^{1}, Michał Przybytek

^{2}, Jacek Komasa

^{3}, James B. Mehl

^{4}, Bogumił Jeziorski

^{2}and Krzysztof Szalewicz

^{1,5}

### Abstract

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations *R* smaller than 4 bohrs and for the distance *R* = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near *R* = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller *R* (all *R* in the case of a QED component) and optimized orbital bases up to the cardinal number *X* = 7 for larger *R*. Near-exact asymptotic constants were used to describe the large-*R* behavior of all components. The fitted potential, exhibiting the minimum of −10.996 ± 0.004 K at *R* = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound ^{4}He_{2} dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 Å relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α^{2} and α^{3}, respectively. The remaining retardation effect, of the order of α^{4} and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.

This work was supported by a NIST Precision Measurement grant, by the NSF Grant No. CHE-0848589 and by the NCN Grants No. N-N204-182840 and N-N204-015338.

I. INTRODUCTION

II. METHODOLOGY OF EXPLICITLY CORRELATED CALCULATIONS

A. Electronic wave functions

B. Adiabatic corrections

C. Relativistic corrections

D. Quantum electrodynamics corrections

E. Regularization of singular operators

III. METHODOLOGY OF ORBITAL CALCULATIONS

IV. ASYMPTOTIC CONSTANTS

V. RETARDATION EFFECTS

VI. NUMERICAL CALCULATIONS

A. Calculations employing explicitly correlated basis sets

B. Calculations employing orbital basis sets

1. Basis sets

2. Extrapolation schemes

3. Adiabatic correction

4. Relativistic corrections

C. Calculations of asymptotic constants

D. Casimir-Polder potential

E. Comparison of ECG and orbital results

F. Analytic fits of potential components

VII. COMPARISON WITH LITERATURE

VIII. IMPORTANCE OF POTENTIAL COMPONENTS AT DIFFERENT SEPARATIONS

IX. BOUND STATE OF HELIUM DIMER

X. THERMOPHYSICAL PROPERTIES OF HELIUM

A. Convergence of density virial calculations

B. Density virial coefficient

C. Acoustic virial coefficient

D. Viscosity and thermal conductivity

E. Overall comparison of theory with experiment

XI. SUMMARY AND CONCLUSIONS

### Key Topics

- Quantum electrodynamic effects
- 109.0
- Basis sets
- 28.0
- Relativistic corrections
- 25.0
- Wave functions
- 23.0
- Viscosity
- 17.0

## Figures

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 d*X*Z 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.

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 d*X*Z 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.

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 d*X*Z family of basis sets with *X* = 3, …, 8. The line represents the best fit obtained from linear regression of all the results.

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 d*X*Z family of basis sets with *X* = 3, …, 8. The line represents the best fit obtained from linear regression of all the results.

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 d*X*Z 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.

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 d*X*Z 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.

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.

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.

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.

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.

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.

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.

Helium dimer (^{4}He_{2}) bond length ⟨*R*⟩ (left axis) and dissociation energy *D* _{0} (right axis) at different levels of theory, see text for the acronyms.

Helium dimer (^{4}He_{2}) bond length ⟨*R*⟩ (left axis) and dissociation energy *D* _{0} (right axis) at different levels of theory, see text for the acronyms.

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 .

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 .

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}.

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}.

Density virial of ^{4}He. 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.

Density virial of ^{4}He. 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.

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.

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.

Comparison of the differences between *B*(*T*) from the measurements of Gaiser and Fellmuth^{134} 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.

Comparison of the differences between *B*(*T*) from the measurements of Gaiser and Fellmuth^{134} 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.

Acoustic virial of ^{4}He. Top: β_{a}(*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.

Acoustic virial of ^{4}He. Top: β_{a}(*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.

Viscosity of ^{4}He. Top: η(*T*). Center: Relative differences between viscosities calculated with and the results of Hurly and Mehl^{125} 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 .

Viscosity of ^{4}He. Top: η(*T*). Center: Relative differences between viscosities calculated with and the results of Hurly and Mehl^{125} 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

The asymptotic information used to construct the potential.

The asymptotic information used to construct the potential.

Comparison of the ECG and orbital values of the adiabatic correction for ^{4}He_{2} (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 σ_{ECG} and σ_{orb}.

Comparison of the ECG and orbital values of the adiabatic correction for ^{4}He_{2} (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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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 σ_{ECG} and σ_{orb}.

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.

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.

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.

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.

Convergence of the relativistic corrections (in kelvin) computed using the d*X*Z bases. The columns “extr” contain the values of Δ*Y* ^{CCSD(T)} extrapolated using Eq. (58), with *X* given in the first column and with the exponent *n* given in the table.

Convergence of the relativistic corrections (in kelvin) computed using the d*X*Z bases. The columns “extr” contain the values of Δ*Y* ^{CCSD(T)} extrapolated using Eq. (58), with *X* given in the first column and with the exponent *n* given in the table.

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

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

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.

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.

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.

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.

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*].

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*].

Measured^{139,140} and calculated viscosity of ^{4}He at 298.15 K. Viscosity reported in Ref. 127 was calculated using the potential of Ref. 44.

Measured^{139,140} and calculated viscosity of ^{4}He at 298.15 K. Viscosity reported in Ref. 127 was calculated using the potential of Ref. 44.

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