VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995 

Ab Initio Calculations for Helium: A Standard for Transport Property Measurements 

Ronald A. Aziz and Alec R. Janzen 

Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3Gi, Canada 

Michael R. Moldover 

Thermophysics Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, 

Gaithersburg, Maryland 20899 

(Received 29 September 1994) 

For helium, the accuracy of calculated transport properties and virial coefficients based on an accurate 

ab initio potential now exceeds that of the best measurements. The ab initio results should be used to 

calibrate measuring apparatus. 

PACS numbers: 51.30.+i, 05.60.+w, 05.70.Ce, 34.20.Cf 

Traditionally, accurate measurements of second virial 

coefficients and transport properties have, in part, been 

used to determine parameters in evermore-refined models 

for interatomic and intermolecular potentials [1-4]. It 

is the purpose of this Letter to point out that, in the 

case of helium, calculations of virial coefficients and 

transport properties based on an accurate state-of-theart ab initio potential are now sufficiently accurate that 

the traditional practices can be usefully reversed. If the 

new theoretical results for helium are used to calibrate 

apparatus that are used to measure virial coefficients and 

transport properties, the accuracy of helium-based gas 

thermometry will be improved, as will measurements of 

the viscosity and thermal conductivity of gases of more 

complex atoms and molecules. The present suggestion 

resolves some controversy about the quality of data from 

different laboratories and it circumvents the unfortunate 

circumstance that the apparatus that was used to obtain 

the most accurate viscosity data for gases near ambient 

temperature is no longer in operation. 

An "ab initio" standard is now possible because, after many years of effort, very accurate point-by-point 

ab initio values of the ground state potential have become available. First, van Mourik and van Lenthe [5] 

recently provided full configuration-interaction (FCI) calculations of the potential at separations of 4.0ao, 5.0ao, 

5.6ao, 6.0ao, 6.5ao, 8.5ao, and 12.0ao with very tight error bars. (ao = Bohr radius = 0.0529177249 nm.) Second, Ceperley and Partridge [6] have employed quantum 

Monte Carlo methods to determine the exact BornOppenheimer interaction energy of two helium atoms with 

separations between 1.0ao and 3.0ao. Finally, accurate 

and self-consistent calculations of the dispersion coefficients have been provided by Thakkar [7] and Koide, 

Meath, and Allnatt [8] which define the long-range part 

of the potential. 

We constructed an analytical representation of the potential using the HFD-B (Hartree-Fock-dispersion) form 

[3] and adjusted parameters so that the representation runs 

nearly through the ab initio points. (For example, the 

representation misses the FCI result by 0.073%, where 

±0.052% is the standard uncertainty in the FCI result at 

6ao. The corresponding values at 6.5ao are -0.071 % and 

±0.029%.) In this way, we characterized the interaction 

energy without experimental input. The analytical HFD-B 

potential is dubbed the "HFD-B3-FCII" and its parameters are presented in Table I. It is found to support a very 

weakly bound state with a binding energy Eh of 1.59 mK. 

To calculate the viscosity and thermal conductivity, we 

used fourth-order Chapman-Cowling approximation expressions because fourth- and fifth-order calculations of 

the viscosity near ambient temperature differed by only 

0.0005%. (If we had used the second-order Kihara approximation [1], the viscosity would have differed by 

only 0.027%.) The most important aspect in the calculation is the evaluation of accurate quantum collision 

integrals. Quantum temperature-dependent reduced cross 

sections are calculated using an adaptation [9] of the classical reduced collision integral program of 0'Hara and 

Smith [10]. In this approach, a Chebyshev approximation 

is found for the quantal cross sections which are calculated using standard formulas [11] including the appropriTABLE I. Parameters for HFD-B3-FCII He-He potential. 

(Extra digits are displayed to avoid round off error.) 

A* 1.86924404 X 105 

a* 10.5717543 

C6 1.35186623 

Cs 0.41495143 

CIO 0.17151143 

C6 (J nm6) 1.39868078 X 10-25 

Cs (J nmS) 3.78266373 X 10-- 27 

C IO (J nm 10) 1.377 555 54 X 10-2H 

f3* -2.077 58779 

f3 (nm -2) -23.5800000 

D 1.43800000 

E/kB (K) 10.9560000 

r m (nm) 0.29683000 

(J' (nm) 0.264 138 13 

1586 0031-9007/95/74(9)/1586(4)$06.00 © 1995 The American Physical Society 
VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995 

ate spin and statistical effects [12]. The required phase 

shifts are obtained using the quantal phase shift routine 

of LeRoy [13] which uses a numerical integration and a 

Gaussian quadrature of the WKB correction to the phase 

shift. The Clenshaw-Curtis [14] quadrature was used to 

perform the energy integration in the collision integrals to 

a specified uncertainty of 0.01 %. 

Curtiss maintains that bound state trajectories in both 

the bound and unbound regions lead to significant "corrections" to low-density transport properties. For gaseous 

argon he calculates corrections [15] on the order of 1% 

when T* = T /(E/kB ) = 1. (Here, E is the Lennard-Jones 

energy parameter.) However, at ambient temperature, 

where we recommend helium be used as a standard, 

T* 29 and the effects of bound states are negligible. 

Furthermore, in contrast with argon, helium has a single 

very weakly bound state (Eb = 1.59 X 10-3 K) and no 

quasibound states. 

To check our evaluation of the quantal collision integrals, we used an independent procedure due to Hanson [16]. The results from the two procedures differed 

at most by 0.0025% of the collision integrals necessary 

for the evaluation of viscosity and thermal conductivity at 

298.15 K. 

For calibrations, Wakeham, Nagashima, and Sengers 

[17] recommend experimentally derived primary reference values for the thermal conductivity A and viscosity YJ near ambient temperature and pressure. Thus, we 

shall pay particular attention to near-ambient conditions 

as we compare the ab initio results with experiments. 

Near 300 K, the computed values of A and 'Y} are most 

sensitive to the value of the ab initio potential at 4.0ao, 

where VCr) is known especially accurately: V(4.0ao) = 

249.90 ± 0.3 K. The 0.3 K standard uncertainty in VCr) 

propagated into a standard uncertainty of only ±0.02% in 

the values of A and 'Y} and ±0.004 cm3 mol- 1 in the value 

of the second virial coefficient B at 298.15 K. This uncertainty in the theoretical results was the dominant one 

affecting the properties near ambient temperature. Our 

results at 298.15 K are A = 154.81 ± 0.08 mWm- 1 K- 1 

and'Y} = 19.800 ± 0.010 ,uPa s. For B, an additional possible uncertainty of 0.004 cm3 mol- 1 due to thermal averaging [18] is included. This results in B = 11.846 ± 

0.008 cm3 mol-l . 

Probably the most accurate measurement of thermal 

conductivity is the hot-wire value of Assael et al. [19] 

at 308.15 K. Remarkably, this value is only 0.05% larger 

than the ab initio value and the difference is well within 

the experimental uncertainty of ±0.2%. At nearly the 

same temperature (298.15K), Kestin, Ro, and Wakeham 

[20] measured YJ = 19.859 ± 0.020 ttPa s. Their result 

exceeds the ab initio value by 0.059 ttPa s or 0.3%, an 

amount that exceeds their estimated experimental uncertainty of ±0.1 %. 

Recall that both A and YJ depend on the same collision 

integral in the first-order Chapman-Cowling approximation and hence a potential which predicts an accurate 

value for one should also predict an accurate value for 

the other in fourth order. At 298.15 K, the ab initio 

value for the dimensionless Eucken ratio 'Y}m/ AC v = 

1.0037. This ratio is not sensitive to details of the 

potential. For hard spheres, 'Y} m / Acv == 1. (Here m is 

the mass of a helium atom and C v is constant-volume 

heat capacity per atom.) The Eucken ratio obtained by 

combining data from [19] and [20] is 1.0013 ± 0.0027. 

We conjecture that the uncertainty in the viscosity data 

from [20] was underestimated and that, perhaps by 

chance, the experimental value of A is more accurate than 

the experimental value of 'Y}. 

To further test the ab initio potential, we made extensive comparisons with thermophysical-properties data 

at other temperatures and with scattering data. Representative comparisons for YJ and A are shown in Figs. 1 

and 2 and representative comparisons for B are made in 

Tables II and III. 

As shown in Fig. 1, Vogel's viscosity results [21] 

track the temperature dependence of the ab initio results with extraordinary precision (±0.05% from 298 

to 623 K). However, Vogel's viscometer [21] was calibrated with the primary data of Kestin, Ro, and Wakeham 

[20]. This accounts for most of the 0.22% deviation of 

Vogel's results from the ab initio results at 298 K. The 

data of Clarke and Smith [22] and of Johnston and Grilly 

[23] agree with the ab initio results within their respective 

experimental uncertainties. As published, the low tem100 

a... 10 

10 100 1000 

x 

o 

in----+-------il-----r-------j 

x 

x 

x 

ot-----

x 

-1 

10 100 1000 

TEMPERATURE / K 

FIG. 1. Ab initio viscosity 1] and deviations of the measured 

1] from the ab initio YJ. Data sources: X Becker and Misenta 

[25] rescaled at 77.2 K; D Coremans et at. [24] rescaled at 

20.4 K; + Johnston and Grilly [23]; <> Clarke and Smith [22]; 

• Vogel [21]; 0 Kestin, Ro, and Wakeham [20]. 

1587 
VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995 

. 

g 100 . 

E 

c-< 10 

10 100 1000 

1.5 cP 0 

0 

1.0 0 0 c< 

--. 0 c< 

<1 0.5 0 0 Cb 0 

0.0 

-0.5 

10 100 1000 

TEMPERATURE / K 

FIG. 2. Ab initio thermal conductivity A and deviations of 

the measured A from the ab initio A. Data sources: 0 Acton 

and Kellner [27]; • Assael et alA [19]; X Haarman [26]; 0 

Wakeham, Nagashima, and Sengers [17]. 

perature results of Coremans et al. [24) and of Becker and 

Misenta [25] disagree with the ab initio results. However, 

both groups used calibration data that must be questioned. 

Thus, prior to plotting the data from [24], we scaled them 

to the ab initio results at 20.4 K. Similarly, we rescaled 

the data from [25] at 77.2 K. After rescaling, these data 

parallel the ab initio results within their uncertainties. 

Figure 2 displays the thermal conductivity data recommended by Wakeham, Nagashima, and Sengers [17] and 

the data of Haarman [26], of Assael et alA [19], and of Acton and Kellner [27]. In all of these cases, the data and 

the ab initio results are in satisfactory agreement. 

Tables II and III compare the ab initio results with the 

accurate second virial coefficient data from the various 

standards laboratories. The data were painstakingly acquired to account for the nonideality of the helium used in 

the gas thermometers that are used to define the ITS-90 

temperature scale from 3.0 to 24.5561 K. The method 

of calculating B (taking into account the binding energy 

contribution in the case of 4He) is described in Aziz and 

Slaman [3). The theoretical potential predicts the 3 He 

second virials of Matacotta et alA [28] (third virial corrected) to 0.1 cm3 mol- l and Berry's smoothed 4He 

virials [29] within ±O.3 cm3 mol-I. Almost all of the 

measured values of Kemp et al. [30] are predicted to 

within their combined random and systematic experimental uncertainties even after ignoring the influence 

of the systematic uncertainty due to the chosen reference point (Berry's value at 20.271 K). All the Gammon 

[31] points from 123 to 423 K are predicted to within 

±O.04 cm3 mol- l . The data of Blancett, Hall, and CanTABLE II. The measured second virial coefficient Bexpt of 

4He, its uncertainty ~B, and the difference between the 

experimental and calculated values in units of cm3 mol-I. 

T(K) Bexpt Bexpt - Beale Ref. 

2.600 -142.295 1.0 0.053 [29] 

2.750 -133.230 0.7 0.002 [29] 

3.330 -106.118 0.7 0.060 [29] 

4.220 -79.392 0.5 0.254 [29] 

7.200 -38.764 0.7 0.344 [29] 

13.800 -11.830 0.4 0.076 [29] 

20.271 -2.472 0.2 -0.003 [29] 

27.100 2.505 0.2 0.041 [29] 

27.097 2.470 0.06a 0.26b 0.008 [30] 

43.794 7.800 0.11 a 0.31 b 0.029 [30] 

54.358 9.280 0.07a 0.27b -0.024 [30] 

63.150 10.010 0.09a 0.29b -0.117 [30] 

83.804 11.130 0.11 a 0.31 b -0.108 [30] 

172.011 12.150 0.14a 0.26b -0.062 [30] 

98.15 11.522 0.05 -0.128 [31] 

123.15 11.984 0.05 -0.041 [31] 

148.15 12.174 0.05 -0.002 [31] 

173.15 12.228 0.05 0.016 [31] 

198.15 12.209 0.05 0.024 [31] 

223.15 12.148 0.05 0.025 [31] 

248.15 12.065 0.05 0.024 [31] 

273.15 11.968 0.05 0.021 [31] 

298.15 11.863 0.05 0.018 [31] 

323.15 11.756 0.05 0.014 [31 ] 

348.15 11.647 0.05 0.010 [31 ] 

373.15 11.539 0.05 0.007 [31] 

398.15 11.433 0.05 0.003 [31] 

423.15 11.329 0.05 0.001 [31] 

223.15 12.16 0.03 0.037 [32] 

273.15 11.94 0.03 -0.006 [32] 

323.15 11.76 0.03 0.018 [32] 

298.15 11.83 0.03 -0.016 [33] 

623.15 10.61 0.01 0.012 [33] 

3.00 -120.522 1.0 -0.314 [39] 

4.00 -85.200 0.7 -0.126 [39] 

5.00 -64.316 0.7 -0.002 [39] 

8.00 -33.283 0.5 0.112 [39] 

10.00 -23.020 0.7 0.104 [39] 

15.00 -9.435 0.4 0.050 [39] 

20.00 -2.723 0.2 0.014 [39] 

27.00 2.412 0.3 0.001 [39] 

aRandom and systematic uncertainties only. 

bRandom, systematic, and reference point uncertainties. 

field [32] and the data of Kell, McLaurin, and Whalley 

[33] at 298.15 and 623 K are predicted within experimental uncertainty. 

It is reassuring that the HFD-B3-FCI1 potential, which 

had no experimental input, is able to predict the "standard" 3 He and 4He virials with significantly smaller deviations than any of the recent "experimental" or model 

potentials [3,4,9,34-36]. 

We have verified that the ab initio results reproduce 

both the total [37] and differential [38] 4He-4 He scatter1588 
VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995 

TABLE III. The measured second virial coefficient Bexpt of 

3He, its uncertainty ~B, and the difference between the 

experimental and calculated values in units of cm3 mol-I. 

T (K) Bexpt ~B B expt - Beale Ref. 

1.47 -173.58 0.5 -0.021 [28] 

1.50 -170.54 0.5 -0.008 [28] 

1.55 -165.68 0.5 0.004 [28] 

1.73 -149.99 0.4 0.051 [28] 

1.90 -137.37 0.4 0.081 [28] 

2.20 -119.12 0.5 0.076 [28] 

3.00 -86.31 0.5 0.010 [28] 

4.30 -57.15 0.4 -0.021 [28] 

9.00 -19.75 0.3 -0.008 [28] 

13.80 -7.33 0.2 -0.079 [28] 

20.30 0.24 0.2 -0.125 [28] 

ing cross section data as well as or better than the model 

or semiempirical potentials that have been used in the past 

[4,9]. These comparisons will be discussed elsewhere. 

We thank T. van Mourik and J. H. van Lenthe for permission to use their ab initio results prior to publication. We thank Professor R. J. LeRoy for the use of his 

phase-shift-time-delay program and J. S. Carley, E. Hanson, and M. J. Slaman for help in coding the quantum collision integral programs. This research was supported in 

part by a grant in aid of research from the Natural Sciences and Engineering Council of Canada (R. A. A.). 

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