Hydrogen molecular ion in a magnetic field

The energy of the ground electronic state of $\mathrm{H}_2^{}{}_{}{}^{+}$ is studied as a function of the internuclear separation $R_{12}$, the angle, $\theta$, between the molecular axis and the magnetic field, and the field strength $B$. For small $B$ the molecular diamagnetism reaches its maximum value when $\theta =\frac{\pi }{2}$ and $R_{12}\cong 5$ Bohr radii. This maximum value is about 50% greater than the diamagnetism of an isolated H atom. At large $B$ the molecule shrinks due to magnetic compression of the electron wave function, and the molecular vibration frequencies increase substantially. A strong diamagnetic torque appears which tends to align the molecular axis along the field. This gives rise to a zero-point rotational oscillation about $\theta =0\mathrm{}$ whose energy can substantially exceed that of the zero-point vibrational oscillation. The calculations presented indicate that even if the protons had infinite mass, the molecule would become unstable to dissociation at $\theta =\frac{\pi }{2}$ in fields $\gtrsim 1.6\times {10}^{11}$ G.