Abstract
We perform highly accurate calculations of the leading-order QED correction to the ground electronic state of molecular hydrogen. Numerical results are obtained for a grid of the internuclear distances $R=0--10$ a.u. with the relative precision of about ${10}^{\ensuremath{-}8}$. The major numerical uncertainty of previous QED results [K. Piszczatowski, G. \L{}ach, M. Przybytek, J. Komasa, K. Pachucki, and B. Jeziorski, J. Chem. Theory Comput. 5, 3039 (2009)] has been eliminated. Nevertheless, the discrepancy with measurements in HD at the level of 1.9 $\ensuremath{\sigma}$ persists.
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