puchalski_2016#

M. Puchalski, K. Piszczatowski, J. Komasa, B. Jeziorski and K. Szalewicz, “Theoretical determination of the polarizability dispersion and the refractive index of helium”, Phys. Rev. A 93, 032515 (2016)

Source record · Catalogue · Review register

DOI: 10.1103/PhysRevA.93.032515

Source kind: theory

Recorded technique: Ab initio dynamic dipole polarizability of the helium atom as the power series alpha(omega) = alpha_0 + alpha_2 omega^2 + alpha_4 omega^4 + ... (Eq. 2), alpha_k = 2 <psi| z R^(k+1) z |psi> (Eq. 10), with explicitly correlated Slater-geminal bases (600-term fully optimized and 800-term stochastically optimized, independent codes). Nonrelativistic coefficients with the finite nuclear mass of 4He included exactly; relativistic (Breit-Pauli, 1/c^2) corrections for the infinite nuclear mass and the relativistic recoil 1/(M c^2) for k = 0, 2; QED 1/c^3 corrections (the static one complete, the dispersion ones in the Pachucki-Sapirstein approximation, 10 % uncertainty), the one-loop 1/c^4 term and the finite nuclear size for the static value (pp. 3-10). Comparison with experiment and a first-principles virial expansion of n (pp. 10-13).

Materials in structured entries: He

Dataset quantities: cauchy_moment, molar_refractivity, polarizability, refractivity_virial_coefficient

Contents: 13 datasets; 3 formulas.

Curation notes#

Units and states (pp. 2-3, 7): atomic units a0^3 for the polarizability and t0 = hbar^3/(m_e e^4) for time, ‘we never use the reduced units in the present work’; M = m_alpha/m_e = 7294.2995361, c = 137.0359991. alpha_k^(0) (Table II) are for 4He with the nuclear mass included exactly (nonrelativistic Hamiltonian with the mass-polarization term, Eq. 12); alpha_k^(20) (Tables III, IV, VIII) and alpha_k^(30) (Table VIII) are for infinite nuclear mass; alpha_k^(21) (Table IV) are the 4He recoil corrections of order 1/(M c^2). The totals of Table VII (static and at 6329.908 A) are for 4He with all of these. Wavelengths are vacuum wavelengths: lambda = f/omega with f = 5e7/R_inf (cm^-1) = 455.633525275 (lambda in A, omega in 1/t0), p. 2.

Uncertainties: basis-set ones are half the difference of the two largest SO bases (‘very conservative’, p. 7); the static total’s 0.14 u a0^3 (0.1 ppm) is dominated by a 25 % allowance on the one-loop 1/c^4 term (p. 3); the dispersion QED terms carry 10 %. The level is not stated (estimates). The static and the 6329.908 A totals share the 0.14 u a0^3 uncertainty (fully correlated).

Recommended alpha(omega) (not printed as one series; see the formula ‘recommended’): the static total of Table VII plus sum_k [alpha_k^(0) + alpha_k^(20) + alpha_k^(21) + alpha_k^(30)] omega^k for k = 2, 4, 6, plus alpha_k^(0) omega^k for k = 8-14. It reproduces Table VII’s 1.39181197(14) at 6329.908 A (checked 2026-09-24).

Relativistic and QED components (all in u a0^3 t0^k = 1e-6 a0^3 t0^k): Table IV alpha_0^(20) = -80.35991(1), alpha_0^(21) = -0.0935(1), alpha_2^(20) = -33.06576(2), alpha_2^(21) = -0.144(1), alpha_4^(20) = 5.05105(5), alpha_6^(20) = 62.8500(5); Table VIII totals of the 1/c^3 terms alpha_0^(30) = 30.47381, alpha_2^(30) = 61.9014, alpha_4^(30) = 116.988, alpha_6^(30) = 216.89, and the one-loop 1/c^4 component alpha_0^(40) = 0.563 89, alpha_2^(40) = 1.156 5, alpha_4^(40) = 2.196, alpha_6^(40) = 4.08 (only alpha_0^(40) enters the recommended value). Bethe logarithm ln k0 = 4.37016022306(2) (Korobov 2012). Bhatia and Drachman’s relativistic values (Table IV, ‘BD’, converted from reduced Rydberg units): -80.062, -32.629, 4.814. Tables I, III and V (basis-set convergence; Table V is 3He, not a species) are not extracted; their largest-basis values are those of Tables II and IV.

Table VII components at 6329.908 A (a0^3): static nonrelativistic 1.3838099864, 1/c^2 -0.0000803599, 1/(Mc^2) -0.0000000935(1), 1/c^3 without the field derivative of the Bethe logarithm 0.0000304738(1), that derivative’s term 0.000000183(1), 1/c^4 0.00000056(14), finite nuclear size 0.0000000217(1), total 1.38376077(14); lambda^-2 terms 0.007995798, -0.000000171, -0.000000001, 1/c^3 0.00000032(3), total 0.00799595(3); lambda^-4 0.000054836, 0.0000000001, 0.0000000031(3), total 0.000054840; lambda^-6 0.000000408; lambda^-8 0.0000000032; alpha(lambda) - alpha_0 = 0.00805120(3) (Bhatia and Drachman, corrected: 0.008050871; their Eq. 15 as printed: 0.008052951). The lambda^-10 term is 2.5e-11.

Magnetizability (diamagnetic susceptibility) chi_0 of the helium atom: refractometry measures alpha + chi, n = 1 + 2 pi (alpha + chi) rho for the ideal gas (Eq. 1, p. 1; a_n = 2 pi (alpha + chi), Eq. 44, and n^2 = eps_r mu_r with the magnetic analogue of Clausius-Mossotti, Eq. 40, p. 11). The paper prints (p. 11, checked on the pdf text): ‘The static nonrelativistic value of A_mu is -0.000007922 4(4) cm3/mol, converted from chi = -0.000021194(1) a0^3 computed by Bruch and Weinhold [47] with an account of the nuclear motion. We assigned the uncertainty assuming that the relativistic corrections enter at the fifth significant digit.’ So chi_0 = -0.000021194(1) in atomic units (a0^3, the Gaussian volume convention of alpha, so that alpha + chi is meaningful), reproduced from L. W. Bruch and F. Weinhold, J. Chem. Phys. 117, 3243 (2002); Erratum 119, 638 (2003) (not in the library); A_mu = 4 pi N_A chi/3 = -0.0000079224(4) cm^3/mol = -1.53e-5 A_eps, so a measured helium refractivity is 1.5e-5 below 2 pi alpha rho. The frequency dependence of chi is neglected (‘negligible for the helium atom’, Table X note a). No dataset: the format has no quantity for a magnetizability (polarizability would be misread). The same chi is printed by piszczatowski_2015 (Table II note b); schmidt_2007 uses A_mu = 4 pi chi_0/3 = -0.000 008 03 (17) cm^3/mol (Table I, from Bruch and Weinhold 2000). The datasets here are alpha and A_eps only (the package’s molar refractivity N_A alpha/(3 eps0) is A_eps).

Table X (not extracted: the format has no quantity for a_r = A_eps + A_mu or for the virial coefficients of n itself), cm^3/mol, cm^6/mol^2, cm^9/mol^3, lambda in A: a_r(0) present 0.51724620(6), expt. [8] (Schmidt et al. 2007) 0.5172455(47); a_r(6329.908) present 0.52025577(6), expt. [41,42] 0.5213(1), expt. [43] 0.5220(3); b_n(0) present 273.16 K 0.0244(3); b_n(6329.908) present 273.16 K 0.0238(3), 303 K 0.0184(3), 323 K 0.0150(3), expt. [42] 0.000(15); c_n(0) present 273.16 K -0.96(25). Here n = 1 + a_n rho + b_n rho^2 + c_n rho^3 (Eq. 41, rho molar density), a_r = (2/3) a_n; the Lorentz-Lorenz second refractivity virial coefficient (the package’s B_R) is b_r = (2/3) b_n - a_r^2/4 (Eq. 49), e.g. -0.0518 cm^6/mol^2 at 273.16 K and 6329.908 A (computed here, not printed). b_n and c_n follow from Eqs. (45)-(46) with the dielectric virial coefficients b_eps = B_eps/A_eps of Rizzo et al. (J. Chem. Phys. 117, 2609 (2002)): b_eps(273.16 K) = -0.0978(4) cm^3/mol static, -0.0996(4) at 6329.908 A, b_eps(303 K) = -0.1065(4), b_eps(323 K) = -0.1108(4) at 6329.908 A (pp. 12-13), and c_eps = -1.38 cm^6/mol^2 (Heller and Gelbart), uncertainty from Huot and Bose’s -1.7(9) (p. 13). The static B_eps(273.16 K) is extracted (dataset dielectric_virial_rizzo). Also quoted (p. 11): Kirouac and Bose (1976) A_eps = 0.5210(2), Huot and Bose (1991) 0.5196(2) cm^3/mol, dielectric measurements the authors call incompatible with theory; not extracted (low accuracy, originals not in the library).

Schmidt et al.’s Eq. (4) lacks a factor 2 in front of the A_eps^2 b rho^2 term (p. 12). Bhatia and Drachman’s A_k (Phys. Rev. A 58, 4470, Eq. 15) were converted from reduced Rydberg units with (1 + m_e/m_alpha)^(2k) instead of (1 + m_e/m_alpha)^k, and their lambda^-4 coefficient has a typographical error (Table VI notes c-e); see bhatia_1998. The abstract of Ref. [13] (Piszczatowski et al. 2015) misprints alpha(6329.908 A) (Table VII note e); see piszczatowski_2015.

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

Datasets#

puchalski_2016/alpha#

{
  "id": "alpha",
  "material": "He",
  "file": "He/puchalski_2016__alpha.csv",
  "table": "tbl_f97b1bb8eafc",
  "quantity": "polarizability",
  "unit": "au",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table VII totals for 4He, all corrections (nonrelativistic with finite mass, 1/c^2, 1/(M c^2), 1/c^3, one-loop 1/c^4, finite nuclear size): the static polarizability 1.38376077(14) (row ‘static’, wavelength inf) and alpha(6329.908 A) = 1.39181197(14) (abstract and conclusions, p. 13). The two uncertainties are the same 0.14 u a0^3 (the 1/c^4 allowance), fully correlated. Level not stated. A check, not primary: the same information as cauchy_total (its row -2 is this static total, and its series gives 1.391811967 at 6329.908 A), which the helium fit uses; fitting both would count the theory twice.

puchalski_2016/alpha_piszczatowski_2015#

{
  "id": "alpha_piszczatowski_2015",
  "material": "He",
  "file": "He/puchalski_2016__alpha_piszczatowski_2015.csv",
  "table": "tbl_f97b1bb8eafc",
  "quantity": "polarizability",
  "unit": "au",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "piszczatowski_2015"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table VII ‘Ref. [13]’: 1.39181164(14), from Table I of Piszczatowski et al. (2015), which neglected the QED dispersion term alpha_2^(30) omega^2 = 0.32 u a0^3; superseded by this paper’s 1.39181197(14), ‘slightly outside the uncertainty range given in Ref. [13]’ (p. 2).

puchalski_2016/alpha_bhatia_drachman#

{
  "id": "alpha_bhatia_drachman",
  "material": "He",
  "file": "He/puchalski_2016__alpha_bhatia_drachman.csv",
  "table": "tbl_f97b1bb8eafc",
  "quantity": "polarizability",
  "unit": "au",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "bhatia_1998"
  ]
}

Table VII ‘BD’: 1.391780800, computed by the authors from Bhatia and Drachman’s (1998) coefficients correctly converted from reduced Rydberg units, with BD’s static value 1.383729929 (note f); not printed by Bhatia and Drachman. No QED; 22 ppm below this paper’s value (p. 10).

puchalski_2016/cauchy_nonrelativistic#

{
  "id": "cauchy_nonrelativistic",
  "material": "He",
  "file": "He/puchalski_2016__cauchy_nonrelativistic.csv",
  "table": "tbl_047bf9b3647c",
  "quantity": "cauchy_moment",
  "unit": "au",
  "abscissa": "order",
  "abscissa_unit": "1",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table II ‘Present’: nonrelativistic dispersion coefficients alpha_k^(0), k = 0, 2, 4, 6, of 4He (finite nuclear mass included exactly), atomic units a0^3 t0^k (not reduced units), stored as the Cauchy moments S(-k-2) (alpha_k = 2<z R^(k+1) z> = S(-k-2)): rows -2, -4, -6, -8. Uncertainties in the last digit, half the difference of the two largest SO bases (p. 7). A check, not primary: these omit the relativistic and QED corrections (static: -80.45 + 30.66 + 0.56 u a0^3, i.e. -3.6e-5 relative); the corrected totals are the dataset cauchy_total.

puchalski_2016/cauchy_nonrelativistic_higher#

{
  "id": "cauchy_nonrelativistic_higher",
  "material": "He",
  "table": "text",
  "quantity": "cauchy_moment",
  "unit": "au",
  "abscissa": "order",
  "abscissa_unit": "1",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

4 inline points in the source record.

p. 7 (Sec. VI A): ‘Using the SO procedure, we also calculated the next four coefficients: alpha_8^(0) = 4.39500532(1), alpha_10^(0) = 6.7725956(1), alpha_12^(0) = 10.622083(1) and alpha_14^(0) = 16.86118(1) a.u.’ (checked on the pdf text), nonrelativistic 4He, stored as S(-10) ... S(-16). The same values are printed in Piszczatowski et al. (2015), p. 3.

puchalski_2016/cauchy_total#

{
  "id": "cauchy_total",
  "material": "He",
  "table": "text",
  "quantity": "cauchy_moment",
  "unit": "au",
  "abscissa": "order",
  "abscissa_unit": "1",
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

8 inline points in the source record.

The total 4He dispersion coefficients with every correction the paper gives, as the Cauchy moments S(-k-2) = alpha_k in atomic units (a0^3 t0^k): COMPUTED HERE from printed components (2026-09-24), not printed as such. Row -2: the static total of Table VII, 1.38376077(14) (nonrelativistic 4He + 1/c^2 + 1/(M c^2) + 1/c^3 incl. the field derivative of ln k0 + one-loop 1/c^4 + finite nuclear size), its printed uncertainty. Rows -4, -6, -8: alpha_k = alpha_k^(0) (Table II, 4He finite mass) + alpha_k^(20) (Table IV, infinite mass) + alpha_k^(21) (Table IV, 4He recoil, k = 2 only; the paper takes it negligible for k >= 4, p. 5) + alpha_k^(30) (Table VIII ‘Total’ 1/c^3 rows, infinite mass): alpha_2 = 1.54321081882 + (-33.06576 - 0.144 + 61.9014)e-6 = 1.54323951046; alpha_4 = 2.0426550150 + (5.05105 + 116.988)e-6 = 2.04277705405; alpha_6 = 2.9304069980 + (62.8500 + 216.89)e-6 = 2.9306867380 (exact decimal sums, digits kept; the uncertainty is in the sixth decimal). Their sigmas add in quadrature the printed basis-set uncertainties (Table II: 1e-11, 1e-10, 7e-10; Table IV: 2e-11, 1e-9 (recoil), 5e-11, 5e-10) and 10 % of alpha_k^(30) (‘the uncertainties of alpha_k^(30), k = 2, 4, 6 are assumed to be 10 %’, p. 10): 6.19e-6, 1.170e-5, 2.169e-5, rounded to 0.0000062, 0.0000117, 0.0000217; the QED term dominates. The one-loop 1/c^4 dispersion components of Table VIII (alpha_2^(40) = 1.1565, alpha_4^(40) = 2.196, alpha_6^(40) = 4.08 u a0^3 t0^k) are, like the paper’s recommended value, NOT included and not added to the sigma (each is below 20 % of the stated QED uncertainty). Rows -10 ... -16: the nonrelativistic alpha_8 ... alpha_14 (p. 7; 4.39500532(1), 6.7725956(1), 10.622083(1), 16.86118(1)), uncorrected; their sigma adds to the printed last-digit uncertainty the size of the omitted relativistic + QED correction estimated from the lower orders: the relative total correction r_k = (alpha_k^(20) + alpha_k^(21) + alpha_k^(30))/alpha_k^(0) is 1.86e-5, 5.97e-5 and 9.55e-5 for k = 2, 4, 6, extrapolated linearly in k from k = 4, 6 to 1.31e-4, 1.67e-4, 2.03e-4, 2.38e-4 for k = 8-14, times alpha_k^(0): 5.8e-4, 1.13e-3, 2.15e-3, 4.02e-3, rounded to 0.00058, 0.0011, 0.0022, 0.0040. These corrections are positive for k = 2-6, so the four uncorrected higher moments are probably low by about their sigma (a bias, not only a spread). Check: with omega = 455.633525275/6329.908 the series of these eight moments gives alpha(6329.908 A) = 1.391811967, Table VII’s 1.39181197(14). Correlations: all rows share the static-to-dispersion QED model; row -2’s 0.14 u is the 1/c^4 allowance, independent of the others.

puchalski_2016/cauchy_bhatia_drachman#

{
  "id": "cauchy_bhatia_drachman",
  "material": "He",
  "file": "He/puchalski_2016__cauchy_bhatia_drachman.csv",
  "table": "tbl_047bf9b3647c",
  "quantity": "cauchy_moment",
  "unit": "au",
  "abscissa": "order",
  "abscissa_unit": "1",
  "use": "reproduced",
  "reproduced_from": [
    "A. K. Bhatia and R. J. Drachman, J. Phys. B 27, 1299 (1994)",
    "A. K. Bhatia and R. J. Drachman, Can. J. Phys. 75, 11 (1997)"
  ]
}

Table II ‘BD’: Bhatia and Drachman’s nonrelativistic alpha_k^(0) converted by the authors from reduced Rydberg to atomic units; rows -2 and -4 (alpha_0, alpha_2) from J. Phys. B 27, 1299 (1994) (note a), rows -6 and -8 (alpha_4, alpha_6) from Can. J. Phys. 75, 11 (1997) (note b). Stored as S(-k-2). Their alpha_4 ‘is accurate only to about four digits’ (p. 7).

puchalski_2016/molar_polarizability#

{
  "id": "molar_polarizability",
  "material": "He",
  "file": "He/puchalski_2016__molar_polarizability.csv",
  "table": "tbl_a088e2718911",
  "quantity": "molar_refractivity",
  "unit": "cm3/mol",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table IX ‘present’: A_eps = 4 pi N_A alpha/3 (the package’s molar refractivity), static 0.51725413(6) (uncertainties from alpha, N_A = 6.02214129(27)e23 and a0 added in squares, note a) and at 6329.908 A 0.52026369(6). The same numbers as dataset ‘alpha’ converted (reproduced to 1e-9 with CODATA 2010, checked 2026-09-24): a check, not fitted beside it. Converted back with today’s N_A (CODATA 2018, 8.8e-8 smaller) the loader’s alpha is 1e-7 above Table VII’s.

puchalski_2016/molar_lach_2004#

{
  "id": "molar_lach_2004",
  "material": "He",
  "file": "He/puchalski_2016__molar_lach_2004.csv",
  "table": "tbl_a088e2718911",
  "quantity": "molar_refractivity",
  "unit": "cm3/mol",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "lach_2004"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table IX ‘Ref. [2]’: static A_eps 0.517 254 19(10) of Lach, Jeziorski and Szalewicz (2004) (who print 0.517 254 19(9)(4)).

puchalski_2016/molar_schmidt_2007#

{
  "id": "molar_schmidt_2007",
  "material": "He",
  "file": "He/puchalski_2016__molar_schmidt_2007.csv",
  "table": "tbl_a088e2718911",
  "quantity": "molar_refractivity",
  "unit": "cm3/mol",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "schmidt_2007"
  ],
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  }
}

Table IX ‘expt. [8]’: Schmidt et al.’s (2007) microwave-resonator A_eps 0.5172535(47) (their magnetic term subtracted), ‘eight times smaller than the experimental uncertainty’ from theory (p. 11).

puchalski_2016/molar_achtermann#

{
  "id": "molar_achtermann",
  "material": "He",
  "file": "He/puchalski_2016__molar_achtermann.csv",
  "table": "tbl_a088e2718911",
  "quantity": "molar_refractivity",
  "unit": "cm3/mol",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "achtermann_1991",
    "H. J. Achtermann, J. G. Hong, G. Magnus, R. A. Aziz and M. J. Slaman, J. Chem. Phys. 98, 2308 (1993)"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table IX ‘expt. [41,42]’: 0.5213(1). Note b (checked on the pdf text; the library’s Markdown loses part of it): ‘The value listed in Refs. [41,42] is A_eps(6329.9 A) + A_mu(6329.9 A) at T = 323 K. Since A_mu(0) = -0.0000079224 cm3/mol, see text, the quoted value is independent of this correction.’ The discrepancy from theory is ‘ten times larger than their uncertainty’ (p. 11).

puchalski_2016/molar_birch_1991#

{
  "id": "molar_birch_1991",
  "material": "He",
  "file": "He/puchalski_2016__molar_birch_1991.csv",
  "table": "tbl_a088e2718911",
  "quantity": "molar_refractivity",
  "unit": "cm3/mol",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "birch_1991"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Table IX ‘expt. [43]’: 0.5220(3). Note c: ‘We took the polarizability listed in Table 1 of Ref. [43] (at the wavelength of 6329.9 A) and assumed it has the same relative uncertainty as the refractivity in Table 2 of that reference.’ (Birch prints 0.8668e-30 m^3 per molecule, i.e. 0.52200 cm^3/mol; the uncertainty is Puchalski’s, not Birch’s.)

puchalski_2016/dielectric_virial_rizzo#

{
  "id": "dielectric_virial_rizzo",
  "material": "He",
  "table": "text",
  "quantity": "refractivity_virial_coefficient",
  "unit": "cm6/mol2",
  "abscissa_unit": "angstrom",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "A. Rizzo, C. Haettig, B. Fernandez and H. Koch, J. Chem. Phys. 117, 2609 (2002)"
  ],
  "parameters": {
    "temperature_k": 273.16
  }
}

1 inline points in the source record.

p. 12: ‘Interpolating between the values at T = 261 and 274 K, we get B_eps(273.16 K) = -0.0506 cm6/mol2’, the static second dielectric virial coefficient B_eps = A_eps b_eps of Rizzo et al. (quantum statistical mechanics), from which b_eps = -0.0978(4) cm3/mol; the assumed uncertainty of b_eps is 0.4 % (p. 12). Row id is the temperature.

Formulas#

puchalski_2016/nonrelativistic_series#

{
  "id": "nonrelativistic_series",
  "form": "alpha_series",
  "terms": {
    "c": [
      "1.383809986408",
      "1.54321081882",
      "2.0426550150",
      "2.9304069980",
      "4.39500532",
      "6.7725956",
      "10.622083",
      "16.86118"
    ]
  },
  "range_nm": [
    150.0,
    "inf"
  ],
  "material": "He",
  "equation": "Eq. (2), p. 2, with the Table II coefficients (p. 7) and alpha_8..alpha_14 from the text (p. 7)",
  "printed": "alpha(omega) = alpha_0 + alpha_2 omega^2 + alpha_4 omega^4 + ...",
  "variable": "omega: angular frequency in atomic units (1/t0, t0 = hbar^3/(m_e e^4); not reduced units), omega = 455.633525275/lambda with lambda the vacuum wavelength in A; alpha in a0^3",
  "state": "none (per atom)",
  "density": "none",
  "range": "The paper evaluates the series only at 6329.908 A. The series converges for omega below the first resonance (1 1S - 2 1P, lambda = 58.4 nm); with the eight printed terms the truncation (next term about 1.6 times the last coefficient times omega^16) is below 1e-7 relative for lambda > 150 nm: that lower limit is this record's choice, not the paper's.",
  "check": "At 6329.908 A the terms are 1.383809986408, 0.0079957979, 0.0000548363, 0.00000040760, 0.0000000031674: Table VII's nonrelativistic rows 1.3838099864, 0.007995798, 0.000054836, 0.000000408, 0.0000000032 to all printed digits (checked 2026-09-24)."
}

Nonrelativistic 4He only (finite nuclear mass exact): it lies 3.6e-5 above the recommended static value (the relativistic and QED corrections of the formula ‘recommended’ are missing). Kept as the structured nonrelativistic model; for accuracy use ‘recommended’.

puchalski_2016/table6_total#

{
  "id": "table6_total",
  "material": "He",
  "equation": "Eq. (3), p. 2, with Table VI 'Present, total' (p. 9)",
  "printed": "alpha(lambda) = A_0 + A_2 lambda^-2 + A_4 lambda^-4 + A_6 lambda^-6, A_0 = 1.3837295330(1), A_2 = 3.2036661813(3) x 10^5, A_4 = 8.803569264(2) x 10^10, A_6 = 2.6219915496(7) x 10^16",
  "variable": "lambda: vacuum wavelength in A; alpha in a0^3; A_k in a0^3 A^k",
  "state": "none (per atom)",
  "density": "none",
  "range": "As 'nonrelativistic_series'",
  "coefficients": {
    "A0": "1.3837295330",
    "A2": "3.2036661813e5",
    "A4": "8.803569264e10",
    "A6": "2.6219915496e16"
  },
  "check": "A_k = f^k alpha_k with f = 455.633525275 reproduces all Table VI entries from Tables II and IV (A_2 relativistic -6.894409 including the recoil -0.0299 of note b; checked 2026-09-24)."
}

4He, nonrelativistic plus 1/c^2 relativistic (with the 1/(M c^2) recoil for A_0 and A_2), no QED: 3e-5 below the recommended static value; a comparison with Bhatia and Drachman, not the recommended model. No form fits: it is a polarizability series in lambda^-2 with lambda in A (a form ‘alpha_wavelength_series’, alpha[a0^3] = sum_k A_k lambda^(-2k), lambda in A, would take it and Bhatia and Drachman’s Eq. 15). Table VI’s BD column (A_0 1.383809991 / -0.000080062 / 1.383729929; A_2 3.2037352e5 / -0.0000677e5 / 3.2036675e5; A_4 8.8032133e10 / 0.0000207e10 / 8.8032341e10; A_6 2.621934e16 total) is Bhatia and Drachman’s Eqs. (12)-(13) correctly converted; see bhatia_1998.

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.