Electron-nucleus Araki-Sucher correction in the hydrogen molecule isotopologues

TL;DR

This work computes the electron–nucleus Araki–Sucher QED correction for rovibrational states of molecular hydrogen and its isotopologues within the Born–Oppenheimer framework. Using NRQED and an explicitly correlated Gaussian basis, the authors build a distance-dependent potential $\mathcal{E}^{(5)}_ extrm{AS,en}(R)$, focusing on the electron-nucleus term of $E^{(5)}_ extrm{AS}$. They regularize nearly singular $1/r^3$ operators, reduce the problem to a finite set of elementary integrals, and perform careful numerical convergence studies to obtain $E^{(5)}_ extrm{AS,en}(R)$ across internuclear separations, including an accurate asymptotic form. The resulting corrections to the dissociation energy are on the order of $10^{-5}$ cm$^{-1}$, significant for achieving sub-MHz agreement with modern measurements, and highlight the necessity of including this term in high-precision molecular QED. The work sets the stage for incorporating nonadiabatic effects in future refinements.

Abstract

The quantum electrodynamic Araki-Sucher correction arising from the interaction between electrons and nuclei is calculated for rovibrational energy levels of the hydrogen molecule and its isotopologues. The corresponding expectation value $\langle r_{en}^{-3}\rangle$ is evaluated across a wide range of internuclear distances using the Born-Oppenheimer approximation. The electronic wave function is represented as a linear combination of explicitly correlated Gaussian basis functions. This correction contributes approximately tenths of a megahertz (about $10^{-5}$cm$^{-1}$) to the dissociation energy of rovibrational levels and to the transitions between them. Given recent spectroscopic measurements with an accuracy of 10 kHz, this correction is necessary to achieve sub-MHz agreement between theory and experiment.

Figures (1)

• Figure 1: Plot of the $\mathcal{E}^{(5)}_\mathrm{AS,en}(R)$ potential (in a.u.). The curve exhibits minimum at $R=2.894$ a.u. and a horizontal asymptote at $\frac{14\,\ln{2}}{6\,\pi\,\mu_\mathrm{n}}\approx 0.001121604$ a.u.