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

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.