hajigeorgiou_2013#
P. G. Hajigeorgiou, “The potential energy function of the ground electronic state of 16O2”, J. Chem. Phys. 138, 014309 (2013)
Source record · Catalogue · Review register
DOI: 10.1063/1.4773285
Source kind: fit
Recorded technique: Direct fit of an analytical potential energy function (a Morse-like form with a 16-term exponent polynomial in the reduced variable y_p, p = 4.52, and an inverse-power inner wall below 0.925 A) and an N-dependent radial function alpha(r) in the centrifugal term to 420 rotational term values of the X 3Sigma_g- state of 16O2 (v = 0-31, every second N up to N_max), themselves computed from Yu et al.’s (J. Chem. Phys. 137, 024304 (2012)) Dunham coefficients for B_v, D_v, H_v and from the author’s Dunham fit of literature vibrational energies (v = 0-35); unweighted nonlinear least squares, eigenvalues from the radial Schroedinger equation on 0.80-6.80 A (step 0.0005 A), standard deviation 0.0031 cm^-1. The constants of Table III are the quantum-mechanical expectation values of the fitted functions (a modified CDIST program), pp. 014309-2 to 014309-5.
Materials in structured entries: O2
Dataset quantities: rotational_constant, vibrational_term_value
Contents: 12 datasets; 1 formula.
Curation notes#
Pages are the article’s (014309-1 to 014309-8); pdf page 1 is the publisher’s download cover, so p. 014309-n = pdf page n + 1. Identity checked against the first page. The library’s warnings (unmatched figures on the cover page; one unmapped glyph on each of pdf pages 7-8, in the Omega symbols of the discussion) touch no value used.
MATERIAL AND STATE: 16O2, the X 3Sigma_g- ground state. Hund’s case (b): Eq. (1), p. 014309-2, E_vN = G_v + B_v N(N+1) - D_v [N(N+1)]^2 + H_v [N(N+1)]^3, N the rotational quantum number without spin. SPIN: the paper does not use the words ‘spin-free’. It says (p. 014309-1): ‘It is not the objective of the present work to carry out an analysis of the spin-spin and spin-rotation structure in the ground state, as these effects have been characterized very precisely in the work of Yu et al.’, whose Dunham fit gave ‘the rotational constants B_v and centrifugal distortion constants D_v and H_v, as well as ... the spin-spin and spin-rotation parameters’ (p. 014309-2). So the term values fitted here, and the B_v, D_v, H_v of Tables I and III, are those of the N(N+1) ladder of Eq. (1) with the spin-spin (lambda) and spin-rotation (gamma) splittings of each N into J = N, N +- 1 excluded (the effective ‘mechanical’ constants, what solprop calls spin-free); they include no spin-spin constants. Also (p. 014309-7): ‘the potential curve determined in this work is associated with the Omega = 1 spin component’, because most of the data come from Schumann-Runge spectra (an adiabatic avoided-crossing treatment of the X-b spin-orbit coupling; relevant near 2 A, far above the levels that matter for Raman).
TABLE I (p. 014309-2, ‘Vibrational energies, rotational constants, and centrifugal distortion constants (cm^-1) employed in calculating term values’; note a: ‘Y00 = 0.331 962 cm^-1 must be added to all calculated term values’): the INPUT of the fit. B_v, D_v, H_v are evaluated from Yu et al.’s Dunham coefficients (fitted there over v = 0-31): reproduced. G_v is the author’s own Dunham fit of ‘literature vibrational energies with their respective relative uncertainties over the range v = 0-35’ (refs. 4, 5, 9-15; p. 014309-2), because Yu et al.’s G_v series covers v = 0-23 only: this paper’s fit, a check. The printed G_v are Dunham sums without Y00: G_v + Y00 is measured from the potential minimum (G_0 + Y00 = 787.321962 against Table III’s 787.3223). N_max (the highest N of the experimental data per v) stays in ‘printed’. Yu et al.’s H_v are all positive; Table III’s are all negative: Yu’s Y03 ‘was constrained rather than fitted’, thought to be the source of the discrepancy, which ‘in no way undermines the quality of the term values’ (p. 014309-5).
TABLE III (p. 014309-5, ‘Quantum-mechanical vibrational energies, rotational constants, and centrifugal distortion constants (cm^-1) calculated from the radial functions V(r) and alpha(r)’): this paper’s result, v = 0-38. B_v = (hbar^2/2 mu) <psi_v|[1 + alpha(r)]/r^2|psi_v> (Eq. 14), with analogous definitions for D_v, H_v. The library’s cells for this table are column-scrambled: transcribed in the script from the PDF text layer and the image (see tools/extract/hajigeorgiou_2013.py). No uncertainty is stated for the constants; the fit reproduces the term values with a standard deviation of 0.0031 cm^-1 (p. 014309-4), and ‘The potential energy function ... is strictly speaking reliable up to 90% of the dissociation limit’ (v <= 31), perhaps to 99 % (v = 38) (p. 014309-8). The G_v of Table III are identical to Table V’s (checked by the script) and are stored once, from Table V (table5_G_O2).
TABLE V (p. 014309-8): G_v from the potential minimum (V(r_e) = 0, p. 014309-4) and the inner and outer turning points (A; no quantity, kept in ‘printed’), v = 0-38; ‘More significant figures have been provided for vibrational levels with v > 31 than are justified by the expected uncertainties’ (p. 014309-7).
TABLE IV (p. 014309-7): G_v - G_0 and B_v for v = 29-38 from Yang and Wodtke (YW; J. Chem. Phys. 90, 7114 (1989)): note b ‘Experimental value with uncertainty +-40 cm^-1’ (v = 29-35), note d ‘Extrapolated value with unknown uncertainty’ (v = 36-38), note c (all B_v) ‘Value calculated from hybrid potential with uncertainty +-0.01 cm^-1’; and, in parentheses beside the present work’s values, the ab initio values of Bytautas, Matsunaga and Ruedenberg (Ref. 7, J. Chem. Phys. 132, 074307 (2010)). The present work’s columns repeat Table III rounded (G_v - G_0 = 35 582 for v = 29 is 36 369.0317 - 787.3223 = 35 581.7): not taken. Levels above v = 31 lie beyond the fitted data.
REPRODUCTION (2026-09-26, scratch script): the radial Schroedinger equation with the Table II potential (Eqs. 2-6 as printed, hbar^2/2 = 16.857 629 20 cm^-1 amu A^2, mu = 7.997 457 309 78 amu, grid 0.80-6.80 A, three-point differences at steps 0.0005 and 0.00025 A with Richardson extrapolation) gives Table III’s G_v to <= 0.0001 cm^-1 for v = 0-11, <= 0.0004 for v <= 26, and within 0.0065 cm^-1 up to v = 38; Eq. (14) gives B_0-B_3 to <= 2e-8 cm^-1 (the printed rounding is 5e-10). D_v and H_v were not recomputed. FINDING: with Table II’s A and q as printed, the inner wall A/r^q (55 664 cm^-1 at 0.925 A) does not join the Morse-like branch (53 943 cm^-1 there), although the text says A and q ‘were determined by considering the values of the potential energy and its first derivative at r = 0.925 A’ (p. 014309-3); a continuous join (A = 15 543.5, q = 15.960) spoils the reproduction of G_v for v >= 26 by up to 0.22 cm^-1, so the printed A and q are those used. Whether the published form differs from the one programmed near 0.925 A cannot be told; it matters only above v ~ 25.
OTHER TEXT VALUES (not stored): r_e = 1.207 526 69(34) A (this work) against Yu et al.’s 1.207 527 63(17) A (p. 014309-5); D_e = 42 055.6 cm^-1 = Cosby and Huestis’s D_0 = 41 268.6 cm^-1 plus G_0, held fixed (p. 014309-4); the equilibrium constants of the Pekeris check, omega_e = 1579.833, omega_e x_e = 11.6938, B_e = 1.445 611 834, alpha_e = 0.015 894 6 cm^-1, ‘a subset of the Dunham coefficients employed’ (Yu et al.’s; reproduced there, not stored; yu_2012’s Table I gives these B_e and alpha_e, but Y_10 = 1580.1037 and -Y_20 = 11.9147 cm-1, so the omega_e and omega_e x_e here are not Yu et al.’s), giving Y00 = 0.335 cm^-1 against the fitted 0.332 (p. 014309-4). The reduced mass 7.997 457 309 78 amu (p. 014309-4) is solprop’s O2 reduced mass.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
hajigeorgiou_2013/table1_G_O2#
{
"id": "table1_G_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table1_G_O2.csv",
"table": "tbl_4bb1a4a3289b",
"quantity": "vibrational_term_value",
"unit": "cm-1",
"abscissa": "label",
"use": "check",
"parameters": {
"reference": "minimum",
"electronic_state": "X 3Sigma_g-"
}
}
Table I (p. 014309-2): G_v, v = 0-31 (the row), ‘employed in calculating term values’: the author’s Dunham fit of literature vibrational energies (refs. 4, 5, 9-15). CAUTION on the zero: these are Dunham G_v without Y00; note a: ‘Y00 = 0.331 962 cm^-1 must be added to all calculated term values’ to measure them from the potential minimum. Differences G_v - G_0 are unaffected. An input, superseded by Table V.
hajigeorgiou_2013/table1_B_O2#
{
"id": "table1_B_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table1_B_O2.csv",
"table": "tbl_4bb1a4a3289b",
"quantity": "rotational_constant",
"unit": "cm-1",
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"yu_2012"
],
"parameters": {
"constant": "B",
"electronic_state": "X 3Sigma_g-"
}
}
Table I (p. 014309-2): B_v of Eq. (1) (the N(N+1) ladder, spin structure excluded), v = 0-31 (the row), evaluated from Yu et al.’s Dunham coefficients. Re-pointed to yu_2012 (2026-09-27): its Table I (the X-state Dunham constants, formula x_state_dunham) reproduces this column to the printed digits for v = 0-31 (max |difference| 1.4e-4 MHz in B, 5e-14 cm-1 in D, 5e-17 cm-1 in H, c = 29979.2458 MHz/cm-1).
hajigeorgiou_2013/table1_D_O2#
{
"id": "table1_D_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table1_D_O2.csv",
"table": "tbl_4bb1a4a3289b",
"quantity": "rotational_constant",
"unit": "cm-1",
"scale": 1e-06,
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"yu_2012"
],
"parameters": {
"constant": "D",
"electronic_state": "X 3Sigma_g-"
}
}
Table I (p. 014309-2): D_v (x 1e-6) of Eq. (1), sign convention -D_v [N(N+1)]^2, v = 0-31 (the row), from Yu et al.’s Dunham coefficients. Re-pointed to yu_2012 (2026-09-27): its Table I (the X-state Dunham constants, formula x_state_dunham) reproduces this column to the printed digits for v = 0-31 (max |difference| 1.4e-4 MHz in B, 5e-14 cm-1 in D, 5e-17 cm-1 in H, c = 29979.2458 MHz/cm-1).
hajigeorgiou_2013/table1_H_O2#
{
"id": "table1_H_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table1_H_O2.csv",
"table": "tbl_4bb1a4a3289b",
"quantity": "rotational_constant",
"unit": "cm-1",
"scale": 1e-12,
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"yu_2012"
],
"parameters": {
"constant": "H",
"electronic_state": "X 3Sigma_g-"
}
}
Table I (p. 014309-2): H_v (x 1e-12) of Eq. (1), sign convention +H_v [N(N+1)]^3, v = 0-31 (the row), from Yu et al.’s Dunham coefficients (all positive; Yu’s Y03 was constrained, p. 014309-5, and Table III’s H_v are all negative). Re-pointed to yu_2012 (2026-09-27): its Table I (the X-state Dunham constants, formula x_state_dunham) reproduces this column to the printed digits for v = 0-31 (max |difference| 1.4e-4 MHz in B, 5e-14 cm-1 in D, 5e-17 cm-1 in H, c = 29979.2458 MHz/cm-1).
hajigeorgiou_2013/table5_G_O2#
{
"id": "table5_G_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table5_G_O2.csv",
"table": "tbl_02f68c84e070",
"quantity": "vibrational_term_value",
"unit": "cm-1",
"abscissa": "label",
"use": "primary",
"parameters": {
"reference": "minimum",
"electronic_state": "X 3Sigma_g-"
}
}
Table V (p. 014309-8), identical to Table III’s G_v column (checked by the script): the quantum-mechanical vibrational energies (N = 0 eigenvalues) of the fitted potential, measured from its minimum (V(r_e) = 0), v = 0-38 (the row). G_1 - G_0 = 1556.3906 cm^-1 (the band origin). No uncertainty stated (fit standard deviation 0.0031 cm^-1); reliable to v = 31 (‘90% of the dissociation limit’), extra digits above v = 31 ‘not justified’ (p. 014309-7). The turning points are kept in ‘printed’.
hajigeorgiou_2013/table3_B_O2#
{
"id": "table3_B_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table3_B_O2.csv",
"table": "tbl_a6413e37f185",
"quantity": "rotational_constant",
"unit": "cm-1",
"abscissa": "label",
"use": "primary",
"parameters": {
"constant": "B",
"electronic_state": "X 3Sigma_g-"
}
}
Table III (p. 014309-5): B_v, v = 0-38 (the row), the expectation value of Eq. (14) over the fitted V(r) and alpha(r); the rotational constant of Eq. (1)’s N(N+1) ladder, spin-spin and spin-rotation structure excluded. B_0 = 1.437 674 521 cm^-1 (solprop’s O2 B). Transcribed from the text layer and image (the library cells are scrambled; see the script). No uncertainty stated.
hajigeorgiou_2013/table3_D_O2#
{
"id": "table3_D_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table3_D_O2.csv",
"table": "tbl_a6413e37f185",
"quantity": "rotational_constant",
"unit": "cm-1",
"abscissa": "label",
"use": "primary",
"parameters": {
"constant": "D",
"electronic_state": "X 3Sigma_g-"
}
}
Table III (p. 014309-5): D_v, v = 0-38 (the row), printed ‘m x 10^-6’ (x 10^-5 for v = 36-38) and written as ‘me-6’; sign convention -D_v [N(N+1)]^2 (Eq. 1). D_0 = 4.839 482e-6 cm^-1 (solprop’s O2 D). Computed with CDIST including alpha(r). Not recomputed here. No uncertainty stated.
hajigeorgiou_2013/table3_H_O2#
{
"id": "table3_H_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table3_H_O2.csv",
"table": "tbl_a6413e37f185",
"quantity": "rotational_constant",
"unit": "cm-1",
"abscissa": "label",
"use": "primary",
"parameters": {
"constant": "H",
"electronic_state": "X 3Sigma_g-"
}
}
Table III (p. 014309-5): H_v, v = 0-38 (the row), printed ‘m x 10^-k’ and written as ‘me-k’; sign convention +H_v [N(N+1)]^3 (Eq. 1); all negative, unlike Table I’s (p. 014309-5). No uncertainty stated.
hajigeorgiou_2013/table4_G_yang_wodtke_O2#
{
"id": "table4_G_yang_wodtke_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table4_G_yang_wodtke_O2.csv",
"table": "tbl_9bf6778409d5",
"quantity": "vibrational_term_value",
"unit": "cm-1",
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"X. Yang and A. M. Wodtke, J. Chem. Phys. 90, 7114 (1989)"
],
"excluded_rows": {
"36": "note d: 'Extrapolated value with unknown uncertainty'",
"37": "note d: 'Extrapolated value with unknown uncertainty'",
"38": "note d: 'Extrapolated value with unknown uncertainty'"
},
"parameters": {
"reference": "v0",
"electronic_state": "X 3Sigma_g-"
},
"uncertainty": {
"column": true,
"level": "not stated"
}
}
Table IV (p. 014309-7), YW column: G_v - G_0, v = 29-38 (the row), from Schumann-Runge emission. v = 29-35: note b, ‘Experimental value with uncertainty +-40 cm^-1’ (the sigma column; level not stated).
hajigeorgiou_2013/table4_B_yang_wodtke_O2#
{
"id": "table4_B_yang_wodtke_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table4_B_yang_wodtke_O2.csv",
"table": "tbl_9bf6778409d5",
"quantity": "rotational_constant",
"unit": "cm-1",
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"X. Yang and A. M. Wodtke, J. Chem. Phys. 90, 7114 (1989)"
],
"parameters": {
"constant": "B",
"electronic_state": "X 3Sigma_g-"
},
"uncertainty": {
"absolute": 0.01,
"level": "not stated"
}
}
Table IV (p. 014309-7), YW column: B_v, v = 29-38 (the row), note c: ‘Value calculated from hybrid potential with uncertainty +-0.01 cm^-1’ (level not stated).
hajigeorgiou_2013/table4_G_bytautas_O2#
{
"id": "table4_G_bytautas_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table4_G_bytautas_O2.csv",
"table": "tbl_9bf6778409d5",
"quantity": "vibrational_term_value",
"unit": "cm-1",
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"L. Bytautas, N. Matsunaga and K. Ruedenberg, J. Chem. Phys. 132, 074307 (2010)"
],
"parameters": {
"reference": "v0",
"electronic_state": "X 3Sigma_g-"
}
}
Table IV (p. 014309-7), the values in parentheses (‘Values in parentheses from Ref. 7’): ab initio G_v - G_0, v = 29-38 (the row), a theory calculation (core-valence correlation and relativistic contributions, p. 014309-1), ‘consistent with experimental energies within +-13 cm^-1, on average’ (p. 014309-7).
hajigeorgiou_2013/table4_B_bytautas_O2#
{
"id": "table4_B_bytautas_O2",
"material": "O2",
"file": "O2/hajigeorgiou_2013__table4_B_bytautas_O2.csv",
"table": "tbl_9bf6778409d5",
"quantity": "rotational_constant",
"unit": "cm-1",
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"L. Bytautas, N. Matsunaga and K. Ruedenberg, J. Chem. Phys. 132, 074307 (2010)"
],
"parameters": {
"constant": "B",
"electronic_state": "X 3Sigma_g-"
}
}
Table IV (p. 014309-7), in parentheses: ab initio B_v, v = 29-38 (the row), theory.
Formulas#
hajigeorgiou_2013/potential#
{
"id": "potential",
"material": "O2",
"equation": "Eqs. (2)-(6), (7), (9)-(11), (14) and Table II, pp. 014309-2 to 014309-5",
"printed": "V(r) = D_e [1 - exp(-beta(y_p) z)]^2 (Eq. 2); beta(y_p) = sum_i beta_i y_p^i (Eq. 3); y_p = (r^p - r_e^p)/(r^p + r_e^p) (Eq. 4); z = y_1 = (r - r_e)/(r + r_e) (Eq. 5); for r <= 0.925 A, V(r) = A/r^q (Eq. 6). Centrifugal term f_cent(r) = hbar^2/(2 mu r^2) [1 + alpha(r)] N(N+1) (Eq. 7), alpha(r) = sum_{i>0} alpha_i x^i / (1/2 + (1/2)(r/r_alpha)^m) (Eq. 11), x = (r - r_e)/r_e (Eq. 10). B_v = (hbar^2/2 mu) <psi_v|[1 + alpha(r)]/r^2|psi_v> (Eq. 14).",
"variable": "r: internuclear distance in A; V in cm^-1 from the minimum; N the rotational quantum number (Hund's case (b))",
"range": "0.80-6.80 A (the integration range; 'It is not expected, however, that the range of validity of the radial functions is that radially extensive'); reliable to v = 31 (90 % of D_e), perhaps v = 38",
"coefficients": {
"D_e": "42 055.6",
"p": "4.52",
"A": "16 591.891 596",
"q": "15.525 821",
"r_e": "1.207 526 689 4",
"beta_0": "6.407 585 460 1",
"beta_1": "1.703 616 263 0",
"beta_2": "9.664 250 463 3e-1",
"beta_3": "1.103 036 912 3",
"beta_4": "7.320 575 031 2e-2",
"beta_5": "6.276 327 564 7",
"beta_6": "-6.187 264 040 3",
"beta_7": "-5.094 846 393 9e+1",
"beta_8": "2.148 609 529 0e+2",
"beta_9": "-6.096 967 020 8e+1",
"beta_10": "-1.079 524 182 4e+3",
"beta_11": "2.031 515 960 7e+3",
"beta_12": "2.094 346 198 8e+2",
"beta_13": "-4.453 811 356 7e+3",
"beta_14": "5.004 031 447 4e+3",
"beta_15": "-1.812 194 155 8e+3",
"r_alpha": "2.10",
"m": "15",
"alpha_5": "1.501 199 512 8",
"alpha_6": "-7.308 811 142 2",
"alpha_7": "7.633 557 922 3",
"mu": "7.997 457 309 78",
"hbar2_over_2": "16.857 629 20"
},
"coefficient_units": {
"D_e": "cm^-1",
"A": "cm^-1 A^q",
"r_e": "A",
"r_alpha": "A",
"mu": "amu (p. 014309-4)",
"hbar2_over_2": "cm^-1 amu A^2 (p. 014309-4)"
},
"check": "Reproduces Table III/V: G_v to <= 0.0001 cm^-1 for v = 0-11, <= 0.0004 cm^-1 for v <= 26, <= 0.0065 cm^-1 to v = 38; B_0-B_3 (Eq. 14) to <= 2e-8 cm^-1 (three-point finite differences, steps 0.0005 and 0.00025 A, Richardson-extrapolated; checked 2026-09-26). D_v, H_v not recomputed."
}
Table II (p. 014309-4): ‘Fitted and constrained parameters ... Parameters in square brackets were constrained’ (D_e, p, A, q, r_alpha, m); the standard deviations of the others: r_e 3.43e-7 A, beta_0 ... beta_15 1.69e-5, 4.86e-4, 1.37e-3, 2.08e-2, 4.00e-2, 3.26e-1, 6.70e-1, 2.30, 5.68, 9.22, 2.16e1, 3.19e1, 3.09e1, 5.54e1, 5.08e1, 1.64e1; alpha_5-alpha_7 6.62e-2, 1.94e-1, 1.43e-1 (‘Many more significant figures have been provided ... than warranted by the standard deviations in order to eliminate round-off error’). alpha_1-alpha_4 are excluded (zero). The printed A and q do not join the outer branch continuously at 0.925 A (see the record’s notes), yet they are the ones that reproduce Table III. Term values are fitted after raising them by |Y00| = 0.332 cm^-1, so V(r_e) = 0.
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 review register for the checks and decisions.