shelton_1987 ============ D\. P\. Shelton, "Vibrational contributions to the hyperpolarizabilities of homonuclear diatomic molecules", Mol\. Phys\. 60, 65 \(1987\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1080/00268978700100051 `__ **Source kind:** theory **Recorded technique:** Perturbation theory \(Orr & Ward's expression for gamma, Eq\. 1\) for the off\-resonance vibrational hyperpolarizability gamma^v of a homonuclear diatomic molecule, with each pair of electronic sums replaced by the static Raman transition polarizability \(Eqs\. 3\-4; exact in the static limit, underestimating gamma^v at optical frequencies by about 4 %, p\. 67\), the isotropic and rotational averages done with spherical tensors \(Eqs\. 5\-17\), and a simplified form \(Eq\. 18\) for the ground vibrational state, J\-averaged transition polarizabilities, neglected rotational frequencies and J \>\> 1\. Evaluated for H2, D2 \(ab initio transition polarizabilities of Hunt, Poll & Wolniewicz 1984\), N2 and O2 \(Raman intensities and depolarisation ratios\) \(pp\. 66\-74\)\. **Materials in structured entries:** D2, H2, N2, O2 **Dataset quantities:** hyperpolarizability, raman\_shift, rotational\_constant **Contents:** 40 datasets; 3 formulas. Curation notes -------------- Pages are the journal's \(65\-76\); pdf page 1 is the publisher's cover, so p\. = pdf page \+ 63\. The library's validation notes an unmatched figure \(the cover\) and a table without caption \(Table 3, whose caption, 'Table 3\. Calculated values of the static gamma^v at 293 K', and unit line are extracted as text before it\); the tables agree with their images, except that Table 2's D2 row mark \(a double dagger on the image\) is read as a dagger\. DEFINITIONS\. gamma^v is 'the sum of all terms which may exhibit vibrational resonances' \(p\. 66\), the pure vibrational part \(not the zero\-point vibrational averaging of gamma^e, which is part of the observed gamma^e, p\. 67\)\. The gamma of Orr & Ward, in the convention p = \.\.\. \+ gamma E^3/6 used by Shelton & Rice \(shelton\_1994 Table VIII reproduces these values in au\: static N2 81 = 0\.506e\-62/6\.2354e\-65\)\. The lab\-frame components are those of the isotropic \(orientation\-averaged\) tensor\: gamma\_xxxx is the parallel \(mean\) gamma, recorded as component 'parallel'; the dc Kerr gamma is 3/2\[gamma\_xxxx \- gamma\_xyyx\]\(\-w; 0, 0, w\) \(Eq\. 23\), component 'K'\. Units C^4 m^4 J^\-3 \(the SI unit of gamma in optmat\)\. TABLE 2 \(p\. 72\), 'Molecular parameter values used to evaluate gamma^v'\: nu\_01 and B\_0/2 pi c \(datasets\), the nuclear\-spin weights g\_ns\(J\) for J even/odd \(H2 1/3, D2 2/1, N2 2/1, O2 0/1\) and the Raman transition polarizabilities \(\(alpha\_par \+ 2 alpha\_perp\)/3\)\_01 and \(alpha\_par \- alpha\_perp\)\_01 in 10^\-42 C^2 m^2 J^\-1\: H2 12\.21 and 10\.09, D2 10\.18 and 8\.28, N2 6\.13 and 4\.04, O2 5\.89 and 5\.89\. The transition polarizabilities are not stored as datasets\: no quantity of the vocabulary holds a Raman transition polarizability \(a matrix element \<0\|alpha\|1\>, not a derivative; FORMAT GAP\)\. Sources\: H2 'Ab initio values for the J = 1 state', D2 'for the J = 2 state', from Hunt, Poll & Wolniewicz, Can\. J\. Phys\. 62, 1719 \(1984\); N2 and O2 'From Raman intensity measurements of \[11\] \(Klöckner & Schrötter 1979, curated as schrotter\_1979\) and depolarization ratios of \[21\] \(Murphy, Holzer & Bernstein 1969\)'\. TABLE 3 \(p\. 72\)\: static gamma^v\_xxxx at 293 K from Eq\. \(17\) and Eq\. \(18\) \(datasets\); the ratio Eq\. \(18\)/Eq\. \(17\) \(0\.993, 0\.994, 1\.000, 1\.000\: 'a measure of the size of quantum effects'\), the ratio \(4/45\)\(alpha\_par \- alpha\_perp\)\_01^2/\(\(alpha\_par \+ 2 alpha\_perp\)/3\)\_01^2 \(0\.061, 0\.059, 0\.039, 0\.089\) and gamma^v/gamma^e static \(0\.292, 0\.286, 0\.094, 0\.141; gamma^e from Mizrahi & Shelton 1985\) are in these notes only\. Checked \(2026\-09\-26\)\: Eq\. \(21\) with Table 2 reproduces the Eq\. \(18\) column\: 1\.149, 1\.109 \(printed 1\.110\), 0\.506, 0\.734, and the ratio column 0\.061, 0\.059, 0\.039, 0\.089\. TABLE 4 \(p\. 74\)\: gamma^v at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'; a wavenumber, so vacuum\) for the dc Kerr effect, DFWM \(the Kerr effect gamma\(\-w; w, w, \-w\)\), CARS at nu\_2 = nu\_1 \- 2500 cm^\-1, ESHG and THG, from Eq\. \(18\); 'For comparison, gamma^e ~ 5 x 10^\-62 C^4 m^4 J^\-3 for the molecules considered'\. ERRATA\: 'Table 4 of Ref\. 9 contains several numerical errors\. Line 2 should read D2, 2988, \+0\.007\*, \+0\.737, \+2\.464, \-0\.031, \-0\.010\. Column 6 should read ESHG, \-0\.064, \-0\.031, \-0\.008, \-0\.005' \(Shelton, Phys\. Rev\. A 36, 3461 \(1987\), ref\. 10 = shelton\_1987b; page image checked\)\. Corrected by errata below\: D2 CARS, and the four ESHG values; D2's nu\_01, printed 2998 in Table 4 \(2988 in Table 2\), is not stored from Table 4\. CHECKED \(2026\-09\-26\) by recomputing Table 4 from Table 2 with Eqs\. \(21\)\-\(23\) at nu\_1 = 15800 cm^\-1\: dc Kerr \-0\.032, \+0\.007, \+0\.003, \+0\.038; DFWM 0\.759, 0\.736, 0\.336, 0\.489; CARS 1\.194, 2\.462 \(2\.464 with Table 3's 1\.110\), \-2\.245 \(printed \-2\.231, 0\.6 %, the CARS difference frequency being 170 cm^\-1 from N2's resonance\), \-0\.309; ESHG \-0\.0637, \-0\.0308, \-0\.0084, \-0\.0054; THG \-0\.0202, \-0\.0100, \-0\.0028, \-0\.0018\. So the corrected values are right; the printed ESHG column is not reproduced by any single factor, and the printed D2 CARS 2\.426 is what nu\_01 = 2998 gives \(2\.416\-2\.427\)\: the misprinted frequency was used\. The Eq\. \(17\) dc Kerr values of note \* are the text\_dc\_kerr\_eq17\_\ datasets\. Reproductions elsewhere\: shelton\_1994 Table VIII gives the N2 and O2 gamma^v in au 'with corrections given in ref 133' \(shelton\_1987b\)\: static 81 and 118, dc Kerr 0\.5 and 6\.1, DFWM 54 and 78 at 632\.8 nm \(= this paper's 0\.506, 0\.734, 0\.003, 0\.038, 0\.336, 0\.489 x 10^\-62 / 6\.2354e\-65\), ESHG \-1\.6 and \-1\.0 and THG \-0\.6 and \-0\.4 at 694\.3 nm \(Eq\. 22 at 694\.3 nm gives \-1\.63 and \-1\.04, \-0\.54 and \-0\.34 au\: recomputed at the other wavelength\)\. H2 and D2 in that table come from another calculation \(its note e\)\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- shelton\_1987/table2\_nu01\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_nu01_H2", "material": "H2", "file": "H2/shelton_1987__table2_nu01_H2.csv", "table": "tbl_6c3a7b517193", "quantity": "raman_shift", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "transition": "v = 0 -> 1 (nu_01)", "assumed": true } } Table 2 \(p\. 72\)\: 'Molecular parameter values used to evaluate gamma^v'\: the fundamental vibrational frequency nu\_01 \(v = 0 \-\> 1\) of H2, in cm^\-1, an input of the calculation \(its source is not stated; the table's footnote on this row marked with a dagger \('Ab initio values for the J = 1 state, from \[20\]', Hunt, Poll & Wolniewicz 1984\) concerns the transition polarizabilities\)\. Table 4 reprints it \(D2 there as 2998, a misprint\: see the record's errata notes\)\. shelton\_1987/table2\_B0\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_B0_H2", "material": "H2", "file": "H2/shelton_1987__table2_B0_H2.csv", "table": "tbl_6c3a7b517193", "quantity": "rotational_constant", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "constant": "B", "vibrational_state_v": 0, "assumed": true } } Table 2 \(p\. 72\)\: B\_0/2 pi c of H2, the ground\-state rotational constant in cm^\-1, an input of Eq\. \(20\) \(source not stated\)\. shelton\_1987/table2\_nu01\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_nu01_D2", "material": "D2", "file": "D2/shelton_1987__table2_nu01_D2.csv", "table": "tbl_6c3a7b517193", "quantity": "raman_shift", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "transition": "v = 0 -> 1 (nu_01)", "assumed": true } } Table 2 \(p\. 72\)\: 'Molecular parameter values used to evaluate gamma^v'\: the fundamental vibrational frequency nu\_01 \(v = 0 \-\> 1\) of D2, in cm^\-1, an input of the calculation \(its source is not stated; the table's footnote on this row marked with a double dagger \('Ab initio values for the J = 2 state, from \[20\]', Hunt, Poll & Wolniewicz 1984; the extraction reads the mark as a dagger\) concerns the transition polarizabilities\)\. Table 4 reprints it \(D2 there as 2998, a misprint\: see the record's errata notes\)\. shelton\_1987/table2\_B0\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_B0_D2", "material": "D2", "file": "D2/shelton_1987__table2_B0_D2.csv", "table": "tbl_6c3a7b517193", "quantity": "rotational_constant", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "constant": "B", "vibrational_state_v": 0, "assumed": true } } Table 2 \(p\. 72\)\: B\_0/2 pi c of D2, the ground\-state rotational constant in cm^\-1, an input of Eq\. \(20\) \(source not stated\)\. shelton\_1987/table2\_nu01\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_nu01_N2", "material": "N2", "file": "N2/shelton_1987__table2_nu01_N2.csv", "table": "tbl_6c3a7b517193", "quantity": "raman_shift", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "transition": "v = 0 -> 1 (nu_01)", "assumed": true } } Table 2 \(p\. 72\)\: 'Molecular parameter values used to evaluate gamma^v'\: the fundamental vibrational frequency nu\_01 \(v = 0 \-\> 1\) of N2, in cm^\-1, an input of the calculation \(its source is not stated; the table's footnote on this row marked with a section sign \('From Raman intensity measurements of \[11\] and depolarization ratios of \[21\]', Klöckner & Schrötter 1979; Murphy, Holzer & Bernstein 1969\) concerns the transition polarizabilities\)\. Table 4 reprints it \(D2 there as 2998, a misprint\: see the record's errata notes\)\. shelton\_1987/table2\_B0\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_B0_N2", "material": "N2", "file": "N2/shelton_1987__table2_B0_N2.csv", "table": "tbl_6c3a7b517193", "quantity": "rotational_constant", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "constant": "B", "vibrational_state_v": 0, "assumed": true } } Table 2 \(p\. 72\)\: B\_0/2 pi c of N2, the ground\-state rotational constant in cm^\-1, an input of Eq\. \(20\) \(source not stated\)\. shelton\_1987/table2\_nu01\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_nu01_O2", "material": "O2", "file": "O2/shelton_1987__table2_nu01_O2.csv", "table": "tbl_6c3a7b517193", "quantity": "raman_shift", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "transition": "v = 0 -> 1 (nu_01)", "assumed": true } } Table 2 \(p\. 72\)\: 'Molecular parameter values used to evaluate gamma^v'\: the fundamental vibrational frequency nu\_01 \(v = 0 \-\> 1\) of O2, in cm^\-1, an input of the calculation \(its source is not stated; the table's footnote on this row marked with a section sign \('From Raman intensity measurements of \[11\] and depolarization ratios of \[21\]', Klöckner & Schrötter 1979; Murphy, Holzer & Bernstein 1969\) concerns the transition polarizabilities\)\. Table 4 reprints it \(D2 there as 2998, a misprint\: see the record's errata notes\)\. shelton\_1987/table2\_B0\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table2_B0_O2", "material": "O2", "file": "O2/shelton_1987__table2_B0_O2.csv", "table": "tbl_6c3a7b517193", "quantity": "rotational_constant", "unit": "cm-1", "abscissa": "label", "use": "check", "parameters": { "constant": "B", "vibrational_state_v": 0, "assumed": true } } Table 2 \(p\. 72\)\: B\_0/2 pi c of O2, the ground\-state rotational constant in cm^\-1, an input of Eq\. \(20\) \(source not stated\)\. shelton\_1987/table3\_eq17\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq17_H2", "material": "H2", "file": "H2/shelton_1987__table3_eq17_H2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel", "temperature_k": 293.0 } } Table 3 \(p\. 72\), 'Calculated values of the static gamma^v at 293 K', gamma^v\_xxxx\(0; 0, 0, 0\) / 10^\-62 C^4 m^4 J^\-3, column 'Equation \(17\)'\: the explicit sum over rotational levels \(the ab initio J\-dependent transition polarizabilities of \[20\], note \|\|\), fundamental v = 0 \-\> 1 only\. The pure vibrational static gamma \(the isotropic gamma\_xxxx, i\.e\. the mean gamma; the static limit has one independent component, Eq\. 19\)\. Theory; no uncertainty stated \(Eq\. 4, the transition\-polarizability approximation, is exact in the static limit, p\. 67\)\. shelton\_1987/table3\_eq18\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq18_H2", "material": "H2", "file": "H2/shelton_1987__table3_eq18_H2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 3 \(p\. 72\), column 'Equation \(18\)'\: the static gamma^v\_xxxx of H2 from the approximate Eq\. \(18\) = Eq\. \(21\) \(ground vibrational state, J\-averaged transition polarizabilities of Table 2, rotational frequencies neglected, J \>\> 1\: 'essentially the classical limit', note dagger\), in 10^\-62 C^4 m^4 J^\-3\. Reproduced here from Table 2 by Eq\. \(21\) \(see the formula eq21\_static\)\. shelton\_1987/table3\_eq17\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq17_D2", "material": "D2", "file": "D2/shelton_1987__table3_eq17_D2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel", "temperature_k": 293.0 } } Table 3 \(p\. 72\), 'Calculated values of the static gamma^v at 293 K', gamma^v\_xxxx\(0; 0, 0, 0\) / 10^\-62 C^4 m^4 J^\-3, column 'Equation \(17\)'\: the explicit sum over rotational levels \(the ab initio J\-dependent transition polarizabilities of \[20\], note \|\|\), fundamental v = 0 \-\> 1 only\. The pure vibrational static gamma \(the isotropic gamma\_xxxx, i\.e\. the mean gamma; the static limit has one independent component, Eq\. 19\)\. Theory; no uncertainty stated \(Eq\. 4, the transition\-polarizability approximation, is exact in the static limit, p\. 67\)\. shelton\_1987/table3\_eq18\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq18_D2", "material": "D2", "file": "D2/shelton_1987__table3_eq18_D2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 3 \(p\. 72\), column 'Equation \(18\)'\: the static gamma^v\_xxxx of D2 from the approximate Eq\. \(18\) = Eq\. \(21\) \(ground vibrational state, J\-averaged transition polarizabilities of Table 2, rotational frequencies neglected, J \>\> 1\: 'essentially the classical limit', note dagger\), in 10^\-62 C^4 m^4 J^\-3\. Reproduced here from Table 2 by Eq\. \(21\) \(see the formula eq21\_static\)\. shelton\_1987/table3\_eq17\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq17_N2", "material": "N2", "file": "N2/shelton_1987__table3_eq17_N2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel", "temperature_k": 293.0 } } Table 3 \(p\. 72\), 'Calculated values of the static gamma^v at 293 K', gamma^v\_xxxx\(0; 0, 0, 0\) / 10^\-62 C^4 m^4 J^\-3, column 'Equation \(17\)'\: the explicit sum over rotational levels \(the J\-averaged transition polarizabilities of Table 2, note pilcrow\), fundamental v = 0 \-\> 1 only\. The pure vibrational static gamma \(the isotropic gamma\_xxxx, i\.e\. the mean gamma; the static limit has one independent component, Eq\. 19\)\. Theory; no uncertainty stated \(Eq\. 4, the transition\-polarizability approximation, is exact in the static limit, p\. 67\)\. shelton\_1987/table3\_eq18\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq18_N2", "material": "N2", "file": "N2/shelton_1987__table3_eq18_N2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 3 \(p\. 72\), column 'Equation \(18\)'\: the static gamma^v\_xxxx of N2 from the approximate Eq\. \(18\) = Eq\. \(21\) \(ground vibrational state, J\-averaged transition polarizabilities of Table 2, rotational frequencies neglected, J \>\> 1\: 'essentially the classical limit', note dagger\), in 10^\-62 C^4 m^4 J^\-3\. Reproduced here from Table 2 by Eq\. \(21\) \(see the formula eq21\_static\)\. shelton\_1987/table3\_eq17\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq17_O2", "material": "O2", "file": "O2/shelton_1987__table3_eq17_O2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel", "temperature_k": 293.0 } } Table 3 \(p\. 72\), 'Calculated values of the static gamma^v at 293 K', gamma^v\_xxxx\(0; 0, 0, 0\) / 10^\-62 C^4 m^4 J^\-3, column 'Equation \(17\)'\: the explicit sum over rotational levels \(the J\-averaged transition polarizabilities of Table 2, note pilcrow\), fundamental v = 0 \-\> 1 only\. The pure vibrational static gamma \(the isotropic gamma\_xxxx, i\.e\. the mean gamma; the static limit has one independent component, Eq\. 19\)\. Theory; no uncertainty stated \(Eq\. 4, the transition\-polarizability approximation, is exact in the static limit, p\. 67\)\. shelton\_1987/table3\_eq18\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table3_eq18_O2", "material": "O2", "file": "O2/shelton_1987__table3_eq18_O2.csv", "table": "tbl_d9a1dad537c6", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "nm", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "static", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 3 \(p\. 72\), column 'Equation \(18\)'\: the static gamma^v\_xxxx of O2 from the approximate Eq\. \(18\) = Eq\. \(21\) \(ground vibrational state, J\-averaged transition polarizabilities of Table 2, rotational frequencies neglected, J \>\> 1\: 'essentially the classical limit', note dagger\), in 10^\-62 C^4 m^4 J^\-3\. Reproduced here from Table 2 by Eq\. \(21\) \(see the formula eq21\_static\)\. shelton\_1987/table4\_H2\_dc\_kerr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_H2_dc_kerr", "material": "H2", "file": "H2/shelton_1987__table4_H2_dc_kerr.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of H2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), dc Kerr, 3/2\[gamma^v\_xxxx\(\-w; 0, 0, w\) \- gamma^v\_xyyx\(\-w; 0, 0, w\)\], Eq\. \(23\) \(note dagger\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. 'The dc Kerr hyperpolarizability is particularly sensitive to the approximations made in order to obtain equation \(18\) from equation \(17\)' \(note \*\)\: the Eq\. \(17\) value is text\_dc\_kerr\_eq17\_H2 \(use that one\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_H2\_dfwm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_H2_dfwm", "material": "H2", "file": "H2/shelton_1987__table4_H2_dfwm.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "kerr", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of H2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), DFWM, gamma^v\_xxxx\(\-w; w, w, \-w\), Eq\. \(22\) \(the nonlinear refractive index\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_H2\_cars ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_H2_cars", "material": "H2", "file": "H2/shelton_1987__table4_H2_cars.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "cars", "convention": "T", "part": "vibrational", "component": "parallel", "stokes_nm": 751.88 } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of H2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), CARS, gamma^v\_xxxx\(\-2w1 \+ w2; w1, w1, \-w2\), Eq\. \(22\), with nu\_2 = nu\_1 \- 2500 cm^\-1 = 13300 cm^\-1 \(stokes\_nm = 1e7/13300 = 751\.88 nm, vacuum, computed here\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_H2\_eshg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_H2_eshg", "material": "H2", "file": "H2/shelton_1987__table4_H2_eshg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "eshg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of H2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), ESHG, gamma^v\_xxxx\(\-2w; w, w, 0\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Printed \-0\.042; corrected to \-0\.064 \(erratum below\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_H2\_thg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_H2_thg", "material": "H2", "file": "H2/shelton_1987__table4_H2_thg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "thg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of H2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), THG, gamma^v\_xxxx\(\-3w; w, w, w\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_D2\_dc\_kerr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_D2_dc_kerr", "material": "D2", "file": "D2/shelton_1987__table4_D2_dc_kerr.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of D2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), dc Kerr, 3/2\[gamma^v\_xxxx\(\-w; 0, 0, w\) \- gamma^v\_xyyx\(\-w; 0, 0, w\)\], Eq\. \(23\) \(note dagger\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. 'The dc Kerr hyperpolarizability is particularly sensitive to the approximations made in order to obtain equation \(18\) from equation \(17\)' \(note \*\)\: the Eq\. \(17\) value is text\_dc\_kerr\_eq17\_D2 \(use that one\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_D2\_dfwm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_D2_dfwm", "material": "D2", "file": "D2/shelton_1987__table4_D2_dfwm.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "kerr", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of D2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), DFWM, gamma^v\_xxxx\(\-w; w, w, \-w\), Eq\. \(22\) \(the nonlinear refractive index\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_D2\_cars ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_D2_cars", "material": "D2", "file": "D2/shelton_1987__table4_D2_cars.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "cars", "convention": "T", "part": "vibrational", "component": "parallel", "stokes_nm": 751.88 } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of D2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), CARS, gamma^v\_xxxx\(\-2w1 \+ w2; w1, w1, \-w2\), Eq\. \(22\), with nu\_2 = nu\_1 \- 2500 cm^\-1 = 13300 cm^\-1 \(stokes\_nm = 1e7/13300 = 751\.88 nm, vacuum, computed here\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Printed \+2\.426; corrected to \+2\.464 by Shelton's Phys\. Rev\. A 36, 3461 ref\. 10 \(erratum below\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_D2\_eshg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_D2_eshg", "material": "D2", "file": "D2/shelton_1987__table4_D2_eshg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "eshg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of D2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), ESHG, gamma^v\_xxxx\(\-2w; w, w, 0\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Printed \-0\.020; corrected to \-0\.031 \(erratum below\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_D2\_thg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_D2_thg", "material": "D2", "file": "D2/shelton_1987__table4_D2_thg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "thg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of D2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), THG, gamma^v\_xxxx\(\-3w; w, w, w\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_N2\_dc\_kerr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_N2_dc_kerr", "material": "N2", "file": "N2/shelton_1987__table4_N2_dc_kerr.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of N2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), dc Kerr, 3/2\[gamma^v\_xxxx\(\-w; 0, 0, w\) \- gamma^v\_xyyx\(\-w; 0, 0, w\)\], Eq\. \(23\) \(note dagger\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. 'The dc Kerr hyperpolarizability is particularly sensitive to the approximations made in order to obtain equation \(18\) from equation \(17\)' \(note \*\)\: the Eq\. \(17\) value is text\_dc\_kerr\_eq17\_N2 \(use that one\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_N2\_dfwm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_N2_dfwm", "material": "N2", "file": "N2/shelton_1987__table4_N2_dfwm.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "kerr", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of N2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), DFWM, gamma^v\_xxxx\(\-w; w, w, \-w\), Eq\. \(22\) \(the nonlinear refractive index\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_N2\_cars ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_N2_cars", "material": "N2", "file": "N2/shelton_1987__table4_N2_cars.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "cars", "convention": "T", "part": "vibrational", "component": "parallel", "stokes_nm": 751.88 } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of N2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), CARS, gamma^v\_xxxx\(\-2w1 \+ w2; w1, w1, \-w2\), Eq\. \(22\), with nu\_2 = nu\_1 \- 2500 cm^\-1 = 13300 cm^\-1 \(stokes\_nm = 1e7/13300 = 751\.88 nm, vacuum, computed here\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_N2\_eshg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_N2_eshg", "material": "N2", "file": "N2/shelton_1987__table4_N2_eshg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "eshg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of N2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), ESHG, gamma^v\_xxxx\(\-2w; w, w, 0\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Printed \-0\.006; corrected to \-0\.008 \(erratum below\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_N2\_thg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_N2_thg", "material": "N2", "file": "N2/shelton_1987__table4_N2_thg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "thg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of N2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), THG, gamma^v\_xxxx\(\-3w; w, w, w\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_O2\_dc\_kerr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_O2_dc_kerr", "material": "O2", "file": "O2/shelton_1987__table4_O2_dc_kerr.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "check", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of O2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), dc Kerr, 3/2\[gamma^v\_xxxx\(\-w; 0, 0, w\) \- gamma^v\_xyyx\(\-w; 0, 0, w\)\], Eq\. \(23\) \(note dagger\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. 'The dc Kerr hyperpolarizability is particularly sensitive to the approximations made in order to obtain equation \(18\) from equation \(17\)' \(note \*\)\: the Eq\. \(17\) value is text\_dc\_kerr\_eq17\_O2 \(use that one\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_O2\_dfwm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_O2_dfwm", "material": "O2", "file": "O2/shelton_1987__table4_O2_dfwm.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "kerr", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of O2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), DFWM, gamma^v\_xxxx\(\-w; w, w, \-w\), Eq\. \(22\) \(the nonlinear refractive index\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_O2\_cars ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_O2_cars", "material": "O2", "file": "O2/shelton_1987__table4_O2_cars.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "cars", "convention": "T", "part": "vibrational", "component": "parallel", "stokes_nm": 751.88 } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of O2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), CARS, gamma^v\_xxxx\(\-2w1 \+ w2; w1, w1, \-w2\), Eq\. \(22\), with nu\_2 = nu\_1 \- 2500 cm^\-1 = 13300 cm^\-1 \(stokes\_nm = 1e7/13300 = 751\.88 nm, vacuum, computed here\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/table4\_O2\_eshg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_O2_eshg", "material": "O2", "file": "O2/shelton_1987__table4_O2_eshg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "eshg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of O2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), ESHG, gamma^v\_xxxx\(\-2w; w, w, 0\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Printed \-0\.004; corrected to \-0\.005 \(erratum below\)\. Theory; no uncertainty stated\. shelton\_1987/table4\_O2\_thg ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table4_O2_thg", "material": "O2", "file": "O2/shelton_1987__table4_O2_thg.csv", "table": "tbl_a48923e63b94", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "thg", "convention": "T", "part": "vibrational", "component": "parallel" } } Table 4 \(p\. 74\), 'Vibrational contributions to the hyperpolarizability at optical frequencies', gamma^v of O2 in 10^\-62 C^4 m^4 J^\-3 at nu\_1 = 15800 cm^\-1 \('lambda = 633 nm'\), THG, gamma^v\_xxxx\(\-3w; w, w, w\), Eq\. \(22\), calculated with Eq\. \(18\) \(the J\-averaged, classical\-limit expression\) and Table 2's inputs; the fundamental v = 0 \-\> 1 only\. Theory; no uncertainty stated\. shelton\_1987/text\_dc\_kerr\_eq17\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_dc_kerr_eq17_H2", "material": "H2", "table": "text", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K", "temperature_k": 293.0 } } 1 inline points in the source record. Table 4 note \* \(p\. 74\)\: 'Using equation \(17\) one obtains gamma^v / 10^\-62 C^4 m^4 J^\-3 = \-0\.041, \+0\.001, \+0\.003 and \+0\.038 for H2, D2, N2 and O2 respectively, at 293 K\. The calculation uses the ab initio alpha\_vJ,v'J' and omega\_vJ,v'J' of \[20\] for H2 and D2 and the J\-averaged transition polarizabilities given in table 2 for N2 and O2\.' The dc Kerr gamma^v = 3/2\[gamma^v\_xxxx \- gamma^v\_xyyx\]\(\-w; 0, 0, w\) of H2 \(\-0\.041\) at nu\_1 = 15800 cm^\-1, the full rotational sum, which the paper prefers for the dc Kerr effect \('if an accurate estimate of gamma^v\_dcKerr is desired one must use equation \(17\)', p\. 73\)\. shelton\_1987/text\_dc\_kerr\_eq17\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_dc_kerr_eq17_D2", "material": "D2", "table": "text", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K", "temperature_k": 293.0 } } 1 inline points in the source record. Table 4 note \* \(p\. 74\)\: 'Using equation \(17\) one obtains gamma^v / 10^\-62 C^4 m^4 J^\-3 = \-0\.041, \+0\.001, \+0\.003 and \+0\.038 for H2, D2, N2 and O2 respectively, at 293 K\. The calculation uses the ab initio alpha\_vJ,v'J' and omega\_vJ,v'J' of \[20\] for H2 and D2 and the J\-averaged transition polarizabilities given in table 2 for N2 and O2\.' The dc Kerr gamma^v = 3/2\[gamma^v\_xxxx \- gamma^v\_xyyx\]\(\-w; 0, 0, w\) of D2 \(\+0\.001\) at nu\_1 = 15800 cm^\-1, the full rotational sum, which the paper prefers for the dc Kerr effect \('if an accurate estimate of gamma^v\_dcKerr is desired one must use equation \(17\)', p\. 73\)\. shelton\_1987/text\_dc\_kerr\_eq17\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_dc_kerr_eq17_N2", "material": "N2", "table": "text", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K", "temperature_k": 293.0 } } 1 inline points in the source record. Table 4 note \* \(p\. 74\)\: 'Using equation \(17\) one obtains gamma^v / 10^\-62 C^4 m^4 J^\-3 = \-0\.041, \+0\.001, \+0\.003 and \+0\.038 for H2, D2, N2 and O2 respectively, at 293 K\. The calculation uses the ab initio alpha\_vJ,v'J' and omega\_vJ,v'J' of \[20\] for H2 and D2 and the J\-averaged transition polarizabilities given in table 2 for N2 and O2\.' The dc Kerr gamma^v = 3/2\[gamma^v\_xxxx \- gamma^v\_xyyx\]\(\-w; 0, 0, w\) of N2 \(\+0\.003\) at nu\_1 = 15800 cm^\-1, the full rotational sum, which the paper prefers for the dc Kerr effect \('if an accurate estimate of gamma^v\_dcKerr is desired one must use equation \(17\)', p\. 73\)\. shelton\_1987/text\_dc\_kerr\_eq17\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_dc_kerr_eq17_O2", "material": "O2", "table": "text", "quantity": "hyperpolarizability", "unit": "SI", "scale": 1e-62, "abscissa_unit": "cm-1", "use": "primary", "uncertainty": { "level": "not stated" }, "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "K", "temperature_k": 293.0 } } 1 inline points in the source record. Table 4 note \* \(p\. 74\)\: 'Using equation \(17\) one obtains gamma^v / 10^\-62 C^4 m^4 J^\-3 = \-0\.041, \+0\.001, \+0\.003 and \+0\.038 for H2, D2, N2 and O2 respectively, at 293 K\. The calculation uses the ab initio alpha\_vJ,v'J' and omega\_vJ,v'J' of \[20\] for H2 and D2 and the J\-averaged transition polarizabilities given in table 2 for N2 and O2\.' The dc Kerr gamma^v = 3/2\[gamma^v\_xxxx \- gamma^v\_xyyx\]\(\-w; 0, 0, w\) of O2 \(\+0\.038\) at nu\_1 = 15800 cm^\-1, the full rotational sum, which the paper prefers for the dc Kerr effect \('if an accurate estimate of gamma^v\_dcKerr is desired one must use equation \(17\)', p\. 73\)\. Formulas -------- shelton\_1987/eq21\_static ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq21_static", "equation": "Eq. (21), p. 72", "printed": "gamma^v_xxxx(0; 0, 0, 0) = (6/(hbar omega_01)) [(4/45)(alpha_par - alpha_perp)_01^2 + ((alpha_par + 2 alpha_perp)/3)_01^2]", "variable": "omega_01 = 2 pi c nu_01, the fundamental vibrational frequency; (alpha)_01 the static Raman transition polarizabilities (Table 2)", "process": "static", "convention": "T", "part": "vibrational", "range": "the static limit; H2, D2, N2, O2 with Table 2's inputs", "coefficients": { "nu01_H2": "4156", "nu01_D2": "2988", "nu01_N2": "2331", "nu01_O2": "1555", "mean01_H2": "12.21", "mean01_D2": "10.18", "mean01_N2": "6.13", "mean01_O2": "5.89", "aniso01_H2": "10.09", "aniso01_D2": "8.28", "aniso01_N2": "4.04", "aniso01_O2": "5.89" }, "coefficient_units": { "nu01": "cm^-1", "mean01": "1e-42 C^2 m^2 J^-1 ((alpha_par + 2 alpha_perp)/3)_01", "aniso01": "1e-42 C^2 m^2 J^-1 (alpha_par - alpha_perp)_01" }, "check": "Reproduces Table 3's Eq. (18) column (checked 2026-09-26): 1.149, 1.109 (printed 1.110), 0.506, 0.734 x 10^-62 C^4 m^4 J^-3 for H2, D2, N2, O2." } The static limit of Eq\. \(18\), for the four molecules of Table 2 \(the formula applies to each; the coefficients are Table 2's\)\. Its D^2 \(anisotropy\) terms carry the approximations of Eq\. \(18\) \(\+8, \+5, \+0\.5, \+0\.3 % against Eq\. \(17\), p\. 73\)\. shelton\_1987/eq22\_optical ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq22_optical", "equation": "Eq. (22), p. 73", "printed": "gamma^v_xxxx(-2 omega_1 + omega_2; omega_1, omega_1, -omega_2) = gamma^v_xxxx(0; 0, 0, 0) (1/3){2 [1 - ((omega_1 - omega_2)/omega_01)^2]^-1 + [1 - (2 omega_1/omega_01)^2]^-1}", "variable": "omega_1, omega_2 the optical frequencies (CARS as written; ESHG: omega_2 = 0; THG: omega_2 = -omega_1; DFWM: omega_2 = omega_1); omega_01 the vibrational frequency; gamma^v(0) from Eq. (21)", "process": "other", "convention": "T", "part": "vibrational", "range": "frequencies well below electronic resonance; not for the dc Kerr effect (Eq. 23)", "check": "With Table 2 and Eq. (21), reproduces Table 4's DFWM, CARS and THG columns and the corrected ESHG column of shelton_1987b ref. 10 at nu_1 = 15800 cm^-1 (checked 2026-09-26; N2 CARS -2.245 against the printed -2.231)." } The classical\-limit \(Eq\. 18\) frequency dependence of gamma^v for every process whose frequencies pair as written; Table 4's source\. shelton\_1987/eq23\_dc\_kerr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq23_dc_kerr", "equation": "Eq. (23), p. 73", "printed": "gamma^v_dcKerr = (3/2)[gamma^v_xxxx(-omega; 0, 0, omega) - gamma^v_xyyx(-omega; 0, 0, omega)] = (6/(hbar omega_01)) {((alpha_par - alpha_perp)_01^2/45) [3 + [1 - (omega/omega_01)^2]^-1] + ((alpha_par + 2 alpha_perp)/3)_01^2 [1 - (omega/omega_01)^2]^-1}", "variable": "omega the optical frequency; omega_01 the vibrational frequency; (alpha)_01 as in Table 2", "process": "dc_kerr", "convention": "T", "part": "vibrational", "range": "below electronic resonance; 'particularly sensitive to the approximations' of Eq. (18): use Eq. (17) for an accurate value (p. 73)", "check": "With Table 2, reproduces Table 4's dc Kerr column at nu_1 = 15800 cm^-1 (checked 2026-09-26): -0.032, +0.007, +0.003, +0.038." } A small difference of large terms; at optical frequencies the anisotropy term dominates \(p\. 73\)\. The Eq\. \(17\) values are in note \* of Table 4 \(text\_dc\_kerr\_eq17\_\\)\. Corrections ----------- .. code-block:: text [ { "dataset": "table4_D2_cars", "row": "cars", "column": "value", "printed": "2.426", "corrected": "2.464", "reason": "Shelton's own correction: 'Table 4 of Ref. 9 contains several numerical errors. Line 2 should read D2, 2988, +0.007*, +0.737, +2.464, -0.031, -0.010' (Phys. Rev. A 36, 3461 (1987), ref. 10). The printed value follows from the misprinted nu_01 = 2998 of Table 4's line 2 (Table 2 gives 2988).", "evidence": "shelton_1987b p. 3463, ref. 10 (page image checked). Recomputed here with Eq. (22), Table 2 and Table 3's Eq. (18) gamma^v(0) = 1.110: 2.464 with nu_01 = 2988; 2.42-2.43 with 2998. Checked 2026-09-26." }, { "dataset": "table4_H2_eshg", "row": "eshg", "column": "value", "printed": "-0.042", "corrected": "-0.064", "reason": "Shelton's own correction: 'Column 6 should read ESHG, -0.064, -0.031, -0.008, -0.005' (Phys. Rev. A 36, 3461 (1987), ref. 10).", "evidence": "shelton_1987b p. 3463, ref. 10 (page image checked). Recomputed here with Eq. (22) (omega_2 = 0), Table 2 and Eq. (21): -0.0637. Checked 2026-09-26." }, { "dataset": "table4_D2_eshg", "row": "eshg", "column": "value", "printed": "-0.020", "corrected": "-0.031", "reason": "Shelton's own correction: 'Column 6 should read ESHG, -0.064, -0.031, -0.008, -0.005' (Phys. Rev. A 36, 3461 (1987), ref. 10).", "evidence": "shelton_1987b p. 3463, ref. 10 (page image checked). Recomputed here with Eq. (22), Table 2 (nu_01 = 2988) and Eq. (21): -0.0308. Checked 2026-09-26." }, { "dataset": "table4_N2_eshg", "row": "eshg", "column": "value", "printed": "-0.006", "corrected": "-0.008", "reason": "Shelton's own correction: 'Column 6 should read ESHG, -0.064, -0.031, -0.008, -0.005' (Phys. Rev. A 36, 3461 (1987), ref. 10).", "evidence": "shelton_1987b p. 3463, ref. 10 (page image checked). Recomputed here with Eq. (22), Table 2 and Eq. (21): -0.0084. Checked 2026-09-26." }, { "dataset": "table4_O2_eshg", "row": "eshg", "column": "value", "printed": "-0.004", "corrected": "-0.005", "reason": "Shelton's own correction: 'Column 6 should read ESHG, -0.064, -0.031, -0.008, -0.005' (Phys. Rev. A 36, 3461 (1987), ref. 10).", "evidence": "shelton_1987b p. 3463, ref. 10 (page image checked). Recomputed here with Eq. (22), Table 2 and Eq. (21): -0.0054. Checked 2026-09-26." } ] 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.