Properties of the ground state of the hydrogen molecular ion

The ground-state energies of the ions ${\mathrm{H}}_2^{+}$ and ${\mathrm{D}}_2^{+}$ have been calculated without making use of the Born-Oppenheimer approximation. Instead, the ions are treated as three-body systems whose ground states are spherically symmetric. The wave function of the ground state is taken to be a generalized Hylleraas form, but it is necessary to use high powers of the internuclear coordinate to simulate the localized motion of the nuclei. We obtain good values of the ground-state energies and compare them with those obtained from earlier calculations. Expectation values are calculated for various operators, the Fermi contact parameter, and the permanent quadrupole moment. The cusp condition is also calculated. The results are compared with the results of other calculations, where available.