Dielectric Virial Coefficients Research Articles

Comprehensive quantum calculation of the first dielectric virial coefficient of water.

We present a complete calculation, fully accounting for quantum effects and for molecular flexibility, of the first dielectric virial coefficient of water and its isotopologues. The contribution of the electronic polarizability is computed from a state-of-the-art intramolecular potential and polarizability surface from the literature, and its small temperature dependence is quantified. The dipolar polarizability is calculated in a similar manner with an accurate literature dipole-moment surface; it differs from the classical result both due to the different molecular geometries sampled at different temperatures and due to the quantization of rotation. We calculate the dipolar contribution independently from spectroscopic information in the HITRAN2020 database and find that the two methods yield consistent results. The resulting first dielectric virial coefficient provides a complete description of the dielectric constant at low density that can be used in humidity metrology and as a boundary condition for new formulations for the static dielectric constant of water and heavy water.

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 200K, we observed a significant discrepancy between the classical results obtained using superposition approximation and the ab initio computed polarizability. For temperatures from 10K up to 200K, 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.

Path-integral calculation of the third dielectric virial coefficient of noble gases.

We present a rigorous framework for fully quantum calculation of the third dielectric virial coefficient Cɛ(T) of noble gases, including exchange effects. The quantum effects are taken into account with the path-integral Monte Carlo method. Calculations employing state-of-the-art pair and three-body potentials and pair polarizabilities yield results generally consistent with the few scattered experimental data available for helium, neon, and argon, but rigorous calculations with well-described uncertainties will require the development of surfaces for the three-body nonadditive polarizability and the three-body dipole moment. The framework, developed here for the first time, will enable new approaches to primary temperature and pressure metrology based on first-principles calculations of gas properties.

Perspective on the Refractive-Index Gas Metrology Data Landscape

The redefinition of the kelvin has increased focus on thermometry techniques that use the newly fixed value of the Boltzmann constant to realize thermodynamic temperature. One such technique that has advanced considerably in recent years is refractive-index gas thermometry. Generalized as refractive-index gas metrology (RIGM), this also includes a range of applications outside of temperature realizations, such as pressure standards and measurements of the physical properties of gases. Here, the current data situation in the field is reviewed, encompassing the latest developments and remaining challenges, in order to suggest possible approaches for reducing RIGM uncertainties and improving RIGM applications. New analyses of existing experimental literature data are presented for the second density virial coefficient Bρ of helium, neon, argon, and nitrogen; the third density virial coefficient Cρ of nitrogen; and the third dielectric virial coefficient Cε of helium, neon, and argon. A need is identified for more accurate reference-quality datasets to be measured or calculated in several areas, with robust uncertainty budgets, to support future RIGM advancements. The most urgent of these are the bulk modulus of copper; thermodynamic accuracy of the International Temperature Scale of 1990; molar optical refractivity AR of neon, argon, and nitrogen; diamagnetic susceptibility χ0 of neon and argon; second density virial coefficient Bρ of argon; third dielectric virial coefficient Cε of helium, neon, and argon; and third optical refractivity virial coefficient CR of helium and neon.

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.

Path-Integral Calculation of the Second Dielectric and Refractivity Virial Coefficients of Helium, Neon, and Argon.

We present a method to calculate dielectric and refractivity virial coefficients using the path-integral Monte Carlo formulation of quantum statistical mechanics and validate it by comparing our results with equivalent calculations in the literature and with more traditional quantum calculations based on wavefunctions. We use state-of-the-art pair potentials and polarizabilities to calculate the second dielectric and refractivity virial coefficients of helium (both 3He and 4He), neon (both 20Ne and 22Ne), and argon. Our calculations extend to temperatures as low as 1 K for helium, 4 K for neon, and 50 K for argon. We estimate the contributions to the uncertainty of the calculated dielectric virial coefficients for helium and argon, finding that the uncertainty of the pair polarizability is by far the greatest contribution. Agreement with the limited experimental data available is generally good, but our results have smaller uncertainties, especially for helium. Our approach can be generalized in a straightforward manner to higher-order coefficients.

Accurate second dielectric virial coefficient of helium, neon, and argon from ab initio potentials and polarizabilities

The state-of-the-art ab initio potentials and interaction-induced isotropic polarizabilities in the literature are used to calculate the second dielectric virial coefficient of helium-4, helium-3, neon, and argon. The statistical-mechanics formulations with quantum corrections up to second order as well as the [1/1] Padé approximant are employed to compute the dielectric virial coefficient down to the temperature of 25 K. The uncertainties of the calculated dielectric virial coefficient values are evaluated from the corresponding estimated uncertainties of the potentials and the polarizabilities. In addition, the second dielectric virial coefficient of helium-4 is determined by the principle of analogy for the temperature further down to 5 K. The comparisons with the available experimental and theoretical data demonstrate that our results given in the supplementary materials are more accurate and can be used as recommended values in scientific studies relating to the dielectric virial coefficient of helium, neon, and argon.

Vacuum metrology based on refractive index of gas

With the development of vacuum technology, subject to the influence of directional flow and uneven temperature, the thermodynamic equilibrium state is destroyed. In this case, the pressure reference is not suitable for characterizing the vacuum state. To ensure the long-term stability and reproducibility of the measurement system, vacuum metrology will be characterized by gas density. The precisive measurement of gas refractive index based on a Fabry-Perot cavity can be used to derive the gas density. This kind of an optical measurement of vacuum links macroscopic dielectric constants of gases with microscopic polarization parameters of atoms and molecules. It replaces the physical standard based on the mercury pressure gauge with the quantum standard. In this paper, we discuss the reverse process from refractive index to gas pressure, and use the laser-locked Fabry-Perot cavity method to measure the refractive index of argon gas. The contribution of related parameters to the uncertainty of determined gas pressure is analyzed. The influences of material parameters and experimental parameters such as gas molar susceptibility, molar susceptibility, dielectric second Virial coefficient and temperature on gas pressure accuracy are analyzed. The result shows that the uncertainty in our measurement of argon within 1 atm is <inline-formula><tex-math id="M2">\begin{document}$u = \sqrt {{{(6\;{\rm{mPa}})}^2} + {{(73 \times {{10}^{ - 6}}p)}^2}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200706_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200706_M2.png"/></alternatives></inline-formula>. Currently, the uncertainty mainly comes from the measurement deviation of gas temperature inside the cavity. After repeating the measurement a few times, the results show that the statistical uncertainty of refractive index is within 100 ppm, which is limited by the accuracy of the pressure gauge used here. In addition, we compare the dipole calculated by the <i>ab initio</i> method with that by the DOSD method. The results show that the dynamic polarizability obtained by the <i>ab initio</i> method is consistent with our experimental results. In conclusion, these experimental results show that the measurement of gas pressure based on the gas refractive index has high repeatability and accuracy. If the temperature control and corresponding measurement accuracy of the gas are further improved, this method can also be used to obtain high-precision microscopic parameters such as the polarizabilities of atoms and molecules. In the future work, we will focus on improving the temperature control and the design of the cavity to reduce cavity leakage and deflation. It is possible that the measurement accuracy of the gas pressure will be increased to 10 ppm level, which is the same level as the current standard pressure gauge and will become a new standard for pressure measurement in the future.

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: BDCGT He0.01 ○C=11.925715 cm3/mol, CDCGT He0.01 ○C=113.4958 cm6/mol2; neon: BDCGT Ne0.01 ○C=10.994528 cm3/mol, CDCGT Ne0.01 ○C=215.815 cm6/mol2; argon: BDCGT Ar0.01 ○C=-21.233144 cm3/mol, CDCGT 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.

Dielectric permittivity, polarizability and dipole moment of refrigerants R1234ze(E) and R1234yf determined using a microwave re-entrant cavity resonator

Abstract Dielectric permittivities (er) have been measured using a microwave re-entrant resonator at temperatures from (248 to 365) K and pressures up to 11 MPa for the alternative refrigerants 1,3,3,3-tetrafluoropropene (R1234ze(E)) and 2,3,3,3-tetrafluoropropene (R1234yf) in the liquid and vapour phases. Values of er were also measured for liquid CO2 at temperatures below 274 K, extending the data available for this fluid. Molecular polarizability, dipole moment, and the dielectric virial coefficients of each hydrofluoroolefin (HFO) refrigerant were determined by applying the dielectric virial expansion and Kirkwood equation to the measured permittivity data. The experimental results for the vapour phase lead to dipole moments with relative standard uncertainties around 5% that are in reasonable agreement with values from molecular modelling. The measured dipole moment for R1234ze(E) of 1.13 D is about 12% smaller than predicted, while the measured dipole moment for R1234yf of 2.24 D is about 11% smaller than predicted. In the compressed liquid region, the two refrigerants exhibited quite different dependencies on temperature and density. The polarizability of liquid R1234ze(E) could be predicted within 4.2% using the gas phase parameters (measured dipole moment and molecular polarizability). In contrast, the temperature dependence of the polarizability of R1234yf was twice as large in the liquid phase as in the gas phase. The density dependence of the polarizability for liquid R1234yf is at least 10 times that of liquid R1234ze(E). Liquid phase data were also used to estimate effective Kirkwood correlation factors, g, by assuming them to be temperature independent over the range of measurements. The results indicate g ≈ 2.4 for R1234yf and g ≈ 1.2 for R1234ze(E).

Dielectric virial expansion of polarizable dipolar spheres.

The dielectric virial expansion is developed for composite systems with embedded interacting dielectric dipolar spheres. Introducing a multiple-scattering expansion for the polarization energy in the presence of an external field enables the derivation of a virial expansion for the polarizability. Substituting the polarizability into the Clausius-Mossotti relation yields the virial series for the effective medium permittivity. When the dipole moment of the particles or inclusions vanishes, the leading-order term in the series reduces to the Maxwell-Garnett mixing rule, whereas the higher-order terms provide corrections that become important at higher densities. The dielectric virial coefficients are readily evaluated by replacing the surface charge contributions with image lines. Numerical data are presented for the second virial coefficients to illustrate the effects of polarization.

Accurate High-Pressure Measurements of Carbon Monoxide's Electrical Properties.

Accurate measurements of carbon monoxide's electrical properties were carried out at high pressure for the first time enabling stringent comparisons with theoretical values calculated ab initio. Dielectric permittivity measurements were conducted utilising a microwave re-entrant cavity resonator over the temperature range from (255 to 313) K and at pressures up to 8 MPa with a relative combined expanded uncertainty (k=2) less than or equal to 52 ppm. The new data enable carbon monoxide's molar polarizability to be correlated within 0.5 %, significantly improving upon existing literature data, which have a relative scatter of about 10 %. The measured molecular polarizability and electric dipole moment of carbon monoxide were determined to be 2.176×10-40 C2 m2 J-1 and 0.107 D. Literature values from ab initio calculations for these properties are within 0.28 % and 3.9 %, respectively, of the measured quantities. Moreover, our measurement of the electric dipole moment at finite temperature agrees within 2.2 % with the value derived from accurate spectroscopic measurements for the ground rovibrational state. The second dielectric virial coefficient of carbon monoxide was determined experimentally for the first time to be bϵ =(1.015±0.044) cm3 mol-1 , which compares reasonably with ab initio estimates.

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.

Evaluation of second and third dielectric virial coefficients for non-polarisable molecular models

ABSTRACTThe dielectric constant, ε, of a dilute vapour can be estimated from the dielectric virial equation of state (VEOS), but the long-ranged nature of the electrostatic interactions complicates the evaluation of coefficients of this series. We propose a formulation of the second and third dielectric coefficients of a general non-polarisable molecular model that permits their reliable calculation using Mayer sampling Monte Carlo. We demonstrate for three models: dipolar hard spheres, dipolar Lennard–Jones, and TIP4P water. The coefficients are used to compute ε for each model as a function of density, which are compared to molecular-simulation data. The form of the VEOS relating ε to density depends on the dielectric constant ε′ of the embedding medium. Three choices are examined: vacuum (ε′ = 1), self-consistent (ε′ = ε) and tin foil (ε′ = ∞). The vacuum-boundary form is found to be unreliable, losing accuracy at low density and yielding divergent results for ε at moderate densities. In contrast, the series formulated using the tin-foil boundary produces accurate and stable values of ε for almost all conditions and models examined here, even when truncated at second order (which itself is shown to be a large improvement over the first-order Clausius–Mossotti–Debye formula).

Concept of a new technique for the combined determination of dielectric and density virial coefficients3

For the combined determination of dielectric and density virial coefficients, a new measurement concept based on dielectric-constant gas thermometry (DCGT) has been elaborated. The developed apparatus design allows conventional DCGT measurements as well as gas expansion experiments to enable a distinction between density and dielectric virial coefficients. The new technique presented in this technical design note offers an independent method to obtain experimental data, which are highly desirable for the improvement of thermodynamic property models for pure gases and gas mixtures.

Theoretical determination of the polarizability dispersion and the refractive index of helium

The refractive index $n$ of gaseous helium can be measured by optical interferometry so accurately that it can be used to establish a pressure standard which is expected to be superior to the current standard based on the height of a mercury column. The new standard requires knowledge of the dynamic polarizability of helium atom with accuracy significantly higher than obtainable in the best experiments, but possible to achieve computationally. Calculations of this quantity are presented at relativistic and quantum electrodynamics levels of theory including relativistic nuclear recoil effects. The uncertainties of the results are carefully estimated. Our recommended value of the dynamic polarizability at the He-Ne laser wavelength of 6329.908 \AA{}, equal 1.39181197(14) a.u., has uncertainty at least two orders of magnitude smaller than that of recent measurements and is sufficiently accurate to establish a new pressure standard. Purely ab initio values of the refraction coefficient $n$ are computed using our polarizability, literature values of magnetic susceptibility, and dielectric virial coefficients. It is shown that $n\ensuremath{-}1$ can be predicted by theory as a function of density and temperature with uncertainty of 1 ppm for pressures up to 3 MPa.

Dielectric-Constant Gas Thermometry and the Relation to the Virial Coefficients

At PTB new dielectric-constant gas thermometry (DCGT) measurements were performed at the temperature of the triple point of water. As discussed recently in an accompanying paper, the main goal was the determination of the Boltzmann constant \(k\) as a contribution to the international efforts directed to a new definition of the base unit kelvin via fixing the value of \(k\). Besides the linear term in the series expansion used for fitting the results of measurements of DCGT isotherms that reveals \(k\), in this paper the higher-order terms are analyzed. For retrieving highly accurate virial coefficients of helium from the data obtained at gas pressures up to 7 MPa, an extended DCGT working equation is developed. Applying this equation, information is deduced on the viral coefficients up to the fourth density virial coefficient. Finally, comparisons with the latest ab initio calculations for the second and third density virial coefficients as well as the second dielectric virial coefficient are performed.

New basis sets for the evaluation of the CO–Ne van der Waals complex interaction induced electric dipole moment and polarizability surfaces

Using the coupled cluster singles and doubles method together with the recently developed LPol-n (n = ds, dl, fs, fl), the aug-pc-2, the SVPD, the TZVPD and Dunning's x-aug-cc-pVXZ basis sets, we calculate the interaction induced electric dipole moment and polarizability of the CO–Ne van der Waals complex. We consider the effect of extending the bases with different sets of mid-bond functions, and after a systematic basis set study carried out at a representative set of intermolecular geometries, we select the aug-cc-pVTZ, the aug-pc-2, the LPol-fs and the TZVPD bases with a set of 3s3p2d1f1g mid-bond functions placed in the middle of the van der Waals bond for the evaluation of the whole interaction induced property surfaces. After having determined the optimal parameters of appropriate analytical functions fitting the interaction induced properties, the resulting surfaces are used in semiclassical calculations of the dielectric and refractivity second virial coefficients of the system. All through this study the results obtained with Dunning's basis set hierarchy and reported in Phys. Chem. Chem. Phys., 11, 9871 (2009) are taken as reference.

Ab initio pair potential energy curve for the argon atom pair and thermophysical properties for the dilute argon gas. II. Thermophysical properties for low-density argon

A recent argon–argon interatomic potential energy curve determined from quantum-mechanical ab initio calculations and described with an analytical representation [B. Jäger, R. Hellmann, E. Bich, and E. Vogel, Mol. Phys. 107, 2181 (2009); 108, 105 (2010)] was used to calculate the most important thermophysical properties of argon governed by two-body interactions. Second pressure, acoustic, and dielectric virial coefficients as well as viscosity and thermal conductivity in the limit of zero density were computed for natural argon from 83 to 10,000 K. The calculated values for the different thermophysical properties are compared with available experimental data and values computed for other argon–argon potentials. This extensive analysis shows that the proposed potential is superior to all previous ones and that the calculated thermophysical property values are accurate enough to be applied as standard values for the complete temperature range of the calculations.

Reference measurements of hydrogen's dielectric permittivity

We used a quasi-spherical cavity resonator to measure the relative dielectric permittivity ε r of H2 at frequencies from 2.4 to 7.3 GHz, at pressures up to 6.5 MPa, and at the temperatures 273 K and 293 K. The resonator was calibrated using auxiliary measurements of ε r(p, T) for helium together with the accurate ab initio values of helium's dielectric permittivity. The measurements determine accurate values of hydrogen's molecular electric polarizability and its temperature dependence. At 273 K, we obtained = (8.9568 ± 0.0008) × 10−41 F m2 (all uncertainties reported here are one standard uncertainty), which agrees with the value (8.9566 ± 0.0026) × 10−41 F m2 that was obtained by combining Rychlewski's ab initio calculations with estimates of rotational level populations. Our results yield the temperature dependence = × (1.1 ± 0.3) × 10−5 K−1, which agrees with the calculated value × 1.0 × 10−5 K−1. Our data also determine hydrogen's second dielectric virial coefficient, = (0.03 ± 0.05) cm3 mol−1, a property that has not yet been calculated.