  We present high-precision non-relativistic variational calculations of bound
vibrational-rotational state energies for the $H_2^+$ and $D_2^+$ molecular
ions in each of the lowest electronic states of $\Sigma_g$, $\Sigma_u$, and
$\Pi_u$ symmetry. The calculations are carried out including coupling between
$\Sigma$ and $\Pi$ states but without using the Born-Oppenheimer or any
adiabatic approximation. Convergence studies are presented which indicate that
the resulting energies for low-lying levels are accurate to about $10^{-13}$.
Our procedure accounts naturally for the lambda-doubling of the $\Pi_u$ state.
