Thermophysical Properties of Krypton-Helium Gas Mixtures From

Thermophysical properties of krypton- helium gas mixtures from ab initio pair potentials

Cite as: J. Chem. Phys. 146, 214302 (2017); https://doi.org/10.1063/1.4984100 Submitted: 28 February 2017 . Accepted: 11 May 2017 . Published Online: 05 June 2017

Benjamin Jäger , and Eckard Bich

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J. Chem. Phys. 146, 214302 (2017); https://doi.org/10.1063/1.4984100 146, 214302

Thermophysical properties of krypton-helium gas mixtures from ab initio pair potentials Benjamin Jager¨ a) and Eckard Bichb) Institut fur¨ Chemie, Universitat¨ Rostock, D-18059 Rostock, Germany (Received 28 February 2017; accepted 11 May 2017; published online 5 June 2017)

A new potential energy curve for the krypton-helium atom pair was developed using supermolecular ab initio computations for 34 interatomic distances. Values for the interaction energies at the complete basis set limit were obtained from calculations with the coupled-cluster method with single, double, and perturbative triple excitations and correlation consistent basis sets up to sextuple-zeta quality augmented with mid-bond functions. Higher-order coupled-cluster excitations up to the full quadruple level were accounted for in a scheme of successive correction terms. Core-core and core-valence correlation effects were included. Relativistic corrections were considered not only at the scalar relativistic level but also using full four-component Dirac–Coulomb and Dirac–Coulomb–Gaunt calculations. The fitted analytical pair potential function is characterized by a well depth of 31.42 K with an estimated standard uncertainty of 0.08 K. Statistical thermodynamics was applied to compute the krypton-helium cross second virial coefficients. The results show a very good agreement with the best experimental data. Kinetic theory calculations based on classical and quantum-mechanical approaches for the underlying collision dynamics were utilized to compute the transport properties of krypton-helium mixtures in the dilute-gas limit for a large temperature range. The results were analyzed with respect to the orders of approximation of kinetic theory and compared with experimental data. Especially the data for the binary diffusion coefficient confirm the predictive quality of the new potential. Furthermore, inconsistencies between two empirical pair potential functions for the krypton-helium system from the literature could be resolved. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4984100]

I. INTRODUCTION included relativistic effects beyond the scalar relativistic level and a correction for the coupled-cluster method with up to For the pure monatomic gases helium through krypton, full iterative quadruple excitations, CCSDTQ.12,13 Unexpect- thermodynamic and transport properties at low densities can be edly, the ab initio pair potentials of Waldrop et al. and Jager¨ predicted from ﬁrst principles with an accuracy that is superior et al. resulted in almost indistinguishable values for the low- or at least equal to that of the best experimental techniques.1 density transport properties of krypton. Jager¨ et al. showed Highly accurate ab initio pair potentials have been developed that the perfect agreement of the viscosity data computed for in the last years for helium by Hellmann et al.2 and Cencek the potential of Waldrop et al. with the most accurate exper- et al.,3 for neon by Hellmann et al.,4 and for argon by Jager¨ imental data was partly due to a fortuitous cancellation of et al.5,6 as well as by Patkowski and Szalewicz.7 Recently, errors. two ab initio pair potentials for krypton have been presented For interactions between unlike noble gas atoms, ab initio by Waldrop et al.8 and by Jager¨ et al.9 Waldrop et al. used pair potentials have not been developed at the same level of frozen-core (FC) explicitly correlated coupled-cluster calcu- accuracy as for the like interactions. Partridge et al.14 calculations to obtain interaction energies for the krypton atom pair lated interaction energies for systems containing helium and at the complete basis set (CBS) limit. They applied a cor- different ground state atoms using the frozen-core CCSD(T) rection for electronic excitations beyond the coupled-cluster method and suitable quadruple order quintuple-zeta basis sets method with single, double, and perturbative triple excita- augmented by a small set of mid-bond functions. Later, Haley tions, CCSD(T),10 using the CCSDT(Q) method of Bomble and Cybulski15 studied various combinations of noble gas et al.,11 which accounts for full triple and perturbative quadru- atoms at the same level of theory employing correlation con- ple excitations. Furthermore, they included core-core and core- sistent basis sets (aug-cc-pVXZ,16–18 abbreviated as aVXZ) valence correlation as well as scalar relativistic effects. Using of up to quintuple-zeta quality and mid-bond functions. We a new basis set of sextuple-zeta quality and standard orbital chose krypton-helium as a model system because of its large CCSD(T) computations, Jager¨ et al. calculated CBS limiting mass difference, which causes a rather strong composition values for the Kr–Kr interaction energy that are somewhat dif- dependence of the binary and thermal diffusion coefﬁcients ferent from the results of Waldrop et al. Furthermore, they compared to the neon-helium and argon-helium systems. Thus, a meaningful test of the higher orders of approxima- a)Electronic mail: [email protected] tion to the kinetic theory can be conducted. Calculations for b)Electronic mail: [email protected] xenon-helium are not feasible at the same level of accuracy

0021-9606/2017/146(21)/214302/15/$30.00 146, 214302-1 Published by AIP Publishing. 214302-2 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017) due to the lack of suitable correlation consistent basis sets. In Sec.II, we present the details of the new ab initio Moreover, the available ab initio and empirical pair poten- pair potential for the krypton-helium atom pair. Section III tials for krypton-helium exhibit a significant disagreement is concerned with the assessment of the analytical potential with respect to their well depths. The ab initio potentials energy curve by the comparison of calculated and experimen- of Partridge et al. and Haley and Cybulski as well as the tally based values of the cross second virial coefficient. In empirical potential of Keil et al.19 are characterized by sim- Sec.IV, the kinetic theory methodology is explained, and the ilar well depths of 29.56 K, 29.84 K, and 29.45 K, respec- computed results for the coefficients of viscosity and thermal tively, whereas the earlier empirical potential of Danielson conductivity as well as of binary and thermal diffusion are 20 and Keil is considerably deeper with ε/kB = 30.95 K (kB is compared with experimental data. Boltzmann’s constant). Note that Keil et al. already observed that their new potential shows, compared to the older version II. KRYPTON-HELIUM PAIR POTENTIAL of Danielson and Keil, a worse agreement of the computed values for the cross second virial coefficient with the best We computed interaction energies for 34 interatomic experimental data. A later ab initio potential of Bouazza and distances from 1.3 Å to 9.0 Å using the supermolecular Bouledroua21 is not based on any improved quantum-chemical approach including the full counterpoise correction by Boys 27 interaction energies but on the results from the two earlier and Bernardi. For helium, the correlation consistent basis studies (Refs. 14 and 15) and is therefore not considered any sets [aVXZ, X = D(2), T(3), Q(4), 5, 6] developed by Dun- 16 17 further. ning as well as by Woon and Dunning were employed. Transport property calculations for noble gas mixtures by The corresponding basis sets for krypton were introduced by 18 means of the kinetic theory of gases were limited to Lennard- Wilson et al. (aVXZ with X = D, T, Q, 5) and by Jager¨ 9 Jones-type or other simple potential models (see, for example, et al. (aV6Z). Basis sets for calculations that incorporate core- Refs. 22–24) for a long time. In the late 1980s, such computa- core and core-valence correlation effects were developed by 28 tions were performed for the more realistic but still empir- DeYonker et al. (wCVXZ with X = D, T, Q, 5; augmenting ical Hartree–Fock-dispersion-type (HFD) potentials.19,20 diffuse functions were taken from the standard aVXZ basis Partridge et al.14 were the first to use ab initio pair poten- sets). The exponents of additional diffuse functions leading tials for the interactions between unlike atoms. They computed to doubly augmented basis sets (daVXZ) were obtained in the necessary collision integrals quantum-mechanically for all an even-tempered manner from the two most diffuse basis atom pairs that include helium and applied up to fourth-order functions of each type of the corresponding aVXZ basis kinetic theory expressions for the transport properties taken sets. from the work of Mason.23,24 Unfortunately, they analyzed At the center between the krypton and the helium atom, their results only marginally regarding the influence of differ- additional bond functions were placed to improve the conver- ent orders of approximation to the kinetic theory solutions. gence behavior towards the CBS limit as in our studies on the 2,4,5,9 Furthermore, their results for the krypton-helium binary diffu- pure-component potentials. The large set of mid-bond functions denoted by (44 332) is characterized by exponents sion coefficient (for xHe → 1) deviate by about 1% from the experimental reference values (as tabulated in Ref. 19), which of 0.06, 0.18, 0.54, and 1.62 for s and p functions, 0.15, 0.45, have an estimated uncertainty of only ±0.3%, indicating that and 1.35 for d and f functions, and 0.3 and 0.9 for g functions. their krypton-helium potential is far from being an accurate The exponents of the smaller set denoted by (3321) are 0.1, representation of the true pair potential. 0.3, and 0.9 for s and p functions, 0.25 and 0.75 for d functions, 25 and 0.45 for a single f function. Later, Song et al. used the ab initio pair potential of 29 Haley and Cybulski15 for the calculation of binary and thermal The two-point scheme recommended by Halkier et al., 3 3 diffusion coefficients of the krypton-helium system. Unfortu- Vcorr,X X − Vcorr,X−1(X − 1) nately, they did not include a quantum-mechanical treatment Vcorr,CBS = , (1) X3 − (X − 1)3 of the collision behavior and neglected the effects beyond the second-order approximation of the kinetic theory. A recent was utilized to extrapolate the correlation parts of the interac- study by Sharipov and Benites26 is concerned with transport tion energies for two successive cardinal numbers X 1 and X properties for the argon-helium system, where results were to the CBS limit. The general additive scheme of computing reported for the 5th-, 12th-, and 20th-order approximations. the total interaction energy at any interatomic distance is given No effects could be observed beyond the 12th-order approxi- by mation for any of the investigated properties. Significant differ- tot V = V + V + V − + V + V , (2) ences between the 5th- and 12th-order results were observed SCF corr post CCSD(T) core rel only for the thermal diffusion factor (up to 0.5%). However, where V SCF is the Hartree–Fock interaction energy, V corr the measurement uncertainties for this property are about ten is the CBS-extrapolated correlation energy at the frozen- times larger. Moreover, the effects of the successive steps up to core (FC) CCSD(T) level of theory, V post CCSD(T) represents the 12th order were not discussed in detail, and one can suspect the contributions due to higher-order coupled-cluster meth- that already a lower order of approximation gives converged ods, V core is the correction for the influence of core elec- results. It is to note that Sharipov and Benites used classical trons on the correlation energy, and V rel accounts for the collision integrals only and that the ab initio argon-helium pair relativistic effects. We adopted the specific approaches for potential (taken from Ref. 15) is limited in accuracy as in the these contributions largely from our study on the Kr–Kr pair case of krypton-helium. potential.9 214302-3 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

The CFOUR program30 was used for most of the coupled- The results for the last term in Eq. (3), which accounts cluster calculations. For the CCSDT(Q) and CCSDTQ com- for the difference between the CCSDTQ and CCSDT(Q) putations, the MRCC package was applied.31 Four-component levels of theory, were distinctly smaller in magnitude com- relativistic calculations were conducted using the DIRAC pared to the ﬁrst two terms. At R = 3.7 Å, the com- 32 aVDZ+(3321) aVTZ+(3321) program. putations yielded VQ−(Q) = 0.004 K and VQ−(Q) = −0.015 K; the results were not extrapolated to the CBS A. Quantum-chemical calculations limit. The effects beyond the CCSDTQ level of theory were tested using the CCSDTQ(P)35 approach. Although the results SCF interaction energies for the krypton-helium atom pair for the aVDZ+(3321) basis set indicate that the (P) Q dif- were computed using daVXZ+(44 332) basis sets with X = Q, ference can be as large as the Q (Q) correction (0.016 K 5, 6. For the near-minimum distance R = 3.7 Å, we obtained daVQZ+(44332) daV5Z+(44332) for R = 3.7 Å), we neglected this correction due to the VSCF = 28.572 K, VSCF = 28.565 K, and daV6Z+(44332) large computational requirements and accounted for it in VSCF = 28.563 K. The results for the quintuple- and the uncertainty estimation. The resulting post-CCSD(T) cor- sextuple-zeta basis set levels are in almost perfect agreement rection according to Eq. (3) signiﬁcantly deepens the pair so that no extrapolation to the CBS limit was required, and the potential; for the near-minimum distance R = 3.7 Å, it daV6Z+(44 332) values were used as reference values. amounts to −0.642 K. All post-CCSD(T) results are given The correlation energy contributions to the CCSD(T)/FC along with the SCF and CCSD(T)/FC interaction energies in interaction energies for the same sequence of basis sets at TableI. daVQZ+(44332) daV5Z+(44332) R = 3.7 Å are Vcorr = −58.262 K, Vcorr To account for the correlation effects beyond the FC daV6Z+(44332) 9 = −58.447 K, and Vcorr = −58.531 K. Extrapolation approximation, we employed a two-step approach daV5Z+(44332) to the CBS limit according to Eq. (1) results in Vcorr,CBS = −58.641 K for the QZ-5Z pair of correlation energies and in Vcore = VIFC−FC + VAE−IFC. (4) V daV6Z+(44 332) = −58.648 K for 5Z-6Z, corresponding to a per- corr,CBS While FC corresponds to a level of theory where only the 4s fect level of convergence with respect to the CBS limit. Sum- and 4p valence electrons of the krypton atom are involved in the ming up the latter value and the sextuple-zeta SCF result yields treatment of electron correlation, the IFC approximation leaves a CBS limiting value at the CCSD(T)/FC level of theory of only an inner core (1s, 2s, and 2p electrons) unconsidered. V daV6Z+(44332) = −30.085 K. Haley and Cybulski15 obtained CCSD(T)/FC,CBS The difference between these two approaches covers most of a value of −29.828 K for the same interatomic distance the core-core and core-valence correlation effects9 so that the from CCSD(T)/FC computations with an aV5Z+(33 221) basis second term of Eq. (4), which accounts for additional electron set. correlation at the all-electron (AE) level, can be expected to The correction for higher-order excitations within the FC be only a small correction. coupled-cluster approach was calculated according to For the computation of V IFC FC, we employed the Vpost−CCSD(T) = VT−(T) + V(Q)−T + VQ−(Q). (3) CCSD(T) method in combination with the dawCVXZ+(3321) series of basis sets for krypton, which is based on the IFC Adopting the methodology used by Patkowski and Szalewicz7 optimized wCVXZ basis sets of DeYonker et al.,28 and the for argon and by Jager¨ et al. for krypton, we extrapolated standard daVXZ sets for helium. At R = 3.7 Å, the cor- the difference V T (T) between the interaction energies for the rections to the Kr–He interaction energy were obtained as coupled-cluster method with up to full iterative triple exci- dawCVTZ+(3321) dawCVQZ+(3321) V − = −0.722 K, V − = −0.666 K, tations33,34 (CCSDT) and the CCSD(T) approach as well IFC FC IFC FC and V dawCV5Z+(3321) = −0.664 K. The perfect agreement as the difference V between those for the CCSDT(Q) IFC−FC (Q) T between quadruple- and quintuple-zeta results is coincidental and CCSDT levels of theory to the CBS limit by means of and was not found for other interatomic separations. Again, Eq. (1). we extrapolated this correction to the CBS limit according As for the Kr–Kr pair potential, daVXZ+(3321) basis sets to Eq. (1) and obtained V dawCVQZ+(3321) = −0.625 K and were employed for the computation of the T–(T) correction IFC−FC,CBS dawCV5Z+(3321) − term. For R = 3.7 Å, the results for the subsequent sets with VIFC−FC,CBS = 0.662 K for the test geometry. daVTZ+(3321) daVQZ+(3321) The computations for V AE IFC at the IFC and AE levels of X = T, Q, 5 are V − = −0.585 K, V − T (T) T (T) electron correlation were performed using fully uncontracted = −0.533 K, and V daV5Z+(3321) = −0.506 K. The negative val- T−(T) aVXZ basis sets (denoted by unc-aVXZ). For R = 3.7 Å, the ues of this correction concur with the results for He–He and unc−aVDZ+(3321) unc−aVTZ+(3321) values of V = −0.001 K and V Ne–Ne; for Ar–Ar and Kr–Kr this term is generally positive. AE−IFC AE−IFC = 0.000 K are vanishingly small due to a zero-crossing. CBS extrapolation according to Eq. (1) yields V daVQZ+(3321) T−(T),CBS Although this correction gives only a minor contribution to − daV5Z+(3321) − = 0.495 K and VT−(T),CBS = 0.478 K. the interaction energy also for any other interatomic distance, The correction for perturbative quadruple excitations was we included the triple-zeta results in our total pair potential for computed using aVXZ+(3321) basis sets of up to quadruple- completeness. aVDZ+(3321) zeta quality. For the test geometry, we obtained V(Q)−T The treatment of the relativistic correction to the inter- = −0.085 K, V aVTZ+(3321) = −0.131 K, V aVQZ+(3321) = action energy was adopted from our study on the Kr–Kr pair (Q)−T (Q)−T potential9 − aVTZ+(3321) 0.141 K, and CBS extrapolated values of V(Q)−T,CBS aVQZ+(3321) − − V = V / +V − +V − +V . (5) = 0.150 K and V(Q)−T,CBS = 0.149 K. rel DPT2 FC,CBS DPT2,AE FC 4cDC DPT2 Gaunt 214302-4 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

TABLE I. SCF and CCSD(T)/FC nonrelativistic interaction energies as well as post-CCSD(T) corrections for Kr–He in Kelvin as a function of the interatomic distance.

daV6Z+(44332) daV6Z+(44332) daV5Z+(3321) aVQZ+(3321) aVTZ+(3321)

R/Å VSCF Vcorr,CBS VCCSD(T)/FC,CBS VT−(T),CBS V(Q)−T,CBS VQ−(Q) V post CCSD(T)

1.3 126 905.050 −6166.421 120 738.629 −18.071 −17.655 0.428 −35.298 1.5 68 655.518 −4537.334 64 118.184 −10.878 −11.384 0.056 −22.206 1.7 36 052.420 −3132.321 32 920.100 −9.169 −7.068 −0.064 −16.301 1.9 18 513.066 −2107.531 16 405.534 −7.780 −4.435 −0.102 −12.317 2.1 9 347.890 −1401.787 7 946.103 −6.357 −2.853 −0.108 −9.318 2.3 4 658.114 −928.484 3 729.630 −4.966 −1.887 −0.098 −6.951 2.5 2 296.402 −614.828 1 681.574 −3.727 −1.276 −0.083 −5.086 2.7 1 122.099 −408.039 714.060 −2.721 −0.876 −0.067 −3.664 2.9 544.246 −271.990 272.255 −1.947 −0.608 −0.052 −2.606 3.0 378.125 −222.587 155.538 −1.638 −0.507 −0.045 −2.191 3.1 262.336 −182.513 79.823 −1.376 −0.424 −0.039 −1.839 3.2 181.767 −149.989 31.778 −1.153 −0.355 −0.033 −1.542 3.3 125.791 −123.569 2.222 −0.966 −0.298 −0.029 −1.292 3.4 86.958 −102.084 −15.126 −0.809 −0.250 −0.024 −1.083 3.5 60.053 −84.587 −24.535 −0.678 −0.210 −0.021 −0.909 3.6 41.433 −70.314 −28.881 −0.569 −0.177 −0.018 −0.763 3.7 28.563 −58.648 −30.085 −0.478 −0.149 −0.015 −0.642 3.8 19.675 −49.090 −29.415 −0.402 −0.126 −0.013 −0.541 3.9 13.543 −41.240 −27.697 −0.339 −0.107 −0.011 −0.457 4.0 9.316 −34.775 −25.460 −0.287 −0.091 −0.009 −0.386 4.1 6.404 −29.435 −23.032 −0.243 −0.077 −0.008 −0.328 4.2 4.400 −25.011 −20.611 −0.206 −0.066 −0.007 −0.279 4.3 3.021 −21.333 −18.311 −0.176 −0.057 −0.006 −0.238 4.4 2.074 −18.264 −16.191 −0.150 −0.049 −0.005 −0.204 4.5 1.423 −15.696 −14.273 −0.129 −0.042 −0.004 −0.175 4.7 0.669 −11.718 −11.049 −0.095 −0.032 −0.003 −0.130 4.9 0.314 −8.870 −8.556 −0.072 −0.024 −0.002 −0.098 5.2 0.101 −5.984 −5.883 −0.048 −0.016 −0.001 −0.065 5.5 0.032 −4.143 −4.111 −0.032 −0.011 −0.001 −0.045 6.0 0.005 −2.360 −2.355 −0.018 −0.006 −0.001 −0.025 6.5 0.001 −1.416 −1.415 −0.011 −0.004 0.000 −0.015 7.0 0.000 −0.887 −0.887 −0.007 −0.002 0.000 −0.009 8.0 0.000 −0.385 −0.385 −0.003 −0.001 0.000 −0.004 9.0 0.000 −0.186 −0.186 −0.002 −0.001 0.000 −0.002

As the main contribution to the relativistic correction, Spin-(own)-orbit contributions to the Kr–He interaction V DPT2/FC,CBS was determined from second-order direct pertur- energy are accounted for by the third term in Eq. (5). bation theory36–38 (DPT2) computations at the CCSD(T)/FC We obtained this correction as the difference between the dawCVTZ+(3321) − results for the relativistic correction determined from FC level of theory, yielding values of VDPT2/FC = 0.076 K, dawCVQZ+(3321) dawCV5Z+(3321) four-component Dirac–Coulomb (4cDC) computations (with V = −0.069 K, and V = −0.061 K DPT2/FC DPT2/FC explicit calculation of the two-electron (SS|SS) integrals over for R = 3.7 Å. The small magnitude of these results is due to the small component as implemented in the DIRAC pro- a zero-crossing of this term close to the test configuration. gram32) and values computed with the DPT2/CCSD(T)/FC For the shortest interatomic distance, R = 1.3 Å, considerable approach. Using unc-aVXZ+(3321) basis sets, values of contributions as large as −1163.597 K were obtained using V unc−aVDZ+(3321) = 0.065 K and V unc−aVTZ+(3321) = 0.061 K the quintuple-zeta basis set. CBS extrapolation at R = 3.7 Å 4cDC−DPT2 4cDC−DPT2 resulted for R = 3.7 Å. The results for the triple-zeta basis yields V dawCVQZ+(3321) = −0.064 K and V dawCV5Z+(3321) DPT2/FC,CBS DPT2/FC,CBS set were used without CBS extrapolation as the V − 4cDC DPT2 = 0.052 K. reference values. With V DPT2,AE FC, the first term in Eq. (5) is corrected for The last term in Eq. (5), accounting for spin-other- core-core and core-valence correlation effects. The computa- orbit effects, was determined as the difference between the tions at the AE and FC levels of theory were performed using results from four-component calculations with the Dirac– again the unc-aVXZ+(3321) basis sets. For R = 3.7 Å, this cor- Coulomb–Gaunt and Dirac–Coulomb Hamiltonians, both rection is negligibly small (V unc−aVTZ+(3321) = −0.002 K), and DPT2,AE−FC obtained within the molecular mean-field approximation (see even for the shortest interatomic distances, the values are two Refs.9 and 39 for further details). For the krypton-helium orders of magnitude smaller than the corresponding values for atom pair at R = 3.7 Å, this difference amounts only to V DPT2/FC,CBS. unc−aVDZ+(3321) unc−aVTZ+(3321) VGaunt = 0.014 K and VGaunt = 0.007 K. 214302-5 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

This small magnitude is again related to a nearby zero-crossing B. Uncertainty budget of V . However, the summed spin-orbit correction terms Gaunt We followed our study on the krypton dimer9 and cal- are of at least the same or even larger relative importance than culated the combined standard uncertainty of the ab initio for the pure Kr–Kr interaction. The results for all contribu- interaction energies as the square root of the sum of the squared tions beyond the nonrelativistic FC level of theory that were standard uncertainties resulting from the individual contribu- considered for the total pair potential are listed in TableII. tions, which are estimated for the CBS-extrapolated terms As suggested by one of the reviewers of this manuscript, as the systematic error due to the Born–Oppenheimer approximation, which was applied for all quantum chemical calculations, 1 was investigated. We computed the so-called diagonal Born– daV6Z+(44332) − daV6Z+(44332) u(Vcorr) = Vcorr,CBS Vcorr , Oppenheimer correction (DBOC) to the interaction energy at 2 the CCSD/FC level of theory,40 as implemented in the CFOUR 1 daV5Z+(3321) daV5Z+(3321) u(VT−(T)) = V − − V − , program, for all interatomic distances using the aVDZ+(3321) 2 T (T),CBS T (T) and aVTZ+(3321) basis sets. At R = 1.3 Å, the correction 1 aVQZ+(3321) aVQZ+(3321) u(V(Q)−T) = V − − V − , amounts to 28.878 K and 29.191 K for DZ and TZ basis sets, 2 (Q) T,CBS (Q) T respectively, whereas −0.012 K and −0.013 K result for the 1 dawCV5Z+(3321) dawCV5Z+(3321) u(VIFC−FC) = V − V , near minimum distance R = 3.7 Å. Further results can be 2 IFC−FC,CBS IFC−FC found in the supplementary material. The values for the DBOC 1 dawCV5Z+(3321) dawCV5Z+(3321) u(VDPT2/FC,CBS) = V − V obtained with the aVTZ+(3321) basis set were added to the 2 DPT2/FC,CBS DPT2/FC total interaction energies and are therefore included within the results for V tot in TableII. and for the remaining contributions as

TABLE II. Corrections to the Kr–He interaction energy for core-core and core-valence correlation and relativistic effects as well as the total interaction energy [Eq. (2)], its estimated combined standard uncertainty, and the ﬁtted pair potential. Here, “bf” denotes the (3321) set of bond functions. All energies are in Kelvin.

dawCV5Z+bf unc−aVTZ+bf dawCV5Z+bf unc−aVTZ+bf unc−aVTZ+bf unc−aVTZ+bf tot tot tot R /Å VIFC−FC,CBS VAE−IFC VDPT2/FC,CBS VDPT2,AE−FC V4cDC−DPT2 VGaunt V uc(V ) V (ﬁtted) 1.3 −741.513 3.994 −1157.090 −14.306 −89.991 65.540 118 799.156 67.004 118 826.480 1.5 −578.233 −1.276 −979.118 −6.639 −41.466 53.786 62 556.780 37.509 62 553.026 1.7 −385.786 −2.328 −630.953 −2.972 −15.616 36.857 31 909.752 23.180 31 902.412 1.9 −234.874 −1.989 −357.420 −1.330 −4.104 22.785 15 819.640 14.299 15 819.640 2.1 −135.199 −1.388 −187.621 −0.598 0.257 13.209 7 627.085 8.534 7 628.000 2.3 −74.886 −0.873 −93.328 −0.267 1.459 7.305 3 562.860 5.039 3 563.145 2.5 −40.297 −0.511 −44.424 −0.119 1.450 3.882 1 596.813 2.947 1 596.797 2.7 −21.183 −0.282 −20.291 −0.055 1.095 1.985 671.805 1.682 671.771 2.9 −10.916 −0.146 −8.860 −0.026 0.725 0.970 251.442 0.927 251.432 3.0 −7.784 −0.102 −5.735 −0.018 0.570 0.664 140.964 0.680 140.961 3.1 −5.528 −0.070 −3.646 −0.012 0.440 0.446 69.619 0.496 69.619 3.2 −3.911 −0.046 −2.266 −0.009 0.333 0.293 24.627 0.360 24.627 3.3 −2.757 −0.029 −1.364 −0.007 0.248 0.186 −2.802 0.261 −2.802 3.4 −1.938 −0.017 −0.783 −0.005 0.181 0.112 −18.671 0.189 −18.671 3.5 −1.358 −0.009 −0.416 −0.003 0.130 0.062 −27.051 0.137 −27.052 3.6 −0.949 −0.004 −0.189 −0.002 0.090 0.029 −30.682 0.100 −30.682 3.7 −0.662 0.000 −0.052 −0.002 0.061 0.007 −31.387 0.075 −31.388 3.8 −0.461 0.002 0.026 −0.001 0.040 −0.006 −30.369 0.057 −30.369 3.9 −0.321 0.004 0.067 −0.001 0.024 −0.014 −28.406 0.044 −28.406 4.0 −0.224 0.004 0.087 0.000 0.013 −0.018 −25.995 0.036 −25.995 4.1 −0.157 0.005 0.093 0.000 0.005 −0.020 −23.443 0.030 −23.443 4.2 −0.110 0.005 0.091 0.000 0.000 −0.020 −20.934 0.026 −20.934 4.3 −0.078 0.004 0.085 0.000 −0.004 −0.020 −18.569 0.022 −18.569 4.4 −0.056 0.004 0.077 0.000 −0.006 −0.019 −16.400 0.020 −16.399 4.5 −0.041 0.004 0.069 0.000 −0.007 −0.017 −14.446 0.017 −14.446 4.7 −0.023 0.003 0.053 0.000 −0.008 −0.014 −11.172 0.014 −11.172 4.9 −0.015 0.003 0.040 0.000 −0.007 −0.011 −8.648 0.011 −8.648 5.2 −0.009 0.002 0.026 0.000 −0.006 −0.008 −5.946 0.008 −5.946 5.5 −0.007 0.001 0.017 0.000 −0.004 −0.006 −4.156 0.006 −4.156 6.0 −0.005 0.001 0.009 0.000 −0.002 −0.003 −2.382 0.003 −2.382 6.5 −0.003 0.000 0.005 0.000 −0.001 −0.002 −1.432 0.002 −1.432 7.0 −0.002 0.000 0.003 0.000 −0.001 −0.001 −0.898 0.001 −0.898 8.0 −0.001 0.000 0.001 0.000 0.000 −0.001 −0.390 0.001 −0.390 9.0 −0.001 0.000 0.000 0.000 0.000 0.000 −0.189 0.000 −0.189 214302-6 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

dispersion coefficients were obtained simultaneously utilizing u(V ) = V daV6Z+(44332) − V daV5Z+(44332), SCF SCF SCF the approximate recursion formula41 aVTZ+(3321) − aVDZ+(3321) u(VQ−(Q)) = VQ−(Q) VQ−(Q) , !3 unc−aVTZ+(3321) − unc−aVDZ+(3321) C2n−2 u(VAE−IFC) = VAE−IFC VAE−IFC , C2n = C2n−6 , n ≥ 6. (7) C2n−4 unc−aVTZ+(3321) − unc−aVDZ+(3321) u(V4cDC−DPT2) = V4cDC−DPT2 V4cDC−DPT2 , unc−aVTZ+(3321) − unc−aVDZ+(3321) tot u(VGaunt) = VGaunt VGaunt . Except for the shortest interatomic distance, all values of V are represented by the analytical function within ±0.01%. In Due to the small magnitude of V , no DPT2,AE FC Table III, all parameters are listed along with ε/kB and Rε. uncertainty contribution was considered for this term. For An additional set of potential parameters was obtained from post-CCSDTQ contributions, we conservatively allocated a U tot a fit to the values V = V + uc to allow a propagation of standard uncertainty of u(V post CCSDTQ) = 2u(V Q (Q)). The the uncertainty of the pair potential to the thermophysical uncertainty of the DBOC contribution was neglected. By prop- properties. Due to the small magnitude of uc, the error prop- agating the uncertainty, a combined standard uncertainty of tot agation can be expected to be symmetric for V + uc and u = 0.075 K was obtained for R = 3.7 Å, where the total tot c V uc. interaction energy, summed up according to Eq. (2), amounts The relative deviation of the C coefficient for the pair tot 6 to V = − 31.387 K. The values of the total interaction potential of the present work from the value of Kumar and tot energy and of the combined standard uncertainty, uc(V ), are Meath42 obtained from the dipole-oscillator strength distri- listed for all 34 interatomic distances R in TableII. Further bution (DOSD) is about 0.5% and thus smaller than the quantum-chemical results are tabulated in the supplementary estimated uncertainty of the DOSD value of ±1%. The com- material. tot bined standard uncertainty of the well depth for V (ε/kB = U 31.416 K) follows from V as uc(ε/kB) = 0.079 K. The distance of the minimum results is Rε = (3.6822 ± 0.0012) Å. III. ANALYTICAL POTENTIAL FUNCTION Regarding ε/k and R , none of the literature pair poten- OF THE KRYPTON-HELIUM ATOM PAIR B ε tials is in agreement with the potential of the present work within its uncertainty estimates. However, the agreement A modified Tang–Toennies type function41 was employed of the well depth is considerably closer for the empirical to represent the 34 total interaction energies potential of Danielson and Keil20 than for the later poten- 2 −1 −2 19 V(R) = A exp(a1R + a2R + a−1R + a−2R ) tial of Keil et al. and the ab initio potential of Haley and Cybulski.15 X8 C X2n (bR)k − 2n 1 − exp(−bR) . (6) R2n k! n=3  k=0    IV. CROSS SECOND VIRIAL COEFFICIENT A least-squares procedure was used to determine the parameters A, a1, a2, a 1, a 2, b, C6, C8, and C10. The higher A. Theory

The cross second virial coefficient, B12, is used for the assessment of the quality of the ab initio potential since tot TABLE III. Parameters of the ab initio pair potential, V , and characteristic it corresponds to that part of the first-order non-ideal ther- parameters of the pair potentials by Danielson and Keil,20 by Keil et al.,19 and 15 modynamic behavior of gaseous krypton-helium mixtures by Haley and Cybulski as well as a literature value for the C6 coefficient by Kumar and Meath.42 Not all figures displayed are significant but are given to which emerges from the interaction between krypton and avoid round-off errors. helium. For the calculation of B12 as a function of the temper- Unit Value Lit. values Ref. ature T, a semi-classical approach was employed, i.e., B12 A K 0.264 217 158 3 × 108 was obtained as a sum of the classical contribution and first- 1 1 a1 Å 0.341 325 355 2 × 10 through third-order quantum corrections 2 −1 a2 Å 0.767 439 136 4 × 10 1 a 1 Å 0.114 653 419 3 × 10 cl qm,1 2 qm,2 3 qm,3 2 B12(T) = B12(T) + λB12 (T) + λ B12 (T) + λ B12 (T). (8) a 2 Å 0.144 719 927 7 b Å 1 0.304 506 839 4 × 101 6 5 5 C6 KÅ 0.934 294 308 4 ×10 0.929 16 × 10 42 In the case of the interaction between two unlike particles, 2 −1 0.953 43 × 105 15 λ= ~ β/24µ12, β = (kBT) , µ12 = m1m2/(m1 + m2), and 8 6 C8 KÅ 0.497 022 115 8 × 10 ~ is Planck’s constant divided by 2π; m1 = 83.798 u and 10 7 C10 KÅ 0.380 133 757 3 × 10 m2 = 4.0026 u are the average atomic masses of naturally ε/kB K 31.416 2 30.95 20 occurring krypton and helium, respectively. The expressions 29.45 19 cl qm,3 for the contributions B12(T),...,B12 (T) are the same as in 29.84 15 the homoatomic case and can be found elsewhere (see, for R Å 3.682 20 3.702 20 ε example, Ref. 43). Standard numerical integration was used 3.693 19 3.709 15 for the computation of the required integrals. The results are converged to within 0.001 cm3 mol−1. 214302-7 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

B. Results virial coefficients of pure krypton are characterized by considerable deviations from the computed values.9 Moreover, The cross second virial coefficient for krypton-helium Fig. 1(b) illustrates that neglecting the quantum corrections was calculated for the temperature range between 50 K and would deteriorate the agreement significantly. Dillard et al.47 5000 K employing the pair potential functions of the present used the Burnett method to derive B values with a low exper- work V tot as well as for the pair potentials of Danielson 12 imental uncertainty of only 0.3 cm3 mol−1. Their data are and Keil,20 Keil et al.,19 and Haley and Cybulski.15 The in a very good agreement with our theoretical values, thus standard uncertainty u(B ) was computed as the differ- 12 confirming the accuracy of the pair potential of the present ence between the results obtained for V U and V tot. For work. T = 50 K and the pair potential of the present work, the Since the empirical correlation of Dymond et al.44 was individual contributions according to Eq. (8) resulted as Bcl 12 fitted to the discussed experimental data, it is not surprising − 3 −1 qm,1 3 −1 2 qm,2 = 34.648 cm mol , λB12 = 4.797 cm mol , λ B12 that only small deviations from the theoretical predictions of − 3 −1 3 qm,3 3 −1 3 −1 = 0.349 cm mol , and λ B12 = 0.050 cm mol , thus less than 0.5 cm mol were found for the range of validity confirming that the semi-classical approach yields reliably of the correlation. However, the inappropriate extrapolation converged values with respect to the quantum-mechanical behavior to temperatures below and above this range limits limit. At ambient temperature, the total quantum correction its general applicability. The B12 values calculated for the amounts to 0.27 cm3 mol−1, with the second- and third-order empirical pair potential of Danielson and Keil20 deviate from quantum corrections being negligibly small. Results for the those for V tot by less than 0.14 cm3 mol−1 for the complete classical and the semi-classical approaches are listed along temperature range. This is due to the fact that the experimen- with the standard uncertainty in the supplementary material. tal cross second virial coefficients were used as a substantial In Fig.1, the theoretical predictions for B12 are com- source of information within the multi-property fitting pro- pared with experimental data, values computed for the pair cedure. However, the empirical potential presented by Keil 19 potentials from the literature, and the empirical correlation et al. yields significantly larger B12 values, with deviations of Dymond et al.44 The data of Brewer,45 obtained from of about 0.6 cm3 mol−1 at room temperature and more than measurements of the pressure change on mixing the pure 4 cm3 mol−1 for T = 50 K in accordance with the shallower gases, are consistent with the computed values within the well of the pair potential. The reason for this disagreement will rather large experimental uncertainty of 2 cm3 mol−1 except be discussed below. For the ab initio pair potential of Haley and for one datum at 275 K. Kate et al.46 determined their Cybulski,15 the deviations are similar to those for the poten- 19 low-temperature B12 values from measurements of the tial of Keil et al. due to the close agreement of the well krypton-helium vapor-solid equilibrium with estimated uncer- depths of these two pair potentials. In this case, the deviations tainties of 0.6 to 1.3 cm3 mol−1. The good agreement with our can clearly be attributed to the neglect of several quantum- theoretical results is quite remarkable, especially considering chemical contributions that lower the interaction energy the fact that the experimentally based low-temperature second considerably.

FIG. 1. Experimentally based and theoretically calculated values for the cross second virial coefﬁcient of the krypton-helium atom pair. (a) Absolute values; (b) absolute deviations from values calculated for the pair potential of the present work including quantum corrections. Experimental data: H, Brewer;45 , Kate et al.;46 N, Dillard et al.47 Experimental correlation: gray line, Dymond et al.44 Calculated values: green line, empirical potential of Danielson and Keil;20 red line, empirical potential of Keil et al.;19 blue line, ab initio potential of Haley and Cybulski;15 ———, classical calculation using the potential of the present work; ··········, range of the standard uncertainty. 214302-8 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

V. TRANSPORT PROPERTIES system of linear equations is given by A. Theory ∞ ! ! X 1s 0 1s 0 S˜ X1s + S˜ X1s " 1s0 AB 1s0 BB# Using the Chapman–Enskog solution to the Boltz- s0=0 AA AB mann equation for small perturbations of the thermody- xAmA 10 namic equilibrium is a well-established strategy for computing = δs0 CA , xAmA + xBmB transport properties of low-density noble gases and their mix- ∞ ! ! X 1s 0 1s 0 tures.1 Within this approach, the intermolecular potentials S˜ X1s + S˜ X1s " 1s0 AB 1s0 BB# and the transport coefficients are related through temperature- s0=0 BA BB (l,m) dependent collision integrals Ω introduced by Chapman xAmA 10 48 24 = −δs0 CB , and Cowling. Mason derived explicit expressions for the xAmA + xBmB computation of the transport coefficients as a function of the s = 0, ... , ∞, (12) collision integrals (fourth-order approximation to the binary 10 1/2 diffusion coefficient, third-order approximation to the thermal where Cα = (kBT/mα) , mα is the atomic mass of the com- 1s diffusion factor, and first-order approximations to viscosity ponent α, and the coefficients S˜ 0 are obtained from those 1s αβ and thermal conductivity; second-order formulae for viscosity defined in Eq. (10) as and thermal conductivity were given in Ref. 49). ! ! !1/2 ! In our study on pure krypton,9 we introduced a generalized 1s 1s xβ mβ 1s ˜ 0 50–53 S 0 = S 0 − δs0δs 0 S 0 . (13) formalism based on the kinetic theory of molecular gases. 1s αβ 1s αβ xα mα 1s αα In this approach, temperature-dependent generalized cross sections σ characterize the binary collision dynamics. Gen- The product of molar density and binary diffusion coefficient, eral expressions were derived for the collision between any two ρmD, is given as spherically symmetric particles α and β [Eqs. (16) and (17) of 2 (xAmA + xBmB) 10 10 Ref.9] as well as for the relation between the generalized cross ρmD = CA XAB, (14) 0 ps 00 ps NAmAmB sections σ 0 and σ 0 and the collision integrals ps αβ ps αβ where NA is Avogadro’s constant. Ω(l,m) [Eq. (18) of Ref.9]. These relations can be employed αβ The thermal diffusion factor αT and the thermal conduc- for the dynamics of unlike particles without modification and tivity λ are obtained from the system of linear equations are therefore not repeated here. ∞ ! ! The viscosity η of the binary noble gas mixture A–B (in X 1s 0 1s 0 S˜ X1s + S˜ X1s = δ C11, our case, A = Kr and B = He) is obtained from the system of 0 A 0 B s1 A 0 " 1s AA 1s AB # linear equations s =0 ∞ ! ! X 1s 0 1s 0 ∞ ! ! S˜ X1s + S˜ X1s = δ C11, X 0 0 0 A 0 B s1 B 2s 2s 2s 2s 20 " 1s 1s # S X + S X = xAδs0C , s0=0 BA BB " 2s0 A 2s0 B # s0=0 AA AB s = 0, ... , ∞, (15) ∞ ! ! X 2s 0 2s 0 2s 2s 20 / S X + S X = xBδs0C , where C11 = (5k T/2m )1 2. The product of the molar density " 2s0 A 2s0 B # α B α s0=0 BA BB T and the thermal diffusion coefficient, ρmD , and the thermal s = 0, ... , ∞. (9) diffusion factor result as

2s0 2s0 T xAmA + xBmB 10 10 Here, δij is the Kronecker delta, XA and XB are the solutions ρmD = xA CA XA , (16) NAmB of the equation√ system, xA and xB are the mole fractions, and 20 2s C = 2. The coefficients S 0 are given as T 2s αβ D αT = . (17) ! ! xAxBD ps X ps S = δ x x hvi σ0 56 ps0 αβ α γ αγ ps0 Muckenfuss and Curtiss showed that two limiting cases for αβ γ=A,B αγ the thermal conductivity of gaseous mixtures can be defined. ! ps By assuming a uniform mixture, one obtains λ , correspond- + x x hvi σ00 , (10) 0 α β αβ 0 ing to a state prior to any temperature-induced change of ps αβ the concentration. In contrast, λ∞ denotes the steady-state 1/2 where hviαβ = (8kBT/πµαβ) is the average relative thermal thermal conductivity, which results after the (thermal) dif- speed. The viscosity of the mixture results as fusion driven by the temperature gradient is ceased and a time-independent concentration gradient is formed. Unfortu- 1 η = k TC20 x X20 + x X20 . (11) nately, Muckenfuss and Curtiss only reported some theoretical 2 B A A B B results for ternary mixtures (e.g., He–Ar–Xe, where λ0 and Equations (9)–(11) are simplified versions of Eqs. (10)–(12) λ∞ differ by up to 1.8%), and no detailed study of this effect in Ref. 54, which were given for a binary mixture of molecular for binary mixtures is available in the literature. Moreover, gases. An error in Eq. (12) of Ref. 54 was corrected in Ref. 55. there is no experimental setup for the determination of λ0; A similar derivation can be made from Eqs. (14)–(16) even the so-called transient hot-wire (THW) technique yields in Ref. 54 to obtain the formalism for binary diffusion. The steady-state results for λ.57 Following Monchick et al.,58 it 214302-9 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017) can be shown that both λ0 and λ∞ are obtained from a simi- integrals for He–He and Kr–He, which show an almost iden- lar system of linear equations [Eq. (15), the same as for αT] tical behavior and are tabulated in the supplementary material as for the complete temperature range, can be regarded as fully 11 11 11 11 converged. λ = kB(xACA XA + xBCB XB ). (18) In Fig.2, theoretical results obtained for the pair poten- 0 tial of the present work, V tot, are compared with experimental The only difference is that for λ0, the indices s and s start with 0 data and values for the pair potentials from the literature. The zero as indicated in Eq. (15), whereas s, s 1 for λ∞. > 63 For s, s0 6 n, one obtains the (n + 1)th-order approxi- room temperature data of Thornton, obtained with a modi- mations for viscosity and binary diffusion and the nth-order fied Rankine capillary viscometer, agree with our values within approximation for thermal diffusion and conductivity. We the estimated experimental uncertainty of 1%. Kestin and co- 64,65 computed the transport coefficients for different orders of workers determined viscosity values of krypton-helium approximation with n 6 6 by numerically solving the sys- mixtures by means of relative measurements with oscillating- tems of linear equations. The averaged atomic masses of disk viscometers. Their claimed uncertainty of 0.1% is cer- helium and krypton were used throughout (see Sec.IVB). tainly too optimistic due to an outdated calibration value and The effects beyond this approximation were tested by treat- a temperature measurement error as thoroughly discussed, for ing krypton as a mixture of its naturally occurring isotopes example, in Ref.9. In accordance with the findings for pure 9 and were found to be negligibly small. All collision integrals krypton, the room-temperature data deviate from the theoret- for the Kr–Kr atom pair were computed classically using our ical results by less than 0.4%, whereas the deviations increase recent Kr–Kr ab initio pair potential9 and a modified version to up to 1.0% at T = 473.15 K and to 0.6% at T = 873.15 K. of the program code developed by O’Hara and Smith.59,60 For Figure2 illustrates that the difference between the vis- Kr–He and He–He, we employed both the classical and the cosity values obtained from classical collision integrals for quantum-mechanical (phase shift) approach. The theory and Kr–He and He–He and those from the corresponding quantum- application of the latter approach are described in detail in Ref. mechanical treatment does not exceed 0.2% at ambient tem- 61 and references therein. The quantum-mechanical collision perature. The results computed for the empirical potentials of 20 19 integrals were computed using an optimized program code Danielson and Keil and Keil et al. deviate from the val- originally developed by Hurly et al.62 Note that the quantum- ues for the potential of the present work by less than 0.05%, mechanical treatment is limited to n 6 4. The ultra accurate ab whereas the results for the ab initio potential of Haley and initio pair potential of Cencek et al.3 was used for the computation of the He–He collision integrals. For comparison, we calculated η and ρmD also for the Kr–He pair potentials from Refs. 15, 19, and 20. All results obtained for the pair potential of the present work V tot in the temperature range from 70 K to 5000 K are listed in the electronic supplementary material.

B. Viscosity In TableIV, results for the mixture viscosity at T = 300 K computed from classical collision integrals illustrate the convergence behavior of the kinetic theory solutions with respect to the order of approximation, (n + 1). The second- order correction, i.e., the difference between the results for n = 1 and n = 0, increases almost linearly from 0.05% for pure krypton to 0.6% for pure helium, whereas the higher-order corrections are characterized by the maxima between xHe = 0.6 and xHe = 0.7. Sixth- and seventh-order contributions are always smaller than 0.001%. Therefore, the 5th-order results based on the quantum-mechanical collision

TABLE IV. Viscosity η of krypton-helium mixtures at T = 300 K for different orders of approximation, (n + 1), of the kinetic theory based on classically computed collision integrals in 10−6 Pa s. FIG. 2. Relative deviations of experimental and theoretical values for the viscosity of low-density krypton-helium mixtures from those calculated using the pair potential of the present work (pair potentials for He–He and Kr– xHe n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 Kr of Cencek et al.3 and Jager¨ et al.,9 respectively). Experimental data: ⊕, 63 0.0 25.4078 25.4203 25.4207 25.4208 25.4208 25.4208 25.4208 T = 291.3 K, Thornton; , T = 293.15 K; , T = 303.15 K, Kestin 64 0.2 25.9560 26.0025 26.0077 26.0088 26.0090 26.0091 26.0091 et al.; , T = 298.15 K; N, T = 372.15 K; H, T = 473.15 K; ^, T = 573.15 K; ◦, T = 675.15 K; 4, T = 773.15 K; O, T = 873.15 K, Kalelkar 0.4 26.4847 26.5656 26.5748 26.5765 26.5769 26.5770 26.5770 and Kestin.65 Calculated values for T = 291.3 K: green line, empirical poten- 0.6 26.7748 26.8862 26.8976 26.8996 26.9000 26.9001 26.9001 tial of Danielson and Keil;20 red line, empirical potential of Keil et al.;19 blue 0.8 25.9762 26.1047 26.1154 26.1170 26.1173 26.1173 26.1173 line, ab initio potential of Haley and Cybulski;15 ———, classical calcula- 1.0 19.7569 19.8736 19.8801 19.8807 19.8808 19.8808 19.8808 tion using the potential of the present work; ··········, range of the standard uncertainty. 214302-10 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

Cybulski15 are characterized by negative deviations of up to 0.39%. It is to note that the mixture viscosity is dominated by the inﬂuence of the He–He and Kr–Kr interactions, which were modeled by the same pair potential functions in all calculations. The viscosity values computed for the potential V U, which includes the combined standard uncertainties of the ab initio interaction energies, differ by less than 0.02% from the values for V tot for the complete range of temperature and composition. Using this result and the computed standard uncertainty for the pure-krypton viscosity,9 we estimate the standard uncertainty of the krypton-helium mixture viscosity values to be 0.14%. This value represents only a global and conservative estimate for the standard uncertainty. Therefore, the uncertainty for the viscosity of helium-rich mixtures can be expected to be considerably smaller due to the high accuracy of the He–He interaction potential.

C. Thermal conductivity Classical results for the thermal conductivity at 300 K are listed in TableV. In contrast to the behavior of the viscosity, already the second-order correction is characterized by FIG. 3. Relative deviations of experimental data for the thermal conductivity λ a pronounced maximum with about 2.5%, whereas the cor- ∞ of krypton-helium mixtures from values calculated using the pair potential of the present work. Experimental data: , T = 302.15 K; , T = 793.15 K, rection amounts to 0.08% and 0.9% for pure krypton and Mason and von Ubisch;67 ⊕, Thornton;63 , T = 308.15 K; 4, T = 323.15 K; helium, respectively. The third- and fourth-order corrections H, T = 343.15 K; ♦, T = 363.15 K, Gambhir and Saxena.68 Uncertainty range contribute up to 0.5% and 0.13% to the mixture thermal con- is not shown to improve readability. ductivity, the former value being consistent with the third-order results for argon-helium and xenon-helium mixtures reported In Fig.3, the theoretical results are compared with the by Assael et al.66 Fifth- and sixth-order results for T = 300 K few available experimental data for the thermal conductiv- are characterized by differences of less than 0.03% and 0.01%, ity of krypton-helium mixtures. The room-temperature data respectively. The cumulated correction for effects beyond n of Mason and von Ubisch67 as well as of Thornton63 are in = 4 is always smaller than 0.1% so that the fourth-order results agreement with the computed values within ±2.1% and ±4.2%, based on the quantum-mechanical collision integrals can be respectively, whereas the results for T = 793.15 K from Ref. 67 regarded as converged, considering also the limited accuracy deviate by up to 10.4%. The data reported by Gambhir and Sax- of the experimental techniques for the determination of λ. The ena68 show a good agreement for the krypton-rich mixtures but difference between λ0 and λ∞ amounts to up to 2.3% at 300 K. increasing deviations with increasing xHe. This considerable effect is due to the large mass difference of We do not discuss the theoretical results for the ther- krypton and helium. The theoretical computations of this dif- mal conductivity with respect to the different pair potentials ference by Muckenfuss and Curtiss56 performed for the ternary since the relative deviations are essentially the same as for mixture He-Ar-Xe yielded a similar order of magnitude. the viscosity. Taking into account the slower convergence

TABLE V. Thermal conductivity λ of krypton-helium mixtures at T = 300 K for different orders of approximation n of the kinetic theory based on classically computed collision integrals in 10−3 W m−1 K−1.

xHe n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

Uniform-mixture thermal conductivity, λ0 0.0 9.453 39 9.460 55 9.460 86 9.460 92 9.460 92 9.460 92 0.2 19.924 2 20.361 4 20.450 3 20.475 4 20.483 4 20.485 9 0.4 34.375 3 35.274 7 35.441 9 35.485 8 35.498 9 35.502 9 0.6 55.598 6 56.949 4 57.168 5 57.220 2 57.234 4 57.238 4 0.8 89.781 8 91.436 5 91.649 9 91.692 4 91.702 5 91.705 1 1.0 153.902 155.302 155.406 155.420 155.422 155.422

Steady-state thermal conductivity, λ∞ 0.0 9.453 39 9.460 55 9.460 86 9.460 92 9.460 92 9.460 92 0.2 19.612 7 19.998 7 20.075 3 20.096 8 20.103 7 20.105 9 0.4 33.714 6 34.516 6 34.662 6 34.700 7 34.712 1 34.715 5 0.6 54.593 3 55.818 2 56.013 1 56.058 8 56.071 4 56.074 9 0.8 88.640 2 90.185 8 90.381 9 90.420 6 90.429 9 90.432 2 1.0 153.902 155.302 155.406 155.420 155.422 155.422 214302-11 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

TABLE VI. Product of the molar density and the binary diffusion coefficient, is observed for xHe → 0 with subsequent corrections of 3.1%, ρmD, of krypton-helium mixtures at T = 300 K for different orders of approx- 0.55%, 0.15%, 0.046%, 0.015%, and 0.005% for n = 1 through imation of the kinetic theory based on classically computed collision integrals n = 6. The cumulated sixth- and seventh-order corrections are in 10−4 mol m−1 s−1. always smaller than 0.03%. xHe n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 In Fig.4, the theoretical results computed from the new ab initio Kr–He pair potential are compared with selected experi- 0.0 25.7490 26.5350 26.6818 26.7215 26.7338 26.7378 26.7391 19,69–75 0.2 25.7490 26.4781 26.6060 26.6387 26.6484 26.6514 26.6522 mentally based diffusion data and with results obtained 15,19,20 0.4 25.7490 26.3996 26.5042 26.5292 26.5361 26.5381 26.5387 for the pair potentials from the literature. Where neces- 0.6 25.7490 26.2842 26.3602 26.3766 26.3808 26.3819 26.3822 sary, the experimental data of ρmD were calculated from the 0.8 25.7490 26.0983 26.1390 26.1466 26.1484 26.1488 26.1489 measured values for the diffusion coefficient and the pressure 1.0 25.7490 25.7500 25.7500 25.7500 25.7500 25.7500 25.7500 assuming ideal gas conditions. We applied this procedure to be in accordance with the common approach that was used to obtain diffusion data at atmospheric pressure from that at of the series of approximations for the thermal conductivity experimental conditions (see, for example, Ref. 69). The esti- compared to that for viscosity, we assign a slightly higher mated experimental uncertainty was usually in the order of standard uncertainty of 0.2% to the calculated results for λ0 (1–2)%, except for the values of Staker and Dunlop73 and and λ∞. Keil et al.,19 for which 0.2% and 0.3% were reported. We estimated the relative standard uncertainty of our calculated D. Binary diffusion values for ρmD, obtained from quantum-mechanical phase Among the mixture transport properties, the binary dif- shift calculations and the fifth-order approximation of kinetic fusion coefficient exhibits the strongest dependence on the theory, as follows. The first uncertainty contribution stems unlike-particle pair potential. As can be seen from TableVI, the from the difference between the results for V tot and for V U. I first-order approximation yields a composition-independent It was found to be ur(ρmD) = 0.03%, showing almost no value for ρmD. Only the second- and higher-order corrections dependence on temperature. The second contribution to the contain contributions from the He–He and Kr–Kr potentials. uncertainty was estimated to be the difference between the Our calculations confirm the typical behavior for a binary fourth- and fifth-order results, having in mind that the cumu- mixture with a large mass difference. Negligible higher-order lated higher-order contributions (beyond n = 4) are always corrections result for xHe → 1, whereas a slower convergence smaller than this difference. We observed that the second

FIG. 4. Experimental and theoretical values for the product of the molar density and the binary diffusion coefficient, ρmD. (a) Absolute values of ρmD for T = 300 K; (b) relative deviations of ρmD values at T = 300 K from those obtained for the pair potential of the present work (fifth-order approximation of kinetic theory); (c) relative deviations of ρmD values from those obtained for the pair potential of the present work as a function of temperature. Experimental data: H, Srivastava and Barua;69 O, Srivastava and Paul;70 ⊕, Fedorov et al.;71 ?, Van Heijningen et al.;72 ♦, Staker and Dunlop;73 , Cain and Taylor;74 4, Keil et al.19 Experimental correlation: gray line, Arora et al.75 Calculated values: (a) black lines correspond from bottom to top to first-order through fifth-order results using the pair potential of the present work, further theoretical results are shown for the first-order approximation; (b) black lines correspond from bottom to top to first-order through fifth-order results using the pair potential of the present work, further theoretical results are shown for the fifth-order approximation; (c) theoretical results are shown for the fifth-order approximation; green line, empirical potential of Danielson and Keil;20 red line, empirical potential of Keil et al.;19 blue line, ab initio potential of Haley and Cybulski;15 ———, classical calculation using the potential of the present work; ··········, range of the standard uncertainty (not shown in (a) and (c) for clarity). 214302-12 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017) uncertainty contribution is again almost independent of tem- temperatures to negative deviations at higher ones can be II perature and can be represented by ur (ρmD) = 0.06% × xKr. observed. Since these experimental diffusion data were used The quadratic propagation of the two contributions results in as a key quantity within the multi-property fitting procedure, combined relative standard uncertainties ranging from 0.03% the empirical pair potential of Keil et al.19 reproduces this to 0.07%. trend, thereby leading to even larger deviations from our pre- In Fig. 4(a), absolute values for the product of the molar dicted values of about 1% at 100 K and 0.9% at 1200 K. density and the binary diffusion coefficient are shown for The values computed for the earlier empirical pair poten- T = 300 K as a function of the mixture composition. The tial of Danielson and Keil20 differ by less than ±0.2% from straight lines correspond to the theoretical results for the first- the values obtained for the potential of the present work. order approximation using the quantum-chemical phase shifts, Furthermore, Fig. 4(c) shows that quantum effects on the whereas the dashed line corresponds to the results based on diffusion coefficient increase from 0.3% at 300 K to 1.2% classical collision integrals. At this particular temperature, the at 100 K. values obtained for the empirical pair potentials of Danielson and Keil20 and Keil et al.19 are in close agreement with those E. Thermal diffusion calculated for the ab initio potential of the present work. This In Table VII, classically computed thermal diffusion fac- is due to the fact that the accurate diffusion data from the Dun- tors for krypton-helium mixtures at 300 K are collected for lop group19,73,75 were considered within the multi-property different orders of approximation of the kinetic theory. The fitting procedure with large weights. The third- through fifth- results show that the convergence behavior of the thermal order results are in agreement with the experimentally based diffusion factor is strongly dependent on the composition of correlation of Arora et al.75 within its stated uncertainty of the mixture. For x → 0, the largest higher-order contri- 0.3%. Figure 4(b) shows relative deviations of diffusion data He butions were observed. The higher orders of approximations at ambient temperature from the values calculated for the pair successively increase α by 5.6%, 1.4%, 0.45%, 0.15%, and potential of the present paper (quantum-mechanically, fifth- T 0.05%. Moreover, it is important to note that these contribu- order approximation). The fifth-order results for the ab initio tions have the largest relative effect of all transport properties. potential of Haley and Cybulski15 deviate by about 0.9% However, the measurement of the thermal diffusion factor from those for the current potential and disagree with the is afflicted with the largest experimental uncertainty of all correlation of Arora et al. The values for the empirical pair transport properties, too. In Fig.5, the theoretical results, potentials deviate by less than 0.15% from our ab initio obtained for the new pair potential using quantum-mechanical data. phase shifts, are compared with the available experimen- Staker and Dunlop73 reported highly accurate pressure tal data.77–83 Most of the data are characterized by a large dependent diffusion coefficients measured with a Loschmidt scatter within each of the data series except for the data cell at one mixture composition, x = 0.1, at 300 K. They Kr of Atkins et al.,77 which differ systematically by 5% to performed an extrapolation to the zero-density limiting value 11% from our fourth-order results, and the data of Trengove for ρ D, which was assessed to have an uncertainty of 0.2% m et al.,83 which show deviations of 3.7% and 1.2%. Figure and is in perfect agreement with our theoretical prediction. 5(b) illustrates that the thermal diffusion factor exhibits a Figure 4(c) shows the comparison with further experimental clear maximum, a feature which is not reflected by the data, covering a range from 111 K to 1194 K. All the data of experimental data. It is worth mentioning that quantum effects Srivastava and Barua,69 Srivastava and Paul,70 Fedorov et al.,71 contribute up to 1.3% to the thermal diffusion factor at 300 K. Van Heijningen et al.,72 and Cain and Taylor74 were deter- Therefore, the theoretical results obtained for the fourth-order mined at equimolar composition with estimated uncertainties approximation using quantum-mechanical phase shifts can be of (1–2)%. They are characterized by deviations of less than expected to be more reliable than the sixth-order classical the reported uncertainties from the corresponding theoretical results. Since the relative magnitudes of the contributions for values. Further experimental diffusion data with larger devia- n > 4 strongly depend on the composition of the mixture tions have already been discussed by Danielson and Keil (see and also on temperature, we give only an upper bound to the Ref. 20 and references therein). Recently, Kugler et al.76 mea- standard uncertainty of our calculated fourth-order α values sured gas diffusion coefficients for various mixtures with the T of 0.5%. help of a Loschmidt cell combined with holographic interfer- ometry. Their data, reported for x = 0.5, temperatures from 293 K to 353 K, and p = 2 bars, deviate from the current TABLE VII. Thermal diffusion factor αT of krypton-helium mixtures at theoretical values by 2.2% to 1.4%. These differences are T = 300 K for different orders of approximation of the kinetic theory based on classically computed collision integrals. close to the estimated experimental standard uncertainties [due to the large deviations, we omitted the data of Kugler et al. in xHe n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 Fig. 4(c)]. The data of Keil et al.19 deserve closer attention. The 0.0 0.264 143 0.279 007 0.282 978 0.284 236 0.284 659 0.284 795 authors derived ρmD limiting values for xKr → 0 from composition dependent measurements in the temperature range 0.2 0.305 789 0.321 690 0.325 693 0.326 895 0.327 279 0.327 397 0.4 0.363 093 0.380 101 0.384 054 0.385 163 0.385 495 0.385 590 220 T/ K 400 with an estimated uncertainty of 0.3%. 6 6 0.6 0.446 902 0.465 061 0.468 820 0.469 778 0.470 042 0.470 112 Figure 4(c) illustrates that our computed values are in 0.8 0.581 200 0.600 654 0.603 964 0.604 688 0.604 863 0.604 904 agreement with their measurements within this uncertainty 1.0 0.831 342 0.853 937 0.856 339 0.856 708 0.856 775 0.856 788 range. However, a clear trend of positive deviations at lower 214302-13 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017)

FIG. 5. Experimental and theoretical values for the thermal diffusion factor, αT. (a) Relative deviations of αT values from those obtained for the pair potential of 77 78 the present work (fourth-order approximation of kinetic theory); (b) absolute values of αT for xHe → 1. Experimental data: , Atkins et al.; , Velds et al.; N, Annis et al.;79 ?, Santamar´ıa et al.;80 , Titov and Suetin;81 ♦, T = 247.15 K; O, T = 337.15 K; 4, T = 493.15 K, Taylor;82 , Trengove et al.83 Calculated values: (a) black lines correspond from bottom to top to ﬁrst-order through fourth-order results using the pair potential of the present work for T = 337.15 K; (b) black line corresponds to fourth-order results using the pair potential of the present work.

VI. SUMMARY AND CONCLUSIONS disagreement was found with the results for the other pair potentials. Ab initio supermolecular calculations for the krypton- Transport properties of low-density krypton-helium mix- helium atom pair were carried out for 34 interatomic tures were calculated by means of the kinetic theory of gases separations. Especially, effects beyond the nonrelativistic using both classical and quantum-mechanical formulations frozen-core CCSD(T) level of theory were investigated care- for the collision integrals. The classical computations were fully. Contrary to the homoatomic cases Ar–Ar and Kr–Kr, conducted for the first-order through seventh-order of approx- where higher-order excitation effects beyond the CCSD(T) imation for the shear viscosity and the product of the molar approach almost cancel each other, the T–(T) and (Q)–T cor- density and the binary diffusion coefficient. For the thermal rections always have the same sign for Kr–He, a behavior conductivity and the thermal diffusion factor, the sixth-order similar to that of the He–He and Ne–Ne atom pairs. The approximation was the upper limit. The results show that all post-CCSD(T) corrections increase the depth of the poten- transport properties converge smoothly and sufficiently to the tial by about 2%. The effect of core-core and core-valence infinite-order limiting values within the seven or sixth steps correlation on the interaction energies was found to be of non- presented here. We believe that 12th- or 20th-order approx- negligible magnitude. The corresponding correction terms imations as discussed by Sharipov and Benites26 for argon- deepen the well of the potential by about 2%. As expected, helium mixtures would give no further benefit. The limited the relativistic correction is dominated by the scalar relativis- achievable accuracy of the experimental methods for the deter- tic contribution obtained from DPT2 computations, whereas mination of transport properties of gaseous mixtures leads to the spin-orbit correction terms are of distinctly minor impor- the conclusion that already fifth-order results (for viscosity and tance. The total interaction energies were used to determine binary diffusion) and fourth-order results (for thermal conduc- the parameters of an analytical potential function. The result- tivity and thermal diffusion) can be regarded as sufficiently ing potential energy curve is characterized by a considerably converged. deeper well of 31.42 K compared to the ab initio potential The computed mixture viscosity values are in good agree- of Haley and Cybulski15 and the empirical potential of Keil ment with the available experimental data at room temperature et al.19 with values of 29.84 K and 29.45 K, respectively. in accordance with the investigations on the pure components However, the well depth of the earlier empirical pair potential krypton and helium. Only the comparison of calculated val- of Danielson and Keil20 differs by less than 0.5 K from our ues with experimental data for the binary diffusion coefficient result. allows us to assess the quality of the empirical and ab initio The calculated values for the cross second virial coeffi- pair potentials. The results for the ab initio potential of Haley cient reflect the differences between the pair potential func- and Cybulski15 differ considerably from the best experimental tions. We found a very good agreement of our calculated data and from our computed values, whereas the values for 20 B12 values with the results for the potential of Danielson and the earlier empirical potential of Danielson and Keil show Keil and with the most accurate experimental data, whereas a very good agreement. Finally, it became obvious that the 214302-14 B. Jager¨ and E. Bich J. Chem. Phys. 146, 214302 (2017) later empirical potential of Keil et al.19 involved an unjustified J. D. Watts and the integral packages molecule (J. Almlof¨ and P. R. Taylor), overestimation of the reliability of the binary diffusion data props (P. R. Taylor), abacus (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, reported in the same work. This leads not only to differences and J. Olsen), and ECP routines by A. V. Mitin and C. van Wullen,¨ for the current version, see http://www.cfour.de. between the results for the binary diffusion coefficient at low 31MRCC, a string-based general coupled cluster program suite written by and high temperatures but is also the cause for the disagreement M. Kallay,´ see also M. Kallay´ and P. R. Surjan,´ J. Chem. Phys. 115, 2945 between the results for the cross second virial coefficients. (2001) as well as: http://www.mrcc.hu. 32DIRAC, a relativistic ab initio electronic structure program, Release DIRAC13, written by L. Visscher, H. J. Aa. Jensen, R. Bast, and T. Saue, SUPPLEMENTARY MATERIAL with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekstrom,¨ E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, A. S. P.Gomes, See supplementary material for details of the quantum- T. Helgaker, J. K. Lærdahl, Y. S. Lee, J. Henriksson, M. Ilias,ˇ Ch. R. Jacob, chemical results for the krypton-helium interaction energies S. Knecht, S. Komorovsky,´ O. 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