zhao_2017#
S.-F. Zhao, J.-K. Li, G.-L. Wang, P.-C. Li and X.-X. Zhou, “Accurate Structure Parameters for Tunneling Ionization Rates of Gas-Phase Linear Molecules”, Commun. Theor. Phys. 67, 289 (2017)
Source record · Catalogue · Review register
DOI: 10.1088/0253-6102/67/3/289
Source kind: theory
Recorded technique: Single-active-electron model potentials for linear molecules from density functional theory in the modified Leeuwen-Baerends (LB-alpha) model of Schipper et al. (ref. 54): alpha times the LDA exchange potential, the LDA correlation potential (Perdew and Wang, ref. 56) and the LB gradient correction (Eq. (12)), with beta = 0.01 and one alpha per molecule, ‘optimized to obtain the accurate ionization potential by comparing with the corresponding experimental vertical ionization potential or the calculated results with GAUSSIAN code’ (p. 292; the alpha are in Table 1). The potential is built in a single-centre expansion (Eq. (4); ‘l_max = 40 [see Eq. (4)] for CO’, Fig. 1) from the density of the occupied orbitals ‘obtained from GAUSSIAN’ (p. 290; GAUSSIAN 03, ref. 47; method and basis set not stated); the one-electron Schrodinger equation is solved with B-splines (N_r = 80 radial and N_xi = 30 angular, Eqs. (18)-(23), p. 291); the radial functions of the single-centre expansion are fitted to Eq. (26) in the asymptotic region for the MO-ADK coefficients C_lm (‘We followed the fitting method proposed in Ref. [38]’, Kjeldsen and Madsen, J. Phys. B 37, 2033 (2004); p. 293).
Materials in structured entries: CO, CO2, H2, H2+, N2, N2O, O2
Dataset quantities: ionization_energy, tunnelling_structure_coefficient
Contents: 35 datasets; 0 formulas.
Curation notes#
Journal pages 289 to 300 = PDF pages 2 to 13 (PDF page 1 is the publisher’s cover sheet). Title, authors, journal, volume, year, pages and DOI checked on the first page of the article (PDF p. 2) on 2026-09-28. The library’s status carries warnings for an unmatched figure caption on PDF p. 5 only. A tabulation of the HOMO of 123 linear molecules by the group of zhao_2010, with a different potential (LB-alpha with correlation and a fitted alpha, beta = 0.01, in place of zhao_2010’s exchange-only LDA + LB with alpha = 1.0, beta = 0.05). The datasets are alternatives to the defaults: all are ‘check’ or ‘reproduced’.
EXTRACTED, for the curated species H2+, H2, N2, O2, CO, CO2 and N2O only. Table 1 (pp. 292-293, four page fragments): the LB-alpha ionisation energy and the ‘experimental vertical’ I_p (eV); R and alpha are kept in the printed cells and here: H2+ R 1.057 A, no alpha (printed ‘-’); H2 0.742, alpha 1.27; N2 1.098, 1.12; O2 1.208, 1.08; CO 1.128, 1.15; CO2 1.162, 1.05; N2O R_NN 1.128 and R_NO 1.184, 1.06. Table 2 (pp. 294-297, four page fragments): the C_lm of the HOMO, this work’s row and the rows the table reprints, one dataset per molecule and row. Every C_lm file carries Table 1’s R as the column internuclear_distance (N2O: internuclear_distance_nn and internuclear_distance_no), on the reprinted rows too, whose own papers may have used another R. Not curated species, not taken: the other 116 molecules, NO among them. Table 3 is quoted below.
CHECKED on 2026-09-28 against the table images, cell by cell: the rows of the seven molecules in Table 1 (images of tbl_3e9e6c6c77bf, tbl_8aa77f541a54, tbl_eeca443820ae) and every row written from Table 2 (images of tbl_5cfe79479e68, tbl_a21cdf242a86, tbl_3ac1855eb657, and the page image of p. 294); the CO and N2O blocks of Table 3 (images of tbl_ba018d0ab114, tbl_b6a5c0b28eba). The extracted cells agree with the images in every one of them. The rows of the other molecules were not checked.
STRUCTURAL FIXES of the script. (1) O2’s rows run over the page break between pp. 294 and 295: the first fragment ends with O2’s own row and its [30] row, the second begins with the rows [53] and [33] without a molecule, which are O2’s (F2 follows). (2) The cells of the second and third fragments of Table 2 merge the columns C_5m ... C_9m and the reference mark into one cell; the rows taken from them (O2’s two reprinted rows and CO’s three rows) print nothing beyond C_4m, and the script raises otherwise. (3) In the first and last fragments the reference mark is in the cells’ last column, under the header C_9m; on the page it stands to the right of C_9m. (4) Table 1’s N2O entry has two rows, ‘1.128 (R_{NN})’ and ‘1.184 (R_{NO})’. (5) The library’s cells print the minus sign as a hyphen; the values are written with ‘-’.
THE COEFFICIENTS (p. 291). Eq. (24): Psi_n^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), ‘where Y_lm(theta_e, chi_e) is the spherical harmonic functions’; Eq. (25): F_lm(r) = int Psi_n^(m)(r) Y*_lm sin theta_e d theta_e d chi_e; ‘the accurate structure parameters C_lm can be extracted by fitting these calculated radial functions to the following form in the asymptotic region’, Eq. (26): F_lm(r -> inf) -> C_lm r^((Z_c/kappa) - 1) e^(-kappa r). With Eq. (1) (p. 290): ‘Z_c ... the charge of the molecular residue (Z_c = 1 for neutral molecules)’, ‘kappa = sqrt(2 I_p), I_p is the ionization potential for the given valence orbital’. This is the form of tong_2002’s Eqs. (1)-(2) and of zhao_2010’s Eqs. (19)-(20). The rate is tong_2002’s, Eqs. (1)-(3) here, with Q(l, m’) = (-1)^((m’ + |m’|)/2) sqrt((2l + 1)(l + |m’|)!/(2(l - |m’|)!)). The m of an orbital is not stated in this paper; Table 2 labels each molecule with the symmetry of its HOMO (sigma, pi), and m = 0 for sigma and m = 1 for pi is taken from zhao_2010’s and Wang et al. 2014’s captions (the same group and method).
NOT STATED: the range of r of the fit (Fig. 2 shows the H2+ radial functions and the fitted form over r = 5 to 20 a.u.); which I_p enters kappa in the fit (the LB-alpha energy or the experimental one); any uncertainty of the C_lm; r_max and the order of the B-splines; the method and basis set of the GAUSSIAN orbitals; l_max for molecules other than CO; the source of the internuclear distances of Table 1 and of its experimental I_p (no reference is attached to any I_p but the calculated ones, marked a to d).
ACCURACY, as the paper states it. ‘In Table 1, one can see that our binding energies agree very well with the experimental ones in deed. We thus expect our structure parameters are more accurate than those obtained from other methods mentioned above’ (p. 294). ‘We should address that there are some discrepancies between the present fitted structure parameters and those tabulated in previous references. The discrepancies are originated from the different qualities of asymptotic tail of molecular wavefunctions’ (p. 294). The validation quoted is that of the earlier parameters: ‘By using our fitted structure parameters, the MO-ADK results are improved significantly by comparing with the TDSE and the TDDFT simulations for H2+, H2, N2, O2, F2, and C2H2 (see Fig. 2 in Ref. [35] and Fig. 4 in Ref. [40])’ (p. 297; ref. 35 is zhao_2010), and ‘We also expect that these structure parameters tabulated in this paper are checked when the more accurate simulations and the experimental data become available in the near future’ (p. 297). No rate, yield or alignment dependence is computed in this paper with the new coefficients.
AGAINST zhao_2010 (the same method with another potential). Table 2 reprints the rows of tong_2002 [30], Kjeldsen and Madsen 2005 [33], Abu-samha and Madsen 2010 [53], Zhao et al. 2009 [34] and Le et al. 2007 [60], but not zhao_2010’s Table III (ref. 35), and the text does not discuss the change from it. This work against zhao_2010: H2+ 4.52, 0.62, 0.03, unchanged; H2 1.72, 0.12 against 1.78, 0.11, 0.00; N2 3.39, 1.40, 0.05 against 2.68, 1.10, 0.06; O2 0.64, 0.04 against 0.52, 0.03; CO -2.78, 2.17, -1.22, 0.28, -0.09 against 2.32, 1.62, 0.82, 0.17, 0.05; CO2 1.74, 0.33, 0.03 against 1.97, 0.40, 0.04. The sum of C_l^2 changes by -7 % (H2), +60 % (N2), +52 % (O2), +61 % (CO) and -22 % (CO2) (computed 2026-09-28 from the printed values). Table 1 against zhao_2010’s Table II: R of H2+ 1.057 (1.058) and of CO2 1.162 (1.163); I_p of H2+ 30.00 (29.99), N2 15.60 (15.58), CO2 13.80 (13.78); H2, O2 and CO as there.
THE SIGNS FOR CO AND THE ORIENTATION. In this work’s CO row and in the reprinted row of Kjeldsen and Madsen [33] the even and odd partial waves have opposite signs; in tong_2002’s and zhao_2010’s CO rows all are positive. Wang et al., Mol. Phys. 112, 1102 (2014) (ref. 51; the same group, the same CO and N2O rows) say that the sign of tong_2002’s and zhao_2010’s coefficients ‘is opposite to that of the present C_lm and of C_lm from [Kjeldsen and Madsen] for even partial waves’ and that ‘the sign of the present structure parameters and of those from [Kjeldsen and Madsen] are correct for the CO molecule’ (their p. 1110). The orientation is fixed by Table 3 (the appendix, pp. 297-299), ‘The z coordinates (in Angstroms) of atoms for several selected gas-phase linear molecules at equilibrium distance calculated from the GAUSSIAN packages’, quoted here for the curated polar molecules (the numbers are printed in groups of three digits, ‘-0.644 571’): CO: C -0.644571, O 0.483429; N2O: N -1.199845, N -0.071645, O 1.112555. The oxygen atom is on +z in both. The differences give R = 1.128000 (CO), 1.128200 (N-N) and 1.184200 (N-O), Table 1’s R. The origin is not stated; the coordinates put the centre of nuclear charge at z = 0 (6 z_C + 8 z_O = 0.000 and 7 z_N + 7 z_N + 8 z_O = 0.000 to the printed digits; HF’s printed -0.825300 and 0.091700 exclude the centre of mass; computed 2026-09-28). With Y_l0 along the axis, sum_l C_l sqrt((2l + 1)/2) is -0.90 towards +z (O) and, with (-1)^l, -7.27 towards -z (C) for this work’s CO row, against 5.35 and 0.74 for zhao_2010’s: the two rows put the larger amplitude on opposite ends of the axis (computed 2026-09-28). zhao_2010 and tong_2002 do not state their orientation of CO.
THE IONISATION ENERGIES. Table 1’s caption: ‘ionization potentials calculated with the present LB-alpha model and experimental vertical ionization potentials ... The calculated ionization energies with GAUSSIAN code for several molecules are also given when the experimental results are unavailable’; none of the seven curated rows carries a mark of a calculated value. kind is recorded as the caption says, vertical. No reference is given for them; Wang et al. 2014 give Turner et al. (1970) for the same N2O and CO values. H2’s 15.47 eV and O2’s 12.03 eV are tong_2002’s Table I values, which tong_2002’s text says are about 0.88 and 0.33 eV below the vertical energies it used. H2+’s 30.00 eV at R = 1.057 A is the fixed-nuclei binding energy of its electron. The LB-alpha energies are the HOMO energies of the model, with alpha fitted to bring them to the experimental value: they are not independent predictions.
DOUBTS. (1) Table 2 prints identical rows for F2 and S2 (C_2m 1.26, C_4m 0.20; image of tbl_a21cdf242a86), which have different R, I_p and alpha in Table 1: probably a copying error in one of them (neither is a curated species). (2) The CO2 row reprinted as [34] (Zhao et al. 2009), 1.27, 0.188, 0.014, and the CS2 row reprinted as [53], 1.86, 0.79, 0.188, 0.03, are the only ones with three decimals; the originals are not in the library and the reprints were not checked against them. (3) The reprinted rows of tong_2002 agree with tong_2002’s Table I, except that H2’s l = 4 (0.00) is not reprinted and that CO has an l = 4 (0.00) which tong_2002 does not print (as in zhao_2010’s reprint). (4) The CO and N2O rows, and Table 1’s entries for them, are those of Wang et al. 2014, where they were first published; other rows common to the two papers differ (CS2 3.01, 1.25, 0.27, 0.05 here against 3.47, 1.63, 0.43, 0.09 there, with alpha 1.10 against 1.05 and LB-alpha 10.13 against 10.45 eV; MgO, KCl, HBr and BN also), without comment.
FIGURES (not digitised): Fig. 1 the partial-wave components of the potential of CO; Fig. 2 the radial functions of H2+ (l = 0, 2, 4) and the fitted form.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
zhao_2017/table1_lbalpha_H2+#
{
"id": "table1_lbalpha_H2+",
"material": "H2+",
"file": "H2+/zhao_2017__table1_lbalpha_H2+.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "1s sigma_g (printed 'sigma_g')"
}
}
Table 1, ‘LB-alpha/eV’: the binding energy of the 1s sigma_g electron in this paper’s calculation at the fixed R of the table; the alpha cell is printed ‘-’ (one electron). Kind not stated.
zhao_2017/table1_ip_H2+#
{
"id": "table1_ip_H2+",
"material": "H2+",
"file": "H2+/zhao_2017__table1_ip_H2+.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "1s sigma_g (printed 'sigma_g')"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. H2+’s value is the fixed-nuclei binding energy of its one electron at R = 1.057 A (zhao_2010 and tong_2002 print 29.99 eV at 1.058 A).
zhao_2017/table1_lbalpha_H2#
{
"id": "table1_lbalpha_H2",
"material": "H2",
"file": "H2/zhao_2017__table1_lbalpha_H2.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "HOMO sigma_g"
}
}
Table 1, ‘LB-alpha/eV’: the HOMO binding energy of this paper’s LB-alpha potential (beta = 0.01) at the fixed geometry of the table, with alpha fitted to bring it to the experimental I_p (p. 292): not an independent prediction. Kind not stated.
zhao_2017/table1_ip_H2#
{
"id": "table1_ip_H2",
"material": "H2",
"file": "H2/zhao_2017__table1_ip_H2.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "HOMO sigma_g"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. The value is tong_2002’s Table I value for the hydrogen row, which tong_2002’s text says is about 0.88 eV below the vertical energy it used.
zhao_2017/table1_lbalpha_N2#
{
"id": "table1_lbalpha_N2",
"material": "N2",
"file": "N2/zhao_2017__table1_lbalpha_N2.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "HOMO sigma_g"
}
}
Table 1, ‘LB-alpha/eV’: the HOMO binding energy of this paper’s LB-alpha potential (beta = 0.01) at the fixed geometry of the table, with alpha fitted to bring it to the experimental I_p (p. 292): not an independent prediction. Kind not stated.
zhao_2017/table1_ip_N2#
{
"id": "table1_ip_N2",
"material": "N2",
"file": "N2/zhao_2017__table1_ip_N2.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "HOMO sigma_g"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. zhao_2010 and tong_2002 print 15.58 eV.
zhao_2017/table1_lbalpha_O2#
{
"id": "table1_lbalpha_O2",
"material": "O2",
"file": "O2/zhao_2017__table1_lbalpha_O2.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "HOMO pi_g"
}
}
Table 1, ‘LB-alpha/eV’: the HOMO binding energy of this paper’s LB-alpha potential (beta = 0.01) at the fixed geometry of the table, with alpha fitted to bring it to the experimental I_p (p. 292): not an independent prediction. Kind not stated.
zhao_2017/table1_ip_O2#
{
"id": "table1_ip_O2",
"material": "O2",
"file": "O2/zhao_2017__table1_ip_O2.csv",
"table": "tbl_3e9e6c6c77bf",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "HOMO pi_g"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. The value is tong_2002’s Table I value, which tong_2002’s text says is about 0.33 eV below the vertical energy it used.
zhao_2017/table1_lbalpha_CO#
{
"id": "table1_lbalpha_CO",
"material": "CO",
"file": "CO/zhao_2017__table1_lbalpha_CO.csv",
"table": "tbl_8aa77f541a54",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "HOMO sigma"
}
}
Table 1, ‘LB-alpha/eV’: the HOMO binding energy of this paper’s LB-alpha potential (beta = 0.01) at the fixed geometry of the table, with alpha fitted to bring it to the experimental I_p (p. 292): not an independent prediction. Kind not stated.
zhao_2017/table1_ip_CO#
{
"id": "table1_ip_CO",
"material": "CO",
"file": "CO/zhao_2017__table1_ip_CO.csv",
"table": "tbl_8aa77f541a54",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "HOMO sigma"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. The value of zhao_2010’s Table II and tong_2002’s Table I; Wang et al. 2014 (the same group) give Turner et al. (1970) for it.
zhao_2017/table1_lbalpha_CO2#
{
"id": "table1_lbalpha_CO2",
"material": "CO2",
"file": "CO2/zhao_2017__table1_lbalpha_CO2.csv",
"table": "tbl_eeca443820ae",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "HOMO pi_g"
}
}
Table 1, ‘LB-alpha/eV’: the HOMO binding energy of this paper’s LB-alpha potential (beta = 0.01) at the fixed geometry of the table, with alpha fitted to bring it to the experimental I_p (p. 292): not an independent prediction. Kind not stated.
zhao_2017/table1_ip_CO2#
{
"id": "table1_ip_CO2",
"material": "CO2",
"file": "CO2/zhao_2017__table1_ip_CO2.csv",
"table": "tbl_eeca443820ae",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "HOMO pi_g"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. zhao_2010 prints 13.78 eV (Table II) and 13.8 eV (Table IV, Turner et al.).
zhao_2017/table1_lbalpha_N2O#
{
"id": "table1_lbalpha_N2O",
"material": "N2O",
"file": "N2O/zhao_2017__table1_lbalpha_N2O.csv",
"table": "tbl_eeca443820ae",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "not stated",
"orbital": "HOMO pi"
}
}
Table 1, ‘LB-alpha/eV’: the HOMO binding energy of this paper’s LB-alpha potential (beta = 0.01) at the fixed geometry of the table, with alpha fitted to bring it to the experimental I_p (p. 292): not an independent prediction. Kind not stated.
zhao_2017/table1_ip_N2O#
{
"id": "table1_ip_N2O",
"material": "N2O",
"file": "N2O/zhao_2017__table1_ip_N2O.csv",
"table": "tbl_eeca443820ae",
"quantity": "ionization_energy",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "vertical",
"orbital": "HOMO pi"
}
}
Table 1, ‘I_p/eV’, ‘experimental vertical ionization potentials’ (caption); no reference is given. Kind as the caption says. Wang et al. 2014 (the same group) give Turner et al. (1970) for it.
zhao_2017/table2_H2+#
{
"id": "table2_H2+",
"material": "H2+",
"file": "H2+/zhao_2017__table2_H2+.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "1s sigma_g (printed 'sigma_g')",
"m": 0,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘H_2^+ (sigma_g)’, this work: the ground state at R = 1.057 A; C_00, C_20, C_40 = 4.52, 0.62, 0.03 are also given in the text with Fig. 2 (p. 294). The same values as zhao_2010’s Table III. The internuclear_distance column is Table 1’s R.
zhao_2017/table2_tong_H2+#
{
"id": "table2_tong_H2+",
"material": "H2+",
"file": "H2+/zhao_2017__table2_tong_H2+.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"tong_2002"
],
"parameters": {
"model": "moadk",
"orbital": "1s sigma_g (printed 'sigma_g')",
"m": 0,
"normalisation": "Tong et al. 2002 Eqs. (1)-(2): Psi^m = sum_l C_l F_l(r) Y_lm, F_l(r -> inf) ~ r^(Z_c/kappa - 1) exp(-kappa r), kappa = sqrt(2 I_p), atomic units"
}
}
Table 2, the row of H2+ marked [30]: Tong et al. 2002, Table I; the printed values agree with tong_2002. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_H2#
{
"id": "table2_H2",
"material": "H2",
"file": "H2/zhao_2017__table2_H2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘H_2 (sigma_g)’, this work: C_0m and C_2m (no l = 4 is printed). zhao_2010 (exchange-only LDA + LB): 1.78, 0.11, 0.00. The internuclear_distance column is Table 1’s R.
zhao_2017/table2_tong_H2#
{
"id": "table2_tong_H2",
"material": "H2",
"file": "H2/zhao_2017__table2_tong_H2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"tong_2002"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "Tong et al. 2002 Eqs. (1)-(2): Psi^m = sum_l C_l F_l(r) Y_lm, F_l(r -> inf) ~ r^(Z_c/kappa - 1) exp(-kappa r), kappa = sqrt(2 I_p), atomic units"
}
}
Table 2, the row of H2 marked [30]: Tong et al. 2002, Table I, whose row is printed ‘D_2’; C_0 and C_2 agree with tong_2002, whose C_4 = 0.00 is not reprinted. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_kjeldsen_H2#
{
"id": "table2_kjeldsen_H2",
"material": "H2",
"file": "H2/zhao_2017__table2_kjeldsen_H2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"T. K. Kjeldsen and L. B. Madsen, Phys. Rev. A 71, 023411 (2005)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of H2 marked [33]: Kjeldsen and Madsen 2005, from Hartree-Fock orbitals of ‘the program X2DHF’ (p. 294). internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_N2#
{
"id": "table2_N2",
"material": "N2",
"file": "N2/zhao_2017__table2_N2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘N_2 (sigma_g)’, this work. zhao_2010 (exchange-only LDA + LB): 2.68, 1.10, 0.06; the sum of C_l^2 is 60 % larger here. The paper does not comment on the change. The internuclear_distance column is Table 1’s R.
zhao_2017/table2_tong_N2#
{
"id": "table2_tong_N2",
"material": "N2",
"file": "N2/zhao_2017__table2_tong_N2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"tong_2002"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "Tong et al. 2002 Eqs. (1)-(2): Psi^m = sum_l C_l F_l(r) Y_lm, F_l(r -> inf) ~ r^(Z_c/kappa - 1) exp(-kappa r), kappa = sqrt(2 I_p), atomic units"
}
}
Table 2, the row of N2 marked [30]: Tong et al. 2002, Table I; the printed values agree with tong_2002. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_abusamha_N2#
{
"id": "table2_abusamha_N2",
"material": "N2",
"file": "N2/zhao_2017__table2_abusamha_N2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"M. Abu-samha and L. B. Madsen, Phys. Rev. A 81, 033416 (2010)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of N2 marked [53]: Abu-samha and Madsen 2010, among the references in which ‘molecular wavefunctions are obtained by solving the time-independent Schrodinger equation (TIDSE) where the electron correlation has been neglected in the molecular potentials’ (p. 294). internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_kjeldsen_N2#
{
"id": "table2_kjeldsen_N2",
"material": "N2",
"file": "N2/zhao_2017__table2_kjeldsen_N2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"T. K. Kjeldsen and L. B. Madsen, Phys. Rev. A 71, 023411 (2005)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma_g",
"m": 0,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of N2 marked [33]: Kjeldsen and Madsen 2005, from Hartree-Fock orbitals of ‘the program X2DHF’ (p. 294). internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_O2#
{
"id": "table2_O2",
"material": "O2",
"file": "O2/zhao_2017__table2_O2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘O_2 (pi_g)’, this work: C_2m and C_4m. zhao_2010 (exchange-only LDA + LB): 0.52, 0.03; the sum of C_l^2 is 52 % larger here. The paper does not comment on the change. The internuclear_distance column is Table 1’s R.
zhao_2017/table2_tong_O2#
{
"id": "table2_tong_O2",
"material": "O2",
"file": "O2/zhao_2017__table2_tong_O2.csv",
"table": "tbl_5cfe79479e68",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"tong_2002"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "Tong et al. 2002 Eqs. (1)-(2): Psi^m = sum_l C_l F_l(r) Y_lm, F_l(r -> inf) ~ r^(Z_c/kappa - 1) exp(-kappa r), kappa = sqrt(2 I_p), atomic units"
}
}
Table 2, the row of O2 marked [30]: Tong et al. 2002, Table I; the printed values agree with tong_2002. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_abusamha_O2#
{
"id": "table2_abusamha_O2",
"material": "O2",
"file": "O2/zhao_2017__table2_abusamha_O2.csv",
"table": "tbl_a21cdf242a86",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"M. Abu-samha and L. B. Madsen, Phys. Rev. A 81, 033416 (2010)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of O2 marked [53]: Abu-samha and Madsen 2010 (see table2_abusamha_N2). The row is at the top of the table’s second page (p. 295), without a molecule: O2’s rows run over the page break. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_kjeldsen_O2#
{
"id": "table2_kjeldsen_O2",
"material": "O2",
"file": "O2/zhao_2017__table2_kjeldsen_O2.csv",
"table": "tbl_a21cdf242a86",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"T. K. Kjeldsen and L. B. Madsen, Phys. Rev. A 71, 023411 (2005)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of O2 marked [33]: Kjeldsen and Madsen 2005, from Hartree-Fock orbitals of ‘the program X2DHF’ (p. 294). The row is at the top of the table’s second page (p. 295), without a molecule: O2’s rows run over the page break. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_CO#
{
"id": "table2_CO",
"material": "CO",
"file": "CO/zhao_2017__table2_CO.csv",
"table": "tbl_a21cdf242a86",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma",
"m": 0,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘CO (sigma)’, this work: C_0m ... C_4m, the even and odd partial waves of opposite sign, with the oxygen atom on +z (Table 3: C -0.644571 A, O 0.483429 A). zhao_2010: 2.32, 1.62, 0.82, 0.17, 0.05, all positive (see the record’s notes on the signs). The row was first published in Wang et al. 2014’s Table 2. The internuclear_distance column is Table 1’s R.
zhao_2017/table2_tong_CO#
{
"id": "table2_tong_CO",
"material": "CO",
"file": "CO/zhao_2017__table2_tong_CO.csv",
"table": "tbl_a21cdf242a86",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"tong_2002"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma",
"m": 0,
"normalisation": "Tong et al. 2002 Eqs. (1)-(2): Psi^m = sum_l C_l F_l(r) Y_lm, F_l(r -> inf) ~ r^(Z_c/kappa - 1) exp(-kappa r), kappa = sqrt(2 I_p), atomic units"
}
}
Table 2, the row of CO marked [30]: Tong et al. 2002, Table I; C_0 ... C_3 agree with tong_2002, which prints no l = 4: the reprint adds C_4m = 0.00 (as zhao_2010’s reprint does). internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_kjeldsen_CO#
{
"id": "table2_kjeldsen_CO",
"material": "CO",
"file": "CO/zhao_2017__table2_kjeldsen_CO.csv",
"table": "tbl_a21cdf242a86",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"T. K. Kjeldsen and L. B. Madsen, Phys. Rev. A 71, 023411 (2005)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO sigma",
"m": 0,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of CO marked [33]: Kjeldsen and Madsen 2005, from Hartree-Fock orbitals of ‘the program X2DHF’ (p. 294): the even and odd partial waves of opposite sign, as in this work’s row. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_CO2#
{
"id": "table2_CO2",
"material": "CO2",
"file": "CO2/zhao_2017__table2_CO2.csv",
"table": "tbl_3ac1855eb657",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘CO_2 (pi_g)’, this work: C_2m, C_4m, C_6m. zhao_2010 (exchange-only LDA + LB): 1.97, 0.40, 0.04. The internuclear_distance column is Table 1’s R.
zhao_2017/table2_abusamha_CO2#
{
"id": "table2_abusamha_CO2",
"material": "CO2",
"file": "CO2/zhao_2017__table2_abusamha_CO2.csv",
"table": "tbl_3ac1855eb657",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"M. Abu-samha and L. B. Madsen, Phys. Rev. A 81, 033416 (2010)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of CO2 marked [53]: Abu-samha and Madsen 2010: ‘Madsen et al. determined structure parameters of the highest occupied molecular orbital (HOMO) for CO2 with wave functions calculated by propagating the TDSE in imaginary time’ (p. 290). internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_zhao2009_CO2#
{
"id": "table2_zhao2009_CO2",
"material": "CO2",
"file": "CO2/zhao_2017__table2_zhao2009_CO2.csv",
"table": "tbl_3ac1855eb657",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"S.-F. Zhao, C. Jin, A.-T. Le, T. F. Jiang and C. D. Lin, Phys. Rev. A 80, 051402(R) (2009)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of CO2 marked [34]: Zhao et al. 2009 (the group’s first determination for CO2). Printed with three decimals, 1.27, 0.188, 0.014; zhao_2010’s Table III has 1.97, 0.40, 0.04 for CO2 and is not reprinted. Not checked against the original. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_le_CO2#
{
"id": "table2_le_CO2",
"material": "CO2",
"file": "CO2/zhao_2017__table2_le_CO2.csv",
"table": "tbl_3ac1855eb657",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance": "angstrom"
},
"use": "reproduced",
"reproduced_from": [
"A.-T. Le, X. M. Tong and C. D. Lin, J. Mod. Opt. 54, 967 (2007)"
],
"parameters": {
"model": "moadk",
"orbital": "HOMO pi_g",
"m": 1,
"normalisation": "as reprinted in Zhao et al. 2017's Table 2 beside coefficients defined by its Eqs. (24)-(26), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r); the original paper's definition and phase convention were not checked (the paper is not in the library)"
}
}
Table 2, the row of CO2 marked [60]: Le, Tong and Lin 2007, from GAUSSIAN orbitals, whose ‘asymptotic behavior ... is not correct’ (p. 294); the row zhao_2010 reprints as its [17]. internuclear_distance: this paper’s Table 1 R, not the original’s.
zhao_2017/table2_N2O#
{
"id": "table2_N2O",
"material": "N2O",
"file": "N2O/zhao_2017__table2_N2O.csv",
"table": "tbl_3ac1855eb657",
"quantity": "tunnelling_structure_coefficient",
"abscissa": "label",
"columns": {
"internuclear_distance_nn": "angstrom",
"internuclear_distance_no": "angstrom"
},
"use": "check",
"parameters": {
"model": "moadk",
"orbital": "HOMO pi",
"m": 1,
"normalisation": "Zhao et al. 2017 Eqs. (24)-(26), p. 291 (the form of Tong et al. 2002 and of zhao_2010): Psi^(m)(r) = sum_l F_lm(r) Y_lm(theta_e, chi_e), F_lm(r -> inf) -> C_lm r^(Z_c/kappa - 1) exp(-kappa r), Z_c the charge of the molecular residue, kappa = sqrt(2 I_p), atomic units; one-centre expansion, the molecule along z (the atoms of a polar molecule at Table 3's z), m along the molecular axis"
}
}
Table 2, ‘N_2O (pi)’, this work: C_1m ... C_6m, with the oxygen atom on +z (Table 3: N -1.199845, N -0.071645, O 1.112555 A). The row was first published in Wang et al. 2014’s Tables 2 and 4 (2 pi, the HOMO). The two internuclear distances of Table 1 are the columns internuclear_distance_nn and internuclear_distance_no.
Formulas#
No formulas are recorded.
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.