  This work reports quantum mechanical and semiclassical WKB calculations for
energies and wave functions of high-lying $^2\Sigma$ states of H$_2^+$ in
atomic units. The high-lying states presented lie in an unexplored regime,
corresponding asymptotically to H $(n\leq 145)$ plus a proton, with $R\leq
120,000$ $a_0$. We compare quantum mechanical energies, spectroscopic
constants, dipole matrix elements, and phases with semiclassical results and
demonstrate a good level of agreement. The quantum mechanical phases are
determined by using Milne's phase-amplitude procedure. Our semiclassical
energies for low-lying states are compared with those published previously in
the literature.
