Here is the present status of the problem. The Schrödinger equation for the hydrogen molecule ion (within the Born-Oppenheimer approximation) can be separated in prolate spheroidal coordinates (xi, eta, phi) into 3 ODEs. No analytic solutions have as yet been found for the xi and eta differential equations, although very accurate asymptotic expansions can be derived. The energy eigenvalues, which are related to the separation constant, can be expressed as an infinite series or an infinite continued fraction, which can be evaluated to arbitrary accuracy. A good update is given in http://arxiv.org/pdf/physics/0607081v1.pdf