  The mass-correction functions in the second-order non-adiabatic Hamiltonian
are computed for the $^4$He$^+_2$ molecular ion using the variational method,
floating explicitly correlated Gaussian functions, and a general
coordinate-transformation formalism. When non-adiabatic rovibrational energy
levels are computed using these (coordinate-dependent) mass-correction
functions and a highly accurate potential energy and diagonal Born-Oppenheimer
correction curve, significantly improved theoretical results are obtained for
the nine rotational and two rovibrational intervals known from high-resolution
spectroscopy experiments.
