bishop_1990 =========== D\. M\. Bishop, "Molecular vibrational and rotational motion in static and dynamic electric fields", Rev\. Mod\. Phys\. 62, 343 \(1990\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/revmodphys\.62\.343 `__ **Source kind:** compilation **Recorded technique:** Review of the theory of vibrational and rotational contributions to static and dynamic \(hyper\)polarizabilities \(clamped\-nucleus, vibrational averaging, pure vibrational and rotational terms; perturbation\-theory, finite\-field and numerical methods\), with sample results quoted from the literature\. **Materials in structured entries:** H2 **Dataset quantities:** hyperpolarizability **Contents:** 9 datasets; 0 formulas. Curation notes -------------- Journal pages 343\-374 = pdf pages 1\-32 \(page = pdf page \+ 342\)\. A review\: its values are quoted from other papers\. Only H2 among the curated gases has tabulated gamma values \(Tables VI and X\); the other tables concern OH, OH\+, OH\-, FH, LiH and H2\+ \(not curated\) or give frequency factors \(Tables IX, XI\)\. DEFINITIONS \(Eq\. 2, p\. 344; also Eq\. 32, p\. 351\)\: E = E0 \- mu\_a F\_a \- \(1/2\) alpha\_ab F\_a F\_b \- \(1/6\) beta\_abg F\_a F\_b F\_g \- \(1/24\) gamma\_abgd F\_a F\_b F\_g F\_d \- \.\.\., 'used irrespective of whether we are considering simply the electronic energy or the total molecular energy'; the dynamic gamma\(\-w\_s; w1, w2, w3\) reduce to these at zero frequency\. 'The reader is cautioned that Ward and co\-workers \(e\.g\., Orr and Ward, 1971\) define beta and gamma in such a way that they are 1/2 and 1/6, respectively, of the above definitions' \(p\. 345\)\: this review, and Bishop's papers, use the Taylor\-series \(T\) convention; Ward's gamma is gamma\_T / 6\. Atomic units \(Table I\: gamma 6\.23538e\-65 C^4 m^4 J^\-3\); 'For high\-precision calculations on atoms, the atomic units may be modified by using the reduced nuclear mass' \(p\. 345\)\. THEORY RECORDED FOR LATER USE \(not datasets\)\. Vibrational component with quantum averaging, Eq\. \(173\)\: gamma^v = sum\_J sum\_\(v \!= 0\) \[A0 \<0\(J\)\|alpha\-bar\|v\(J\)\>^2 \+ A1 \<0\(J\)\|Delta alpha\|v\(J\+2\)\>^2 \+ A2 \<0\(J\)\|Delta alpha\|v\(J\)\>^2 \+ A3 \<0\(J\+2\)\|Delta alpha\|v\(J\)\>^2\], with rho\(J\)\-weighted frequency factors f, f' of Table IX \(for example f\(x\) for Kerr\-par = 2\[1 \+ 2\(1 \- z^2\)^\-1\]/x, ESHG\-par = 2\[\(1 \- 4z^2\)^\-1 \+ 2\(1 \- z^2\)^\-1\]/x, z = hbar w / x\) \(pp\. 366\-367\)\. Rotational component, Eq\. \(178\), and its high\-temperature forms \(p\. 367\)\: kT gamma\_par^r\(K\) = \(2/3\)\(mu\)\(beta\-bar\)\(1 \+ a\) \+ \(1/15\) \(Delta alpha\)^2 \(1 \+ 2b\) \+ kT gamma\_par^dist, kT gamma\_par^dist = \(Delta alpha\)^2 / 45; kT gamma\_perp^r\(K\) = \(2/9\)\(mu\)\(beta\-bar\)\(1 \+ a\) \+ \(1/30\)\(Delta alpha\)^2 \(3b \- 1\) \+ kT gamma\_perp^dist, kT gamma\_perp^dist = \-\(Delta alpha\)^2 / 90; kT gamma\_par^r\(E\) = \(1/3\) \(mu\)\(beta\-bar\)\(1 \+ 2a \+ c\) \+ \(1/15\)\(Delta alpha\)^2 \(2b \+ d\); kT gamma\_perp^r\(E\) = \(1/9\) \(mu\)\(beta\-bar\)\(1 \+ 2a \+ c\) \+ \(1/30\)\(Delta alpha\)^2 \(3b \- d\), with a = \[1 \- \(w / w\_\(J\+1,J\)\)^2\]^\-1, b = \[1 \- \(w / w\_\(J\+2,J\)\)^2\]^\-1, c = \[1 \- 4 \(w / w\_\(J\+1,J\)\)^2\]^\-1, d = \[1 \- 4 \(w / w\_\(J\+2,J\)\)^2\]^\-1 \(Eqs\. 179\-188; K = dc Kerr, E = ESHG\)\. For the dc Kerr gamma^K = \(3/2\)\(gamma\_par \- gamma\_perp\) \(p\. 368, Eqs\. 191\-194\)\: kT gamma\_r^K = \(2/3\) \(mu\)\(beta\-bar\)\(1 \+ a\) \+ \(1/20\)\(Delta alpha\)^2 \(3 \+ b\) \+ kT gamma\_dist^K, kT gamma\_dist^K = \(1/20\)\(Delta alpha\)^2; at w = 0, kT gamma\_r^K = \(4/3\)\(mu\)\(beta^K\) \+ \(1/4\) \(Delta alpha\)^2, and for w \>\> 0 \(a = b = 0\), kT gamma\_r^K = \(2/3\)\(mu\)\(beta^K\) \+ \(1/5\) \(Delta alpha\)^2\. For a nonpolar linear molecule \(N2, O2, H2, CO2\) only the \(Delta alpha\)^2 terms remain\: the rotational \(orientational\) contribution to the dc Kerr gamma at optical frequency is \(Delta alpha\)^2 / \(5 kT\) \(with Delta alpha the static anisotropy x the optical one in the dc Kerr case; see Shelton and Rice's Eq\. 9\)\. The distribution term gamma^dist was 'neglected by Shelton \(1987b\)' \(Bishop and Shelton 1988\)\. The dispersive factors b, d show that the rotational ESHG contribution vanishes at optical frequencies far above the rotational transitions \(b, d \-\> 0\), while the dc Kerr one does not \(the 3 \+ b factor\)\. DATASETS \(H2\)\. Table VI \(p\. 358\)\: 'total \(electronic and vibrational\)' gamma\_zzzz of H2 \(v = 0\), static, by finite field \(Adamowicz 1988a\: 1088\), derivative Numerov\-Cooley \(Augspurger and Dykstra 1988\: 1086\), perturbation theory \(Bishop and Lam 1987b\: 1099 = the electronic gamma\_zzzz\(R\_e\) of Maroulis and Bishop 1986 plus gamma\_zzzz^v 416, footnote b\) and 'Exact' \(Bishop, Pipin and Rerat 1990\: 854\.5, 'a thermal average and with respect to space\-fixed axes', footnote c, i\.e\. gamma\_par = 670\.8 \+ 183\.7 of Table X\)\: reproduced\. Table X \(p\. 367\)\: gamma\_par^e, gamma\_par^v, gamma\_perp^e, gamma\_perp^v of H2 at infinite wavelength and at 4880 A for Kerr and ESHG, 'From Bishop et al\. \(1990\)' \(Bishop, Pipin and Rerat, J\. Chem\. Phys\. 92, 1902 \(1990\), not in the library\)\. The electronic values are superseded by bishop\_1991 \(682\.7 static, 1\.3\-1\.8 % higher\), and bishop\_1991 reproduces the same vibrational values at 13 frequencies\: all of Table X is kept but excluded\. 'The vibrational component compared with the electronic component is less significant at optical frequencies and only about 1 % for ESHG' \(p\. 367\)\. Shelton's approximations change the vibrational gamma^K of H2 at 6328 A from \-0\.041 to \-0\.032 \(x 10^\-62 C^4 m^4 J^\-3\) \(p\. 367\)\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- bishop\_1990/table6 ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "table6", "material": "H2", "file": "H2/bishop_1990__table6.csv", "table": "tbl_9c908a3a7ef7", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "nm", "use": "reproduced", "reproduced_from": [ "L. Adamowicz (1988a), as cited in the review", "D. W. Augspurger and C. E. Dykstra (1988), as cited", "D. M. Bishop and B. Lam (1987b), as cited", "bishop_1990b" ], "parameters": { "process": "static", "convention": "T", "part": "electronic_vibrational", "component": "zzzz" } } Table VI\: static total \(electronic \+ vibrational\) gamma of H2 in the v = 0 state; rows A, AD, BL are the molecular\-frame gamma\_zzzz \(v = 0\), row BPR the thermal space\-fixed gamma\_par \(295 K\)\. Row ids are the table's reference codes\. bishop\_1990/dc\_kerr\_par\_e ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dc_kerr_par_e", "material": "H2", "file": "H2/bishop_1990__dc_kerr_par_e.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "dc_kerr", "convention": "T", "part": "electronic", "component": "parallel", "temperature_k": 295 } } EXCLUDED\: superseded by bishop\_1991 \(extended Delta basis, 1\.3\-1\.8 % higher\)\. Table X, gamma\_par^e for the dc Kerr process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-Kerr'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/dc\_kerr\_par\_v ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dc_kerr_par_v", "material": "H2", "file": "H2/bishop_1990__dc_kerr_par_v.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "parallel", "temperature_k": 295 } } EXCLUDED\: the same pure vibrational values as bishop\_1991's \(reproduced there at 13 frequencies\)\. Table X, gamma\_par^v for the dc Kerr process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-Kerr'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/dc\_kerr\_perp\_e ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dc_kerr_perp_e", "material": "H2", "file": "H2/bishop_1990__dc_kerr_perp_e.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "dc_kerr", "convention": "T", "part": "electronic", "component": "perpendicular", "temperature_k": 295 } } EXCLUDED\: superseded by bishop\_1991 \(extended Delta basis, 1\.3\-1\.8 % higher\)\. Table X, gamma\_perp^e for the dc Kerr process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-Kerr'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/dc\_kerr\_perp\_v ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dc_kerr_perp_v", "material": "H2", "file": "H2/bishop_1990__dc_kerr_perp_v.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "dc_kerr", "convention": "T", "part": "vibrational", "component": "perpendicular", "temperature_k": 295 } } EXCLUDED\: the same pure vibrational values as bishop\_1991's \(reproduced there at 13 frequencies\)\. Table X, gamma\_perp^v for the dc Kerr process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-Kerr'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/eshg\_par\_e ~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eshg_par_e", "material": "H2", "file": "H2/bishop_1990__eshg_par_e.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "eshg", "convention": "T", "part": "electronic", "component": "parallel", "temperature_k": 295 } } EXCLUDED\: superseded by bishop\_1991 \(extended Delta basis, 1\.3\-1\.8 % higher\)\. Table X, gamma\_par^e for the ESHG process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-ESHG'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/eshg\_par\_v ~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eshg_par_v", "material": "H2", "file": "H2/bishop_1990__eshg_par_v.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "eshg", "convention": "T", "part": "vibrational", "component": "parallel", "temperature_k": 295 } } EXCLUDED\: the same pure vibrational values as bishop\_1991's \(reproduced there at 13 frequencies\)\. Table X, gamma\_par^v for the ESHG process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-ESHG'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/eshg\_perp\_e ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eshg_perp_e", "material": "H2", "file": "H2/bishop_1990__eshg_perp_e.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "eshg", "convention": "T", "part": "electronic", "component": "perpendicular", "temperature_k": 295 } } EXCLUDED\: superseded by bishop\_1991 \(extended Delta basis, 1\.3\-1\.8 % higher\)\. Table X, gamma\_perp^e for the ESHG process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-ESHG'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. bishop\_1990/eshg\_perp\_v ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eshg_perp_v", "material": "H2", "file": "H2/bishop_1990__eshg_perp_v.csv", "table": "tbl_b41995b8424a", "quantity": "hyperpolarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "excluded", "parameters": { "process": "eshg", "convention": "T", "part": "vibrational", "component": "perpendicular", "temperature_k": 295 } } EXCLUDED\: the same pure vibrational values as bishop\_1991's \(reproduced there at 13 frequencies\)\. Table X, gamma\_perp^v for the ESHG process\: rows 'infinity' \(static; the same row is in the dc Kerr and ESHG datasets\) and '4880\-ESHG'\. From Bishop, Pipin and Rerat \(1990\), 295 K thermal average\. 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 :doc:`review register ` for the checks and decisions.