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 `__ · :doc:`Catalogue ` · :doc:`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 \ \(Eq\. 10\), 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 ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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\ = 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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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/recommended ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "recommended", "material": "He", "equation": "Eq. (2), p. 2, with Tables II, IV, VII and VIII (pp. 7-10)", "printed": "alpha(omega) = alpha_0 + alpha_2 omega^2 + alpha_4 omega^4 + ..., alpha_k = alpha_k^(0) + alpha_k^(20) + alpha_k^(21) + alpha_k^(30) + ... (Eqs. 4-5)", "variable": "omega: angular frequency in atomic units (1/t0), omega = 455.633525275/lambda with lambda the vacuum wavelength in A; alpha in a0^3", "state": "none (per atom)", "density": "none", "range": "As 'nonrelativistic_series' (the paper evaluates it at 6329.908 A only)", "coefficients": { "alpha_0": "1.38376077", "alpha_2_nonrel": "1.54321081882", "alpha_2_rel": "-33.06576e-6", "alpha_2_recoil": "-0.144e-6", "alpha_2_qed": "61.9014e-6", "alpha_4_nonrel": "2.0426550150", "alpha_4_rel": "5.05105e-6", "alpha_4_qed": "116.988e-6", "alpha_6_nonrel": "2.9304069980", "alpha_6_rel": "62.8500e-6", "alpha_6_qed": "216.89e-6", "alpha_8": "4.39500532", "alpha_10": "6.7725956", "alpha_12": "10.622083", "alpha_14": "16.86118" }, "check": "Summed (computed here, not printed): alpha_2 = 1.54323951046, alpha_4 = 2.04277705405, alpha_6 = 2.930686738 a0^3 t0^k. With alpha_0 = 1.38376077 (Table VII static total) and alpha_8..alpha_14 nonrelativistic, the series gives 1.391811967 at 6329.908 A (Table VII: 1.39181197(14)) and alpha(lambda) - alpha_0 = 0.008051197 (Table VII: 0.00805120(3)). Checked 2026-09-24." } Not a printed series\: the coefficients are sums of printed components \(4He finite\-mass nonrelativistic Table II; infinite\-mass relativistic Table IV; 4He recoil Table IV, k = 0, 2 only; infinite\-mass QED 1/c^3 Table VIII 'Total' rows; the static one\-loop 1/c^4 and finite\-size terms only in alpha\_0\)\. The alpha\_series form would take it if its coefficients were given as these sums \(a form accepting a list of components per coefficient, or precomputed sums\)\. The dispersion QED terms carry a 10 % uncertainty \(p\. 10\); the dispersion 1/c^4 one\-loop terms \(Table VIII, alpha\_2^\(40\) = 1\.1565 u a0^3 t0^2\) are not included by the paper\. puchalski\_2016/table6\_total ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 :doc:`review register ` for the checks and decisions.