets was highlighted in Ramirez & Kaltenegger (2017) in the context of rocky planets with $H_2$-dominated atmospheres. Furthermore, the impact of CIA on the radiative budget has been demonstrated in the atmospheres of Earth and Super-Earth mass exoplanets with $H_2$-enriched atmospheres (Pierrehumbert & Gaidos 2011). Other molecular pairs, including $CO_2$ have been studied to assess the impact of CIA on the surface temperature and to draw conclusion on the climate and habitability of early Mars (Turbet et al. 2019; Godin et al. 2020; Turbet, Boulet & Karman 2020). These CIA signatures are expected in spectra of hot Jupiters (Bell et al. 2024) too. During their campaign of measurements, Turbet et al. (2019) showed the disagreement between experimental data and the theoretical ab initio calculations of Wordsworth et al. (2017). Sufficient wavenumber coverage and appropriate temperature ranges for the data are necessary for the CIA
### 5.4.2 Continuum

Anisman et al. (2022b) modelled the effects of including vs not including continuum data for H₂O on examples of low-mass temperate exoplanet transmission spectra using Tau-REx3 (Al-Refaie et al. 2021). The effect of continuum cross-sections exhibit an inverse relationship with temperature, so are most important for use in modelling atmospheres with temperatures at or close to Earth's.

As previously mentioned, one aspect of using continuum models such as MT_CKD is that they rely on specific line wings being used for computations of corresponding molecular cross-sections. For example, the MT_CKD model assumes that a line wing cutoff of 25 cm⁻¹ is used, otherwise there can be double counting of the wing contribution (see, for example, Gharib-Nezhad et al. 2024).

### 5.5 What's being worked on?

Collision induced absorption (CIA) coefficients of H₂ and H₂-He have been recently been measured in the 0.4 - 8 µm spectral range in order to characterise Jupiter's atmosphere (Stefani et al. in press). These are shown in Figure 4, covering pressure values from 10 mbar to 70 bar and temperature values of 100 - 550 K.

CIA laboratory measurements have also been published recently from groups in Grenoble (Fleurbaey et al. 2022). There is a simulation chamber for absorption spectroscopy in planetary atmospheres in INAF (Snels et al. 2021); see Fig. 5. In terms of species, molecular systems related to Venus and Early Mars are mostly being targeted, i.e. CO₂-H₂O and CO₂-CO₂. Theoretical works are planned for molecular systems related to Jupiter-like planets.

## 5.6 Data needs: What's missing and urgent?

### 5.6.1 Collision induced absorption (CIA)

(i) **Data for additional CIA species:** CIA data is currently unavailable for many pairs of species that are thought to be important in exoplanetary atmospheres, for example N₂ is currently the only available species for collisions with H₂O (see Table 10).

(ii) **Wavelength coverage:** much of the data that is available is limited to bands at wavelength ranges outside the window observed by Ariel (see Table 10 for an overview of current coverage of CIA data).

(iii) **Temperature dependence:** data is often available only at room temperature, and the extension to higher temperatures is desirable, especially for modelling hot Jupiters and lava planets. We note some work has been done towards this, e.g. Stefani et al. (2018), along with data for H₂-H₂, H₂-H, H₂-He, He-H, as detailed in Table 10.

(iv) **Isotopologues:** CIA data coverage does not currently extend far in the consideration of isotopologues; generally the main isotopologue is taken for theoretically derived data, while the terrestrial ratios are used for experimentally derived data. The potential impact of a variation in isotopologues for CIA is small though it is potentially relevant for the HD-H₂ pair (Karman et al. 2019).

### 5.6.2 Continuum

(i) **Species other than H₂O:** although the MT_CKD database contains some other species such as CO₂, O₃, N₂, and O₂, the main focus of continuum absorption has been on H₂O.

(ii) **Temperature dependence:** continuum absorption data for H₂O is currently only available below temperatures of 500 K, so there is not available data to use for modelling exoplanets with higher temperature atmospheres. We do note, however, that the impact of the H₂O continuum absorption on spectra decreases as the temperature increases (Anisman et al. 2022b), so this is not thought to be of urgent concern for the community.

## 6 AEROSOLS AND SURFACES

It is well known that the types of cloud or haze particles, collectively known as aerosols, in the atmospheres of exoplanets will have an effect on their observed transmission or emission spectra (Wakeford & Sing 2015; Pinhas & Madhusudhan 2017; Min et al. 2020; Ma et al. 2023; Taylor & Parmentier 2023), and their albedo (Muñoz & Isaak 2015; Fraine et al. 2021; Barrientos et al. 2023; Chubb et al. 2023), through the cloud particle's absorption and scattering properties. Generally, haze particles are smaller than cloud particles and are formed via photochemistry, which is a disequilibrium process with stellar UV leading to the formation of solid particles. Clouds, on the other hand, are formed via equilibrium processes which involve condensation and rainout.

Gao et al. (2021) give an overview of the state of knowledge (as of February 2021) of exoplanet aerosols as determined from observations, modeling, and laboratory experiments. Wavelength-dependent, and ideally also temperature-dependent, refractive indices, also known as optical constants, are required to compute the scattering and absorption coefficients of clouds and hazes (aerosols) for use in exo-planetary atmosphere models. Thermochemical data are also required for predicting which aerosol species are expected to be present at various altitudes and horizontally across a given exoplanet atmosphere. Typically the outputs from global circulation models (GCMs), such as pressure-temperature profiles as a function of longitude and latitude across the planet (see, e.g. Lee et al. 2021; Cho et al. 2021; Schneider et al. 2022; Skinner & Cho 2022; Skinner et al. 2023; Changeat et al. 2024, for details on GCMs and complex atmospheric dynamics), are used as a basis for kinetic cloud models which then predict which particles will condense at different locations of the atmosphere (see, e.g. Ackerman & Marley 2001; Helling et al. 2008; Gao et al. 2020; Roman et al. 2021; Rooney et al. 2022; Helling et al. 2023). Usually some assumptions need to be made, such as initial elemental abundances available for condensing, and the strength of vertical atmospheric mixing, for example. Due to the importance of optical and UV data for understanding aerosol properties in exoplanet atmospheres, it has been suggested by Fairman et al. (2024) and Wakeford et al. (2020) to complement JWST observations with HST observations.

Similarly, planetary surfaces can be modelled by a simple reflectance model (for instance using linear mixing) (Cui et al. 2018), but also by radiative transfer modelling using optical constants, similar to the approach for Solar System planets (Hapke 1993; Andrieu et al. 2015). In both contexts,
effect to be included. When it comes to symmetric molecules, such as H₂ or O₂, the CIA contribution is the only way to quantify the density of these molecules, bearing in mind that their absorption coefficients depend on the square density (see e.g. Richard et al. 2012).

## 5.2 Continuum

Continuum absorption is a process, and therefore data set, which is typically presented as distinct from CIA. There are two main contributions which make up continuum absorption data: (1) excess opacity due to far line wings (see Section 3), and (2) dimer absorption, i.e. the absorption due to molecules which collide and become loosely bonded; to date this is most commonly associated with water (Ptashnik, Shine & Vigasin 2011a; Shine et al. 2016).

There is potentially some community consensus lacking on how the continuum absorption and CIA processes are distinguished in experiment and calculations, and how they are treated numerically in radiative transfer codes. In some cases (see Section 5.3.2) continuum data contains some CIA absorption also, so care needs to be taken to not double count these contributions. We note the temperature coverage of continuum data is limited.

Probably the most well used source of continuum data, particularly for terrestrial applications and Earth-like planets, is the water vapour continuum. The continuum is a source of opacity which should be included in addition to line-by-line molecular absorption of H₂O (and other species, where necessary and available). It is typically tailored to be used alongside a particular opacity source; e.g. the MT_CKD continuum data (Mlawer et al. 2012, 2023) is designed to work alongside opacities computed from the HITRAN line list with a 25 cm⁻¹ Voigt line wing cut-off. This is due to the continuum data making up for the excess opacity due to far line wings.

## 5.3 State-of-the-art – data availability

### 5.3.1 Collision induced absorption (CIA)

The primary source for CIA data is the HITRANonline website,<fnref n="32" /> where coverage for the number of pairs of species is greatest. Another compilation of CIA data can be found via the Laboratoire de Météorologie Dynamique Generic GCM (LMDG) online repository<fnref n="33" /> (see e.g. Charnay et al. 2021), which can be found via the Exo-k website.<fnref n="34" /> For hot Jupiters, with a H₂- and He-dominated atmosphere, H₂-H₂, H₂-He, and He-He are the most important sources of CIA to be included. However for Earth-like or heavier atmospheres, a wider range of species will need to be considered.

Table 11 summarizes the current status of CIA data available in the HITRAN data base. An overview of the transitions and pairs of species included at the time of the 2019 update of the HITRAN CIA parameters can be found in Karman et al. (2019), with updates for the 2020 release of the HITRAN data base outlined in Gordon et al. (2022). Table 11 shows the coverage and limitations of the current state of CIA data for the species where data exists and also the wavenumber regions and temperatures for which this data is applicable. It can be seen that the coverage of CIA data for heavy molecules is seriously lacking, which could have an impact on the

modelling efforts of super-Earth atmospheres. CIA data for H₂-H₂ and H₂-He were originally motivated by the Solar System gas giants (Fletcher, Gustafsson & Orton 2018), but the wide increase in temperature coverage was due to the interest in higher temperature exoplanet atmospheres such as hot Jupiters (Borysow 2002). Light molecular systems are generally less complicated to model and measure, which could also explain the more complete wavelength and temperature coverage of data for lighter species. Higher mass molecular systems with many degrees of freedom, on the other hand, are more challenging to understand. Classical molecular dynamics simulations were used for CH₄-CO₂ (Fakhardji et al. 2022) and compared to measurements (Turbet et al. 2019). The agreement was good but the limitations of the model were highlighted.

The accurate spectroscopic parameters obtained from the comb-referenced spectroscopy measurements of O₂ by Fleurbaey et al. (2021) in Table 9, combined with other CRDS measurements (Mondelain, Kassi & Campargue 2019; Kassi et al. 2021), were recently utilized (Adkins et al. 2023) for fitting a parametrized representation of the theoretical calculations by Karman et al. (2018) for O₂-O₂ and O₂-N₂ collision-induced absorption.

### 5.3.2 Continuum

There are various data bases and literature which provide continuum data, such as the MT_CKD (Mlawer-Tobin-Clough-Kneizys-Davies) data base<fnref n="35" /> (Mlawer et al. 2012; Mlawer et al. 2019). Both self (due to the interaction of the species with itself) and foreign (due to interaction with the species with another species in air) continuum data are available. The MT_CKD water vapour continuum was recently made available through HITRANonline (Mlawer et al. 2023). H₂O continuum data is also available from the CAVIAR project (Ptashnik et al. 2011b; Shine et al. 2016). Anisman et al. (2022a) performed a recent analyses of the impact of continuum data in exoplanet atmospheres and showed that its inclusion led to potential observable shifts in the transit depth in the region of 60 ppm for planets with water-rich atmospheres. Original sources of continuum data measured in the laboratory include works such as Baranov et al. (2008) and Odintsova et al. (2020). As well as H₂O, the MT_CKD continuum model also includes contributions from the so-called foreign continuum involving carbon dioxide, ozone, nitrogen, and oxygen (Mlawer et al. 2012). It includes a number of collision-induced continuum absorption bands, so care needs to be taken not to double count when including both continuum and CIA data (Mlawer et al. 2012).

Although we do not go into detail here, we note that other continuum opacities, such as H⁻, can be important for characterizing hot gaseous exoplanet atmospheres. For example, opacity data were computed and implemented into retrieval codes used in the observational study of Lewis et al. (2020) using data from John (1988). Retrieval analysis found evidence for significant amounts of H⁻ in the atmosphere of hot Jupiter HAT-P-41b, thought to be produced through a combination of photochemical and collisional processes (Lewis et al. 2020). Evidence for H⁻ was also found by Mikal-Evans et al. (2022) in the atmosphere of ultrahot gas giant exoplanet WASP-121b and by Arcangeli et al. (2018) for very hot gas giant exoplanet WASP-18b.

<footnote n="32">³²https://hitran.org/cia</footnote>

<footnote n="33">³³https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/continuum_data</footnote>

<footnote n="34">³⁴https://perso.astrophy.u-bordeaux.fr/~jleconte/exo-k-doc/where_to_find_data.html</footnote>

<footnote n="35">³⁵https://github.com/AER-RC/MT_CKD</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 4.** Left: H₂ CIA coefficients at different pressures (8, 10, 12 bars) and room temperature (296 K). Right: H₂-He CIA coefficients at different pressures (10, 13 bars) and temperatures (201, 299, 401 K).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 5.** The absorption spectra of 16.03 amagat of carbon dioxide at 243 and 323 K, demonstrating a number of CIA bands, recorded with PASSxS, after subtraction of the strong contributions of the $v_3$ fundamental band at 2350 cm⁻¹ and a weaker Fermi doublet ($v_3 + v_1, v_3 + 2v_2$) around 3700 cm⁻¹, as well as contributions of the ¹⁶O¹²C¹⁸O isotopomer (Snels et al. 2021).</figcaption></figure>

already existing databases contain relevant and useful sources but in many cases new measurements would be desirable (see Section 6.4). Here we outline some typical sources of currently available data for these different applications.

## 6.1 State of the art - Data availability

### 6.1.1 Refractive Indices

Codes which compute the scattering and absorption properties of a particle or a surface require the refractive index to be known. The real and imaginary part of the refractive index relates to the scattering and absorption properties of the material, respectively. If modelling an atmospheric layer consisting of different types of aerosol materials, or modelling mixed-material aerosol particles, then the refractive indices are often mixed using effective medium theory (see, for example, Mishchenko et al. 2016).

As well as there being variation with wavelength, there is also some temperature dependence with refractive indices,

although data for high temperatures are typically not available for many species. Table 11 gives the wavelength and temperature range of the available refractive indices of various materials from different sources. We note this table is not exhaustive, and it is useful to search in the various databases and compilations mentioned below for additional species of interest. Studies which predict which cloud or haze species are present in various exoplanet atmospheres (see, for example, Ackerman & Marley 2001; Helling et al. 2008; Gao et al. 2020; Rooney et al. 2022; Helling et al. 2023) are useful as references for species to include in atmospheric models.

Databases which compile the optical properties of various materials include the Heidelberg-Jena-St.Petersburg-Database of Optical Constants (HJPDOC)<fnref n="38" /> (Henning et al. 1999; Jäger et al. 2003), the Database of Optical Constants for Cosmic Dust<fnref n="39" />, the Aerosol Refractive Index Archive (ARIA) <fnref n="40" />, the HITRAN2020 database<fnref n="41" /> (Gordon et al. 2022), the Optical Constants database OCdb<fnref n="42" /> (see, e.g. Drant et al. 2024), the recent open-source *refractiveindex.info* database of optical constants<fnref n="43" /> (Polyanskiy 2024), and Texas A&M University dust 2020 (TAMUdust2020) (Saito et al. 2021) which compiles the optical properties of irregularly shaped aerosol particles. Original sources of refractive index data include those of works such as Henning et al. (1995); Jäger et al. (2003); Dorschner et al. (1995); Zeidler et al. (2011); Drant et al. (2024). A number of species are also included in Palik's handbook of optical constants (Palik 2012).

Refractive indices as a function of wavelength for different species thought to be present in exoplanetary atmospheres have been compiled in various works, such as Lavvas & Koskinen (2017); Kitzmann & Heng (2017); Min et al. (2020); Lietzow & Wolf (2022). Kitzmann & Heng (2017) provide a rather extensive compilation of wavelength-dependent refractive indices (real and imaginary components; see their Table 1 for species and sources) which are stored at the as-

<footnote n="38">³⁸ https://www2.mpi-a-hd.mpg.de/HJPDOC/</footnote>

<footnote n="39">³⁹ https://www.astro.uni-jena.de/Laboratory/OCDB/</footnote>

<footnote n="40">⁴⁰ http://eodg.atm.ox.ac.uk/ARIA/</footnote>

<footnote n="41">⁴¹ https://hitran.org/aerosols/</footnote>

<footnote n="42">⁴² https://ocdb.smce.nasa.gov</footnote>

<footnote n="43">⁴³ https://refractiveindex.info/</footnote>