Generalized spheroidal wave equation for real and complex valued parameters. An algorithm based on the analytic derivatives for the eigenvalues

This paper presents a new approach for the computation of eigenvalues of the generalized spheroidal wave equations. The novelty of the present method is in the use of the analytical derivatives of the eigenvalues to minimize losses in accuracy. The derivatives are constructed in the form of three-term recurrent relations within the method of continued fractions associated with the corresponding spheroidal wave equation. Very accurate results for the eigenvalues are obtained for a wide range of the parameters of the problem. As an illustrative example, the electronic energies and the separation constants are computed for various electronic states and geometries of selected ($\rm{H}_2^{+}$, $\rm{HeH}^{2+}$, and $\rm{BH}^{5+}$) quasimolecular systems. The computations for high lying ${}^{2}Σ$ electronic states of $\rm{H}_2^{+}$ up to very large internuclear separations ($ \leq 1.7 \times 10^5$ au) are presented. Also presented the computations for the eigenvalues of the generalized spheroidal wave equations with complex valued parameters. The agreement between the obtained results and the results of other authors is discussed.