The gas models

Read Dispersion models for selection and Fitting dispersion models for construction. This chapter records the supporting evidence and coefficients for each gas; Gas dispersion: accuracy and limitations interprets the uncertainty. Start with Quick start for a first calculation, and use Source catalogue to inspect the cited records.

Every gas has one fitted model of its refractive index (fit/<gas>), recommended by default, and each published formula of the curated sources beside it for comparison (Dispersion models). The fit and its model are described on Fitting dispersion models. This page says what each model rests on, how far the formulas in common use are from it, and what the curation found about the data along the way.

What each model rests on

The backbone of a model is the data that set its level and its dispersion: ab initio theory, precision metrology or a dataset of stated accuracy. The rest either constrain the shape without their own level (“level free”: the dataset’s scale is fitted) or are validators, compared with the fit but not fitted. The range is the vacuum-wavelength range over which the model is declared valid. The uncertainty is the fit’s statistical one-sigma uncertainty of \(n - 1\) (the level) at 800 nm; Gas dispersion: accuracy and limitations gives the uncertainties of the dispersion, which matter more for propagation, with the working estimates.

Gas	What the model rests on	Range (nm)	\(n - 1\), \(1\sigma\) at 800 nm	Reduced \(\chi^2\)
He	Puchalski et al. 2016’s ab initio dispersion coefficients with every relativistic and QED correction; Pachucki & Puchalski’s retardation term; Schmidt et al. 2007’s microwave static polarizability. The experiments are tested against it, not needed by it.	60-static	0.1 ppm	0.58
Ne	Lesiuk et al. 2020’s ab initio moments; Gaiser & Fellmuth’s dielectric-constant gas thermometry; the dispersion of Cuthbertson (level free), Bideau-Mehu and Shelton’s He/Ne ratio.	76-static	8.6 ppm	2.63
Ar	Lesiuk & Jeziorski 2023’s ab initio moments; dielectric-constant gas thermometry; Egan et al.’s refractometry; the shapes of Peck & Fisher (level free), Bideau-Mehu and Gordon.	108-static	2 ppm	0.50
Kr	Dielectric static values (Orcutt & Cole, Huot & Bose); Birch and Hohm & Kerl at 633 nm; Burns et al. at 458-633 nm; Smith et al. 1976 in the ultraviolet; Bideau-Mehu; Shelton’s Ar/Kr ratio; Hohm’s 325-633 nm dispersion.	125-static	78 ppm	1.23
Xe	Huot & Bose’s dielectric static value; Burns et al.; Hohm & Kerl at 633 nm; Abjean et al. 1971 and Bideau-Mehu to 147 nm; Shelton’s Ar/Xe ratio; Hohm’s 325-633 nm dispersion.	148-static	1.7e+02 ppm	0.96
H2	Raj et al. 2018’s ab initio average of normal hydrogen (0.1 % allowed); the shapes of Peck & Hung and Smith et al.; Shelton’s He/H2 ratio.	112-static	1.3e+02 ppm	0.22
D2	Raj et al. 2018’s ab initio average of normal deuterium; Shelton’s D2/H2 and H2/D2 ratios.	112-static	1.4e+02 ppm	0.14
N2	Peck & Khanna; the 633 nm metrology (Birch, Stone, Achtermann, Egan); static values; Griesmann’s vacuum ultraviolet (level free); Shelton’s He/N2 ratio.	145-static	15 ppm	1.16
O2	Egan et al. 2026 at 633 and 1542 nm; Birch; Smith et al. 1976 above the Schumann-Runge bands; Hohm’s 325-633 nm dispersion; Shelton’s N2/O2 ratio.	195-static	5.4 ppm	0.23
CO	Smith et al.’s 167-288 nm interferometry; the Cuthbertsons’ visible dispersion; Watson’s green line; Zhao et al.’s 1566 nm refractometry; HITRAN’s bands.	167-2288 + 2 intervals to 12500	42 ppm	0.37
CO2	Old, Gentili & Peck; the Cuthbertsons and Watson & Ramaswamy (level free); the 633 nm metrology; Egan et al. 2026; Hohm’s dispersion; Shelton’s ratios; Marchetti’s 10 um values as validators; HITRAN’s bands; Chan et al.’s two weak bands below 10 eV as lines.	250-1400 + 7 intervals to 8941	8.1 ppm	4.01
CH4	The Cuthbertsons’ and Watson & Ramaswamy’s dispersion (level free); Achtermann; Burns et al.; Zhao et al.’s 1566 nm value; Shelton’s 14 ratios; Hohm’s 325-633 nm dispersion (level free); HITRAN’s bands.	240-1123 + 4 intervals to 12088	44 ppm	0.68
SF6	Swift; Vukovic et al.; Achtermann; Burns et al.; Shelton’s 8 ratios; HITRAN’s bands.	230-5000	3.3e+02 ppm	4.07
N2O	Hohm’s 325-633 nm dispersion and the Cuthbertsons’ 1914 dispersion (both level free); Birch; Egan et al.; HITRAN’s bands; the Thomas-Reiche-Kuhn sum as a prior.	320-1516 + 3 intervals to 7048	58 ppm	3.01
O3	The measured absorption spectrum by Kramers-Kronig (Serdyuchenko et al. 2014 at 293 K and three others); one fitted strength above 12 eV; Swift et al., the Cuthbertsons’ level and the ab initio static value with Naves et al.’s vibrational correction.	185-3800	0.96 %	0.61

Range:

• The atoms extend to the static limit. An atom has no infrared resonance, so between the near infrared and zero frequency its polarizability is fixed by its dipole sums \(S(-2)\) and \(S(-4)\), which the backbones determine. The mid-infrared points of Rusch (argon, 6.76 µm) and Marchetti (helium, 10.57 µm) agree within their few-per-mille uncertainties.

• The infrared-active molecules (CO, CO2, CH4, N2O) hold on several intervals (OEP 3; Fitting dispersion models). Their bands are Gaussian rules of HITRAN’s lines.

  • Their optical range is split only where the pure gas absorbs. CH4 now ends at 1313 nm and has gaps at 1123-1137, 1149-1177 and 1187.4-1187.9 nm; N2O has gaps at 1516-1518 nm and from 1950 nm.

  • Beyond the optical range they hold again in the windows between the bands, where the bands’ representation meets the tolerance. CO2’s last window is 8.4-8.9 µm: its laser bands at 9.4 and 10.4 µm are kept, so 8.9-12.5 µm, where Marchetti’s values lie, is not a qualified interval (TC-797).

  • The range warning names each gap’s reason. The column gives the first interval, and how many follow and where the last ends.

• Hydrogen and deuterium rest on theory; the experiments’ levels scatter by about 1e-3 about it (TC-95).

• Ozone is modelled differently. Its measured absorption enters by Kramers–Kronig, so the model is complex, and its uncertainty is set by the structure of the absorption spectrum rather than by the fit (see Gas dispersion: accuracy and limitations).

Temperature: the molecules’ polarizabilities refer to 293.15 K (N2, O2, CO2, N2O, H2, D2). Egan et al.’s 303 K values that set the levels of CO2 and O2 are brought there with their own measured coefficients, which moves the models’ \(n - 1\) by -40 ppm (CO2) and -21 ppm (O2). CH4 and CO rest on 297.5 K refractometry and SF6 mostly on 373 K values, whose temperature coefficients are not curated. The atoms need no reference temperature (see the conventions page, TC-730).

The molecules’ qualified intervals

CO, CO2, CH4 and N2O carry their infrared bands as Gaussian rules of HITRAN’s lines, and are qualified on several intervals (OEP 3; Fitting dispersion models, The curated data). For each interval:

• the largest smoothed absorption of the pure gas at 101325 Pa and 296 K in it, at a resolution of 1 cm⁻¹;

• the largest single-line Lorentz peak, so that a user can judge their own path length, pressure and bandwidth;

• the reason for the gap that follows it, as the derivation labels the gap over its whole span.

Inside a fit’s optical range only absorption splits it (TC-749). There the gap begins where the absorption reaches 0.1 m⁻¹, whatever the gap’s label. Beyond the range, the representation of the bands decides (representation; kept band where a band is kept as poles rather than folded), and width marks a window narrower than 50 cm⁻¹.

The models are the background at that resolution, not the resolved structure of the weak lines. A picosecond pulse near a weak band sees structure the model does not hold. The representation bound is a deterministic bound, given beside the statistical sigma and not combined with it (FittedModel.representation_bound_au).

Gas	Interval (µm)	Largest \(a_\sigma\) (m⁻¹)	Largest line peak (m⁻¹)	The gap that follows
CO	0.1673-2.288	0.023	0.27	representation: kept band
CO	2.565-4.34	2.6e-09	4.7e-08	absorption
CO	5.191-12.5	3.5e-05	0.00032	—
CO2	0.2504-1.4	0.0086	0.019	representation
CO2	2.302-2.38	1.2e-06	1.8e-06	representation
CO2	2.381-2.475	1.3e-05	3.5e-05	representation
CO2	2.541-2.59	0.00024	0.0003	representation
CO2	3.203-3.35	3.5e-05	0.00012	representation
CO2	3.357-3.593	0.00011	0.00023	representation
CO2	5.549-6.946	1e-05	9.5e-06	representation: kept band
CO2	8.435-8.941	6.7e-07	1.6e-06	—
CH4	0.2401-1.123	0.09	0.21	width
CH4	1.137-1.149	0.1	0.39	absorption
CH4	1.177-1.187	0.099	0.2	absorption
CH4	1.188-1.313	0.097	0.13	representation
CH4	11.12-12.09	2e-06	5e-06	—
N2O	0.3202-1.516	0.098	0.028	absorption
N2O	1.518-1.95	0.098	0.18	representation
N2O	4.814-4.956	0.0013	0.003	representation
N2O	6.752-7.048	0.0056	0.0098	—

Pure CO2 is not qualified at its laser lines: its 9.4 and 10.4 µm bands are kept, and the last window ends at 8.94 µm.

The intervals are what OEP 3 proposed and implemented: its theory, decisions and phase reports are the record.

The compact coefficients

Each molecule’s compact model (Fitting dispersion models, Compact infrared terms; OEP 2) as printed coefficients, from optmat.gas.models.compact.coefficients and format_coefficients, to eleven significant digits: the rounding then changes the polarizability, \(D_2\) and \(D_3\) by at most 0.002 of the fit’s sigma. The polarizability form goes through the package’s density conversion (Lorentz-Lorenz, the real-gas density and the refractivity virial; Conventions), which keeps the stated tolerance at any density. The Sellmeier form is \(n - 1\) in the dilute limit at 101325 Pa and 293.15 K, the temperature the molecular polarizabilities refer to, with its error against the full model stated: about \(10^{-4}\) of \(n - 1\), mostly the Lorentz-Lorenz term it drops. Each table holds only in its intervals; between and beyond them the gas absorbs, and the compact poles are effective ones, not band centres.

compact/CO, the polarizability form; qualified at 167.3-2000.0 nm. \(\alpha = \sum_k f_k / (E_k^2 - \omega^2)\) in atomic units, \(E_k = hc / \lambda_k\).

Term	Kind	\(f_k\)	\(\lambda_k\) (µm)
effective 2	effective	4.2653378498e+00	7.3643744199e-02
effective 1	effective	1.8512815886e-01	1.4715780853e-01
compact 3	infrared	8.2279160419e-08	2.3394546499e+00
compact 2	infrared	1.6094654462e-08	2.7915109723e+00
compact 1	infrared	1.1641060935e-05	4.6649107831e+00
rotational 0.0	rotational	2.0923184033e-08	2.5412095195e+02

compact/CO, the Sellmeier form at 101325 Pa and 293.15 K (\(N\) = 2.5045083675e+25 m\(^{-3}\)); qualified at 167.3-2000.0 nm. \(n - 1 = \sum_k B_k \lambda^2 / (\lambda^2 - C_k)\); against the full model, at most 8.7e-05 relative in \(n - 1\) and 0.00021 of the largest \(\beta_2\).

Term	Kind	\(B_k\)	\(C_k\) (µm²)
effective 2	effective	2.5983614722e-04	5.4234010597e-03
effective 1	effective	4.5031199478e-05	2.1655420612e-02
compact 3	infrared	5.0581711831e-09	5.4730480591e+00
compact 2	infrared	1.4087526484e-09	7.7925335087e+00
compact 1	infrared	2.8454676784e-06	2.1761392615e+01
rotational 0.0	rotational	1.5176902074e-05	6.4577458222e+04

compact/CO2, the polarizability form; qualified at 250.4-1400.0 nm. \(\alpha = \sum_k f_k / (E_k^2 - \omega^2)\) in atomic units, \(E_k = hc / \lambda_k\).

Term	Kind	\(f_k\)	\(\lambda_k\) (µm)
effective 2	effective	6.4907086934e+00	6.4889749029e-02
effective 1	effective	7.7236988164e-01	1.0667109688e-01
CO2 1Pi_g (1pi_g -> 3s sigma_g), about 9.30 eV	line	7.6027031505e-03	1.3332000000e-01
CO2 1Delta_u (1pi_g -> pi*), about 8.38 eV	line	5.9884165655e-03	1.4795000000e-01
compact 5	infrared	2.1372528019e-09	1.4319392695e+00
compact 4	infrared	8.2792836701e-10	1.4334495406e+00
compact 3	infrared	8.0538606278e-07	2.2958630257e+00
compact 2	infrared	1.1273466434e-04	4.1495064626e+00
compact 1	infrared	1.4322012500e-05	1.3101376123e+01
rotational 0.0	rotational	2.1672907678e-16	6.5691891555e+02

compact/CO2, the Sellmeier form at 101325 Pa and 293.15 K (\(N\) = 2.5169069720e+25 m\(^{-3}\)); qualified at 250.4-1400.0 nm. \(n - 1 = \sum_k B_k \lambda^2 / (\lambda^2 - C_k)\); against the full model, at most 8.8e-05 relative in \(n - 1\) and 0.00017 of the largest \(\beta_2\).

Term	Kind	\(B_k\)	\(C_k\) (µm²)
effective 2	effective	3.0850584292e-04	4.2106795291e-03
effective 1	effective	9.9205999884e-05	1.1378722910e-02
CO2 1Pi_g (1pi_g -> 3s sigma_g), about 9.30 eV	line	1.5253788847e-06	1.7774222401e-02
CO2 1Delta_u (1pi_g -> pi*), about 8.38 eV	line	1.4796569170e-06	2.1889202501e-02
compact 5	infrared	4.9467980443e-11	2.0504500716e+00
compact 4	infrared	1.9203335482e-11	2.0547775854e+00
compact 3	infrared	4.7919816272e-08	5.2709870326e+00
compact 2	infrared	2.1911366006e-05	1.7218403883e+01
compact 1	infrared	2.7749615169e-05	1.7164605631e+02
rotational 0.0	rotational	1.0557468691e-12	4.3154246161e+05

compact/CH4, the polarizability form; qualified at 240.1-1070.0 nm. \(\alpha = \sum_k f_k / (E_k^2 - \omega^2)\) in atomic units, \(E_k = hc / \lambda_k\).

Term	Kind	\(f_k\)	\(\lambda_k\) (µm)
effective 2	effective	4.1267565855e+03	1.5360712677e-03
effective 1	effective	2.5306220540e+00	1.0154430067e-01
compact 5	infrared	6.8182074197e-09	1.1249545641e+00
compact 4	infrared	2.1158851236e-08	1.1576252434e+00
compact 3	infrared	7.4914852172e-08	1.3729279715e+00
compact 2	infrared	1.0832835868e-05	2.8784429688e+00
compact 1	infrared	9.2912506578e-06	6.6302255555e+00
rotational 0.0	rotational	4.8380075997e-11	1.0171149383e+02

compact/CH4, the Sellmeier form at 101325 Pa and 293.15 K (\(N\) = 2.5081503359e+25 m\(^{-3}\)); qualified at 240.1-1070.0 nm. \(n - 1 = \sum_k B_k \lambda^2 / (\lambda^2 - C_k)\); against the full model, at most 0.00012 relative in \(n - 1\) and 0.00021 of the largest \(\beta_2\).

Term	Kind	\(B_k\)	\(C_k\) (µm²)
effective 2	effective	1.0953100472e-04	2.3595149395e-06
effective 1	effective	2.9352417863e-04	1.0311244999e-02
compact 5	infrared	9.7061200818e-11	1.2655227714e+00
compact 4	infrared	3.1895803645e-10	1.3400962041e+00
compact 3	infrared	1.5884330706e-09	1.8849312149e+00
compact 2	infrared	1.0096315379e-06	8.2854339244e+00
compact 1	infrared	4.5944789867e-06	4.3959890917e+01
rotational 0.0	rotational	5.6300480167e-09	1.0345227977e+04

compact/N2O, the polarizability form; qualified at 320.2-1500.0 nm. \(\alpha = \sum_k f_k / (E_k^2 - \omega^2)\) in atomic units, \(E_k = hc / \lambda_k\).

Term	Kind	\(f_k\)	\(\lambda_k\) (µm)
effective 2	effective	2.0888813178e+01	3.3472221509e-02
effective 1	effective	1.1188454137e+00	1.2507109255e-01
compact 5	infrared	7.5438533852e-10	1.5159040405e+00
compact 4	infrared	3.1374048071e-09	1.5174692101e+00
compact 3	infrared	4.6537983265e-07	2.2819960154e+00
compact 2	infrared	6.3167428291e-05	4.2594630608e+00
compact 1	infrared	1.8062801923e-05	8.5288772204e+00
rotational 0.0	rotational	1.0534861604e-08	5.3412336540e+02

compact/N2O, the Sellmeier form at 101325 Pa and 293.15 K (\(N\) = 2.5179981372e+25 m\(^{-3}\)); qualified at 320.2-1500.0 nm. \(n - 1 = \sum_k B_k \lambda^2 / (\lambda^2 - C_k)\); against the full model, at most 8.3e-05 relative in \(n - 1\) and 0.00018 of the largest \(\beta_2\).

Term	Kind	\(B_k\)	\(C_k\) (µm²)
effective 2	effective	2.6429574227e-04	1.1203896127e-03
effective 1	effective	1.9764748866e-04	1.5642778191e-02
compact 5	infrared	1.9576902942e-11	2.2979650600e+00
compact 4	infrared	8.1586379407e-11	2.3027128035e+00
compact 3	infrared	2.7368100121e-08	5.2075058145e+00
compact 2	infrared	1.2942263447e-05	1.8143025567e+01
compact 1	infrared	1.4838027503e-05	7.2741746641e+01
rotational 0.0	rotational	3.3940584367e-05	2.8528776946e+05

compact/SF6, the polarizability form; qualified at 230.0-5000.0 nm. \(\alpha = \sum_k f_k / (E_k^2 - \omega^2)\) in atomic units, \(E_k = hc / \lambda_k\).

Term	Kind	\(f_k\)	\(\lambda_k\) (µm)
effective 2	effective	1.7687344964e+01	5.9307116398e-02
effective 1	effective	5.8154314433e-03	1.8225339665e-01
compact (the parent’s) 3	infrared	3.0068868641e-06	6.3390105109e+00
compact (the parent’s) 2	infrared	2.2162382890e-04	1.0581503679e+01
compact (the parent’s) 1	infrared	1.5200997054e-05	1.6244212056e+01

compact/SF6, the Sellmeier form at 101325 Pa and 293.15 K (\(N\) = 2.5338300116e+25 m\(^{-3}\)); qualified at 230.0-5000.0 nm. \(n - 1 = \sum_k B_k \lambda^2 / (\lambda^2 - C_k)\); against the full model, at most 0.00021 relative in \(n - 1\) and 0.00026 of the largest \(\beta_2\).

Term	Kind	\(B_k\)	\(C_k\) (µm²)
effective 2	effective	7.0697761300e-04	3.5173340554e-03
effective 1	effective	2.1951419072e-06	3.3216300590e-02
compact (the parent’s) 3	infrared	1.3730597305e-06	4.0183054258e+01
compact (the parent’s) 2	infrared	2.8199449285e-04	1.1196822012e+02
compact (the parent’s) 1	infrared	4.5582566469e-05	2.6387442531e+02

Compared with the formulas in common use

• Helium: the largest difference for propagation is its dispersion. Several published or legacy helium formulas give a group-velocity dispersion in the visible and near infrared 5–14 % away from the ab initio value. The fit, Stone & Stejskal’s formula and Puchalski et al.’s own series agree to four digits.

• Argon and krypton: the common formulas are good in the visible, but their vacuum-ultraviolet extensions (unpublished fits distributed with propagation codes) err by 5–15 % near the resonances, where resonant dispersive waves are phase-matched.

• CH4, SF6 and N2O: the published formulas (He et al. 2021) have no infrared term.

  • They miss Shelton’s near-infrared phase-matching ratios by 3 % (CH4) and 28 % (SF6) at 1319 nm.

  • They overstate SF6’s group-velocity dispersion at 800 nm by 55 %.

The Gas dispersion: accuracy and limitations page gives the statistical and empirical uncertainties. The dispersion gallery compares named formulas and fits over their stated ranges.

What the curation found

• A reference state misread. Ermolov et al.’s helium formula is a fit at 1 atm, although the paper says 1 bar. At 1 bar it lies 1.3 % above every source (TC-5).

• Hohm’s absolute levels. His 1994 polarizabilities (Mol. Phys. 81) lie 0.04-0.47 % above the fits (Kr +0.20 %, Xe +0.47 %), and his own 1990 values of the same gases are 0.1-0.3 % lower. His shapes agree: argon’s with theory to about 4e-5 rms (7e-5 at 325 nm). The fits take his dispersion with the level free (TC-400).

• The classical absolute levels are off:

  • argon: Peck & Fisher +0.035 %, Zhang et al. +0.026 % (TC-73);

  • the light gases: the old refractometry is high by 0.1–0.3 %, consistent with traces of heavier gases (TC-90, TC-93).

  The fits take these datasets’ shapes, not their levels.

• Mansfield & Peck’s helium dispersion departs from theory by 1e-4 (TC-89).

• Neon’s ab initio dispersion is 2.5 sigma below every experiment (TC-92). This is the one open physics question of the noble-gas models.

• The magnetic term and the virial term matter at the 1e-5 level, and bring the metrology into agreement: Stone’s helium refractivity at 633 nm closes to 0.8 ppm, and Egan et al.’s argon value to 20 ppm of Lesiuk’s theory.

• Misprints, each corrected with its evidence (107 errata and 13 corrected formula coefficients in all):

  • Bideau-Mehu’s neon and xenon formulas;

  • Börzsönyi’s xenon \(C_1\) (a factor of 1000);

  • Mansfield & Peck’s theoretical formulas;

  • many table values.

  Bideau-Mehu’s krypton formula does not reproduce its own table and is kept as printed.

• Tensions in the molecular fits. The SF6, CO2 and N2O fits carry reduced chi-squared of 4.1, 3.3 and 3.0:

  • SF6’s two sets of Shelton ratios prefer HITRAN’s band strengths times 1.036 (TC-146);

  • N2O’s two visible dispersions, Hohm’s and the Cuthbertsons’, differ in shape by about 1e-4, and the fit follows Hohm’s (TC-178).

Cross-checks

• Shelton’s phase-matching ratios. These density ratios tie the near-infrared dispersion of pairs of gases together to 0.02–0.3 %. The fits use them, with the partner gas held at its recommended model, and every fitted ratio lies within 1.9 sigma.

• Argon, the one inconsistency. Shelton’s set is self-consistent, but the argon model’s He/Ar ratio is 0.3 % below his (TC-96). Fitting his N2/Ar ratio would pull nitrogen onto that difference, so it is a validator.

• Börzsönyi et al.’s group-delay dispersions at 800 nm. The formulas in common use were fitted to these. They scatter about the fits by −12 % (neon) to +21 % (helium, within its 16 % uncertainty), and by +10 % for nitrogen (12 sigma); they are validators (TC-97).

The gallery page Cross-checks shows both.

Robustness

Each fit is refitted without each of its datasets (gallery page Robustness):

• The backbones move their models by many sigma when dropped, by design:

  • helium’s ab initio moments;

  • neon’s and argon’s dielectric values and moments;

  • hydrogen’s and deuterium’s theory.

• Every other dataset (100 in all): 80 move their model by less than 2 sigma anywhere in range and 93 by less than 4. The seven above 4 sigma are each the only data over part of a model’s range, so the shift appears where the model would otherwise be extrapolated:

  • the ultraviolet of CO (Smith et al. 1976, 63 sigma), N2O (Hohm’s 325 nm point, 34), O2 and Kr (Smith et al. 1976, 13 and 5.4) and CH4 (Shelton’s 1986 harmonics, 5.8);

  • CO’s near infrared (Zhao et al.’s 1566 nm value, 7.7);

  • N2O’s level (Egan et al. 2019, 5.6).

• Hydrogen’s level once rested on one 633 nm metrology value. It is now on theory, and three ab initio calculations agree to 0.15 % (TC-95).

• Oxygen’s level, which rested on Birch alone, is now fixed as well by Egan et al.’s 2026 refractometry, in agreement (TC-147).

• Near-degenerate minima. A far effective oscillator enters only through \(f/E^2\), so the fits’ cost has near-degenerate minima. The ones a wider search finds differ from the stored fits by at most 0.44 sigma anywhere in range. CO2’s two effective oscillators were the exception: with the lower one free down to 202 nm, the minimum followed its bound and the ultraviolet dispersion moved by up to 30 % at 250 nm (TC-725). Its two weak bands below 10 eV are now lines with priors from the measured photoabsorption, and the effective oscillators start at 10 eV (TC-726).

The compact coefficients are what OEP 2 proposed and implemented: its theory, decisions and phase reports are the record.