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New study of the line profiles of sodium perturbed by H2

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that new unified sodium–H2 line profiles, computed from improved ab initio potentials and valid to H2 densities of $10^{21}\,\mathrm{cm}^{-3}$, should replace Lorentzian and older pseudo-potential treatments of sodium…

desk verdict A genuinely useful update to the Na–H2 opacity tables, but the two-orientation, fixed-H2 potentials carry an unquantified systematic error that a referee should push on. read the letter →

arxiv 1908.01989 v1 pith:YY3JGWF6 submitted 2019-08-06 astro-ph.SR astro-ph.EP

N.F. Allard , F. Spiegelman , T. Leininger , P. Molliere This is my paper · ORCID
classification astro-ph.SRastro-ph.EP
keywords sodiumresonancelinesmolecularhydrogenpressurebroadeningunifiedlineprofilesabinitiopotentialsspin-orbitcouplingopacitytablesbrowndwarfatmosphereshotJupitertransmissionspectra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to replace the standard Lorentzian and older pseudo-potential treatments of the sodium resonance lines perturbed by molecular hydrogen with unified line profiles computed from improved ab initio Na–H2 potentials and transition dipole moments. It argues that only such calculations, carried to high enough order in the density expansion, are valid for the H2 densities, up to $10^{21}\,\mathrm{cm}^{-3}$, found in brown dwarfs and hot Jupiter atmospheres. The central quantitative result is a pair of power laws: the full width at half maximum is linear in H2 density and scales as $T^{0.33}$ for the D1 line and $T^{0.39}$ for the D2 line. If correct, the new profiles move the blue satellite closer to line center than the older RP85-based profiles, changing the optical pseudo-continuum opacity and the temperature structure of hot Jupiter atmospheres.

What carries the argument

The load-bearing object is the unified line-shape machinery: the absorption profile is the Fourier transform of a dipole autocorrelation function, $I(\Delta\omega)=\frac{1}{\pi}\mathrm{Re}\int_0^\infty \Phi(s)e^{-i\Delta\omega s}\,ds$, with $\Phi(s)=e^{-n_p g(s)}$ for a perturber density $n_p$. The calculation uses ab initio Na–H2 potential energy surfaces in the linear ($C_{\infty v}$) and T-shaped ($C_{2v}$) geometries, with the H2 bond frozen at its equilibrium length, together with transition dipole moments and a spin-orbit coupling treatment that separates the D1 and D2 components. To reach high densities the paper expands the autocorrelation function in powers of density, splitting $g(s)$ into averaged and oscillating parts; this is what allows opacity tables valid to $n_{\mathrm{H_2}}=10^{21}\,\mathrm{cm}^{-3}$ rather than the previous $10^{19}\,\mathrm{cm}^{-3}$.

What would settle it

A laboratory measurement of the sodium D1 and D2 absorption wings in a cell with a known density of H2 at roughly 1000–1500 K, looking for the predicted blue satellite near 5170 Å and the density-dependent shoulder near 4800 Å, would settle the central claim: if the satellite position or wing shape deviates strongly from the unified profiles, the potentials are wrong.

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Extended reading notes

Core claim

The core discovery, stated on the paper's own terms, is that the improved potentials make the Na–H2 resonance line widths obey simple density-linear, temperature-power-law forms, with $w_{\mathrm{D1}}=0.169\times10^{-20} n_{\mathrm{H_2}} T^{0.33}$ and $w_{\mathrm{D2}}=0.242\times10^{-20} n_{\mathrm{H_2}} T^{0.39}$ (w in cm$^{-1}$, $n_{\mathrm{H_2}}$ in cm$^{-3}$, $T$ in K), valid from 500 to at least 3000 K. The unified profiles also place the first blue satellite near 5170 Å and, at $n_{\mathrm{H_2}}=10^{21}\,\mathrm{cm}^{-3}$, a second shoulder near 4800 Å arising from multiple-perturber effects. The blue wing is markedly less extended than in the older RP85 profiles while the red wing remains similar. The paper concludes that Lorentzian profiles are not appropriate for these line wings and that unified profiles should be incorporated into spectral models.

Load-bearing premise

The load-bearing premise is that the Na–H2 potential energy surfaces are realistic, yet they are computed for only two geometries with the H2 bond length fixed and are never compared to molecular data, only to sodium atomic transition energies.

Editorial extensions

If this is right

  • The D1 and D2 line widths can be computed directly from the two power-law formulas in the 500–3000 K range, replacing interpolation of full profiles for the core width.
  • Opacity tables now reach $n_{\mathrm{H_2}}=10^{21}\,\mathrm{cm}^{-3}$, two orders of magnitude higher than the previous $10^{19}\,\mathrm{cm}^{-3}$ tables.
  • The blue wing is less extended than in RP85-based profiles, so sodium pseudo-continuum opacity in the blue optical is weaker in brown dwarf and hot Jupiter models.
  • In the hot Jupiter model presented, the new profiles produce more greenhouse heating in deep atmospheric layers and slightly cooler upper layers than the old profiles.
  • Lorentzian profiles are unsuitable for the sodium doublet line wings; unified profiles are needed to reproduce the satellite position and wing shape.
  • The blue satellite lies closer to line center than in RP85 profiles, shifting the optical pseudo-continuum compared with previous model spectra.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not test the potential surfaces against molecular data; a natural next step is to repeat the calculation with a flexible H2 bond and an average over approach angles, which would show how much the frozen-bond, two-geometry choice affects the satellite position and widths.
  • The same machinery should transfer to K–H2, since potassium wings shape the Y-band flux of self-luminous planets; by analogy with sodium, updated K profiles could shift the pseudo-continuum and the inferred temperature structure.
  • If the power-law widths are imported into retrieval codes, they carry a validity limit: near $10^{20}$–$10^{21}\,\mathrm{cm}^{-3}$ the far blue wing is nonlinearly density-dependent, so a pure line-center treatment cannot be extrapolated.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents unified, semi-classical line profiles for the Na I 3s–3p doublet perturbed by H2, computed from new ab initio S17 potential energy surfaces that include spin-orbit coupling and transition dipole moments. It reports temperature- and density-dependent widths via power laws (Eqs. 7–8), a blue satellite at 5170 Å, opacity tables claimed to be valid up to n_H2 = 10^21 cm^-3, and applications to petitCODE models of self-luminous atmospheres and hot Jupiters. The intended contribution is to replace Lorentzian and older RP85-based wing treatments with profiles based on improved molecular data.

Significance. If substantiated, these profiles would be a practical improvement for brown dwarf and exoplanet opacity modeling, since Na–H2 wings are a major pseudo-continuum source and current tables are based on older potentials. The inclusion of spin-orbit coupling for the doublet, the diabatization of the 3p/4s crossing, and the explicit unified-theory treatment at densities up to 10^21 cm^-3 are clear advances. The paper also delivers ready-to-use parameterizations and states that tables and generation code will be archived at the CDS, which is valuable for the community. The width parameterizations are derived from the potentials rather than fitted to observed line widths, which is a strength. However, the accuracy claim rests on the molecular potentials and on an underspecified treatment of the H2 orientation, so the significance is conditional on the issues below.

major comments (3)
  1. [Sect. 2 and Sect. 3, Eq. (11)] The manuscript does not specify how the two computed Na–H2 geometries are used in the line-shape calculation. The S17 potentials are computed only for C∞v and C2v with the H2 bond frozen at r_e = 1.401 a.u., yet Eq. (11) uses a single pair autocorrelation function g(s). For a non-spherical perturber, g(s) must include a thermal average over the H2 orientation angle, or an equivalent justified approximation. The text contains no such average, no weighting of the linear and T-shaped geometries, and no estimate of the error from omitting intermediate orientations. Because the blue satellite position and the wing shapes are controlled by the difference potential ΔV(R, θ), this missing step directly affects the numerical results in Eqs. (7)–(8), Fig. 7, and the opacity tables. Please either document the orientation averaging or justify that one geometry dominates over the relevant R and T range.
  2. [Sect. 3.1 and Table 1] The only validation reported is the sodium atomic transition energies, with errors up to 25 cm^-1. No comparison is made between the S17 molecular potential energy surfaces and experimental or high-level reference data, and no error bars are propagated to the line widths or satellite position. Since the paper's central claim is that these profiles are accurate enough to replace previous treatments, the manuscript should include at least one molecular-level validation, such as a comparison with an experimental Na–H2 absorption spectrum or with independent high-level electronic structure calculations, or an explicit uncertainty analysis. This is a load-bearing gap rather than a cosmetic one.
  3. [Sect. 3.3] The statement that the new opacity tables are 'constructed to a higher order' of the density expansion does not specify the truncation order, and no convergence test is shown for the claimed validity up to n_H2 = 10^21 cm^-3. The density expansion in Eq. (14) can break down if n_p g_osc is not small; a brief comparison of the retained order with the next order, or with the exact Fourier transform of Eq. (11) at high density, is needed to support the table's quoted range.
minor comments (4)
  1. [Fig. 6 caption] The Fig. 6 caption lists the D2 power-law exponent as 0.32, while Eq. (8) and the abstract give 0.39; please correct this inconsistency.
  2. [Fig. 1 bottom panel] The label 'triangular-NaH2' is not consistent with the text's 'T-shape (C2v)' terminology; please unify the nomenclature for the same geometry.
  3. [Eq. (12)] The Gaussian A(s) is introduced without defining its width or the criterion used to separate g_av from g_osc; a one-sentence definition would improve reproducibility.
  4. [Eq. (16)] The oscillator strength f is used in Eq. (16) without an explicit definition or specification of which transition it refers to; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Na-H2 line profiles, widths, and opacity tables are numerical outputs of ab initio potentials and a general unified line-shape theory, not re-imported inputs.

full rationale

The paper's derivation chain is: S17 ab initio Na-H2 potentials and transition dipole moments (Sect. 2) are inputs to the unified line-shape formalism of Allard et al. (1999), producing the computed D1/D2 profiles, the line widths summarized by the power laws in Eqs. (7)-(8), the blue satellite at 5170 Å, and the density-expansion opacity tables. None of these outputs is used to define the potentials or the theory. The power laws in Eqs. (7)-(8) are explicitly fits to the numerically computed widths ('These expressions accurately represent the numerical results as shown in Fig. 6'), not fits to observed line widths or atmospheric spectra, so they are not fitted inputs renamed as predictions. The self-citations to Allard et al. (1999), Allard et al. (2003), Allard et al. (2012b), and earlier papers establish methodological lineage and provide the unified theory and previous profile versions, but the theory itself is parameter-free with stated assumptions (stationary radiator, independent perturbers, adiabatic scalar-additive potentials) and does not contain the Na-H2 line-width result. No uniqueness theorem is imported from the authors' prior work to forbid alternative potentials, and no ansatz is smuggled in via citation: the potentials are newly computed ab initio in this paper. The astrophysical applications with petitCODE are comparisons of the new versus old profiles, not inversions that assume the quoted widths. The absence of an independent benchmark for the molecular potential energy surfaces is a legitimate accuracy and robustness concern, but it is not a circularity: the derivation does not reduce to its own output by construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central calculation depends on the adopted Na core-polarization parameters (from prior literature), the fixed H2 bond length and two symmetry geometries, the semi-classical line-shape assumptions, and an unspecified Gaussian split in the density expansion. No new physical entities are introduced.

free parameters (5)
  • Na core-polarization parameters (alpha, rho_c) = alpha=0.997 a0^3, rho_c=0.62
    Adopted from Müller et al. (1984) for the sodium ECP/CPP; the potential energy surfaces depend on these values, which are not refit here.
  • Line-width power-law coefficients for D1 = 0.169e-20, exponent 0.33
    Fitted to the computed D1 widths in Fig. 6; presented as a convenient formula (Eq. 7), not a first-principles prediction.
  • Line-width power-law coefficients for D2 = 0.242e-20, exponent 0.39
    Fitted to the computed D2 widths in Fig. 6; same status as the D1 fit (Eq. 8).
  • Gaussian splitting width A(s) = not specified
    The density expansion in Eqs 12-13 splits g(s) into averaged and oscillating parts using a Gaussian A(s) whose width is not given; the truncated expansion can depend on this choice.
  • Order of density expansion = not specified; 'higher order'
    Previous tables were third order; the new tables are described only as 'higher order', with no explicit order given, which affects accuracy at n_H2=1e21 cm^-3.
assumptions (4)
  • domain assumption H2 bond length fixed at r_e = 1.401 a.u. and only C2v (T-shape) and C_infinity v (linear) Na-H2 geometries are computed.
    Sect. 2. Assumes these two rigid geometries adequately represent the collision average that determines the line profile; no angular averaging or vibrational sampling is described.
  • domain assumption Semi-classical unified line shape assumptions: stationary radiator, independent perturbers, adiabatic scalar-additive potentials.
    Sect. 3.3, Eqs 9-11. The Fourier transform of N-th power of one-pair autocorrelation neglects interperturber correlations; this may be strained at n_H2 up to 1e21 cm^-3.
  • domain assumption Cohen-Schneider atom-in-molecule spin-orbit approximation: molecular SO matrix elements are replaced by asymptotic atomic values, with a 3p/4s diabatization.
    Sect. 2, Eq. 2. The SO coupling is assumed transferable from the atom except for the diabatic mixing angle; errors are not quantified.
  • domain assumption MRCI/CAS single-active-electron plus core-polarization calculation yields accurate Na-H2 potential energy surfaces.
    Sect. 2. Only atomic transition energies are benchmarked (Table 1, errors <25 cm^-1); molecular PES accuracy is asserted, not demonstrated against experiment or independent theory.

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Pith. "Pith review of New study of the line profiles of sodium perturbed by H2." pith.science (2026). https://pith.science/paper/YY3JGWF6

@misc{pith2026190801989,
  author       = {Pith},
  title        = {Pith review of: New study of the line profiles of sodium perturbed by H2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YY3JGWF6}},
  note         = {Machine review of arXiv:1908.01989}
}
read the original abstract

The opacity of alkali atoms, most importantly of Na and K, plays a crucial role in the atmospheres of brown dwarfs and exoplanets. We present a comprehensive study of NaH2 collisional profiles at temperatures from 500 to 3000 K, the temperatures prevailing in the atmosphere of brown dwarfs and Jupiter-mass planets.The relevant H2 perturber densities reach several 10^19 cm^-3 in hot Teff > 1500 K Jupiter-mass planets and can exceed 10^20 cm^-3 for more massive or cooler objects. Accurate pressure-broadened profiles that are valid at high densities of H2 should be incorporated into spectral models. Unified profiles of sodium perturbed by molecular hydrogen were calculated in the semi-classical approach using up-to-date molecular data.New NaH2 collisional profiles and their effects on the synthetic spectra of brown dwarfs and hot Jupiters computed with petitCODE are presented.

Figures

Figures reproduced from arXiv: 1908.01989 by the authors.

Figure 1
Figure 1. Potential curves of the Na–H2 molecule for the C∞v (top) and C2v symmetries (bottom). For the C2v case, the symmetry labeling cor￾responds to the convention of the reference plane as that containing the molecule and may be different from that of previous publications. We note that states 12A2 and 42A1 correlated with the 3d asymptote are su￾perimposed at the scale of the figure. Article number, page 2 of 9 [PITH_FU… view at source ↗
Figure 2
Figure 2. Long-range potential curves of NaH2 correlated with the 3p1/2 and 3p3/2 asymptotes in C∞v symmetry. 16900 16920 16940 16960 16980 17000 5 10 15 20 25 30 C2v Na-H2 with SO coupling long range 3p asymptote 3p 2P1/2 3p 2P3/2 ENERGY (cm-1 ) R (a0 ) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Coefficients (cosθ and sinθ)) of the 3p/4p diabatization in C∞v (top) and C2v symmetries (bottom) [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: Transition dipole moment for the B-X (full line), A 2P3/2-X (dot￾ted line), and A 2P1/2-X (dashed line) transitions of the Na-H2 molecule for the C∞v (red curves) and C2vsymmetries (black curves); B-X (full line), A 2P3/2 − X (dotted line), and A 2P1/2 − X (dashed line…
Figure 4
Figure 4. Figure 4: We obtain a 8×8 spin-orbit coupling matrix for the 3p/4s states, which is given in the Appendix. The spin-orbit energy splitting 2P3 2 − 2 P1 2 of the 3p levels of sodium is 17.19 cm−1 (= 3 2 ζ) (Moore 1971). No such diaba￾tization was considered for the 4p configurati…
Figure 6
Figure 6. Figure 6: Variation with temperature of the half-width of the D2 (blue curves) and D1 (red curves) lines of Na I perturbed by H2 collisions. New ab initio potentials (full line), pseudo-potentials of Rossi & Pas￾cale (1985) (dashed lines), and the van der Waals potential (black …
Figure 8
Figure 8. Figure 8: Variation of the absorption cross sections of the 3s￾3p D2 line component with temperature. (from top to bottom T =2500, 1500, 1000, and 600 K) for nH2 = 1021 cm−3 . The Lorentzian profile for 1000 K is overplotted (black full line). 5000 6000 7000 8000 9000 10000 1100…
Figure 9
Figure 9. Figure 9: Variation of the absorption cross sections of the 3s￾3p D1 line component with temperature. (from top to bottom T =2500, 1500, 1000, and 600 K) for nH2 = 1021 cm−3 . The Lorentzian profile for 1000 K is overplotted (black full line). The spectrum I(∆ω) can be written a…
Figure 10
Figure 10. Figure 10: Emission spectra for cloud-free, self-luminous objects (exoplanets or Brown Dwarfs) at solar composition and varying effective temper￾ature, calculated with petitCODE (Mollière et al. 2015, 2017). The atmospheric surface gravity was set to log(g) = 4.5, with g in unit…
Figure 11
Figure 11. Figure 11: Transmission radii for cloud-free, hot-Jupiter exoplanets at solar composition for a planetary effective temperature of 1800 K, calculated with petitCODE (Mollière et al. 2015, 2017). The planet mass and radius were chosen to be identical to the values of Jupiter, and…

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