Accurate reference spectra of HD in an H 2 –He bath for planetary applications

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Introduction
Hydrogen deuteride (HD), the second most abundant isotopologue of molecular hydrogen, is an important tracer of deuterium in the Universe.The small and constant primordial fraction of deuterium to hydrogen, D/H ((2.8 ± 0.2) × 10 −5 , Pettini et al. (2008)), is one of the key arguments supporting the Big Bang theory.Measurements of the D/H ratio in the Solar System provide information about planetary formation and evolution.The standard ratio on Earth is considered to be Vienna Standard Mean Ocean Water (VSMOW; D/H = 1.5576 × 10 −4 , Araguás-Araguás et al. (1998)).With that Donahue et al. (1982) reported the D/H ratio in Venusian atmosphere to be (1.6±0.2)×10−2 (i.e. two orders of magnitude higher than VSMOW).This higher ratio is attributed to the evaporation of oceans and subsequent photodissociation of H 2 O in the upper parts of the atmosphere (Donahue & Pollack 1983).Comparison of the D/H ratio on Earth and Mars with the values determined for various comets indicate the role of the latter in the volatile accretion on these two planets (Drake & Righter 2002;Hartogh et al. 2011).The two largest gas giants, Jupiter and Saturn, are incapable of nuclear fusion, thus planetary models predict that the D/H ratio in their atmospheres should be close to the primordial value for the Solar System (Lellouch et al. 2001) or slightly larger, due to the accretion of deuterium-rich icy grains and planetesimals (Guillot 1999).The abundance of deuterium is larger by a factor of 2.5 in the atmosphere of Uranus and Neptune (Feuchtgruber et al. 2013), owing to their ice-rich interior (Guillot 1999).Additionally, the determination of the D/H ratio in comets and moons provides information about the formation of ices in the early Solar System (Hersant et al. 2001;Gautier & Hersant 2005;Horner et al. 2006).
The D/H ratio in planets, moons, or comets can be derived from either in situ mass spectrometry (Eberhardt et al. 1995;Mahaffy et al. 1998;Niemann et al. 2005;Altwegg et al. 2015) or spectroscopic observations of various molecules and their deuterated isotopologues in millimeter and infrared range (Bockelée-Morvan et al. 1998;Meier et al. 1998;Crovisier et al. 2004;Fletcher et al. 2009;Pierel et al. 2017; Krasnopolsky we use the scattering S-matrices obtained in this way to determine the collision-induced line-shape parameters.We use the cavity ring-down spectroscopy for accurate validation of our theoretical methodology, demonstrating sub-percent-level accuracy that considerably surpasses the accuracy of any previous theoretical (Schäfer & Monchick 1992) and experimental studies of line-shape parameters of pure rotational lines in HD (Ulivi et al. 1989;Lu et al. 1993;Sung et al. 2022), and offers valuable input for the HITRAN database (Gordon et al. 2022).This work presents significant methodological and computational progress; calculations at this level of theory and accurate experimental validation have been formerly performed for a molecule-atom system (3D potential energy surface) (Słowiński et al. 2020;Słowiński et al. 2022), while in this work, we extend it to a molecule-molecule system (6D potential energy surface).All the reported parameters (and their temperature dependences) are consistent with the HITRAN database format, hence allowing for the use of HAPI (HITRAN Application Programming Interface) (Kochanov et al. 2016) for generating the beyond-Voigt spectra of HD for any H 2 /He perturbing gas composition and thermodynamic conditions.

Ab initio calculations of the line-shape parameters
In recent years, the methodology for accurate ab initio calculations of the line-shape parameters (including the beyond-Voigt parameters) was developed and experimentally tested for Heperturbed H 2 and HD rovibrational lines, starting from accurate ab initio H 2 -He potential energy surface calculations (Bakr et al. 2013;Thibault et al. 2017), through the state-of-the-art quantum scattering calculations and line-shape parameter determination (Thibault et al. 2017;Jóźwiak et al. 2018), up to accurate experimental validation (Słowiński et al. 2020;Słowiński et al. 2022) and using the results for populating the HITRAN database (Wcisło et al. 2021;Stankiewicz et al. 2021).
In this work, we extend the entire methodology to a much more complex system of a diatomic molecule colliding with another diatomic molecule.First, we calculated an accurate 6D H 2 -H 2 PES.Second, we performed state-of-the-art quantumscattering calculations.Third, we calculated the full set of the six line-shape parameters in a wide temperature range.Finally, we report the results in a format consistent with the HITRAN database.
The 6D PES was obtained using the supermolecular approach based on the level of theory similar to that used to calculate the 4D H 2 -H 2 surface (Patkowski et al. 2008).The crucial contributions involve interaction energy calculated at the Hartree-Fock (HF) level, the correlation contribution to the interaction energy calculated using the coupled-cluster method with up to perturbative triple excitations, CCSD(T) (with the results extrapolated to the complete basis set (CBS) limit (Halkier et al. 1999)), electron correlation effects beyond CCSD(T) up to full configuration interaction (FCI), and the diagonal Born-Oppenheimer correction (DBOC) (Handy et al. 1986).The details regarding the basis sets used in calculations of each contribution and the analytical fit to the interaction energies are given in Appendix A. The PES is expected to be valid for intramolecular distances r i ∈ [0.85, 2.25] a 0 .
For the purpose of performing quantum scattering calculations, the 6D PES is expanded over a set of appropriate angular functions and the resulting 3D numerical function in radial coordinates is then expanded in terms of rovibrational wave functions of isolated molecules, see Appendix B for details.The closecoupling equations are solved in the body-fixed frame using a renormalized Numerov's algorithm, for the total number of 3 014 energies (E T = E kin + E v 1 j 1 + E v 2 j 2 , where E kin is the relative kinetic energy of the colliding pair, and E v 1 j 1 and E v 2 j 2 are the rovibrational energies of the two molecules at large separations) in a range from 10 −3 cm −1 to 4000 cm −1 .Calculations are performed using the quantum scattering code from the BIGOS package developed in our group.The scattering S-matrix elements are obtained from boundary conditions imposed on the radial scattering function.Convergence of the calculated S-matrix elements is ensured by a proper choice of the integration range, propagator step, and the size of the rovibrational basis, see Appendix C for details.
Next, we calculate the generalized spectroscopic crosssections, σ q λ (Monchick & Hunter 1986;Schäfer & Monchick 1992), which describe how collisions perturb the shape of molecular resonance.Contrary to the state-to-state cross-sections, which give the rate coefficients (see, for instance, Wan et al. (2019)), the σ q λ cross-sections are complex.For λ = 0, real and imaginary parts of this cross-section correspond to the pressure broadening and shift cross-section, respectively.For λ = 1, the complex cross-section describes the collisional perturbation of the translational motion and is crucial for the proper description of the Dicke effect.The index q is the tensor rank of the spectral transition operator and equals 1 for electric dipole lines considered here.
We use the σ 1 0 and σ 1 1 cross-sections to calculate the six lineshape parameters relevant to collision-perturbed HD spectra, the collisional broadening and shift the speed dependences of collisional broadening and shift and the real and imaginary parts of the complex Dicke parameter where v r is the relative (absorber to perturber) speed of the colliding molecules, ⟨v r ⟩ is its mean value at temperature T , vp is the most probable speed of the perturbed distribution, M a = m a m a +m p , m a and m a and m p are the masses of the active and perturbing molecules, respectively (Wcisło et al. (2021)).We estimate the uncertainty of the calculated line-shape parameters in Appendix C. The six line-shape parameters define the modified Hartmann-Tran (mHT) profile (see Appendix D), which encapsulates the relevant beyond-Voigt effects.To make the outcome of this work consistent with the HI-TRAN database (Gordon et al. (2022)), we provide temperature dependences of the calculated line-shape parameters within the double-power-law (DPL) format (Gamache & Vispoel (2018); Stolarczyk et al. (2020)): where T ref = 296 K.
3. Results: line-shape parameters for the R(0), R(1) and R(2) lines in HD perturbed by a mixture of H 2 and He In Figure 1, we show the main result of this work, i.e., all the six line-shape parameters for the R(0), R(1), and R(2) 0-0 lines in H 2 -perturbed HD calculated as a function of temperature;  2017)), for our full temperature range, 20 to 1000 K, see the Supplementary Material.In Figure 1, we also recall the corresponding He-perturbed data calculated with the same methodology at the same accuracy level (Stankiewicz et al. (2021)).The difference between the two perturbers is not negligible, and for many cases the line-shape parameters differ by a factor of 2 or even more.The data shown in Fig. 1 are given in Table 1 in a numerical form within the HITRAN DPL format, see Eqs. ( 4).The accuracy of our ab initio line-shape parameters is within 1% of the magnitude of each parameter (see Appendix C for details).The DPL approximation of the temperature dependences introduces additional errors.For γ 0 (T ) and νr opt (T ), the DPL error is negligible, for δ 0 (T ) the DPL error is at the 1 % level, and for other line-shape parameters it can be even higher, but their impact on the final line profile is much smaller, see Appendix E for details.For applications that require the full accuracy of our ab initio line-shape parameters, we provide the line-shape parameter values explicitly on a dense temperature grid in the Supplementary Information.
The set of parameters in Table 1 contains all the information necessary to simulate the collision-perturbed shapes of the three HD lines at a high level of accuracy at any conditions relevant to the atmospheres of giant planets (pressure, temperature, and He/H 2 relative concentration).In Figure 2, we show an example of simulated spectra based on the data from Table 1.It should be emphasized that at the conditions relevant to giant planets, the shapes of the HD lines may considerably deviate from the simple Voigt profile.In Figure 2 (a), we show the difference between the Voigt profile and a more physical profile, which includes the relevant beyond-Voigt effects such as Dicke narrowing and speed dependence of broadening and shift (the modified Hartmann-Tran profile, see Appendix D).For the moderate pressures, the error introduced by the Voigt-profile approximation can reach almost 70% (the orange, yellow, red, and black lines in Fig. 2 (a) are the temperature-pressure profiles for Jupiter, Saturn, Uranus, and Neptune, respectively).We illustrate this with spectra simulations for the case of Neptune's atmosphere, see Figs.Voigt-profile approximations.The (b)-(d) cases illustrate three different regimes.The (b) case is the low-pressure regime with a small collisional contribution in which the line shape collapses to Gaussian, hence the beyond-Voigt effects are small.Case (d) is the opposite, the shape of a resonance is dominated by the collisional effects, but the simple Lorentzian broadening dominates over other collisional effects and again the beyond-Voigt effects are small.Case (c) illustrates the nontrivial situation in which the shape of resonance is dominated by the beyond-Voigt effects, see the green horizontal ridges in Fig. 2 (a).The discrepancies between the beyond-Voigt line-shape model and the Voigt profile are also clearly seen as a difference between the blue lines and blue shadows in Fig. 2 (c).In the context of the giant planet studies, it should be noted that the beyond-Voigt regions marked in Fig. 2 (a) (the green horizontal ridges) coincide well with the maxima of the monochromatic contribution functions for these three HD lines, see Fig. 1 in (Feuchtgruber et al. (2013)) for the case of the atmospheres of Neptune and Uranus.
Figures 2 (b)-(d) also illustrate the influence of atmosphere composition on the collision-induced shapes of the HD lines for the example of Neptune atmosphere.Black and red lines in Figs. 2 (b)-(d) correspond to He-and H 2 -perturbers, respectively.The gray lines correspond to the 20% He and 80% H 2 mixture that is relevant to Neptune atmosphere.The differences between black and red curves are negligible in the low-pressure regime, Figs. 2 (b), since at these conditions the line shape is mainly determined by thermal Doppler broadening.At moderate-and high-pressure ranges, Figs. 2 (c) and (d), the profiles differ at the peak by a factor of 2, hence including both perturbing species is important for spectra analysis of the atmospheres of giant planets, especially for the R(0) line, whose contribution function dominates at moderate and high pressures (Feuchtgruber et al. (2013)).
The data reported in this article, see Table 1, account for three factors that are necessary for reaching sub-percent-level accuracy: 1) separate ab initio data for both perturbers (that allows one to simulate perturbation by any H 2 /He mixture), 2) accurate representation of temperature dependences, and 3) parametrization of the beyond-Voigt line-shape effects.In general, simulating the beyond-Voigt line-shape profiles is a complex task, see Appendix D. In this work, we used the HITRAN Application Programming Interface (HAPI) (Kochanov et al. (2016)) to generate the beyond-Voigt spectra shown in Fig. 2  The combination of the data reported in Table 1 and the Pythonbased HAPI constitutes a powerful tool that allows one to efficiently generate accurate HD spectra (based on advanced beyond-Voigt model, the mHT profile) for arbitrary temperature, pressure, and mixture composition.At low temperatures, relevant for studies of giant planet atmospheres and the chemistry and dynamics of the interstellar medium and protoplanetary discs, the spin isomer (para/ortho) concentration ratio of H 2 at thermal equilibrium (eq-H 2 ) deviates from 1:3 (the ratio of so-called normal H 2 , n-H 2 ).Moreover, various processes, such as diffusion between atmospheric layers in gas giants, might result in the sub-equilibrium distribution of H 2 .These non-trivial para/ortho distributions play a key role in atmospheric models that involve collisional induced absorption (Karman et al. (2019)) and spectral features originating from hydrogen dimers (Fletcher et al. (2018)), as well as in isotope chemistry of the interstellar medium, where para/ortho ratio controls the deuterium fractionation process (Flower et al. (2006); Nomura et al. (2022)).In Figure 3, we show the influence of the spin isomer concentration on the line-shape parameters.Spin isomer concentration has a large impact at low temperatures.All the line-shape parameters reported in this work are calculated for the thermal equilibrium spin isomer concentration.The role of the beyond-Voigt effects and bath mixture composition on collision-perturbed spectra of HD at conditions relevant for giant planet atmospheres.(a) Relative error of the Voigt-profile approximation as a function of pressure and temperature, shown as the relative difference between the Voigt and mHT profiles at profile maximum.The panels are arranged to correspond to the R(0), R(1), and R(2) lines, from top to bottom, respectively.(b), (c), (d) Simulations of the HD spectra (blue lines) at conditions relevant for the Neptune atmosphere (the perturbing bath consists of a mixture of 80% H 2 and 20% He).The spectra are generated with the mHT profile using HAPI based on the DPL temperature parametrization.As a reference, we show the same lines for the cases of pure H 2 and pure He perturbers, see the red and black lines, respectively.The blue shadows show the same simulations as the blue lines but generated with a simple Voigt profile.The three panels, (b), (c), (d), correspond to the three points, (b), (c), (d), shown in the temperature-pressure maps in panel (a) (the three selected points lay on the Neptune temperaturepressure line).The three cases illustrate three different line-shape regimes.The first one, (b), is the low-pressure case in which the lines are broadened mainly by the Doppler effect, and the pressure-induced collisional effects do not dominate the line shapes.The intermediate-pressure case, panel (c), illustrates the extreme non-Voigt regime (the differences between the blue curves and blue shadows reach almost a factor of 2), see also the green ridge in the maps in the bottom panel.The third case, panel (d), illustrates the high-pressure regime at which the HD lines are well described by the simple Voigt profile (the blue shadows almost overlap with the blue lines), but setting a proper composition of the perturber gas components plays an important role.
Table 1.DPL parameterization of the temperature dependences of the line-shape parameters of HD perturbed by He and H Fig. 3. Spin isomer concentration ratio (x = n p /n o ) dependence of pressure broadening, γ 0 (solid lines), and shift, δ 0 (dashed lines), parameters for the H 2 -perturbed R(0) line in HD, at different temperatures.The grey vertical lines correspond to the value of x for normal H 2 , x = 1/3, and x eq (T ), as determined by the Boltzmann distribution at T = 50, 195, and 296 K.

Experimental validation
In Figure 4, we show a comparison between our ab initio calculations (black lines) and the experimental data available in the literature.Fourier-transformed scans from the Michelson interferometer were used to obtain the high-pressure spectra reported in the works of Ulivi et al. (1989) and Lu et al. (1993), see the green and red points, respectively; the spectra were collected in a temperature range from 77 to 296 K. Recently, the same lines were measured at low pressures (< 1 bar) with the Fourier transform spectrometer coupled to the Soleil-synchrotron far-infrared source (Sung et al. (2022)) in a temperature range from 98 to 296 K, see the olive lines in Fig. 4. The discrepancy between these experimental data is by far too large to test our theoretical results at the one percent level.
To validate our ab initio calculations at the estimated accuracy level, we performed accurate measurements using a frequency-stabilized cavity ring-down spectrometer (FS-CRDS) linked to an optical frequency comb (OFC), referenced to a primary frequency standard (Cygan et al. (2016(Cygan et al. ( , 2019)); Zaborowski et al. (2020)).Our 73.5-cm-long ultrahigh finesse (F = 637 000) optical cavity operates in the frequency-agile, rapid scanning spectroscopy (FARS) mode (Truong et al. (2013); Cygan et al. (2016Cygan et al. ( , 2019))); see Zaborowski et al. (2020) for details regarding the experimental setup.Since our spectrometer operates at 1.6 µm, we chose the S(2) 2-0 line in the H 2 -perturbed D 2 (we repeated all the ab initio calculations for this case).From the perspective of theoretical methodology, the H 2 -perturbed D 2 and H 2 -perturbed HD are equivalent and either can be used for validating the theoretical methodology (for both cases two distinguishable diatomic molecules are considered and the PES is the same except for the almost negligible DBOC term, see Appendix A).We used a sample of 2% D 2 and 98% of H 2 mixture and collected the spectra at four pressures (0.5, 1, 1.5, and 2 atm) and two temperatures (296 and 330 K), see the black dots in Fig. 5.The corresponding theoretical spectra are the red curves.The methodology for simulating the collisionperturbed shapes of molecular lines (based on the line-shape parameters calculated from Eqs. ( 1)-( 3)) is described in our previ-ous works (Wcisło et al. 2018;Słowiński et al. 2020;Słowiński et al. 2022).The two sets of residuals depicted in Fig. 5 show comparisons with two line-shape models, the speed-dependent billiard-ball profile (SDBB profile) and the mHT profile.The SDBB profile (Shapiro et al. 2002;Ciuryło et al. 2002) is the state-of-the-art approach that gives the most realistic description of the underlying collisional processes.As expected it gives the best agreement with experimental spectra (the mean residuals are 0.65%, see Fig. 5), but it is computationally very expensive (Wcisło et al. (2013)).The mHT profile is slightly less accurate (the mean residuals are 1.23%) but it is highly efficient from a computational perspective and, hence, well suited for practical spectroscopic applications.In conclusion, the ab initio lineshape parameters reported in this work (Fig. 1 and Table 1) lead to profiles that are in excellent agreement with accurate experimental spectra, and the theory-experiment comparison is limited by a choice of a line-shape model used to simulate the experimental spectra.Lu 1993 Sung 2022 Fig. 4. Comparison of the experimental and theoretical values of the pressure broadening and shift, γ 0 and δ 0 , parameters (in the units of 10 −3 cm −1 atm −1 ).Black curves correspond to the ab initio calculations performed in this work, while green and red points report the experimental measurements from (Ulivi et al. (1989)) and (Lu et al. (1993), respectively.The olive curves are the single-power-law (for γ 0 ) and linear (for δ 0 ) temperature dependences retrieved from the measurements of (Sung et al. (2022)).

Conclusion
We computed accurate collision-induced line-shape parameters for the three pure rotational HD lines (R(0), R(1), and R(2)), which are currently employed for the analysis of giant planets' atmospheres.To this end, we investigated HD-H 2 collisions using coupled channel quantum scattering calculations on a new, highly accurate ab initio PES.Scattering S-matrices determined from these calculations allowed us to obtain the collisional width and shift, as well as their speed dependences and the complex Dicke parameter of H 2 -perturbed HD lines.By integrating data from our previous work on the HD-He system (Stankiewicz et al. 2020(Stankiewicz et al. , 2021)), we provide comprehensive results covering a wide range of thermodynamic conditions, including temperature, pressure, and H 2 /He concentration, relevant to the atmospheres of giant planets.The theoretical methodology was validated through cavity ring-down spectroscopy, demonstrating sub-percent-level accuracy that surpasses the accuracy of pre-   vious theoretical and experimental studies of line-shape parameters in HD.
All the reported line-shape parameters and their temperature dependences are consistent with the HITRAN database format.Utilizing the HITRAN Application Programming Interface (HAPI), we demonstrated how our results can be applied to simulate HD spectra under various conditions pertinent to giant planets in the solar system.
Until now, the analysis of observed collision-perturbed spectra in astrophysical studies has predominantly relied on the simple Voigt profile.We have introduced a methodology and provided a comprehensive dataset that enables the simulation of beyond-Voigt shapes for hydrogen deuteride in H 2 /He atmospheres.Our work demonstrates that accounting for the speed dependence of collisional width and shift, along with the complex Dicke parameter, is crucial.These factors can alter the effective width and height of HD lines by up to a factor of 2, which in turn could significantly impact the HD abundance inferred from astrophysical observations.the I 4 4 8 (θ 1 , θ 2 , ϕ) bispherical harmonic) and an expanded set of expansion coefficients describing higher anisotropies of the system (up to the I 6 6 12 (θ 1 , θ 2 , ϕ) term).The results are gathered in Tab.C.1.
The number of partial waves (or equivalently, blocks with given total angular momentum J) necessary to converge the scattering equations was determined based on a criterion of stability in the calculated cross-sections.We solved the coupled equations for an increasing number of J-blocks until four consecutive J-blocks contributed to the largest elastic and inelastic state-to-state crosssections by less than 10 −4 Å 2 .The convergence criterion ensured that the estimated error introduced by the number of partial waves was smaller than the smallest uncertainty attributable to the other parameters in our study.This implies that the uncertainty in the number of partial waves did not significantly contribute to the overall uncertainty in our results, thus we do not consider this factor in Tab.C.1.
The size of the rotational basis set is a critical factor in quantum scattering calculations, and it was chosen with great care to ensure a consistent level of accuracy across different rotational states.For each calculation, we checked that the basis set included all energetically accessible (open) levels of the colliding pair, as well as a certain number of asymptotically energetically inaccessible (closed) levels.We gradually increased the size of the basis set until the calculated cross-sections did not show appreciable differences, identifying a fully converged basis set.We then determined the smallest basis set that ensured convergence to better than 1% with respect to the fully converged basis.This was done for each initial state of the HD-H 2 system in a way that the estimated error for all transitions (R( j HD ), j HD =0, 1, 2), including those involving rotationally excited states, remained within the specified limit.
The tests were conducted separately for collisions with para-H 2 (which involves only even rotational quantum numbers) and ortho-H 2 (which involves only odd rotational quantum numbers).In the case of para-H 2 , all rotational levels of HD and H 2 with j ≤ j max = 4 were consistently included in the calculations.For specific calculations with H 2 initially in the j H 2 = 4 state, or HD initially in the j HD = 3 state, the basis set was expanded to incorporate all rotational levels of HD and H 2 with j ≤ 6.For ortho-H 2 , the basis set consistently included all rotational levels of HD and H 2 with j ≤ j max = 5.We extended the basis set to cover j max HD = 6 and j max H 2 = 5 for cases where HD and H 2 were initially in the ( j HD , j H 2 ) = (0,3), (1,3), or (2,1) states.The largest basis set, with j max HD = j H 2 = 7, was employed in all calculations involving HD and H 2 in ( j HD , j H 2 ) = (0,5), (1,5), (2,3), (2,5), (3,1), (3,3), and (3,5) states.
Assuming the uncertainties associated with each parameter as independent, we estimated the maximum total uncertainty of each line-shape parameter using the root-sum-square method.The results are gathered in the last line of Tab.C.1.

Appendix D: Modified Hartmann-Tran (mHT) profile
This appendix describes the modified Hartmann-Tran profile, which we used to simulate the spectra.We consider the mHT profile as the best compromise between the accurate but computationally-demanding speed-dependent billiard ball (SDBB) profile and the simple Voigt profile (VP).The mHT profile can be expressed as a quotient of two quadratic speed-dependent Voigt (qSDV) profiles, The line-shape profile uses the parameters in the pressure-dependent form, i.e., is the Maxwell-Boltzmann distribution of the active molecule velocity, v m is its most probable speed and v z is one of the three Cartesian components of the v vector.f , f 0 , and f D are the frequency of light, the central frequency of the transition, and the Doppler frequency, respectively.
The hard-collision model of the velocity-changing collisions, which is used in the mHT profile, suffices to describe the velocitychanging line-shape effects (such as the Dicke narrowing) in the majority of the molecular species.However, in the cases with a significant Dicke narrowing, such as molecular hydrogen transitions, the hard-collision model does not reproduce the line shapes at the required accuracy level.To overcome this problem, a simple analytical correction (the β correction function) was introduced (Wcisło et al. (2016); Konefał et al. (2020)), which mimics the behavior of the billiard ball model and, hence, considerably improves the accuracy of the mHT profile for hydrogen, at negligible numerical cost.The correction is made by replacing the ν r opt with β α (χ)ν r opt , where α is the perturber-to-absorber mass ratio and χ = ν r opt /Γ D (where Γ D is the Doppler width, see Konefał et al. (2020) for details).It should be emphasized that the β correction does not require any additional transition-specific parameters (it depends only on the perturber-to-absorber mass ratio α).The β correction was applied every time the mHT profile was used in this work.Temperature dependences of the six collisional line-shape parameters, γ 0 , δ 0 , γ 2 , δ 2 , νr opt and νi opt of the R(0) 0-0, R(1) 0-0 and R(2) 0-0 lines of HD perturbed by H 2 .The red and black curves are the ab initio results and DPL approximations, respectively.The small panels show the residuals from the DPL fits.The vertical axes for all the panels (including residuals) are in 10 −3 cm −1 atm −1 .

Fig. 1 .
Fig.1.Ab initio temperature dependences of the collisional line-shape parameters (in the units of 10 −3 cm −1 atm −1 ) of the first three electric dipole lines of HD perturbed by H 2 (red curves) and He (black curves).
Fig.2.The role of the beyond-Voigt effects and bath mixture composition on collision-perturbed spectra of HD at conditions relevant for giant planet atmospheres.(a) Relative error of the Voigt-profile approximation as a function of pressure and temperature, shown as the relative difference between the Voigt and mHT profiles at profile maximum.The panels are arranged to correspond to the R(0), R(1), and R(2) lines, from top to bottom, respectively.(b), (c), (d) Simulations of the HD spectra (blue lines) at conditions relevant for the Neptune atmosphere (the perturbing bath consists of a mixture of 80% H 2 and 20% He).The spectra are generated with the mHT profile using HAPI based on the DPL temperature parametrization.As a reference, we show the same lines for the cases of pure H 2 and pure He perturbers, see the red and black lines, respectively.The blue shadows show the same simulations as the blue lines but generated with a simple Voigt profile.The three panels, (b), (c), (d), correspond to the three points, (b), (c), (d), shown in the temperature-pressure maps in panel (a) (the three selected points lay on the Neptune temperaturepressure line).The three cases illustrate three different line-shape regimes.The first one, (b), is the low-pressure case in which the lines are broadened mainly by the Doppler effect, and the pressure-induced collisional effects do not dominate the line shapes.The intermediate-pressure case, panel (c), illustrates the extreme non-Voigt regime (the differences between the blue curves and blue shadows reach almost a factor of 2), see also the green ridge in the maps in the bottom panel.The third case, panel (d), illustrates the high-pressure regime at which the HD lines are well described by the simple Voigt profile (the blue shadows almost overlap with the blue lines), but setting a proper composition of the perturber gas components plays an important role.

Fig. 5 .
Fig. 5. Direct validation of the ab initio quantum-scattering calculations on the accurate experimental spectra of the S(2) 2-0 line of D 2 perturbed by collisions with H 2 molecules (see the text for details).The black dots are the experimental spectra and the red lines are the ab initio profiles.Below each profile, we show the absolute residuals of two models: the speed-dependent billiard-ball (SDBB) profile and the modified Hartmann-Tran (mHT) profile.To quantify how well theory agrees with experiments, we report the relative (with respect to the profile peak value) root mean square errors (rRMSE) of the experiment-theory differences calculated within the ±FWHM range around the line center, see the numbers (in percent) below the residuals.The mean rRMSE are also summarized for each of the models, see the numbers on the right side of the figure.
(based on the DPL parameters from Table1):