Adiabatic formulation of heteronuclear hydrogen molecular ion

We revisit the ${\mathrm{HD}}^{+}$ molecular ion by starting from a definition of the nonrelativistic adiabatic Hamiltonian that recovers the isotopic splitting. The Hamiltonian is written in prolate spheroidal coordinates with the origin at the nuclear center of mass. This prescription for the adiabatic Hamiltonian implicitly contains the symmetry-breaking effects arising from the mass difference between the proton and the deuteron, and gives the correct asymptotic atomic reduced masses. We calculate potential-energy curves, vibrational energies, transition dipole moments, and coupling matrix elements for ${\mathrm{HD}}^{+}.$ In our adiabatic approximation, the isotopic splitting is recovered to nine significant digits at a value of ${R=750a}_0.$ We investigate the dependence of the vibrational energies on the number of coupled electronic states, and compare with other calculations.