Accurate Born-Oppenheimer potentials for excited $\Sigma^{+}$ states of the hydrogen molecule

We report on highly accurate calculations of Born-Oppenheimer potentials for excited $n\,\Sigma^{+}$ states of the hydrogen molecule for all possible combinations of singlet/triplet and gerade/ungerade symmetries up to $n=7$. A relative accuracy of $10^{-10}$ (0.00002 cm$^{-1}$) or better is achieved for all the internuclear distances and all the excited states under consideration -- an improvement with respect to the best results available in the literature by at least 6 orders of magnitude. Presented variational calculations rely on efficient evaluation of molecular integrals with the explicitly correlated exponential basis in arbitrary precision arithmetics.