From First-Principles to Quantum Electrodynamics: Pushing the Limits of Theory with the Hydrogen Molecule

Modern spectroscopic techniques enable the determination of the spacing between rovibrational levels of H₂ with a relative accuracy of approximately 10^-11. At this extreme level of precision, subtle quantum electrodynamic (QED) effects, such as electron self-interaction and vacuum polarization, are probed. A theoretical model aiming to achieve similar accuracy must precisely describe not only these relatively small QED effects but also the more significant contributions related to electron correlation, coupling between electronic and nuclear motions, and relativistic effects. Although the hydrogen molecule exhibits most of the phenomena found in larger molecules, it is simple enough to meet the requirements mentioned above. In this article, we report on enhancements to the current capabilities of quantum mechanical calculations for the hydrogen molecule. We present a method based on exponential functions that fully captures electron correlation or, more broadly, interparticle correlation, enabling a comprehensive description of effects related to nuclear motion. Specifically, we solve the four-particle Schrödinger equation without invoking commonly used approximations such as the one-electron or the Born-Oppenheimer approximation. The only source of nonrelativistic energy error comes from the finite size of the basis set. The explicitly correlated nonadiabatic wave function used here is then employed to determine the relativistic and QED effects. As a result, the dissociation energy for the lowest rovibrational levels in the electronic ground state of H₂ has been obtained with a relative accuracy of 7 × 10^-10, while the frequencies of intervals between these levels have been determined with sub-MHz accuracy, corresponding to a relative accuracy of 3 × 10^-9. In consequence, the discrepancies between the highest precision measurements and earlier theoretical predictions have been resolved.