OEP 2 — Compact infrared terms for the molecular gases#

OEP

2

Title

Compact infrared terms for the molecular gases

Author

John Travers

Drafting agent

Opus 5.5

Status

Final

Review status

Reviewed

Reviewers

GPT-6 (AI); John Travers (human)

Type

Feature

Created

2026-09-29

Resolution

2026-09-30: accepted by the author, John Travers, after the human and AI reviews; merged directly to main

Requires

OEP 3: the parents’ qualified intervals and representation bound

Supersedes

—

Superseded-by

—

Implementation

Opus 5.5 (AI), for John Travers. Phases 1 and 2 merged from feature/oep-0002-compact

Manual pages

Fitting dispersion models (Compact infrared terms), The gas models (The compact coefficients)

Part I — Proposal#

1. Abstract#

The fitted models of the infrared-active molecules carry their vibrational bands from HITRAN as fixed poles, one per band: 4296 for CO2, 2518 for N2O, 295 for CH4 and 30 for CO. The model is correct, but it cannot be printed, typed into another code or checked by hand. This OEP proposes a compact model beside each molecular fit, compact/<gas>. It keeps the fit’s electronic terms and covariance unchanged, and replaces the fixed poles by a few effective infrared poles. These are derived from the stored fit by a constrained least-squares fit to its infrared contribution over every interval in which the fit is qualified. The fit keeps two moments of the bands exactly: the optical-side limit and the static contribution.

This OEP is sequenced after OEP 3, which extends the parent fits into the infrared windows between their bands, with several qualified intervals per model and a more accurate representation of each band. The compact models follow whatever intervals the parents qualify. Each compact pole is kept inside an absorbing region, outside every qualified interval.

A feasibility calculation against today’s parents, each qualified on one interval from the ultraviolet to 1.4-2 µm, finds the following:

  • 2 (CO), 3 (CH4), 4 (N2O) and 6 (CO2) poles reproduce the full model everywhere in its range to within 0.06 of its own statistical uncertainty;

  • the check covers the polarizability, the group-velocity dispersion and the third-order dispersion;

  • each molecule then needs 4 to 10 Sellmeier-type terms in all;

  • the reproduction extends to every dataset and validator of the parent fits inside that range, and to its static values, which the constrained moment preserves exactly.

With OEP 3’s windows the number of poles will grow with the band systems that bound them; Phase 1 recomputes it.

The compact models are derived and never fitted to data. They are never a default. They are regenerated and tested whenever their parent fit changes. Success is judged by one stated tolerance, met at every wavelength of every qualified interval. The proposal also defines how the coefficients are exported for print: the polarizability poles with the density conversion, and, as a convenience, an \(n - 1\) Sellmeier form at a stated state, with its error against the full model stated.

2. Motivation and scope#

The use. Three readers need the molecules’ dispersion in a closed form:

  • a publication of optmat’s gas models, whose tables must let a reader compute \(n(\lambda)\) without the package;

  • propagation codes that do not call optmat at run time and embed a formula;

  • anyone checking a number by hand against a published formula.

The alternatives in the literature for these gases have no infrared term. He et al.’s UV-visible formulas for CH4, SF6 and N2O [10] overstate SF6’s group-velocity dispersion at 800 nm by 55 % and miss Shelton’s near-infrared phase-matching ratios by 3-28 % (docs/gas_models.md, TC-104). Air formulas with infrared terms exist, but not for the pure molecular gases (§3).

In scope:

  • compact models for CO, CO2, CH4 and N2O, derived from their stored fits and qualified over each fit’s qualified intervals (Model.intervals, OEP 3; today a single interval, Model.wavelength_range);

  • SF6, whose three poles are already compact, carried through unchanged so that every molecule has the same interface;

  • the derivation tool, the stored compact models and their tests;

  • an export of the coefficients in two printed forms, and a manual section with the tables.

Out of scope:

  • Qualifying the parents in the infrared windows. That is OEP 3.

    • Today’s parents declare one interval each; CO2’s is 250-1400 nm. Its 8-12 µm window appears only in its configuration’s description and on the accuracy page, and is not qualified.

    • There, the parent’s one pole per band puts 224 poles among the lines, which alone contribute a median of 2.5 times its statistical sigma (§8). No few-pole model can follow that, and the parent is not accurate there either.

    • OEP 3 replaces those poles, folds the weak in-window bands and qualifies the windows. The compact models then cover them.

    • Until then, the compact CO2 model warns in the window, like the parent, and differs from it by up to 3 % in the polarizability.

  • Data wavelengths outside every qualified interval. Some parent fits use data beyond their declared range today: CO2 uses Old et al.’s points to 1529 nm and Egan et al.’s 1542 nm, and CH4 uses Zhao’s 1566 nm. The compact model reproduces these as reported numbers (§8), without a promise. After OEP 3 most of them lie in qualified windows.

  • Ozone. Its model takes the measured absorption by Kramers-Kronig and is complex (docs/gas_models.md); it is not a sum of fixed poles.

  • Changing the full models. The compact model represents the stored fit. It does not refit the electronic terms or the HITRAN strengths. The band representation is OEP 3’s (§11, §13 Q3).

  • The absorption region. Like the full model, the compact one describes the index away from the bands. It is not a model of the infrared absorption.

  • Temperature dependence. HITRAN’s strengths refer to 296 K, and the molecular polarizabilities to 293.15 K (TC-730), as in the full models.

  • Air and water vapour. optmat has no air or water model (OEP 1 proposes an air model; the compact terms would compose with it).

3. Context#

What optmat does today.

  • optmat.gas.infrared reads the band totals of data/external/hitran/<gas>_bands.csv. Per band these are the sum of the line intensities \(S\) and of \(S/\tilde\nu^2\), and the span of its lines (min_cm, max_cm). Each band becomes one pole that keeps the band’s total strength and its static contribution. Pure rotational bands are left out (TC-156). See docs/data.md, Infrared bands (HITRAN).

  • A fit configuration’s [model] infrared_max_wavenumber adds the bands up to that wavenumber as fixed trailing terms of the fitted model, outside its covariance (docs/fitting.md, FittedModel, n_fixed).

  • The molecular fits (docs/gas_models.md), as stored on 2026-09-30:

    Fit

    Electronic terms

    Fixed poles

    Highest pole

    Declared range

    Its lowest frequency

    fit/CO

    2

    30

    4262.0 cm⁻¹

    167-2000 nm

    5000 cm⁻¹

    fit/CO2

    4 (2 lines, 2 effective)

    4296

    6996.7 cm⁻¹

    250-1400 nm

    7142.9 cm⁻¹

    fit/CH4

    2

    295

    4499.7 cm⁻¹

    240-1400 nm

    7142.9 cm⁻¹

    fit/N2O

    2

    2518

    4499.5 cm⁻¹

    320-2000 nm

    5000 cm⁻¹

    fit/SF6

    2

    3 (cross-sections)

    1577.5 cm⁻¹

    230-5000 nm

    2000 cm⁻¹

    Every declared range lies above every fixed pole. The closest case is CO2: its highest pole is 2.0 % below the range’s lowest frequency. OEP 3 changes this inventory: every band of the tables kept, several poles per band, and several qualified intervals with bands between them.

Register entries.

  • TC-104: the HITRAN strengths are checked against Shelton’s phase-matching ratios.

  • TC-146: the CH4, SF6 and N2O fits with the infrared bands.

  • TC-148: the CO2 fit and its range.

  • TC-156: N2O’s rotational (orientation) term is left out.

  • TC-720: Voronin & Zheltikov’s generalized Sellmeier equation for air, the closest precedent (below).

  • TC-721 and TC-722: Mathar’s line-by-line models of humid air.

  • TC-723: Tang’s bounds on one-term formulas, and what they say of the per-band pole.

  • TC-725 and TC-726: the refits of fit/CO2, and its electronic form (two weak bands as lines, two effective oscillators from 10 eV).

  • TC-730: the molecules’ reference temperature, 293.15 K.

Prior work (curated as method records):

  • Voronin & Zheltikov [1] represent air’s infrared (HITRAN 2012, about 650,000 lines) by two poles per band, 15 bands and 30 poles in all. Only each band’s total strength is constrained; the pole wavelengths are fitted. \(n - 1\) is linear in the partial densities, without Lorentz-Lorenz, over 0.3-13 µm. Their CO2 terms are the only separate infrared terms for CO2 in the literature. Against optmat’s HITRAN 2024 bands they are 1.7 % low in strength and 1.8 % low in static contribution, a match to the principal isotopologue alone (TC-720; an inference, not stated by them). Their per-species electronic terms are tuned as a set to Ciddor’s air formula and are not per-species polarizabilities (argon 6.5 % low; TC-720).

  • Mathar sums every HITRAN line of water and CO2, with ultraviolet pseudospectra and no fitting [2]. His later fits of humid air are polynomials in wavenumber, one per window [3]. He reports that a two-resonance Sellmeier form fitted better at a fixed state, but that expanding its parameters in temperature, pressure and humidity “did not achieve competitive stability” ([3], p. 473; TC-722).

  • Tang [4] proves bounds on one-term dispersion formulas for positive oscillator distributions; Tang & Poon [5] generalise them to several moments. §4 applies them to the infrared poles.

What is new here.

  • Per molecule rather than per mixture.

  • A polarizability with optmat’s density conversion (Lorentz-Lorenz, real-gas density, refractivity virial), rather than \(n - 1\) linear in density.

  • Few poles for all the bands together, fitted on the qualified intervals, with two moments exact and the error stated against the model’s own uncertainty.

  • All isotopologues, and electronic terms fitted to measurements with a covariance.

Mathar’s instability does not arise: the poles are properties of the molecule, and the state enters only through the density.

Related OEPs. OEP 3 (Draft) extends the parents into the infrared windows and is to be implemented first (§12). OEP 1 (Draft) declares the propagation-code interface.

Roadmap. None of the informal items covers this; the draft is listed under Draft enhancement proposals.

4. Theory#

The infrared terms. In SI, the electronic and vibrational oscillators of a molecule give the dynamic dipole polarizability

\[ \alpha(\omega) = \frac{e^2}{m_e}\sum_k \frac{f_k}{\omega_k^2 - \omega^2} \]

(C m² V⁻¹; in atomic units \(\alpha = \sum_k f_k/(E_k^2 - \omega^2)\), the form the code uses), with dimensionless oscillator strengths \(f_k\). A vibrational line of integrated intensity \(S\) (HITRAN’s definition, cm/molecule at 296 K, which includes the lower-state population and stimulated emission) has \(f = S/(\pi r_e)\), with \(r_e\) the classical electron radius in cm (optmat.gas.infrared). Write the infrared part as \(\alpha_\mathrm{IR}(\omega) = (e^2/m_e)\sum_{b} f_b/(\omega_b^2 - \omega^2)\), summed over the \(B\) fixed poles of the stored fit, and its moments

\[ M_k = \sum_b f_b\,\omega_b^{k}. \]

Above every band (\(\omega > \max_b \omega_b\), which holds throughout today’s declared ranges, §3)

\[ \alpha_\mathrm{IR}(\omega) = -\frac{e^2}{m_e}\left(\frac{M_0}{\omega^2} + \frac{M_2}{\omega^4} + \frac{M_4}{\omega^6} + \cdots\right), \]

and below every band \(\alpha_\mathrm{IR}(\omega) = (e^2/m_e)(M_{-2} + M_{-4}\omega^2 + \cdots)\). \(M_0\) is the vibrational Thomas-Reiche-Kuhn sum and fixes the optical-side limit. \((e^2/m_e) M_{-2}\) is the vibrational static polarizability. In OEP 3’s windows, bands lie on both sides of the frequency, and neither expansion holds. The reduction below does not rely on either: it fits the parent’s terms directly on the qualified intervals.

The parent’s representation of a band. Today one pole per band, a lower bound. OEP 3 replaces it by a Gaussian rule of a few poles per band, built from the lines. The reduction below represents whatever the parent stores, and inherits its representation without adding to it. Today’s pole for band \(b\) takes \(f_b = \sum_{\ell\in b} f_\ell\) and \(\omega_b^2 = \sum f_\ell / \sum (f_\ell/\omega_\ell^2)\), keeping the band’s \(M_0\) and \(M_{-2}\). That is Tang’s construction [4] applied to one band: below the band the pole bounds the band’s contribution from below, and above it the pole bounds its magnitude from below. Above the band, the pole’s magnitude falls short of the band’s by, to leading order,

\[ \frac{e^2}{m_e}\,\frac{M_2^{(b)} - \left(M_0^{(b)}\right)^2/M_{-2}^{(b)}}{\omega^4} \;\ge\; 0 , \]

non-negative by the Cauchy-Schwarz inequality. For CO2’s 4.3 µm band it is about \(6\times10^{-6}\) of the band’s contribution at 633 nm (TC-723). At the edges of the infrared windows the error is far larger, which is why OEP 3 replaces it.

The compact representation. Replace the \(B\) poles by \(K \ll B\) poles \((F_j, \Omega_j)\), \(F_j \ge 0\), chosen to minimise

\[ \chi^2_\mathrm{red} = \sum_{i}\sum_{m\in\{0,2,3\}} \left(\frac{D_m[\alpha_\mathrm{IR}^{K}](\omega_i) - D_m[\alpha_\mathrm{IR}](\omega_i)} {\sigma_m(\omega_i)}\right)^2 \quad\text{subject to}\quad \sum_j F_j = M_0,\;\; \sum_j \frac{F_j}{\Omega_j^2} = M_{-2}, \]

over a grid \(\omega_i\) spanning every qualified interval, where \(D_0[\alpha] = \alpha\) and \(D_m[\alpha] = \partial^m(\omega\alpha)/\partial\omega^m\) for \(m = 2, 3\). These are proportional to the group-velocity dispersion \(\beta_2\) and the third-order dispersion \(\beta_3\) in the dilute limit (dispersion_sigma_relative()). The two equalities keep the optical-side limit and the static contribution exactly, so the compact model has the parent’s static polarizability. The strengths enter linearly. For given \(\Omega_j\) they follow from an equality-constrained non-negative linear least-squares problem (variable projection, §7). Only the \(K\) energies are searched, each within one absorbing region (below).

The weights. \(\sigma_m(\omega)\) is the parent’s absolute one-sigma statistical uncertainty of \(D_m\) of the full polarizability, \(\sqrt{g_m^\mathsf{T}\,\Sigma\,g_m}\), with \(g_m\) the gradient of \(D_m\) with respect to the fitted parameters and \(\Sigma\) their covariance. The fixed infrared terms carry no statistical uncertainty.

  • OEP 3’s representation bound on the parent’s infrared terms (representation_bound_au) is a deterministic bound, not a standard deviation. It is not combined with \(\sigma_m\).

  • The compact model reproduces its parent. Its error against OEP 3’s line-sum reference is at most the parent’s bound plus the compact model’s own residual, and both are stated.

  • OEP 3’s rotational band system is carried unchanged, outside the reduction. The moments \(M_0\) and \(M_{-2}\) constrained here are those of the vibrational systems. The covariance is scaled by the reduced chi-squared where that exceeds one, the Birge convention of the parent’s own uncertainties (birge=True, the default of polarizability_sigma_au()). The absolute sigma is finite where \(D_2\) crosses zero: at SF6’s zero-dispersion wavelength, 1271.2 nm, it is 0.0284 au, against a largest \(|D_2|\) of 2576 au over the range. No floor is needed. For \(m \ge 2\) this sigma equals the package’s relative uncertainty times \(|D_m|\) wherever \(D_m \ne 0\).

Where the compact model is valid. The compact model is qualified on the parent’s qualified intervals. The frequencies outside them divide into absorbing regions:

  • below the lowest interval;

  • between two adjacent intervals, where a band system absorbs.

Only regions that contain parent poles count. Each \(\Omega_j\) is bounded to lie within one such region, between its lowest and highest parent pole, and \(K\) is divided among the regions. So no compact pole can fall inside a qualified interval. This follows from the bounds, independent of the sampling grid, and the tests assert it directly.

On today’s parents there is one region, below the single interval: every declared range lies above \(\max_b \omega_b\) (§3).

The density conversion. The tolerance is stated on the polarizability and its derivatives, which do not depend on the density; \(\beta_2\) and \(\beta_3\) are proportional to \(D_2\) and \(D_3\) in the dilute limit. At a finite density the parent and the compact model pass through the same conversion (Lorentz-Lorenz, the real-gas density, the refractivity virial). To first order in the difference, the relative error in \(n - 1\) is then the relative error in \(\alpha\), and the ratio of error to sigma is unchanged, up to a relative correction of the order of the susceptibility. §8 measures it: \(7\times10^{-5}\) at 1 bar and \(8\times10^{-4}\) at 10 bar, of itself. The tolerance is therefore labelled a dilute-limit tolerance, and a test repeats the comparison at 10 bar.

Why not moment matching alone. A \(K\)-point Gaussian quadrature of the band distribution in the variable \(x = \omega_b^2\), with weights \(f_b\), has positive weights and matches \(2K\) moments in \(x\): \(M_0, M_2, \ldots, M_{4K-2}\), the coefficients of the optical-side expansion above. Its error is governed by the next term of that expansion in \((\omega_b/\omega)^2\). The expansion converges slowly at the long-wavelength end of a range: CO2’s highest pole, at 6996.7 cm⁻¹, lies 2.0 % below the range’s end at 1.4 µm (7142.9 cm⁻¹). The feasibility calculation (§8) confirms this. A 6-node rule on the current CO2 parent misses by 23 sigma in \(D_2\) and 7320 sigma in \(D_3\), and a 10-node rule by 1.1 and 469 sigma. Moment matching is kept as a starting point of the search.

The printed forms. The compact polarizability is a finite sum of poles, so the package’s density conversion applies unchanged (Conventions). For print, a dilute-limit Sellmeier form at a stated state \((p, T)\) follows from \(f/(\omega_k^2 - \omega^2) = (f/\omega_k^2)\,\lambda^2/(\lambda^2 - \lambda_k^2)\):

\[ n - 1 \approx \frac{N\alpha}{2\varepsilon_0} = \sum_k B_k \frac{\lambda^2}{\lambda^2 - C_k},\qquad B_k = \frac{N e^2 f_k}{2\varepsilon_0 m_e \omega_k^2},\quad C_k = \lambda_k^2 , \]

with \(\lambda\) the vacuum wavelength and \(N\) the real-gas number density at the stated state (optmat.gas.eos). The infrared terms have \(\lambda < \lambda_k\) and contribute negatively. The form drops Lorentz-Lorenz, whose leading relative correction to \(n - 1\) is \((n-1)/6\), and the refractivity virial. The export does not rely on that estimate. It states the form’s error as the largest relative difference in \(n - 1\) and in \(\beta_2\) from the full parent model at that state, over the qualified intervals, computed directly. For the four molecules at 101325 Pa and 293.15 K the difference in \(n - 1\) is 0.5 to \(1.2\times10^{-4}\), the same order as the parent’s statistical sigma (§8). The Sellmeier form is therefore a convenience at its state. The polarizability form carries the stated tolerance.

Conventions. SI throughout, vacuum wavelengths, and the refractive polarizability as defined on the conventions page: electric plus \(\chi_0\), with \(\chi_0 = 0\) for the molecules here. The effective infrared poles are not band centres, and are not to be read as spectroscopy.

5. Primary sources and data#

  • No new data. The compact models derive from the stored fits (data/gas/models/fitted/<gas>.json), which already contain the HITRAN 2024 [6] band poles (data/external/hitran/, TC-104, TC-148). No transcription, extraction or vocabulary quantity is needed.

  • The tolerance is set by the parent fit’s covariance (§4, §7), not by external data.

  • Reproduction checks. The compact model is a representation of its parent, so its checks test the reproduction, not the parent’s physics.

    • Every fitted dataset and validator of the parent is re-evaluated with the compact infrared terms, through the fit’s own observation code, at its own wavelengths (harmonics included). The parent’s fitted scales and partner models stay fixed.

    • The static validators test the constrained \(M_{-2}\): the CH4 and CO2 dielectric values and CO’s total static polarizability.

    • Marchetti & Simili’s CO2-laser values [7] and Shelton’s ratios [8, 9] enter the parent fits, so they are reproduction checks, not independent validation.

  • Validation of the physics is the parent’s (its validators, TC-104, TC-146, TC-148, and OEP 3’s infrared validators). The compact model inherits it within the qualified intervals, and §8 shows the difference there is negligible against every datum’s uncertainty.

  • Methods cited: Voronin & Zheltikov [1], Mathar [2, 3], Tang [4] and Tang & Poon [5], all curated.

  • Uncertainty. The compact model carries the parent’s covariance of the electronic terms. Its infrared poles are fixed, as the HITRAN poles are. The reduction error is stated per gas as the largest residual, in units of the parent’s sigma, of \(\alpha\), \(D_2\) and \(D_3\) over the qualified intervals. It is recorded in the stored compact model and quoted in the manual.

6. Proposed interface#

  • Models. compact/<gas> for CO, CO2, CH4, N2O and SF6.

    • Each is loaded by optmat.gas.models.compact.compact_models() and joins optmat.gas.models.registry beside the fitted, published and comparison models. It is listed by registry.models(gas) and returned by registry.model("compact/CO2"), and never by registry.default_model.

    • It is a FittedModel with the parent’s leading terms and covariance and \(K\) fixed trailing terms. So polarizability, dispersion_au, polarizability_sigma_au and dispersion_sigma_relative work unchanged.

    • Its qualified intervals are the parent’s: intervals (OEP 3) and the envelope wavelength_range. Outside every interval, ModelRangeWarning, as for the parent. Today each parent has a single interval.

    • Its record names the parent (fit/<gas>), the SHA-256 of the stored parent record, the parent’s config_sha256, the digest of the reduction configuration, the tolerance and the achieved residuals.

  • Storage and configuration.

    • The stored models are data/gas/models/compact/<gas>.json, written by tools/fit/compact.py <gas> from the stored parent.

    • The reduction’s settings are a separate file, data/gas/compact/<gas>.toml: \(K\), the constraints, the grid and the tolerance.

    • The parent’s configuration is not touched, so its digest, stored fit and robustness record do not change (§13 Q11).

  • Export.

    from optmat.gas.models.compact import coefficients
table = coefficients("CO2")                  # the polarizability form
table = coefficients("CO2", form="sellmeier", pressure=101325.0, temperature=293.15)

    • coefficients returns a frozen CompactCoefficients:

      • the model and parent names and their digests;

      • form, and for the Sellmeier form the state (Pa, K) and the number density used (m⁻³, from optmat.gas.eos);

      • the term labels, and kinds (line, effective or infrared);

      • for the polarizability form, strengths \(f_k\) (dimensionless) and wavelengths \(\lambda_k\) (m), and the constant term \(\chi_0\) (C m² V⁻¹; zero for these molecules);

      • for the Sellmeier form, B (dimensionless) and C (m²);

      • the intervals (m) in which the coefficients are qualified. A printed table states them, and says that between them the gas absorbs;

      • the reduction’s achieved residuals, and for the Sellmeier form its stated error (§4): the largest relative differences in \(n - 1\) and in \(\beta_2\) from the full parent at that state, over those intervals;

      • the covariance of the electronic terms, transformed to the printed parameters.

    • Terms are ordered by descending energy (ascending \(\lambda_k\)): electronic, then infrared.

    • Pressure is in Pa and temperature in K, both within the range where the gas’s equation of state is valid.

    • A formatter writes the printed tables with 10 significant digits, the number the round-trip test requires (§8).

  • Refused. compact/<gas> for a gas without fixed infrared terms (atoms, H2, D2, N2, O2) and for O3: OptmatError.

7. Implementation strategies#

Where. In optmat.gas: a module optmat.gas.models.compact (loading, export) and the tool tools/fit/compact.py (derivation). The registry’s _all_models gains the compact models. There are no core changes, no vocabulary changes and no new dependencies (NumPy and SciPy’s least_squares, already used by the fits).

The derivation. Recommended: constrained least squares on the qualified intervals.

  • The inner problem. For fixed energies \(\Omega\), minimise \(\lVert A F - b\rVert^2\) subject to \(C F = d\) (the \(M_0\) and \(M_{-2}\) rows) and \(F \ge 0\).

    • Solve it exactly by enumerating the supports \(S \subseteq \{1..K\}\). On each support, solve the equality-constrained least squares by the null-space method (a particular solution of \(C_S F_S = d\), plus the least-squares solution in the null space of \(C_S\)).

    • Keep the supports whose solution is non-negative and meets the equalities to a relative \(10^{-10}\), and take the one of lowest cost.

    • The optimum of this convex problem is the equality-constrained minimiser on its own support, so the enumeration finds it. The \(2^K\) supports (\(\le 1024\) for \(K \le 10\)) cost milliseconds.

    • For larger \(K\), which OEP 3’s windows may need, an active-set solver of the same problem replaces the enumeration. It is held to the same moment tolerance, and checked against the enumeration at \(K \le 10\).

    • Strengths are scaled by \(M_0\) for conditioning.

    • Penalty rows are not used: they do not make a moment exact.

  • Feasibility. With both equalities, a solution with \(F \ge 0\) exists exactly when \(\min_j \Omega_j \le \bar\omega \le \max_j \Omega_j\), with \(\bar\omega = (M_0/M_{-2})^{1/2}\). The search guarantees this by its bounds.

    • At least one energy is bounded to parent poles below \(\bar\omega\), and one to parent poles above it, each within its absorbing region.

    • The others lie anywhere within their regions.

    • For \(K = 1\) the pole is fixed at \(\bar\omega\) (Tang’s pole for all the bands together). This is possible only when one region holds every parent pole.

  • The outer problem. Bounded nonlinear least squares in \(\log\Omega\), on the weighted residual vector of the inner solution.

  • Starting points (all tried, with a recorded seed), within each absorbing region for its share of \(K\):

    • the Gaussian rule in \(x = \omega_b^2\) of the region’s poles (§4);

    • the energies of the region’s strongest bands at least 5 % apart;

    • two log-spaced spreads over the region;

    • for CO2, twelve log-uniform random starts. Without them the \(K = 6\) search stopped in a local minimum with a pole of zero strength.

  • Selection. Among the converged candidates, those meeting every tolerance on the check grid are preferred, and of those the lowest cost is kept. The lowest cost alone can have a worse largest residual and fail a threshold that another candidate passes. If none passes, \(K\) is raised.

  • \(K\), and its division among the regions, is the smallest that meets the tolerance. It is found region by region and recorded in the reduction configuration.

  • The derivation is deterministic: a fixed grid, fixed starts and seed, and a fixed selection rule.

The tolerance. Recommended:

  • the reduction: every residual below 0.1 of the parent’s sigma in \(\alpha\), \(D_2\) and \(D_3\) (§4), at every point of the check grid over every qualified interval. The sigma is the parent’s statistical one (§4). In OEP 3’s windows the parent’s representation bound is stated beside it;

  • rounding: the printed coefficients, read back from their decimal strings, change the same quantities by less than 0.01 sigma;

  • reproduction: every datum of the parent’s datasets and validators whose wavelengths all lie in qualified intervals, or at shorter wavelengths than the first (harmonics in the ultraviolet), moves by less than 0.1 of its own sigma. Its chi-squared change is reported rather than bounded, since that change, \(2 r^\mathsf{T}\delta + \delta^\mathsf{T}\delta\), depends on the parent’s residuals \(r\);

  • static values: equal to the parent’s to a relative \(10^{-10}\) (the constrained \(M_{-2}\)).

Alternative: a fraction of the working estimate. It is looser and not computable, since the working estimates are judgements.

The grid. Geometric in wavelength over each qualified interval: 400 points fitted per interval, and the residual checked on four times as many. The per-datum checks of §5 add the data wavelengths.

Option B, moment matching alone. Deterministic and parameter-free, with guaranteed positivity. It fails the tolerance near the long-wavelength end (§4, §8), and has no meaning in a window with bands on both sides. It is kept as a starting point, not the method.

Option C, a joint refit. Fit the compact infrared poles together with the electronic terms to the data. Rejected: it makes a second, different model with its own uncertainty. The compact model should be a faithful representation of the published fit.

Cost. On today’s parents, seconds per gas for CO, CH4, N2O and SF6. CO2, with its 4296 poles on the grid and its extra starts, takes about 30 s. OEP 3’s parents have more poles and more intervals: minutes per gas are acceptable, since the derivation runs only when a parent changes. Checking a stored compact model’s residuals and its parent’s digest takes well under a second and runs in the default test lane. The re-derivation runs in the fits lane.

8. Verification and validation#

  • Feasibility on today’s parents (computed 2026-09-30 against the stored fits after TC-730, with the method and tolerance of §7; one qualified interval each). It shows that the method works and what it costs. Phase 1 recomputes it on OEP 3’s parents, with their windows. The last three columns are the worst residual over the declared range, on the check grid, in units of the parent’s sigma:

    Gas

    Parent terms

    \(K\)

    Poles (cm⁻¹)

    Terms in all

    \(\alpha\)

    \(D_2\)

    \(D_3\)

    CO

    2 + 30

    2

    2144.29, 4261.20

    4

    0.0005

    0.006

    0.057

    CH4

    2 + 295

    3

    1309.87, 3026.07, 4290.76

    5

    < 0.0001

    < 0.0001

    0.0002

    N2O

    2 + 2518

    4

    999.58, 2222.49, 3521.16, 4410.52

    6

    0.0005

    0.0002

    0.006

    CO2

    4 + 4296

    6

    718.72, 2383.86, 4140.41, 6288.96, 6916.97, 6968.31

    10

    0.0002

    0.0002

    0.011

    SF6

    2 + 3

    3

    615.60, 945.05, 1577.53 (the parent’s, unchanged)

    5

    0

    0

    0

    • One pole fewer fails in every case, worst in \(D_3\): CO \(K = 1\) by 2700 sigma, CH4 \(K = 2\) by 2.2, N2O \(K = 3\) by 2.6 and CO2 \(K = 5\) by 0.26.

    • \(M_0\) holds to a relative \(10^{-11}\) and \(M_{-2}\) to \(2\times10^{-15}\). No energy lies on a bound, and every energy lies below its range’s lowest frequency (§3).

    • Parents (SHA-256 of data/gas/models/fitted/<gas>.json):

      • CO bebaa3c0a2524636469a306ca6134f3f04813887796b8662fdc9fc9040b2f9be;

      • CH4 45136078db914ef910dd5d60ff2e3e762ebb7e27f38f6ff2276c4e848193ebbd;

      • N2O 89b74841e95bbf7fa9e7c4457fdbf15a6a1bc8a1d0a06e410de5ac35ae914576;

      • CO2 aef7f296e07513017a19d99e04f1a0cf95053105026dc60d4da17c14b881f17b;

      • SF6 91a123f67c5a599d499672e58508db7235e11963606a9404ee6d21773d5b88c5.

    • The infrared coefficients, \(F_j\) and \(\Omega_j\) in hartree, to evaluate the residuals independently:

      Gas

      \(F_j\)

      \(\Omega_j\) (hartree)

      CO

      1.165240679e-05, 8.649074412e-08

      9.770118942e-03, 1.941543689e-02

      CH4

      6.018792862e-06, 1.296480419e-05, 8.835528366e-07

      5.968208577e-03, 1.378776856e-02, 1.955015025e-02

      N2O

      1.051371077e-05, 6.867801594e-05, 2.361823676e-06, 7.821345467e-08

      4.554409104e-03, 1.012639765e-02, 1.604360791e-02, 2.009579522e-02

      CO2

      1.232725699e-05, 1.141932460e-04, 1.340842468e-06, 1.597703545e-09, 1.450522007e-10, 1.782683831e-09

      3.274751732e-03, 1.086165586e-02, 1.886509979e-02, 2.865459354e-02, 3.151604114e-02, 3.174995254e-02

    • The sigmas were checked against the package’s own polarizability_sigma_au and dispersion_sigma_relative times \(|D_m|\), which they equal.

  • Reproduction of the parents’ data (the compact infrared terms in the fit’s own observation code, fitted scales and partner models fixed):

    • inside each declared range, the largest change of any datum is 0.0002 of its sigma (CO 0.00015, CH4 0.00002, N2O 0.00018, CO2 0.00017);

    • the static validators are equal to a relative \(4\times10^{-16}\);

    • beyond the range (reported, not asserted):

      • CH4 at 1566 nm, 0.00003 sigma;

      • CO2 at 1476-1529 nm (Old et al.), 1.3 sigma, and at 1542 nm (Egan et al. 2026), 0.15 sigma;

      • CO2 at 9.2-10.7 µm (Marchetti, Willetts, Simmons), 1.2 to 10 sigma, up to 3 % in \(\alpha\).

  • CO2’s window on today’s parent (the grounds of §2, and one motivation of OEP 3): 224 parent poles lie in 833-1250 cm⁻¹, carrying \(1.45\times10^{-5}\) of \(M_0\). The lines of 258 kept bands reach into the window and together cover all of it. On a 4001-point grid, those 224 poles alone contribute a median of 2.5 sigma, and more than 0.1 sigma at 99.5 % of the points.

  • Rounding. Coefficients rounded to \(d\) significant digits (the electronic and infrared \(f_k\) and \(\lambda_k\)) change the worst residual by the following amounts.

    • CO2: 16 sigma at 6 digits, 0.32 at 8, 0.034 at 9 and 0.0034 at 10. \(D_3\) near 1.4 µm needs the digits.

    • CO, CH4 and N2O: below 0.01 sigma from 7 or 8 digits.

    • So the printed tables use 10 significant digits.

  • Finite density. At 293.15 K and 1 and 10 bar, the relative error in \(n - 1\) of the compact model against the parent equals the relative error in \(\alpha\) to \(7\times10^{-5}\) and \(8\times10^{-4}\) of itself (§4).

  • The Sellmeier form at 101325 Pa and 293.15 K, against the full parent (with Lorentz-Lorenz, the real-gas density and the virial). The relative error in \(n - 1\) is:

    Gas

    Relative error in \(n - 1\)

    CO

    \(-8.7\) to \(-5.1\times10^{-5}\)

    CH4

    \(-1.2\) to \(-1.1\times10^{-4}\)

    N2O

    \(-8.4\) to \(-7.6\times10^{-5}\)

    CO2

    \(-8.8\) to \(-8.0\times10^{-5}\)

    This is mainly Lorentz-Lorenz. At 273.15 K it is about 8 % larger.

  • Invariants:

    • \(M_0\) and \(M_{-2}\) are exact to a relative \(10^{-10}\);

    • every \(F_j \ge 0\);

    • every \(\Omega_j\) lies within its absorbing region, outside every qualified interval: an analytic check, independent of the grid;

    • the compact model equals the parent at every electronic term and in its covariance.

  • Freshness: a default-lane test checks each stored compact model’s parent SHA-256 and reduction digest against the files, and its recorded residuals on the check grid. A fits-lane test re-derives each compact model and compares it with the stored one, as test-fits does for fits.

  • Tolerance: a test evaluates the residuals on the check grid and asserts the tolerance of §7, and repeats the \(n - 1\) comparison at 10 bar.

  • Reproduction: a test asserts the per-datum and static tolerances of §7, through the fit’s observation code at the data’s own wavelengths, harmonics included.

  • Round trip: a test formats both printed forms, parses the decimal strings back and asserts the rounding tolerance. For the Sellmeier form it asserts the stated error against the full parent at its state.

  • Parity: solprop does not consume these models today, so there is no parity fixture.

9. Documentation#

  • docs/fitting.md: a section Compact infrared terms with the method and the tolerance. docs/data.md links to it from Infrared bands (HITRAN).

  • docs/gas_models.md: a table of the compact coefficients per gas, generated by the formatter and tested against the stored compact models like the page’s other tables. It states the qualified intervals, and that the gas absorbs between them.

  • A gallery example comparing each compact model with its parent: residuals in sigma units over the qualified intervals, and the infrared share of the dispersion.

  • Docstrings with units and ranges on coefficients, CompactCoefficients and the loader.

10. Compatibility and impact#

  • Nothing changes for existing users. The defaults, the parent fits, their configurations, digests, stored results and robustness records are untouched: the reduction has its own configuration (§6).

  • registry.models(gas) lists one more model per molecule. Code or pages that iterate over every model (the gallery’s comparisons) will include the compact models. That is intended, but it must be reviewed.

  • Every refit of a molecular parent requires regenerating its compact model. The freshness test fails until that is done, and AGENTS.md’s definition of done gains the step in Phase 1.

  • If OEP 1 is accepted, the compact models carry its identity and capability flags like every other model.

  • Sequencing. OEP 3 changes every molecular parent: its bands, its poles and its intervals. Deriving compact models before it would mean deriving them twice. So this OEP is implemented after OEP 3’s last phase. Nothing here depends on OEP 3’s details beyond the parents’ intervals and representation bound; with one interval and no bound, it reduces to the calculation of §8.

  • No register entries are superseded.

11. Alternatives considered#

  • Two poles per band (Voronin & Zheltikov [1]). It keeps more structure near each band, but for CO2 it needs 8 poles for 4 bands and grows with the bands kept. It does not reduce across bands, and nothing here needs the band structure.

  • Store more moments per band (Tang [4], Tang & Poon [5], TC-723). Each band’s \(M_2\), one more column from the same line lists, gives two-sided bounds on the band’s contribution. An exact two-point rule per band needs four moments of the measure \(f/\omega^2\): \(M_{-2}\), \(M_0\), \(M_2\) and \(M_4\). With \(M_2\) alone, a two-point rule needs one node fixed, at a band edge (a Radau rule). Either improves the full model near the bands, not the compactness. OEP 3 takes this up, as a Gaussian rule of several nodes per band built from the cached line lists (§13 Q3).

  • Window polynomials (Mathar [3]). Valid only inside each window, with no static or high-frequency limit and no physical interpretation. Rejected.

  • Line-by-line sums at run time (Mathar [2]). Exact, but the opposite of compact.

  • Leaving the infrared out of printed formulas (He et al. [10]). This is the status quo that the motivation shows to be wrong by up to 55 % in SF6’s GVD.

12. Phases#

Both phases follow OEP 3’s Phase 3, the refitted molecular parents.

  1. The compact models.

    • tools/fit/compact.py, optmat.gas.models.compact (the loader) and the registry integration.

    • The reduction configurations data/gas/compact/<gas>.toml and the stored models for CO, CO2, CH4, N2O and SF6.

    • Tests for freshness, tolerance, invariants and reproduction.

    • The AGENTS.md regeneration step for the molecular fits.

    • Register entries for the method and tolerance, and for each gas’s \(K\), bounds and achieved residuals.

    • The Compact infrared terms section of docs/fitting.md.

    • Verify: the gate, and test-fits with the re-derivation.

  2. The printed forms.

    • coefficients and CompactCoefficients in both forms, and the formatter.

    • The coefficient table on docs/gas_models.md (tested) and the gallery example.

    • Verify: the round trip of both forms, the Sellmeier form’s stated error against the full model at its state, and the gate.

13. Open questions#

Each question keeps its text as raised. Its disposition follows it (2026-09-30).

  1. The tolerance (§7): 0.1 sigma in \(n - 1\) and \(\beta_2\), 0.5 in \(\beta_3\). Is that the right bar, or should it follow a fraction of the working estimate?

    Disposition: decided by the author (2026-09-30). Tighten to 0.1 sigma in \(\alpha\), \(D_2\) and \(D_3\) alike. Add 0.01 sigma for rounding, 0.1 of each datum’s sigma for the reproduction, and exact static values (§7). Every gas meets it with margin (worst: CO’s \(D_3\), 0.057; §8). The working-estimate alternative stays rejected, because it is not computable.

  2. CO2’s parent. TC-725 put one of its electronic oscillators on a 202 nm bound; TC-726 settled it (the two weak bands below 10 eV as lines). The feasibility numbers of §8 were computed against the TC-725 fit and are to be recomputed in Phase 1.

    Disposition: resolved. §8 is recomputed against the current parents (after TC-726 and TC-730), with their digests. Phase 1 recomputes only if a parent changes again, which the freshness test detects.

  3. Per-band \(M_2\) (§11): take it up as its own change, before or after this OEP?

    Disposition: decided by the author (2026-09-30): moved to OEP 3.

    • A band’s higher moments matter at the edges of the infrared windows, which is OEP 3’s subject. There, one pole per band misses the GVD by as much as the whole electronic GVD.

    • OEP 3 generalises the question to a Gaussian rule of several nodes per band, built from the lines. Its \(n = 2\) case is the two-point rule from \(M_2\) and \(M_4\).

    • The compact models represent whatever the parent stores, so they need no change for it.

  4. The printed state for the Sellmeier form: 273.15 K and 101325 Pa (the refractometry convention), or 293.15 K and 100 kPa (the accuracy page’s), or both.

    Disposition: decided by the author (2026-09-30). Use 293.15 K and 101325 Pa (optmat.conditions.ONE_ATM_20C). The molecular polarizabilities refer to 293.15 K (TC-730), so a table at 273.15 K would put the reference-temperature polarizability at another temperature’s density without saying so. The export takes any state, so this choice fixes only the manual’s and the paper’s table.

  5. The name. compact/<gas>, or fit/<gas>/compact, which keeps the parent in the name?

    Disposition: decided by the author (2026-09-30). Use compact/<gas>, a fourth family beside fit/<gas> (e.g. fit/CO2), <source>/<formula> (e.g. old_1971/sellmeier) and refractiveindex.info/<gas>/<entry> (e.g. refractiveindex.info/CO2/Old).

    • fit/<gas> means a stored fit with a configuration and a robustness record.

    • Three-part names are already used by the refractiveindex.info/<gas>/<entry> comparison models.

    • The parent is named in the record and in CompactCoefficients.

The following questions record GPT-6’s review of Part I on 2026-09-29. They require resolution, or an explicit disposition, before acceptance; they do not change the parent models or select a new physical approximation.

  1. CO2’s window contains poles (§1, §3, §7, §8). The stored parent has 224 fixed poles strictly inside 8-12 µm, including a pole at 9.41273 µm with strength 1.01815e-9. At a relative frequency detuning of 1e-4 that term alone contributes about 0.217 au, against an electronic statistical sigma of 1.497e-4 au: about 1451 sigma. The residual diverges at every parent pole that the compact model does not preserve. Thus six poles cannot satisfy a finite error bound at every wavelength of that entire interval. A geometric grid, even four times finer, cannot establish the stated guarantee. Which pole neighbourhoods are excluded, retained exactly, or treated by a separately justified smooth target? Define that domain and its provenance before recomputing feasibility. The existing manual’s broad window claim does not establish a pole-free interval. The snapshot checked is models/fitted/CO2.json, SHA-256 d0be2569a263b73e853cac9e807216736735934cce1acf886a685ed537558fe3.

    Disposition: decided by the author (2026-09-30): the windows go to OEP 3, and this OEP follows the parents’ qualified intervals.

    • The finding is confirmed on the current parent (§8). The window has no line-free interval at all, and the in-window poles contribute more than 0.1 sigma almost everywhere in it. The parent itself is not accurate there.

    • OEP 3 defines the parents’ windows and their representation:

      • an \(n\)-node rule per band, from the lines;

      • the weak in-window bands folded into band systems outside;

      • a representation bound measured against the line sum. It also takes up the description’s and docs/accuracy.md’s “valid again between the bands, 8-12 µm”.

    • The compact models are qualified on whatever intervals the parent qualifies, with no pole inside any of them (§4, §7), and weighted by the parent’s stated uncertainty there. On today’s parents that is one interval, 250-1400 nm for CO2.

  2. Representing the domain and constraining new poles (§6, §7). Model.wavelength_range and FittedModel currently carry one pair of endpoints; CO2’s stored pair is 250-1400 nm. Copying it warns throughout the proposed second interval, while expanding it to 12000 nm declares the intervening bands valid. Specify how the compact model records and checks separate qualified intervals and Q6’s exclusions, and how the export carries them. Bounds spanning all parent energies also allow a new effective pole inside a qualified interval. Require a pole-free domain check independent of the sampling grid, and constrain energies accordingly. Locate the registry integration at optmat.gas.models.registry.models, the existing entry point; §6 currently names an export that optmat.gas.models does not provide.

    Disposition: resolved.

    • Separate intervals are OEP 3’s Model.intervals, with wavelength_range as their envelope. The compact model takes the parent’s, and CompactCoefficients.intervals carries them (§6). Q6’s exclusions become the gaps between intervals.

    • Each compact energy is bounded within one absorbing region, between two intervals or below the first, so no compact pole can enter a qualified interval. That is proved from the bounds, not sampled, and asserted by a test (§4, §8).

    • The registry integration is named in §6 and §7.

  3. Exact constraints in the inner solve (§4, §7, §8). Plain non-negative least squares solves the positivity constraint, but not sum F = M0 or sum F / Omega**2 = M-2. Specify an equality-constrained non-negative solve or an equivalent parameterisation, its numerical moment tolerance, and the treatment of trial energies for which those equalities are infeasible. Penalty rows do not make a moment exact. The feasibility table explicitly omits the moment constraints, so it does not yet demonstrate the recommended method or its pole counts. Retain reproducible inputs, full coefficients, grid and parent digests for the revised calculation. When choosing among starts, prefer a candidate meeting every stated tolerance: the lowest summed square error can have a worse maximum residual and fail a threshold that another candidate passes.

    Disposition: resolved.

    • §7 specifies the exact solve (support enumeration with a null-space solve), the moment tolerance (\(10^{-10}\) relative; achieved \(10^{-11}\)) and feasibility by the bounds.

    • Both moments are now constrained for every gas, which also settles Q10’s static validators.

    • §8 is recomputed with the constraints and records the parent digests, the grid, the starts and the full coefficients.

    • The selection rule prefers candidates that pass.

  4. Absolute sigma and the density convention (§4, §5, §7). A zero of beta2 does not imply a zero of its absolute uncertainty. The current SF6 model crosses zero at about 1.27122 µm, where the absolute sigma of D2 is about 0.02835 in atomic units, with the parent’s usual Birge scaling. Compute sigma directly as sqrt(g.T @ covariance @ g); using the relative-uncertainty method at an exact zero and then multiplying by the dispersion produces an undefined product. Flooring this finite sigma by a fraction of the dispersion elsewhere is not justified by the zero crossing and changes the advertised statistical metric. State the Birge convention explicitly. Also distinguish the density-independent tolerances on alpha and D2, D3 from tolerances on the exact finite-density refractivity and dispersion: Lorentz-Lorenz is nonlinear, and the latter need a specified state domain and propagated uncertainties, or a dilute-limit label.

    Disposition: resolved.

    • §4 now defines the absolute sigma from the covariance with the Birge convention stated, and drops the floor. SF6’s value is confirmed: 1271.2 nm, 0.0284 au.

    • The tolerance is labelled dilute-limit.

    • The finite-density effect is bounded analytically to first order and measured (\(8\times10^{-4}\) of itself at 10 bar), and a test repeats it at 10 bar.

  5. Observable-specific checks and static validators (§5, §7, §8). A pointwise error in units of model sigma does not define a tolerance on a dataset’s chi-squared or on a phase-matching ratio. For fixed data-normalised residuals r and prediction changes delta, the chi-squared change is 2 * r.T @ delta + delta.T @ delta; it depends on the original residuals and the dataset size. Define the allowed changes per observable, the handling of dataset scales and partner models, and tests at the actual harmonic wavelengths. The parent’s validators also include static dielectric values for CH4 and a total static polarizability for CO. These depend on M-2 even though those compact models are qualified only at optical wavelengths. Either constrain or separately test that moment, or explicitly exclude those extrapolation checks from the preservation promise. Marchetti’s CO2 data and several Shelton ratios already enter the parent fits: they are useful reproduction checks, but are not independent validation of the parent’s physical accuracy.

    Disposition: resolved.

    • The reproduction tolerance is per datum: 0.1 of its own sigma, inside the declared range. The fitted scales and partner models stay fixed, and the checks run at the data’s own wavelengths, harmonics included. Chi-squared changes are reported, not bounded (§7).

    • \(M_{-2}\) is constrained, so the static validators are exact (§8).

    • §5 now calls Marchetti’s values and Shelton’s ratios reproduction checks.

  6. Configuration digest and regeneration (§6, §10, §12). load_config hashes the complete TOML bytes. Adding [compact], or later changing its grid or pole count, changes the digest in both the stored parent and its robustness record; tests/test_stored_fits.py checks both against the current configuration. Therefore §10’s promise to leave the parent records untouched is incompatible with the proposed storage without a further design choice. Prefer separate reduction configuration, or specify an explicit digest migration and which records must be regenerated. Put the regeneration rule in Phase 1, when the compact records and freshness checks first become part of the repository.

    Disposition: resolved. The reduction configuration is a separate file, data/gas/compact/<gas>.toml, whose digest the compact record carries. The parent’s configuration, digest and records are untouched (§6, §10). The regeneration rule and the freshness test are in Phase 1 (§12).

  7. The quadrature construction (§4, §7, §11). State the quadrature variable and measure. Gaussian quadrature in x = omega_b**2 with weights f_b matches the even moments governing the optical expansion; quadrature in omega_b matches a different sequence. Grouping bands and preserving each group’s M0 and M-2 is not that Gaussian construction. An independent six-node Gaussian calculation in squared frequency on the current CO2 parent does fail the optical tolerance (on 1600 geometric points, about 24 sigma in D2 and 7266 in D3), supporting the motivation for an optical fit, but not the claimed equivalence of the two trials. In §11, the three moments M-2, M0, M2 do not determine a unique unrestricted two-pole rule: identify the fourth moment, or a fixed endpoint and the corresponding quadrature construction, before calling it an exact two-point quadrature.

    Disposition: resolved.

    • §4 states the variable (\(x = \omega_b^2\)) and the measure (\(f_b\)), and the moments matched. It drops the grouped-band trial, and quotes the Gaussian rule’s failure, reproduced independently: 23 sigma in \(D_2\) and 7320 in \(D_3\) on the current parent.

    • §11 now says that an exact two-point rule needs \(M_4\) as well, or one fixed node (Radau).

  8. Qualifying the printed coefficients (§6, §8, §9). Specify the return type, coefficient ordering and units, qualified intervals, number density used at the printed state, parent identity and error metadata. A printed table needs enough significant digits to preserve the error bound after its decimal strings are read back; testing the stored binary coefficients alone does not test the main publication use case. Round-trip both printed forms and budget rounding error together with reduction error. Define whether the Sellmeier form’s stated error is an absolute or relative error, over which wavelengths and for which observables; test the complete difference against the parent, including the omitted Lorentz-Lorenz and virial contributions. The leading (n-1)/6 estimate is an expansion, not a bound over every state allowed by an equation of state.

    Disposition: resolved.

    • §6 specifies CompactCoefficients: its fields, ordering, units, qualified range, number density, parent identity and error metadata.

    • The rounding budget is 0.01 sigma, and 10 significant digits meet it (CO2 needs them; §8).

    • Both forms are round-tripped from their decimal strings.

    • The Sellmeier form’s error is the largest relative difference in \(n - 1\) and \(\beta_2\) from the full parent at the stated state over the qualified intervals, computed directly rather than from the \((n-1)/6\) estimate.

  9. Reconcile the feasibility claims with the parent (§1, §3, §8, Q2). Q2 acknowledges the old CO2 fit, but the abstract still presents its pole counts and errors as the proposed method’s result. The current CO2 model has four leading electronic terms, so six infrared terms make ten terms, not the stated maximum of eight. Its fixed poles reach about 6996.69 cm^-1, beyond the 6972 cm^-1 used in the edge argument. Update the parent inventory, separate provisional unconstrained results from qualified results, and recompute after Q6 and Q8 are settled. The present review could not reproduce the original feasibility table because its full coefficients and calculation are not recorded in the proposal or a public derivation tool.

    Disposition: resolved.

    • §3 has the current inventory: electronic terms, fixed poles, the highest pole and the range.

    • The abstract and §8 give the constrained results on the current parents, with 4 to 10 terms in all. The edge argument uses 6996.7 cm⁻¹.

    • The unconstrained trial table is withdrawn.

    • §8 records the full coefficients and digests.

14. References#

  1. A. A. Voronin and A. M. Zheltikov, “The generalized Sellmeier equation for air,” Sci. Rep. 7, 46111 (2017). doi:10.1038/srep46111 (voronin_2017)

  2. R. J. Mathar, “Calculated refractivity of water vapor and moist air in the atmospheric window at 10 µm,” Appl. Opt. 43, 928 (2004). doi:10.1364/AO.43.000928 (mathar_2004)

  3. R. J. Mathar, “Refractive index of humid air in the infrared: model fits,” J. Opt. A: Pure Appl. Opt. 9, 470 (2007). doi:10.1088/1464-4258/9/5/008 (mathar_2007)

  4. K. T. Tang, “One-term Sellmeier formula for dispersion of dilute gases,” J. Opt. Soc. Am. 62, 644 (1972). doi:10.1364/JOSA.62.000644 (tang_1972)

  5. K. T. Tang and K. K. Poon, J. Opt. Soc. Am. 64, 1582 (1974). doi:10.1364/JOSA.64.001582 (tang_1974)

  6. I. E. Gordon, L. S. Rothman, R. J. Hargreaves et al., “The HITRAN2024 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 353, 109807 (2026). doi:10.1016/j.jqsrt.2026.109807. A data service, not a curated source record; the provenance of the band tables is data/external/hitran/PROVENANCE.md.

  7. S. Marchetti and R. Simili, Infrared Phys. Technol. 50, 78 (2007). doi:10.1016/j.infrared.2006.07.003 (marchetti_2007)

  8. D. P. Shelton, Phys. Rev. A 42, 2578 (1990). doi:10.1103/PhysRevA.42.2578 (shelton_1990)

  9. D. P. Shelton and V. Mizrahi, Chem. Phys. Lett. 120, 318 (1985). doi:10.1016/0009-2614(85)87065-2 (shelton_1985_sf6); D. P. Shelton, J. Chem. Phys. 85, 4234 (1986). doi:10.1063/1.451794 (shelton_1986_co2)

  10. Q. He, Z. Fang, O. Shoshanim, S. S. Brown and Y. Rudich, Atmos. Chem. Phys. 21, 14927 (2021). doi:10.5194/acp-21-14927-2021 (he_2021)

Part II — Implementation record#

Before Phase 1 — the scope on OEP 3’s windowed parents#

  • Finding (TC-798): OEP 3’s Phase 3 gives the molecular parents windows beyond their optical ranges. There the fixed infrared terms carry no statistical uncertainty, so §7’s tolerance, a tenth of the parent’s sigma, would demand the band rules themselves. A feasibility calculation on the refitted CO, with §7’s method:

    • five poles per absorbing region miss by 56 sigma in \(D_2\) and 2300 in \(D_3\);

    • with the representation bound as the tolerance in the windows, about 2 sigma at best;

    • on the optical range alone, three poles meet the tolerance in \(\alpha\) and \(D_2\).

  • Decision (the author, 2026-10-01): a deviation from §4 and §6, recorded here as OEP 0 requires.

    • The compact models are qualified on the parents’ optical intervals only: their qualified intervals inside the optical range (range_nm), with §7’s tolerance.

    • The windows beyond are left to the full model, and a compact model’s range warning names the parent for them.

    • Compact poles are kept outside every optical interval; they may lie in the parents’ windows.

Phase 1 — The compact models#

  • Scope: Phase 1 of §12, 2026-10-01, implemented by Opus 5.5 (AI) for John Travers, on feature/oep-0002-compact.

  • Delivered:

    • optmat.gas.models.compact: the reduction (derive), the residuals, the reproduction check and the loader, compact_models(), joined to the registry;

    • tools/fit/compact.py;

    • the configurations data/gas/compact/<gas>.toml and the stored models for CO, CO2, CH4, N2O and SF6;

    • FittedModel.dispersion_sigma_au, the absolute sigma of §4;

    • tests of freshness, tolerance, invariants, reproduction and the finite density, and the re-derivation in the fits lane;

    • the section Compact infrared terms of Fitting dispersion models, with a tested table;

    • the regeneration step in AGENTS.md.

  • Deviations from the proposal:

    • The intervals are the parents’ optical ones (Part II, Before Phase 1). CH4’s and N2O’s are shortened further, to 240-1070 nm and 320-1500 nm, as decided with the author (below; TC-800).

    • The absorbing regions include one above the intervals: OEP 3’s fold puts a share of each top band system on a node beyond the parent’s short-wavelength end, at 107-148 nm. No compact model gives it a pole; the other poles take its share within both moments.

    • A region given as many poles as it holds keeps the parent’s poles unchanged; SF6’s three are carried through this way, reordered by energy.

    • The absolute sigma of §4 is a new public method, FittedModel.dispersion_sigma_au; dispersion_sigma_relative now divides it by the dispersion.

  • Decisions made:

    • Poles per region: CO 3, CO2 5, CH4 5, N2O 5 below the intervals; none above.

    • Starting points: §7’s, with four seeded random starts (twelve for CO2).

    • The compact model’s status is comparison, with the reason that it is derived for print and never a default.

    • The reproduction counts a static value apart from a dataset’s dynamic ones.

  • Results (TC-799):

    Model

    Interval

    Compact poles

    \(\alpha\)

    \(D_2\)

    \(D_3\)

    compact/CO

    167-2000 nm

    3

    0.0017

    0.0057

    0.049

    compact/CO2

    250-1400 nm

    5

    0.0022

    0.0015

    0.046

    compact/CH4

    240-1070 nm

    5

    0.0007

    0.0006

    0.010

    compact/N2O

    320-1500 nm

    5

    0.0006

    0.0002

    0.069

    compact/SF6

    230-5000 nm

    3 (kept)

    0

    0

    0

    • One pole fewer fails in every reduced case.

    • \(M_0\) and \(M_{-2}\) hold to \(10^{-15}\).

    • Every datum inside the intervals moves by at most 0.003 of its sigma, and the static values are exact to \(6\times10^{-16}\).

    • At 10 bar the finite-density correction is \(10^{-3}\) of the reduction error.

  • Register entries: TC-799 (the method and results), TC-800 (the finding on CH4 and N2O).

  • Verification:

    • The full gate, and the fits lane’s re-derivation of all five.

    • The golden values of every existing model are unchanged; the compact models are added.

  • Sources checked: none new; the method’s references are curated (§14).

  • Limitations:

    • The compact models do not cover the parents’ infrared windows (Part II), nor, for CH4 and N2O, the near infrared beyond 1070 and 1500 nm (Q15).

    • The printed forms, coefficients and the coefficient table are Phase 2.

  • Decided with the author: CH4’s and N2O’s compact intervals (TC-800). Their optical ranges are split by narrow absorbing gaps, whose band-rule poles lie within 0.2 to 3 cm⁻¹ of the interval edges; there a single parent pole contributes up to \(10^{16}\) of the parent’s sigma in \(D_3\), and no few poles follow it. The options were:

    • end the compact intervals at 1070 nm (CH4, 5 poles, worst 0.010 sigma) and 1500 nm (N2O, 5 poles, 0.069), leaving the near infrared beyond to the full models;

    • keep the parents’ gap poles unchanged, at 36 (CH4) and 20 (N2O) poles and more beyond the last interval: hundreds of terms, not a printable model;

    • leave CH4 and N2O without a compact model.

    Decided (the author, 2026-10-01): the shortened intervals.

Phase 2 — The printed forms#

  • Scope: Phase 2 of §12, 2026-10-01, implemented by Opus 5.5 (AI) for John Travers, on feature/oep-0002-compact.

  • Delivered:

    • coefficients, CompactCoefficients and format_coefficients in optmat.gas.models.compact, in both forms;

    • the section The compact coefficients of The gas models, both forms for each molecule, tested against the formatter;

    • the gallery example The compact models against their parents.

  • Deviations from the proposal:

    • Eleven significant digits, not ten: on the present parents ten change CO2’s \(D_3\) by 0.0101 of its sigma, above the 0.01 budget; eleven give at most 0.0017.

    • The Sellmeier form’s \(\beta_2\) error is stated relative to the largest \(|\beta_2|\) over the intervals: \(\beta_2\) changes sign inside the intervals of CO, CO2, N2O and SF6, where a pointwise relative difference is unbounded.

    • The terms’ kinds include rotational, OEP 3’s rotational term.

  • Decisions made:

    • The printed covariance’s parameters are the electronic terms’ \(f_k\) (or \(B_k\)) in printed order, then the effective terms’ \(\lambda_k\) (or \(C_k\)), Birge-scaled as the parent’s.

    • The page prints the Sellmeier form at 101325 Pa and 293.15 K (Q4).

  • Results (TC-801): the Sellmeier form’s error against the full parent at that state is 8.3e-5 to 2.1e-4 in \(n - 1\) and 1.7e-4 to 2.6e-4 of the largest \(\beta_2\); the printed covariance reproduces the parent’s sigma to rounding.

  • Register entries: TC-801.

  • Verification: the full gate; the round trip of both forms from their decimal strings; the Sellmeier form’s stated error recomputed from the printed values.

  • Sources checked: none new.

  • Limitations: as Phase 1’s. With both phases implemented, Final awaits the author.

Part III — History#

  • 2026-09-29: Drafted.

  • 2026-09-29: Q2 updated: CO2’s parent settled by TC-726.

  • 2026-09-29: GPT-6 (AI) reviewed all of Part I against the current models, configurations, fitting and uncertainty code, tests, manual and register. Recorded unresolved findings as Q6-Q14; the recommendation is to revise before acceptance. Numerical checks counted CO2’s poles inside the proposed window, evaluated their contribution relative to the covariance, propagated SF6’s uncertainty at its zero-dispersion wavelength, and tested Gaussian quadrature in squared frequency. Primary-source pages checked: Tang 1972, p. 645 (the one-term bound); Mathar 2007, p. 473 (the instability of state-dependent Sellmeier parameters); Voronin and Zheltikov 2017, pp. 2-3 (the dilute oscillator model and transparency restriction). HITRAN’s official citation page confirms reference [6]; its DOI is 10.1016/j.jqsrt.2026.109807. SciPy’s nnls documentation confirms that its constraints are non-negativity only. Peck’s original, the measurement papers, and the HITRAN line lists were not freshly checked; no independent reproduction of the original feasibility table is claimed. No data, fit or physical convention was changed.

  • 2026-09-30: Revised by the drafting agent (Opus 5.5) in response to GPT-6’s review. Every question now has a disposition. Q2 and Q7-Q14 are resolved in the text. Q1, Q3, Q4 and Q5, and the scope narrowing of Q6, are recommendations for the author to confirm. The changes:

    • Scope: the compact models are qualified on each parent’s declared range only, and CO2’s 8-12 µm window is out.

    • Method: both \(M_0\) and \(M_{-2}\) are exact for every gas, through an exactly constrained inner solve. Sigmas are absolute from the covariance, Birge-scaled, with no floor.

    • Tolerance: 0.1 sigma in \(\alpha\), \(D_2\) and \(D_3\), plus rounding, per-datum and static tolerances.

    • Storage and interface: a separate reduction configuration, and a specified CompactCoefficients.

    • Recomputed against the parents stored after TC-730, with digests and full coefficients recorded. The unconstrained 2026-09-29 table is withdrawn.

    • Also checked on the current parents: CO2’s window, per datum for every parent dataset and validator, rounding, finite density and the Sellmeier form.

    • Peck’s reference, which was not curated and served only as the format Mathar compares with, is dropped.

    • The sigmas were checked against the package’s own methods. GPT-6’s SF6 and Gaussian-quadrature numbers were reproduced.

    • This revision is not an independent review: the author’s human review is still required for Reviewed. No data, fit or physical convention was changed.

  • 2026-09-30: The author decided Q1, Q3, Q4, Q5 and Q6.

    • Q3 and CO2’s window move to the new OEP 3 (the molecular gases between their infrared bands), which is to be implemented first.

    • This OEP now requires OEP 3, and is written to follow the parents’ qualified intervals: compact poles are bounded to the absorbing regions between them, the weights include OEP 3’s representation uncertainty, and CompactCoefficients carries the intervals.

    • The feasibility numbers are labelled as those of today’s single-interval parents, to be recomputed in Phase 1.

    • The author’s decisions on these questions are not a full review of Part I, and the review status is unchanged.

  • 2026-09-30: Aligned with OEP 3’s revision for its review. OEP 3’s representation uncertainty is now a representation bound, which is stated beside the statistical sigma and not combined with it in quadrature (§4, §7). The weights are the parents’ statistical sigma. OEP 3’s rotational band system is carried outside the reduction. No number changes on today’s parents, where the bound is absent.

  • 2026-09-30: John Travers (human) reviewed Part I. With GPT-6’s review, the review status is Reviewed.

  • 2026-09-30: Accepted by John Travers. Part I is frozen.

  • 2026-10-01: The author decided the scope on OEP 3’s windowed parents before Phase 1: the optical intervals only (Part II; TC-798).

  • 2026-10-01: Phase 1 implemented (TC-799). The author decided CH4’s and N2O’s compact intervals: 240-1070 nm and 320-1500 nm (Part II; TC-800).

  • 2026-10-01: Phase 2 implemented (TC-801).

  • 2026-10-01: The author checked the compact models and their tables. Final.