sundholm_1994#
D. Sundholm, A. Rizzo and P. Jørgensen, “Multiconfiguration self-consistent field quadratic response calculations of the two-photon transition probability rate constants for argon”, J. Chem. Phys. 101, 4931-4935 (1994)
Source record · Catalogue · Review register
DOI: 10.1063/1.467415
Source kind: theory
Recorded technique: MCSCF quadratic response (the implementation of Hettema et al., J. Chem. Phys. 97, 1174 (1992), hettema_1992): the two-photon transition moment S_ab of the ground state to the excited state as the residue of the quadratic response function, from two linear response vectors and the eigenvector of the linear response eigenvalue equation (Eqs. 12-15), in the length form, at the SCF (HF) level and with a complete active space 2s1p//2s2p1d(8) (3s, 3p, 4s, 4p, 3d active; 1s, 2s, 2p inactive). Basis (p. 4931): the Cartesian components of an uncontracted <17s13p6d3f1g> Gaussian set (137 functions), Rahman, Rizzo and Yeager’s with one f (0.80) and one g (0.1) added; HF energy -526.810 238 au (7.275 mH above the HF limit), CAS -526.986 730 au (7.653 mH above the CAS limit). Non-relativistic, LS coupling. Two pairs of frequencies: omega_1 = omega_2 = omega(1De)/2 with the method’s own excitation energy, and omega_1 = 0.3855 au (the experiment of McCann et al.), omega_2 = omega(1De) - omega_1.
Materials in structured entries: Ar
Dataset quantities: oscillator_strength
Contents: 2 datasets; 0 formulas.
Curation notes#
Journal pp. 4931-4935 = PDF pages 2-6 (PDF page 1 is the AIP download cover); journal page = PDF page + 4929. Identity checked on the first page (PDF p. 2: ‘J. Chem. Phys. 101 (6), 15 September 1994’, ‘0021-9606/94/101(6)/4931/5’, Dage Sundholm (Helsinki), Antonio Rizzo (Pisa), Poul Jørgensen (Aarhus); received 24 February 1994, accepted 12 May 1994) on 2026-10-08.
THE QUANTITIES (p. 4932). ‘K_{i-f} = a0^4 t0 (2 pi)^3 alpha^2 omega_1 omega_2 delta^TP’ (Eq. 2; a0 = 0.529 177 26e-8 cm, t0 = 2.418 884 4e-17 s, alpha = 7.297 353 0e-3), the rate constant of W_if = K I_1 I_2 with I the photon fluxes (Eq. 1), a delta function line shape; delta^TP = F delta_F + G delta_G + H delta_H with delta_F = sum S_aa S_bb* / 30, delta_G = sum S_ab S_ab* / 30, delta_H = sum S_ab S_ba* / 30 (Eqs. 7-10), ‘For linearly polarized laser beams with parallel polarization planes we have F = G = H = 2’ (after Monson and McClain 1970; the general case Moccia and Rizzo 1985, moccia_1985). The paper treats one final level, 3p5 4p 1De, and prints its tensor element (Table III, ‘Total’, a.u.: HF 9.499 and 20.467, CAS 10.424 and 27.367 at the two pairs of frequencies) and its K (Table IV). THE CONVENTION, established by reproduction (2026-10-08): every K of Table IV is Eq. (2) with delta^TP = (4/15) S^2 and S the Table III total (0.370, 1.19, 0.426, 1.91 x 10^-51 cm4 s reproduced as 0.3701, 1.1873, 0.4261, 1.9107; worst -0.22 %, within the three printed figures), omega_2 from Table I’s energy of the method (HF 108 760, CAS 106 347 cm-1). So S is the one non-zero element S_xy = S_yx of one real component (xy) of the 1De level, and delta^TP is the F = G = H = 2 rotational average of that one component, (4/15) S_xy^2. Summed over the level’s five components the average is (4/3) S_xy^2 (henriksson_2005’s convention, TC-888; it is also the square of the zz element of the M = 0 component, Moccia and Rizzo’s |M|^2, moccia_1984). The tables therefore hold neither the level’s delta^TP nor any quantity of the vocabulary: the strengths and the K are not taken (below).
TAKEN (text, p. 4933): the oscillator strength of the 1Se -> 1Po transition, ‘The HF value ... is 0.3154 a.u. In the CAS calculation it became 0.3358 a.u.’ (one inline dataset per method, use = check). hattig_1998b’s Table II reproduces both as ‘SCF’ and ‘CAS’ (Ref. 12 there), not taken there (TC-887): the values agree as printed.
CHECKED (2026-10-08, by direct computation from the printed numbers). Table III: each column’s four contributions sum to its total (all four exactly). Table I against hattig_1998b’s Table I (SCF and MCSCF columns, footnote b, Ref. 12): 98 396, 95 301 (1Po), 108 760, 106 347 (1De), 107 083 (1Pe), 111 063, 108 361 cm-1 (1Se) are 12.1995, 11.8158, 13.4845, 13.1853, 13.2766, 13.7701, 13.4351 eV, which hattig_1998b prints as 12.200, 11.816, 13.485, 13.185, 13.277, 13.770, 13.435. The frequencies: 0.247 77 and 0.242 28 au are half of 108 760 and 106 347 cm-1; Ref. 17’s 0.242 05 is half of moccia_1984’s experimental 0.4841 au. hattig_1998b’s argon 1De SCF and MCSCF S^0f (90.23, 418.90, 108.66, 748.95) are the squares of Table III’s totals (90.231, 418.898, 108.660, 748.953) and its K (0.370, 1.19 (printed ‘1.19 x 20^-51’), 0.426, 1.91 x 10^-51) are Table IV’s. The CAS oscillator strength is 2 omega mu^2 with Table I’s 1Po energy and Table II’s mu (0.3358); the HF one is not (2 x 0.448 32 x 0.5948^2 = 0.3172 against 0.3154). The text’s ratios: CAS/HF 1.097 (‘10%’), A[2]B 1.453 (‘45%’), 27.367/10.424 = 2.63 (‘2.6 times’), its square 6.9 (‘a factor of about 7’), 27.367/20.467 = 1.34 (‘33%’), 2.05/1.91 = 1.07 (‘7%’); the closure estimate with the quadrupole moment 1.3220 au and omega_average 0.502 au, Q [1/(omega_av - omega_1) + 1/(omega_av - omega_2)], 10.40 at omega_1 = omega_2 (‘10.4’) and 14.72 at omega_1 = 0.3855 (printed ‘14.8’, 0.5 % higher; ‘25% smaller than the HF’ moment: 14.8/20.467 is 28 % smaller). The experimental sum 1.4 + 0.53 + 0.12 = 2.05 x 10^-51 cm4 s.
FINDING: THE PREVIOUS CALCULATIONS IN TABLE IV ARE QUOTED TEN TIMES TOO SMALL. ‘HF (omega_1 = 0.242 05) 0.216 x 10^-51, Ref. 17’ (Moccia and Rizzo, Mol. Phys. 51, 1095 (1984), moccia_1984): that paper’s Table 4 prints 2.16(-51) cm4 s (EMM, length form, 1De), which its Eq. (6) reproduces from its amplitude 12.149 (2.167e-51). ‘VM 0.288 x 10^-51, Ref. 29’ (Lebedev and Yakovleva, ‘omega_1 = omega_2 = 0.242 is assumed’): the amplitude 14.0 that moccia_1984’s Table 2 reproduces from Lebedev and Yakovleva gives 2.88e-51 by the same Eq. (6) at 0.242 au. So the text’s ‘The present HF value ... is in good agreement with previously calculated rate constants’ rests on two values a factor of 10 low. In a common convention the comparison is HF 5 x 0.370 = 1.85e-51 (the level, at 0.247 77 au) against moccia_1984’s 2.16e-51 (RPA with experimental energies, 0.242 05 au).
FINDING: THE COMPARISON WITH EXPERIMENT. The computed K are one component’s (above), the experimental 2.05e-51 cm4 s the sum of the rates to the three J = 2 levels of 3p5 4p with 1De character (Payne et al. and McCann et al., Refs. 2, 5, 28). The level’s rate constant is 5 K: HF 1.85 and 5.94, CAS 2.13 and 9.55 x 10^-51 cm4 s (omega_1 = omega_2, and 0.3855 au), so the ‘excellent agreement’ of CAS at 0.3855 au (1.91 against 2.05, 7 %) becomes a factor 4.7 when both refer to the level. The same holds for hattig_1998b’s CCSD 1.83e-51 (TC-887).
FINDING: THE EXPERIMENTAL VALUE IS McCANN ET AL.’S T2 TIMES t0, WITH ONE TERM LOW. The text calls the three experimental values ‘transition probability rates [see Eq. (4)]’, Eq. (4) being T2 = K/t0, the laser-independent ‘two-photon constant’ in cm4 of McCann et al. (Ref. 5, mccann_1988), with W = T2 G I1 I2 and G = 1/Delta, Delta = (Delta1^2 + Delta2^2)^(1/2) the lasers’ combined FWHM in rad/s (Eqs. 3, 5, 6): a G that is not the peak of a normalised line shape (mccann_1988’s notes). mccann_1988’s Table II (image checked, 2026-10-09) prints T2 = 6.2 x 10^-35 (4p[5/2]2, 105 611 cm-1), 2.2 x 10^-35 (4p[3/2]2, 106 235) and 5.1 x 10^-36 cm4 (4p’[3/2]2, 107 287), the levels of the text’s ‘1.4, 0.53, 0.12 x 10^-51 cm4 s, respectively’. T2 t0 (t0 = 2.418 884 4e-17 s, Eq. 2) is 1.500, 0.532 and 0.123 x 10^-51: the second and third are the printed values, the first is not (1.4 printed, 7 % lower), and their sum is 2.16 x 10^-51 (2.15 with the printed 0.53 and 0.12), not 2.05. hattig_1998b quotes the 2.05 (TC-887). The measurement is non-degenerate (a 118.2 nm VUV photon, 0.3855 au, with a visible one), which is why the paper’s second pair of frequencies is omega_1 = 0.3855 au. Payne et al.’s Refs. 2 and 28 (Payne_1982_ApplicationsResonanceIonization, Payne_1981_ResonanceIonizationSpectroscopy) print no argon rate constants (searched, 2026-10-09).
DOUBTS. Table II’s S_1Po, the 1Po term of Eq. (11), are not mu(1S-1P) mu(1P-1D) [1/(omega_P - omega_1) + 1/(omega_P - omega_2)] with Table I’s energies (13.36, 25.29, 14.43, 32.55 against the printed 11.222, 24.422, 11.945, 31.592; implied factors 0.840, 0.966, 0.828, 0.971): the angular factor and the energies used are not stated, not resolved. The text says the 1Po contribution is ‘13%-16% larger than the final two-photon transition moment’: Tables II and III give 18.1 % and 19.3 % (HF), 14.6 % and 15.4 % (CAS). Uncertainty: ‘the uncertainty in the calculated transition probability rate constant is less than 5%’ (p. 4934), a judgement, for the CAS value.
NOT TAKEN. Table III (S_xy, one tensor element of one component: no quantity holds it; the level’s delta^TP would be (4/3) S_xy^2, a derived value) and Table IV (rate constants in cm4 s, one component’s; no quantity). Table I: computed excitation energies (non-relativistic model values) and Moore’s experimental energies (C. E. Moore, NBS Circ. 467, not in the library; ‘the J-component with the largest J’). Table II’s dipole transition moments mu (no quantity). The experimental rate constants (2.05e-51 and its parts 1.4, 0.53, 0.12 x 10^-51 cm4 s; secondary, from McCann et al.’s T2, mccann_1988, above). The experimental oscillator strength 0.3312 (Chan et al. 1992, chan_1992).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
sundholm_1994/f_Ar_HF#
{
"id": "f_Ar_HF",
"material": "Ar",
"table": "text",
"quantity": "oscillator_strength",
"abscissa": "label",
"use": "check",
"parameters": {
"method": "SCF (Hartree-Fock) linear response, uncontracted <17s13p6d3f1g> Cartesian Gaussian basis (137 functions)"
}
}
1 inline points in the source record.
p. 4933: ‘The HF value for the oscillator strength of the 1Se to 1Po transition is 0.3154 a.u.’ (the lowest 1Po level, 3p5 4s; LS coupling, no spin-orbit), computed. Reproduced in hattig_1998b’s Table II as ‘SCF’.
sundholm_1994/f_Ar_CAS#
{
"id": "f_Ar_CAS",
"material": "Ar",
"table": "text",
"quantity": "oscillator_strength",
"abscissa": "label",
"use": "check",
"parameters": {
"method": "MCSCF linear response, complete active space 2s1p//2s2p1d(8) (3s, 3p, 4s, 4p, 3d active), uncontracted <17s13p6d3f1g> Cartesian Gaussian basis (137 functions)"
}
}
1 inline points in the source record.
pp. 4933-4934: ‘In the CAS calculation it became 0.3358 a.u.’ (the 1Se to 1Po transition, the lowest 1Po level, 3p5 4s; LS coupling, no spin-orbit), computed; 2 omega mu^2 with Table I’s CAS 1Po energy (95 301 cm-1) and Table II’s mu = 0.6218 gives 0.3358. Reproduced in hattig_1998b’s Table II as ‘CAS’.
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.