mizrahi_1985_h2d2#
V. Mizrahi and D. P. Shelton, Phys. Rev. A 32, 3454 (1985)
Source record · Catalogue · Review register
Source kind: measurement
Recorded technique: Electric-field-induced second-harmonic generation (ESHG) with periodic phase matching (150 electrode pairs, 2.69 mm spacing), cw argon-ion and dye lasers at 457.9-700 nm, the gas density scanned through the phase-match peak; H2 measured against He and against D2 in triplets; pressures by capacitance manometer (0.15 %) and Bourdon gauge (He, 0.1 % of 160 bar), densities with virial corrections, T = 22 C (pp. 3455-3456).
Materials in structured entries: D2, H2, He
Dataset quantities: hyperpolarizability, hyperpolarizability_ratio
Contents: 9 datasets; 6 formulas.
Curation notes#
No linear-dispersion data: nothing is extracted. The paper measures ratios of the third-order susceptibility chi(3)(-2w; w, w, 0), H2/He at 19 and H2/D2 at 9 wavelengths (Table I, raw and corrected for the vibrational resonance), extrapolates them to w = 0 with A(1 + B w^2) (B = 9.01(22), 8.31(21), 0.47(13), 0.24(13) x 10^-10 cm^2 for the four columns, Table I caption) and compares chi(3)el(0) of H2 (647 +- 4 a.u., with chi_He(3)(0) = 42.8 a.u.) and the H2/D2 ratio (1.025 +- 0.004) with theory (Table III). The phase-match densities, which would measure n(2w) - n(w), are not tabulated (they enter Eq. (4) only), and the phase-match pressure is given only as ‘of order 1 atm (T = 22 C), except for helium ... of order 100 atm’ (p. 3455). The linear polarizability appears only through alpha(R) plotted in Fig. 4 (Kolos & Wolniewicz, Ref. 19) and the statement <R>_H2 = 1.01 <R>_D2 (p. 3459). The D2/H2 phase-matching density ratios of this group are in shelton_1986 (Table III) and shelton_1990 (Table I).
For the later chi(3) work: Table I (tbl_d9c17b1ffdb1, JSON) gives lambda in air (nm) and the vacuum wavenumber (cm^-1) of each line; Table II (tbl_79b8cb3720a5) the ab initio transition polarizabilities (Hunt et al.) used for the vibrational correction (H2 alpha_01,11 = 0.74027, gamma_01,11 = 0.61175, omega = 0.01894 a.u.; D2 alpha_02,12 = 0.61765, gamma_02,12 = 0.50224, omega = 0.01361); Eq. (8) is Sitz & Yaris’s chi_He(3)(w) as 42.6 {1 + [w/(58300 cm^-1)]^2} a.u. Accuracy of the ratios 0.3-1.2 % (p. 3457).
Duplicate: the library’s second copy Mizrahi_1985_NonlinearSusceptibilityH2_2b7b33 (another PDF of the same article, 7 pages) has cell-for-cell identical Tables I-III (checked 2026-09-25); not used.
THIRD-ORDER DATA (added 2026-09-26; supersedes “nothing is extracted” above, which concerned linear dispersion). DEFINITIONS (Sec. II, p. 3454): ‘ESHG in gases may, for our experimental geometry (optical-field polarization parallel to the static field), be described by P_z(2 omega) = (3/2) chi_zzzz(3)(-2 omega; omega, omega, 0) E_z(omega) E_z(omega) E_z(0)’ (Eq. 1), in the notation of Bogaard & Orr, which ‘has the desirable property that chi(3)(0; 0, 0, 0) = lim chi(3)(-2 omega; omega, omega, 0) without the need for additional numerical factors’; the macroscopic chi_zzzz(3) = (1/6) rho L <Phi_za Phi_zb Phi_zc Phi_zd> gamma_abcd(-2 omega; omega, omega, 0) (Eq. 2), L the local-field factor (< 1 % from unity here); the ‘microscopic nonlinear susceptibility’ chi(3)(omega) is the isotropic average <Phi Phi Phi Phi> gamma, and chi(3)(0) = (1/5) gamma_xixietaeta (Eq. 3): Shelton & Rice’s gamma_par (Taylor convention), called chi(3) here. Ratios: chi_A/chi_B = (S_A/S_B)^(1/2) (n_B rho_B)/(n_A rho_A) (Eq. 4). Expansion chi(3)(omega) = chi(3)(0)(1 + a omega^2 + b omega^4 + ...) (Eq. 5), fitted as A(1 + B omega^2) with omega the vacuum wavenumber (cm^-1; not nu_L^2). PARTS: chi(3) = chi(3)el + chi(3)vib (Eq. 6); the vibrational resonance contribution is calculated (Eq. 7) from the v = 1, J = 1 (H2) or J = 2 (D2) Q(J) transition only (Delta J = 0; rotational Raman ignored), with Hunt, Poll & Wolniewicz’s matrix elements (Table II: H2 alpha_01,11 = 0.74027, gamma_01,11 = 0.61175, omega = 0.01894 au = 4156 cm^-1; D2 alpha_02,12 = 0.61765, gamma_02,12 = 0.50224, omega = 0.01361 au = 2988 cm^-1; here gamma is the transition anisotropy alpha_par - alpha_perp, not a hyperpolarizability), and subtracted: ‘The corrections for H2 vary between 1.7% and 0.5%, while the corrections for D2 are about half as large’ (p. 3458). CALIBRATION: Sitz & Yaris’s chi_He(3)(omega) = 42.6 {1 + [omega/(58300 cm^-1)]^2} au (Eq. 8) for the dispersion, and chi_He(3)(0) = 42.8 au (‘about midrange’ of four calculations: 43.10, 42.81, 42.8, 42.6 au) for the static result 647 +- 4 au (conversion 1 au = 6.2353e-65 C^4 m^4 J^-3 = 5.0366e-40 esu, Table III note a). ACCURACY: ‘estimated accuracies ranging between 0.3% and 1.2%’ (p. 3457); Ward & Miller’s 694.3 nm ratio agrees with the smoothed raw value; Shelton & Buckingham’s 514.5 nm value (shelton_1982) is 1.9 % lower, attributed to beam steering (p. 3457). Other statements: footnote 18 (p. 3460) quotes Buckingham & Dunmur’s dc Kerr gamma_He = 53.6 au +- 7 % and gamma_Ar / gamma_He = 21.8 +- 2.2 (632.8 nm), judged to carry a systematic error of 20 % or more (both reproduced in buckingham_1968’s record, not repeated here); footnote 22: Jaszunski & Roos’s Table 7 value ‘668.8 ... should be 688.8 as in their Table 6’ (a misprint in another paper). The J = 0 averages 691.2 (JR), 689.5 (JR recomputed) and 667.0 au (BW) precede the thermal averages of Table III (p. 3459).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
mizrahi_1985_h2d2/H2_over_He_raw#
{
"id": "H2_over_He_raw",
"material": "H2",
"file": "H2/mizrahi_1985_h2d2__H2_over_He_raw.csv",
"table": "tbl_d9c17b1ffdb1",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"abscissa_unit": "cm-1",
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"reference_material": "He",
"process": "eshg",
"convention": "T",
"part": "total"
}
}
Table I (p. 3456): chi_H2(3)/chi_He(3) (per molecule, ESHG, zzzz), raw data, 19 lines 700.0-457.9 nm (row: air wavelength; abscissa: vacuum wavenumber); parenthesised last-digit uncertainties as sigma (0.3-1.2 %). Total gamma of H2 (vibration included). The extrapolated omega = 0 row is the formula table1_H2_over_He_raw. shelton_1990 Table III(a) re-expresses these on Bishop & Pipin’s He as absolute gamma (reproduced there).
mizrahi_1985_h2d2/H2_over_He_corrected#
{
"id": "H2_over_He_corrected",
"material": "H2",
"file": "H2/mizrahi_1985_h2d2__H2_over_He_corrected.csv",
"table": "tbl_d9c17b1ffdb1",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"abscissa_unit": "cm-1",
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"reference_material": "He",
"process": "eshg",
"convention": "T",
"part": "electronic"
}
}
Table I (p. 3456): chi_H2(3)el/chi_He(3), the raw ratios with the calculated vibrational-resonance contribution (Eq. 7, Table II) removed: the electronic part of H2 (the same measurements as H2_over_He_raw, in another part). Check at 632.8 nm: chi_vib = 184 au x [2/(1 - x^2) + 1/(1 - 4x^2)]/3 with x = 15798/4156 gives -10.20 au; divided by chi_He = 42.6 (1 + (15798/58300)^2) = 45.73 au it adds 0.223 to 17.88, giving 18.10 as printed (checked 2026-09-26).
mizrahi_1985_h2d2/H2_over_D2_raw#
{
"id": "H2_over_D2_raw",
"material": "H2",
"file": "H2/mizrahi_1985_h2d2__H2_over_D2_raw.csv",
"table": "tbl_d9c17b1ffdb1",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"abscissa_unit": "cm-1",
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"reference_material": "D2",
"process": "eshg",
"convention": "T",
"part": "total"
}
}
Table I (p. 3456): chi_H2(3)/chi_D2(3), raw data, 9 lines 694.3-476.5 nm, measured directly H2 against D2; parenthesised last-digit uncertainties as sigma. Total gammas.
mizrahi_1985_h2d2/H2_over_D2_corrected#
{
"id": "H2_over_D2_corrected",
"material": "H2",
"file": "H2/mizrahi_1985_h2d2__H2_over_D2_corrected.csv",
"table": "tbl_d9c17b1ffdb1",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"abscissa_unit": "cm-1",
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"reference_material": "D2",
"process": "eshg",
"convention": "T",
"part": "electronic"
}
}
Table I (p. 3456): chi_H2(3)el/chi_D2(3)el, vibrationally corrected (both molecules), 9 lines; the electronic ratio (the same measurements as H2_over_D2_raw).
mizrahi_1985_h2d2/vibrational_static_H2#
{
"id": "vibrational_static_H2",
"material": "H2",
"file": "H2/mizrahi_1985_h2d2__vibrational_static_H2.csv",
"table": "tbl_79b8cb3720a5",
"quantity": "hyperpolarizability",
"unit": "au",
"abscissa_unit": "nm",
"use": "primary",
"uncertainty": {
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "T",
"part": "vibrational"
}
}
Table II (p. 3458): chi(3)vib_H2(0) = (6 alpha_01,11^2 + (8/15) gamma_01,11^2)/omega_01,11 = 184 au, the static vibrational (Q(1) fundamental, Delta J = 0) contribution from Hunt et al.’s ab initio matrix elements (check: (6 x 0.74027^2 + 0.5333 x 0.61175^2)/0.01894 = 184.1). ‘nearly 30% of chi(3)el(0)’ (p. 3458). Shelton & Rice 1994 Table VIII gives 183.7 (bishop_1991) for the static gamma^v of H2.
mizrahi_1985_h2d2/vibrational_static_D2#
{
"id": "vibrational_static_D2",
"material": "D2",
"file": "D2/mizrahi_1985_h2d2__vibrational_static_D2.csv",
"table": "tbl_79b8cb3720a5",
"quantity": "hyperpolarizability",
"unit": "au",
"abscissa_unit": "nm",
"use": "primary",
"uncertainty": {
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "T",
"part": "vibrational"
}
}
Table II (p. 3458): chi(3)vib_D2(0) = 178 au (Q(2) fundamental; check: (6 x 0.61765^2 + 0.5333 x 0.50224^2)/0.01361 = 178.1). ‘the near equality of chi(3)vib(0) for H2 and D2 is not a coincidence’ (Placzek scaling, p. 3458).
mizrahi_1985_h2d2/table3_static_H2#
{
"id": "table3_static_H2",
"material": "H2",
"table": "tbl_fd6b909622b1",
"quantity": "hyperpolarizability",
"unit": "au",
"abscissa_unit": "nm",
"use": "check",
"reproduced_from": [
"M. Jaszunski and B. O. Roos, Mol. Phys. 52, 1209 (1984)",
"R. M. Berns and P. E. S. Wormer, Mol. Phys. 44, 1215 (1981)"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "T",
"part": "electronic"
}
}
3 inline points in the source record.
Table III (p. 3459), ‘thermally averaged (at T = 22 C)’: chi(3)el_H2(0) from experiment, 647 +- 4 au (‘Obtained assuming chi_He(3)(0) = 42.8 a.u. The assigned uncertainty does not include a contribution due to the uncertainty of chi_He(3)(0)’; = 15.13 x 42.8 = 647.6), and from the calculations of Jaszunski & Roos (693) and Berns & Wormer (671) as isotropically and internuclear-averaged by the authors. Inline (a three-cell table, one mark per cell). The experimental row is the static intercept of formula table1_H2_over_He_corrected; the theory rows are reproduced (hence the dataset is ‘check’).
mizrahi_1985_h2d2/table3_static_ratio#
{
"id": "table3_static_ratio",
"material": "H2",
"table": "tbl_fd6b909622b1",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"abscissa_unit": "nm",
"use": "check",
"reproduced_from": [
"M. Jaszunski and B. O. Roos, Mol. Phys. 52, 1209 (1984)",
"R. M. Berns and P. E. S. Wormer, Mol. Phys. 44, 1215 (1981)"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"reference_material": "D2",
"process": "static",
"convention": "T",
"part": "electronic"
}
}
2 inline points in the source record.
Table III (p. 3459): chi(3)el_H2(0)/chi(3)el_D2(0), experiment 1.025 +- 0.004 (the intercept 1.0249(43) of the corrected H2/D2 fit, Table I) and theory 1.029 (both calculations, thermally averaged with Poll’s J-dependent moments). ‘The experimental ratio is in good agreement with the theoretical value ... Note that the experimental results agree only after the vibrational correction has been made’ (p. 3459).
mizrahi_1985_h2d2/text_static_He_theory#
{
"id": "text_static_He_theory",
"material": "He",
"table": "text",
"quantity": "hyperpolarizability",
"unit": "au",
"abscissa_unit": "nm",
"use": "reproduced",
"reproduced_from": [
"A. D. Buckingham and P. G. Hibbard, Symp. Faraday Soc. 2, 41 (1968)",
"M. N. Grasso, K. T. Chung and R. P. Hurst, Phys. Rev. 167, 1 (1968)",
"R. Klingbeil, Phys. Rev. A 7, 48 (1973)",
"P. Sitz and R. Yaris, J. Chem. Phys. 49, 3546 (1968)"
],
"uncertainty": {
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "T",
"part": "electronic"
}
}
4 inline points in the source record.
p. 3458: ‘The values of chi_He(3)(0) obtained from the four best calculations (those employing explicitly correlated wave functions) are 43.10 a.u. (Buckingham and Hibbard), 42.81 a.u. (Grasso, Chung, and Hurst), 42.8 a.u. (Klingbeil), and 42.6 a.u. (Sitz and Yaris).’ The later benchmark is bishop_1989 (43.104 au).
Formulas#
mizrahi_1985_h2d2/table1_H2_over_He_raw#
{
"id": "table1_H2_over_He_raw",
"material": "H2",
"equation": "Table I and its caption, p. 3456",
"printed": "f(omega) = A(1 + B omega^2); A = 14.72(9) (the omega = 0 row), B = 9.01(22) x 10^-10 cm^2 (caption)",
"variable": "omega: vacuum wavenumber of the fundamental in cm^-1 (not nu_L^2)",
"process": "eshg",
"convention": "T",
"part": "total",
"range": "14282-21831 cm^-1",
"coefficients": {
"A": "14.72",
"B": "9.01e-10"
},
"coefficient_units": {
"A": "1 (chi_H2/chi_He)",
"B": "cm^2"
},
"check": "Gives 17.42 at 14282 cm^-1 and 21.04 at 21831 cm^-1 against 17.53(11) and 21.07(25). The same A, B as mizrahi_1985_arn2o2's table2_H2. Checked 2026-09-26."
}
Weighted least-squares fit to the raw ratios.
mizrahi_1985_h2d2/table1_H2_over_He_corrected#
{
"id": "table1_H2_over_He_corrected",
"material": "H2",
"equation": "Table I and its caption, p. 3456",
"printed": "f(omega) = A(1 + B omega^2); A = 15.13(9), B = 8.31(21) x 10^-10 cm^2",
"variable": "omega: vacuum wavenumber of the fundamental in cm^-1",
"process": "eshg",
"convention": "T",
"part": "electronic",
"range": "14282-21831 cm^-1",
"coefficients": {
"A": "15.13",
"B": "8.31e-10"
},
"coefficient_units": {
"A": "1 (chi_H2^el/chi_He)",
"B": "cm^2"
},
"check": "Gives 17.69 at 14282 cm^-1 and 21.12 at 21831 cm^-1 against 17.82(11) and 21.18(25). The intercept is '2.7% higher than would have been obtained by extrapolating the raw data' (15.13/14.72 = 1.028). Checked 2026-09-26."
}
Fit to the vibrationally corrected (electronic) ratios; times chi_He(0) = 42.8 au it gives 647 au (Table III).
mizrahi_1985_h2d2/table1_H2_over_D2_raw#
{
"id": "table1_H2_over_D2_raw",
"material": "H2",
"equation": "Table I and its caption, p. 3456",
"printed": "f(omega) = A(1 + B omega^2); A = 1.0125(43), B = 0.47(13) x 10^-10 cm^2",
"variable": "omega: vacuum wavenumber of the fundamental in cm^-1",
"process": "eshg",
"convention": "T",
"part": "total",
"range": "14399-20981 cm^-1",
"coefficients": {
"A": "1.0125",
"B": "0.47e-10"
},
"coefficient_units": {
"A": "1 (chi_H2/chi_D2)",
"B": "cm^2"
},
"check": "Gives 1.0224 at 14399 cm^-1 and 1.0334 at 20981 cm^-1 against 1.0229(35) and 1.0307(50). Checked 2026-09-26."
}
Fit to the raw H2/D2 ratios.
mizrahi_1985_h2d2/table1_H2_over_D2_corrected#
{
"id": "table1_H2_over_D2_corrected",
"material": "H2",
"equation": "Table I and its caption, p. 3456",
"printed": "f(omega) = A(1 + B omega^2); A = 1.0249(43), B = 0.24(13) x 10^-10 cm^2",
"variable": "omega: vacuum wavenumber of the fundamental in cm^-1",
"process": "eshg",
"convention": "T",
"part": "electronic",
"range": "14399-20981 cm^-1",
"coefficients": {
"A": "1.0249",
"B": "0.24e-10"
},
"coefficient_units": {
"A": "1 (chi_H2^el/chi_D2^el)",
"B": "cm^2"
},
"check": "Gives 1.0300 at 14399 cm^-1 and 1.0357 at 20981 cm^-1 against 1.0310(35) and 1.0335(50). Checked 2026-09-26."
}
Fit to the corrected ratios; the intercept is Table III’s 1.025 +- 0.004.
mizrahi_1985_h2d2/vibrational_dispersion_H2#
{
"id": "vibrational_dispersion_H2",
"material": "H2",
"equation": "Eq. (7), p. 3457; Table II, p. 3458",
"printed": "chi(3)vib(omega) = omega_0J,1J (2 alpha_0J,1J^2 + (8/45) gamma_0J,1J^2) [2/(omega_0J,1J^2 - omega^2) + 1/(omega_0J,1J^2 - 4 omega^2)]; chi(3)vib(x) = chi(3)vib(0) [2 (1 - x^2)^-1 + (1 - 4 x^2)^-1]/3, x = omega/omega_0J,1J; H2: chi(3)vib(0) = 184, omega_01,11 = 0.01894 (= 4156 cm^-1)",
"variable": "x = omega/omega_01,11: the fundamental's frequency over the Q(1) vibrational frequency (atomic units, or cm^-1 with 4156 cm^-1)",
"process": "eshg",
"convention": "T",
"part": "vibrational",
"range": "optical frequencies (x >> 1), v = 1, J = 1 only, Delta J = 0",
"coefficients": {
"chi_vib0": "184",
"omega_v": "0.01894",
"omega_v_cm": "4156"
},
"coefficient_units": {
"chi_vib0": "au",
"omega_v": "au (hartree)",
"omega_v_cm": "cm^-1"
},
"check": "Reproduces Table I's raw-to-corrected shift of H2/He at 632.8 nm (0.223, see H2_over_He_corrected). Checked 2026-09-26."
}
In atomic units (Eq. 7). Here gamma_0J,1J is the transition polarizability anisotropy, not a hyperpolarizability.
mizrahi_1985_h2d2/vibrational_dispersion_D2#
{
"id": "vibrational_dispersion_D2",
"material": "D2",
"equation": "Eq. (7), p. 3457; Table II, p. 3458",
"printed": "chi(3)vib(x) = chi(3)vib(0) [2 (1 - x^2)^-1 + (1 - 4 x^2)^-1]/3, x = omega/omega_02,12; D2: chi(3)vib(0) = 178, omega_02,12 = 0.01361 (= 2988 cm^-1)",
"variable": "x = omega/omega_02,12",
"process": "eshg",
"convention": "T",
"part": "vibrational",
"range": "optical frequencies, v = 1, J = 2 only, Delta J = 0",
"coefficients": {
"chi_vib0": "178",
"omega_v": "0.01361",
"omega_v_cm": "2988"
},
"coefficient_units": {
"chi_vib0": "au",
"omega_v": "au (hartree)",
"omega_v_cm": "cm^-1"
},
"check": "Not checked against the table (the D2/He ratio is not tabulated)."
}
As for H2.
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.