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 H2 and related low-energy tests of fundamental interactions. 12.20.Ds, The hydrogen atom and various hydrogenic systems like positronium, muonium, muonic hydrogen, and He + , due to highly accurate theoretical predictions [1], are considered for the determination of fundamental physical constants [2] and for the low-energy tests of the Standard Model [3,4]. However, they are limited by uncertainties in the nuclear structure or natural life-time of the system. The 1S − 2S transition in H is the best example, where the precision of the measurement f (1S − 2S) = 2 466 061 413 187 035(10) Hz [5] exceeds by orders of magnitude any theoretical predictions. This is because of the relatively large theoretical uncertainties in the proton structure and resulting inaccuracies in fundamental constants. The lack of another sharp transition in the hydrogen makes the determination of the Rydberg (R ∞ ) constant, which transforms atomic units to inverse of the transition wavelength, much less accurate than it would be if another such transition was available. Here we point out that the dissociation energy of H 2 can serve this purpose, as it is stable in the ground electronic state and can be calculated with sufficient precision. So having two accurate and calculable transitions the two unknowns R ∞ and r p can be determined, which among others, would help resolve the proton charge radius puzzle. Another alternative systems for which high precision calculations are possible, include the helium ion He + [6], heavy hydrogen like ions [7], and the hydrogen molecular ion [8,9].The calculations for the hydrogen molecule have never been considered to be as accurate as for the hydrogen atom due to the lack of an analytic solution of the Schrödinger equation. However, the numerical solution of this equation, as has been shown recently [10], can be as accurate as 10 −12 , and thus it will not limit the accuracy of theoretical predictions. There are obviously various corrections, such as relativistic and quantum electrodynamic (QED) ones. So far, they have been calculated up to α 5 m order [11], and only in the adiabatic approximation. Beyond this approximation, namely the combined nonadiabatic and relativistic effects, have not yet been obtained and they will limit the accuracy of current predictions. Here we calculate one of the most difficult, the α 6 m correction, using the so-called nonrelativistic QED approach. Next, we point out that when the higher order α 7 m correction is determined, energies of the hydrogen molecule can be obtained almost as accurately as those of the hydrogen atom alone, and thus may be used for determination of the R ∞ constant. Meanwhile, on the basis of the α 6 m correction obtained herein...
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 a very good agreement with the parallel measurement [Hölsch et al., Phys. Rev. Lett. 122, 103002 (2019)]. Molecular hydrogen has thus become a cornerstone of ultraprecise quantum chemistry, which opens perspectives for determination of fundamental physical constants from its spectra. 01 −a2r 2 02 −a3r 2 03 −a4r 2 12 −a5r 2 13 −a6r 2 23 . (7)
We present an accurate theoretical determination of rovibrational energy levels of the hydrogen molecule and its isotopologues in its electronic ground state. We consider all significant corrections to the Born-Oppenheimer approximation, obtained within nonadiabatic perturbation theory, including the mixed nonadiabatic-relativistic effects. Quantum electrodynamic corrections in the leading α 5 m and the next-to-leading α 6 m orders, as well as finite nuclear size effect, are also taken into account but within the Born-Oppenheimer approximation only. Final results for the transition wavelength between rovibrational levels achieve accuracy of the order of 10 −3 -10 −7 cm −1 , and are provided by simple to use computer code.
We calculate the nonadiabatic relativistic correction to rovibrational energy levels of H2, D2, and HD molecules using the nonadiabatic perturbation theory. This approach allows one to obtain nonadiabatic corrections to all the molecular levels with the help of a single effective potential. The obtained results are in very good agreement with the previous direct calculation of nonadiabatic relativistic effects for dissociation energies and resolve the reported discrepancies of theoretical predictions with recent experimental results.
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