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.

Higher-Order QED Corrections to the Hyperfine Splitting in

We present a calculation of the hyperfine splitting of the 2^{3}S state in the ^{3}He atom with inclusion of all QED effects up to α^{3}E_{F}, where E_{F} is the Fermi splitting. Using the experimental value of the 1S hyperfine splitting in ^{3}He^{+}, we eliminate uncertainties from the nuclear structure and obtain the theoretical prediction for ^{3}He of ν_{hfs}=-6 739 701 181(41) Hz, which is in perfect agreement with the experimental value -6 739 701 177(16) Hz [S. D. Rosner and F. M. Pipkin, Phys. Rev. A 1, 571 (1970)PLRAAN0556-279110.1103/PhysRevA.1.571]. This result constitutes a 40-fold improvement in precision as compared to the previous value and is the most accurate theoretical prediction ever obtained for a nonhydrogenic system.

QED Theory of the Nuclear Recoil with Finite Size.

We investigate the modification of the transverse electromagnetic interaction between two pointlike particles when one particle acquires a finite size. It is shown that the correct treatment of such interaction cannot be accomplished within the Breit approximation but should be addressed within the QED. The complete QED formula is derived for the finite-size nuclear recoil, exact in the coupling strength parameter Zα. Numerical calculations are carried out for a wide range of Z and verified against the (Zα)^{5} contribution. The comparison with the Zα expansion identifies the contribution of order (Zα)^{6}, which is linear in the nuclear radius and numerically dominates over the lower-order (Zα)^{5} term.

QED Effect on the Nuclear Magnetic Shielding of

The leading quantum electrodynamic corrections to the nuclear magnetic shielding in one- and two-electron atomic systems are obtained in a complete form, and the shielding constants of ^{1}H, ^{3}He^{+}, and ^{3}He are calculated to be 17.735 436(3)×10^{-6}, 35.507 434(9)×10^{-6}, and 59.967 029(23)×10^{-6}, respectively. These results are orders of magnitude more accurate than previous ones, and, with the ongoing measurement of the nuclear magnetic moment of ^{3}He^{+} and planned ^{3}He^{2+}, they open the window for high-precision absolute magnetometry using ^{3}He NMR probes. The presented theoretical approach is applicable to all other light atomic and molecular systems, which facilitates the improved determination of magnetic moments of any light nuclei.

Long-range asymptotics of exchange energy in the hydrogen molecule.

The exchange energy, i.e., the splitting ΔE between gerade and ungerade states in the hydrogen molecule, has proven very difficult in numerical calculation at large internuclear distances R, while the known results are sparse and highly inaccurate. On the other hand, there are conflicting analytical results in the literature concerning its asymptotics. In this work, we develop a flexible and efficient numerical approach using explicitly correlated exponential functions and demonstrate highly accurate exchange energies for internuclear distances as large as 57.5 a.u. This approach may find further applications in calculations of inter-atomic interactions. In particular, our results support the asymptotics form ΔE ∼ R5/2e-2R, but with the leading coefficient being 2σ away from the analytically derived value.

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.

Nuclear Charge Radii of

The first laser spectroscopic determination of the change in the nuclear charge radius for a five-electron system is reported. This is achieved by combining high-accuracy ab initio mass-shift calculations and a high-accuracy measurement of the isotope shift in the 2s^{2}2p ^{2}P_{1/2}→2s^{2}3s ^{2}S_{1/2} ground state transition in boron atoms. Accuracy is increased by orders of magnitude for the stable isotopes ^{10,11}B and the results are used to extract their difference in the mean-square charge radius ⟨r_{c}^{2}⟩^{11}-⟨r_{c}^{2}⟩^{10}=-0.49(12) fm^{2}. The result is qualitatively explained by a possible cluster structure of the boron nuclei and quantitatively used to benchmark new ab initio nuclear structure calculations using the no-core shell model and Green's function Monte Carlo approaches. These results are the foundation for a laser spectroscopic determination of the charge radius of the proton-halo candidate ^{8}B.

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.

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 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 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.

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 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.

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.

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.

Precision Test of Many-Body QED in the Be+ 2p Fine Structure Doublet Using Short-Lived Isotopes.

Absolute transition frequencies of the 2s 2S{1/2}→2p2P{1/2,3/2} transitions in Be^{+} were measured for the isotopes ^{7,9-12}Be. The fine structure splitting of the 2p state and its isotope dependence are extracted and compared to results of ab initio calculations using explicitly correlated basis functions, including relativistic and quantum electrodynamics effects at the order of mα(6) and mα(7) ⁢ln α. Accuracy has been improved in both the theory and experiment by 2 orders of magnitude, and good agreement is observed. This represents one of the most accurate tests of quantum electrodynamics for many-electron systems, being insensitive to nuclear uncertainties.

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.

Quantum electrodynamics corrections to the 2P fine splitting in Li.

We consider quantum electrodynamics (QED) corrections to the fine splitting E(2P_{3/2})-E(2P_{1/2}) in the Li atom. We derive complete formulas for the mα^{6} and mα^{7}lnα contributions and calculate them numerically using highly optimized, explicitly correlated basis functions. The obtained results are in agreement with the most recent measurement, helping to resolve discrepancies between former ones and lay the foundation for the investigation of QED effects in light, many-electron atoms.

Ground state hyperfine splitting in 6,7Li atoms and the nuclear structure.

Relativistic and QED corrections are calculated for a hyperfine splitting of the 2S1/2 ground state in 6,7Li atoms with a numerically exact account for electronic correlations. The resulting theoretical predictions achieve such a precision level that, by comparison with experimental values, they enable determination of the nuclear properties. In particular, the obtained results show that the 7Li nucleus, having a charge radius smaller than 6Li, has about a 40% larger Zemach radius. Together with known differences in the electric quadrupole and magnetic dipole moments, this calls for a deeper understanding of the Li nuclear structure.