bishop_1996#
D. M. Bishop and D. W. De Kee, “The frequency dependence of nonlinear optical processes”, J. Chem. Phys. 104, 9876 (1996)
Source record · Catalogue · Review register
DOI: 10.1063/1.471752
Source kind: theory
Recorded technique: Sum rules (the hyperpolarizability counterparts of Cauchy moments) for the diagonal components of the dynamic X^3 = gamma and X^5 = epsilon, derived from the sum-over-states expressions (Eq. 4) by expanding the frequency denominators to fourth order; evaluated for H and He with near-exact wave functions (He: explicitly correlated Hylleraas-type functions as in bishop_1989, 140 S, 140 P, 190 D and 190 F states) (pp. 9877-9885).
Materials in structured entries: He
Dataset quantities: None recorded.
Contents: 0 datasets; 5 formulas.
Curation notes#
Identity: pdf page 1 is the publisher’s cover; journal pages 9876-9887 = pdf pages 2-13 (page = pdf page + 9874). Only He is curated (H is the hydrogen atom, not a curated species).
THE GENERALISED DISPERSION FORMULA (abstract; Eq. 22, p. 9879; Eq. 67, p. 9884): X^n(-w_s; w1, ..., wn) = X^n(0) + A W2 + B W2^2 + B’ W4 + ..., with W2 = w_s^2 + w1^2 + ... + wn^2 (‘the same as omega_L^2 in the previous literature’) and W4 = w_s^4 + w1^4 + ... + wn^4; A, B and B’ are process-independent sum rules. Unlike Shelton’s relative A, B, these coefficients are ABSOLUTE: ‘the units of A, B, B’, a, and b in Tables XI-XIV are those of the hyperpolarizabilities themselves’, with the frequencies in hartree (p. 9885). So Shelton’s A = A / X(0) / 219474.6^2 (cm^2) and Shelton’s B for a given process = (B + B’ W4/W2^2) / X(0) / 219474.6^4: the fourth-order coefficient is process-dependent through W4/W2^2 (1/2 for dc Kerr and ESHG, 1/4 for IDRI, 7/12 for THG, computed here). ‘It is apparent that B’ is generally smaller than B and this accounts for the reasonable success of the Shelton-Bishop dispersion formula’ (abstract). ‘For both the Kerr effect and ESHG W4 = (1/2) W2^2 and consequently an expansion in W2 and W2^2 alone will furnish the same coefficients for these two processes’ (p. 9884).
CONVENTION: X^n by the SOS formula Eq. (4) with hbar^-n and the sum over the (n+1)! permutations, ‘in conventional notation X^1 = alpha, X^2 = beta, X^3 = gamma’ (abstract), the same definition as bishop_1989: T. Components: diagonal (zzzz), which for atoms is the isotropic value (p. 9885). Part electronic. Processes of Table XII: X^3(-w; w, 0, 0) (dc Kerr), (-2w; w, w, 0) (ESHG), (-3w; w, w, w) (THG), (-w; w, -w, w) (‘the intensity-dependent refractive index’, process kerr).
FORMULAS: he_sum_rules (Table XI) and he_<process> (Table XII); all images checked. Checks (2026-09-26): A = A(+) + A(-) = 106.1183 - 11.74475 = 94.37355, B = 165.7406, B’ = -30.6705 (printed); A / 43.104 = 2.1894, bishop_1989 Table V’s 2.189; Table XII’s a, b follow from A, B, B’ with W2/w^2 = 2, 6, 12, 4 and W4/w^4 = 2, 18, 84, 4 (e.g. THG b = 144 x 165.7406 - 84 x 30.6705 = 21290, printed 2.1290 (4)); and the Table XII series reproduce bishop_1989 Table III: dc Kerr at 0.10 a.u. 45.052 (45.054), ESHG at 0.05 44.553 (44.554), THG at 0.05 46.068 (46.074), DFWM at 0.05 44.064 (44.064). The authors: ‘at least three-figure agreement’ with directly calculated values for He when hbar w <= 0.19 (Kerr), 0.11 (ESHG, IDRI), 0.07 E_h (THG) (p. 9885). X^3(0) = 43.104 ‘taken from Ref. 55’ (the bishop_1989 value); not recorded again.
NOT RECORDED: the fourth hyperpolarizability X^5 (epsilon) sum rules and a, b for He and H (Tables XIII, XIV: He X^5(0) = 1.4937e4 a.u.; fifth-harmonic, second-order Kerr and self-induced IDRI) - no vocabulary quantity for fifth-order response; noted for the higher-order-Kerr discussion (these are bound-electron values, positive, far below resonance). The H values (not curated).
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
No datasets are recorded.
Formulas#
bishop_1996/he_sum_rules#
{
"id": "he_sum_rules",
"material": "He",
"equation": "Eq. (67), p. 9884, with Table XI (tbl_75a4ba038074), He column",
"printed": "X^3_zzzz(w) = X^3_zzzz(0) + A W2 + B W2^2 + B' W4; He: A(+) 1.061 183 (2), A(-) -1.174 475 (1), A 9.437 355 (1), B(+) 1.707 919 (2), B(-) -5.051 324 (0), B 1.657 406 (2), B'(+) -1.864 023 (1), B'(-) -1.203 027 (1), B' -3.067 050 (1) [numbers in parentheses: powers of ten]",
"variable": "W2 = w_s^2 + w1^2 + w2^2 + w3^2 and W4 = w_s^4 + w1^4 + w2^4 + w3^4, the frequencies in hartree; X^3 = gamma_zzzz of He in atomic units (T convention, electronic, the isotropic value for an atom); any process",
"state": "none (per atom)",
"density": "none",
"range": "a fourth-order expansion; three-figure agreement with direct values for hbar w <= 0.19 (dc Kerr), 0.11 (ESHG, IDRI), 0.07 E_h (THG) (p. 9885)",
"coefficients": {
"X0": "43.104",
"A": "9.437355e1",
"B": "1.657406e2",
"B_prime": "-3.067050e1"
},
"check": "A, B, B' equal the sums of their (+) and (-) parts; with X(0) = 43.104 they reproduce bishop_1989 Table III to three figures in the stated ranges (source notes). Checked 2026-09-26."
}
Absolute coefficients (units of gamma, frequencies in hartree), not Shelton’s relative ones: Shelton A = A / 43.104 / 219474.6^2 cm^2 = 0.4545e-10 cm^2. X(0) = 43.104 is quoted from Ref. 55 (= bishop_1989’s value), infinite-nuclear-mass atomic units.
bishop_1996/he_dc_kerr#
{
"id": "he_dc_kerr",
"material": "He",
"equation": "Eq. (68), p. 9884, with Table XII (tbl_ed5ea5559158), row X^3(-w; w, 0, 0), He",
"printed": "X^3(w) = X^3(0) + a w^2 + b w^4; X^3(0) 4.3104 (1), a 1.8875 (2), b 6.0162 (2)",
"variable": "w: laser photon energy in hartree; X^3 = gamma_zzzz(-w; w, 0, 0) of He, the dc Kerr process, atomic units, T convention",
"state": "none (per atom)",
"density": "none",
"range": "hbar w <= 0.19 E_h for three-figure accuracy (p. 9885)",
"coefficients": {
"X0": "4.3104e1",
"a": "1.8875e2",
"b": "6.0162e2"
},
"check": "a = 2A = 188.75 and b = 4B + 2B' = 601.62 from he_sum_rules; at w = 0.10 a.u. 45.052 against bishop_1989 Table III 45.054. Checked 2026-09-26."
}
‘The values in Table XII have been verified by fitting Eq. (68) to independently-determined values of X^3(w)’ (p. 9884).
bishop_1996/he_eshg#
{
"id": "he_eshg",
"material": "He",
"equation": "Eq. (68), p. 9884, with Table XII, row X^3(-2w; w, w, 0), He",
"printed": "X^3(w) = X^3(0) + a w^2 + b w^4; X^3(0) 4.3104 (1), a 5.6624 (2), b 5.4146 (3)",
"variable": "w: laser photon energy in hartree; X^3 = gamma_zzzz(-2w; w, w, 0) of He (ESHG), atomic units, T convention",
"state": "none (per atom)",
"density": "none",
"range": "hbar w <= 0.11 E_h for three-figure accuracy (p. 9885)",
"coefficients": {
"X0": "4.3104e1",
"a": "5.6624e2",
"b": "5.4146e3"
},
"check": "a = 6A, b = 36B + 18B' = 5414.6 from he_sum_rules; at w = 0.05 a.u. 44.553 against bishop_1989 Table III 44.554. Checked 2026-09-26."
}
As he_dc_kerr.
bishop_1996/he_thg#
{
"id": "he_thg",
"material": "He",
"equation": "Eq. (68), p. 9884, with Table XII, row X^3(-3w; w, w, w), He",
"printed": "X^3(w) = X^3(0) + a w^2 + b w^4; X^3(0) 4.3104 (1), a 1.1325 (3), b 2.1290 (4)",
"variable": "w: laser photon energy in hartree; X^3 = gamma_zzzz(-3w; w, w, w) of He (THG), atomic units, T convention",
"state": "none (per atom)",
"density": "none",
"range": "hbar w <= 0.07 E_h for three-figure accuracy (p. 9885)",
"coefficients": {
"X0": "4.3104e1",
"a": "1.1325e3",
"b": "2.1290e4"
},
"check": "a = 12A, b = 144B + 84B' = 21290 from he_sum_rules; at w = 0.05 a.u. 46.068 against bishop_1989 Table III 46.074. Checked 2026-09-26."
}
As he_dc_kerr.
bishop_1996/he_kerr#
{
"id": "he_kerr",
"material": "He",
"equation": "Eq. (68), p. 9884, with Table XII, row X^3(-w; w, -w, w), He",
"printed": "X^3(w) = X^3(0) + a w^2 + b w^4; X^3(0) 4.3104 (1), a 3.7749 (2), b 2.5292 (3)",
"variable": "w: laser photon energy in hartree; X^3 = gamma_zzzz(-w; w, -w, w) of He, the intensity-dependent refractive index (process kerr), atomic units, T convention",
"state": "none (per atom)",
"density": "none",
"range": "hbar w <= 0.11 E_h for three-figure accuracy (p. 9885)",
"coefficients": {
"X0": "4.3104e1",
"a": "3.7749e2",
"b": "2.5292e3"
},
"check": "a = 4A, b = 16B + 4B' = 2529.2 from he_sum_rules; at w = 0.05 a.u. 44.064 against bishop_1989 Table III (DFWM) 44.064. Checked 2026-09-26."
}
The Kerr coefficient of He (SPM/XPM gamma) as a two-term series; with X(0) = 43.104 infinite-mass atomic units (bishop_1989 notes).
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.