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 <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/bishop_1996.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1063/1\.471752 <https://doi.org/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
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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 :doc:`review register </register>` for the checks and decisions.