lang_2026
=========

J\. Lang, G\. Garberoglio, M\. Przybytek and M\. Lesiuk, "Determination of fundamental properties of nitrogen from first principles\. III\. Temperature and frequency dependence of the molecular polarizability and magnetic susceptibility", arXiv\:2607\.21261v2 \[physics\.chem\-ph\] \(2026\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/lang_2026.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.48550/arxiv\.2607\.21261 <https://doi.org/10.48550/arxiv.2607.21261>`__

**Source kind:** theory

**Recorded technique:** Ab initio isotropic static polarizability alpha\_0\(R\), Cauchy coefficients alpha\_2\(R\), alpha\_4\(R\), alpha\_6\(R\) \(alpha\(omega, R\) = alpha\_0 \+ alpha\_2 omega^2 \+ alpha\_4 omega^4 \+ alpha\_6 omega^6, Eq\. \(2\)\) and isotropic magnetic susceptibility chi\_0\(R\) of 14N2 \(X 1Sigma\_g\+\) on a grid of internuclear distances R = 1\.75\-2\.60 bohr\. Polarizability\: composite coupled\-cluster scheme, Eq\. \(32\)\: frozen\-core CCSD \(with the Hartree\-Fock part; aug\-cc\-pVXZ to X = 7\) \+ CC3 \(X = 6\) \+ full CCSDT \(X = 4\) \+ CCSDTQ \(aug\-cc\-pVDZ, 50 % uncertainty\) \+ all\-electron CC3 \(aug\-cc\-pCVXZ, X = 4\) and CCSDT \(X = 3\) core corrections \+ one\-electron scalar relativistic \(DKH2, all\-electron CCSD, X = 4\); Riemann extrapolation to the basis\-set limit with random\-walk uncertainty; alpha\_6 at CCSD \+ CC3 only \(flat 10 % uncertainty\)\. QED, diagonal Born\-Oppenheimer and two\-electron relativistic terms neglected \(p\. 7\)\. The Cauchy coefficients of the post\-CC3 terms by fitting a 27\-frequency grid \(p\. 7\)\. Magnetic susceptibility\: diamagnetic plus paramagnetic \(Van Vleck\) terms, Eq\. \(40\), frozen\-core CC3 response with aug\-cc\-pV5Z \(gauge origin at the molecular centre; 5 % uncertainty; no relativistic or post\-CC3 terms; frequency dependence neglected, pp\. 10\-11\)\. Thermal averages 50\-2000 K by rovibrational averaging over 345 levels of the Paper II potential with the 14N2 nuclear\-spin weights, Eqs\. \(18\)\-\(22\), and independently by path integral Monte Carlo \(Eqs\. \(24\)\-\(31\)\); the two agree within the PIMC statistics \(p\. 13\)\.

**Materials in structured entries:** N2

**Dataset quantities:** cauchy\_moment, magnetic\_susceptibility, polarizability

**Contents:** 9 datasets; 3 formulas.

Curation notes
--------------

The library copy is the arXiv preprint 2607\.21261v2 \(dated 9 September 2026, 18 pdf pages\); not peer reviewed; no journal reference\. Page numbers are the preprint's pdf pages\. Atomic units throughout\: polarizability and magnetic susceptibility in a0^3 \(the magnetic term enters the Lorentz\-Lorenz relation as alpha \+ chi, Eq\. \(1\)\), omega in hartree; 1 cm^3/mol = 2\.675205 a\.u\. \(p\. 2; the package's \(4 pi/3\) N\_A a0^3 = 0\.3738031 cm^3/mol gives 2\.675205\), CODATA 2022\.

Papers I \(the nitrogen atom's polarizability, the same Cauchy expansion\) and II \(the N2 ground\-state potential, its rovibrational levels, D\_0 = 9\.76 eV\) of the series are referred to only as 'Paper I' and 'Paper II' \(pp\. 1, 2, 4\-5, 7\)\: the reference list does not cite them\. Neither is in the library; neither holds a quantity of the package for N2 \(Paper II's potential enters here only through the averaging\)\. The supplementary material \(Zenodo, doi 10\.5281/zenodo\.22662189\: alpha\_n\(T\) and chi\_0\(T\) on a 1 K grid, the Eq\. \(52\) parameters, the uncertainty fits, the full R grid\) is not in the library\: the temperature dependence exists here only at 303 K \(Eq\. \(53\)\) and, semi\-empirically, at 273\.16 K \(p\. 15\)\. Figures 2 and 3 \(alpha\_n\(T\), chi\_0\(T\), raster\) are not digitised\: the numbers are published in the supplement \(TC\-699\)\.

ELECTRIC POLARIZABILITY\. The alpha\_n here are the electric polarizability alone; the refractive \(package\) polarizability is alpha \+ chi\_0 \(Eq\. \(54\)\)\. Every dataset below says which it holds\.

UNCERTAINTIES\. alpha\_0, alpha\_2, alpha\_4 at 303 K\: expanded, k = 2 \('evaluated at 2 sigma confidence level', p\. 14; 'approximately a 95% confidence level', p\. 6\), including the potential's propagated uncertainty\. alpha\_6 and chi\_0 carry flat 10 % and 5 % estimates \(pp\. 9, 11, 13\), with no stated level \('extremely unlikely to exceed 5%', p\. 11, for chi\_0\)\.

CHECKS \(2026\-09\-29\)\. Eq\. \(55\) with the Eq\. \(53\) values reproduces the paper's theoretical molar polarizabilities 11\.743\(13\) at 1542\.383 nm and 11\.875\(13\) at 632\.9908 nm \(computed 11\.743019, 11\.874645\)\. The semi\-empirical alpha\_0 = 11\.735962 and alpha\_2 = 30\.086359 with the theoretical alpha\_4, alpha\_6, chi\_0 reproduce the measured 11\.762120 and 11\.894428 to 1e\-6 \(computed 11\.762119, 11\.894427\)\. Eq\. \(50\) with Table III reproduces Table I's totals at R = 1\.750, 2\.070, 2\.600 bohr \(formula alpha\_n\_of\_R\); Eq\. \(51\) reproduces Table II's aug\-cc\-pV5Z diamagnetic \+ paramagnetic sums \(formula chi0\_of\_R\)\. Eq\. \(53\) and the p\. 15 values checked on the pdf text\.

COMPARISON WITH fit/N2 \(computed 2026\-09\-29; fit/N2 has chi0\_au = 0, so its polarizability is the refractive one\)\. Ab initio alpha \+ chi\_0 at 303 K is 1606 ppm below the fit at the static limit, 1615, 1638 and 1656 ppm below at 1542, 800 and 633 nm \(2\.9\-3\.0 sigma of the theory's k = 2 uncertainty halved\); its GVD \(d^2\(omega alpha\)/d omega^2\) is 0\.54, 0\.52, 0\.51 % below the fit's at 1550, 800, 633 nm \(theory sigma 0\.13 %\: about 4 sigma\); S\(\-4\) is 0\.55 % below the fit's 30\.0938, S\(\-6\) 0\.20 % above its 100\.423, S\(\-8\) 2\.8 % above its 372\.66 \(within the 10 %\)\. The semi\-empirical values \(below\) agree with the fit to \+9\.9 ppm \(static, with chi\_0\), \+9\.3 \(1542 nm\), \+8\.0 \(800 nm\), \+7\.0 ppm \(633 nm\), and in GVD to \-0\.02 %; fit/N2 rests on data near 293 K, and the theory's temperature coefficient \(below\) moves the 303 K values by \-11 to \-12 ppm to 293 K, leaving the fit 1\.5 \(1542 nm\) to 4\.6 ppm \(633 nm\) above them\.

TEMPERATURE COEFFICIENT \(computed from the two semi\-empirical static values, p\. 15\)\: \(11\.735962 \- 11\.735585\)/29\.84 K = 1\.263e\-5 a\.u\./K = 1\.077e\-6 per K of alpha\_0 = 47\.2e\-7 \(cm^3/mol\)/K, 273\.16\-303 K; egan\_2019 lists Buldakov et al\. 48\(1\)e\-7 \(optical, theory\), Sharipov et al\. 51e\-7 \(static, theory\), Schmidt and Moldover 62\(11\)e\-7 \(static\); Egan and Yang \(2023, Int\. J\. Thermophys\. 44, 181, p\. 17; in the library, not curated\) measure A\_theta = 1\.18\(4\)e\-6 per K at 632\.9908 nm over 293\-433 K\. N2's first refractivity coefficient changes by about 30 ppm between 273 and 303 K, four times its pair term at 1 atm \(Achtermann's B\_R rho/A\_R = 8e\-6\)\.

MAGNETIC SUSCEPTIBILITY\. chi\_0\(303 K\) = \-1\.75\(9\) x 10^\-4 a\.u\. \(Eq\. \(53\)\), diamagnetic \-3\.46 x 10^\-4 and paramagnetic \+1\.70 x 10^\-4 at R = 2\.07 bohr \(Table II\)\: the paramagnetic term 'is substantial and cannot be omitted' \(p\. 15\)\. \-14\.9 ppm of alpha\_0\. The four measurements the paper quotes \(datasets below\) are 5\-31 % smaller in magnitude; the authors 'cannot explain the origin of this discrepancy' \(p\. 15\)\. The same measurer's neon agrees with the ab initio neon to 1\.1 % and his argon is 6\.3 % low \(lesiuk\_2023b, Table X\)\: scaled by argon's 6\.3 %, Havens' N2 would still be 18 % below this theory\. The centre\-of\-mass coupling \(Eq\. \(49\)\) and the frequency dependence of chi \(below 1 %, p\. 11\) are negligible\. The conclusions \(p\. 16\) speak of 'the single experimental result'; p\. 15 quotes four\.

Also in the text \(p\. 15\)\: Egan and Yang's own static value 11\.735909 a\.u\. \(the yang\_2025 A\_eps\_N2, 4\.38692 cm^3/mol, with older alpha\_4, alpha\_6, chi\_0\) and 'Egan and collaborators' alpha\_2 30\.109 a\.u\. \(yang\_2025's n2\_cauchy, S\(\-4\) = 30\.1086\)\: not extracted \(in yang\_2025\)\. The kinetic\-energy correction to Eq\. \(10\), about \-2 x 10^\-4 a\.u\. for v = J = 0 from the b 1Pi\_u state alone \(p\. 4\), is neglected by the paper\.

Byproduct \(computed, not printed\)\: Eq\. \(50\) gives d alpha\_0/dR = 5\.98 a0^2 at R = 2\.07 bohr \(1\.675 angstrom^2\), the derivative of the mean polarizability with the bond length; murphy\_1969's reproduced N2 d alpha/dr is 1\.75 angstrom^2 \(4 % higher\)\.

Not extracted\: Table I's components by R \(not a quantity\: a polarizability at fixed R is not the molecule's\), Table II's basis\-set sequence \(its X = 5 sums are the check of chi0\_of\_R\), Fig\. 1 \(the partition\-function exchange ratio\)\.

**Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

lang\_2026/cauchy\_303K
~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "cauchy_303K",
     "material": "N2",
     "table": "text",
     "quantity": "cauchy_moment",
     "unit": "au",
     "abscissa": "order",
     "abscissa_unit": "1",
     "use": "check",
     "uncertainty": {
       "column": true,
       "level": "2 sigma"
     },
     "parameters": {
       "temperature_k": 303.0
     }
   }

4 inline points in the source record.

Eq\. \(53\), p\. 14\: alpha\_0 = 11\.717\(13\), alpha\_2 = 29\.928\(80\), alpha\_4 = 100\.62\(82\), alpha\_6 = 383\(38\) a\.u\. of 14N2 at 303 K, rovibrationally averaged \(PIMC uncertainty propagation\), stored as S\(\-2\), S\(\-4\), S\(\-6\), S\(\-8\) \(alpha\_2k = S\(\-2k\-2\), Eq\. \(6\)\)\. ELECTRIC polarizability\: chi\_0 not included\. Uncertainties k = 2 for the first three; alpha\_6's is the flat 10 % \(p\. 9\), level not stated \(stored under the same level\)\. A check, not fitted\: a preprint; the uncertainty of S\(\-2\), 0\.11 % \(k = 2\), is 100 times the metrology's; S\(\-2\) is 1600 ppm and S\(\-4\) 0\.53 % below the semi\-empirical values \(2\.9 and 4 sigma\), which the authors themselves recommend over the ab initio ones \(p\. 15\)\. S\(\-6\) and S\(\-8\) \(100\.42 and 372\.66 in fit/N2\) are the only ab initio N2 moments of their order in the library with uncertainties\.

lang\_2026/chi0\_303K
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "chi0_303K",
     "material": "N2",
     "table": "text",
     "quantity": "magnetic_susceptibility",
     "unit": "au",
     "scale": 0.0001,
     "abscissa": "none",
     "use": "primary",
     "uncertainty": {
       "column": true,
       "level": "not stated"
     },
     "parameters": {
       "temperature_k": 303.0
     }
   }

1 inline points in the source record.

Eq\. \(53\), p\. 14 \(and p\. 15\)\: chi\_0 = \-1\.75\(9\) x 10^\-4 a\.u\., the isotropic magnetic susceptibility \(magnetizability\) of 14N2 at 303 K, rovibrationally averaged, frozen\-core CC3/aug\-cc\-pV5Z, static \(the frequency dependence 'most likely smaller than 1%', p\. 11\)\. The uncertainty is the flat 5 % goal \(pp\. 11, 13\), level not stated\. \-14\.9 ppm of alpha\_0\. The only calculation of N2's susceptibility in the library; 31 % larger in magnitude than havens\_1933 \(\-1\.338\) and 5\-31 % than every measurement \(source notes\)\. The temperature dependence is weak \(Fig\. 3; numbers in the supplement, not in the library\)\. A preprint\: primary as the value to adopt for fit/N2's chi0\_au, with the unresolved disagreement with experiment stated\.

lang\_2026/hector\_1924
~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "hector_1924",
     "material": "N2",
     "table": "text",
     "quantity": "magnetic_susceptibility",
     "unit": "au",
     "scale": 0.0001,
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "L. G. Hector, Phys. Rev. 24, 418 (1924)"
     ],
     "uncertainty": {
       "relative": 0.02,
       "level": "not stated"
     }
   }

1 inline points in the source record.

p\. 15\: Hector's measurement as converted by Lang et al\., chi\_0 = \-1\.34 x 10^\-4 a\.u\., 'stating the uncertainty at 2%' \(level not stated\); the value the CRC Handbook cites \(\-12\.0 x 10^\-6 cm^3/mol in cgs molar units is \-1\.345 x 10^\-4 a\.u\., computed\)\. Temperature not stated\. The original is not in the library\.

lang\_2026/vaidyanathan\_1928
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "vaidyanathan_1928",
     "material": "N2",
     "table": "text",
     "quantity": "magnetic_susceptibility",
     "unit": "au",
     "scale": 0.0001,
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "V. Vaidyanathan, Lond. Edinb. Dubl. Phil. Mag. J. Sci. 5, 380 (1928)"
     ],
     "uncertainty": {
       "relative": 0.05,
       "level": "bound"
     },
     "parameters": {
       "temperature_k": 303.0
     }
   }

1 inline points in the source record.

p\. 15\: \-1\.45 x 10^\-4 a\.u\. at 303 K, 'estimating the error at no more than 5%' \(a bound\)\. As converted by Lang et al\.; the original is not in the library\.

lang\_2026/bitter\_1930
~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "bitter_1930",
     "material": "N2",
     "table": "text",
     "quantity": "magnetic_susceptibility",
     "unit": "au",
     "scale": 0.0001,
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "F. Bitter, Phys. Rev. 36, 1648 (1930)"
     ],
     "parameters": {
       "temperature_k": 298.0
     }
   }

1 inline points in the source record.

p\. 15\: \-1\.66 x 10^\-4 a\.u\. at 298 K, 'without explicit uncertainty estimation'; the measurement closest to the theory \(5 % smaller in magnitude\)\. As converted by Lang et al\.; the original is not in the library\.

lang\_2026/havens\_1933
~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "havens_1933",
     "material": "N2",
     "table": "text",
     "quantity": "magnetic_susceptibility",
     "unit": "au",
     "scale": 0.0001,
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "G. G. Havens, Phys. Rev. 43, 992 (1933)"
     ],
     "uncertainty": {
       "column": true,
       "level": "not stated"
     }
   }

1 inline points in the source record.

p\. 15\: \-1\.338\(2\) x 10^\-4 a\.u\. 'at room temperature' \(level not stated\), as converted by Lang et al\. The same work's Ne and Ar values are lesiuk\_2023b/havens\_1933\_Ne \(1\.1 % from theory\) and havens\_1933\_Ar \(6\.3 % low\)\. Soné \(1920\) and Hammar \(1926\), 'roughly two times smaller' \(gas purity\), print no number here\. The original is not in the library\.

lang\_2026/semi\_empirical\_303K
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "semi_empirical_303K",
     "material": "N2",
     "table": "text",
     "quantity": "cauchy_moment",
     "unit": "au",
     "abscissa": "order",
     "abscissa_unit": "1",
     "use": "check",
     "uncertainty": {
       "relative_rows": {
         "-2": 1e-05
       },
       "level": "bound"
     },
     "parameters": {
       "temperature_k": 303.0
     }
   }

2 inline points in the source record.

p\. 15\: the static polarizability alpha\_0\(303 K\) = 11\.735962 a\.u\. and the second Cauchy coefficient alpha\_2 = 30\.086359 a\.u\., solved from Egan and Yang's two molar polarizabilities \(dataset egan\_yang\_303K\) with this paper's alpha\_4, alpha\_6 and chi\_0 at 303 K, Eq\. \(55\)\. ELECTRIC \(chi\_0 removed\)\. alpha\_0 'no worse than 10ppm' \(a bound, row \-2\); alpha\_2 printed without uncertainty \(the theory's alpha\_4 uncertainty alone moves it by 0\.0025, 83 ppm; the two measurements, if uncorrelated, by 0\.012, 0\.04 %\: computed\)\. Not independent of the Egan and Yang measurements \(the refractivities of egan\_yang\_303K; yang\_2025's A\_eps\_N2 is the same data reduced with older theory, 4\.5 ppm lower\)\: a check\. Against fit/N2\: S\(\-2\) \+ chi\_0 = 11\.735787 is 9\.9 ppm above the fit's static value; S\(\-4\) is 0\.025 % below the fit's 30\.0938 \(0\.6 sigma\), and yang\_2025's 30\.1086 0\.05 % above it\.

lang\_2026/semi\_empirical\_273K
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "semi_empirical_273K",
     "material": "N2",
     "table": "text",
     "quantity": "polarizability",
     "unit": "au",
     "abscissa_unit": "nm",
     "use": "check",
     "parameters": {
       "temperature_k": 273.16
     }
   }

1 inline points in the source record.

p\. 15\: alpha\_0\(273\.16 K\) = 11\.735585 a\.u\., semi\_empirical\_303K's alpha\_0 moved to the triple point of water with this paper's theoretical alpha\_0\(303 K\) \- alpha\_0\(273\.16 K\) \(the supplement's grid, not in the library\)\: 32\.1 ppm lower than at 303 K\. ELECTRIC \(static\)\. No uncertainty printed\. Schmidt and Moldover's capacitance A\_eps at 273\.16 K \(schmidt\_2003/A\_eps\_tau\_N2, 4\.38748 cm^3/mol = 11\.73731 a\.u\.\) is 147 ppm higher \(their budget about 60 ppm\)\.

lang\_2026/egan\_yang\_303K
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "egan_yang_303K",
     "material": "N2",
     "table": "text",
     "quantity": "polarizability",
     "unit": "au",
     "abscissa_unit": "nm",
     "wavelength_medium": "vacuum",
     "use": "reproduced",
     "reproduced_from": [
       "egan_2023",
       "P. F. Egan and Y. Yang, Int. J. Thermophys. 45, 120 (2024)",
       "yang_2025"
     ],
     "uncertainty": {
       "column": true,
       "level": "1 sigma"
     },
     "parameters": {
       "temperature_k": 303.0
     }
   }

2 inline points in the source record.

p\. 15\: 'the corresponding experimental results', the molar polarizability alpha \+ chi \(the package's refractive polarizability\) of N2 at 303 K from the NIST refractometry of Egan and Yang \(Refs\. 85\-87, not assigned per wavelength\), standard uncertainties \(1 sigma, p\. 15\)\. The 632\.9908 nm value is Egan and Yang 2023's A\_303 = 4\.446175\(14\) cm^3/mol \(p\. 17 of that paper; in the library as Egan\_2023\_Optical90Measurement, not curated\), 11\.894429 a\.u\. by the paper's factor; the 1542\.383 nm value's original \(presumably Egan and Yang 2024\) is not in the library \(yang\_2025 prints only the reduced A\_eps and A\_2; its laser is 194\.36862 THz = 1542\.3912 nm, 0\.008 nm from the wavelength quoted here, 1e\-9 of alpha\)\. Against fit/N2\: \-9\.4 ppm \(1542 nm\) and \-7\.1 ppm \(633 nm\); moved to 293 K with the theory's temperature coefficient, \+1\.5 and \+4\.6 ppm \(source notes\)\. The best copy in the library of these two numbers\.

Formulas
--------

lang\_2026/cauchy\_theory\_303K
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "cauchy_theory_303K",
     "form": "alpha_series",
     "terms": {
       "c": [
         "11.717",
         "29.928",
         "100.62",
         "383"
       ]
     },
     "range_nm": [
       455.6,
       "inf"
     ],
     "material": "N2",
     "equation": "Eq. (2) with Eq. (53), p. 14",
     "printed": "alpha(omega, T) = alpha_0(T) + alpha_2(T) omega^2 + alpha_4(T) omega^4 + alpha_6(T) omega^6; T = 303 K: alpha_0 = 11.717(13), alpha_2 = 29.928(80), alpha_4 = 100.62(82), alpha_6 = 383(38)",
     "variable": "omega: angular frequency in atomic units (hartree); alpha in a0^3",
     "state": "none (per molecule, 303 K)",
     "density": "none",
     "range": "'wavelengths longer than 400nm or omega < 0.1 in atomic units' (p. 2): the two limits differ (0.1 a.u. is 455.6 nm); range_nm takes the stricter.",
     "coefficients": {
       "alpha_0": "11.717",
       "alpha_2": "29.928",
       "alpha_4": "100.62",
       "alpha_6": "383"
     },
     "check": "With chi_0 = -1.75e-4 added (Eq. (55)) the series gives 11.743019 at 1542.383 nm and 11.874645 at 632.9908 nm (omega = 0.0295409, 0.0719811 a.u.): the paper's 11.743(13) and 11.875(13) (p. 15). Checked 2026-09-29."
   }

The paper's ab initio series at 303 K\. ELECTRIC polarizability\: chi\_0 = \-1\.75\(9\)e\-4 a\.u\. \(15 ppm\) is not included; add it for the refractive index \(Eq\. \(54\)\)\. 1600 ppm below the measured values; the semi\-empirical alpha\_0, alpha\_2 \(dataset semi\_empirical\_303K\) with this series' alpha\_4, alpha\_6 reproduce the measurements\.

lang\_2026/alpha\_n\_of\_R
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "alpha_n_of_R",
     "material": "N2",
     "equation": "Eq. (50), p. 13, with Table III, p. 12",
     "printed": "alpha_n(R) = exp(-a_n^(1) R) sum_{i=-1}^{1} c_in^(1) R^i + exp(-a_n^(2) R) sum_{i=0}^{2} c_in^(2) R^i, n = 0, 2, 4, 6",
     "variable": "R: internuclear distance in bohr; alpha_n in atomic units (a0^3 hartree^-n)",
     "state": "none (per molecule, clamped nuclei)",
     "density": "none",
     "range": "R in [1.75, 2.60] bohr only (p. 13: not to be extrapolated)",
     "coefficients": {
       "c_m1_0": "3973.239843",
       "c_0_0": "-2550.566901",
       "c_1_0": "480.40394",
       "d_0_0": "-17636.209308",
       "d_1_0": "6178.632412",
       "d_2_0": "-7984.197341",
       "a1_0": "0.72712153",
       "a2_0": "3.06212416",
       "c_m1_2": "3121512.910661",
       "c_0_2": "-1565760.084872",
       "c_1_2": "213071.693344",
       "d_0_2": "-12511573.39832",
       "d_1_2": "4306441.860778",
       "d_2_2": "-4591578.708124",
       "a1_2": "1.57852725",
       "a2_2": "3.56319374",
       "c_m1_4": "2.9522897",
       "c_0_4": "-1.06250446",
       "c_1_4": "0.10234389",
       "d_0_4": "-8.65155509",
       "d_1_4": "2.18269184",
       "d_2_4": "-1.76012369",
       "a1_4": "-6.04767072",
       "a2_4": "-4.59332841",
       "c_m1_6": "19.85061337",
       "c_0_6": "-7.34704556",
       "c_1_6": "0.72912448",
       "d_0_6": "-59.44131003",
       "d_1_6": "15.28369979",
       "d_2_6": "-12.65835287",
       "a1_6": "-5.57635061",
       "a2_6": "-4.08555911"
     },
     "check": "At R = 1.750, 2.070, 2.600 bohr: alpha_0 = 9.8260, 11.6427, 14.8590 (Table I totals 9.8255(80), 11.642(12), 14.856(40)); alpha_2 = 21.1215, 29.4793, 50.6091 (21.118(21), 29.471(70), 50.58(41)); alpha_4 = 60.414, 97.999, 242.65 (60.43(19), 98.09(57), 242.5(51)); within the printed uncertainties. Table III image checked against the cells. Computed 2026-09-29."
   }

Coefficient names\: c\_i\_n = c\_in^\(1\) \(c\_m1 for i = \-1\), d\_i\_n = c\_in^\(2\), a1\_n = a\_n^\(1\), a2\_n = a\_n^\(2\), n the Cauchy order\. The purely electronic, clamped\-nucleus alpha\_n\(R\) that the thermal averages are built from; not a molecular polarizability\. Its slope, d alpha\_0/dR = 5\.98 a0^2 at 2\.07 bohr \(computed\), is the bond\-length derivative of the mean polarizability \(source notes\)\. The uncertainty fits are in the supplement \(not in the library\)\.

lang\_2026/chi0\_of\_R
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "chi0_of_R",
     "material": "N2",
     "equation": "Eq. (51), p. 13, parameters in the text, p. 13",
     "printed": "chi_0(R) = exp(-b^(1) R) sum_{i=0}^{1} d_i^(1) R^i + exp(-b^(2) R^2) sum_{i=0}^{1} d_i^(2) R^i; d_0^(1) = -0.0022528, d_1^(1) = 0.0012629, d_0^(2) = 0.0012288, d_1^(2) = -0.0008785, b^(1) = 0.729884, b^(2) = 0.195461 (atomic units)",
     "variable": "R: internuclear distance in bohr; chi_0 in a0^3",
     "state": "none (per molecule, clamped nuclei)",
     "density": "none",
     "range": "R in [1.75, 2.60] bohr only",
     "coefficients": {
       "d1_0": "-0.0022528",
       "d1_1": "0.0012629",
       "d2_0": "0.0012288",
       "d2_1": "-0.0008785",
       "b1": "0.729884",
       "b2": "0.195461"
     },
     "check": "At R = 1.750, 2.070, 2.600 bohr: -1.815e-4, -1.754e-4, -1.270e-4 a.u.; Table II aug-cc-pV5Z diamagnetic + paramagnetic: -1.82e-4, -1.76e-4, -1.28e-4 (to the printed digits). Checked 2026-09-29 against the pdf text (the superscripts are lost in the text layer; the order is that of the equation)."
   }

Note the Gaussian R^2 in the second exponent\. The thermal average at 303 K, \-1\.75e\-4 \(dataset chi0\_303K\), is close to the value at R = 2\.07 bohr\.

Corrections
-----------

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register </register>` for the checks and decisions.