Wrong-way recursion yields more accurate eigenparameters for the hydrogen-molecule ion

In the power-series solution of the separated two-center equations for ${\mathrm{H}}_2^{+}$, determining eigenparameters by recursion in the stable direction (from high to low order) encounters severe numerical limitations in the case of large internuclear distance and high spatial dimension or, equivalently, high angular momentum. We find this can be overcome by use of recursion in the unstable direction (from low to high order), thereby greatly improving the accuracy of the eigenparameters obtained. For instance, for the ground electronic state with angular momentum ‖m‖=0 and dimension D=67 (equivalent to ‖m‖=32 for D=3), the improvement in accuracy is about five orders of magnitude for the energy and about eight orders for the separation constant related to the Runge-Lenz vector. The source of this seemingly paradoxical behavior is revealed by examining the divergent solution for each recursion direction. The method is useful for determining the eigenparameters only; recursion in the stable direction must still be used to determine the eigenfunctions.