The excited electronic states with the magnetic quantum numbers 4 less
than or equal to\m\less than or equal to 10 of the hydrogen molecular
ion are investigated. In the field-free case the lowest electronic st
ates with \m\greater than or equal to 5 possess purely repulsive poten
tial-energy curves and are, therefore, unbound. In the presence of a s
trong magnetic field the shape of these electronic states changes dras
tically For parallel internuclear and magnetic-field axes we observe w
ell-pronounced potential wells that contain many vibrational states; t
he corresponding electronic states are, therefore, strongly bound. The
coupling of the electronic degrees of freedom to the ionic center-of-
mass motion is estimated.