Method for One Particle Bound to Two Identical Fixed Centers: Application to H2+

A method is presented for the quantum-mechanical problem of one particle moving in the field of two identical fixed centers. An equation for the problem is derived in both position and momentum space as a special limiting case of our general method for the three-body problem. When applied to the $\mathrm{H}_{2}^{}{}_{}{}^{+}$ problem, using the Coulomb-Sturmian set as an expansion basis, the method gives an infinite secular equation for the energy eigenvalues which can be solved exactly in the limits as the internuclear distance goes to zero and to infinity. Numerical results are also reported for the energy as a function of internuclear distance for the $1{\ensuremath{\sigma}}_{g}$, $1{\ensuremath{\sigma}}_{u}$, $2{\ensuremath{\sigma}}_{g}$, and $2{\ensuremath{\sigma}}_{u}$ states of $\mathrm{H}_{2}^{}{}_{}{}^{+}$.