Completeα6 mCorrections to the Ground State ofH2

Abstract

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 full infrared spectrum of molecular hydrogen

“…The higher order E (6) QED correction was evaluated in the non-recoil limit by Puchalski et al (2016). The accuracy of this correction is limited by two factors that contribute errors of the same order-the unknown finite-nuclear-mass corrections and the numerical convergence.…”

Precision measurement of the fundamental vibrational frequencies of tritium-bearing hydrogen molecules: T2, DT, HT

“…On the theoretical side level energy calculations of the molecular hydrogen four-body system have undergone equally great improvements, producing highly accurate level energies of H 2 and D 2 18 as well as for HD 19 . These methods were based on 20 , and separate approaches for adiabatic 21 and non-adiabatic 22 corrections, as well as computations of relativistic 23 and quantum electrodynamic effects 24 . This development has led to the comprehensive approach of nonadiabatic perturbation theory (NAPT) 25,26 , which now enables the rapid computation of all bound rovibrational states in the ground electronic manifold of the hydrogen isotopologues.…”

Isotope dependence of thec3Πu−b3Σ${}_{u}^{+}$ andD1Π${}_{u}^{+}$−B′1Σ${}_{u}^{+}$ predissociation rates of molecular hydrogen

“…The effect of the coupling with other states is treated as a small non-adiabatic correction (Wolniewicz 1993, 1995, Pachucki and Komasa 2015. When relativistic (Puchalski et al 2017) and radiative (Piszczatowski et al 2009, Puchalski et al 2016 corrections are considered, the calculated and measured X g 1 S + energies agree within ∼0.001 cm −1 for low (v, J) levels and ∼0.005 cm −1 for high (v, J) levels (Komasa et al 2011, Pachucki and Komasa 2010, Niu et al 2014, Trivikram et al 2016, Cozijn et al 2018. Since excited states are not well-separated, the coupled Schrödinger equation method, usually for a few close low-lying states, is used (Senn et al 1988).…”