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The mass-correction function is evaluated for selected excited states of thehydrogen molecule within a single-state non-adiabatic treatment. Itsqualitative features are studied under the avoided crossing of the $EF$ withthe $GK$ state and also for the outer well of the $H\bar{H}$ state. For the$H\bar{H}$ state, a negative mass correction is obtained for the vibrationalmotion near the outer minimum, which accounts for most of the deviation betweenexperiment and earlier theoretical work.

Non-adiabatic mass correction for excited states of molecular hydrogen: improvement for the outer-well $H\bar{H}\ ^1\Sigma_\mathrm{g}^+$ term values