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Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium
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10.1063/1.4712218
/content/aip/journal/jcp/136/22/10.1063/1.4712218
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/22/10.1063/1.4712218

Figures

Image of FIG. 1.
FIG. 1.

Basis-set convergence of the atomic expectation value of the Cowan-Griffin operator to the exact value equal to −114.317553 × 10−6 (Ref. 76). Computations were performed at the FCI level of theory using the dXZ family of basis sets with X = 3, …, 8. The line represents the best fit obtained from linear regression of the results for basis sets with X ⩾ 5.

Image of FIG. 2.
FIG. 2.

Basis-set convergence of the atomic expectation value of the two-electron Darwin operator to the exact value equal to −17.790950 × 10−6 (Ref. 76). Computations were performed at the FCI level of theory using the dXZ family of basis sets with X = 3, …, 8. The line represents the best fit obtained from linear regression of all the results.

Image of FIG. 3.
FIG. 3.

Basis-set convergence of the atomic expectation value of the Breit operator to the exact value equal to −7.406981 × 10−6 (Ref. 76). Computations were performed at the FCI level of theory and using the dXZ family of basis sets with X = 3, …, 8. The line represents the best fit obtained from linear regression of the results for basis sets with X ⩾ 5.

Image of FIG. 4.
FIG. 4.

Difference between the fit of the complete potential and the sum of the fits of individual components: V BO, V ad, V CG, V D2, V Br, and V QED. The gray area shows the range [− σ(R), +σ(R)], where σ(R) is the analytic fit of the total potential uncertainty from Ref. 70.

Image of FIG. 5.
FIG. 5.

Potential components at short and intermediate distances R. The ordinate scale is proportional to , which is approximately linear for small V and proportional to for large |V|. Top panel: the potential V of Eq. (2). The potential would be optically indistinguishable. Bottom panel: the post-BO components of V and the residual retardation correction V ret. The analytic fit of the uncertainty σ of the potential V is also shown.

Image of FIG. 6.
FIG. 6.

Comparison of the large-R behavior of the Born-Oppenheimer (V BO), adiabatic (V ad), Breit (V Br), and Araki-Sucher (V AS) contributions to the helium pair potential. The total relativistic (V rel), total QED (V QED), corrections would be optically indistinguishable from V Br and V AS, respectively. The retardation corrections appropriate for the nonrelativistic BO (), relativistic (), and QED (V ret) levels of theory are also shown.

Image of FIG. 7.
FIG. 7.

Helium dimer (4He2) bond length ⟨R⟩ (left axis) and dissociation energy D 0 (right axis) at different levels of theory, see text for the acronyms.

Image of FIG. 8.
FIG. 8.

Phase shifts for E = 10−5 computed with potentials V(R) (pluses) and (squares). Alternate points have been omitted for ℓ > 20; the lines show the Born approximation for potentials −C 3/R 3, −C 6/R 6, and .

Image of FIG. 9.
FIG. 9.

Absolute values of the phase shifts for E = 0.001 computed with potentials V(R) and . Negative phase shifts are plotted as points; positive phase shifts are plotted as lines to avoid excessive point densities. Other lines show the Born approximation for potentials −C 3/R 3 and −C 6/R 6.

Image of FIG. 10.
FIG. 10.

Density virial of 4He. Top: B(T). Center: Uncertainty σ B due to uncertainty of potential , differences between B(T) calculated with and the results of Hurly-Mehl (HM) and Bich et al. (BHV). Bottom: Effects of the post-BO contributions to the potential. See main text for definitions.

Image of FIG. 11.
FIG. 11.

Measured (squares)27 and calculated (line) values of B(T); the uncertainty of the calculated values is smaller than the width of the plotted line and much smaller than the uncertainty of the measurements.

Image of FIG. 12.
FIG. 12.

Comparison of the differences between B(T) from the measurements of Gaiser and Fellmuth134 and our values computed with with the sum of all post-BO effects, i.e., the difference . The theoretical uncertainties ±σ B due to the potential uncertainty are also shown.

Image of FIG. 13.
FIG. 13.

Acoustic virial of 4He. Top: βa(T), this work. Center: Effects of post-BO potential terms. Bottom: Differences between measurements of Pitre et al. 12 and Gavioso et al. 138 and calculations with , as well as the uncertainty due to the uncertainty in the potential.

Image of FIG. 14.
FIG. 14.

Viscosity of 4He. Top: η(T). Center: Relative differences between viscosities calculated with and the results of Hurly and Mehl125 and Bich et al. 127 Bottom: Effects of the post-BO contributions to the potential, the use of nuclear rather than atomic masses, and the uncertainty estimated from the uncertainty in .

Tables

Generic image for table
Table I.

The asymptotic information used to construct the potential.

Generic image for table
Table II.

Comparison of the ECG and orbital values of the adiabatic correction for 4He2 (in kelvin) computed for a set of internuclear distances R. The ECG results obtained with two largest bases, and , are shown (results obtained with two largest orbital bases are given in the supplementary material).112 See text for the definition of the extrapolated values and . The last column lists the absolute difference between the extrapolated values divided by the sum of their uncertainties σECG and σorb.

Generic image for table
Table III.

Comparison of the ECG and orbital values of the one-electron Darwin correction (in kelvin) computed for a set of internuclear distances R. The ECG results obtained with two largest bases, and , are shown (results obtained with two largest orbital bases are given in the supplementary material).112 See text for the definition of the extrapolated values and . The last column lists the absolute difference between the extrapolated values divided by the sum of their uncertainties σECG and σorb.

Generic image for table
Table IV.

Comparison of the ECG and orbital values of the two-electron Darwin correction (in kelvin) computed for a set of internuclear distances R. The ECG results obtained with two largest bases, and , are shown (results obtained with two largest orbital bases are given in the supplementary material).112 See text for the definition of the extrapolated values and . The last column lists the absolute difference between the extrapolated values divided by the sum of their uncertainties σECG and σorb.

Generic image for table
Table V.

Comparison of the ECG and orbital values of the Cowan-Griffin correction (in kelvin) computed for a set of internuclear distances R. The ECG results obtained with two largest bases, and , are shown (results obtained with two largest orbital bases are given in the supplementary material).112 See text for the definition of the extrapolated values and . The last column lists the absolute difference between the extrapolated values divided by the sum of their uncertainties σECG and σorb.

Generic image for table
Table VI.

Comparison of the ECG and orbital values of the Breit correction (in kelvin) computed for a set of internuclear distances R. The ECG results obtained with two largest bases, and , are shown (results obtained with two largest orbital bases are given in the supplementary material).112 See text for the definition of the extrapolated values and . The last column lists the absolute difference between the extrapolated values divided by the sum of their uncertainties σECG and σorb.

Generic image for table
Table VII.

ECG values of the Araki-Sucher correction (in kelvin) for a set of internuclear distances R. See text for the description of the extrapolations and definition of uncertainty σ.

Generic image for table
Table VIII.

Convergence of the adiabatic correction (in kelvin) at selected internuclear distances R. The column labeled “extr” contains the values extrapolated using Eq. (58), with the exponent n = 3 and X shown in the first column.

Generic image for table
Table IX.

Convergence of the relativistic corrections (in kelvin) computed using the dXZ bases. The columns “extr” contain the values of ΔY CCSD(T) extrapolated using Eq. (58), with X given in the first column and with the exponent n given in the table.

Generic image for table
Table X.

Parameters of the analytic fit of the post-BO corrections, see Eqs. (60) and (61). The symbol A(p) means A × 10 p .

Generic image for table
Table XI.

Root mean square errors (RMSE) and maximum and average ratios of the fit errors to the data point uncertainties for the fits of the individual components.

Generic image for table
Table XII.

Parameters of the analytic fit to the BO potential V BO. The constants C n for n = 6, 8, 10 were taken from Ref. 120, whereas the remaining ones from Ref. 111.

Generic image for table
Table XIII.

Comparison of the potential from Ref. 44 (denoted here as V HBV) with the present results (all energies are in kelvins). V is the total potential as defined by Eq. (2) with the uncertainties given in parentheses and V′ is the sum of the components which have been considered in Ref. 44, i.e., V′ = V BO + V ad + V CG. The symbol A +ret denotes the sum of a potential A and of an appropriate retardation correction [given by Eq. (44) for V HBV and by Eq. (46) for V].

Generic image for table
Table XIV.

Measured139,140 and calculated viscosity of 4He at 298.15 K. Viscosity reported in Ref. 127 was calculated using the potential of Ref. 44.

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/content/aip/journal/jcp/136/22/10.1063/1.4712218
2012-06-11
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/22/10.1063/1.4712218
10.1063/1.4712218
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