Lightmatter interactions lie at the heart of our exploration of exoplanetary atmospheres. Interpreting data obtained by remote sensing is enabled by meticulous time and resourceconsuming work aiming at deepening our understanding of such interactions i.e. opacity models. Recently citetNiraula2022 pointed out that due primarily to limitations on our modeling of broadening and farwing behaviors opacity models needed a timely update for exoplanet exploration in the JWST era and thus argued for a scalable approach. In this Letter we introduce an endtoend solution from ab initio calculations to pressure broadening and use the perturbation framework to identify the need for precision to a level of sim10. We focus on the CO2H2 system as CO2 presents a key absorption feature for exoplanet research primarily driven by the observation of gas giants at sim4.3mum and yet severely lack opacity data. We compute elastic and inelastic crosssections for the collision of orthoH2 with CO2 in the ground vibrational state and at the coupledchannel fully converged level. For scattering energies above sim20cm1 moderate precision intermolecular potentials are indistinguishable from high precision ones in crosssections. Our calculations agree with the currently available measurement within 7 i.e. well beyond the precision requirements. Our proofofconcept introduces a computationally affordable way to compute fulldimensional interaction potentials and scattering quantum dynamics with a precision sufficient to reduce the modellimited biases originating from the pressure broadening and thus support instrumentlimited science with JWST and future missions.
['Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Physics - Atomic Physics']
Ab initio quantum dynamics as a scalable solution to the exoplanet opacity challenge A case study of CO2 in hydrogen atmosphere
{'No Header': "Ab initio quantum dynamics as a scalable solution to the exoplanet opacity challenge: A case study of CO 2 in hydrogen atmosphere  \n<!-- image -->  \nLaurent Wiesenfeld , 1, 2 Prajwal Niraula , 2 Julien de Wit , 2 Nejmeddine Ja¨ıdane, 3, 1 Iouli E. Gordon , 4 and Robert J. Hargreaves 4  \n1 Universit'e Paris-Saclay, CNRS, Laboratoire Aim'e-Cotton, 91405 Orsay, France  \n2 Department of Earth, Atmospheric and Planetary Sciences, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 3 Universit'e Tunis El Manar, Faculty of Sciences, Tunis, Tunisia 4 Harvard-Smithsonian Center for Astrophysics, Atomic and Molecular Physics Division, Cambridge, MA, USA", 'ABSTRACT': 'Light-matter interactions lie at the heart of our exploration of exoplanetary atmospheres. Interpreting data obtained by remote sensing is enabled by meticulous, time- and resource-consuming work aiming at deepening our understanding of such interactions (i.e., opacity models). Recently, Niraula et al. (2022) pointed out that due primarily to limitations on our modeling of broadening and far-wing behaviors, opacity models needed a timely update for exoplanet exploration in the JWST era, and thus argued for a scalable approach. In this Letter, we introduce an end-to-end solution from ab initio calculations to pressure broadening, and use the perturbation framework to identify the need for precision to a level of ∼ 10%. We focus on the CO 2 -H 2 system as CO 2 presents a key absorption feature for exoplanet research (primarily driven by the observation of gas giants) at ∼ 4.3 µ m and yet severely lack opacity data. We compute elastic and inelastic cross-sections for the collision of ortho-H 2 with CO 2 , in the ground vibrational state, and at the coupled-channel fully converged level. For scattering energies above ∼ 20 cm -1 , moderate precision inter-molecular potentials are indistinguishable from high precision ones in cross-sections. Our calculations agree with the currently available measurement within 7%, i.e., well beyond the precision requirements. Our proof-of-concept introduces a computationally affordable way to compute full-dimensional interaction potentials and scattering quantum dynamics with a precision sufficient to reduce the model-limited biases originating from the pressure broadening and thus support instrument-limited science with JWST and future missions.  \nKeywords: Astronomy databases(83); Astronomy data analysis(1858); Spectral line lists(2082); Laboratory astrophysics(2004); James Webb Space Telescope(2291); Infrared spectroscopy(2285); Transmission spectroscopy(2133)', '1. INTRODUCTION': "Amongst the JWST molecular observations, carbon dioxide is almost universally observed in exoplanetary atmospheres. From hot Jupiters such as WASP-39 b (Rustamkulov et al. 2023; Niraula et al. 2023) to temperate sub-Neptunes such as K2-18 b (Madhusudhan et al. 2023), transmission spectra have shown the characteristic spectral feature at 4.3 microns. In fact, as this feature  \nEmail: laurent.wiesenfeld@universite-paris-saclay.fr  \nEmail: pniraula@mit.edu  \nfalls in a spectral sweet spot where the overall noise budget and the effect of cloud and/or hazes are optimal it is also ideal for the detection of temperate terrestrial atmospheres and assessing their (in)habitability (e.g., Triaud et al. 2023; TRAPPIST-1 JWST Community Initiative et al. 2024).  \nYet, despite these numerous favorable aspects, exploring exoplanet atmospheres using CO 2 can be challenging due to the current state of its opacity model. Niraula et al. (2022) (henceforth, N22) pointed out that the current limitations of opacity models (primarily due to broadening and far-wing parameterizations) can yield an accuracy wall on the inferred atmospheric properties at  \nFigure 1. Flowchart for exoplanet atmosphere retrieval focusing on broadening parameters, as a key source of limitations in current opacity models. Flowchart adapted from N22. Ab initio calculations are performed with the MOLPRO code (Werner et al. 2010), fits: Rist (Rist & Faure (2012) and YUMI for scattering and dynamics code (Ja¨ıdane et al. in prep.).  \n<!-- image -->  \nthe level of ∼ 1 dex, orders of magnitude above JWST's capabilities (see Figure 1 for general framework). Niraula et al. (2023) subsequently highlighted that the aforementioned 4.3 µ m feature of CO 2 , while dominant in most exoplanet spectra, can be challenging to translate into robust atmospheric inferences for a large range of exoplanets as these will harbor a hydrogen-dominated atmosphere, and very scarce data exists for broadening of CO 2 lines by collisions with H 2 , especially at the higher temperatures (Hanson & Whitty 2014; Padmanabhan et al. 2014). This is partly because the experiments on carbon dioxide and molecular hydrogen mixtures, especially at elevated temperatures, are very challenging from a safety perspective, adding extra complexity and expense to already existing difficulties in accurate measurements of broadening parameters. Due to a lack of data, the HITRAN spectroscopic database Gordon et al. (2022); Tan et al. (2022), for instance, uses a very approximate solution of scaling the well-known airbroadening data based on limited hydrogen-broadening data. Since the dependence of air and hydrogen broadening data on the type of the transition is very different, the errors associated with this approach can be quite large.  \nIn the light of the hurdles in providing a solution to the opacity challenge on a timescale relevant to the JWST mission, this Letter presents the viability of a new-generation framework for ab initio calculations with a focus on pressure broadening coefficients (typically defined as half-widths at half maximum (HWHM) Gordon et al. (2022)). Different theoretical methods for calculating the pressure broadening employing classical dynamics/semi-classical quantizing exist, with var-  \nious degrees of approximation (Hartmann et al. 2018; Ngo et al. 2021). For example, Buldyreva et al. (2024) have used classical phase-shift theory to provide some data/estimates for the UV/Visible range. There is also a large series of classical computations using very approximate potentials, but capable of computing the whole lineshape, including the far wings (Hartmann et al. 2002). Also, Guest et al. (2024) have investigated machine learning to provide broadening data based on HITRAN parameters. This is one avenue that would be considered state-of-the-art, but is limited by the learning set and will struggle to predict broadening for molecules/systems not included. In other words, while the ab initio is not the only way to address the opacity challenge using theoretical approaches, it is certainly the most accurate (see, for instance, Serov et al. (2021); Gancewski et al. (2021)) and, as shown here, can provide a cost-effective and thus scalable solution that can be deployed in a timescale relevant for ongoing missions such as JWST.  \nFirst, we assess the precision requirement on the pressure broadening calculations given JWST's capabilities in section 2. Then, in section 3, we describe our method for calculating the pressure broadening parameters for CO 2 +H 2 . Our results are presented in section 4. We discuss implications (incl., scalability) and conclude in section 5. The necessary background pertaining to the pressure broadening calculations is included in the Appendix B.", '2. PRECISION REQUIREMENTS': "In this section, we aim to provide a more granular understanding of the pressure-broadening contribution to  \nFigure 2. Perturbation analysis via injection-retrieval to assess the precision required on the broadening parameters in the JWST era. Top Left: Synthetic transmission spectrum of a warm-Jupiter spanning NIRSpec and MIRI bands used for the retrieval. The blue line represents the best-fit model obtained when cross-retrieved with the CSd3.00 cross-sections. Bottom Left: Deviation of the residual corresponding to the best fit presented on the upper panel. No structure is seen due to parameter compensation as highlighted in N22. Top Right: Posterior probability distributions for the mixing ratio of CO 2 for 15 different cross-sections showing increased levels of biases with increased perturbation. The dotted black line represents the 'truth' used for generating the synthetic spectrum. Overall variation of ∼ 1 dex is seen among the retrieved molecular abundance, consistent with N22. Bottom Right: Same for H 2 O. A similar variation of ∼ 1 dex overall, though leading to more significant biases due to the narrower distributions than for CO 2 .  \n<!-- image -->  \nthe uncertainty budget in exoplanet atmosphere analysis than N22 in order to set the precision requirement on the pressure broadening calculations. We similarly define that requirement to those introduced in Berardo et al. (2024); Rackham & de Wit (2024), namely such that the contribution of the model of interest (here the opacity model) to the overall uncertainty budget is equal or lower than the measurement's contribution (i.e., that we reach a regime of instrument-limited science).  \nTo this end, we perform cross-retrievals with perturbed cross-sections on the two synthetic planetary systems following the framework introduced in N22-one super-Earth (1.1 R ⊕ , 1.1 M ⊕ around a 0.1R ⊙ M-dwarf star) and a warm Jupiter (1.0 R Jup , 1.0 M Jup around a 0.55R ⊙ K-dwarf star). We use pandexo (Batalha et al. 2017) to model instrumental precision using the same setup as in N22. As we focus here on the influence of perturbations on pressure broadening coefficients to test the sensitivity of atmospheric inferences on that single parameter, we use a simpler synthetic atmosphere than in N22, considering only four molecular species (hydrogen, helium, carbon dioxide, and water). For the super-Earth scenario, we consider a high metallicity (MR H 2 O=0.1, MR CO 2 =0.1) isothermal (T=300 K) cloudless atmosphere, whereas for the warm-Jupiter scenario we con-  \nlow-metallicity atmosphere (MR H 2 O=2 × 10 -4 , MR CO 2 =1 × 10 -4 ) and slightly hotter cloudless atmosphere (T=400 K).  \nTo estimate the perturbation level leading to biases of amplitude equal to the 1σ confidence interval on atmospheric inference given by the instrumental precision, we generate an ensemble of perturbed cross-sections as denoted by CSxYYY where x is either m (for multiplied) or d (for divided) and YYY goes from 3 to 1. We generated fifteen relevant cross-sections: CSd3.00 (i.e., all the broadening parameters are one-third of those reported as nominal), CSd2.00, CSd1.50, CSd1.25, CSd1.10, CSd1.05, CSd1.01, CS1.00 (i.e., unperturbed), CSm1.01, CSm1.05, , CSm1.10, CSm1.25, CSm1.50, CSm2.00, and CSm3.00 (i.e., all the broadening parameters are three times those reported as nominal). As in N22 we use HITRAN (Gordon et al. 2022)/ HITEMP (Rothman et al. 2010) for our opacity model inputs. We then performed the injection-retrieval tests, generating the synthetic spectra using the unperturbed crosssection (CS1) and performing the retrieval with all crosssections. We ran our atmospheric models with emcee for a step-size of 5000, ensuring our walkers are fully converged.  \nFigure 2 shows the typical results derived here for the warm Jupiter scenario. As found in N22, even for large perturbations on the cross-sections, a good fit to the data can be found (no structure in the residuals) owing to compensation in the dimensions of the inverse problem associated with the atmospheric parameters. These compensations translate into biased inferences up to ∼ 1 dex, for example, on the abundance of absorbers such as carbon dioxide and water (right panels of Figure 2). While both water and carbon dioxide showcase biases up to ∼ 1 dex, the statistical significance of the biases for water appears more prominent due to the narrower posterior probability distribution, which results from the higher information content associated with water (primarily due to the wider wavelength range associated with its absorption features).  \n<!-- image -->  \n|  \n|  \nFigure 3. Greater than 5 σ biases observed among the retrieved water and carbon-dioxide abundances in our retrievals, with larger perturbations inducing stronger biases. The green region represents an acceptable bias range from -1 σ to +1 σ , demonstrating a need for precision of 10-15% in the pressure broadening parameter for JWST observations.  \nWe summarize our precision-requirement tests in Figure 3. As expected (and as shown in the right panels of Figure 2), it shows that the biases on the atmospheric inferences decrease with the perturbation level. The biases in Figure 3 are reported in significance level ( σ ), i.e., the ratio between the bias and the width of the distribution which is solely driven by the measurement uncertainty. It shows that biases become marginal (i.e., below 1 σ ) when the perturbation on the broadening parameters drops below ∼ 10%. In other words, our precision requirement for broadening parameters to ensure instrument-limited science in the JWST era (i.e., avoid model-limited inferences) is 10%.", '3. METHODS': 'With the well-justified precision requirement in hand, we can now turn to the series of calculations required to  \nderive ab initio estimates of the broadening parameters. In this section, we provide an overview of the steps. For further details, refer to the Appendix B.', '3.1. Potential Energy Surface': 'We begin by computing the electronic interaction energy of the two molecules taken both as rigid rotors, employing the ab initio MOLPRO code (Werner et al. 2015). We solve the electronic Schrodinger equation with fixed nuclei, using a quantum chemistry method (CCSD(T) that allows for a good description of the electronic correlations, even for two molecules separated by a sizeable distance (here, up to 25 ˚ A). The potential energy surface (PES) for the CO 2 -H 2 interaction was computed for some 500 relative orientations of the molecules and for 29 intermolecular distances ranging from approximately 1.75 ˚ A to 25 ˚ A. For shorter distances, the two molecules repel so strongly ( E interaction ≳ 10 , 000 cm -1 ) that the approximation of two isolated molecules does not hold. For distances larger than about 25 ˚ A, the interaction is weak ( E interaction ≲ 1 cm -1 ) and becomes neither reliable nor relevant for our computations. Note that because of the symmetry of both molecules ( D ∞ h ), the potential is π -periodic in the dihedral angle ϕ and π/ 2-periodic in both polar angles θ 1 and θ 2 (see Figure B1).  \nWe then fit the ab initio isolated points onto a functional form according to the geometrical conventions of the angles ϕ, θ 1 , θ 2 , see equation (A9) of (Green 1975) and Figure B1. We performed several fits of various precision, with 38 up to 158 terms, and found that an acceptable precision ( | V fit -V ab initio | / | V fit + V ab initio | ≤ a few %) is reached as soon as the orientation of the H 2 molecules is described with at least l 2 = 0 , 2 , 4 Y l 2 ,m 2 ( θ 2 , ϕ ) spherical harmonics. Ultimately, we used a 149 terms expansion, up to l 1 = 24 , l 2 = 6 terms in Y l 1 ,m 1 ( θ 1 , 0) Y l 2 ,m 2 ( θ 2 , ϕ ). 2-D cuts of the potential energy surface are presented in Figure 4, emphasizing the change of the shape of the PES as the orientation of H 2 changes.', '3.2. Dynamics: Broadening Coefficient, Collision Rates, etc.': "With the PES fits in hand, we can solve the quantum scattering dynamics in the time-independent domain, using the close-coupling formalism (Green 1975; Valiron et al. 2008). We use a home-written code (YUMI, Ja¨ıdane et al., in prep.), specifically designed for taking advantage of massively parallel computing. The code has been extensively tested against the existing stateof-the-art MOLSCAT code (Hutson & Le Sueur 2019), and will be described in detail in a forthcoming paper (Ja¨ıdane et al., in prep.).  \nFigure 4. Three 2-dimensional cuts of the 4-dimensional interaction potential CO 2 -H 2 . Planar configurations: panel (a) : θ 2 = 0 , ϕ = 0; panel (b) θ 2 = π/ 2 , ϕ = 0; non-planar configuration: panel (c) θ 2 = π/ 2 , ϕ = π/ 2. Blue lines represent positive values of the potential, red lines negative values, and the black lines zero values. Note that the potential is only asymptotically isotropic.  \n<!-- image -->  \nThe quantum computations of the pressurebroadening cross-section σ PB ( E ) and the rate coefficient γ ( T ) are performed in the approximation of isolated binary collisions. It allows for no overlap between the different spectral lines: These hypotheses remain valid up to about one amagat of density (about 10 19 particles/cm 3 ) and for P or R branches of the IR spectrum, not for the Q branches if any present. The isolated collision hypothesis allows for computing first the S ( E ) matrices of the collision and derive from these the desired cross sections, whether elastic, inelastic or pressure broadening and pressure shift (Green 1977; Drouin & Wiesenfeld 2012b; Faure et al. 2013; Selim et al. 2023). The cross sections are computed at given collision energies (the relative kinetic energy of the two molecules), then averaged over a Maxwellian distribution of kinetic energies to yield the relevant rates.  \nFor this proof-of-concept application, we focus on reproducing the few experimentally known values, measured at different temperatures (Hanson & Whitty 2014; Padmanabhan et al. 2014), which connects an asymmetric stretch excited state with the ground state. The Hanson & Whitty (2014) measurement targets P(24) transition of the 20012-00001 band, whereas Padmanabhan et al. (2014) covers P(16)-P(34) transitions of the 3001200001 band. (Spectroscopic notations, see section A). Based on selection rules, the only allowed transitions are v ' 3 = 1 , j ' 1 odd ↔ v '' 3 = 0 , j '' 1 = j ' 1 ± 1. The nuclear spin statistics dictates that only even rotational levels in the ground vibrational state are populated because 12 C 16 O 2 is a symmetric molecule with zero nuclear spin values for all atoms, hence exist only for even j 1 in the ground state and odd j 1 in the v 3 = 1 case. The v 1 excitation does not interfere with symmetry rules, merely acting as a spectator.  \nThe experimental values for the P24 ( m = -24), in cm -1 /atm, are γ = 0 . 112, for (Hanson & Whitty 2014) and γ = 0 . 113 ± 0 . 006, for (Padmanabhan et al. 2014), emphasizing the weak dependence on the exact vibra-  \nnal transition. Errors bars for Hanson & Whitty (2014) are of the order of a few %.  \nIt has recently been shown (Wiesenfeld 2022; Yang et al. 2021) that computing inelastic scattering crosssections between two different ro-vibrational states is doable, but requires an extremely large computational resources, which would render the scalability of our approach questionable. Fortunately, the pressure broadening effect arises through quasi-elastic collision amplitudes, connecting the j ' 2 , v ' 3 or j '' 2 , v '' 3 levels with themselves, but for different orbital angular momenta, see section B.4. The experimental consequence is a very weak dependence of the pressure broadening with the molecular vibrations, except for some very anharmonic vibrations of modes containing atom(s) of hydrogen. The theoretical consequence is that we are allowed to consider the interaction potential and the dynamics as depending only on the average geometry of the molecules -rigid rotors approximation-, and not on the vibrational dynamics, making the whole computation scalable. If selection rule allows (like for CO or HCN molecules, of symmetry C ∞ h ), one simply considers the pure rotational transitions as a proxy for the actual IR transition (see e.g. Dumouchel et al. (2010); Green (1989); Luo et al. (2001)). The case of D ∞ h includes selection rules different for rotational transition, similar to Raman S, Q, T branches, and different from IR P, R branches. The same approximations are made, without taking into account the selection rules of the transition (Thibault et al. 2011), or taking into account these rules, but not the full dynamics (this work). Calculations by Rosenmann et al. (1988) show that there is practically no difference between the broadening of IR and Raman transitions of CO 2 for several different projectile molecules, and they also did not observe the vibrational dependence. In general, in examining experimental data for linear molecules in the HITRAN database (Gordon et al. 2022; Tan et al. 2022; Hashemi et al. 2020), weak or no dependence on vibrational modes is typically observed for linear molecules.  \nFigure 5. Theoretical pressure broadening coefficient γ ( T ), for the m = -24 (P(24)) transition. The results for collisions with ortho -H 2 , j 2 = 1 is shown in black, for para -H 2 , j 2 = 0 in red. (a) Experimental point and temperature dependence from Hanson & Whitty (2014); these experiments were performed from 300K to 640K. (b) Experimental point from (Padmanabhan et al. 2014).  \n<!-- image -->", '4. RESULTS': 'As previously mentioned, our goal via this proof-ofconcept of a scalable ab initio framework for broadeningparameter generation is to reproduce the only experimentally known values (Hanson & Whitty 2014; Padmanabhan et al. 2014), measured at different temperatures.  \nThe cross-sections σ ( E coll. ; m = -24) (elastic, inelastic, and pressure broadening, see appendix) were computed for about 100 collision energies, ranging from 10 to 1,500 cm -1 .  \nWe present our finding in Figure 5, which shows that the precision requirements are achieved. For 296 K, the theory ( γ theo = 0 . 120cm -1 / atm . ) slightly overshoots the experimental values, γ exp. = 0 . 112 or 0 . 113cm -1 /atm). The difference (6 . 9%) is better than the precision requirement defined in section 2.', '5.1. Performance Tests, Caveats and Previous State-of-the-art': 'Computations of the pressure broadening coefficients for these transitions are performed through a series of codes that enables a full quantum dynamical approach to the problem, with no adjustable free parameter (details in Appendix B). While the underlying theoretical framework was presented decades ago (e.g., Baranger 1958; Peach 1981; Green 1977), it remains used to provide accurate calculations today (Hutson &  \nLe Sueur (2019), YUMI code description, Ja¨ıdane et al., in prep.). Indeed, while we assessed that a 10% of precision in the line broadening parameters is required for state-of-the-art applications in exoplanetary sciences, we found that for, the transition computed, the agreement with existing measurements well within this requirement ( ∼ 7%). While our calculations do not reach the accuracy achieved, for instance, in the Torun group Serov et al. (2021); Gancewski et al. (2021) where the non-Voigt parameters are also computed, they are much faster (i.e., scalable), while maintaining degree of accuracy required by targeted application.  \nYet, a few remarks are in order. The weak dependence of our result on any parameter -but for the convergence of the quantum base, which remains paramount- concerns this particular transition, where the σ PB ( E ) function displays practically no structure nor dependence of the para or ortho character of H 2 . This is plausibly because of the occurrence of large quantum numbers, dramatically increasing the number of channels, reducing quantum interferences, and a semi-classical behavior of all the angular algebra.  \nThe complete ab initio computation of pressure broadening coefficients has been undertaken, mostly with atomic colliders (Thibault et al. 2000; Godard Palluet et al. 2022; Serov et al. 2021) but also molecules (Thibault et al. 2011; Wiesenfeld & Faure 2013). All these ab initio calculations only require physical inputs for the experimental geometry of the target and projectile molecules and their experimental rotational constants, possibly from ground and vibrational excited states. No attempts have been made to explicitly include vibrational motions.', '5.2. Scalability': "A fundamental advantage of the solution to the exoplanet opacity challenge presented here relates to its scalability. YUMI , our FORTRAN-based quantum scattering code, takes advantage of the CPUarchitectures, in particular, the extensive parallelism and memory hierarchies. The code has been optimized using an OpenMP framework, and streamlined for simultaneous multiple jobs submission leveraging toolkits such as LLTriple (Byun et al. 2015) developed for handling thousands of jobs in parallel. Our code performed at ∼ 30 GigaFlops during the speed benchmark test performed in a MIT SuperCloud node, which consists of Intel(R) Xeon(R) Platinum 8260 CPU@2.40GHz with 48 cores. The current speed bottleneck is the matrix inversion, which involves a recursive algorithm implemented from Ingemarsson & Gustafsson (2015), built on top of Intel's MKL library. Our algorithm scales at  \nO ( N ∼ 2 . 6 ), i.e. smaller than 3, with N , the matrix size. For CO 2 -H 2 , the typical matrix size is ∼ 1000, and the typical calculation takes less than ∼ 1 hour for a single transition at a particular energy. For a single transition, such calculations needs to be performed roughly at 100 different energies, and for more complex molecules such as water, matrix sizes reach ∼ 10,000. The related calculations would thus take ∼ 400 times more computational time (i.e. ∼ 2 weeks), significantly longer but still within our computational reach with the aforementioned supercomputers. Leveraging the symmetry within the molecule, GPUs, and possibly machine-learning techniques in future applications will further help us enhance the scalability of our approach.", '5.3. Future Prospects': 'The results presented in this Letter provide a proof-ofconcept for a scalable solution to a key limitation behind the exoplanet opacity challenge, namely the modeling of broadening. While our application focuses on the datascarce CO 2 -H 2 collisional system due to its importance in most of the exoplanet spectra acquired with JWST and the scarcity of data available for the system, it can now be applied to the hundreds of other collisional systems of relevance for JWST and future missions.  \nWhile the CO 2 -H 2 system could be investigated with sufficient funding and careful experimental works on an adequate timescale ( ∼ 1 yr), it is just one collisional system out of thousands of molecules of relevance. Unfortunately, the measurements of broadening parameters with sufficient accuracy are challenging, and often rendered impractically complex and expensive due to safety issues (as a reminder, the primary targets of exoplanetary sciences remain hydrogen-rich high-temperature worlds). Therefore, the methodology presented here provides a timely and realistic approach to fulfilling the opacity needs of the exoplanet community, and helps pave the way toward robust atmospheric inferences reaching accuracy driven by the instrument itself (rather than opacity models).  \nSoftware: emcee (Foreman-Mackey et al. 2013), tierra (Niraula et al. 2022), MOLPRO (Werner et al. 2015), Matlab', 'ACKNOWLEDGEMENTS': "JdW and LW thank the MIT Global Seed Funds and MIT-France for their support towards this study. NJ thanks Universit'e Paris-Saclay for a visiting fellowship. IG and RH acknowledge NASA PDAR grant 80NSSC24K0080. LW and NJ thank Olivier Dulieu for fruitful discussions. LW, PN, and JdW thank Jeremy Kepner, Deborah Woods, and Cooper Loughlin for their  \ninsights regarding optimal deployment on supercomputers, scalability, etc. which will be pivotal in the next steps of this project. Part of these computations were performed on the Jean-Zay IDRIS-CNRS supercomputer, contract A0140810769. Other parts were computed on the MIT-Supercloud/Lincoln-MIT computers. The authors acknowledge IDRIS and the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing resources (HPC, software expertise, database, consultation) that have contributed to the research results reported here.", 'A. NOTATIONS': "We used in this work the following conventions and notations:  \nQuantum numbers and spectroscopy: To avoid any ambiguity between commonly used notation for angular momenta, we use the following notation: j 1 , j 2 , angular momentum resp. of the CO 2 or H 2 molecule; l , orbital angular momentum of the collision; J , total angular momentum of the collision (a conserved quantity for a collision). The running index m is defined as follows m = -j '' 1 for P branch, m = + j '' 1 for he Q branch, and m = + j '' 1 +1 for the R branch. Therefore for the transition studied here m =-24 corresponds to P(24). Primed and double primed quantities are lower and and upper levels, respectively. If one looks at the HITRAN file for carbon dioxide they will see branch designation and the j 1 ' value for unique identification.  \nVibrational levels of CO 2 are given (in HITRAN notation) by [ v 1 v 2 l 2 v 3 q ], where v i are the three normal modes, i.e. symmetric stretch, bend, and asymmetric stretch, l 2 is the rotational vibration quantum number and q labels the levels of the same polyad.  \nEnergy: Energies are defined as follows: (i) E is the total energy of the collision, (ii) collision energy E coll. = E -E rotation (CO 2 ) -E rotation (H 2 ).", 'B. QUANTUM DYNAMICS': "Our aim is to compute pressure broadening coefficients for the 4.3 µ transition of 12 C 16 O 2 , that is the IR lines that correspond to v '' 3 = 1 , j '' 1 = j ' 1 ± 1 ← v ' 3 = 0 , j ' 1 , broadened by collisions with H 2 . We aim at a full quantum ab initio scheme, based on the close coupling (CC) formalism (Arthurs & Dalgarno 1960; Blatt & Biedenharn 1952; Baranger 1958). In order to do so, one needs successively: (i) a series of ab initio potential energy points V ( R n ) describing the interaction of CO 2 with H 2 ( R , the 4 coordinates setting the geometry of CO 2 with respect to H 2 ); (ii) a fit of these V ( R n ) points onto a suitable functional form valid ; (iii) solving the quantum dynamical equation of the collision, in order to get the S -matrix connecting the entrance and outgoing channels of the collision (usually, solving the Schrodinger equation); (iv) from the knowledge of the S -matrix, constructing the pressure broadening cross-sections and rate coefficients. This section describes the details of all four steps.", 'B.1. Potential Energy ab initio points': "We compute the pressure broadening by H 2 of Infra-Red transitions of CO 2 . Hence, it would seem sensible to allow for CO 2 vibrational modes to be included in the Potential Energy Surface (PES). However, vibrational motion influences very weakly the pressure broadening coefficients (Gordon et al. 2022; Tan et al. 2022), at least in the temperature regimes that we aim at, 50 ≤ T ≤ 1000 K. Note that the same does not seem to be true for line shifts. We are thus entitled to restrain our computations to rigid body interaction. We use the ground state vibrational averaged geometries for CO 2 and H 2 , with r (CO) = 2 . 1944 bohr (Godard Palluet et al. 2022) and r (HH) = 1 . 44836 bohr (Sahnoun et al. 2020). The PES has 4 degrees of freedom, namely, the R distance between the center of masses of the two molecules, the θ 1 and θ 2 angles that describe the orientation of the two linear molecules with the intermolecular axis and ϕ , the dihedral angle, see figure B1.  \nIn the vibrational ground state, both molecules display D ∞ h symmetry; the ground state intermolecular PES must reflect these symmetries. With the geometry defined in figure B1, the whole PES is described by 0 ≤ θ 1 < π , 0 ≤ θ 2 < π/ 2, and 0 ≤ ϕ < π/ 2.  \nAs the atoms are relatively light, and as we have to deal with 22 electrons only (among those, 6 core electrons), it is possible to compute the PES in multiple ways, of various precision. This allows us to study the influence of the PES precision on the observables, elastic and inelastic cross sections, and pressure broadening rate coefficients in particular.  \nB.1.1. Ab initio quantum chemical computations  \nThe interaction energy is computed via the super-molecule approach, including systematically the basis set superposition errors (BSSE, (Kodrycka & Patkowski 2019), see also our previous works (Khalifa et al. 2019; Sahnoun et al.  \nFigure B1. The 4 degrees of freedom ( R , θ 1 , θ 2 , ϕ ) of the CO 2 -H 2 complex, in the rigid bodies approximation.  \n<!-- image -->  \n2018) for a detailed description). All computations are performed at the CCSD(T) level, with the 2020 version of the MOLPRO code Werner et al. (2015). The interaction energy is defined as:  \nE AB interaction = E AB Dimer AB -E AB Monomer A -E AB Monomer B , (B1)  \nwere the superscript AB denotes the full molecular basis A ⊕ B , A ≡ CO 2 and B ≡ H 2 .  \nIn order to evaluate the precision needed for computing elastic and inelastic cross section for the collision CO 2 - H 2 , we used four precision levels for computing the interaction (all described in the documentation of Molpro, (Werner et al. 2015): (1) basis set aug-cc-pVTZ (denoted in the following by avtz ), alongside with a bonding function center (Shaw & Hill 2018) at 2/3 of the distance between the center of mass of H 2 and the nearest atom of the CO 2 molecule (The PES depends very weakly to the actual position of the bonding function, as long as it is not too nearby any actual atom). (2) The same formalism but with a basis set aug-cc-pVQZ (denoted in the following by avqz ). (3) An extrapolation to the complete basis set, with the avtz and avqz values at each ab initio point, and an extrapolation formula (Varandas 2021):  \nV cbs = 3 3 · V avtz -4 3 · V avqz 3 3 -4 3 (B2)  \nThe ab initio PES obtained via formula (B2) is called hereafter avcbs .  \nThe last formalism (4) is the so-called 'gold standard', widely used for computing intermolecular interaction, the CCSD(T)-F12a formalism (Kodrycka & Patkowski 2019). We used exclusively the avqz basis for this computation (even if avtz could yield very similar results). Using a bonding function proved unreliable and was not used. We noted the well-known problem of size inconsistency of the explicitly correlated F12 family of methods. We duly subtracted the value of the interaction at infinity (here, 50 bohr), which was isotropic within less than 1%.  \nSome points (see below) were computed at the aug-cc-pV5Z level ( av5z ), in order to have some hints of the convergence of the PES, on the one hand, and to see if the CCSD(T)-F12 methods are reliable, especially so for overshooting the minimum of the PES. Results around the different stationary points are shown in table B1. As briefly explained beforehand, convergence at the wave-number scale is essentially achieved for av5z and the more economical avcbs . avqz method with the F12 acceleration slightly overshoots the values, and avtz method is clearly less precise. However, all methods agree one with another within 4-5% and with the earlier computation of Li et al. (2010).", 'B.2. Geometries and fit': "While some formalism allows for computing the PES with a limited number of points (Ben Khalifa et al. 2020; Ajili et al. 2022), we decided to compute the PES on a dense grid of ab initio points, like we did repeatedly in previous cases, like Sahnoun et al. (2020); Valiron et al. (2008); Rist & Faure (2012); Khalifa et al. (2020). To summarize, we compute ab-initio points for 500 sets angles { θ 1 , θ 2 , ϕ } per distance R . These angles are randomly chosen with 0 ≤ θ 1 , 2 < π and 0 ≤ ϕ < 2 π . In doing so, we do not impose the C ∞ h symmetries of H 2 and CO 2 . The subsequent fit of the potential should reveal the symmetries (see below). We computed the potential for 29 distances, between 4.20 bohr and 50.00 bohr. Note that the random set of angles is the same for each distance, facilitating the fit.  \nWe fit the PES distance by distance, with the following formula adapted from Green (1975).  \nV ( R,θ 1 , θ 2 , ϕ ) = ∑ l 1 ,l 2 ,L V l 1 ,l 2 ,L ( R ) X l 1 ,l 2 ,L ( θ 1 , θ 2 , ϕ ) (B3)  \nTable B1. Minimum V min of the CO 2 -H 2 potential energy surface, at intermolecular distance R min . The geometry is always θ 1 = 90 · , θ 2 = 90 · and ϕ = 0 · . The partial av5z computations are included. Not that our computations are at average ground state geometries and Li et al. (2010) are at equilibrium geometries.  \nwith the angular functions X equal to :  \nX l 1 ,l 2 ,L ( θ 1 , θ 2 , ϕ ) = 2 (4 π ) 3 / 2 ( -1) ( l 1 -l 2 ) {( l 1 l 2 L 0 0 0 ) P l 1 0 ( θ 1 ) P l 2 0 ( θ 2 ) + min( l 1 ,l 2 ) ∑ m =1 2( -1) m [( l 1 l 2 L m -m 0 ) P l 1 m ( θ 1 ) P l 2 m ( θ 2 ) cos( mϕ ) ]} (B4)  \nwith ( l 1 + l 2 + L ) even and ( ... ... ) are Wigner 3j symbols.The normalization is:  \n∫ π 0 P lm ( θ ) P l ' m ( θ ) sin θ d θ = δ ll ' (B5)  \nWe took great care to have the same expansions and normalizations of the angular functions P lm ( θ ) in this expansion and in the scattering code. To avoid any confusion, the explicit relations between the P lm ( θ ) functions, the associated Legendre functions, and the spherical harmonics are given in Zare (1988), section 1.3.  \nWe tried three bases for the fit, indexed by the the maximum allowed values of the l 1 , l 2 and L indices in equation (B3), the so called :  \n- (i) precise expansion, with l 1 ≤ 24, l 2 ≤ 6, and L ≤ 26, resulting in 158 terms in the expansion (B3). Because of limited relevance, not all terms allowed by angular algebra are included.\n- (ii) intermediate expansion, with l 1 ≤ 20, l 2 ≤ 4, and L ≤ 22, resulting in 76 terms.\n- (iii) imprecise expansion with l 1 ≤ 20, l 2 ≤ 2, and L ≤ 22, resulting in 38 terms.  \nIn all cases, only even values of angular momenta are allowed, because of the symmetries of the molecules.", 'B.2.1. Potential surface': 'We computed all 12 combinations { basis set } × { fit precision } , even if some combinations make little sense. For the precise fit, comparison of some stationary points of the V ( R,θ 1 , θ 2 , ϕ ) potential functions are given in Table B1. Figure B2 depicts errors in the fit for the three levels.', 'B.3. Dynamics': 'In order to solve for the S - matrix characterizing the scattering, we resorted to the well-documented time-independent close-coupling (CC) formalism (see e.g. Arthurs & Dalgarno (1960); Hutson & Le Sueur (2019); Alexander et al. (2023)). In order to accelerate computations and take advantage of the large memory available for each computing core, as well as of the massive now possible on modern supercomputers, we used a new home-written CC code. The main differences between the widely available MOLSCAT code and the present one are the native use of OpenMP capabilities - readily extensible to GPU-based OpenACC capabilities 1 , as well as memory management capable to take full advantage of the hierarchy of RAM in the cores and the nodes. A full description of the code will be available in a future publication.  \nFigure B2. Different fits of the ab initio potential. The angle averaged absolute value potential, thick black line. Relative fit errors in red, scale on the right. Precise fit, filled upper triangles; intermediate fit, open diamonds; imprecise fit, empty lower triangles.  \n<!-- image -->', 'B.3.1. Cross sections': 'Once S or T matrices are available, it is simple to sum up all relevant contributions and get the elastic/inelastic cross sections at a given collision energy, σ ij ( E coll ), with i ≷ j . These computations are much more sensitive to the details  \nFigure B3. Comparison between two ab initio bases, the less precise ( avtz basis), continuous line, and the most precise ( avcbs ), diamonds. Large fit, with 156 terms. Elastic cross sections for the collision CO 2 -ortho H 2 . j 1 is the CO 2 rotational quantum number, and v 3 = 0.  \n<!-- image -->  \nof the PES than their pressure broadening counterpart, as the next section will show. An example of the sensitivity to PES precision is given in figure B3.', 'B.4. Pressure Broadening computation': "The knowledge of the S or T matrices allows for the computing of the collision energy dependent pressure broadening σ j '' 1 v '' b ← j ' 1 v ' b ( E PB ). The formalism dates from (Baranger 1958) and (Schaefer & Monchick 1987).  \nThe full analytical formula, eq.(1) of (Schaefer & Monchick 1987) was used in order to compute σ PB ( E coll ), for the transition v '' 3 = 0 , j '' 1 = 24 ← v ' 3 = 1 , j ' 1 = 25, at about 100 different collision energies E coll . The degeneracy factor (2 j ' 2 +1) was taken into account, see remarks in Green (1975); Wiesenfeld & Faure (2013). At each energy, 4 events were computed: collision with H 2 , j 2 = 0, j ' 1 = 24 or 25, and the same with H 2 , j 2 = 1. Results are presented subsection B.4. The remarkable smoothness of the results contrasts with the multiple resonances observed in elastic cross sections, like figure B3.  \nIt must be underlined, though, that neither the smoothness of results nor the similarity of ortho - and para -H 2 pressure broadening cross sections need to remain true for all regimes. Because of the presence of high angular momentum for CO 2 , the collision studied here has a marked semi-classical character, witnessed by the behavior of 3 -j and 6 -j symbols. The behavior at low j 1 is expected to be much more quantum-like, as was observed repeatedly, for quadrupole dominated-collision e.g. Thibault et al. (2011), and even more so, for dipole-dominated collision Mengel et al. (2000); Drouin & Wiesenfeld (2012). From the knowledge of the σ PB ( E coll ), sections, we define the Maxwell-averaged cross-  \nFigure B4. Pressure broadening cross-sections for the collision CO 2 -H 2 , σPB ( E coll ). Ortho and para H 2 yield very similar results, except at very low energies (see inset).  \n<!-- image -->  \n[h!]  \nsection and pressure broadening coefficient in the usual manner:  \nσ PB ( T ) = 1 ( k B T ) 2 ∫ ∞ 0 σ PB ( E coll ) E coll e ( -E coll /k B T ) dE coll. (B6)  \nγ ( T ) = ¯ vσ PB ( T ) 2 π k B T (B7)  \nwhere the kinetic average relative speed is ¯ v = √ 8 k B T/π M , M being the reduced mass of the collisional system.", 'B.4.1. Precision analysis': "Given the acceptable error for retrievals (of the order of ± 10 %), we found it useful to compare the results shown in figures B.4 and 5 with alternative methods, more economical in computer time.  \nThe first test we made was to apply the Random-Phase Approximation (RPA) to the σ PB ( E coll ) computations, neglecting all interferences terms between the different channels (Baranger 1958; Drouin & Wiesenfeld 2012a; Faure et al. 2013). It is known that this approximation (see e.g. the H 2 O-H 2 case, (Faure et al. 2013) ) is valid only for collision energies E coll ≫ | V min | , the minimum value of the potential. For the deep potential we have here ( V min ≃ -220 cm -1 , see section B1), this approximation should not be not valid, and was indeed tested as being way too imprecise.  \nThe second series of tests tried to disentangle the necessity of precise quantum mechanical methods for determining the potentials and precise fits of the potential points. In order to shorten computation, we only computed σ P B ( E coll ), for E coll = 200cm -1 (about 300 K). Precision study is not expected to change significantly form a full-fledged γ ( T ) computation, except for low temperature results, which are much more potentially dependent, because of the occurrence of scattering resonances (Drouin & Wiesenfeld 2012; Bergeat et al. 2020)(see figure B3). Remarkably, the σ P B ( E coll ) value does not change at the 0.1% level of precision for the different methods of computing the PES ( avtz up to avcbs ).  \nChanges in the fitting procedure precision do have a limited influence on the results, of at most 2%. Fits with very few coefficients may sometimes make the dynamics diverge, and lead to nonphysical results; a compromise is necessary.  \nHowever, it must be underlined that pressure shifts, inelastic and elastic cross sections are much more sensitive to the details of the PES and fit. Also, there is now no indication if smaller values of | m | behave the same way (Thibault et al. 2011). Consequently, all data aside from pressure broadening should be carefully analyzed for convergence, and influence of the vibrational motion. Recall also that the inelastic cross sections averaged over the Maxwellian distribution of collision energy yield the population transfer rate coefficients, Arthurs & Dalgarno (1960); Roueff & Lique (2013), useful for level population determination.  \nThe relative insensitivity of σ PB ( E coll ) (hence of σ PB ( T )) to the details of the quantum computation comforts the idea of scalability of this computational scheme, thanks to the possibility of using properly tailored computations. It may even be envisioned, for lower quality results, to resort to approximate quantum scattering dynamical methods, based on Born first or higher order methods, and resorting to specific tests to validate approaches. 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