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

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

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

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

Pair Potential with Submillikelvin Uncertainties and Nonadiabatic Treatment of the Halo State of the Helium Dimer.

The pair potential for helium is 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 are obtained from large full configuration interaction calculations. Nonadiabatic perturbation theory is used to predict the properties of the halo state of the helium dimer. Its binding energy and the average value of the interatomic distance are found to be 138.9(5) neV and 47.13(8) Å. The binding energy agrees with its first experimental determination of 151.9(13.3) neV [Zeller et al., Proc. Natl. Acad. Sci. U.S.A. 113, 14651 (2016)PNASA60027-842410.1073/pnas.1610688113].

Frequency-dependent polarizability of helium including relativistic effects with nuclear recoil terms.

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 1.39181141 a.u. has uncertainty of 0.1 ppm that is 2 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.

Exchange splitting of the interaction energy and the multipole expansion of the wave function.

The exchange splitting J of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula Jsurf[Φ], the volume-integral formula of the symmetry-adapted perturbation theory JSAPT[Φ], and a variational volume-integral formula Jvar[Φ]. The calculations are based on the multipole expansion of the wave function Φ, which is divergent for any internuclear distance R. Nevertheless, the resulting approximations to the leading coefficient j0 in the large-R asymptotic series J(R) = 2e(-R-1)R(j0 + j1R(-1) + j2R(-2) + ⋯) converge with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the Jvar[Φ], Jsurf[Φ], and JSAPT[Φ] formulas are used, respectively. Additionally, we observe that also the higher jk coefficients are predicted correctly when the multipole expansion is used in the Jvar[Φ] and Jsurf[Φ] formulas. The symmetry adapted perturbation theory formula JSAPT[Φ] predicts correctly only the first two coefficients, j0 and j1, gives a wrong value of j2, and diverges for higher jn. 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.

On the large interelectronic distance behavior of the correlation factor for explicitly correlated wave functions.

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(r(ij)). While the short-distance behavior of this factor is well understood, little is known about the form of f(r(ij)) at large r(ij). 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 r(ij) as r12(ρ)e(Br12) and obtained simple analytic expressions for the system dependent values of ρ 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)e(-ßr), n = 0,...,N, 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.

Onset of Casimir-Polder retardation in a long-range molecular quantum state.

Two 4He atoms form a diatomic molecule with a significant vibrational wave function amplitude at interatomic separations R>100 Å, where the retardation switches the London R(-6) decay of the potential to the Casimir-Polder R(-7) form. It has been assumed that this effect of retardation on the long-range part of the potential is responsible for the 2 Å (4%) increase of the bond length of 4He2. We show that is, unexpectedly, insensitive to the potential at R>20 Å and its increase is due to quantum electrodynamics effects computed by us from expressions valid at short R--beyond the validity range of Casimir-Polder theory--that seamlessly extend this theory to distances relevant for properties of long molecules.

Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium.

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations R smaller than 4 bohrs and for the distance R = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near R = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller R (all R in the case of a QED component) and optimized orbital bases up to the cardinal number X = 7 for larger R. Near-exact asymptotic constants were used to describe the large-R behavior of all components. The fitted potential, exhibiting the minimum of -10.996 ± 0.004 K at R = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound (4)He(2) dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 Å relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α(2) and α(3), respectively. The remaining retardation effect, of the order of α(4) and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.

Physical mechanisms of intermolecular interactions from symmetry-adapted perturbation theory.

Symmetry-adapted perturbation theory (SAPT) is a method for computational studies of noncovalent interactions between molecules. This method will be discussed here from the perspective of establishing the paradigm for understanding mechanisms of intermolecular interactions. SAPT interaction energies are obtained as sums of several contributions. Each contribution possesses a clear physical interpretation as it results from some specific physical process. It also exhibits a specific dependence on the intermolecular separation R. The four major contributions are the electrostatic, induction, dispersion, and exchange energies, each due to a different mechanism, valid at any R. In addition, at large R, SAPT interaction energies are seamlessly connected with the corresponding terms in the asymptotic multipole expansion of interaction energy in inverse powers of R. Since such expansion explicitly depends on monomers' multipole moments and polarizabilities, this connection provides additional insights by rigorously relating interaction energies to monomers' properties.

Complete Basis Set Extrapolation of Electronic Correlation Energies Using the Riemann Zeta Function.

In this article, we demonstrate the effectiveness of the method of complete basis set (CBS) extrapolation of correlation energies based on the application of the Riemann zeta function. Instead of fitting the results obtained with a systematic sequence of one-electron bases with a certain functional form, an analytic resummation of the missing contributions coming from higher angular momenta, l, is performed. The assumption that these contributions vanish asymptotically as an inverse power of l leads to an expression for the CBS limit given in terms of the 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 method is assessed by comparing the results with very 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 sequence of bases. A recipe for the estimation of the residual extrapolation error is also proposed.

Symmetry-adapted perturbation theory utilizing density functional description of monomers for high-spin open-shell complexes.

We present an implementation of symmetry-adapted perturbation theory (SAPT) to interactions of high-spin open-shell monomers forming high-spin dimers. The monomer spin-orbitals used in the expressions for the electrostatic and exchange contributions to the interaction energy are obtained from density functional theory using a spin-restricted formulation of the open-shell Kohn-Sham (ROKS) method. The dispersion and induction energies are expressed through the density-density response functions predicted by the time-dependent ROKS theory. The method was applied to several systems: NH...He, CN...Ne, H2O...HO2, and NH...NH. It provides accuracy comparable to that of the best previously available methods such as the open-shell coupled-cluster method with single, double, and noniterative triple excitations, RCCSD(T), with a significantly reduced computational cost.

Quantum Electrodynamics Effects in Rovibrational Spectra of Molecular Hydrogen.

The dissociation energies from all rovibrational levels of H2 and D2 in the ground electronic state are calculated with high accuracy by including relativistic and quantum electrodynamics (QED) effects in the nonadiabatic treatment of the nuclear motion. For D2, the obtained energies have theoretical uncertainties of 0.001 cm(-1). For H2, similar uncertainties are for the lowest levels, while for the higher ones the uncertainty increases to 0.005 cm(-1). Very good agreement with recent high-resolution measurements of the rotational v = 0 levels of H2, including states with large angular momentum J, is achieved. This agreement would not have been possible without accurate evaluation of the relativistic and QED contributions and may be viewed as the first observation of the QED effects, mainly the electron self-energy, in a molecular spectrum. For several electric quadrupole transitions, we still observe certain disagreement with experimental results, which remains to be explained.

Theoretical Determination of the Dissociation Energy of Molecular Hydrogen.

The dissociation energy of molecular hydrogen is determined theoretically with a careful estimation of error bars by including nonadiabatic, relativistic, and quantum electrodynamics (QED) corrections. The relativistic and QED corrections were obtained at the adiabatic level of theory by including all contributions of the order α(2) and α(3) as well as the major (one-loop) α(4) term, where α is the fine-structure constant. The computed α(0), α(2), α(3), and α(4) components of the dissociation energy of the H2 isotopomer are 36 118.7978(2), -0.5319(3), -0.1948(2), and -0.0016(8) cm(-1), respectively, while their sum amounts to 36 118.0695(10) cm(-1), where the total uncertainty includes the estimated size (±0.0004 cm(-1)) of the neglected relativistic nonadiabatic/recoil corrections. The obtained theoretical value of the dissociation energy is in excellent agreement with the most recent experimental determination 36 118.0696(4) cm(-1) [J. Liu et al. J. Chem. Phys. 2009, 130, 174 306]. This agreement would have been impossible without inclusion of several subtle QED contributions which have not been considered, thus far, for molecules. A similarly good agreement is observed for the leading vibrational and rotational energy differences. For the D2 molecule we observe, however, a small disagreement between our value 36 748.3633(9) cm(-1) and the experimental result 36 748.343(10) cm(-1) obtained in a somewhat older and less precise experiment [Y. P. Zhang et al. Phys. Rev. Lett. 2004, 92, 203003]. The reason of this discrepancy is not known.

Dispersion energy from density-fitted density susceptibilities of singles and doubles coupled cluster theory.

A new method of calculation of the second-order dispersion energy is proposed. It is based on the Longuet-Higgins formula [Faraday Discuss. Chem. Soc. 40, 7 (1965)], which describes the dispersion interaction in terms of frequency-dependent density susceptibilities of monomers. In this study, the density susceptibilities are obtained from the coupled cluster theory at the singles and doubles level. Density fitting is applied in order to reduce the computational effort for the evaluation of density susceptibilities. It is shown that density fitting improves the scaling of the computational resources with molecular size by one order of magnitude without affecting the accuracy of the resulting dispersion energy. Numerical results are presented for several van der Waals molecules to illustrate the performance of the new approach.

Accurate pair interaction energies for helium from supermolecular Gaussian geminal calculations.

Nonrelativistic clamped-nuclei pair interaction energy for ground-state helium atoms has been computed for 12 interatomic separations ranging from 3.0 to 9.0 bohr. The calculations applied the supermolecular approach. The major part of the interaction energy was obtained using the Gaussian geminal implementation of the coupled-cluster theory with double excitations (CCD). Relatively small contributions from single, triple, and quadruple excitations were subsequently included employing the conventional orbital coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and the full configuration interaction (FCI) method. For three distances, the single-excitation contribution was taken from literature Gaussian-geminal calculations at the CCSD level. The orbital CCSD(T) and FCI calculations used very large basis sets, up to doubly augmented septuple- and sextuple-zeta size, respectively, and were followed by extrapolations to the complete basis set limits. The accuracy of the total interaction energies has been estimated to be about 3 mK or 0.03% at the minimum of the potential well. For the attractive part of the well, the relative errors remain consistently smaller than 0.03%. In the repulsive part, the accuracy is even better, except, of course, for the region where the potential goes through zero. For interatomic separations smaller than 4.0 bohr, the relative errors do not exceed 0.01%. Such uncertainties are significantly smaller than the expected values of the relativistic and diagonal Born-Oppenheimer contributions to the potential.

Pair potential for helium from symmetry-adapted perturbation theory calculations and from supermolecular data.

Symmetry-adapted perturbation theory (SAPT) was applied to the helium dimer for interatomic separations R from 3 to 12 bohrs. The first-order interaction energy and the bulk of the second-order contribution were obtained using Gaussian geminal basis sets and are converged to about 0.1 mK near the minimum and for larger R. The remaining second-order contributions available in the SAPT suite of codes were computed using very large orbital basis sets, up to septuple-zeta quality, augmented by diffuse and midbond functions. The accuracy reached at this level was better than 1 mK in the same region. All the remaining components of the interaction energy were computed using the full configuration interaction method in bases up to sextuple-zeta quality. The latter components, although contributing only 1% near the minimum, have the largest uncertainty of about 10 mK in this region. The total interaction energy at R=5.6 bohrs is -11.000+/-0.011 K. For R< or =6.5 bohrs, the supermolecular (SM) interaction energies computed by us recently turned out to be slightly more accurate. Therefore, we have combined the SM results for R< or =6.5 bohrs with the SAPT results from 7.0 to 12 bohrs to fit analytic functions for the potential and for its error bars. The potential fit uses the best available van der Waals constants C(6) through C(16), including C(11), C(13), and C(15), and is believed to be the best current representation of the Born-Oppenheimer (BO) potential for helium. Using these fits, we found that the BO potential for the helium dimer exhibits the well depth D(e)=11.006+/-0.004 K, the equilibrium distance R(e)=5.608+/-0.012 bohrs, and supports one bound state for (4)He(2) with the dissociation energy D(0)=1.73+/-0.04 mK, and the average interatomic separation R=45.6+/-0.5 A.

One-electron properties and electrostatic interaction energies from the expectation value expression and wave function of singles and doubles coupled cluster theory.

One-electron density matrices resulting from the explicitly connected commutator expansion of the expectation value were implemented at the singles and doubles coupled cluster (CCSD) level. In the proposed approach the one-electron density matrix is obtained at a little extra cost in comparison to the calculation of the CCSD correlation energy. Therefore, in terms of the computational time the new method is significantly less demanding than the conventional linear-response CCSD theory which requires additionally an expensive calculation of the left-hand solution of the CCSD equations. The quality of the new density matrices was investigated by computing a set of one-electron properties for a series of molecules of varying sizes and comparing the results with data obtained using the full configuration interaction method or higher level coupled cluster theory. It has been found that the results obtained using the new approach are of the same quality as those predicted by the linear-response CCSD method. The novel one-electron density matrices have also been applied to study the energy of the electrostatic interaction for a number of van der Waals complexes, including the benzene and azulene dimers.

Third-order interactions in symmetry-adapted perturbation theory.

We present an extension of many-body symmetry-adapted perturbation theory (SAPT) by including all third-order polarization and exchange contributions obtained with the neglect of intramonomer correlation effects. The third-order polarization energy, which naturally decomposes into the induction, dispersion, and mixed, induction-dispersion components, is significantly quenched at short range by electron exchange effects. We propose a decomposition of the total third-order exchange energy into the exchange-induction, exchange-dispersion, and exchange-induction-dispersion contributions which provide the quenching for the corresponding individual polarization contributions. All components of the third-order energy have been expressed in terms of molecular integrals and orbital energies. The obtained formulas, valid for both dimer- and monomer-centered basis sets, have been implemented within the general closed-shell many-electron SAPT program. Test calculations for several small dimers have been performed and their results are presented. For dispersion-bound dimers, the inclusion of the third-order effects eliminates the need for a hybrid SAPT approach, involving supermolecular Hartree-Fock calculations. For dimers consisting of strongly polar monomers, the hybrid approach remains more accurate. It is shown that, due to the extent of the quenching, the third-order polarization effects should be included only together with their exchange counterparts. Furthermore, the latter have to be calculated exactly, rather than estimated by scaling the second-order values.

Bounds for the scattering length of spin-polarized helium from high-accuracy electronic structure calculations.

We developed a series of correlation-consistent, polarized multiple zeta basis sets optimized specifically for the energy of the 2 3S state of helium atom. These basis sets were subsequently augmented with diffuse functions optimized for the van der Waals constants C6 through C14 which determine the asymptotic behavior of the second-order dispersion interaction between 2 3S helium atoms at large interatomic separation R. The resulting bases were applied to compute the Born-Oppenheimer (BO) potential for the lowest 5Sigmag+ state of the helium dimer. The coupled cluster and the full configuration-interaction techniques were employed to account for the electron correlation effects. The cardinal number extrapolation technique was used to obtain the complete-basis-set limit V(R) for the interaction potential and to find its lower VL(R) and upper VU(R) bounds. The resulting potentials were fitted to an analytical function containing accurate van der Waals constants C6 through C12 (including C11). We found that the complete-basis-set BO potential has a well depth De=1048.24+/-0.36 cm-1. The highest rotationless vibrational level is bound by D14=90.2+/-4.7 MHz, much stronger than the previous most accurate estimation of 15.2 MHz. The error bounds for De and D14 were obtained using the VL(R) and VU(R) potentials. The S-wave scattering length computed using the VL(R), V(R), and VU(R) potentials (assuming atomic masses) is aL=7.41 nm, a=7.54 nm, and aU=7.69 nm, respectively. We also computed the adiabatic, relativistic, and quantum electrodynamics (QED) corrections to the BO potential. When these corrections are taken into account the values of D14 and of a (both computed assuming nuclear masses) are 87.4+/-6.7 MHz and 7.64+/-0.20 nm; the error bounds reflect now also the uncertainty of the included adiabatic, relativistic, and QED corrections. The value of the scattering length resulting from our investigation lies outside the error bounds of all experimental determinations based on the properties of Bose-Einstein condensate of spin-polarized helium atoms.

Intermolecular potentials based on symmetry-adapted perturbation theory with dispersion energies from time-dependent density-functional calculations.

Recently, three of us have proposed a method [Phys. Rev. Lett. 91, 33201 (2003)] for an accurate calculation of the dispersion energy utilizing frequency-dependent density susceptibilities of monomers obtained from time-dependent density-functional theory (DFT). In the present paper, we report numerical calculations for the helium, neon, water, and carbon dioxide dimers and show that for a wide range of intermonomer separations, including the van der Waals and short-range repulsion regions, the method provides dispersion energies with accuracies comparable to those that can be achieved using the current most sophisticated wave-function methods. If the dispersion energy is combined with (i) the electrostatic and first-order exchange interaction energies as defined in symmetry-adapted perturbation theory (SAPT) but computed using monomer Kohn-Sham (KS) determinants, and (ii) the induction energy computed using the coupled KS static response theory, (iii) the exchange-induction and exchange-dispersion energies computed using KS orbitals and orbital energies, the resulting method, denoted by SAPT(DFT), produces very accurate total interaction potentials. For the helium dimer, the only system with nearly exact benchmark values, SAPT(DFT) reproduces the interaction energy to within about 2% at the minimum and to a similar accuracy for all other distances ranging from the strongly repulsive to the asymptotic region. For the remaining systems investigated by us, the quality of the SAPT(DFT) interaction energies is so high that these energies may actually be more accurate than the best available results obtained with wave-function techniques. At the same time, SAPT(DFT) is much more computationally efficient than any method previously used for calculating the dispersion and other interaction energy components at this level of accuracy.