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We develop a surface for the electric dipole moment of three interacting helium atoms and use it, together with state-of-the-art potential and polarizability surfaces, to compute the third dielectric virial coefficient, $C_\varepsilon$, for both $^4$He and $^3$He isotopes. Our results agree with previously published data computed using an approximated form for the three-body polarizability, and are extended to the low-temperature regime by including exchange effects. Additionally, the uncertainty of $C_\varepsilon$ is rigorously determined for the first time by propagating the uncertainties of the potential and polarizability surfaces; this uncertainty is much larger than the contribution from the dipole-moment surface to $C_\varepsilon$. Our results compare reasonably well with the limited experimental data. The first-principles values of $C_\epsilon$ computed in this work will enhance the accuracy of primary temperature and pressure metrology based on measurements of the dielectric constant of helium., Comment: 18 pages, 7 figures

We report theoretical calculations of the static diamagnetic susceptibility, $\chi_0$, of neon and argon atoms. The calculations were performed using a hierarchy coupled-cluster methods combined with application of both the Gaussian and Slater orbital basis sets. We included the complete relativistic correction of order of $\alpha^4$, where $\alpha$ is the fine structure constant, and obtained an estimate of the quantum electrodynamics (QED) contributions. The finite nuclear mass and size corrections were also considered but are found to be small. The final results are $\chi_0=-8.4786(7)\cdot10^{-5}$ $a_0^3$ for the neon and $\chi_0=-22.9545(32)\cdot10^{-5}$ $a_0^3$ for the argon atom, where $a_0$ is the Bohr radius. The uncertainties in the last digits, shown in the parentheses, are primarily due to the errors in the non-relativistic electronic wavefunction, as well as due to the neglected quantum electrodynamics corrections.

We report theoretical calculations of the diamagnetic susceptibility, $\chi_0$, of helium atom. We determined the complete relativistic correction to $\chi_0$ of the order of $\alpha^4$, where $\alpha$ is the fine structure constant, by including all $\alpha^4$ terms originating from the Dirac and Breit equations for a helium atom in a static magnetic field. Finite nuclear mass corrections to $\chi_0$ was also evaluated. To obtain very accurate results and reliable uncertainty estimates we used a sequence of explicitly correlated basis sets of fully optimized Slater geminals. We found that $\chi_0=-2.119\,106(34)\cdot10^{-5}$ $a_0^3$ and $\chi_0=-2.119\,400(34)\cdot10^{-5}$ $a_0^3$ for $^4$He and $^3$He isotopes, respectively, where $a_0$ is the Bohr radius and the uncertainties shown in the parentheses are due entirely to the very conservative estimate of the neglected QED corrections of the order of $\alpha^5$. Our results are compared with the available experimental data and with previous, incomplete theoretical determinations of the $\alpha^4$ contributions to the diamagnetic susceptibility of helium.

Recent advances regarding the interplay between ab initio calculations and metrology are reviewed, with particular emphasis on gas-based techniques used for temperature and pressure measurements. Since roughly 2010, several thermophysical quantities - in particular, virial and transport coefficients - can be computed from first principles without uncontrolled approximations and with rigorously propagated uncertainties. In the case of helium, computational results have accuracies that exceed the best experimental data by at least one order of magnitude and are suitable to be used in primary metrology. The availability of ab initio virial and transport coefficients contributed to the recent SI definition of temperature by facilitating measurements of the Boltzmann constant with unprecedented accuracy. Presently, they enable the development of primary standards of temperature in the range 2.5-552 K and pressure up to 7 MPa using acoustic gas thermometry, dielectric constant gas thermometry, and refractive index gas thermometry. These approaches will be reviewed, highlighting the effect of first-principles data on their accuracy. The recent advances in electronic structure calculations that enabled highly accurate solutions for the many-body interaction potentials and polarizabilities of atoms - particularly helium - will be described, together with the subsequent computational methods, most often based on quantum statistical mechanics and its path-integral formulation, that provide thermophysical properties and their uncertainties. Similar approaches for molecular systems, and their applications, are briefly discussed. Current limitations and expected future lines of research are assessed., Comment: 59 pages, 9 figures

The non-additive three-body interaction potential for helium was computed using the coupled-cluster theory and the full configuration interaction method. The obtained potential comprises an improved nonrelativistic Born--Oppenheimer energy and the leading relativistic and nuclear-motion corrections. The mean absolute uncertainty of our calculations due to the incompleteness of the orbital basis set was determined employing complete-basis-set extrapolation techniques and was found to be 1.2%. For three helium atoms forming an equilateral triangle with the side length of 5.6~bohr our three-body potential amounts to $-$90.6~mK, with an estimated uncertainty of 0.5~mK. An analytic function, developed to accurately fit the computed three-body interaction energies, was chosen to correctly describe the asymptotic behavior of the three-body potential for trimer configurations corresponding to both the three-atomic and the atom-diatom fragmentation channels. For large triangles with sides $r_{12}$, $r_{23}$, and $r_{31}$, the potential takes correctly into account all angular terms decaying as $r_{12}^{-l} r_{23}^{-m} r_{31}^{-n}$ with $l+m+n \le 14$ for the nonrelativistic Born--Oppenheimer energy and $l+m+n \le 9$ for the post-Born--Oppenheimer corrections. We also developed a short-range analytic function describing the local behavior of the total uncertainty of the computed three-body interaction energies. Using both fits we calculated the third pressure and acoustic virial coefficients for helium and their uncertainties for a wide range of temperatures. The results of these calculations were compared with available experimental data and with previous theoretical determinations. The estimated uncertainties of present calculations are 3-5 times smaller than those reported in the best previous work., Comment: 68 pages, 10 figures, 9 tables

In this work we report state-of-the-art theoretical calculations of the dipole polarizability of the argon atom. Frequency dependence of the polarizability is taken into account by means of the dispersion coefficients (Cauchy coefficients) which is sufficient for experimentally relevant wavelengths below the first resonant frequency. In the proposed theoretical framework, all known physical effects including the relativistic, quantum electrodynamics, finite nuclear mass, and finite nuclear size corrections are accounted for. We obtained $\alpha_0=11.0763(19)$ for the static polarizability and $\alpha_2=27.976(15)$ and $\alpha_4=95.02(11)$ for the second and fourth dispersion coefficients, respectively. The result obtained for the static polarizability agrees (within the estimated uncertainty) with the most recent experimental data [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)], but is less accurate. The dispersion coefficients determined in this work appear to be most accurate in the literature, improving by more than an order of magnitude upon previous estimates. By combining the experimentally determined value of the static polarizability with the dispersion coefficients from our calculations, the polarizability of argon can be calculated with accuracy of around $10\,$ppm for wavelengths above roughly $450\,$nm. This result is important from the point of view of quantum metrology, especially for a new pressure standard based on thermophysical properties of gaseous argon. Additionally, in this work we calculate the static magnetic susceptibility of argon which relates the refractive index of dilute argon gas with its pressure. While our results for this quantity are less accurate than in the case of the polarizability, they can provide, via Lorenz-Lorentz formula, the best available theoretical estimate of the refractive index of argon.

We present first-principles theoretical determination of the three-body polarizability and the third dielectric virial coefficient of helium. Coupled-cluster theory and the full configuration interaction procedure were used to perform required electronic structure calculations. The mean absolute relative uncertainty of the trace of the three-body polarizability tensor, resulting from the incompleteness of orbital basis set, was determined using extrapolation techniques. Additional uncertainty due to approximate treatment of triple and the neglect of higher excitations was estimated using full configuration interaction calculations. An analytic function was developed to describe the behavior of the polarizability and its asymptotic decay to three-atomic and atom-diatom fragmentation channels. We also developed an analytic function describing the local behavior of the total uncertainty of our calculations. Using both fits we calculated the third dielectric virial coefficient and its uncertainty using the classical and semiclassical Feynman-Hibbs approaches. The results of our calculations were compared with available experimental data and with recent Path-Integral Monte Carlo (PIMC) calculations employing the so-called superposition approximation of the three-body polarizability. For temperatures above 200 K we observed significant discrepancy between the classical results obtained using either the superposition approximation or the ab initio computed polarizability. The theoretical data reported in this work eliminate the main accuracy bottleneck of the development of optical pressure standard and are expected to facilitate further progress in the field of quantum thermal metrology, Comment: 30 pages, 7 figures

We present the first framework for fully quantum calculation of the third dielectric virial coefficient $C_\varepsilon(T)$ of noble gases, including exchange effects. The quantum effects are taken into account with the path-integral Monte Carlo method. Calculations employing state-of-the-art pair and three-body potentials and pair polarizabilities yield results generally consistent with the few scattered experimental data available for helium, neon, and argon, but rigorous calculations with well-described uncertainties will require the development of surfaces for the three-body nonadditive polarizability and the three-body dipole moment. The framework developed here will enable new approaches to primary temperature and pressure metrology based on first-principles calculations of gas properties., Comment: 13 pages, 4 figures. The following article has been submitted to The Journal of Chemical Physics. After it is published, it will be found at https://aip.scitation.org/journal/jcp

The second virial coefficient and the second acoustic virial coefficient for helium-4 and helium-3 are computed for a wide range of temperatures (0.5 - 1000K) using a highly accurate nonrelativistic interaction potential [M. Przybytek et al., Phys. Rev. Lett. 119, 123401 (2017)] and recalculated relativistic and quantum-electrodynamic components. The effects of the long-range retardation and of the nonadiabatic coupling of the nuclear and electronic motion are also taken into account. The results of our calculations represent at least fivefold improvement in accuracy compared to the previous ab initio work. The computed virial coefficients agree well with the most accurate recent measurements but have significantly smaller uncertainty., Comment: 13 pages, 2 figures, 7 suplemental files; includes changes introduced during the peer review and the proofreading process; published in Phys. Rev. A (copyrighted by American Physical Society): https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.042810

We report theoretical determination of the dipole polarizability of the neon atom, including its frequency dependence. Corrections for the relativistic, quantum electrodynamics, finite nuclear mass, and finite nuclear size effects are taken into account. We obtain the value $\alpha_0=2.66080(36)$ for the static polarizability, and $\alpha_2=2.850(7)$ and $\alpha_4=4.932(14)$ for the first two polarizability dispersion coefficients (Cauchy moments); all values are in atomic units (a.u.). In the case of static polarizability, our result agrees with the best experimental determination [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)], but our estimated uncertainty is significantly larger. For the dispersion coefficients, the results obtained in this work appear to be the most accurate to date overall compared to published theoretical and experimental data. We also calculated the static magnetic susceptibility of the neon atom, needed to obtain the refractive index of gaseous neon. Our result, $\chi_0 = -8.484(19) \cdot 10^{-5}$ a.u., is about 9% larger in absolute value than the recommended experimental value [CRC Handbook of Chemistry and Physics, CRC Press, 2019, p. 4-145].

The QED contribution to the dipole polarizability of the $^4$He atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of the Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of the Bethe logarithm, equal to $0.048\,557\,2(14)$ in atomic units, confirms the small magnitude of this quantity found in the only previous calculation [G. {\L}ach, B. Jeziorski, and K. Szalewicz, Phys. Rev. Lett. 92, 233001 (2004)], but differs from it by about 5\%. The origin of this difference is explained. The total QED correction of the order of {\alpha} 3 in the fine-structure constant {\alpha} amounts to 30.6671(1)$\cdot 10^{-6}$, including the 0.1822$\cdot 10^{-6}$ contribution from the electric-field derivative of the Bethe logarithm and the 0.01112(1)$\cdot 10^{-6}$ correction for the finite nuclear mass, with all values in atomic units. The resulting theoretical value of the molar polarizability of helium-4 is $0.517\,254\,08(5)\,$cm$^3$/mol with the error estimate dominated by the uncertainty of the QED corrections of order $\alpha^4$ and higher. Our value is in agreement with but an order of magnitude more accurate than the result $0.517\, 254\, 4(10)\,$cm$^3$/mol of the most recent experimental determination [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)].

In this communication we present a method of complete basis set (CBS) extrapolation of correlation energies obtained with a systematic sequence of one-electron basis sets. Instead of fitting the finite-basis results with a certain functional form, we perform analytic re-summation of the missing contributions coming from higher angular momenta, $l$. The assumption that they vanish asymptotically as an inverse power of $l$ leads to an expression for the CBS limit given in terms of the Riemann zeta function. This result is turned into an extrapolation method that is very easy to use and requires no "empirical" parameters to be optimized. The performance of the proposed method is assessed by comparing the results with accurate reference data obtained with explicitly correlated theories and with results obtained with standard extrapolation schemes. On average, the errors of the zeta-function extrapolation are several times smaller compared with the conventional schemes employing the same number of points. A recipe for estimation of the residual extrapolation error is also proposed.

Explicitly correlated calculations of interaction energies with wave functions that include all interparticle distances have suffered so far from the lack of size-consistency resulting from the difficulty to define monomer energies corresponding to the applied dimer basis. As a consequence it has not been possible to obtain interaction energies vanishing at infinite intermonomer distance $R$. This has dramatically reduced the accuracy of calculations at distances where the error in the dimer energy was comparable with the interaction energy itself. The same problem occurs in calculations of interaction-induced properties. In this communication we show how to circumvent this difficulty and obtain interaction energies or interaction-induced properties that vanish at large $R$. This is achieved by relaxing the Pauli principle in the diagonalization of the Hamiltonian of noninteracting monomers. The basis functions used for this diagonalization belong to the representation of the permutation group of the dimer induced by the product of representations appropriate for the monomer spin states. Nonlinear parameters of the basis set are optimized only for the dimer in the Pauli-allowed sector of the Hilbert space. In this way, one obtains $R$-dependent energy of noninteracting monomers and the corresponding interaction energy includes a counterpoise correction for the basis set superposition error. The efficiency of this procedure is demonstrated for the interaction of two hydrogen atoms where accurate reference data are known.

The pair potential for helium has been computed with accuracy improved by an order of magnitude relative to the best previous determination. For the well region, its uncertainties are now below 1 millikelvin. The main improvement is due to the use of explicitly correlated wave functions at the nonrelativistic Born-Oppenheimer (BO) level of theory. The diagonal BO and the relativistic corrections were obtained from large full configuration interaction calculations. The nonadiabatic perturbation theory was used to predict the properties of the halo state of helium dimer. Its binding energy and the average value of interatomic distance are found to be 138.9(5) neV and 47.13(8) {\AA}. The binding energy agrees with its first experimental determination of 151.9(13.3) neV [Zeller et al., PNAS 113, 14651 (2016)].

The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surface-integral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances $R$, all three formulas yield the correct expression $-(2/e)Re^{-R}$, but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error falling as $3^{-K}$, where $K$ is the order of the applied polarization expansion. The error of the surface-integral formula decreases exponentially as $a^K/(K+1)!$, where $a=\ln2 -\tfrac{1}{2}$. The variational formula performs best, its error decays as $K^{1/2} [a^{ K}/(K+1)!]^2$. These convergence rates are much faster than those resulting from approximating the wave function through the multipole expansion. This shows the efficiency of the partial resummation of the multipole series effected by the polarization expansion. Our results demonstrate also the benefits of incorporating the variational principle into the perturbation theory of molecular interactions., Comment: 5 pages, 1 figure

The conventional surface integral formula $J_{\rm surf}[\Phi]$ and an alternative volume integral formula $J_{\rm var}[\Phi]$ are used to compute the asymptotic exchange splitting of the interaction energy of the hydrogen atom and a proton employing the primitive function $\Phi$ in the form of its truncated multipole expansion. Closed-form formulas are obtained for the asymptotics of $J_{\rm surf}[\Phi_N]$ and $J_{\rm var}[\Phi_N]$, where $\Phi_N$ is the multipole expansion of $\Phi$ truncated after the $1/R^N$ term, $R$ being the internuclear separation. It is shown that the obtained sequences of approximations converge to the exact results with the rate corresponding to the convergence radius equal to 2 and 4 when the surface and the volume integral formulas are used, respectively. When the multipole expansion of a truncated, $K$th order polarization function is used to approximate the primitive function the convergence radius becomes equal to unity in the case of $J_{\textrm{var}}[\Phi]$. At low order the observed convergence of $J_{\rm var}[\Phi_N]$ is, however, geometric and switches to harmonic only at certain value of $N=N_c$ dependent on $K$. An equation for $N_c$ is derived which very well reproduces the observed $K$-dependent convergence pattern. The results shed new light on the convergence properties of the conventional SAPT expansion used in applications to many-electron diatomics., Comment: 20 pages, 2 figures (incl. graphical abstract), submitted to Molecular Physics

The exchange splitting $J$ of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula $J_{\textrm{surf}}[\varphi]$, the volume-integral formula of the symmetry-adapted perturbation theory $J_{\textrm{SAPT}}[\varphi]$, and a variational volume-integral formula $J_{\textrm{var}}[\varphi]$. The calculations are based on the multipole expansion of the wave function $\varphi$, which is divergent for any internuclear distance $R$. Nevertheless, the resulting approximations to the leading coefficient $j_0$ in the large-$R$ asymptotic series $J(R) = 2 e^{-R-1} R ( j_0 + j_1 R^{-1} + j_2 R^{-2} +\cdots ) $ converge, with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the $J_{\textrm{var}}[\varphi]$, $J_{\textrm{surf}}[\varphi]$, and $J_{\textrm{SAPT}}[\varphi]$ formulas are used, respectively. Additionally, we observe that also the higher $j_k$ coefficients are predicted correctly when the multipole expansion is used in the $J_{\textrm{var}}[\varphi]$ and $J_{\textrm{surf}}[\varphi]$ formulas. The SAPT formula $J_{\textrm{SAPT}}[\varphi]$ predicts correctly only the first two coefficients, $j_0$ and $j_1$, gives a wrong value of $j_2$, and diverges for higher $j_n$. Since the variational volume-integral formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general., Comment: 11 pages, 3 figures, submitted to J. Chem. Phys

Future metrology standards will be partly based on physical quantities computed from first principles rather than measured. In particular, a new pressure standard can be established if the dynamic polarizability of helium can be determined from theory with an uncertainty smaller than 0.2 ppm. We present calculations of the frequency-dependent part of this quantity including relativistic effects with full account of leading nuclear recoil terms and using highly optimized explicitly correlated basis sets. A particular emphasis is put on uncertainty estimates. At the He-Ne laser wavelength of 632.9908 nm, the computed polarizability value of 1.391 811 41 a.u. has uncertainty of 0.1 ppm that is two orders of magnitude smaller than those of the most accurate polarizability measurements. We also obtained an accurate expansion of the helium refractive index in powers of density., Comment: Submitted to Phys. Rev. Lett

In this paper we present results of ab-initio calculations for the beryllium dimer with basis set of Slater-type orbitals (STOs). Nonrelativistic interaction energy of the system is determined using the frozen-core full configuration interaction calculations combined with high-level coupled cluster correction for inner-shell effects. Newly developed STOs basis sets, ranging in quality from double to sextuple zeta, are used in these computations. Principles of their construction are discussed and several atomic benchmarks are presented. Relativistic effects of order ${\alpha}^2$ are calculated perturbatively by using the Breit-Pauli Hamiltonian and are found to be significant. We also estimate the leading-order QED effects. Influence of the adiabatic correction is found to be negligible. Finally, the interaction energy of the beryllium dimer is determined to be 929.0$\,\pm\,$1.9 $cm^{-1}$, in a very good agreement with the recent experimental value. The results presented here appear to be the most accurate ab-initio calculations for the beryllium dimer available in the literature up to date and probably also one of the most accurate calculations for molecular systems containing more than four electrons., Comment: submitted to Phys. Rev. A

The exchange splitting energy $J$ of the lowest \emph{gerade} and \emph{ungerade} states of the H$_2^+$ molecular ion was calculated using a volume integral expression of symmetry-adapted perturbation theory and standard basis set techniques of quantum chemistry. The performance of the proposed expression was compared to the well known surface integral formula. Both formulas involve the primitive function which we calculated employing either the Hirschfelder-Silbey perturbation theory or the conventional Rayleigh-Schr\"odinger perturbation theory (the polarization expansion). Our calculations show that very accurate values of $J$ can be obtained using the proposed volume integral formula. When the Hirschfelder-Silbey primitive function is used in both formulas the volume formula gives much more accurate results than the surface integral expression. We also show that using the volume integral formula with the primitive function approximated by Rayleigh-Schr\"odinger perturbation theory, one correctly obtains only the first four terms in the asymptotic expansion of the exchange splitting energy., Comment: 8 pages, 6 figures, v2: added results of bigger basis sets, added a formal proof of some claims, conclusions unchanged, added journal reference

In currently most popular explicitly correlated electronic structure theories the dependence of the wave function on the interelectronic distance rij is built via the correlation factor f(rij). While the short-distance behavior of this factor is well understood, little is known about the form of f(rij) at large rij. In this work we investigate the optimal form of f(r12) on the example of the helium atom and helium-like ions and several well-motivated models of the wave function. Using the Rayleigh-Ritz variational principle we derive a differential equation for f(r12) and solve it using numerical propagation or analytic asymptotic expansion techniques. We found that for every model under consideration, f(r12) behaves at large r12 as r12^{\rho} exp(B r12) and obtained simple analytic expressions for the system dependent values of {\rho} and B. For the ground state of the helium-like ions the value of B is positive, so that f(r12) diverges as r12 tends to infinity. The numerical propagation confirms this result. When the Hartree-Fock orbitals, multiplied by the correlation factor, are expanded in terms of Slater functions r^n exp(-{\beta}r), the numerical propagation reveals a minimum in f(r12) with depth increasing with N. For the lowest triplet state B is negative. Employing our analytical findings, we propose a new 'range-separated' form of the correlation factor with the short- and long-range r12 regimes approximated by appropriate asymptotic formulas connected by a switching function. Exemplary calculations show that this new form of f(r12) performs somewhat better than the correlation factors used thus far in the standard R12 or F12 theories., Comment: 42 pages, 5 figures, submitted to the Journal of Chemical Physics