Hyperfine Structure of the First Rotational Level in H_{2}, D_{2} and HD Molecules and the Deuteron Quadrupole Moment.

We perform the four-body calculation of the hyperfine structure in the first rotational state J=1 of the H_{2}, D_{2}, and HD molecules and determine the accurate value for the deuteron electric quadrupole moment Q_{d}=0.285 699(15)(18) fm^{2} in significant disagreement with former spectroscopic determinations. Our results for the hyperfine parameters agree very well with the currently most accurate molecular-beam magnetic resonance measurement performed several decades ago by N.F. Ramsey and coworkers. They also indicate the significance of previously neglected nonadiabatic effects. Moreover, a very good agreement with the recent calculation of Q_{d} based on the chiral effective field theory, although much less accurate, indicates the importance of the spin dependence of nucleon interactions in the accurate description of nuclei.

Nonadiabatic QED Correction to the Dissociation Energy of the Hydrogen Molecule.

The quantum electrodynamic correction to the energy of the hydrogen molecule has been evaluated without expansion in the electron-proton mass ratio. The obtained results significantly improve the accuracy of theoretical predictions reaching the level of 1 MHz for the dissociation energy, in very good agreement with the parallel measurement [Hölsch et al., Phys. Rev. Lett. 122, 103002 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.103002]. Molecular hydrogen has thus become a cornerstone of ultraprecise quantum chemistry, which opens perspectives for determination of fundamental physical constants from its spectra.

Nonrelativistic energy levels of D2.

Nonrelativistic energies of the deuterium molecule, accurate to 10-7-10-8 cm-1 for all levels located up to 8000 cm-1 above the ground state, are presented. The employed nonadiabatic James-Coolidge wave functions with angular factors enable the high accuracy to be reached regardless of vibrational or rotational quantum number. The derivative of the energy with respect to the deuteron-to-electron mass ratio is supplied for each level, which makes the results independent of the future changes in this physical parameter and will enable its determination from sufficiently accurate experimental data.

Nuclear Spin-Spin Coupling in HD, HT, and DT.

The interaction between nuclear spins in a molecule is exceptionally sensitive to the physics beyond the standard model. However, all present calculations of the nuclear spin-spin coupling constant J are burdened by computational difficulties, which hinders the comparison to experimental results. Here, we present a variational approach and calculate the constant J in the hydrogen molecule with the controlled numerical precision, using the adiabatic approximation. The apparent discrepancy with experimental result is removed by an analysis of nonadiabatic effects based on the experimental values of the J constant for HD, HT, and DT molecules. This study significantly improves the reliability of the NMR theory for searching new physics in the spin-spin coupling.

Nonadiabatic Relativistic Correction to the Dissociation Energy of H_{2}, D_{2}, and HD.

The relativistic correction to the dissociation energy of H_{2}, D_{2}, and HD molecules has been accurately calculated without expansion in the small electron-nucleus mass ratio. The obtained results indicate the significance of nonadiabatic effects and resolve the discrepancy of theoretical predictions with recent experimental values for H_{2} and D_{2}. While the theoretical accuracy is now significantly improved and is higher than the experimental one, we observe about 3σ discrepancy for the dissociation energy of HD, which requires further investigation.

Nonrelativistic energy levels of HD.

Nonadiabatic exponential functions are employed to solve the four-body Schrödinger equation. Nonrelativistic bound energy levels of the HD molecule are calculated to the relative accuracy of 10-12-10-13, which is the first step toward highly accurate prediction of dissociation and transition energies. Such energies, in connection with equally accurate experimental data, will enable refinement of the physical constant and aid the search for deviations caused by yet unknown interactions at the atomic scale.

Nonadiabatic rotational states of the hydrogen molecule.

We present a new computational method for the determination of energy levels in four-particle systems like H2, HD, and HeH+ using explicitly correlated exponential basis functions and analytic integration formulas. In solving the Schrödinger equation, no adiabatic separation of the nuclear and electronic degrees of freedom is introduced. We provide formulas for the coupling between the rotational and electronic angular momenta, which enable calculations of arbitrary rotationally excited energy levels. To illustrate the high numerical efficiency of the method, we present the results for various states of the hydrogen molecule. The relative accuracy to which we determined the nonrelativistic energy reached the level of 10-12-10-13, which corresponds to an uncertainty of 10-7-10-8 cm-1.

Complete α

We perform the calculation of all relativistic and quantum electrodynamic corrections of the order of α^{6} m to the ground electronic state of a hydrogen molecule and present improved results for the dissociation and the fundamental transition energies. These results open the window for the high-precision spectroscopy of H_{2} and related low-energy tests of fundamental interactions.

Schrödinger equation solved for the hydrogen molecule with unprecedented accuracy.

The hydrogen molecule can be used for determination of physical constants, including the proton charge radius, and for improved tests of the hypothetical long range force between hadrons, which require a sufficiently accurate knowledge of the molecular levels. In this work, we perform the first step toward a significant improvement in theoretical predictions of H2 and solve the nonrelativistic Schrödinger equation to the unprecedented accuracy of 10(-12). We hope that it will inspire a parallel progress in the spectroscopy of the molecular hydrogen.

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.

Leading order nonadiabatic corrections to rovibrational levels of H2, D2, and T2.

An efficient computational approach to nonadiabatic effects in the hydrogen molecule (H2, D2, and T2) is presented. The electronic wave function is expanded in the James-Coolidge basis set, which enables obtaining a very high accuracy of nonadiabatic potentials. A single point convergence of the potentials with growing size of the basis set reveals a relative accuracy ranging from 10(-8) to 10(-13). An estimated accuracy of the leading nonadiabatic correction to the rovibrational energy levels is of the order of 10(-7) cm(-1). After a significant increase in the accuracy of the Born-Oppenheimer and adiabatic calculations, the nonadiabatic results presented in this report constitute another step towards highly accurate theoretical description of the hydrogen molecule.

Accurate adiabatic correction in the hydrogen molecule.

A new formalism for the accurate treatment of adiabatic effects in the hydrogen molecule is presented, in which the electronic wave function is expanded in the James-Coolidge basis functions. Systematic increase in the size of the basis set permits estimation of the accuracy. Numerical results for the adiabatic correction to the Born-Oppenheimer interaction energy reveal a relative precision of 10(-12) at an arbitrary internuclear distance. Such calculations have been performed for 88 internuclear distances in the range of 0 < R ⩽ 12 bohrs to construct the adiabatic correction potential and to solve the nuclear Schrödinger equation. Finally, the adiabatic correction to the dissociation energies of all rovibrational levels in H2, HD, HT, D2, DT, and T2 has been determined. For the ground state of H2 the estimated precision is 3 × 10(-7) cm(-1), which is almost three orders of magnitude higher than that of the best previous result. The achieved accuracy removes the adiabatic contribution from the overall error budget of the present day theoretical predictions for the rovibrational levels.

Relativistic Correction from the Four-Body Nonadiabatic Exponential Wave Function.

We present a method for calculating the relativistic correction in hydrogen molecules that significantly exceeds the accuracy of all the previous literature results. This method utilizes the explicitly correlated nonadiabatic exponential wave function, and thus treats electrons and nuclei equivalently. The proposed method can be applied to any rovibrational state, including highly excited ones. The numerical precision of the relativistic correction reaches several kHz (∼10-7 cm-1), which is below the best experimental accuracy.

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.

The absorption spectrum of H2: CRDS measurements of the (2-0) band, review of the literature data and accurate ab initio line list up to 35000 cm(-1).

Five very weak transitions-O(2), O(3), O(4), O(5) and Q(5)-of the first overtone band of H(2) are measured by very high sensitivity CW-Cavity Ring Down Spectroscopy (CRDS) between 6900 and 7920 cm(-1). The noise equivalent absorption of the recordings is on the order of α(min)≈ 5 × 10(-11) cm(-1) allowing for the detection of the O(5) transition with an intensity of 1.1 × 10(-30) cm per molecule, the smallest intensity value measured so far for an H(2) absorption line. A Galatry profile was used to reproduce the measured line shape and derive the line strengths. The pressure shift of the O(2) and O(3) lines was accurately determined from a series of recordings with pressure ranging between 10 and 700 Torr. From an exhaustive review of the literature data, the list of H(2) absorption lines detected so far has been constructed. It includes a total of 39 transitions ranging from the S(0) pure rotational line near 354 cm(-1) up to the S(1) transition of the (5-0) band near 18,908 cm(-1). These experimental values are compared to a highly accurate theoretical line list constructed for pure H(2) at 296 K (0-35,000 cm(-1), intensity cut off of 1 × 10(-34) cm per molecule). The energy levels and transition moments were computed from high level quantum mechanics calculations. The overall agreement between the theoretical and experimental values is found to be very good for the line positions. Some deviations for the intensities of the high overtone bands (V > 2) are discussed in relation with possible pressure effects affecting the retrieved intensity values. We conclude that the hydrogen molecule is probably a unique case in rovibrational spectroscopy for which first principles theory can provide accurate spectroscopic parameters at the level of the performances of the state of the art experimental techniques.

Rovibrational levels of helium hydride ion.

Dissociation energy (D(0)) of rovibrational levels of (4)HeH(+) has been predicted theoretically to the accuracy of the order of 0.01 cm(-1). The calculations take into account adiabatic and nonadiabatic corrections as well as relativistic and quantum electrodynamics effects. For the ground rovibrational level D(0) = 14,874.215(10) cm(-1) and it differs by several tens of the inverse centimeter from previous theoretical estimations. For a collection of about 50 transition energies measured between dipole connected levels the difference between theory and experiment is of the order of hundredths of cm(-1) or less.

The absorption spectrum of D2: ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24,000 cm(-1).

Eleven very weak electric quadrupole transitions Q(2), Q(1), S(0)-S(8) of the first overtone band of D(2) have been measured by very high sensitivity CW-cavity ring down spectroscopy (CRDS) between 5850 and 6720 cm(-1). The noise equivalent absorption of the recordings is on the order of α(min) ≈ 3 × 10(-11) cm(-1). By averaging a high number of spectra, the noise level was lowered to α(min) ≈ 4 × 10(-12) cm(-1) in order to detect the S(8) transition which is among the weakest transitions ever detected in laboratory experiments (line intensity on the order of 1.8 × 10(-31) cm/molecule at 296 K). A Galatry profile was used to reproduce the measured line shape and derive the line strengths. The pressure shift and position at zero pressure limit were determined from recordings with pressures ranging between 10 and 750 Torr. A highly accurate theoretical line list was constructed for pure D(2) at 296 K. The intensity threshold was fixed to a value of 1 × 10(-34) cm/molecule at 296 K. The obtained line list is provided as supplementary material. It extends up to 24,000 cm(-1) and includes 201 transitions belonging to ten v-0 cold bands (v = 0-9) and three v-1 hot bands (v = 1-3). The energy levels include the relativistic and quantum electrodynamic corrections as well as the effects of the finite nuclear mass. The quadrupole transition moments are calculated using highly accurate adiabatic wave functions. The CRDS line positions and intensities of the first overtone band are compared to the corresponding calculated values and to previous measurements of the S(0)-S(3) lines. The agreement between the CRDS and theoretical results is found within the claimed experimental uncertainties (on the order of 1 × 10(-3) cm(-1) and 2% for the positions and intensities, respectively) while the previous S(0)-S(3) measurements showed important deviations for the line intensities.

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.

Collision-induced dipole polarizability of helium dimer from explicitly correlated calculations.

Large expansions in basis sets of explicitly correlated Gaussian functions and the variation-perturbation technique were used to calculate the static dipole polarizability of the helium dimer at 16 different internuclear separations from 1.0 to 9.0 bohrs. The convergence towards the complete basis set limit was analyzed in order to estimate uncertainties of all the calculated values. The results are significantly more accurate than literature data. Asymptotically correct analytic fits for the trace and anisotropy of collision-induced polarizability were obtained.

Rovibrational levels of HD.

The dissociation energies of all rotation-vibrational states of the molecular HD in the ground electronic state are calculated to a high accuracy by including nonadiabatic, relativistic alpha(2), and quantum electrodynamic alpha(3) effects, with approximate treatment of small higher order alpha(4), and finite nuclear size corrections. The obtained result for the ground molecular state of 36 405.7828(10) cm(-1) is in a small disagreement with the latest most precise experimental value.