yu_2012 ======= S\. Yu, C\. E\. Miller, B\. J\. Drouin and H\. S\. P\. Müller, "High resolution spectral analysis of oxygen\. I\. Isotopically invariant Dunham fit for the X 3Sigma\_g\-, a 1Delta\_g, b 1Sigma\_g\+ states", J\. Chem\. Phys\. 137, 024304 \(2012\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1063/1\.4719170 `__ **Source kind:** fit **Recorded technique:** A global, isotopically invariant Dunham fit \(Pickett's SPFIT\) of 11 944 transition frequencies of all six O2 isotopologues from the microwave to the ultraviolet\: microwave and THz lines of X and a, pure rotational and vibration\-rotation Raman lines of X, the a\-X, b\-X and b\-a systems, and the Schumann\-Runge \(B\-X\), Chamberlain \(A'\-a\) and Herzberg III \(A'\-X\) systems as interconnections; each line weighted by the inverse square of its \(sometimes reassessed\) uncertainty, reduced rms 1\.1\. X 3Sigma\_g\- levels v = 0\-31 of 16O16O \(v = 0\-1 from MW/THz data, 2\-3 and 11\-18 from Schumann\-Runge data to \+\-0\.03\-0\.05 cm\-1, 4\-10 and 19\-23 to \+\-0\.1\-0\.3 cm\-1, 24\-25 no data, 26\-31 \+\-3 cm\-1\), a v = 0\-10, b v = 0\-12 \(pp\. 2\-8\)\. Band\-by\-band fits of v = 0 and 1 of X and b in addition \(Table III\)\. **Materials in structured entries:** O2 **Dataset quantities:** rotational\_constant **Contents:** 18 datasets; 1 formula. Curation notes -------------- Journal pages 024304\-1 to 024304\-20 = PDF pages 2\-21 \(PDF page 1 is the publisher's cover; p\. 024304\-n is PDF page n \+ 1\)\. The library's warnings \(unmatched figure captions on the cover page; single unmapped glyphs on PDF pages 5, 10, 11, 13 and 15, in the text; references reordered on PDF page 20\) touch no value used\: Tables I and III were read from their JSON cells and checked against the images\. Identity checked against the first page\. Page numbers below are the article's \(p\. n = 024304\-n\)\. MATERIAL\: 16O16O, the X 3Sigma\_g\- ground state, is the material O2\. The a 1Delta\_g and b 1Sigma\_g\+ states, the five minor isotopologues \(16O17O, 16O18O, 17O17O, 17O18O, 18O18O\) and the 16O17O hyperfine constants are not transcribed \(see below\)\. THE HAMILTONIAN \(Eq\. 2, p\. 3; Hund's case \(b\), T\_e = 0 for X\)\: H = T\_e \+ G\_v \+ B\_v N^2 \+ D\_v \(N^2\)^2 \+ H\_v \(N^2\)^3 \+ \[lambda\_v \+ lambda\_Dv N^2 \+ lambda\_Hv \(N^2\)^2\] \(2/3\)\(3 S\_z^2 \- S^2\) \+ \[gamma\_v \+ gamma\_Dv N^2 \+ gamma\_Hv \(N^2\)^2\] N\.S \(\+ hyperfine terms for 17O\)\. SIGN\: D\_v enters with a plus sign, so the fitted D\_v is negative \(Table III labels it '\-D\_v'\); the conventional distortion constant \(E = B N\(N\+1\) \- D \[N\(N\+1\)\]^2\) is \+0\.14516 MHz = 4\.8421e\-6 cm\-1 for v = 0\. The spin\-spin constant appears in the tables as 4 lambda \(4 x lambda\)\. The level energies need the diagonalization of Eq\. \(2\) in the case\-\(b\) basis \(N = J, J \+\- 1 mixed for each J\); the paper prints no level energies \(the line list and predictions are in its supplementary material, not in the library\)\. THE DUNHAM EXPANSIONS \(Eqs\. 4\-14, pp\. 4\-5\)\: G\_v = sum\_l Y\_l0 \(v \+ 1/2\)^l, B\_v = sum\_l Y\_l1 \(v \+ 1/2\)^l, D\_v = sum\_l Y\_l2 \(v \+ 1/2\)^l, H\_v = sum\_l Y\_l3 \(v \+ 1/2\)^l; lambda\_v, lambda\_Dv, lambda\_Hv \(and gamma\) = sum\_l X\_l0, X\_l1, X\_l2 \(v \+ 1/2\)^l; Y\_lm = U\_lm \(1/mu\)^\(l/2 \+ m\) \+ U\_lm Delta\_lm m\_e \(1/mu\)^\(1 \+ l/2 \+ m\) \(the Born\-Oppenheimer\-breakdown term\), and for gamma one more power of 1/mu\. Table I prints, per row, 'U\_{l,m} x mu^\-\(l/2\+m\)/2^l' etc\.\: the BO part and the BOB part of Y\_lm divided by 2^l, i\.e\. the coefficients of \(2v \+ 1\)^l \(checked\: the sums reproduce Table III and hajigeorgiou\_2013's Table I, see the formula's check\)\. Atomic masses \(p\. 5\)\: 15\.99491461956\(16\) u \(AME 2003\), 16\.9991317566\(9\) u and 17\.9991596130\(13\) u\. 'BOB corrections were necessary for Y\_1,0, Y\_0,1, lambda\_0,0, and gamma\_0,0 of the X 3Sigma\_g\- state' \(p\. 8\)\. 'The U\_0,3 parameters of all the three states were marginally determinable and were therefore fixed at values calculated using lower order U\_l,m parameters' \(p\. 8; Table I note a\)\. TABLE III \(p\. 8, 'Comparison of 16O16O X 3Sigma\_g\- and b 1Sigma\_g\+ v = 0,1 parameters determined from band\-by\-band fits with different choices of experimental data\: Fit \#1 including all available data; Fit \#2 with repeated less accurate data removed'; MHz\)\: the X\-state constants of v = 0 and 1 are the datasets fit1\_\* \(primary\) and fit2\_\* \(check\), 14 values each\. 'The parameters in the ground states are mainly determined by the common MW, THz, and Raman datasets and are the same from these two fits' \(p\. 8\)\. Not stored\: T\_1 = 46659391\.5\(37\) MHz \(both fits\), the X v = 1 band origin \(1556\.38977 cm\-1 with c = 29979\.2458 MHz/cm\-1; the vocabulary's vibrational term and Raman shift are in cm\-1 only\), and the b 1Sigma\_g\+ rows \(T\_0 393395205\.43\(34\), B\_0 41708\.6100\(33\), \-D\_0 \-0\.1609821\(74\), H\_0 0\.313\(100\)e\-9, T\_1 435508142\.65\(200\), B\_1 41160\.5224\(211\), \-D\_1 \-0\.162250\(63\), H\_1 0\.198\(55\)e\-6 MHz, Fit \#1\)\. The band\-by\-band B, D and H of v = 0 and 1 are not identical to the Dunham ones\: their H\_v is free \(0\.1209e\-6 against the Dunham 3\.18e\-9 MHz\), and D and B move with it \(see the formula's check\)\. UNCERTAINTIES\: the parenthesised digits of Tables I and III are the fit uncertainties in units of the last digit; the paper does not name their level, but speaks of the fits agreeing 'with 1 sigma error' \(p\. 8\)\. Recorded as level 'not stated'\. TABLE I, NOT TAKEN\: the a 1Delta\_g and b 1Sigma\_g\+ columns \(U^Te 237 577 161\(52\) and 395 594 904\(149\) MHz, their Y\_lm; the b column is headed 'b 1Sigma\_g\-' in print, a misprint for Sigma\_g\+ that touches no stored value\); the 16O17O hyperfine constants \(U^a \-213\.18 \(a state\), U^bF \-54\.74911, U^c 140\.0610, 1\.5 U^eQq \-12\.317, U^CI \-0\.01753 MHz\)\. Table II \(BOB coefficients and r\_e\)\: 16O16O X r\_e = 1\.20752763\(17\) A, r\_e^BO = 1\.20745398\(15\) A \(notes only\)\. The a and b states' electronic energies\: U^Te = 7924\.72103\(102\) and 13 195\.62255\(497\) cm\-1 \(p\. 8\)\. RAMAN DATA IN THE FIT \(Sec\. IV B, pp\. 12\-13\)\: 30 pure rotational \(Delta N = 2\) lines of 16O16O v = 0 \(Brown et al\., \+\-0\.005 cm\-1, N'' = 3\-21\) and the 1\-0 band from Rouille et al\., Millot et al\. and Brodersen and Bendtsen \(0\.0004\-0\.016 cm\-1\); 2\-1 and 3\-2 Q\-branch data of Rouille et al\. The Delta N = 0 Raman lines' spin structure was not resolved and each was treated as a blend of three components of equal intensity\. Since 2026\-09\-27 the two Raman sources are curated\: brown\_1992 \(its Table I, 30 ^SS lines S\(J,N\) with lower\-level N = 1\-19, the upper levels N = 3\-21, which this paper writes N'' = 3\-21; \+\-0\.005 cm\-1\) and rouille\_1992 \(its Table II 1\-0 ^SS and ^OO triplets, Table V Q branches of the 1\-0, 2\-1 and 3\-2 bands, and its v = 0\-3 constants\)\. Against this record's Table III Fit \#1 \(MHz / 29979\.2458\)\: rouille\_1992's B\_0 1\.437676476, lambda\_0 1\.984751322, mu\_0 \(= gamma\_0\) \-8\.425390e\-3 and B\_1 1\.42186454 cm\-1 agree with 1\.43767616, 1\.98475119, \-8\.42537e\-3 and 1\.42186433 to 3e\-7 cm\-1\. No dataset here reproduces either paper, so nothing is re\-pointed\. REPRODUCTIONS ELSEWHERE\: hajigeorgiou\_2013's Table I B\_v, D\_v and H\_v \(v = 0\-31\) are evaluated from this Table I \(re\-pointed to yu\_2012 on 2026\-09\-27; see the formula's check\: they agree to the printed digits\)\. Its Pekeris\-check constants \(omega\_e = 1579\.833, omega\_e x\_e = 11\.6938, B\_e = 1\.445611834, alpha\_e = 0\.0158946 cm\-1, 'a subset of the Dunham coefficients employed'\)\: B\_e and alpha\_e are this Table I's Y\_01 and \-Y\_11 \(Y\_01 = \(43343\.5117 \- 5\.1592\)/29979\.2458 = 1\.44561183 cm\-1, \-Y\_11 = 2 x 238\.2546/29979\.2458 = 0\.0158946 cm\-1\), but omega\_e and omega\_e x\_e are not\: this Table I gives Y\_10 = 2 \(23683150 \+ 2009\)/29979\.2458 = 1580\.1037 and \-Y\_20 = 4 x 89298\.8/29979\.2458 = 11\.9147 cm\-1, so Hajigeorgiou's 1579\.833 and 11\.6938 are presumably from his own G\_v fit \(checked 2026\-09\-27\)\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- yu\_2012/fit1\_B\_O2 ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_B_O2", "material": "O2", "file": "O2/yu_2012__fit1_B_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "B, the rotational constant of Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: B\_v \(rows B\_0, B\_1; SPFIT operators 10000, 10101\)\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_minus\_D\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_minus_D_O2", "material": "O2", "file": "O2/yu_2012__fit1_minus_D_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "-D, printed '-D_v': the coefficient of +(N^2)^2 in Eq. (2) (per row: v); the conventional centrifugal distortion constant D_v is its negative", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: '\-D\_v' \(rows \-D\_0, \-D\_1; operators 20000, 20101\)\: the printed values are negative, the coefficient of \+\(N^2\)^2 in Eq\. \(2\); D\_v = \+0\.14516 MHz\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_H\_O2 ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_H_O2", "material": "O2", "file": "O2/yu_2012__fit1_H_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 1e-06, "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "H, the coefficient of +(N^2)^3 in Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: H\_v \(rows H\_0, H\_1; operators 30000, 30101\), x 10^\-6 \(the dataset scale\)\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_4lambda\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_4lambda_O2", "material": "O2", "file": "O2/yu_2012__fit1_4lambda_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "4 lambda, four times the spin-spin constant lambda_v of Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: 4 lambda\_v \(rows 4lambda\_0, 4lambda\_1; operators 11001000000, 11001000101\)\: four times the spin\-spin constant\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_4lambda\_D\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_4lambda_D_O2", "material": "O2", "file": "O2/yu_2012__fit1_4lambda_D_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "4 lambda_D, four times the N^2 coefficient of the spin-spin term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: 4 lambda\_D, printed '4lambda\_D0' with operator 11001019999 \(the 9999 ending\: common to both levels\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_4lambda\_H\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_4lambda_H_O2", "material": "O2", "file": "O2/yu_2012__fit1_4lambda_H_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 1e-06, "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "4 lambda_H, four times the (N^2)^2 coefficient of the spin-spin term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: 4 lambda\_H, printed '4lambda\_H0' with operator 11001029999 \(common to both levels\), x 10^\-6 \(the dataset scale\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_gamma\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_gamma_O2", "material": "O2", "file": "O2/yu_2012__fit1_gamma_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "gamma, the spin-rotation constant of Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: gamma\_v \(rows gamma\_0, gamma\_1; operators 1000000000, 1000000101\)\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_gamma\_D\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_gamma_D_O2", "material": "O2", "file": "O2/yu_2012__fit1_gamma_D_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 0.001, "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "gamma_D, the N^2 coefficient of the spin-rotation term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: gamma\_D, printed 'gamma\_D1' with operator 1000019999 \(common to both levels\), x 10^\-3 \(the dataset scale\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit1\_gamma\_H\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit1_gamma_H_O2", "material": "O2", "file": "O2/yu_2012__fit1_gamma_H_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 1e-09, "abscissa": "label", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "gamma_H, the (N^2)^2 coefficient of the spin-rotation term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#1 \(all A\- and B\-band data\), 16O16O X 3Sigma\_g\-\: gamma\_H, printed 'gamma\_H1' with operator 1000029999 \(common to both levels\), x 10^\-9 \(the dataset scale\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. yu\_2012/fit2\_B\_O2 ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_B_O2", "material": "O2", "file": "O2/yu_2012__fit2_B_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "B, the rotational constant of Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: B\_v \(rows B\_0, B\_1; SPFIT operators 10000, 10101\)\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_minus\_D\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_minus_D_O2", "material": "O2", "file": "O2/yu_2012__fit2_minus_D_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "-D, printed '-D_v': the coefficient of +(N^2)^2 in Eq. (2) (per row: v); the conventional centrifugal distortion constant D_v is its negative", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: '\-D\_v' \(rows \-D\_0, \-D\_1; operators 20000, 20101\)\: the printed values are negative, the coefficient of \+\(N^2\)^2 in Eq\. \(2\); D\_v = \+0\.14516 MHz\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_H\_O2 ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_H_O2", "material": "O2", "file": "O2/yu_2012__fit2_H_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 1e-06, "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "H, the coefficient of +(N^2)^3 in Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: H\_v \(rows H\_0, H\_1; operators 30000, 30101\), x 10^\-6 \(the dataset scale\)\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_4lambda\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_4lambda_O2", "material": "O2", "file": "O2/yu_2012__fit2_4lambda_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "4 lambda, four times the spin-spin constant lambda_v of Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: 4 lambda\_v \(rows 4lambda\_0, 4lambda\_1; operators 11001000000, 11001000101\)\: four times the spin\-spin constant\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_4lambda\_D\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_4lambda_D_O2", "material": "O2", "file": "O2/yu_2012__fit2_4lambda_D_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "4 lambda_D, four times the N^2 coefficient of the spin-spin term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: 4 lambda\_D, printed '4lambda\_D0' with operator 11001019999 \(the 9999 ending\: common to both levels\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_4lambda\_H\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_4lambda_H_O2", "material": "O2", "file": "O2/yu_2012__fit2_4lambda_H_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 1e-06, "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "4 lambda_H, four times the (N^2)^2 coefficient of the spin-spin term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: 4 lambda\_H, printed '4lambda\_H0' with operator 11001029999 \(common to both levels\), x 10^\-6 \(the dataset scale\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_gamma\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_gamma_O2", "material": "O2", "file": "O2/yu_2012__fit2_gamma_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "gamma, the spin-rotation constant of Eq. (2) (per row: v)", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: gamma\_v \(rows gamma\_0, gamma\_1; operators 1000000000, 1000000101\)\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_gamma\_D\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_gamma_D_O2", "material": "O2", "file": "O2/yu_2012__fit2_gamma_D_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 0.001, "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "gamma_D, the N^2 coefficient of the spin-rotation term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: gamma\_D, printed 'gamma\_D1' with operator 1000019999 \(common to both levels\), x 10^\-3 \(the dataset scale\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. yu\_2012/fit2\_gamma\_H\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fit2_gamma_H_O2", "material": "O2", "file": "O2/yu_2012__fit2_gamma_H_O2.csv", "table": "tbl_844a6de53312", "quantity": "rotational_constant", "unit": "MHz", "scale": 1e-09, "abscissa": "label", "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "constant": "gamma_H, the (N^2)^2 coefficient of the spin-rotation term of Eq. (2), common to v = 0 and 1", "vibrational_state_v": "per row", "electronic_state": "X 3Sigma_g-" } } Table III \(p\. 8\), Fit \#2 \(repeated less accurate A\- and B\-band data removed\), 16O16O X 3Sigma\_g\-\: gamma\_H, printed 'gamma\_H1' with operator 1000029999 \(common to both levels\), x 10^\-9 \(the dataset scale\), one row\. The uncertainty in parentheses \(last digits\) is the sigma column; its level is not stated \(SPFIT fit uncertainties\)\. The same as Fit \#1 within its uncertainty \(the ground\-state constants are fixed by the common MW, THz and Raman data, p\. 8\)\. Formulas -------- yu\_2012/x\_state\_dunham ~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "x_state_dunham", "material": "O2", "equation": "Eq. (2), p. 3; Eqs. (4)-(14), pp. 4-5; Table I (tbl_f249a8f9f01f), p. 7", "printed": "H = G_v + B_v N^2 + D_v (N^2)^2 + H_v (N^2)^3 + [lambda_v + lambda_Dv N^2 + lambda_Hv (N^2)^2] (2/3)(3 S_z^2 - S^2) + [gamma_v + gamma_Dv N^2 + gamma_Hv (N^2)^2] N.S (Eq. 2, X 3Sigma_g-, T_e = 0); G_v = sum_l Y_l,0 (v + 1/2)^l, B_v = sum_l Y_l,1 (v + 1/2)^l, D_v = sum_l Y_l,2 (v + 1/2)^l, H_v = sum_l Y_l,3 (v + 1/2)^l; lambda_v, lambda_Dv, lambda_Hv = sum_l lambda_l,0, lambda_l,1, lambda_l,2 (v + 1/2)^l, the same for gamma; Table I (MHz, 16O16O): U_l,m mu^-(l/2+m)/2^l and U_l,m Delta_l,m m_e mu^-(1+l/2+m)/2^l, 4 x U^lambda_l,m ..., U^gamma_l,m mu^-(1+l/2+m)/2^l ...", "variable": "v: the vibrational quantum number of X 3Sigma_g-; N: the rotational angular momentum without spin; S: the electron spin (S = 1). Each coefficient is the printed Table I entry for 16O16O in MHz: Ylm = U_l,m mu^-(l/2+m)/2^l, the Born-Oppenheimer part of the Dunham constant Y_l,m divided by 2^l; Ylm_BOB = U_l,m Delta_l,m m_e mu^-(1+l/2+m)/2^l, its Born-Oppenheimer-breakdown part divided by 2^l (Y10, Y01 only); Llm = 4 x lambda_l,m/2^l (L00_BOB its BOB part); Glm = gamma_l,m/2^l (G00_BOB its BOB part). So, with x = 2v + 1: G_v = sum_l (Yl0 [+ Yl0_BOB]) x^l; B_v = sum_l (Yl1 [+ Yl1_BOB]) x^l; D_v (Eq. 2's, negative) = sum_l Yl2 x^l; H_v = sum_l Yl3 x^l; 4 lambda_v = sum_l (Ll0 [+ L00_BOB]) x^l; 4 lambda_Dv = sum_l Ll1 x^l; 4 lambda_Hv = sum_l Ll2 x^l; gamma_v = sum_l (Gl0 [+ G00_BOB]) x^l; gamma_Dv = G01; gamma_Hv = G02. L03 (4 x U^lambda_0,3 mu^-3) is an (N^2)^3 spin-spin term that Eq. (2) as printed does not show. The keys ending _sigma are the printed uncertainties (last digits) as absolute numbers; Y03 has none (fixed, note a).", "state": "16O16O X 3Sigma_g-, the free molecule (line positions)", "density": "unknown", "range": "v = 0-31 of X (the data: v = 24-25 have none, 26-31 only to +-3 cm-1); predictions for higher v are extrapolations", "coefficients": { "Y10": "23683150", "Y10_sigma": "166", "Y10_BOB": "2009", "Y10_BOB_sigma": "165", "Y20": "-89298.8", "Y20_sigma": "6.7", "Y30": "137.94", "Y30_sigma": "0.99", "Y40": "-1.591", "Y40_sigma": "0.056", "Y50": "6.31e-3", "Y50_sigma": "1.34e-3", "Y60": "-0.2165e-3", "Y60_sigma": "0.0114e-3", "Y01": "43343.5117", "Y01_sigma": "0.0110", "Y01_BOB": "-5.1592", "Y01_BOB_sigma": "0.0091", "Y11": "-238.2546", "Y11_sigma": "0.0158", "Y21": "0.3600", "Y21_sigma": "0.0039", "Y31": "-0.015555", "Y31_sigma": "0.000271", "Y41": "0.4723e-3", "Y41_sigma": "0.0073e-3", "Y51": "-6.602e-6", "Y51_sigma": "0.062e-6", "Y02": "-0.1450753", "Y02_sigma": "0.0000043", "Y12": "-0.0515e-3", "Y12_sigma": "0.0048e-3", "Y22": "0.01241e-3", "Y22_sigma": "0.00112e-3", "Y32": "-0.5865e-6", "Y32_sigma": "0.0226e-6", "Y03": "-1.34e-9", "Y13": "4.52e-9", "Y13_sigma": "0.73e-9", "L00": "237726.55", "L00_sigma": "0.40", "L00_BOB": "31.328", "L00_BOB_sigma": "0.109", "L10": "226.24", "L10_sigma": "0.89", "L20": "23.89", "L20_sigma": "0.46", "L30": "-2.661", "L30_sigma": "0.079", "L40": "0.05798", "L40_sigma": "0.00124", "L01": "0.21107", "L01_sigma": "0.00039", "L11": "0.02681", "L11_sigma": "0.00054", "L21": "-4.500e-3", "L21_sigma": "0.142e-3", "L02": "1.064e-6", "L02_sigma": "0.062e-6", "L03": "0.086e-9", "L03_sigma": "0.039e-9", "G00": "-252.28822", "G00_sigma": "0.00055", "G00_BOB": "8.31e-3", "G00_BOB_sigma": "0.60e-3", "G10": "-0.306488", "G10_sigma": "0.000052", "G01": "-0.243777e-3", "G01_sigma": "0.000283e-3", "G02": "-1.113e-9", "G02_sigma": "0.210e-9" }, "check": "Scratch (2026-09-27), with x = 2v + 1 and the BOB terms added: v = 0: B = 43100.44281 MHz, D (Eq. 2) = -0.14511498, H = 3.180e-9, 4 lambda = 238005.4050, 4 lambda_D = 0.233380, gamma = -252.586398; v = 1: B = 42626.44537, 4 lambda = 238584.4574, gamma = -253.199374; G_1 - G_0 = 46659388.1 MHz. Against Table III's band-by-band Fit #1: B_0 43100.44712(70) (-0.0043 MHz), B_1 42626.4203(228) (+0.025), 4 lambda_0 238005.3746(64) (+0.030), 4 lambda_1 238594.1(285) (-9.6), 4 lambda_D0 0.233476(40) (-0.0001), gamma_0 -252.586311(176) (-0.00009), gamma_1 -252.888(238) (-0.31), T_1 46659391.5(37) (-3.4 MHz): all within the Dunham constants' own uncertainties (Y01 +-0.011, L00 +-0.40 MHz) though B_0 is 6 sigma of Table III's +-0.0007; D_0 -0.1451150 against -0.1451596(32) and H_0 3.2e-9 against 0.121e-6 differ because Table III's band fits free H_v (correlated with D). Against hajigeorgiou_2013's Table I (B_v, D_v, H_v in cm-1 'evaluated from Yu et al.'s Dunham coefficients', v = 0-31): identical to the printed digits, max |difference| 1.4e-4 MHz in B, 5e-14 cm-1 in D, 5e-17 cm-1 in H, with c = 29979.2458 MHz/cm-1 (B_0 = 1.43767602, D_0 = 4.8405146e-6, H_0 = 0.10607e-12 cm-1). Without the BOB term of Y01 B_0 would be 1.43784811 cm-1: the BOB terms must be added." } The global fit of all six isotopologues; only 16O16O's X\-state constants are recorded \(the U's are isotopically invariant, the mu powers fixed for 16O16O\: the printed entries already include them\)\. Hund's case \(b\) levels follow from diagonalizing Eq\. \(2\) with these v\-dependent constants; the conventional D\_v = \-sum Yl2 x^l\. The fit reproduces 11 944 lines with a reduced rms of 1\.1 \(p\. 7\)\. 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 ` for the checks and decisions.