Ab Initio Values of the Thermophysical Properties of Helium as Standards

Recent quantum mechanical calculations of the interaction energy of pairs of helium atoms are accurate and some include reliable estimates of their uncertainty. We combined these ab initio results with earlier published results to obtain a helium-helium interatomic potential that includes relativistic retardation effects over all ranges of interaction. From this potential, we calculated the thermophysical properties of helium, i.e., the second virial coefficients, the dilute-gas viscosities, and the dilute-gas thermal conductivities of 3He, 4He, and their equimolar mixture from 1 K to 104 K. We also calculated the diffusion and thermal diffusion coefficients of mixtures of 3He and 4He. For the pure fluids, the uncertainties of the calculated values are dominated by the uncertainties of the potential; for the mixtures, the uncertainties of the transport properties also include contributions from approximations in the transport theory. In all cases, the uncertainties are smaller than the corresponding experimental uncertainties; therefore, we recommend the ab initio results be used as standards for calibrating instruments relying on these thermophysical properties. We present the calculated thermophysical properties in easy-to-use tabular form.


Introduction
Today, the most accurate values of the thermophysical properties of helium at low densities can be obtained from two, very lengthy, calculations. The first calculation uses quantum mechanics and the fundamental constants to obtain, ab initio , a potential energy (r ) for the helium-helium (He 2 ) interaction at discrete values of the interatomic separation r and also limiting forms of (r ) at large r (see Fig. 1). The second calculation uses standard formulae from quantum-statistical mechanics and the kinetic theory of gases to obtain the thermophysical properties of low-density helium from (r ). Here, we report the results of the second calculation spanning the temperature range 1 K to 10 4 K for the second virial coefficient B (T ), the viscosity (T ), the thermal conductivity (T ), the mass diffusion coefficient D (T ), and the thermal diffusion factor ␣ T (T ) for 3 He, 4 He, and their equimolar mixture. Our results, together with estimates of their uncertainties, are presented in easy-to-use tabular form in Appendix A. For the pure fluids, the statistical-mechanics calculations make negligible contributions to the uncertainties of the tabulated properties; therefore, we estimated the uncertainties of the results by varying (r ) within its uncertainty and examining the consequences. For the equimolar mixture, the results from different orders of approximation in the transport theory are compared to estimate their contribution to the uncertainties.  [15]; Komasa [24]; ᭹ Korona et al. [18]; ᭺ van Mourik and Dunning [21]; ᭛ van de Bovenkamp and van Duijneveldt [20]; ᭝ Gdanitz [19].
The present results can be applied to many problems in metrology; here we mention a few. Low-density helium is used in primary, constant-volume, gas thermometry [1]; primary, dielectric-constant gas thermometry [2]; and in interpolating gas thermometry (required by ITS-90 in the temperature range 3 K to 24.6 K) [3]. These applications require the extrapolation of measurements to zero pressure. If the present values of B (T ) are used for such extrapolations, the results may be more accurate and the probability of detecting systematic errors in the measurements will be increased. Low-density helium can be used to calibrate acoustic resonators for acoustic thermometry and for measuring the speed of sound in diverse gases. Spherical acoustic resonators [4] may be calibrated using the present values of (T ), B (T ), and temperature derivatives dB /dT and d 2 B /dT 2 . The same properties together with (T ) may be used to calibrate cylindrical acoustic resonators [5]. Other instruments that might be calibrated with the help of the present results include the vibrating wire viscometer [6], the Greenspan acoustic viscometer [7], and the Burnett apparatus [8] for making very accurate measurements of the equation of state of moderately dense fluids.
The present work contrasts with a long tradition of using semi-empirical models for (r ) to correlate the thermophysical property data for helium and the other monatomic gases [9,10,11]. These semi-empirical models combined limited ab initio results with critically evaluated and judiciously selected experimental data to determine the function (r ) that correlates as much data as possible. In this work, we did not consider experimental results until all of the calculations were completed as in [12,13]. The ab initio results were then compared to the sets of data that others had selected as inputs to semi-empirical models. In every case that we examined, the ab initio values of the thermophysical properties agreed with the data within plausible estimates of their combined uncertainties.
This manuscript is organized as follows: Sec. 2 reviews the ab initio results for (r ) and our analytic representation of them. Section 3 outlines the steps in calculating the thermophysical properties of helium from (r ). Each step includes a description of the precautions that were taken to insure that imperfections of the numerical methods did not adversely affect the results. Section 4 estimates the uncertainty of the ab initio helium pair potential and how it propagates into the uncertainties of the calculated properties. Section 5 describes the tabulated results and methods for their use. Section 6 compares the calculated properties with selected measurements. Section 7 summarizes the present results and the prospects for future refinements.

Ab Initio Values for the He2 Potential
Energy Functions (r ) Table 1 lists recent ab initio values of (r ) at selected values of r (3.0 bohr, 4.0 bohr, and 5.6 bohr, where 1 bohr = 0.052917721 nm) and, where available, the uncertainties estimated by the original authors. As is conventional in this field, the potential energy is divided by k B K and thus has the unit K ( k B is the Boltzmann constant [14] and K is the unit symbol for the kelvin). The various calculations almost, but not quite, agree within their uncertainties. The discrepancies near  Table 1 is beyond the scope of this paper. Here, we mention the observations that guided our selection among the sources cited in Table 1 to obtain 00 (r ), the function that we used to calculate the thermophysical properties of helium.

Long-Ranges: r > ϳ 8 bohr
The asymptotic long-range attractive behavior of our preferred potential 00 (r ) is represented by the twobody dispersion coefficients C n (n = 6, 8, ...) in the multipole expansion. These coefficients have been calculated, ab initio , by two independent groups [22,23] using a sum-over-states formalism with explicitly electron-correlated wave functions to describe the states. The independent calculations [22,23] differed by less than 1 in the fourth digit. This small difference makes a negligible contribution to the uncertainties of the thermophysical properties calculated from 00 (r ).

Short-Ranges: r < ϳ 3 bohr
Ceperley and Partridge [15] obtained values of (r ) at small r using a quantum Monte Carlo (QMC) method. The QMC method is exact insofar as it requires no mathematical or physical approximations beyond those in the Schrödinger equation and the method yields estimates of the uncertainties of (r ). Komasa [24] used a variational method to obtain rigorous upper bounds to (r ) in the range 0.01 bohr Յ r Յ 15 bohr. At some values of r , the variational values of (r ) are less than the QMC values; however the differences between the values are usually within twice the QMC uncertainties. Thus, we used the variational values to determine 00 (r ) and we have evidence that the QMC uncertainties are reasonable. At smaller values of r the variational and QMC results are inconsistent. For example, at r = 1 bohr (not plotted), Komasa reports (1 bohr) = (286.44 Ϯ 0.03) ϫ 10 3 K, and Ceperley and Partridge report (1 bohr) = (291.9 Ϯ 0.6) ϫ 10 3 K. We are unable to resolve this inconsistency; however, the inconsistency does not affect the thermophysical properties in the temperature range 1 K to 10 4 K.
Komasa provides two values for the well depth at 5.6 bohr, /k B = Ϫ 10.947 K using a 1200-term basis set and /k B = Ϫ 10.978 K using a 2048 term basis set. The second value is 0.3 % lower. Komasa's calculations at other values of r used the 1200-term basis set. We speculate that comparable reductions in (r ) would occur if Komasa's variational calculation were repeated with the larger basis set at all values of r .

Intermediate Ranges: 3 > ϳ r > ϳ 8 bohr
At intermediate ranges, we considered the seven relevant publications cited in Table 1. Anderson et al. [16] report exact QMC results that have relatively large uncertainties. Klopper and Noga [17] used an explicitly correlated coupled cluster [CCSD(T)] method that resulted in the limiting value for the well depth of /k B = Ϫ 10.68 K at 5.6 bohr. Then, they estimated the effects of quadruple substitutions to be Ϫ0.32 K at 5.6 bohr (and Ϫ1.9 K at 4.0 bohr) by comparing their results to the full configuration interaction (FCI) calculation of van Mourik and van Lenthe [25]. This extrapolation to a complete basis set resulted in /k B = Ϫ(11.0 Ϯ 0.03) K, which agrees with the QMC results of Anderson [16].
Korona et al. [18] used symmetry-adapted perturbation theory (SAPT) to calculate values for (r ) with uncertainties that they estimated to be the larger of 0.3 % or 0.03 K in the range 3 bohr Յ r Յ 7 bohr. The SAPT well-depth is /k B = Ϫ(11.06 Ϯ 0.03) K, the lowest of all ab initio results; however, it also agrees with the QMC result [16] within the latter's uncertainty.
While this project was in progress, two groups extended the CCSD(T) calculations of Klopper and Noga [17]. These groups (de Bovenkamp and Duijneveldt [20]; and van Mourik and Dunning [21]) used different techniques to extrapolate the results of Klopper and Noga [17] to an infinite basis set. Gdanitz [19] also published calculations labeled r12-MRACPF in which he extrapolated his results to an infinite basis set by yet another method. These three recent publications and the variational results of Komasa [24] indicate that the SAPT [18] results in the region around r = 4.0 bohr are too attractive by approximately 0.05 K (Fig. 1, lower panel). Nevertheless, we used the SAPT intermediaterange results in determining the potential 00 (r ) and we used the differences between the SAPT and the other results to determine alternative potentials that were used to estimate the uncertainties of the thermophysical properties. Our decisions are based on three observations. First, we recalled that Komasa's [24] variational result at 5.6 bohr decreased 0.3 % upon increasing the basis set from 1200 terms to 2048 terms. If Komasa's result at 4.0 bohr (292.784 K) were decreased by 0.3 %, it would be 291.906 K, in agreement with the SAPT value of 291.64 K. Gdanitz suggested that the decrease at 4.0 bohr might be less than 0.3 % because the variation method is more accurate at smaller separations [26]. Second, we noted that the two extensions of Klopper and Noga' work [17] are not independent. The two decompose (r ) in several components, the largest of which were calculated best by Klopper and Noga. Thus, the uncertainties of these results may be dominated by those of Klopper and Noga. (van Mourik and Dunning [21] state, "It is likely that the corrected curve is the most accurate available to date for He 2 interactions". In effect, they asserted that Klopper and Noga's interaction energies are more accurate than their own complete basis set extrapolated energies.) Third, Bukowski et al. [27] argue that their own Gaussian-type geminals (GTG) computa-tion bounds the larger components of Klopper and Noga's CCSD(T) computations and they suggest that Klopper and Noga's results may be too high by approximately 0.3 K at 4 bohr and by approximately 0.04 K at 5.6 bohr. If Bukowski et al.'s suggestion is correct and if one decreases the CCSD(T) values of (r ) accordingly, then they all would agree with the SAPT results. Ultimately, additional calculations will resolve these issues.

Algebraic Representations of ab initio Values of (r )
We calculated the thermophysical properties of helium six times, each using a different function to represent ab initio values of (r ). We fitted two of these six functions, 00 and B , to our own selections among the published ab initio values. The third function, SAPT , had already been fitted by others to ab initio results and used to calculate thermophysical properties. [18] We fitted the fourth, A , to the same ab initio results used to obtain SAPT ; however, we added one additional fitting parameter. Thus, differences between the thermophysical properties computed from SAPT and A provide one indication of the sensitivity of the properties to the algebraic representation of the ab initio "data". The last two functions are denoted Ϫ A , and + A . To obtain Ϫ A , we decreased the ab initio short-range results [15] by their claimed uncertainties and decreased the intermediaterange SAPT results by 0.1 % and re-fitted them. Then, we increased the ab initio results by their claimed uncertainties and the SAPT results by 0.1 % and fitted them to obtain + A . The differences between the thermophysical properties calculated using A , Ϫ A , + A and SAPT are analogous to the uncertainties of measured values of thermophysical properties conducted in a single laboratory and analyzed using different methods. In the present case, the differences between the thermophysical properties calculated from A , Ϫ A , + A , and SAPT are much smaller than the differences between those calculated from 00 , A , and B .

00
We used 00 to calculate the thermophysical properties tabulated in Appendix A. In our judgement, 00 is the best representation of the ab initio results available at the time of this writing. The subscript "00" identifies 00 by the year in which we began using it. The ab initio results fitted by 00 (r ) come from three sources: (1) at small r (1 < r < 2.5 bohr), the results of the variational calculation from Komasa [24], (2) at intermediate r (3 bohr < r < 7 bohr), the SAPT results from Korona et al. [18], (3) at large r , the asymptotic constants from the "exact" dispersion coefficients of Bishop and Pipin [22] and the higher order dispersion coefficients determined from the approximate relations presented by Thakkar [29]. The algebraic representation of 00 (r ) is a modification of the form given by Tang and Toennies [9]. The representation is the sum of repulsive ( rep ) and attractive ( att ) terms: Equation (1) includes the factor f 2n (r ) that accounts for the relativistic retardation of the dipole-dipole (n = 3) term applied over all r . This factor changes the behavior of the dipole-dipole term from r Ϫ6 to r Ϫ7 at very large r , and it was taken from Jamieson et al. [30]. When the expressions for the retardation of the higher dispersion terms C 8 and C 10 given by Chen and Chung [23] were applied to 00 , the well depth changed by only 0.0014 K out of 11 K. The resulting changes in the calculated thermophysical properties were much smaller than their uncertainties; thus, we used the approximation f 2n (r ) ≡ 1 for n > 3. (Note: retardation is included when calculating the thermophysical properties; however, by convention, it is not included when comparing Eq. (1) to the ab initio results.) We also considered the adiabatic correction of the helium dimer given by Komasa et al. [28]. The effects of this correction were also much smaller than those from the uncertainties in (r ); thus, we omitted this correction. The definition of 00 (r ) in Eq. (1) is broken into two ranges. If this were not done, 00 (r ) would have a spurious maximum at very small values of r . As indicated in Eq. (1), the break-point was set at 0.3 bohr.
The dispersion coefficients (C 6 , C 8 , . . . C 16 ) in Eq. (1) and Table 1 were held fixed [22,29]. The values of the remaining parameters in Table 1 (a Ϫ2 , a Ϫ1 , a 1 , a 2 , and ␦ ) were determined by fitting 00 (r ) to the ab initio results. When fitting 00 the ab initio results were weighted in proportion to the reciprocal of the uncertainty squared, where the uncertainties were taken (when available) from the publications that presented the results. [15,18,20,21,24].

SAPT
Korona et al. fitted their SAPT results and the QMC values of Ceperley and Partridge [15] to the algebraic expression of Tang and Toennies [9] while holding constant the asymptotic dispersion coefficients of Bishop and Pipin [22]. They included higher order dispersion coefficient determined with combining rules of Thakkar [29] and retardation effects of the C 6 dispersion coefficient as given by Jamieson et al. [30]. Janzen and Aziz [11] calculated the thermophysical properties of helium using SAPT and they "judged it to be the most accurate characterization of the helium interaction yet proposed." We believe that 00 is more accurate than SAPT because it uses the recent, accurate variational results of Komasa [24] instead of the earlier short range QMC values of Ceperley and Partridge [15].

A , ؊
A , and +

A
In an attempt to ascertain how uncertainties in the interaction energies propagate into the thermophysical properties we constructed alternative potentials which differed in the choice of ab initio results, and in the form of the algebraic expression. The first alternative, denoted A , was obtained by fitting the exact same ab initio results from [18,22,24,29] as SAPT . The algebraic expression of Tang and Toennies [9] was modified by adding a a 3 r 3 to the exponent of the repulsive term, such that rep = A exp(a 1 r + a 2 r 2 + a 3 r 3 ). The additional a 3 r 3 term enables A to fit the SAPT ab initio results within 0.1 % in two regions r = 3 bohr and at r > 6 bohr where SAPT [18] deviates from the ab initio results slightly greater than 0.1 %.
To obtain Ϫ A , we decreased the ab initio short-range [15] and long-range [22] results by their claimed uncertainties and decreased the intermediate-range SAPT results by 0.1 % and the long-range dispersion coefficients by 0.08 %. Equation (1) was then re-fitted to obtain Ϫ A . We then increased the ab initio results by their claimed uncertainties and the intermediate-range SAPT results by 0.1 % and again fitted them to obtain + A .

B
The potential B , uses the CCSD(T) results of van Mourik and Dunning [21] and of van de Bovenkamp and van Duijneveldt [20] instead of the SAPT results of Korona et al. [18] in the intermediate range of 3 bohr < r < 7 bohr. To fit these values the algebraic expression of Tang and Toennies [9] was modified again by adding a a Ϫ1 r Ϫ1 and a Ϫ2 r Ϫ2 to the exponent of the repulsive term, such that rep. = A exp(a 1 r + a 2 r 2 + a Ϫ1 r Ϫ1 + a Ϫ2 r Ϫ2 ).   [29] the ab initio results. The differences between the thermophysical properties calculated using 00 , SAPT , A , and B are analogous to the differences between measurements of thermophysical properties conducted in different laboratories using different methods and they are used to estimate the uncertainties of the results for pure 3 He and pure 4 He. Table 3 lists some characteristic properties of the potentials that we have used. They include the well depth /k B , the locations of the zero ( ) and of the minimum (r m ) of the potential, and the energy of the bound state (E b ) of a pair of 4 He atoms. Following Janzen and Aziz [31], we estimated the number of Efimov states N E from the scattering length and the effective range with the result N E = 0.77 Ϯ 0.01 for 00 . Because N E < 1 for all potentials in Table 2, Efimov states are unlikely to exist. A discussion of these properties of the interatomic potential for helium can be found in Ref. [31].

Numerical Calculations and Their Uncertainties
Here, we outline the steps required to calculate the thermophysical properties of helium from the inter-atomic potential. We also describe the precautions that were taken to insure that the uncertainties in the results from approximations in statistical mechanics and in the numerical methods were both smaller than the uncertainties results from different choices for (r ).
The initial steps of calculating the thermophysical properties that depend upon pairs of helium atoms are all the same. (1) The Schrödinger equation for the scattering of a helium atom at the energy E in the potential (r ) is separated in spherical coordinates, (2) the radial part of the wave function is expanded in partial waves ᐉ (r ) of angular momentum ᐉ , (3) several nodes of the scattered wave are located far from the scattering atom, and (4) the phase shifts ␦ ᐉ of the scattered wave are determined and (5) summed with appropriate statistics to obtain cross sections. The summations account for large symmetry effects at low temperatures [32]. Thus, separate summations are required for 3 He and 4 He and their mixtures when calculating the second virial coefficient and the transport properties. The final step (6) is an integration over energy that is appropriate to the thermophysical property under consideration.

Integration of the Radial Schrödinger Equation
The Schrödinger equation is separated in spherical coordinates and decomposed into angular momentum states to obtain where ប is Planck's constant [14] divided by 2, is the reduced mass = (m 1 + m 2 )/m 1 m 2 , k = (2E ) 1/2 /ប is the wave number, and E is the energy of the incoming wave. Equation (2) is integrated to obtain the perturbed wave functions ᐉ (k , r ). The location r n of the n th zero (or node) of the wave function ᐉ (k , r ) was found using a five point Aiken interpolation formula with values of ᐉ (k , r ) near the n th node. The integration was performed using Numerov's method [33] as implemented in [34] and [35]. At each energy, r n was recalculated using successively smaller step sizes. The calculation was terminated when halving the step size changed r n less than 10 Ϫ9 ϫ r n . We verified that the tolerance 10 Ϫ9 ϫ r n was sufficiently small that further reductions of the step-size did not change the thermophysical properties beyond the tolerances given in Table 6. The final sizes of the integration steps are listed in Table 4. Table 4. Integration step sizes used in a given energy range to locate the n th zero of ⌿ᐉ (k , r ) The relative phase shifts, ␦ ᐉ (k , n ) of the outgoing partial wave were evaluated from the relation where j ᐉ (k , r n ) and n ᐉ (k , r n ) are the Bessel and Neuman functions for angular momentum quantum number ᐉ and wave number k . In practice, the phase shifts were evaluated at groups of three consecutive nodes. If the phase shift did not change by more than 10 Ϫ8 ϫ ␦ ᐉ (k , n ) between the first and last of the three nodes, it was assumed that n (and r n ) were sufficiently large that additional effects of the potential were negligible, and the calculation was terminated. Otherwise, the calculation was continued to larger values of r , and the test was repeated. We verified that the tolerance 10 Ϫ8 ϫ ␦ ᐉ (k , n ) is consistent with the uncertainties of the thermophysical properties listed in Table 6.

Calculation of the Second Virial Coefficient, B (T )
The second virial coefficient was obtained by adding two or three terms; the first term is a thermal average B th (T ), the second term is that of an ideal gas B ideal (T ), and the third term is the bound state term B bound (T ), which applies to 4 He, but not to 3 He because a bound state exists only for 4 He.

The Thermal Average Term B th (T )
The thermal average term B th (T ) is (4) where k B is the Boltzmann constant, and ␦ ᐉ (k , n Ӎ ϱ) is the phase shift at large enough separation that the potential no longer perturbs the outgoing wave function [32,36]. Equation (4) contains both a sum and an integral with the limits 0 and ϱ. Truncating the sum and the integral at a finite upper bound is a potential source of error. At each value of k , the sum was computed until the addition of six phase shifts did not change the sum by more than 10 Ϫ8 of its value. At the lowest energies, this condition was met after adding seven phase shifts; at the highest energies, hundreds of phase shifts were added.
At this step of the calculation, symmetry effects are incorporated. The unweighted sum [Eq. (4) [32].
The integral in Eq. (4) was evaluated using a standard integration routine, DQAGI [37]. This routine is designed for semi-infinite or infinite intervals and automatically uses nonlinear transformation and extrapolation to achieve user-specified absolute and relative tolerances for a user-specified function. The relative error was set to 10 Ϫ8 . If the integrator could not achieve this accuracy, an error message would have been reported the problem.

The Ideal-Gas and Bound State Terms B ideal (T ) and B bound (T )
The ideal-gas contribution B ideal (T ) is negative for BE and positive for FD, and zero for Boltzmann statistics as given by where N A is the Avagodro constant and ≡ [ប /( k B T )] 1/2 is the "thermal wavelength." The ideal-gas term is important only at low temperatures; it is 1/10 of B (T ) at 5 K and 1/100 of B (T ) at 75 K. The ideal-gas term is a function of fundamental physical constants and the resulting standard uncertainty is on the order of 10 Ϫ6 . For 4 He, the bound state term B bound (T ) is where E b is the energy of the bound state. The bound state term is 1/1000 of B (T ) at 3 K and 1/100 of B (T ) at 0.4 K. E b was determined from integrating the Schrödinger equation; thus, it depended upon the integration step size. Decreasing step sizes were used until consecutive values of E b differed by less than 10 Ϫ6 ϫ E b . This numerical uncertainty is much smaller than the 18 % difference between E b determined from Ϫ A and that determined from + A ( Table 3). The sum of the numerical uncertainties in the calculation of B (T ) is at most 10 Ϫ5 ϫ B (T ). This is insignificant compared with the uncertainty of B (T ) which arises from the uncertainty of the potential (r ). For example, the uncertainty of B (T ) resulting from the uncertainty of (r ) is 0.0022 ϫ B (T ) at 300 K; the relative uncertainties at other temperatures are listed in Table 6.

Calculation of the Transport Properties
In order to calculate the transport properties, we used the numerical methods outlined above to obtain the phase shifts as functions of the wave number and angular momentum quantum number. Then we computed the sums over the phase shift that determine the quantum cross sections, Q (1) , Q (2) , Q (3) ....Q (n) , etc. [38]. The cross sections were integrated with respect to energy to obtain the temperature-dependent collision integrals. Finally, the transport properties were calculated using the appropriate combinations of the collision integrals.

Calculation of the Quantum Cross Sections Q (n)
The quantum cross sections are functions involving the sums of the phase shifts that depend upon the symmetry of the interacting atoms. The sums over the even and the odd values of ᐉ are needed separately: (2ᐉ + 1)sin 2 ␦ ᐉ and then weighted sums are computed. To evaluate Q (1) for Bose-Einstein (BE) or Fermi-Dirac (FD) statistics the sums are weighted with the spin-dependent quotients, as shown in Eq. (5). As for the case of the second virial coefficient, the sums in Eq. (8) extend to ᐉ = ϱ.
The sum was continued until the addition of six more phase shifts changed the cross section by less than 10 Ϫ8 of its value. Cross sections with moments up to n = 6 are required to calculate the collision or omega integrals used in the higher order approximations for the transport properties. The equations for these calculations are given by Ref. [38].

Calculation of the Collision Integrals ⍀ (n,s)
The reduced collision integrals were evaluated from the equation where the superscript ଙ indicates that both the energy and the temperature were scaled by the well-depth of 00 and Q (n)ଙ was scaled by the value Q (n) for a rigid sphere of radius r m , the location of the minimum of 00 (Table  3; See Ref. [32]). In order to evaluate of Eq. (9), the quantum cross sections Q (n)ଙ must be calculated at each energy E used for the quadrature. We calculated a table of Q (n)ଙ as a function of E ଙ and used a 5 point Aiken interpolation to determine values of Q (n)ଙ between tabulated values. The intervals in the table were determined such that the interpolated values had a uncertainty of less that 10 Ϫ6 ϫ Q (n)ଙ . Equation (9) was integrated using the automated quadrature routine DQAGI [37], discussed in Sec. 3.1.3, with the tolerance set to 10 Ϫ8 . The numerical methods used to calculate the collision integrals yielded results with a relative uncertainty of less than 10 Ϫ5 .

Calculation of the Transport Properties From the Collision Integrals
The transport properties of dilute gases are calculated using combinations of the collision integrals in approximations of increasing complexity and accuracy. The viscosity and thermal conductivity of pure 3 He and 4 He were calculated to the 5th order approximation [39]. The equimolar mixture thermal conductivity [40] and thermal diffusion factors [41] were calculate to the 3rd order, and the diffusion coefficient and mixture viscosity were calculated to 2nd order. Figure 2 shows the effects of truncating the order of the calculation. The changes in and for 4 He and equimolar mixtures of 4 He and 3 He are compared at four temperatures upon increasing order of the approximation. The calculations converge very well; 2nd to 3rd order results in less than a 0.1 % change, 3rd to 4th order results in less than a 0.01 % change, and 4th to the 5th order less than 0.001 %. The behavior of the other transport properties ( and for pure 3 He, D 12 , and ␣ T ) is similar to that shown in Fig. 2. Figure 2 shows that the change in and of the equimolar mixture from 1st to 2nd order, is very close to that of pure 4 He. These results show that only calculations of the 2nd order contribute any significant uncertainty to the calculated properties. From these observations, we conclude that the relative uncertainty of and D 12 for the equimolar mixture ranges from 0.01 % to 0.04 % in the temperature range 10 K Յ T Յ 10 4 K. Figure 2, together with the equivalent figure for pure 3 He, suggest that, at T > 100 K, the accuracy of the calculated and D 12 for the equimolar mixture might be improved if one extrapolated from 2nd order to 5th order by following the curves for pure 4 He and 3 He.
The rapid reduction of the uncertainty of the calculated viscosity with increasing order of approximation is not sensitive to (r ); Viehland et al. [39] obtained similar results for the viscosity of rigid atoms that interact via (12-6) Lennard-Jones potentials.

Interpolation as a Function of Temperature
The tables in Appendix A list values of the second virial coefficient, the transport properties, and their first derivatives as functions of temperature. The temperature intervals were chosen so that the errors from linear interpolation would be smaller than the uncertainties propagated from the uncertainties of the interatomic potential. Table 5 lists bounds of the interpolation errors, and the unweighted average over the entire temperature range. Below 10 K, the interpolation errors increase because the temperature derivatives of the properties increase.

Classical Calculation
We made an important check of the entire calculation of each thermophysical property. To do so, we performed the relatively simple classical calculation [32] which is valid at high temperatures where the ratio of the de Broglie wavelength h (2mkT ) Ϫ1/2 to atomic diameter is much less than 1. Figures 3 and 4 show that the classical calculations of the viscosity and of the second virial coefficient asymptotically approach the quantum results.

Uncertainties of the Thermophysical Properties From the Uncertainty of the Potential
We now evaluate how the uncertainty of the ab initio values of (r ) propagates into the uncertainty of calculated thermophysical properties. To do so, we calculated the properties with each of the potentials discussed in Sec. 2 and we plotted the results as deviations from the results obtained for 00 (r ). Figure 3 shows these deviations for the viscosity of 4 He. In Fig. 3, the width of the shaded band surrounding the curve A spans the range Fig. 3. Fractional deviation of the calculated viscosity using the considered potentials. The base line is the viscosity calculated to the 5th approximation for 4 He from 00. The other curves are identified in the figure. The shaded region around the curve for A shows how Ϫ A and + A vary the predicted viscosity. Also shown is the classically calculated viscosity asymptotically approaching the quantum values with increasing temperature. of results obtained with Ϫ A to those obtained with + A . Similar bands could have been placed about the results from those obtained with 00 , B , and SAPT ; they were omitted for clarity.
We took the differences in the alternative potentials as an accurate estimate of the uncertainty in 00 . By comparing the properties calculated from each alternative potential, we estimated the actual uncertainty propagated into each reported thermophysical property. Figure 3 shows that as the temperature is increased from 1 K to 10 K, the relative uncertainty of the viscosity u r ( ) of 4 He decreases from 0.4 % to 0.1 %. In this temperature range, the discrepancies among the potentials are comparable to the uncertainty of each potential, as indicated by the width of the shaded band. In the range 10 K < T < 1000 K, the difference between the results obtained using 00 and the results obtained with B and SAPT lead us to conclude that u r ( ) is approximately 0.08 %. If the discrepancies between the ab initio results around 4.0 bohr could be resolved, then u r ( ) would be reduced by nearly a factor of three in this temperature range. In the range 1000 K < T < 10 4 K, we also conclude u r ( ) ≈ 0.08 %. In this temperature range, the results from 00 and B are more reliable than the results from A and SAPT having been fit to the short-range variational calculations of Komasa [5] as discussed in Sec. 2, above.
The relative uncertainty of the second virial coefficient u r (B ) of 4 He can be judged from Fig. 4. In the range 1 K < T < 10 K, u r (B ) ≈ 1 %. At T ≈ 23.4 K, B (T ) passes through zero. There, u r (B ) diverges; however, the uncertainty of B , u (B ) ≈ 0.3 cm 3 иmol Ϫ1 . In the range 100 K < T < 10 4 K, u r (B ) of 4 He gradually declines from 0.4 % to 0.1 %.
The uncertainties for each property are summarized in Table 6 from comparisons similar to those provided in Figs. 3 and 4 and described in the preceding paragraphs. These uncertainties are much lower than those from measurements; thus, the corresponding values of the properties listed in the Appendices can be used as standards.

Results
The results of the present calculations for 4 He, 3 He, and their equimolar mixture are listed in Tables A1, A2, and A3 in Appendix A. These tables contain the second virial coefficient B for the pure species and the interaction second virial B 12 where B mix = x 2 1 B 11 + 2x 1 x 2 B 12 + x 2 2 B 22 . The zero-density viscosity, thermal conductivity and their equimolar mixture. The diffusion coefficient at 101.325 kPa (one atmosphere), and the thermal diffusion factor. Derivatives with respect to temperature are provided to facilitate interpolation and for use in calculating acoustic virial coefficients. The tables for pure 4 He and 3 He contain the self-diffusion coefficient calculated without symmetry effects ( Boltzmann statistics), and the thermal diffusion factor of a mixture of 99.999 % 4 He or 3 He respectively. The tables span the temperature interval 1 K Յ T Յ 10 4 K. The highest temperature is well below the first excited state of helium (2 ϫ 10 5 K) and well below 2.91 ϫ 10 5 K, the highest value of the ab initio results used to determine A . In order to calculate the thermophysical properties between the tabulated temperatures, we recommend interpolation using the cubic polynomial f (T ) such that where f' = df /dT and ⌬T = T 2 Ϫ T 1 . The calculated values listed in the Tables are accurate to the uncertainties discussed in Sec. 4. Equation (10) contributes an additional uncertainty from the interpolation discussed as in Sec. 3.

Comparison With Measurements
In this section we compare the values of the thermophysical properties calculated using 00 with the best experimental values. In nearly every case, the experimental values agree with the calculated values within their combined uncertainties, and the calculated properties have the smaller uncertainties.    4 He from B00 calculated using 00. Key: G Ref. [42]; ᭺ Ref. [42] adjusted by Ref. [43]; Ref. [43]; ᭡ Ref. [43]; ᭢ Ref. [44]; ᭜ Ref [45]; ᮀ Ref. [46]; ᭝ Ref. [8] Eqs. (37)  At very low temperatures B (T ) is sensitive to the shape of the potential well. Figure 5 shows that the lower well depth of 00 predicted by Korona et al. [18] reproduces the low temperature measurements better than the shallower well depth predicted by Van de Bovenkamp and van Duijneveldt [20] and by Van Mourik and Dunning [21]. To further strengthen this argument, it is known that the low temperature second virial measurements have not been corrected for contributions from the third virial coefficient C (T ). For 4 He [42], the size of this "third virial correction" can be seen in the top panel of Figure 5. In that panel, the solid circles show the values B (T ) before they were corrected in Ref. [43], and the open circles show the values after the correction.

Second Virial Coefficient
The correction for C (T ) lowers the second virial values bringing them further in line with B 00 (T ) and away from B B (T ) indicating a preference for the lower well depth. Table 7 provides two numerical measures of the differences between experimental values of B (T ) and those calculated using 00 . One measure is the mean of the absolute values of the differences B exp Ϫ B 00 and the second is the range of these differences. The final column of Table 7 lists the range of the uncertainties reported by the experimenters. In nearly all cases the experimental uncertainties exceed the differences B exp Ϫ B 00 . Figure 6 compares B exp (T ) of 3 He, deduced from the measurements of Matacotta et al. [47], with B 00 (T ). There is an obvious trend in the deviations which is larger than the experimental uncertainties below 5 K. Probably, the trend would be removed if B exp (T ) was corrected for the for effects of the third virial coefficient of 3 He [48] as discussed above for 4 He [42]. Table 8 compare the zero-density viscosity 00 , calculated using 00 , with measured values from many sources. The experimental results are typically reported at 101.325 kPa where the density dependence is negligible in comparison with experimental uncertainties. Figure 7 shows the viscosity of 3 He and 4 He at low temperatures where the large quantum effects lead to important differences between the isotopes. The 00 (T ) values are in good agreement with the measurements of Becker et al. [49]. Figure 8 displays the fractional deviations of various values of exp of 4 He from 00 . In nearly every case they are smaller than the uncertainties provided by the experimenters. In Fig. 8, the barely visible dashed curve represents A calculated using A , and the dash-dot-dot curve B , calculated from B . The differences between these curves are a measure of the ab initio uncertainties which are much smaller than the reported experimental uncertainties.    Figure 9 and Table 9 compare the values of the zerodensity thermal conductivity of 4 He calculated using 00 with measured values from several sources. The experimental thermal conductivities are typically reported at 101.325 kPa, however the density dependence is negligible compared to the experimental uncertainties. As it was the case for B and , most of the values of exp differ from 00 by an amount comparable to the uncertainty of the measurements. The differences between the values 00 , A , and B calculated using 00 , A , and B are much smaller than the uncertainties of the measurements. Table 9 lists the root mean square of the relative differences ⌬ / 00 ≡ ( exp Ϫ 00 )/ 00 and the range of these relative differences. The final column of Table 9 lists the range of the uncertainties reported by the experimenters. Figure 10 and Table 10 compare the values of the mutual diffusion coefficient for an equimolar mixture of 3 He and 4 He at one atmosphere (101325 Pa). Figure 10 shows the deviations of D 12,exp taken from three sources, from D 12,00 , where D 12,00 was calculated using 00 . Because the diffusion coefficient is difficult to measure, the uncertainties of the experimental values are comparatively large; therefore, the relative deviations of the values calculated using A and B are not visible in Fig.  10. The difference in D 12 on going from the first to the second order approximation is practically the same as seen for the viscosity in Fig. 2. Table 10 lists the rootmean-square of the relative differences ⌬D / D 00 ≡ (D exp Ϫ D 00 )/D 00 and the range of these relative differences as well as the range of the uncertainties reported by the experimenters.

Thermal Diffusion Factor ␣ T
The thermal diffusion factor ␣ T is a complicated function of temperature and concentration and only a few, relatively inaccurate measurements are available. Figure 11 compares ␣ T,exp for an equimolar mixture of 3 He and 4 He to the ␣ T,00 values calculated from 00 . The values of ␣ T,A and ␣ T,B calculated from A and B are also shown, only differing at low temperatures. In the first-order approximate calculation of the transport properties, ␣ T is identically zero; thus, we compared the second-order transport-theory results to the third-order results to estimate the uncertainties of the ab initio results from truncating the transport theory. Going from the second to third order increased ␣ T by 0.56 % at 10 K and by 0.36 % at 10,000 K. The thermal diffusion factor is very difficult to measure the typical relative uncertainties are 4 % to 8 %. Owing to the experimental difficulties, the calculated values would be more accurate than any experimentally determined value. Fig. 11. Thermal diffusion factor ␣T as a function of temperature for an equimolar mixture of 3 He and 4 He. The solid curve represents ␣T,00 calculated using 00. Key: ᭺ Ref. [71]; ᮀ Ref. [72]; ᭝ Ref. [73]; ᭝ Ref. [74]; ᭛ Ref [75]; --Values of ␣T,A calculated using A; -• • -␣T,B calculated using B.

Conclusion
We have reviewed the recent ab initio calculations of (r ) for helium. We represented one of the most accurate ab initio values of (r ) by the algebraic expression 00 (r ) and we estimated its uncertainty by comparing the various ab initio calculations. For the thermophysical properties, the most significant uncertainties occur near 4.0 bohr. Using 00 (r ), we calculated B , , , D 12 , and ␣ T . The numerical methods used in these calculations contributed negligible uncertainty to the results. In all cases, the uncertainties of the calculated thermophysical properties propagated from the uncertainties in 00 (r ) were much less than the uncertainties of published measurements. Therefore, the calculated values should be used as standard reference values.
The large number of recent ab initio calculations of (r ) demonstrate that this is an active field of research. In the near future, ab initio calculations will surely reduce the uncertainty of (r ) near 4.0 bohr, further reducing the uncertainties in the calculated properties. Improved ab initio calculations of the molar polarizabil-ity of helium and of the dielectric virial coefficients are also under way. These may well lead to an ab initio standard of pressure based on measurements of the dielectric constant of helium near 273.16 K [70].

Appendix A. Calculations
The results of the present calculations for 4 He, 3 He, and their equimolar mixture are listed in Tables A1, A2, and A3, respectively. These tables contain the second virial coefficient for the pure species, the interaction second virial coefficient B 12 , the zero-density viscosity and thermal conductivity, the diffusion coefficient, and the thermal diffusion coefficient. Derivatives with respect to temperature are provided to facilitate interpolation and for use in calculating acoustic virial coefficients. The tables for pure 4 He and 3 He contain the self-diffusion coefficient, and the thermal diffusion factor of binary mixtures of 4 He and 3 He with mole fractions of 0.99999 and 0.00001, respectively.