Ab Initio Calculation of Fluid Properties for Precision Metrology

On May 20, 2019, the base SI units of mass (kilogram), electric current (ampere), temperature (kelvin) and amount of substance (mole) were redefined by assigning fixed values to fundamental constants of nature: the Planck constant, the electron charge, the Boltzmann constant, and the Avogadro constant, respectively.^1–3 By decoupling the base units from specific material artifacts, this new redefinition is expected to lead to improved scientific instruments, reducing the degradation in accuracy when measuring quantities at larger or smaller magnitudes than a predefined unit standard. Additionally, the most accurate experimental technique available at each scale can be used to implement a primary standard, resulting in easier calibrations, increased accuracies of measuring devices, and further technological advancements.

At many conditions, gas-based techniques provide unparalleled performance for primary measurements of temperature and pressure. These involve acoustic, dielectric, or refractivity measurements, because frequency and electromagnetic measurements can be made with very high accuracy. A model, typically expressed as the ideal-gas behavior with a series of corrections in powers of density, is used to relate the measured quantity to the temperature or pressure; in the case of dielectric or refractivity measurements, one set of corrections relates the measured quantity to the gas density and the familiar virial expansion is used to relate the density to the pressure and temperature.

These gas-based methods have been greatly facilitated in recent years by the ability to perform ab initio calculations of the thermophysical properties (such as the polarizability and the density, dielectric, and refractivity virial coefficients) of the working gases with no uncontrolled approximations and rigorously defined uncertainties. These calculated properties often have much smaller uncertainties than the best experimental determinations, especially when the gas considered is helium. These techniques have been successfully applied for pressures up to 7 MPa and for thermodynamic temperatures in the range (2.5–552) K (with extension to 1000 K or more progressing⁴).

These achievements have been facilitated by the increase in supercomputing power and advances in numerical techniques for electronic structure calculations. For example, state-of-the-art calculations for up to three He atoms even include relativistic and quantum electrodynamics effects. In particular, these numerical investigations produce pair and three-body potentials, as well as single-atom, pair, and three-body polarizabilities, with unprecedented accuracy.

Building on these results, the exact quantum statistical mechanics formulation enabled rigorous calculations of the coefficients appearing in the density (virial) expansion of the equation of state, the speed of sound, the dielectric constant, and the refractive index. The path-integral Monte Carlo (PIMC) method has been shown to provide sufficient accuracy for these quantities. As a consequence, it has been possible to devise a fully first-principles chain of calculations with rigorous uncertainty propagation to compute virial coefficients of helium gas.

As a result of these endeavors, since about 2010 thermophysical properties of gaseous helium have been known from theory with an accuracy that in most cases surpasses that of the most precise experimental determinations. Currently, the uncertainties of the ab initio second and third virial coefficients of helium are at least one order of magnitude smaller than the experimental ones. The situation is similar for the density dependence of the speed of sound, the dielectric constant, and the refractive index, where it leads to improved accuracy in Acoustic Gas Thermometry (AGT), Dielectric Constant Gas Thermometry (DCGT), and Refractive Index Gas Thermometry (RIGT), respectively.

Section 2 describes these gas-based experimental techniques for temperature and pressure measurement, including their operating principles, temperature and pressure ranges, recent technological improvements, and the sources of uncertainty. We highlight the ways in which theoretical knowledge, in the form of ab initio polarizabilities and virial coefficients, has improved these measurements by reducing significant components of the uncertainty.

First-principles calculations of virial coefficients involve two steps: the ab initio electronic structure calculation of interatomic potentials and/or polarizabilities, followed by the solution of the exact quantum statistical equations describing virial coefficients.

We therefore present in Sec. 3 a critical review of the state of the art of non-relativistic, relativistic, and quantum electrodynamic electronic structure calculations, with particular emphasis on the determination of uncertainties. Our primary focus will be on helium – which is currently the only substance for which computations can be performed that consistently exceed the accuracy of the best experiments – but other noble gases will be briefly covered due to their importance in metrology. For the sake of completeness, we will recall the hierarchy of physical theories involved in quantum chemical calculations, with particular emphasis on the Full Configuration Interaction (FCI) approach, which is exact within a given orbital basis set and is currently feasible for systems with up to ten electrons. Relativistic and quantum electrodynamic effects (expressed as expansions in powers of the fine-structure constant) have been crucial for achieving the extremely low uncertainty of the latest helium calculations, and are also progressively important in describing larger atoms (notably, neon and argon). Additionally, the evaluation of electronic polarizabilities and magnetic susceptibilities will be discussed. All of these theoretical advances will be exemplified for the case of helium, where we will present the current state of the art regarding interaction potentials and many-body polarizabilities.

Knowledge of interaction potentials and polarizabilities enables calculation of the coefficients appearing in the virial expansion of the equation of state, the speed of sound, the dielectric constant, and the refractive index, which are crucial ingredients in the uncertainty budgets of AGT, DCGT, and RIGT. In the past 15 years, the path-integral approach to quantum statistical mechanics has been successfully applied in calculating virial coefficients without uncontrolled approximations. The main features of this method are reviewed in Sec. 4, with particular attention to the question of uncertainty propagation from the potentials and the polarizabilities. In the case of pair properties, an alternative method based on the solution of the Schrödinger equation is available and provides mutual validation of the path-integral results, as well as enabling the calculation of transport properties. Most of this review is focused on thermodynamic properties, but ab initio calculations also provide viscosity and thermal conductivity. We briefly review how this leads to improvements in flow-rate measurements.

Although most efforts have been devoted to noble gases, highly accurate theoretical calculations are also available for molecular systems and have the potential to enable a similar paradigm shift in some metrological applications. We describe in Sec. 5 the present situation in the first-principles calculation of molecular properties, and point out a few areas where computational contributions are expected to have an increasing impact in the near future, namely humidity metrology, measurements of very low pressures, and atmospheric science. We end our review in Sec. 6, where future perspectives and an overview of the status of highly accurate ab initio property calculations will be presented.

Traditionally, accurate measurements of temperature-dependent thermophysical properties of gases [such as: second density virial coefficient B(T), viscosity η(T), thermal conductivity λ(T)] have been used to determine parameters in evermore-refined models for interatomic and intermolecular potentials. This tradition/paradigm can be traced back to the 18th century when “… Bernoulli had proposed that in Boyle’s law the specific volume v be replaced by (v − b), where b was thought to be the volume of the molecules.”⁵ During the past 25 years, the accuracy of the calculated thermophysical properties of the noble gases (particularly helium) has increased dramatically. An example is shown in Fig. 1, which shows how the accuracy of the second virial coefficient B(T) of ⁴He improved with time. The data plotted are for temperatures near T_Ne. (T_Ne ≡ 24.5561 K is the defined temperature of the triple point of neon on the international temperature scale, ITS-90.⁶) Since the year 2012, the uncertainty of B(T_Ne), as calculated ab initio, has been smaller than the uncertainty of the best measurements of B(T_Ne).

The paradigm reversal (replacing measured thermophysical properties of helium with calculated thermophysical properties) applies to zero-density values of the viscosity η(T), thermal conductivity λ(T), and ³He–⁴He mutual diffusion coefficient as well as to the density and acoustic virial coefficients, relative dielectric permittivity (dielectric constant) ɛ_r(p, T), relative magnetic permittivity μ_r(p, T), and refractive index $n(p,T)=εrμr$⁠. For many of these properties, the values calculated for helium are standards that are used to calibrate apparatus that measures the same properties for other gases.

The paradigm reversals for ɛ_r(p, T) and n(p, T) have been combined with technical advances in the measurement of ɛ_r(p, T) and n(p, T) to develop novel pressure standards. One standard operating at optical frequencies and low pressures (100 Pa $≤p≤$ 100 kPa) is more accurate than manometers based on liquid columns (see Sec. 2.3.1 and Ref. 18). Other standards operating at audio and microwave frequencies and higher pressures (100 kPa $≤p≤$ 7 MPa) have enabled exacting tests of mechanical pressure generators based on the dimensions of a rotating piston in a cylinder (see Sec. 2.3.2 and Refs. 19 and 20). At still higher pressures (up to 40 MPa), the values of helium’s density calculated from the virial equation of state (VEOS) have been used to calibrate magnetic suspension densimeters.²¹ A more accurate high-pressure scale may result. In Sec. 2.5, we will comment on ab initio calculations of transport properties and their contribution to improved flow metrology.

During the past 25 years, the accurate calculations of the thermophysical properties of the noble gases have strongly interacted with gas-based measurements of the thermodynamic temperature T. To put this in context, we compare in Fig. 2 the evolution of “consensus” temperature metrology with “thermodynamic” temperature metrology.²² 

In Fig. 2, the squares represent estimates of the relative uncertainties u_r(T_scale) of the consensus temperature scales disseminated by National Metrology Institutes (NMIs). We plot the values of u_r(T_scale) near the boiling point of water at intervals of roughly 20 years. Most of the points are at years when the NMIs agreed to disseminate a new consensus scale that was either a better approximation of thermodynamic temperatures and/or an extension of the consensus scale to higher and lower temperatures. The most recent scale is the “International Temperature Scale of 1990” (ITS-90), and temperatures measured using ITS-90 are denoted T₉₀.⁶ The data underlying ITS-90 are constant-volume gas thermometry (CVGT) and spectral radiation thermometry linked to CVGT.²³ The pre-1990 CVGT was based on the ideal-gas equation of state, as corrected by virial coefficients either taken from the experimental literature or measured during the CVGT. Post-1990 thermometry, together with ab initio calculations of virial coefficients, revealed that the authors of ITS-90 were unaware that errors in ITS-90 exceeded their expanded (k = 2) uncertainty by roughly a factor of two. (See Fig. 3 and the discussion at the end of Sec. 2.2.1).

In Fig. 2, the circles represent the relative uncertainty of determinations of the Boltzmann constant u_r(k_B). To determine k_B, one measures the mean energy k_BT per degree-of-freedom of a system in thermal equilibrium at the thermodynamic temperature T. During the interval 1960–2019, the thermodynamic temperature of the triple point of water was defined as T_TPW ≡ 273.16 K, exactly. Thus, measurements of k_BT that were conducted near T_TPW had a negligible uncertainty from T and u_r(k_BT) was an excellent proxy for u_r(T), the uncertainty of measurements of T under the most favorable conditions.

As displayed in Fig. 2, u_r(T_scale) decreased from $∼10$ to $∼2$ ppm (1 ppm ≡ 1 part in 10⁶) during the 20th century. Also during the 20th century, the relative uncertainty u_r(k_B) decreased from $∼20000$ to $∼2$ ppm. Thus, u_r(k_B) ≫ u_r(T_scale) for most of the 20th century, even though k_B was a “fundamental” constant and, therefore, a worthy challenge for metrology.

Between 1973 and 2017, AGT measurements decreased the uncertainty of u_r(k_B) 100-fold from $∼40$ to $∼0.4$ ppm.^24,25 By 2017, DCGT achieved the uncertainty u_r(k_B) = 1.9 ppm and Johnson noise thermometry achieved u_r(k_B) = 2.7 ppm.²⁵ 

In 1995, Aziz et al.⁷ argued that the values of the thermal conductivity λ(T), viscosity η(T), and second density virial coefficient B(T) of helium, as calculated using ab initio input, were more accurate than the best available measurements of these quantities. Subsequently, helium-based AGT measurements of k_B relied on ab initio values of λ(T) to account for the thermoacoustic boundary layer. Just before the Boltzmann constant was defined in 2019, the lowest-uncertainty measurements of k_B used either the ab initio value of thermal conductivity of helium λ_He(273.16 K) or the value of λ_Ar(273.16 K) that was deduced from ratio measurements using λ_He(273.16 K) as a standard.^26,27

In 2019, the unit of temperature, the kelvin, was redefined by assigning the fixed numerical value 1.380 649 × 10⁻²³ to the Boltzmann constant, k_B, when k_B is expressed in the unit J K⁻¹.^2,3 Thus, the Boltzmann constant can no longer be measured. However, the thermodynamic temperature of the triple point of water now has an uncertainty of a few parts in 10⁷, although the best current value is still 273.16 K.²⁰ 

As discussed in the next section, the techniques for measuring thermodynamic temperatures are evolving rapidly. They are becoming more accurate and easier to implement. We anticipate NMIs will disseminate thermodynamic temperatures instead of ITS-90 at temperatures below 25 K. This would not be possible without the accurate ab initio values of the thermophysical properties of helium.

During the past two decades, AGT has emerged as the most accurate primary thermometry technique over the temperature range 7–552 K, achieving uncertainties as low as 10⁻⁶T. AGT experiments were instrumental in measuring the Boltzmann constant for the redefinition of the kelvin,²⁸ and have revealed small, systematic errors in the ITS-90.^20,23 The construction of ITS-90 and the definition of k_B force T₉₀ and T to be essentially equal at the temperature of the triple point of water T_TPW; however, the derivative dT₉₀/dT ≈ 1.0001 at T_TPW. Figure 3 provides evidence that ITS-90 has errors of $∼25×10−6T$ near water’s boiling point and ∼−35 × 10⁻⁶T near 173 K. This section is necessarily brief; for an in-depth review of AGT, the reader is referred to Ref. 29.

Most modern realizations of primary AGT determine the speed of sound from the resonance frequencies of the acoustic normal modes in a cavity resonator of fixed and stable dimensions. Resonators have been manufactured from copper, aluminum, and stainless steel, with internal volumes between 0.5 and 3 l. Cavity shapes have either been spherical, quasi-spherical (with smooth, deliberate deviations from sphericity), or cylindrical. The use of diamond turning to produce quasi-spherical resonators (QSRs) with extremely accurate forms (∼1 μm) and smooth surfaces (average surface roughness on the order of 3 nm) has significantly improved performance.³² In spherical geometries, the best results are obtained from the radially symmetric acoustic modes, since these possess high quality factors and are relatively insensitive to imperfections in the cavity shape. In cylindrical geometries, the longitudinal plane-wave modes are typically favored.

In diamond-turned QSRs, the relative uncertainty in V from the microwave method can be less than 1 × 10⁻⁶.³⁴ This was made possible by improvements in theory,³⁵ resonator shape accuracy, and studies of small perturbations due to probes.³⁴ Recently, it has been demonstrated that comparable uncertainties can be achieved with low-cost microwave equipment.^36,37 Accurate microwave dimensional measurements have also been performed in cylindrical acoustic resonators.³⁸ 

In both absolute and relative primary AGT, maintaining gas purity is of critical importance. Impurities will shift the average molecular mass of the gas, and hence the speed of sound, by an amount that depends on the mass contrast between the bulk gas and impurity. For example, the speed of sound in helium is ∼16 times more sensitive to water vapor than it is in argon. Impurities can either be present in the gas source or arise from outgassing or leaks in the apparatus itself.

Relative AGT requires only that m remain unchanged between the measurements at T and T_ref. Temperature dependence in m can arise through several mechanisms: impurities such as water, hydrocarbons, or heavy noble gases can be condensed out at low temperatures; higher temperatures (⁠$>500$ K) cause significant outgassing from the walls of steel resonators.³⁹ Gas purity is vastly improved by maintaining a flow of gas (typically <50 μmol/s) through the resonator and supply manifold.

Absolute AGT has more stringent requirements on gas purity than relative AGT. To determine an accurate value for m, both the isotopic abundance of the gas and any residual impurities must be quantified. Reactive impurities, including water, can be removed from the source gas using gas purifiers, and noble gas impurities can be removed from helium using a cold trap.²⁶ The isotopic ratios ³⁶Ar/⁴⁰Ar and ³⁸Ar/⁴⁰Ar in argon, and ³He/⁴He in helium, have been determined by mass spectrometry, and vary significantly from source to source.⁴⁰ Alternatively, isotopically pure ⁴⁰Ar gas can be used, although this is only available in small quantities and at great expense.⁴¹ 

The low uncertainty of the AGT technique arises from the excellent agreement between acoustic theory and experiment. The simplicity of Eq. (2) hides a number of temperature-, pressure-, and mode-dependent corrections that constitute the term Δf_a. The largest of these are the thermoacoustic boundary layer corrections, which arise from an irreversible heat exchange between the oscillating gas and resonator walls.^41,42 This effect both lowers the frequency of the acoustic resonances and broadens them; a valuable cross-check of experiment and theory can be made by comparing the predicted and measured resonance widths. The radial-mode boundary layer correction in QSRs is approximately proportional to the square root of the gas thermal conductivity – in cylinders, the gas viscosity also features in the correction.⁴³ For most temperature ranges, the uncertainty in these parameters can be considered negligibly small for both helium and argon due to improved ab initio calculations (see Sec. 2.5).

AGT measurements are typically conducted on isotherms in a pressure range between 25 and 500 kPa, with the optimum pressure range depending on several factors such as the type of gas, temperature, and particular details of the apparatus.⁴⁴ At low pressures, the accuracy in determining f_a is compromised by weak acoustic signals, interference from neighboring modes due to resonance broadening, and the need to account for details of the interaction of the gas with the resonator’s walls.⁴⁵ At high pressures, higher-order virial terms are required to account for molecular interactions, and the elastic recoil of the resonator walls becomes increasingly significant. The shell recoil effect, which shifts f_a in proportion to gas density,⁴⁶ is difficult to predict in real resonators^47,48 because of the complex mechanical properties of the joint(s) formed when the cavity resonator is assembled.

For this and other reasons, it is not common practice to use Eq. (1) to determine T from w; instead, the measured data are fitted to low-order polynomials that account for the virial coefficients and perturbations that are proportional to pressure. Isotherm measurements have the advantage of data redundancy and reduced uncertainty, but are very slow to execute, with each pressure point taking several hours. Single-state AGT,⁴⁹ which utilizes low-uncertainty ab initio calculations of β_a and γ_a in helium, offers a much faster means of primary thermometry.

Figure 3 compares AGT measurements from five countries with ITS-90. The AGT data indicate that ITS-90 has an error of $∼25×10−6T$ near water’s boiling point and ∼−35 × 10⁻⁶T near 173 K. Near T_TPW, the derivative dT₉₀/dT ≈ 1 + 1.0 × 10⁻⁴. This implies that heat-capacity measurements made using ITS-90 will generate values of the heat capacity that are 0.01% larger than the true heat capacity. However, we are not aware of heat capacity measurement uncertainties as low as 0.01%.

Prior to the AGT publications shown in Fig. 3, Astrov et al. corrected an estimate used in their CVGT. They had used measurements of the linear thermal expansion of a metal sample to estimate the thermal expansion of the volume of their CVGT “bulb.” Using additional expansion measurements, Astrov et al. corrected their T − T₉₀ results. They now agree, within combined uncertainties, with the AGT data.⁵⁶ (Because AGT uses microwave resonances to measure the cavity’s volume in situ, it is not subject to errors from auxiliary measurements of thermal expansion.)

The higher-order terms contain combinations of both the dielectric and density virial coefficients and the compressibility. Equation (10) up to the fourth order can be found in Ref. 61.

DCGT works as a primary thermometer if the molar polarizability A_ɛ and virial coefficients contained in Eq. (10) are known from fundamental principles or independent measurements with sufficient accuracy. The effective compressibility κ_eff is also required. For classical DCGT, where isotherms are measured and the data are extrapolated to zero pressure via least-squares fitting, only A_ɛ and κ_eff are mandatory. This was the way thermodynamic temperature was determined for decades.^57,59,60 Consequently, in classical DCGT, ab initio data on virial coefficients serve as a consistency check or conversely DCGT is used for determination of virial coefficients to check theory.⁶¹ Since the theoretical calculations of the virial coefficients for helium improved drastically, it is now possible to use higher-order virial coefficients from theory to reduce the number of fitting coefficients or even to use the working equation directly without fitting and to determine temperature at each pressure point via the rearranged working equation. Recently, all three approaches have been tested and compared.⁶² Especially, the point-by-point evaluation is a shift of paradigm and at the moment only possible for helium, where the uncertainty of the ab initio calculations, especially of the second density virial coefficient, is small enough. Nevertheless, for other gases not only the virial coefficients but also the molar polarizabilities determined via DCGT have comparable or smaller uncertainties than ab initio calculations.⁶³ This is a field of potential improvement of theory already started with calculations of A_ɛ for neon^64,65 and for argon.⁶⁶ 

DCGT was operated in the temperature range from 2.5 K to about 273 K using helium-3, helium-4, neon,⁶⁷ and argon.⁶⁸ All noble gases have the advantage that the molar polarizability is independent of temperature at a level of precision far beyond that of state-of-the-art experiments.⁶⁹ 

Besides the use of dielectric measurements in primary thermometry, accurate determinations of polarizability and virial coefficients of noble gases and molecules using gas-filled capacitors have a much longer tradition. These setups, very similar to DCGT, use thermodynamic temperature as one of the input parameters. A complete overview of measurements cannot be given here. Already a very broad overview of existing data, partly at radio frequencies, was summarized by NBS in the 1950s.⁷⁰ In the following decades,⁷¹ different institutes with changing teams performed measurements until the early 1990s.⁷² In the year 2000, NIST started measurements on gases using capacitors resulting in the most accurate values for the measured molecules.^73,74 Very recently, PTB established a setup for separate measurement of dielectric and density virial coefficients using a combination of Burnett expansion techniques and DCGT.⁷⁵ The focus of this setup is the determination of properties of energy gases such as hydrogen-methane mixtures in the context of the transition to renewable energy. The setup will also provide lower-uncertainty tests of the ab initio calculations of the dielectric and density virial coefficients of the noble gases.

For primary thermometry, most significant recent improvements in DCGT have been achieved by independent determination of κ_eff using resonant ultrasound spectroscopy around 0 °C and an optimal choice of capacitor materials.⁷⁶ For the Boltzmann experiment with measuring pressures of up to 7 MPa, tungsten carbide was the ideal choice, while at low temperatures beryllium copper was used together with an extrapolation method. Relative uncertainties for κ_eff in terms of temperature on the level of 1 ppm near 0 °C have been achieved. Equally important are the improvements in pressure measurement. In contrast to AGT, where pressure is a second-order effect, in DCGT ɛ_r is directly linked to pressure. Therefore, the relative uncertainty in pressure is transferred to a relative uncertainty in temperature. The major steps here are discussed in Sec. 2.3.2 regarding the mechanical pressure standard developed at PTB in the framework of the Boltzmann constant determination.⁷⁷ These systems with relative uncertainties on the level of 1 ppm at pressures up to 7 MPa have been used to calibrate commercially available systems for pressures up to 0.3 MPa with relative uncertainties between 3 and 4 ppm. The dominant uncertainty component in DCGT measurements is the standard deviation of the capacitance measurement. The typical relative uncertainty in terms of temperature connected to this component is on the order of 5 ppm for the low temperature range but was reduced to the 1 ppm level in the case of the Boltzmann experiment at about 0 °C.⁷⁸ Finally, one problem in DCGT using helium is the very small molar polarizability compared to all other gases and molecules. Therefore, special care must be taken concerning impurities and here an especially severe issue is contamination with water.

The polarizability of water at frequencies of capacitance bridges and microwave resonators (see Sec. 5.1.2) is about a factor of 160 larger than that of helium. At cryogenic temperatures, water contamination in the gas phase is naturally reduced by outfreezing, but especially at room temperature the whole measuring setup as well as the gas purifying system must be highly developed. Furthermore, pollution with other noble gases must be treated carefully because they cannot be extracted by getters and filters. Ideally, a mass-spectrometer should be used for the detection of noble gas impurities to allow for an upper estimate of the uncertainty due to gas purity. In summary, with DCGT in the low temperature range from 4 to 25 K uncertainties near 0.2 mK for thermodynamic temperature are achievable. At around 0 °C, the smallest uncertainty for DCGT was achieved during the determination of the Boltzmann constant.⁷⁸ Converted to an uncertainty for thermodynamic temperature, this becomes about 0.5 mK.

In the intermediate range, the uncertainties are larger (between 1 and 2 mK at 200 K⁶⁸). The main restriction of the present low-temperature setup is the limited pressure range at intermediate temperatures. A measurement of high-pressure isotherms in this range is planned. Together with improved ab initio calculations for the second virial coefficients of argon and neon, a single-state version of DCGT might be possible, in analogy with single-state AGT. This could result in a significant reduction in both uncertainty and measurement time.

The constants B, B_R, C, C_R, etc. that appear in the higher-order terms of Eq. (12) are obtained either from theory or from fitting measurements of n²(p) on isotherms. [DCGT determines T using a version of Eq. (12) in which ɛ_r replaces n².]

Here, we focus on RIGT conducted at microwave frequencies as developed by Schmidt et al.⁸¹ and as recently reviewed by Rourke et al.⁸² These authors determined n from measurements of the microwave resonance frequencies f_m of a gas-filled, metal-walled, quasi-spherical cavity. Typical frequencies ranged from 2.5 to 13 GHz; for this range, the frequency dependence of n in the noble gases is negligible. As discussed in Sec. 2.3.1, RIGT has also been realized at optical frequencies in the context of pressure standards.⁸³ For helium, the corrections of A_ɛ and B_R from zero frequency to optical frequencies have been calculated ab initio.^79,84

Recently, two independent groups explored a two-gas method for measuring κ_eff of assembled RIGT resonators.^17,85 Ideally, two-gas measurements would replace measurements of κ_T of samples of the material comprising the resonator’s wall and also models for the cavity’s deformation under pressure. Both groups relied on new, accurately measured and/or calculated values of the density and refractivity virial coefficients of neon or argon.^61,79 Using helium and argon, Rourke determined κ_eff at T_TPW with the remarkably low uncertainty u_r(κ_eff) = 9.6 × 10⁻⁴.⁸⁵ Madonna Ripa et al. combined helium and neon data to reduce the uncertainty contribution from κ_eff to their determinations of T at the triple points of O₂ (⁠$≈54$ K), Ar (⁠$≈84$ K), and Xe (⁠$≈161$ K).¹⁷ They reported “partial success” and suggested that a revised apparatus using both gases and operating at higher pressures (p > 500 kPa) would obtain lower-uncertainty determinations of T. They also noted that the two-gas method requires twice as much RIGT data, accurate pressure measurements, and dimensional stability between gas fillings.

Rourke’s review of RIGT⁸² noted five groups implementing RIGT using microwave technology. In contrast, we are aware of only one group (at PTB) implementing DCGT.⁶² The relative popularity of RIGT results from the commercial availability of vector analyzers that can measure microwave frequency ratios with resolutions of 10⁻⁹. To our knowledge, using commercially available capacitance bridges, the best attainable capacitance ratio resolution is 70 × 10⁻⁹.⁸⁶ To attain higher resolution for DCGT, PTB developed a unique bridge that measures capacitance ratios with a resolution of order 10⁻⁸ in a 1 s averaging time. To achieve this specification, the PTB bridge must operate at 1 kHz and both the standard (evacuated) capacitor and the unknown (gas-filled) capacitor must have identical construction and be located in the same thermostat.⁸⁷ 

Figure 4 illustrates the several strategies being explored for acquiring RIGT data. Absolute RIGT acquires many (p, n) data on an isotherm and determines T via Eq. (14). This method requires state-of-the-art, absolute pressure measurements; therefore, the pressure gradient between the gas-filled cavity and the manometer (normally at ambient temperature) is required.⁸⁸ Uncertainty budgets for absolute RIGT can be found in Refs. 17 and 85.

Single-pressure RIGT (spRIGT) measures (p, n, T) and (p, n, T_ref) and determines T from $T/Tref≈(nT2−1)/(nTref2−1)$⁠. This strategy entirely avoids accurate pressure measurements; instead, the pressure in the cavity is required to be identical when n is measured at T and T_ref and the pressure (actually, the density of the gas) must be sufficiently low that an approximate pressure is adequate for making the virial corrections. This strategy was used by Gao et al. for RIGT between the triple point of neon (T_ref ≈ 24.5 K) and 5 K.⁸⁹ After establishing T_ref by acoustic thermometry, they claimed the uncertainties of this implementation of RIGT were smaller than the uncertainties of ITS-90.⁹⁰ 

When constant-frequency RIGT (cfRIGT) is implemented, the pressure in the cavity is changed to keep the refractive index constant as the temperature is changed from T_ref to T. In this case, T/T_ref ≈ p(T, n)/p(T_ref, n).⁹¹ This scheme minimizes the frequency-dependent effects of the coaxial cables on the microwave determination of T/T_ref.

To economically search for measurement or modeling errors, one can obtain three redundant values of T/T_ref by measuring microwave frequencies at four judiciously chosen values of (p, n). Two measurements are made on the isotherm T_ref at the values (p₁, n₁) and (p₂, n₂). Two other measurements are on the isotherm T at (p₂, n₃) and (p₃, n₁). spRIGT connects the points (p₂, n₂) and (p₂, n₃). cfRIGT connects the points (p₁, n₁) and (p₃, n₁). All four points are used to approximately implement rRIGT via Eq. (15).

Compared with other forms of gas thermometry, relative RIGT has significant advantages at low temperatures. We have already emphasized the availability of microwave network analyzers and the possibility of avoiding state-of-the art pressure measurements. By measuring several microwave resonance frequencies at each state, certain imperfections of the measurements and modeling can be detected. Comparisons of the frequencies of transverse electric (TE) and transverse magnetic (TM) microwave modes might detect the presence of dielectric films such as oxides, oil deposits, or adsorbed water on the cavity’s walls.⁹² Because relative RIGT relies on microwave frequency ratios, the precise shape of the cavity is unimportant. Cavity shapes other than quasispheres may be advantageous in particular applications.

RIGT is simpler and more rugged than relative AGT (rAGT) because RIGT requires neither delicate acoustic transducers nor acoustic ducts. However, RIGT is unlikely to replace rAGT at ambient and higher temperatures because RIGT is more sensitive to the cavity’s dimensions than rAGT by the factor 1/(ɛ_r − 1), which typically ranges from 200 to 20 000. Furthermore, microwave RIGT is especially sensitive to polar impurities. Adding 1 ppm (mole fraction) of water vapor to dilute argon gas at 293 K will increase the dielectric constant of the gas by 18 ppm and increase the square of the speed of sound by 0.12 ppm. If the water vapor were undetected, these changes would reduce argon’s apparent RIGT temperature by 18 ppm and increase argon’s apparent rAGT temperature by 0.12 ppm. For helium, the corresponding temperatures are reduced by 145 ppm and 4 ppm.

The website of the International Bureau of Weights and Measures includes a document (“Mise en pratique…”) that indicates how the SI base unit, the kelvin, may be realized in practice using four different versions of gas thermometry.⁹³ Surprisingly, this document omits CVGT, the version of gas thermometry that was the primary basis of ITS-90. In this section, we briefly describe the operation of a particular realization of CVGT and the inconsistent results it generated. This may explain why CVGT was omitted from the Mise en pratique. We mention the post-1990 theoretical and experimental developments that suggest an updated realization of CVGT might generate very accurate realizations of the kelvin.

The simplicity of Eq. (16) hides the many complications of CVGT. We mention three examples. (1) During pressure measurements, helium outside the gas bulb was maintained at the same pressure as the helium inside the gas bulb. (2) Thermo-molecular and hydrostatic pressure gradients in the capillary were taken into account. (3) At high temperatures, creep in the gas bulb’s volume was detected by time-dependent pressure changes; the pressure was extrapolated back in time to its value when the bulb was placed in the furnace.

We denote the second most recent realization of NBS/NIST’s relative CVGT by “CV_NBS76.”⁹⁶ Both CV_NIST90 and CV_NBS76 shared apparatus and many procedures. However, Ref. 94 lists 11 significant changes. Here, we mention only one. CV_NIST90’s two cylindrical gas bulbs had been fabricated entirely from sheets of (80 wt. % Pt + 20 wt. % Rh) alloy. The sides and bottom of CV_NBS76’s gas bulb were fabricated from the same alloy; however, the top of the bulb was inadvertently fabricated from (88 wt. % Pt + 12 wt. % Rh) alloy. Perhaps the slight differences in thermal expansions of these alloys led to an anomalous thermal expansion of the volume of CV_NBS76’s gas bulb.

Unfortunately, the results from CV_NIST90 and CV_NBS76 were inconsistent, within claimed uncertainties, in the range of temperature overlap (505 K $≤T≤$ 730 K). An approximate expression for the differences is: T_NIST90 − T_NBS76 ≈ 0.090 × (T/K − 400) mK. This inconsistency was not explained by the authors of CV_NIST90 nor by the authors of CV_NBS76. Furthermore, the authors did not assert the more recent CV_NIST90 results were more accurate than the earlier CV_NBS76 results. The working group that developed ITS-90 had no other data, from NIST or elsewhere, that were suitable for resolving the inconsistency. Therefore, the working group required ITS-90 to be the average of T_NIST90 and T_NBS76 in the overlap range.⁹⁷ 

In the range 2.5–308 K, ITS-90 relied, in part, on another realization of CVGT that had a troubled history. Astrov et al. deduced the thermal expansion of their copper gas bulb’s volume from measurements of the linear thermal expansion of copper samples taken from the block used to manufacture their bulb.⁹⁸ However, the thermal expansion data were inconsistent with other data for copper. Astrov’s group repeated the thermal expansion measurements using another (better) dilatometer. The more recent expansion data, published in 1995, changed the values of T by more than 50 × 10⁻⁶T in the range 130 K $<T<$ 180 K, where the uncertainties had been estimated as $≤26×10−6T$⁠.⁵⁶ 

Recently, a working group of the Consultative Committee for Thermometry reviewed primary thermometry below 335 K.²⁰ Astrov’s revised CVGT values are close to the current consensus, which is primarily based on AGT and DCGT. The working group retained three other low-temperature realizations of CVGT. Post-1990 AGT measurements of T − T₉₀ near 470 and 552 K indicate that CV_NIST90 is indeed more accurate than CV_NBS76.⁵⁰ Despite the fact that CVGT was the primary basis for the ITS-90, the Mise en pratique does not include CVGT. We speculate that no temperature metrology group is pursuing CVGT because: (1) CVGT is complex, (2) Astrov et al.’s thermal expansion problem, (3) unexplained problems with NBS/NIST’s CVGT, and (4) rapid advances in other versions of gas thermometry.

We now ask: is CVGT a viable method of primary thermometry today? The gas bulb of a modern CVGT would incorporate feedthroughs to enable measuring microwave resonance frequencies of the bulb’s cavity. The resonance frequencies would determine the bulb’s volumetric thermal expansion, thereby avoiding auxiliary measurements of linear thermal expansion and also avoiding the assumption of isotropic expansion. If the bulb incorporated a valve and a differential-pressure-sensing diaphragm, the dead-space corrections would vanish. (The diaphragm’s motion could be detected using optical interferometry.) Today, the ab initio values of B(T) would reduce the uncertainty component from B(T) to near zero. A contemporary CVGT could operate at $∼5×$ higher helium densities than published experiments without generating significant uncertainties from either the virial coefficients or from pressure-ratio measurements. The higher density, together with simultaneous pressure and microwave measurements, might enable separation of the bulb’s creep from contamination by outgassing. Most outgassing contaminants affect helium’s dielectric constant, refractivity, and speed of sound much more than they affect helium’s pressure, an advantage of CVGT. However, CVGT inherently uses fixed aliquots of gas. Therefore, CVGT cannot benefit from flowing gas techniques that have been used, for example, in high-temperature AGT.⁵⁰ In summary, contemporary CVGT could be competitive with other forms of primary gas thermometry, with a possible exception at the highest temperatures, where flowing gas might be required to maintain gas purity.

Traditionally, standards based on the realization of the mechanical definition of pressure, the normal force applied per unit area onto the surface of an artifact, include pressure balances and liquid column manometers. The combined overall pressure working range of these instruments extends over seven orders of magnitude, roughly between 10 Pa and 100 MPa. Liquid column manometers achieve their best performance, with relative standard uncertainty as low as 2.5 ppm, near their upper working limit at a few hundred kPa.⁹⁹ With a few notable exceptions, the typical relative standard uncertainty of pressure balances spans between nearly 1 × 10⁻³ at 10 Pa, the lowest end of their utilization range, down to 2–3 ppm in the range between 100 kPa and 3 MPa.^99,100 One such exception is the remarkable achievement of a relative standard uncertainty as low as 0.9 ppm for the determination of helium pressures up to 7 MPa,⁷⁷ though this achievement required the extensive dimensional characterization, and the cross-float comparison, of the effective areas of six piston–cylinder sets manufactured to extraordinarily tight specifications, with a research effort lasting several years. In spite of this outstanding result, the accurate characterization of pressure balances is challenging, due to the complexity of the dimensional characterization of the cross-sectional area of piston–cylinder assemblies, which includes finite-element modeling of their deformation under pressure.^101,102 International comparisons periodically provide realistic estimates of the average uncertainty of realization of primary standards among NMIs. In 1999, a comparison of primary mechanical pressure standards in the range 0.62 MPa $<p<$ 6.8 MPa, involving five NMIs leading in pressure metrology exchanging a selected piston–cylinder set, was completed.¹⁰³ The resulting differences ΔA_eff ≡ 10⁶(A_eff/⟨A_eff⟩ − 1) of the effective area A_eff of the piston from the reference value ⟨A_eff⟩ spanned beyond their combined uncertainties with such significant spread to show that the pressure standards realized by different NMIs were mutually inconsistent.

These inconsistencies strengthened the motivation for the development of standards realizing a thermodynamic definition of pressure by the experimental determination of a physical property of a gas having a calculable thermodynamic dependence on density, combined with accurate thermometry. This possibility was initially proposed in 1998 by Moldover,¹⁰⁴ who envisaged, already at that time, the potential of first-principles calculation to accurately predict the thermodynamics and electromagnetic properties of helium and the maturity of experiments determining the dielectric constant using calculable capacitors. The metrological performance of thermodynamic pressure standards has continuously improved over the last two decades to become increasingly competitive in terms of accuracy, providing important alternatives that may test the exactness of the mechanical standards discussed above and eventually replace some of them. Also, due to their reduced complexity and bulkiness, simplified versions of thermodynamics-based standards may be more flexibly adapted to specific technological and scientific applications of pressure metrology. The best-performing recent realizations of gas-based pressure standards include measurements of the dielectric constant using capacitors and of the refractive index at microwave and optical frequencies, respectively using resonant cavities and Fabry–Pérot refractometers. In Secs. 2.3.1 and 2.3.2, we separately discuss the most notable of these developments depending on the pressure range of their application.

In the low vacuum regime, several experimental methods are available which may provide alternative routes for traceability to the pascal. For the cases involving optical measurements, these methods include: (1) refractometry (interferometry), implemented in various configurations that employ single or multiple cavities or cells with fixed or variable path lengths; (2) line-absorption methods. The achievements and perspectives of all these methods were recently reviewed.¹⁰⁵ 

At present, Fabry–Pérot refractometry with fixed length optical cavities (FLOC) has demonstrated the lowest uncertainty for the realization of pressure standards near atmospheric pressure and down to 100 Pa. In principle, the uncertainty of this method is limited by several optical and mechanical effects, most importantly by the change in the length of the cavity due to compression by the test gas, with the same sensitivity to the imperfect estimate of the compressibility κ_T that affects RIGT. However, this major uncertainty contribution may be drastically reduced, though not completely eliminated, by measuring the pressure-induced length change of a second reference FLOC monolithically built on the same spacer, which is kept continuously evacuated. In 2015, a dual-cavity FLOC achieved an extremely accurate determination of the refractive index of nitrogen at λ = 632.9908 nm, T = 302.9190 K and 100.0000 kPa by reference to the pressure realized by a primary standard mercury manometer, and using refractive index measurements in helium to determine the compressibility.¹⁰⁶ A comparison of the pressures determined by the nitrogen refractometer with the mercury manometer below the primary calibration point at 100 kPa down to 100 Pa showed relative differences within 10 ppm. A direct comparison between laser refractometry with nitrogen and a mercury manometer was realized one year later also at NIST.¹⁸ The comparison showed relative differences between these instruments within 10 ppm over the range between 100 Pa and 180 kPa. The laser refractometer outperforms the precision and repeatability of the liquid manometer and demonstrates a pressure transfer standard below 1 kPa that is more accurate than its current primary realization. Such remarkably low uncertainty also favorably compares to the best dimensional characterization and modeling of non-rotating piston–cylinder assemblies.¹⁰⁷ 

In 2017, more accurate measurements in helium and nitrogen were performed between 320 and 420 kPa using a triple-cell heterodyne interferometer referenced to a carefully calibrated piston gauge, showing relative differences within 5 ppm with uncertainties on the order of 10 ppm.⁸³ Some pressure distortion errors affecting FLOC might in principle be eliminated by refractive index measurement with a variable length optical cavity (VLOC). The realization of this technique requires extremely challenging dimensional measurements, with displacements on the order of 15 cm that must be determined with picometer uncertainty.¹⁰⁸ Gas modulation techniques, with the measuring cavity frequently and repeatedly switched between a filled and evacuated condition, have been recently developed,^109,110 aiming at the reduction of the effects of dimensional instabilities and other short- and long-term fluctuations that affect Fabry–Pérot refractometers. A novel realization of an optical pressure standard, based on a multi-reflection interferometry technique, has also been recently developed, demonstrating the possible realization of the pascal with a relative standard uncertainty of 10 ppm between 10 and 120 kPa.¹¹¹ Optical refractometry for pressure measurement is also being pursued at other NMIs.^112,113

With accurate pressure measurement, these optical methods can yield the thermodynamic temperature, becoming another approach to RIGT. This was demonstrated in Ref. 83, where a refractometer was used to measure the Boltzmann constant (albeit with higher uncertainty than AGT and DCGT measurements) prior to the 2019 SI redefinition.

At microwave frequencies, the realization of a low-pressure standard requires a substantial enhancement in frequency resolution. Recently, it was demonstrated by Gambette et al. that by coating the internal surface of a copper cavity with a layer of niobium, and working at temperatures below 9 K where niobium becomes superconducting, pressures between 500 Pa and 20 kPa can be realized very precisely.^114,115 The overall relative standard uncertainty of this method is currently 0.04%, with the largest contribution from non-state-of-the-art thermometry, which is likely to be substantially reduced in future work.

Differently than initially envisaged, the first realization of a thermodynamic pressure standard was not obtained by capacitance measurements, but using a microwave resonant cavity working in the GHz frequency range, i.e., by a RIGT method. A main motivation for this choice was the development of quasi-spherical microwave resonators, whose internal triaxial ellipsoidal shape slightly deviates from that of a perfect sphere.⁹² This particular geometry resolved the intrinsic degeneracy of microwave modes, allowing enhanced precision in the determination of resonance frequencies.

By 2007, Schmidt et al.⁸¹ demonstrated a pressure standard based on the measurement of the refractive index of helium to achieve overall relative pressure uncertainty u_r(p) within 9 × 10⁻⁶ between 0.8 and 7 MPa. At the upper limit of the pressure range, the uncertainty was dominated by the uncertainty of the isothermal compressibility κ_T of maraging steel, which was determined using resonance ultrasound spectroscopy (RUS).⁷⁶ Recently, Gaiser et al.¹⁹ realized Moldover’s original proposal of a capacitance pressure standard using DCGT techniques that they had refined during their measurements of the Boltzmann constant. They achieved the remarkably low uncertainty u_r(p) = 4.4 × 10⁻⁶ near 7 MPa. Recently, the same experimental data were re-analyzed to take advantage of the increased accuracy of the ab initio calculation of the second density virial coefficient B of He,¹¹ reducing the overall uncertainty of the capacitance pressure standard to u_r(p) = 2.2 × 10⁻⁶.¹¹⁶ 

At pressures below 1 MPa, the uncertainty of the realization of a pressure standard based on DCGT or RIGT with helium is limited by the resolution of relative capacitance or frequency measurements. This limit would be immediately reduced by up to one order of magnitude by using, instead of helium, a more polarizable gas like neon or argon. However, while a significant improvement of the interaction potential, and hence of the ab initio calculated B, has recently been achieved for neon¹¹⁷ and for Ar,¹¹⁸ it is not likely that the best available calculations of the molar polarizability A_ɛ of neon⁶⁵ or argon⁶⁶ can be improved sufficiently to replace experiment in the near future. However, an experimental estimate of A_ɛ of both neon and argon was obtained by comparative DCGT measurements relative to helium, with relative uncertainty of 2.4 ppm,⁶³ and may now be used for the realization of pressure standards with other apparatus. For similar purposes, the ratio of the refractivity of several monatomic and molecular gases, namely Ne, Ar, Xe, N₂, CO₂, and N₂O, to the refractivity of helium was determined at T = 293.15 K, λ ∼ 633 nm, with standard uncertainty within 16 × 10⁻⁶, using interferometry.¹¹⁹ At pressures higher than a few MPa, the imperfect determination of the deformation of the cavity under pressure would impact the overall uncertainty of a pressure standard based on RIGT or DCGT. One way to overcome this limit would be to measure the refractivity of two gases at identical values of an unknown pressure using a single apparatus at a known temperature. If the refractivity of both gases were known (either from ab initio calculations or reference measurements), the two measurements would determine both the effective compressibility κ_T of the apparatus and the unknown pressure. The same strategy is also applied to increase the upper pressure range where refractometry methods like FLOC can be applied, though use of helium for the determination of distortion effects requires correcting for diffusion within the glasses used for the construction of these apparatuses.¹²⁰ 

Up to this point, we have considered interactions between temperature and pressure standards and the rigorously calculated, low-density properties of the noble gases including the polarizability and second and third density and dielectric virial coefficients. We now compare ab initio calculations with measurements at pressures above 7 MPa and at correspondingly higher densities. The literature includes temperature-dependent values of 6 density virial coefficients of helium,¹²¹ 7 acoustic virial coefficients of krypton,¹²² and 6 density virial coefficients of argon.¹²³ These calculations used the best ab initio two-body and nonadditive three-body potentials that were available at the time of publication. Many-body non-additive potentials involving four or more bodies, which are needed for the exact calculation of virial coefficients from the fourth onwards, are not available and are generally neglected, resulting in an uncontrolled approximation. Here, we compare measurements of the density of helium ρ_meas(p, T) with values calculated ab initio. This comparison avoids fitting ρ_meas(p, T) to the VEOS because such fits yield highly correlated values for the separate virial coefficients, each with large uncertainties. Later in this section, we comment on comparisons using speed-of-sound data.

Measurements of gas densities with uncertainties below 0.1% are expensive and rare because they are not required for chemical and mechanical engineering. The uncertainties of most process models are dominated by imperfect models of equipment (heat exchangers, compressors, distillation columns, etc.) and/or imperfect knowledge of the composition of feedstocks and products. An example of a demanding application of gas density and composition measurements is custody transfer of natural gas as it flows through large pipelines near ambient temperature and at high pressures (e.g., 7 MPa). An international comparison among NMIs achieved a k = 2 volumetric flow uncertainty of only 0.22%.¹²⁴ In this context, density and composition measurements with uncertainties of order 0.1% are satisfactory for converting volumetric flows into mass flows and heating values.

In Fig. 5, the remarkable data of McLinden and Lösch-Will are used to test the ab initio VEOS of helium in the ranges 1 MPa $<p<$ 38 MPa and 223 K $<T<$ 323 K.¹²⁵ These data were acquired using a magnetic suspension densimeter. A weigh scale determined the buoyant forces on two “sinkers” immersed in the helium. The data are precise, well-documented, and traced to SI standards with a claimed, k = 2, density uncertainty of 0.015% + 0.001 kg/m³ at the temperature extremes and at the highest density. These features attracted previous comparisons with theory.^21,121,126

For the present comparison, where recently published theoretical values of the virial coefficients are used, we converted the measured temperatures from the ITS-90 to thermodynamic temperatures using Ref. 20 and we converted the measured mass densities to molar densities using the defined value of the universal gas constant and the molar mass for McLinden and Lösch-Will’s helium sample. At densities below $∼4000$ mol/m³, the uncertainties and the values of (ρ_meas/ρ_calc − 1) diverge on isotherms as ρ⁻¹ and/or p⁻¹. (See Fig. 5.) These low-density divergences result from time-dependent drifts in the zeros of the densimeter and/or pressure transducer. Because the divergences contain more information about the apparatus than about helium’s VEOS, we do not discuss them.

The claimed k = 2 uncertainty of ρ_meas is 150 ppm;¹²⁵ the span of the upper panels of Fig. 5 is ±150 ppm. The dashed curves (- -) in the middle panel of Fig. 5 represent upper bounds to the uncertainty of ρ_calc(T, ρ) at 223 K. For these upper bounds, we used the k = 2 uncertainties of the virial coefficients U(B), U(C), … provided by their authors. In Eq. (17) we replaced B with B + U(B); we replaced C with C + U(C), etc. The uncertainties of ρ_calc(T, ρ) are smaller at higher temperatures. We conclude that ρ_calc agrees with ρ_meas well within combined uncertainties.

At densities above $∼4000$ mol/m³, the differences (ρ_meas/ρ_calc − 1) are nearly independent of the density; however, the average densities are 34 ppm larger than their expected values ρ_calc. These offsets are well within the claimed measurement uncertainties (k = 2, $∼150$ ppm). However, as shown in the lower panel of Fig. 5, the offsets have both a random and a systematic dependence on the temperature. The systematic temperature dependence can be treated as a correction to the calibration of the sinkers’ densities ρ_sinker(p, T). Such a correction does not remove the spread (±14 ppm) among the four isotherms at 273 K. Possible causes of this spread are changes between runs of temperature (±3.8 mK) and/or of impurity content (e.g., ±2.3 ppm of N₂). In any case, the offsets are smaller than the claimed uncertainties of ρ_meas(p, T).

Moldover and McLinden²¹ extended McLinden and Lösch-Will’s data¹²⁵ to 500 K. The extended data are a less-stringent test of the VEOS than Fig. 5 because they span the same pressure range (p < 38 MPa) at higher temperatures; therefore, they span a smaller density range. If McLinden’s data could be extended to lower temperatures with comparable uncertainties, they would test helium’s VEOS in greater detail and they might reach a regime where U(ρ_meas) < U(ρ_calc). Schultz and Kofke conducted much more detailed tests of McLinden and Lösch-Will’s data.¹²¹ We agree with their conclusion that the data are consistent with the VEOS calculated ab initio.

It may be possible to significantly reduce the uncertainty of ρ_meas by improving magnetic suspension densimeters, as suggested by Kayukawa et al.¹²⁸ They fabricated sinkers from single crystals of silicon and germanium because these materials have outstanding isotropy, stability, and well-known physical properties. Also, they refined the model and the functioning of their magnetic suspension so that it was independent of the magnetic properties of the fluid under study at the level of 1 ppm. They measured the density of a liquid near ambient temperature and pressure with a claimed k = 1 relative uncertainty of 5.4 × 10⁻⁶. To date, they have not demonstrated this uncertainty far from ambient temperature and pressure. Even if ρ_meas achieved such low uncertainties, tests of the VEOS would have to solve problems arising from impure gas samples and imperfect temperature and pressure measurements.

Alternative methods of measuring equations of state have been reviewed by McLinden.¹²⁹ Several methods require filling a container of known volume V_cont(p₀, T₀) with a known quantity of gas and then measuring the pressure as the temperature is changed. These methods resemble the CVGT method discussed in Sec. 2.2.4. Like CVGT, they require accurate values of V_cont(p, T); however, unlike CVGT, testing a VEOS requires much higher pressures. Determining V_cont(p, T) over wide ranges is complex because: (1) containers comprised of metal alloys have anisotropic elastic and thermal expansions; (2) containers have seals and joints or welds which have complicated mechanical properties; (3) alloys creep and/or anneal under thermal and mechanical stresses. In summary, volumetric methods are unlikely to replace Archimedes-type densimeters because V_cont(p, T) is an assembled object subjected to complicated stresses; in contrast, the densimeter’s sinkers are single objects subjected to hydrostatic pressure.

Remarkably, the Burnett method¹³⁰ of measuring the equation of state requires neither determining V_cont(p, T) nor measuring quantities of gas. This method uses two pressure vessels with stable volumes V_a and V_b. On each isotherm, gas is admitted into V_a and the pressure is measured. The gas is allowed to expand so that it fills both V_a and V_b and the pressure is measured again. V_b is evacuated and the process is repeated several times. The measured pressures on each isotherm are fitted to the VEOS and an apparatus parameter: the volume ratio at zero pressure (V_a,0 + V_b,0)/V_a,0. The pressure dependences of V_a and V_b must also be known. Usually, they are estimated from elastic constants and models of the pressure vessels; therefore, precise estimates encounter complications of estimating V_cont(p, T). Perhaps this explains the large scatter in Burnett determinations of D(T).¹²⁶ A fairly recent Burnett measurement of the equations of state of nitrogen and hydrogen (353–473 K; 1–100 MPa) claimed k = 2 uncertainties of ρ_meas ranging from 0.07% to 0.24%.¹³¹ 

In addition to ρ_meas, measurements of the squared speed of sound w²(p, T) in gases have been used to critically test either the VEOS¹²² of Eq. (7) or its acoustic analog, Eq. (1). Accurate values of w²(p, T) in gases are readily available. At the low gas pressures used for acoustic thermometry, the relative expanded uncertainties U_r(w²(p, T)) measured using quasi-spherical cavity resonators are a few parts in 10⁶ and are dominated by thermometry problems and/or impurities. However, uncertainties grow approximately linearly in pressure because of imperfect models of the recoil of the cavity’s walls in response to the resonating gas. In one study of argon, U_r(w²) ≈ 1.2 × 10⁻⁴(p/20 MPa) except near the critical point.¹³² At pressures above $∼5$ to $∼10$ MPa, pulse-echo techniques achieve uncertainties comparable to or smaller than resonance techniques.^122,133 Remarkably, w² from the two techniques agreed within 60–200 ppm within a range of overlap (argon, 250–400 K, $∼10$ to $∼20$ MPa¹³³).

It is more complex to compare $wmeas2(p,T)$ to a calculated VEOS than to compare ρ_meas to the same VEOS. To calculate the n-th acoustic virial coefficient from the n-th density virial coefficient, one also needs the first and second temperature derivatives of the n-th virial coefficient as well as all the lower-order density virial coefficients and their temperature derivatives. There are several routes to conduct such a comparison, which are completely equivalent. First, the temperature derivatives of the density virial coefficients can be calculated from ab initio potentials using, e.g., the Mayer sampling Monte Carlo (MC) method. Second, the temperature derivatives can be obtained from fits of the theoretically calculated temperature-dependent density virial coefficients. Third, the virial equation of state can be transformed by thermodynamic identities into an acoustic virial equation of state or it can be integrated to formulate a Helmholtz energy equation, from which the speed of sound can be calculated. Speeds of sound calculated by either of the two resulting equations contain contributions from terms with higher acoustic virial coefficients than those used in the density virial equation of state, i.e., it can be expected that the region of convergence of this virial equation of state for the speed of sound extends to higher pressures than that of the acoustic virial equation of state with virial coefficients derived directly from density virial coefficients. These terms describe contributions of configurations of particles which are contained in the low-order density virial coefficients to the higher-order acoustic virial coefficients. Fourth, densities can be calculated from $wmeas2(p,T)$ by the method of thermodynamic integration¹³⁴ and directly compared to the density virial equation of state. As initial conditions for the integration, the density and heat capacity on an isobar must be known. There are subtleties to integrating $wmeas2(p,T)$⁠.¹³⁵ In the first method the uncertainties of the virial coefficients and their temperature derivatives follow from the MC simulation and can be propagated into an uncertainty of the acoustic virial equation of state, while in the other methods the uncertainty of the density virial coefficients or the experimental speeds of sound can be propagated into the acoustic virial equation of state or calculated densities, respectively.

For helium, Gokul et al.¹³⁶ calculated the acoustic virial coefficients through the seventh order by the second method outlined above from density virial coefficients. They used the second density virial coefficients reported by Czachorowski et al.,¹¹ which are based on the pair potential reported in the same work. The higher virial coefficients were taken from the work of Schultz and Kofke.¹²¹ They are based on the pair potential of Przybytek et al.¹³⁷ and the three-body potential of Cencek et al.¹³⁸ Uncertainties in the density virial coefficients were propagated into uncertainties in the acoustic virial coefficients by the MC method recommended in Supplement 1 to the “Guide to the Expression of Uncertainty in Measurement.”¹³⁹ Gokul et al.¹³⁶ formulated the acoustic virial equation of state as expansion in terms of density or pressure. The uncertainty of speeds of sound calculated with the acoustic VEOS was estimated from the uncertainty of the acoustic virial coefficients.

The density expansion of Gokul et al. was compared to the experimental data of Gammon,¹⁴⁰ Kortbeek et al.,¹⁴¹ and Plumb and Cataland.¹⁴² The data of Gammon were measured with a variable-path interferometer operating at 0.5 MHz. They cover the temperature range between 98 and 423 K with pressures up to 15 MPa, and according to the author have an uncertainty of 0.003% of w². For these data, we estimated the expanded (k = 2) relative uncertainty U_r(w²) = 0.000 09 by adding uncertainties of 0.003% (for the distance between the crystals), 0.001% (for the precision), and 0.005% (for sample impurities and/or temperature errors, based on the inconsistencies among the 14 isotherms). Gammon’s data agree with the acoustic virial equation of state within 0.01% with a few exceptions. The data of Kortbeek et al. were measured with a double-path-length pulse-echo technique, cover the temperature range from 98 to 298 K at pressures between 100 MPa and 1 GPa, and, according to the authors, have an uncertainty of 0.08%. They deviate from the acoustic VEOS between a few tenths of a percent at 100 MPa up to about 4% at 298 K and 1 GPa. These rather large deviations are due to the fact that the acoustic virial equation of state is not converged at such high pressures. The measurements of Plumb and Cataland cover the low temperature range between 2.3 and 20 K at pressures up to 150 kPa. They agree with the acoustic virial equation of state of Gokul et al. to within 0.05% except at the lowest measured pressures of about 1.5 kPa, where the deviations reach up to 0.18%. Gokul et al. also assessed the pressure range in which the acoustic VEOS is more accurate than the available experimental data for the speed of sound. At low pressures, they observed that speeds of sound calculated with the acoustic VEOS are more accurate than the experimental data of Gammon. Gokul et al. further noticed that speeds of sound calculated with the acoustic VEOS are more accurate than the experimental data of Kortbeek et al. up to about 300 MPa depending on temperature. At higher pressures, they considered the experimental data of Kortbeek et al. to be more accurate than the computed virial equation of state. This conclusion appears to be too optimistic in light of the low uncertainty of 0.08% in the experimental data and the rather large deviations of up to 2% from the virial equation of state below 300 MPa.

Gokul et al. also examined the convergence behavior of the acoustic virial equation of state more closely for the expansions in density and pressure. They considered a virial equation of state converged if the value of the speed of sound calculated with it agrees with all higher orders of the expansion within a certain tolerance. They observed that the pressure range in which the expansion in density converges is extended when the tolerance is increased. However, the expansion in pressure hits a pressure limit in the supercritical region, above which increasing the tolerance does not extend the region of convergence farther. Above this pressure limit, the expansion in pressure completely fails. Recently, Wedler and Trusler measured the speed of sound in supercritical helium with a dual-path pulse-echo technique in the temperature range between 273 and 373 K up to 100 MPa with an expanded uncertainty (k = 2) of 0.02%–0.04%.¹⁴³ Their data agree with the seventh-order acoustic VEOS in density of Gokul et al.¹³⁶ with a few exceptions in the whole range of the measurements within 0.025%, which shows that this form of the VEOS is converged in the region of the measurements.

The first calculation of the third virial coefficient of argon using a first-principles three-body potential was performed by Mas et al.¹⁴⁴ using the empirical pair potential developed by Aziz.¹⁴⁵ The results agreed almost to within combined uncertainties with the third virial coefficients extracted from experimental data (with theoretical constraints) by Dymond and Alder.¹⁴⁶ Jäger et al. calculated density virial coefficients up to seventh order for argon with their pair and nonadditive three-body potentials.¹²³ The calculated virial coefficients were fitted by polynomials in temperature. The seventh-order VEOS was compared with the very accurate (p, ρ, T) data of Gilgen et al.,¹⁴⁷ which were measured with a magnetic suspension densimeter. These data are characterized by a relative uncertainty (k = 2) in density of 0.02%. Pressures calculated with the theoretical virial equation of state agree with these data at the highest temperature of the measurements, 340 K, within 0.01%.

In further work, Jäger¹⁴⁸ used thermodynamic identities to calculate several properties of argon including the speed of sound from the virial equation of state and compared the results with the accurate experimental data of Estrada-Alexanders and Trusler¹³² and Meier and Kabelac.¹³³ The data of Estrada-Alexanders and Trusler¹³² were measured with a spherical resonator and cover the temperature range between 110 and 450 K at pressures up to 19 MPa, while the data of Meier and Kabelac were measured with a dual-path-length pulse-echo technique and cover the temperature range between 200 and 420 K with pressures between 9 and 100 MPa. The expanded (k = 2) uncertainty of these datasets was estimated to be 0.001%–0.007% and 0.011%–0.036%, respectively. At 300 and 400 K, the calculated speeds of sound agree with both experimental datasets up to 100 MPa within 0.04% and 0.08%, respectively. At the near-critical temperature 146 K and supercritical temperature 250 K, the deviations of the calculated values from the experimental data of Ref. 132 increase with pressure from essentially zero in the ideal-gas limit to about 0.3% at 3.7 MPa and about 0.02% at 12.2 MPa.

In another paper, Jäger et al. presented calculations of the second and third density virial coefficient of krypton.¹⁴⁹ They developed a very accurate pair potential for the krypton dimer, and nonadditive three-body interactions were described by an ab initio extended Axilrod–Teller–Muto potential, which was fitted to quantum chemical calculations of the interaction energy of equilateral triangle configurations of three krypton atoms. El Hawary et al.¹²² calculated density virial coefficients from the fourth to the eighth using the pair potential and extended Axilrod–Teller–Muto potential of Jäger et al. The calculated virial coefficients were fitted to polynomials in temperature, and the virial equation of state was integrated to formulate it as a fundamental equation of state in terms of the Helmholtz energy. Furthermore, El Hawary et al. measured the speed of sound in liquid and supercritical krypton between 200 and 420 K at pressures from 6.1 to 100 MPa with an uncertainty (k = 2) of 0.005%–0.018%. At 240, 320, and 420 K, the seventh-order and eighth-order virial equations of state agree with each other within 0.02% up to 7, 17, and 38 MPa, respectively. In the region where the virial equation of state is sufficiently converged, the calculated speeds of sound are systematically about 0.08% lower than the experimental data. This small difference is probably due to the uncertainty of the pair potential and the simplified treatment of nonadditive three-body interactions with the extended Axilrod–Teller–Muto model.

At high density in the supercritical region where the virial equation of state does not converge and in the liquid region, thermodynamic properties can be calculated by MC or molecular-dynamics (MD) simulations.¹⁵⁰ Since the generation of Markov chains in MC simulations avoids some of the numerical errors of algorithms used to integrate the equations of motion in MD simulations, MC simulations are the preferred method for calculating accurate values for thermodynamic properties. In statistical mechanics, there are eight basic ensembles for performing MC or MD simulations,¹⁵¹ which are characterized by a thermodynamic potential, three independent variables, and a weight factor, which describes the distribution of systems in the ensemble, in which MC simulations of fluids can be performed. Ströker et al.¹⁵² pointed out that the NpT ensemble, in which the number of particles, the pressure, and the temperature are the independent variables, is best suited for the calculation of thermodynamic properties because only ensemble averages involving the enthalpy and volume, but no derivatives of the potential energy with respect to volume, appear in the equations for thermodynamic properties. This means that no derivatives of the potentials are needed in a simulation.

The argon calculations of Mas et al.¹⁴⁴ described above were later extended by performing NVT, NpT, and Gibbs ensemble MC simulations¹⁵³ along the vapor–liquid coexistence curve. The parameters of the critical point agreed with experiments within 0.8% or better.¹⁵⁴ 

Ströker et al.¹⁵² carried out semiclassical MC simulations of thermodynamic properties of argon in the NpT ensemble at the subcritical isotherm 100 K and the supercritical isotherm 300 K at pressures up to 100 MPa. The interactions between argon atoms were described by the pair potential of Jäger et al.¹⁵⁵ and the nonadditive three-body potential of Jäger et al.¹²³ Quantum effects were accounted for by the Feynman–Hibbs corrections to the pair potential. Calculated densities agree with the accurate data measured by Gilgen et al.¹⁴⁷ and Klimeck et al.¹⁵⁶ within less than 0.01%, while calculated speeds of sound agree within less than 0.1% with the accurate experimental data of Estrada-Alexanders and Trusler¹³² at low pressure in the supercritical region and Meier and Kabelac¹³³ at high pressure in the liquid and supercritical region.

Ströker et al.¹⁵⁷ also performed MC simulations for liquid and supercritical krypton. They employed the accurate pair potential and an extended Axilrod–Teller–Muto potential of Jäger et al.¹⁴⁹ to account for nonadditive three-body interactions. Quantum effects were again accounted for semiclassically. Since the potential models for krypton are not as accurate as those for argon, the deviations of the results for the density and speed of sound from experimental data were larger than for argon, about 0.2% and 0.36%, respectively.

In this section, we describe the impact of the ab initio calculations of the zero-density limit of helium’s thermal conductivity λ_He and viscosity η_He. First, we mention the impact of λ_He and λ_Ar on temperature metrology. Then, we describe how accurate values of η_He have been used as standards to reduce the uncertainty of viscosity measurements of many gases by a factor of 10. We conclude by briefly considering the impact of accurate viscosity data on metering process gases, for example, during the manufacture of semiconductor chips.

As discussed in Sec. 2.2.1, AGT requires accurate values of λ of the working gas at low densities to account for the effect of the thermo-acoustic boundary layer on the measured resonance frequencies. For example, in 2010, Gavioso et al. used helium at $∼410$ kPa in a single-state, AGT determination of the Boltzmann constant k_B prior to its definition in 2019.¹⁵⁸ They reported that a relative standard uncertainty u_r(λ_He) = 0.015 generated a relative standard uncertainty of the Boltzmann constant u_r(k_B) = (1–3) × 10⁻⁶.

Today, an uncertainty of (1–3) × 10⁻⁶ would be the largest contributor to a state-of-the-art determination of the thermodynamic temperature T near 273 K. At low temperatures, the uncertainty of measured values of λ_He is much larger. Below 20 K, the λ_He data span a range on the order of ±6%.¹⁵⁹ This large an uncertainty would lead to u_r(T) > 10⁻⁵ for acoustic determinations of T. Fortunately, the values of λ_He calculated ab initio have extraordinarily small uncertainties, e.g., u_r(λ_He) = 9.6 × 10⁻⁶ at 273 K and u_r(λ_He) = 7.3 × 10⁻⁵ at 10 K.¹⁰ In essence, the calculated values of λ_He removed u_r(λ_He) from the uncertainty budgets of acoustic thermometers based on helium-filled quasi-spherical cavities.

Cylindrical, argon-filled cavities are being developed for high-temperature acoustic thermometry.^4,55 These projects require low-uncertainty values of both λ_Ar and η_Ar. Low-uncertainty values of η_Ar were generated from accurate measurements of the ratios η_Ar/η_He in the range 200–653 K and the ab initio values of η_He. Then λ_Ar(T) was obtained by combining the ratio-deduced values of η_Ar(T) with values of the Prandtl number Pr_Ar calculated from model pair potentials. (Pr = C_pη/λ, where C_p is the constant-pressure heat capacity per mass. For the noble gases, Pr is only weakly sensitive to the potential.)^160–162 The measured ratios η_Ar/η_He were consistent, within a few tenths of a percent, with highly accurate measurements made with an oscillating-disk viscometer¹⁶³ and with calculations of η_Ar based on ab initio Ar–Ar potentials.¹⁶⁴ Thus, the needs of argon-based acoustic thermometry are now met at all useable temperatures. To put this achievement in context, we note that measuring the thermal conductivity of dilute gases is difficult, even for noble gases near ambient temperature and pressure. Evidence for this appears in Lemmon and Jacobsen’s correlation of the “best” measurements of λ_Ar and η_Ar near ambient temperature (270–370 K) and pressure.¹⁶⁵ The average absolute deviations of selected measurements from their correlation ranged from 0.24% to 1.0%. Lemmon and Jacobsen estimated the uncertainty of the correlated values of λ_Ar was 2% and the uncertainty of η_Ar was 0.5%. (With the benefit of ab initio calculations and ratio measurements, we now know their correlation overestimated λ_Ar by 0.54% at 270 K and by 0.45% at 370 K.)

In 2012, Berg and Moldover reviewed measurements of the viscosity of 11 dilute gases near 25 °C.¹⁶⁶ These measurements were made using 18 different instruments that used five different operating principles and produced 235 independent viscosity ratios during the years 1959–2012. Using the ab initio value of η_He at 25 °C as a reference, the viscosities of the ten other gases (Ne, Ar, Kr, Xe, H₂, N₂, CH₄, C₂H₆, C₃H₈, SF₆) were determined with low uncertainties u_r(η) ranging from 0.000 27 to 0.000 36. These ratio-derived uncertainties are less than 1/10 the uncertainties claimed for absolute viscosity measurements, such as the measurements of η_Ar correlated by Lemmon and Jacobsen.¹⁶⁵ Now, any one of these gases can be used to calibrate a viscometer within these uncertainties. Such ratio-based calibrations have reduced uncertainties of η for many other gases¹⁶⁷ and they have been extended to a wide range of temperatures.¹⁶⁸ During their study of viscosity ratios, Berg and Moldover observed that the viscosity ratios determined using one instrument (a magnetically suspended, rotating cylinder) were anomalous. Their observation led to an improved theory of the instrument, thereby illustrating the power of combining a reliable standard η_He(T) with precise ratio measurements.¹⁶⁹ 

The initial density expansion of the viscosity has the form η/η₀ = 1 + η₁ρ, where the low-density limit of the viscosity η₀ depends entirely on pair interactions and the virial-like coefficient η₁ depends on the interactions among two and three molecules. Unfortunately, unlike the density and dielectric virial coefficients and η₀, no rigorous theory exists for η₁(T). An approximate theory was developed by Rainwater and Friend,^171,172 who presented quantitative results based on the Lennard-Jones potential. It was later extended with more accurate pair potentials for noble gases.¹⁷³ While the results from the Rainwater–Friend model are in reasonable agreement with the limited experimental data available for the initial density dependence of the viscosity for noble gases,¹⁷³ the error introduced by its approximations is unknown. We note that it is a classical theory, which introduces another source of error for light gases (such as helium) where quantum effects might be important, even at ambient temperatures.

In principle, solutions of the equations of relativistic quantum mechanics, possibly including quantum electrodynamics (QED) corrections, can predict all properties of matter to a precision sufficient for thermal metrology applications. In practice, if the goal is to match or exceed the accuracy of experiments, the range of systems reduces to few-particle ones. The first quantum mechanical calculations challenging experimental measurements for molecules appeared only in the 1960s (e.g., Ref. 174), while the first calculations relevant to metrology were published in the mid-1990s.^7,175,176 Currently, the branches of metrology discussed in this review are becoming increasingly dependent on theoretical input, as discussed in Sec. 2.

Theory improvements leading to results with decreased uncertainties proceed along three main, essentially orthogonal directions: level of physics, truncation of many-electron expansions, and basis set size. There exists an extended hierarchy of approaches in each direction. For the first direction, there exists a set of progressively more accurate physical theories that can be used in calculations relevant for metrology, from Schrödinger’s quantum mechanics for electrons’ motion in the field of nuclei fixed in space to relativistic quantum mechanics and to QED. The second direction is relevant for any many-electron system: one has to choose a truncation of the expansion of the many-electron wave function in terms of virtual-excitation operators at the double, triple, quadruple, etc. level or, equivalently in methods that use explicitly correlated bases (depending explicitly on interelectronic distances), to take into account only correlations of two, three, four, etc. electrons simultaneously. Third, for any given theory and many-electron expansion level, there are several methods of solving quantum equations specific for this level; in particular different types of basis sets are used to expand wave functions, resulting in different magnitudes of uncertainties from such calculations.

The lowest theory level is Schrödinger’s quantum mechanics for electrons moving in the field of nuclei fixed in space, i.e., quantum mechanics in the Born–Oppenheimer (BO) approximation. At the next level, one usually first accounts for the relativistic effects. Post-BO treatment of the Schrödinger equation can be limited to computations of the so-called diagonal adiabatic correction, which is the simplest method of accounting for couplings of electronic and nuclear motions, or it can fully include nonadiabatic effects, i.e., account for the complete couplings of these two types of motion. The highest level of theory applied in calculations relevant to metrology is QED, and it can be implemented at several approximations labeled by powers of the fine-structure constant α.

The third direction determining the accuracy of electronic structure calculations involves the size of the basis sets used to expand wave functions. In the CC and CI approaches, the standard technique is to use products of orbital (one-electron) basis sets. Many such basis sets are available; the ones most often used in metrology-related calculations are the correlation consistent (cc) basis sets introduced by Dunning.¹⁸³ These basis sets are denoted by cc-pVXZ: cc polarized Valence (i.e., optimized using a frozen-core approximation), X-Zeta, where X = D, T, Q, 5, … is the so-called cardinal number determining the maximum angular momentum of orbitals. Such basis sets can be augmented by an additional set of diffused functions and are then denoted as aug-cc-pVXZ, or two such sets: daug-cc-pVXZ. Another option is to use explicitly correlated basis sets in the CC method or to expand the whole many-electron wave function in such a basis set. Explicitly correlated basis sets provide a much faster convergence than products of orbital basis sets, but in most cases require optimizations of a large number of nonlinear parameters.

In order to achieve some target size of uncertainties, one has to choose a proper level in each of the three hierarchies defined earlier. For example, it is possible to perform an FCI calculation for a ten-electron system such as Ne. However, since FCI calculations scale factorially with the number of orbitals, only very small basis sets can be used, resulting in a large uncertainty of the results. Consequently, a better strategy is to use the CCSD(T) method which allows applications of the largest orbital bases available for a system like Ne₂. The computed interaction energy will be accurate to about four significant digits relative to the CCSD(T) limit, but will have a fairly large error, of the order of 1%–2%, with respect to the exact interaction energy at the non-relativistic BO level. In contrast, FCI calculations for Ne₂ employing the smallest sensible basis set of augmented double-zeta size, which would be extremely difficult to perform, would have an error of the order of 40% (such calculations might still be useful in hybrid approaches discussed below).

The orbital basis sets consist of families of bases of varying size. One usually carries out calculations in two or more such basis sets and then performs approximate extrapolations to the complete basis set (CBS) limit. In addition to the standard extrapolations, which assume the X⁻³ decay of errors, extrapolations using very accurate ECG results can be performed.^184–186 CCSD(T)/CBS results may have sufficiently small uncertainties to make calculations of relativistic and diagonal adiabatic corrections necessary, i.e., these corrections may be of the same order of magnitude as the uncertainties of the CCSD(T)/CBS results. To reduce the errors resulting from the truncation of the many-electron expansion, one can follow CCSD(T)/CBS calculations by CCSDT(Q) or FCI ones in smaller basis sets. These effects are then included in an incremental way, i.e., by adding the difference between FCI and CCSD(T) energies computed in the same (small) basis set.

Accurate solutions of quantum equations are followed by estimates of uncertainties, absolutely necessary for metrology purposes. The latter step is often more time-consuming than the former. One should emphasize that theoretical estimates of uncertainties are different from statistical estimates of uncertainties of measurements and in particular one cannot assign a rigorous confidence level to them, although for purposes of metrology one usually assumes that theoretical uncertainties are equivalent to k = 2 expanded uncertainties (95% confidence level).

A theoretical estimate of uncertainty consists of several elements. The most rigorous and reliable estimates are those of basis set truncation errors derived from the observed patterns of convergence in basis set. Much more difficult are estimates of uncertainties resulting from truncations of many-electron expansions. Such estimates can sometimes be made by performing higher-level calculations at a single point on a PES, but one most often uses analogy to similar systems for which higher-level calculations have been performed. The same approach can be used to estimate the neglected physical effects, for example, to estimate the uncertainty due to relativistic effects.

Solutions of the electronic Schrödinger equation for a given nuclear configuration of a dimer or a larger cluster, providing accurate quantum mechanical descriptions of such systems, are only the first step in theoretical work of relevance to metrology, as most measured quantities discussed in this review are either bulk properties or response properties of atoms and molecules. In the former case, i.e., to predict properties of gases or liquids relevant for metrology, one needs to know energies of such systems for a large number configurations, i.e., for different geometries of clusters. This issue is approached by using the many-body expansion, where here the bodies are atoms or molecules forming the cluster, starting from two-body (pair) interactions, followed by three-body (pairwise nonadditive) interactions. The approach can be continued to higher-level many-body interactions, but so far this has not been done. The ab initio energies are usually fitted to analytic forms only for the two- and three-body interactions.

In addition to energies, metrology applications often require knowledge of accurate values of various properties of atoms and molecules, mainly the static and dynamic polarizabilities and magnetic susceptibilities. These quantities can be computed as analytic energy derivatives with respect to appropriate perturbations. Properties of a single atom or molecule change in condensed phases and the so-called interaction-induced corrections to properties of isolated atoms or molecules are of interest to metrology.

As already mentioned above, although Schrödinger’s quantum mechanics at the BO level provides the bulk of the physical values of interest to metrology, computations of various effects beyond this level are often needed to reduce uncertainties of these properties to the magnitude needed for metrology standards. We will refer to these as post-BO effects. It should be stressed that we really have in mind here the post-nonrelativistic-BO level, since both the relativistic and QED corrections for molecules are usually computed using the BO approximation. One goes beyond this approximation when computing adiabatic and nonadiabatic corrections. Any reasonably detailed description of methodologies used in post-BO calculations would be too voluminous for the present review. Therefore, we refer the reader to the original papers, in particular Refs. 10, 11, 84, 137, 177, and 187–192.

Systems of interest to thermodynamics-based precision metrology are mainly noble-gas atoms and their clusters, and this section will be restricted to such systems, with the majority of text devoted to helium. Apart from being the substance whose behavior is closest to the ideal gas, it is also the only system where theory can currently provide values of physical quantities that are generally more accurate than the measured ones. Nevertheless, neon and argon are also of significant interest since they may be used in secondary standards to improve instrument sensitivity or ease of use. Although for most properties computations for neon and argon have larger uncertainties than the best measurements, such results are still useful as independent checks of experimental work and to guide extrapolation beyond the measured range.

The current theoretical results for helium owe their very small uncertainties mostly to the use of explicitly correlated basis sets. The calculations involving helium atoms are probably one of the best examples where an important science problem was solved using these bases. To clearly show where this field would be without the use of such basis sets, we discuss in this subsection numerical comparisons of ECG and orbital calculations for He₂, performed recently in Ref. 193. The majority of molecular electronic structure calculations are carried out using orbital basis sets. This means that many-electron wave functions are expanded in products of orbitals. The simplest example is the CI method discussed earlier, where the wave function is a linear combination of Slater determinants built of orbitals that are usually obtained by solutions of HF equations. However, expansions in orbital products converge slowly due to the difficulty of reproducing the electron cusps in wave functions.

A way around this difficulty is to use bases that depend explicitly on r₁₂ = |r₂ − r₁|, the distance between electrons. Bases of this type are called explicitly correlated. For few-electron systems, such bases are mostly used to directly expand the N-electron wave functions of the nonrelativistic BO approximation. The explicitly correlated bases are also often used for many-electron systems within a perturbative or CC approach.^194,195 For two-electron systems, one uses Hylleraas bases¹⁹⁶ with polynomial-only dependence on r₁₂ or, recently more frequently, expansions in purely exponential functions of r₁₂, called Slater geminals.^84,192 Bases combining both types of dependence on r₁₂ are also sometimes employed.¹⁹⁷ For more than two electrons, integrals needed for such bases become very expensive and bases involving Gaussian correlation factors $e−γrij2$⁠, i.e., ECG bases, are mostly used. For a review of the ECG approach, see Refs. 194 and 198.

Since expansions in explicitly correlated bases of the type described above approach solutions of the Schrödinger equation, the equivalent orbital calculations should be performed at the FCI level. As already mentioned, FCI calculations scale as N! with the number of electrons and therefore are the most expensive of all orbital calculations. Even for He₂, FCI calculations cannot be performed using the largest available orbital basis sets. Therefore, the optimal orbital-based strategy is a hybrid one consisting of performing calculations in the largest basis sets at a lower level of theory, for example, at the CCSD(T) level, and adding to these results FCI corrections computed in smaller basis sets.

The BO energies computed in ECG basis sets in Ref. 177 established a new accuracy benchmark for the helium dimer; see the description of these calculations in Sec. 3.3.1. These ECG interaction energies were compared in Ref. 193 to those computed in orbital bases at the hybrid CCSD(T) plus FCI level. The largest available basis sets were applied. For most points, the CCSD(T)+ΔFCI approach gives errors nearly two orders of magnitude larger than the ECG estimated uncertainties. For a couple of points, the CCSD(T)+ΔFCI results are fairly close to the ECG results, but this is mainly due to the former method overestimating the magnitude of the interaction energy at small R and underestimating at large R. Since these points are near the van der Waals minimum, some previous evaluations of the performance of orbital methods restricted to this region might have been too optimistic. When the whole range of R is considered, CCSD(T)+ΔFCI is no match for the ECG approach. One should also realize that any improvements of accuracy of the CCSD(T)+ΔFCI approach would require a huge effort; in particular, one would have to develop quadruple-precision versions of all needed orbital electronic structure codes.

One of the properties of helium required by precision measurement standards^199–201 is the helium atom polarizability, both static and dynamic (frequency-dependent). Nonrelativistic calculations of the static polarizability date back to the 1930s and reached an accuracy of 0.1 ppb in 1996 calculations using Hylleraas basis sets.²⁰² However, the relativistic correction, which is proportional to α², could be expected to contribute at the 60 ppm level relative to the total polarizability. Unfortunately, the values of these corrections published before 2001 differed significantly from one another. These discrepancies were resolved by accurate calculations of Refs. 187 (using GTGs) and 203 (using Slater geminals) with uncertainties of 20 ppb relative to the total polarizability. This work used the Breit–Pauli operator,²⁰⁴ whose expectation values were computed with the ground-state wave function for the nonrelativistic Hamiltonian.

The authors of Ref. 203 also computed the QED correction of order α³, which turned out to be significant, amounting to about 20 ppm relative to the total polarizability. However, a part of the α³ QED correction to the polarizability, resulting from the so-called Bethe logarithm, was only roughly estimated due to the very difficult to compute second electric-field derivative of a second-order-type perturbation theory expression involving the logarithm of the Hamiltonian. The first complete calculation of Bethe-logarithm contribution to helium polarizability was reported in Ref. 188. As such a calculation had never been done before, the algorithms and their numerical implementations had to be developed from scratch. The term containing the electric-field derivative of Bethe’s logarithm turned out to be unexpectedly small, representing only about 0.6% of the total α³ QED correction. Thus, this correction still makes a contribution of about 20 ppm to the total polarizability.

Further improvement of the accuracy of helium’s static polarizability was achieved in Ref. 192, which concentrated on the second derivative of the Bethe logarithm with respect to the electric field. This quantity can be obtained in a couple of ways, with completely different algorithms. The goal was to achieve agreement between two such approaches and also with Ref. 188. This goal was met, providing a reliable cross-validation for both approaches. The results of Ref. 192, providing currently the most accurate theoretical determination of the polarizability of helium, are shown in Table 1. The value of the α³ QED contribution computed in Ref. 188 differs from the current one by only 0.03% or 7 ppb relative to the total polarizability. This error is much smaller than the current uncertainty of the α⁴ QED contribution, which is estimated to amount to 0.1 ppm; see Table 1.

Calculations of Refs. 187 and 188 were extended to frequency-dependent polarizabilities.^84,191 This polarizability was expanded in inverse powers of the wavelength λ up to λ⁻⁸. Different levels of theory were used for each power of λ: up to α⁴ for the static term, α² for inverse powers 2 through 6 (only even powers contribute), and nonrelativistic for 8. The dynamic polarizability at the He–Ne laser wavelength of 632.9908 nm had an uncertainty of 0.1 ppm. This uncertainty results entirely from the uncertainty of the static polarizability. The latter was reduced compared to Ref. 188 mainly because the work of Ref. 205 has shown that the error of the so-called one-loop approximation used to evaluate the α⁴ terms is smaller than previously expected, amounting to only about 5% when applied to the excitation energies of helium. Another small change in the static polarizability was due to a slightly improved value of the Bethe-logarithm contribution; see also Ref. 192.

The polarizabilities computed in Refs. 84, 187, 188, and 191 had uncertainties orders of magnitude smaller than the best experimental results. However, recently a new, very accurate measurement of this quantity was published.⁶³ The measured value of the molar polarizability, 0.517 254 4(10) cm³/mol, is consistent with the theoretical molar polarizability computed from the atomic one listed in Table 1 and equal to 0.517 254 08(5) cm³/mol. The combined uncertainty is more than three times the difference while the experimental uncertainty is 20 times larger than the theoretical one.

When a helium atom is in a gas or condensed phase, one can expect that its polarizability changes due to interactions with other atoms. More precisely, the polarizability of a helium cluster is not equal to the sum of polarizabilities of helium atoms. This change is often referred to as collision-induced polarizability and for atoms is a function of interatomic distance. For a pair of helium atoms, reliable values of this quantity were computed in Ref. 206, reconciling previously published inconsistent calculations. The results of Ref. 206 were used to compute the second⁷⁹ and third²⁰⁷ dielectric virial coefficients of helium. Very recently, the collision-induced three-body polarizability of helium was computed.²⁰⁸ 

Due to inversion symmetry, a system consisting of one or two helium atoms cannot have a dipole moment in the BO approximation. However, configurations of three or more atoms may have a non-zero dipole moment, which in turn influences the value of the third dielectric virial coefficient.²⁰⁷ Presently, the only ab initio description of the three-body dipole moment of noble gases is the one developed by Li and Hunt.²⁰⁹ However, the results of Ref. 209 apply only at large separations, and do not have associated uncertainties. A dipole-moment surface for the helium trimer with rigorously defined uncertainty is currently being developed.²¹⁰ 

The interest in the helium dimer potential is nearly as old as quantum mechanics. In 1928, Slater²¹¹ developed the first potential for this system, which gave the interaction energy of −8.8 K at the internuclear distance R = 5.6 bohrs (1 bohr $≈52.91772109$ pm). There is a wide range of helium dimer potentials available in the literature; see Ref. 212 for a comparison of bound-state calculations using a large number of potentials. Figure 6 illustrates the remarkable progress in accuracy of predictions achieved since 1979. Empirical potentials dominated the field until the end of the 1980s; the two most widely used ones, HFDHE2²¹³ and HFD-B,²¹⁴ were developed by Aziz et al. The first really successful ab initio one was the LM-2 potential (published only in a tabular form) developed by Liu and McLean.²¹⁵ Those authors performed CI calculations and, by analyzing the configuration space and basis set convergence, obtained extrapolated interaction energies with estimated uncertainties. Although these estimates were rather crude and do not embrace the current best values for most values of R, cf. Fig. 6, they are reasonable.

Aziz and Slaman²¹⁶ used the HFD-B functional form with refitted parameters to “mimic” the behavior of the LM-2 potentials, of the unpublished ab initio data computed by Vos et al.,²¹⁷ and of the small-R Green's-function MC (GFMC) data²¹⁸ to obtain potentials denoted as LM2M1 and LM2M2, differing by assuming, respectively, the smallest and the largest well depth of the LM-2 potential as determined by the estimates of uncertainty. The parameters of these potentials were not fitted directly to ab initio data, but chosen by trial and error to reproduce both theoretical data and measured quantities to within their error bars. The LM2M2 potential was considered to be the best helium potential until the mid-1990s, when purely ab initio calculations took the lead. Among the latter ones, the TTY potential developed by Tang et al.²¹⁹ has a remarkably simple analytical form based on perturbation theory. The HFD-B3-FCI1 potential was obtained by Aziz et al.,⁷ who used the HFD-B functional form with its original parameters adjusted so that the new potential runs nearly through the ab initio data points. These points were GFMC results of Ref. 218 and the FCI results of van Mourik and van Lenthe.²²⁰ No uncertainties were assigned to HFD-B3-FCI1, and Fig. 6 shows that it was about as accurate as LM2M2.

The SAPT96 potential^175,176 opens an era of helium potentials based mostly on calculations with explicitly correlated functions. It was the first fully first-principles He₂ potential with a systematic estimation of uncertainties. The potential was obtained using a two-level incremental strategy. The leading SAPT corrections (the complete first-order and the bulk of the second-order interaction energies) were computed using GTG basis sets. The GTG-based variant of SAPT was developed in Refs. 221–225. Higher-order SAPT corrections were computed using the general SAPT program based on orbital expansions.^226–230 Large orbital basis sets including up to g-symmetry functions and midbond functions (placed between the nuclei)²³¹ were used. The remaining many-electron effects were computed using both SAPT based on FCI-level monomers, with summations to a very high order of perturbation theory (using He₂-specific codes), and supermolecular FCI calculations in small orbital basis sets. It is interesting to note that the actual errors of the SAPT96 potential relative to the current best results turned out to be completely dominated by the residual orbital (rather than GTG) contributions. For instance, at R = 5.6 bohrs, the orbital part constitutes only −1.81 K out of −11.00 K, but its error was −0.05 K out of the total SAPT96 error of −0.06 K. The factor of 2 underestimation of the uncertainties seen in Fig. 6 for R = 5.6 bohrs was entirely due to this issue. SAPT96 is about as accurate as LM2, except for large R where it is more accurate, with SAPT96 overestimating and LM2 underestimating the magnitude of interaction energy. With an added retardation correction, SAPT96 was used (under the name SAPT2) by Janzen and Aziz²³² to calculate properties of helium and found to be the most accurate helium potential at that time.

In 1999, van Mourik and Dunning²³³ calculated CCSD(T) energies in basis sets up to daug-cc-pV6Z, CCSDT − CCSD(T) differences in the daug-cc-pVQZ basis set, and FCI − CCSDT differences in the daug-cc-pVTZ basis set. The CCSD(T) energies were CBS-extrapolated and then refined by adding a correction equal to the R-interpolated differences between highly accurate CCSD(T)-R12 results (available at a few distances in Ref. 234) and the obtained CBS limits. The CC-R12 methods are analogous to CC-GTG methods, but the explicit correlation factor enters linearly.¹⁹⁵ As seen in Fig. 6, the results of Ref. 233 were more accurate than any previously published ones, but no estimates of uncertainties were provided and the computed interaction energies were not fitted.

Supermolecular ECG-based calculations for He₂ started to appear in the late 1990s,^235,236 and were initially aimed at providing upper bounds to the interaction energies (by subtracting essentially exact monomer energies), as the authors did not attempt to extrapolate their results to the CBS limits. Another application of explicitly correlated functions to the He–He interaction was a series of papers by Gdanitz,^237–239 who used the multireference averaged coupled-pair functional method with linear r₁₂ factors, r₁₂-MR-ACPF. The extrapolated results from the last paper of the series, Ref. 239 (denoted “Gdanitz01” in Fig. 6), were among the most accurate results available at that time. However, the reported uncertainties were strongly underestimated at shorter distances (as much as 5 times at 5.6 bohrs and 17 times at 4.0 bohrs).

Another important series of papers was published by Anderson et al.,^240–242 who reported quantum MC energies with progressively reduced statistical uncertainties. Although these results were obtained only for a few internuclear distances, they represented very valuable benchmarks for mainstream electronic structure methods. In fact, until the publication of the CCSAPT07 potential,¹⁸⁵ the result from Ref. 242, −10.998(5) K (see “Anderson04” in Fig. 6), was the most accurate value available at 5.6 bohrs.

In Refs. 243 and 244, a hybrid supermolecular ECG/orbital method was applied to the helium dimer. The bulk of the correlation effect on the interaction energy, at the CCSD level, was evaluated using GTG functions and the method developed in Refs. 245–256. The nonlinear parameters were optimized at the MP2 level. The effects of noniterative triple excitations [the “(T)” contribution], i.e., the differences between CCSD(T) and CCSD energies, were calculated using large orbital basis sets (up to aug-cc-pV6Z with bond functions and daug-cc-pV6Z) and extrapolated to the CBS limits. Finally, the FCI corrections [differences between FCI and CCSD(T) energies] were obtained in basis sets up to aug-cc-pV5Z with bond functions and daug-cc-pV5Z, and also extrapolated. Results for three distances were reported in Ref. 244 (see “Cencek04” in Fig. 6).

Hurly and Mehl (HM) analyzed the best existing ab initio data for the helium dimer and created a new potential⁹ representing a compromise based on uncertainties of existing data and their mutual agreement (for instance, as can be seen in Fig. 6, the result from Ref. 244 was used at R = 7.0 bohrs). The diagonal adiabatic corrections from Ref. 257 were added to the final potential, which was then used to calculate the second virial coefficient, viscosity, and thermal conductivity of helium. HM recommended that the values of these thermophysical properties should serve as standards for measurements.

The CCSAPT07 potential¹⁸⁵ based on the hybrid GTG/orbital method, published in 2007, was a significant improvement over the previous complete potential of this type, i.e., the SAPT96 potential.^175,176 CCSAPT07 combined three different computational techniques, according to the criterion of the lowest uncertainty available for a given internuclear distance. Variational four-electron ECG calculations were used for R ≤ 3.0 bohrs and SAPT+FCI was employed for R > 6.5 bohrs. At intermediate distances, the hybrid supermolecular method developed in Refs. 243 and 244 and described above provided the highest accuracy. Compared to Refs. 243 and 244, several computational improvements were introduced,¹⁸⁴ resulting in significantly reduced uncertainties. The SAPT calculations¹⁸⁵ of CCSAPT07 followed the SAPT96 recipe, but also with larger basis sets and some computational improvements. The uncertainties of this potential were smaller than some effects that are neglected at the nonrelativistic BO level. Calculations of these effects will be discussed in Sec. 3.3.2.

Another highly accurate potential, by Hellmann, Bich, and Vogel (HBV),²⁵⁸ appeared at almost the same time as CCSAPT07. Those authors used very large basis sets (up to daug-cc-pV8Z with added bond functions at the CCSD level, and gradually smaller bases for higher levels of theory up to FCI) followed by CBS extrapolations. After augmenting the HBV potential with adiabatic, approximate relativistic, and retardation corrections, the authors used it to calculate thermophysical properties of helium.²⁵⁹ However, the uncertainties of the HBV potential were not estimated, which restricts its usefulness. A direct accuracy comparison between the pure BO component of HBV and CCSAPT07 is now possible because of the much higher accuracy of the present-day benchmark energies,¹⁷⁷ and we performed such analysis using the values reported in the last column of Table 3 in Ref. 258. Out of 11 distances for which all three energies are available, the largest relative error (with respect to the results of Ref. 177), equal to 0.90%, occurs for the CCSAPT07 energy at 5.0 bohrs, while the error of the HBV energy at this distance is 0.48%. If one excludes this distance, which is close to where the helium potential crosses zero, and calculates the average relative error at the remaining distances, one obtains 0.007% for CCSAPT07 and 0.011% for HBV. Therefore, both potentials exhibit a similar accuracy and represent a significant improvement over all previously published helium dimer potentials.

The current most accurate nonrelativistic BO potential for the helium dimer (labeled as “Przybytek17” in Fig. 6) was published in Ref. 177; see Ref. 193 for details of these calculations. The significant improvement over all previous potentials was achieved by a combination of three factors. First, a pure ECG approach was used, i.e., with all four electrons explicitly correlated and no contributions calculated with orbital methods. Indeed, the residual errors of the older hybrid ECG/orbital potentials, SAPT96^175,176 and CCSAPT07,¹⁸⁵ were dominated by insufficient basis set saturation of the relatively small orbital contributions. Second, the use of the monomer contraction method,^189,260 i.e., the use of the product of helium atoms wave functions as one of the functions in the basis set, dramatically improved the energy convergence with respect to the ECG expansion size. Furthermore, a replacement of the simple product of monomer wave functions by a more compact sum of four-electron functions optimized for two noninteracting helium atoms^206,261 reduced the computational cost at the nonlinear optimization stage. Third, a near-complete optimization of nonlinear parameters in large basis set expansions was possible due to this reduced cost and due to other improvements of the optimization algorithm.

All post-BO corrections were computed using both four-electron ECG basis sets and orbital basis sets (except for the so-called Araki–Sucher part of the QED operator where only ECG functions were used). The calculations with smaller uncertainties were selected for the final potential. Orbital calculations were performed using a combination of CCSD(T) and FCI approaches or FCI alone. For the adiabatic correction, only FCI was used. The calculations of the average values of the operators listed above with ECG and FCI wave functions are straightforward (although regularization techniques have to be used for singular operators). However, the CCSD(T) wave function needed to compute expectation values is not available (not defined) and instead the CCSD(T) linear response method was used.²⁶⁴ The retardation effects of long-range electromagnetic interactions were computed from the Casimir–Polder formula²⁶⁵ by subtracting the retardation part of the α² relativistic and α³ QED corrections.¹⁹⁰ 

The calculations of Ref. 177 significantly improved the accuracy of the helium dimer potential, with uncertainties reduced by an order of magnitude compared to those of Refs. 137 and 185. As already discussed, the main improvement was due to the use of larger and better optimized ECG wave functions at the nonrelativistic BO level of theory for all R ≤ 9 bohrs. Accuracy of the adiabatic and relativistic corrections was also improved by using larger basis sets than in Refs. 10 and 137. A major theoretical advance was the calculation of the properties of the very weak bound state of He₂ (the so-called halo state) with full inclusion of nonadiabatic effects.

The accuracy of relativistic and QED contributions was further improved in Ref. 11. The contributions to the interaction energy at the van der Waals minimum are presented in Table 2. Clearly, with the uncertainty of the BO contribution of 0.000 20 K, all the included post-BO contributions are relevant, except for the retardation contribution, but this contribution does become important at very large separations.¹⁹⁰ One can also see that uncertainties of the adiabatic, relativistic, and QED terms are almost negligible compared to the uncertainty of the BO term. The potential of Ref. 11 was used to compute the second virial coefficient and the second acoustic virial coefficient of helium.

In any fluid, the total interaction energy includes terms beyond pairwise-additive interactions between monomers. These so-called nonadditive contributions begin with three-body nonadditive terms defined as the part of the trimer interaction energy that cannot be recovered by the sum of two-body interactions. The additive and nonadditive interactions form a series called the many-body expansion of interaction energy. Fortunately, for all fluids consisting of monomers interacting via noncovalent forces, this expansion converges very fast and usually it is sufficient to limit calculation to two- and three-body terms. For a review of the many-body expansion, see Ref. 266. For metrology, the three-body potential is needed to calculate the third virial coefficient.

A pairwise-nonadditive potential for helium was developed in Ref. 267 and improved in Ref. 138. In the earlier work, two independent potentials were obtained. One was based on three-body SAPT^268–272 and the other on the supermolecular CCSD(T) approach. Orbital basis sets up to aug-cc-pV5Z were used. The two potentials were in very good agreement. In Ref. 138, the CCSD(T) potential was improved by calculating the FCI correction in an incremental approach and increasing the number of grid points, with CCSD(T) values taken from Ref. 267 except for the new grid points. Near the minimum of the total potential, the three-body contribution is only −0.0885 K, which should be compared to the total interaction energy of about $−33$ K, but the three-body contribution is much larger than the uncertainty resulting from the two-body term, which is 0.0006 K. The uncertainty of the three-body term at the minimum of the total potential was estimated to be 0.002 K.

Recently, the three-body potential for helium was further improved²⁷³ by adding the relativistic and adiabatic corrections, as well as using a new set of correlation-consistent basis sets specifically developed for helium atoms.¹⁷⁷ An improved functional form was also used to analytically represent the potential at large distances. In particular, new terms were developed for the case when two atoms remain close while the third is progressively more distant. These refinements resulted in a reduction of the uncertainty by a factor of about 5 overall. In particular, the uncertainty at the minimum was reduced to 0.5 mK, a factor of 4 smaller than that of the previous work.¹³⁸ 

While theory is superior to experiment for the helium atom and helium clusters, this is not the case for neon, and even less so for argon. The simple reason is the number of electrons per atom: 2, 10, and 18, respectively. While for the helium atom and small helium clusters the N-electrons explicitly correlated bases can reach ppm or smaller uncertainties, and FCI calculations can be performed in fairly large bases, for neon neither type of calculation can be performed in bases large enough to get meaningful results. To quantify this statement, let us examine the most accurate calculations for the neon dimer,¹¹⁷ see Table 3. The calculations at the CCSD(T) level of theory were performed in the largest available basis sets: modified daug-cc-pV8Z with bond functions. The uncertainty of the interaction energy obtained in this way is about 200 ppm, which is only ten times larger than the 20 ppm uncertainty of the He₂ BO interaction energy. However, uncertainties coming from some excitations of higher rank are significantly larger: the pentuple excitation contribution, Δ(P), increases the uncertainty of the total value of interaction energy to about 1000 ppm. The increase of uncertainties is due to the use of smaller and smaller basis sets as the number of excitations increases: at the CCSDTQ(P) level of theory only the daug-cc-pVDZ basis set could be used. Furthermore, based on the results in Table 3, it is not possible to estimate the uncertainty resulting from neglecting excitations beyond (P). The lower part of Table 3 shows the convergence in the rank of excitation. One can see that while the contribution of the triple excitations is very substantial, a 29% increase in the magnitude of interaction energy relative to the CCSD level, the contribution of quadruple excitations is 57 times smaller than that of triple ones. However, the contribution of pentuple excitations breaks this fairly fast convergence: it is of similar magnitude to that of the quadruple excitations. Note that one cannot blame the noniterative character of the pentuple excitations, as for lower-rank excitations the iterated and noniterated values are fairly similar. One may ask if the value of the pentuple contribution computed in Ref. 117 could be a numerical artifact resulting from the use of a rather small basis set. This issue was investigated in Ref. 117, and the results computed in the aug-cc-pVDZ and aug-cc-pVTZ were 0.0227 and 0.1113 K, respectively. While these results may indicate that even the first digit in the pentuple excitations contribution may be uncertain, they also indicate that the order of magnitude will likely remain the same when going to larger basis sets. This would indicate that for Ne₂ the coupled-cluster expansion converges very slowly, whereas for other closed-shell systems investigated in the literature CCSDTQ(P) agrees with FCI very well, indicating that effects of higher excitations are negligible. Unfortunately, FCI calculations would be extremely difficult to perform for Ne₂ even in the aug-cc-pVDZ basis set.

Similar calculations at the limits of the available technology were reported for Ar₂ in Ref. 274 (see also earlier calculations²⁷⁵ with accurate treatment at very small values of interatomic distances R). The value of the interaction energy obtained at the van der Waals minimum is −142.86 K and its uncertainty was estimated at 0.46 K. This uncertainty, representing 3000 ppm (0.3%) of the computed well depth, does not include an estimate of effects beyond CCSDTQ. The results of Ref. 117 for Ne₂ indicate, however, that the post-CCSDTQ contribution may be not negligible.

The first first-principles three-body potential for argon was developed in Ref. 269 using three-body SAPT. It was then used to compute the third virial coefficient of argon¹⁴⁴ and to simulate vapor–liquid equilibria.¹⁵³ An improved three-body potential for argon was developed in Ref. 276 using the CCSDT(Q) level of theory and including core correlation and relativistic effects. Uncertainties of the potential were estimated. The authors of Ref. 276 also computed the third virial coefficient, obtaining good overall agreement with experimental data. In particular, in some regions of temperature, theoretical values exhibited smaller uncertainties than experiment and comparisons with theory allowed evaluation of different experiments. When the experimental data were refitted by a new model that included an approximate fourth virial coefficient,¹²³ the agreement with theory improved, which can be considered to be a validation of the new model. The work of Ref. 276 shows that despite limitations of accuracy, for some properties of argon theory may provide information relevant for metrology and its accuracy may be competitive with experimental accuracy.

Magnetic susceptibilities of noble gas atoms are relevant for RIGT; see Eq. (11). In general, the magnetic susceptibility is several orders of magnitude smaller than its electric counterpart (hence, A_μ is several orders of magnitude smaller than A_ɛ). This means that only modest accuracy for the magnetic susceptibility, on the order of 0.1% or even 1%, is sufficient for it to make a negligible contribution to the uncertainty budget of current or planned refractivity-based thermodynamic metrology. Calculations at the BO level are therefore probably sufficient, but it is still desirable to compute additional effects, at least at lowest order, to verify that they are relatively small.

The first comprehensive calculation of the magnetic susceptibility of the helium atom was performed by Bruch and Weinhold.²⁷⁷ They added corrections for relativistic effects and nuclear motion to an existing high-accuracy calculation at the BO level. However, their calculation included only some of the relativistic corrections that enter at lowest order. Recently, Puchalski et al.⁸⁰ presented a definitive calculation of all effects through order α⁴, along with a more accurately computed value for the nonrelativistic BO limit using Slater geminals. They obtained agreement within mutual uncertainties with the calculations of Bruch and Weinhold for individual terms,²⁷⁷ but included some terms that had been omitted in the previous work. When converted from the atomic units used in the paper, the final result for ⁴He corresponds to A_μ = −7.921 28(13) × 10⁻⁶ cm³ mol⁻¹. The relative uncertainty of this result, primarily due to neglected QED effects that enter at the α⁵ level, was conservatively estimated at 16 ppm. This is far more than sufficient for any conceivable application of refractivity for temperature or pressure measurement, and the agreement with previous work encourages confidence in the result.

As with other properties, the greater number of electrons renders the calculation of magnetic susceptibility much more difficult for neon and especially for argon. The current state-of-the-art calculations for neon⁶⁴ and argon⁶⁶ were performed only at the nonrelativistic BO level, with a rough uncertainty estimate for neglected relativistic effects based on the magnitude of those effects for the electric polarizability. The estimated uncertainty of this calculated quantity was ∼0.2% for neon⁶⁴ and 1% for argon.⁶⁶ The limited experimental information for the magnetic susceptibility is discussed in Sec. 4.5.3.

Virial expansions are exact results from quantum statistical mechanics which enable a systematically improvable evaluation of various thermophysical properties as a power series in density starting from the ideal-gas reference system. The coefficients appearing in the N-th term of the series can be computed from the knowledge of the interaction of clusters of N particles.

Although the focus of this review is on calculations with no uncontrolled approximation, let us briefly discuss the classical limit of the approach we have outlined. Classical expressions can be computed relatively easily, and provide a useful high-temperature check for the more involved calculations described below.