The quantum-mechanical energy splitting for a ground state hydrogen molecular ion has been computed as the complex-valued mean first-passage time across the potential barrier between two wells, by separating the Schrödinger equation using spheroidal coordinates and applying a subsequent transformation so as to obtain three one-dimensional Hermitian operator equations. One of these operators describes a particle oscillating between two wells separated by a potential barrier. The splitting is then calculated as the energy difference between the symmetry-adapted wavefunctions and a reference state whose probability density is concentrated on one side of the barrier. The eigenfunctions are calculated as the solutions of a Volterra integral equation, where the energy difference is the eigenvalue.