shelton_1987b#

D. P. Shelton, “Rotational hyperpolarizability of a homonuclear diatomic molecule”, Phys. Rev. A 36, 3461 (1987)

Source record · Catalogue · Review register

Source kind: theory

Recorded technique: Perturbation theory (Orr & Ward) for the off-resonance rotational hyperpolarizability gamma^r of a homonuclear diatomic molecule, treated fully quantum mechanically: gamma^r collects the terms with a pure rotational level of the ground electronic manifold as intermediate state; each pair of electronic sums is replaced by the static Raman transition polarizability; M sublevels averaged with the Wigner-Eckart theorem and J with a normalised population rho(J), following shelton_1987 (pp. 3461-3462).

Materials in structured entries: None recorded.

Dataset quantities: None recorded.

Contents: 0 datasets; 2 formulas.

Curation notes#

A brief report with no numerical values of gamma: only formulas (recorded below), Table I (the frequency factors 6 D^2(Omega) for the xxxx, xxyy, xyxy and xyyx polarization combinations, in terms of D1, D2, D3 = D(Omega; w1 + w2), D(Omega; w1 + w3), D(Omega; w2 + w3), with D(Omega; w) = 2 Omega/(Omega^2 - w^2), Eq. 6; the same as shelton_1987 Table 1’s 6 D^2 column; image checked) and the correction of shelton_1987’s Table 4. Pages are the journal’s (3461-3463; p. = pdf page + 3460).

gamma^r: only Delta J = +-2 transitions contribute (Delta J = 0 elements are those of JM -> JM, excluded, p. 3462), through alpha_0^(2) = (2/3)^1/2 (alpha_par - alpha_perp). With damping, D(Omega; w) becomes 2 Omega/(Omega^2 - w^2 + Gamma^2) (1 + i y)/(1 + y^2), y = 2 w Gamma/(Omega^2 - w^2 + Gamma^2) (Eq. 7), Gamma the HWHM: significant only for Gamma >~ Omega, e.g. N2 above 300 atm, where the rotational Raman spectrum collapses (p. 3462).

Eq. (10), the classical limit (hbar B/kT << 1, J >> 1, thermal populations, the J dependence of alpha_par - alpha_perp neglected) of the dc Kerr gamma^r: gamma^r_dcKerr = ((alpha_par - alpha_perp)^2/5kT){3/4 + (1/4)[1 - (w/Omega_m)^2]^-1}, Omega_m = 4B(kT/hbar B)^1/2 when w >> Omega. ‘The zero-frequency limit of Eq. (10) agrees exactly with the classical result [(alpha_par - alpha_perp)^2/5kT]. However, at optical frequencies where w >> Omega, Eq. (10) gives 3/4 times the classical result. Thus, Eqs. (9) and (10) disagree with the classical dc Kerr result and also with the previously calculated quantum result for the dc Kerr effect [Buckingham & Orr 1968]’; ‘it appears that Buckingham’s ansatz for calculating the dc Kerr birefringence in terms of a perturbed refractive index is only valid in the static limit’ (pp. 3462-3463). This bears on every dc Kerr gamma obtained by subtracting a Buckingham-Orr orientation term (tammer_1990, tammer_1992, buckingham_1968b, buckingham_1970, gentle_1989); Shelton & Rugar 1993 (shelton_1993) later measured H2 and used the full expression.

Ref. 10 (p. 3463), the erratum to shelton_1987: ‘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’ (page image checked; applied as errata in shelton_1987 and verified there by recomputation).

Reproductions elsewhere: shelton_1994 Table VIII (gamma^v of H2, D2, N2, O2 with ‘the corrections given in ref 133’, and gamma^R). The library’s validation reports references reordered on p. 3 (info only).

Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets#

No datasets are recorded.

Formulas#

shelton_1987b/eq5_rotational#

{
  "id": "eq5_rotational",
  "equation": "Eq. (5), p. 3462",
  "printed": "<gamma^r(-w_sigma; w1, w2, w3)>_abcd = (1/(30 hbar)) sum_J (alpha_par - alpha_perp)_{J,J+2}^2 [rho(J) - ((2J + 1)/(2J + 5)) rho(J + 2)] ((J + 1)(J + 2)/((2J + 1)(2J + 3))) 6 D^2(Omega_{J,J+2})",
  "variable": "rho(J): the normalised rotational population; Omega_{J,J+2}: the J -> J+2 rotational Raman frequency; 6 D^2(Omega): the frequency factor of Table I for the polarization combination abcd, from D(Omega; w) = 2 Omega/(Omega^2 - w^2) (Eq. 6)",
  "process": "other",
  "convention": "T",
  "part": "rotational",
  "range": "off resonance, damping negligible (Gamma << Omega; Eq. 7 otherwise)",
  "check": "No numerical values in the paper to reproduce."
}

The quantum rotational hyperpolarizability of a homonuclear diatomic molecule for any off-resonance third-order process (the process enters through the frequencies in Table I’s D1, D2, D3).

shelton_1987b/eq9_dc_kerr#

{
  "id": "eq9_dc_kerr",
  "equation": "Eq. (9), p. 3462",
  "printed": "gamma^r_dcKerr = sum_J ((alpha_par - alpha_perp)_{J,J+2}^2/(5kT)) (kT/(hbar B)) [rho(J) - ((2J + 1)/(2J + 5)) rho(J + 2)] ((J + 1)(J + 2)/((2J + 1)(2J + 3)^2)) (1/2){3 + [1 - (w/Omega_{J,J+2})^2]^-1}",
  "variable": "gamma_dcKerr = (3/2)[<gamma(-w; 0, 0, w)>_xxxx - <gamma(-w; 0, 0, w)>_xyyx] (Eq. 8); rigid-rotor levels E(J) = J(J + 1) hbar B, Omega_{J,J+2} = (6 + 4J) B; damping ignored",
  "process": "dc_kerr",
  "convention": "T",
  "part": "rotational",
  "range": "off resonance, damping negligible",
  "check": "No numerical values in the paper to reproduce."
}

Its classical limit is Eq. (10) (in the record’s notes): 3/4 of Buckingham’s classical (alpha_par - alpha_perp)^2/5kT at optical frequencies, equal to it in the static limit.

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.