Ab initiopotential energy curve for the helium atom pair and thermophysical properties of the dilute helium gas. II. Thermophysical standard values for low-density helium

Impact of Combination Rules, Level of Theory, and Potential Function on the Modeling of Gas- and Condensed-Phase Properties of Noble Gases

The systems of noble gases are particularly instructive for molecular modeling due to the elemental nature of their interactions. They do not normally form bonds nor possess a (permanent) dipole moment, and the only forces determining their bonding/clustering stems from van der Waals forces─dispersion and Pauli repulsion, which can be modeled by empirical potential functions. Combination rules, that is, formulas to derive parameters for pair potentials of heterodimers from parameters of corresponding homodimers, have been studied at length for the Lennard-Jones 12-6 potentials but not in great detail for other, more accurate, potentials. In this work, we examine the usefulness of nine empirical potentials in their ability to reproduce quantum mechanical (QM) benchmark dissociation curves of noble gas dimers (He, Ne, Ar, Kr, and Xe homo- and heterodimers), and we systematically study the efficacy of different permutations of combination relations for each parameter of the potentials. Our QM benchmark comprises dissociation curves computed by several different coupled cluster implementations as well as symmetry-adapted perturbation theory. The two-parameter Lennard-Jones potentials were decisively outperformed by more elaborate potentials that sport a 25-30 times lower root-mean-square error (RMSE) when fitted to QM dissociation curves. Very good fits to the QM dissociation curves can be achieved with relatively inexpensive four- or even three-parameter potentials, for instance, the damped 14-7 potential (Halgren, J. Am. Chem. Soc. 1992, 114, 7827-7843), a four-parameter Buckingham potential (Werhahn et al., Chem. Phys. Lett. 2015, 619, 133-138), or the three-parameter Morse potential (Morse, Phys. Rev. 1929, 34, 57-64). Potentials for heterodimers that are generated from combination rules have an RMSE that is up to 20 times higher than potentials that are directly fitted to the QM dissociation curves. This means that the RMSE, in particular, for light atoms, is comparable in magnitude to the well-depth of the potential. Based on a systematic permutation of combination rules, we present one or more combination rules for each potential tested that yield a relatively low RMSE. Two new combination rules are introduced that perform well, one for the van der Waals radius σij as ( 1 2 ( σ i 3 + σ j 3 ) ) 1 / 3 and one for the well-depth ϵij as ( 1 2 ( ϵ i - 2 + ϵ j - 2 ) ) - 1 / 2 . The QM data and the fitted potentials were evaluated in the gas phase against experimental second virial coefficients for homo- and heterodimers, the latter of which allowed evaluation of the combination rules. The fitted models were used to perform condensed phase molecular dynamics simulations to verify the melting points, liquid densities at the melting point, and the enthalpies of vaporization produced by the models for pure substances. Subtle differences in the benchmark potentials, in particular, the well-depth, due to the level of theory used were found here to have a profound effect on the macroscopic properties of noble gases: second virial coefficients or the bulk properties in simulations. By explicitly including three-body dispersion in molecular simulations employing the best pair potential, we were able to obtain accurate melting points as well as satisfactory densities and enthalpies of vaporization.

Three-body potential and third virial coefficients for helium including relativistic and nuclear-motion effects

The non-additive three-body interaction potential for helium was computed using the coupled-cluster theory and the full configuration interaction method. The obtained potential comprises an improved nonrelativistic Born-Oppenheimer energy and the leading relativistic and nuclear-motion corrections. The mean absolute uncertainty of our calculations due to the incompleteness of the orbital basis set was determined employing complete-basis-set extrapolation techniques and was found to be 1.2%. For three helium atoms forming an equilateral triangle with the side length of 5.6 bohr - a geometry close to the minimum of the total potential energy surface - our three-body potential amounts to -90.6 mK, with an estimated uncertainty of 0.5 mK. An analytic function, developed to accurately fit the computed three-body interaction energies, was chosen to correctly describe the asymptotic behavior of the three-body potential for trimer configurations corresponding to both the three-atomic and the atom-diatom fragmentation channels. For large triangles with sides r12, r23, and r31, the potential takes correctly into account all angular terms decaying as r-l12 r-m23 r-n21 with l + m + n ≤ 14 for the nonrelativistic Born-Oppenheimer energy and l + m + n ≤ 9 for the post-Born-Oppenheimer corrections. We also developed a short-range analytic function describing the local behavior of the total uncertainty of the computed three-body interaction energies. Using both fits we calculated the third pressure and acoustic virial coefficients for helium and their uncertainties for a wide range of temperatures. The results of these calculations were compared with available experimental data and with previous theoretical determinations. The estimated uncertainties of present calculations are 3-5 times smaller than those reported in the best previous works.

Collision-induced three-body polarizability of helium

We present the first-principles determination of the three-body polarizability and the third dielectric virial coefficient of helium. Coupled-cluster and full configuration interaction methods were used to perform electronic structure calculations. The mean absolute relative uncertainty of the trace of the polarizability tensor, resulting from the incompleteness of the orbital basis set, was found to be 4.7%. Additional uncertainty due to the approximate treatment of triple and the neglect of higher excitations was estimated at 5.7%. An analytic function was developed to describe the short-range behavior of the polarizability and its asymptotics in all fragmentation channels. We calculated the third dielectric virial coefficient and its uncertainty using the classical and semiclassical Feynman-Hibbs approaches. The results of our calculations were compared with experimental data and with recent Path-Integral Monte Carlo (PIMC) calculations [Garberoglio et al., J. Chem. Phys. 155, 234103 (2021)] employing the so-called superposition approximation of the three-body polarizability. For temperatures above 200 K, we observed a significant discrepancy between the classical results obtained using superposition approximation and the ab initio computed polarizability. For temperatures from 10 K up to 200 K, the differences between PIMC and semiclassical calculations are several times smaller than the uncertainties of our results. Except at low temperatures, our results agree very well with the available experimental data but have much smaller uncertainties. The data reported in this work eliminate the main accuracy bottleneck in the optical pressure standard [Gaiser et al., Ann. Phys. 534, 2200336 (2022)] and facilitate further progress in the field of quantum metrology.

Thermodynamic properties of krypton from Monte Carlo simulations using ab initio potentials

Ten different thermodynamic properties of the noble gas krypton were calculated by Monte Carlo simulations in the isothermal-isobaric ensemble using a highly accurate ab initio pair potential, Feynman-Hibbs corrections for quantum effects, and an extended Axilrod-Teller-Muto potential to account for nonadditive three-body interactions. Fourteen state points at a liquid and a supercritical isotherm were simulated. To obtain results representative for macroscopic systems, simulations with several particle numbers were carried out and extrapolated to the thermodynamic limit. Our results agree well with experimental data from the literature, an accurate equation of state for krypton, and a recent virial equation of state (VEOS) for krypton in the region where the VEOS has converged. These results demonstrate that very good agreement between simulation and experiment can only be achieved if nonadditive three-body interactions and quantum effects are taken into account.

Transport coefficients of isotopic mixtures of noble gases based on ab initio potentials

The transport coefficients such as viscosity, thermal conductivity, diffusion and thermal diffusion of neon, argon, krypton, and xenon are computed for a wide range of temperatures taking into consideration their real isotopic compositions. A new concept of isotopic thermal diffusion factor is introduced and calculated. The Chapman-Enskog method based on the 10th order approximation with respect to the Sonine polynomial expansion is applied. Ab initio potentials of interatomic interactions are employed to compute the transport cross-sections as they are part of the coefficient expressions. To study the influence of the isotopic composition, the same transport coefficients have been calculated for the single gases having an average atomic mass. The estimated numerical error of the present results is a function of the temperature and is different for each coefficient. At the room temperature, the relative numerical error of viscosity, thermal conductivity and diffusion coefficient is on the order of 10-6. The numerical error of the thermal diffusion factor affects the fifth decimal digit. The influence of the isotopic composition on viscosity and thermal conductivity depends on the gas species. It is negligible for argon and significant (about 0.02%) for xenon, while neon and krypton are weakly affected by the isotopic composition. The diffusion coefficient for each pair of isotopes differs from the corresponding self-diffusion coefficient by about 3%. The thermal diffusion factor of each isotope differs from the thermal self-diffusion factor in the third decimal digit.

Thermophysical properties of low-density neon gas from highly accurate first-principles calculations and dielectric-constant gas thermometry measurements

New interatomic potential energy and interaction-induced polarizability curves for two ground-state neon atoms were developed and used to predict the second density, acoustic, and dielectric virial coefficients and the dilute gas shear viscosity and thermal conductivity of neon at temperatures up to 5000 K. The potential energy curve is based on supermolecular coupled-cluster (CC) calculations at very high levels up to CC with single, double, triple, quadruple, and perturbative pentuple excitations [CCSDTQ(P)]. Scalar and spin-orbit relativistic effects, the diagonal Born-Oppenheimer correction, and retardation of the dispersion interactions were taken into account. The interaction-induced polarizability curve, which in this work is only needed for the calculation of the second dielectric virial coefficient, is based on supermolecular calculations at levels up to CCSDT and includes a correction for scalar relativistic effects. In addition to these first-principles calculations, highly accurate dielectric-constant gas thermometry (DCGT) datasets measured at temperatures from 24.5 to 200 K were analyzed to obtain the difference between the second density and dielectric virial coefficients with previously unattained accuracy. The agreement of the DCGT values with the ones resulting from the first-principles calculations is, despite some small systematic deviations, very satisfactory. Apart from this combination of two virial coefficients, the calculated thermophysical property values of this work are significantly more accurate than any available experimental data.

Eighth-order virial equation of state and speed-of-sound measurements for krypton

An eighth-order virial equation of state (VEOS) for krypton, valid for temperatures up to 5000 K, was developed using the accurate potential functions proposed by Jäger et al. [J. Chem. Phys. 144, 114304 (2016)] for the pair interactions and nonadditive three-body interactions between krypton atoms. While the second and third virial coefficients were already calculated by Jäger et al., the fourth- to eighth-order coefficients were determined in the present work. A simple analytical function was fitted individually to the calculated values of each virial coefficient to obtain the VEOS in an easy-to-use analytical form. To enable a stringent test of the quality of the new VEOS, we measured the speed of sound in krypton in the temperature range from 200 K to 420 K and at pressures up to 100 MPa with a very low uncertainty (at the 0.95 confidence level) of 0.005%-0.018% employing the pulse-echo technique. In order to verify that the isotopic composition of the krypton sample conforms to that of natural krypton in air, high-precision measurements of krypton isotope ratios using a high-sensitivity noble gas mass spectrometer were performed. The extensive comparison with the new speed-of-sound data as well as with experimental p-ρ-T and speed-of-sound data from the literature indicates that pressures and speeds of sound calculated using our new VEOS have uncertainties (at the 0.95 confidence level) of less than 0.1% at state points at which the VEOS is sufficiently converged.

Highly-accurate density-virial-coefficient values for helium, neon, and argon at 0.01 ○C determined by dielectric-constant gas thermometry

The dielectric-constant gas thermometer of Physikalisch-Technische Bundesanstalt (PTB) developed for measuring the Boltzmann constant with a relative uncertainty of 1.9 parts per million was used for determining the virial coefficients of the three noble gases, helium, neon, and argon, at the triple point of water (0.01 ^○C). For this purpose, isotherms were measured up to a maximum pressure of 7 MPa. The evaluation of the highly accurate data by fitting is required to derive an extended working equation for the dependence of the gas pressure on the dielectric constant. The following values have been obtained for the second B and third C virial coefficients, with the standard uncertainties given in parentheses as a multiple of the last digit, considering literature data for the dielectric virial coefficients: helium: B_DCGT ^He0.01 ^○C=11.925715 cm³/mol, C_DCGT ^He0.01 ^○C=113.4958 cm⁶/mol²; neon: B_DCGT ^Ne0.01 ^○C=10.994528 cm³/mol, C_DCGT ^Ne0.01 ^○C=215.815 cm⁶/mol²; argon: B_DCGT ^Ar0.01 ^○C=-21.233144 cm³/mol, C_DCGT ^Ar0.01 ^○C=1143.339 cm⁶/mol². These values are compared with the results of the latest ab initio calculations of the second and third virial coefficients.

Fully quantum calculation of the second and third virial coefficients of water and its isotopologues from ab initio potentials

Path-Integral Monte Carlo methods were applied to calculate the second, B(T), and the third, C(T), virial coefficients for water. A fully quantum approach and state-of-the-art flexible-monomer pair and three-body potentials were used. Flexible-monomer potentials allow calculations for any isotopologue; we performed calculations for both H2O and D2O. For B(T) of H2O, the quantum effect contributes 25% of the value at 300 K and is not entirely negligible even at 1000 K, in accordance with recent literature findings. The effect of monomer flexibility, while not as large as some claims in the literature, is significant compared to the experimental uncertainty. It is of opposite sign to the quantum effect, smaller in magnitude than the latter below 500 K, and varies from 2% at 300 K to 10% at 700 K. When monomer flexibility is accounted for, results from the CCpol-8sf pair potential are in excellent agreement with the available experimental data and provide reliable B(T) values at temperatures outside the range of experimental data. The flexible-monomer MB-pol pair potential yields B(T) values that are slightly too high compared to experiment. For C(T), our calculations confirm earlier findings that the use of three-body potential is necessary for meaningful predictions. However, due to various uncertainties of the potentials used, especially the three-body ones, we were not able to establish benchmark values of C(T), although our results are in qualitative agreement with available experimental data. The quantum effect, never before included for water, reduces the magnitude of the classical value for H2O by a factor of 2.5 at 300 K and is not entirely negligible even at 1000 K.

State-of-the-art ab initio potential energy curve for the xenon atom pair and related spectroscopic and thermophysical properties

A new ab initio interatomic potential energy curve for two ground-state xenon atoms is presented. It is based on supermolecular calculations at the coupled-cluster level with single, double, and perturbative triple excitations [CCSD(T)] employing basis sets up to sextuple-zeta quality, which were developed as part of this work. In addition, corrections were determined for higher coupled-cluster levels up to CCSDTQ as well as for scalar and spin-orbit relativistic effects at the CCSD(T) level. A physically motivated analytical function was fitted to the calculated interaction energies and used to compute the vibrational spectrum of the dimer, the second virial coefficient, and the dilute gas transport properties. The agreement with the best available experimental data for the investigated properties is excellent; the new potential function is superior not only to previous ab initio potentials but also to the most popular empirical ones.

Accurate virial coefficients of gaseous krypton from state-of-the-art ab initio potential and polarizability of the krypton dimer

We have developed a new krypton-krypton interaction-induced isotropic dipole polarizability curve based on high-level ab initio methods. The determination was carried out using the coupled-cluster singles and doubles plus perturbative triples method with very large basis sets up to augmented correlation-consistent sextuple zeta as well as the corrections for core-core and core-valence correlation and relativistic effects. The analytical function of polarizability and our recently constructed reference interatomic potential [J. M. Waldrop et al., J. Chem. Phys. 142, 204307 (2015)] were used to predict the thermophysical and electromagnetic properties of krypton gas. The second pressure, acoustic, and dielectric virial coefficients were computed for the temperature range of 116 K-5000 K using classical statistical mechanics supplemented with high-order quantum corrections. The virial coefficients calculated were compared with the generally less precise available experimental data as well as with values computed from other potentials in the literature {in particular, the recent highly accurate potential of Jäger et al. [J. Chem. Phys. 144, 114304 (2016)]}. The detailed examination in this work suggests that the present theoretical prediction can be applied as reference values in disciplines involving thermophysical and electromagnetic properties of krypton gas.

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

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.

Dirac bubble potential for He-He and inadequacies in the continuum: Comparing an analytic model with elastic collision experiments

We focus on the long-pending issue of the inadequacy of the Dirac bubble potential model in the description of He-He interactions in the continuum [L. L. Lohr and S. M. Blinder, Int. J. Quantum Chem. 53, 413 (1995)]. We attribute this failure to the lack of a potential wall to mimic the onset of the repulsive interaction at close range separations. This observation offers the explanation to why this excessively simple model proves incapable of quantitatively reproducing previous experimental findings of glory scattering in He-He, although being notorious for its capability of reproducing several distinctive features of the atomic and isotopic helium dimers and trimers [L. L. Lohr and S. M. Blinder, Int. J. Quantum Chem. 90, 419 (2002)]. Here, we show that an infinitely high, energy-dependent potential wall of properly calculated thickness r_c(E) taken as a supplement to the Dirac bubble potential suffices for agreement with variable-energy elastic collision cross section experiments for ⁴He-⁴He, ³He-⁴He, and ³He-³He [R. Feltgen et al., J. Chem. Phys. 76, 2360 (1982)]. In the very low energy regime, consistency is found between the Dirac bubble potential (to which our extended model is shown to reduce) and cold collision experiments [J. C. Mester et al., Phys. Rev. Lett. 71, 1343 (1993)]; this consistency, which in this regime lends credence to the Dirac bubble potential, was never noticed by its authors. The revised model being still analytic is of high didactical value while expected to increase in predictive power relative to other appraisals.

Energy accommodation coefficient extracted from acoustic resonator experiments

We review values of the temperature jump coefficient ζT determined from measurements of the acoustic resonance frequencies facoust of helium-filled and argon-filled, spherical cavities near ambient temperature. We combine these values of ζT with literature data for tangential momentum accommodation coefficient (TMAC) and the Cercignani-Lampis model of the gas-surface interaction to obtain measurement-derived values of the normal energy accommodation coefficient (NEAC). We found that NEAC ranges from 0 to 0.1 for helium and from 0.61 to 0.85 for argon at ambient temperature for several different surfaces. We suggest that measurements of facoust of gas-filled, cylindrical cavities and of the non-radial modes of quasi-spherical cavities might separately determine TMAC and NEAC. Alternatively, TMAC and NEAC could be determined by measuring the heat transfer and momentum transfer between parallel rotating discs at low pressure.

Transport coefficients of helium-argon mixture based on ab initio potential

The viscosity, thermal conductivity, diffusion coefficient, and thermal diffusion factor of helium-argon mixtures are calculated for a wide range of temperature and for various mole fractions up to the 12th order of the Sonine polynomial expansion with an ab initio intermolecular potential. The calculated values for these transport coefficients are compared with other data available in the open literature. The comparison shows that the obtained transport coefficients of helium-argon mixture have the best accuracy for the moment.

Ab initio simulation of transport phenomena in rarefied gases

Ab initio potentials are implemented into the direct simulation Monte Carlo (DSMC) method. Such an implementation allows us to model transport phenomena in rarefied gases without any fitting parameter of intermolecular collisions usually extracted from experimental data. Applying the method proposed by Sharipov and Strapasson [Phys. Fluids 24, 011703 (2012)], the use of ab initio potentials in the DSMC requires the same computational efforts as the widely used potentials such as hard spheres, variable hard sphere, variable soft spheres, etc. At the same time, the ab initio potentials provide more reliable results than any other one. As an example, the transport coefficients of a binary mixture He-Ar, viz., viscosity, thermal conductivity, and thermal diffusion factor, have been calculated for several values of the mole fraction.