  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.
