  We recalculate the leading relativistic corrections for the ground electronic
state of the hydrogen molecule using variational method with explicitly
correlated functions which satisfy the interelectronic cusp condition. The new
computational approach allowed for the control of the numerical precision which
reached about 8 significant digits. More importantly, the updated theoretical
energies became discrepant with the known experimental values and we conclude
that the yet unknown relativistic recoil corrections might be larger than
previously anticipated.
