Algorithms for Solving Two-Center Coulomb Problem

Abstract: New algorithms for solving the two-center Coulomb problem with discrete and continuous spectrum in prolate spheroidal coordinates with separation of independent variables are presented. The eigenvalues of energy and separation constant, as well as the eigenfunctions of the discrete spectrum are calculated by the secant method and the finite element method (FEM) on a suitable grid of a real valued parameter, the distance between the Coulomb centers. At each step of the secant method, the eigensolutions of the discrete spectrum are calculated using the KANTBP 5M program, which implements the FEM in Maple. For the problem with a continuous spectrum at a fixed energy value, it is sufficient to solve the eigenvalue problem for the quasi-angular equation with respect to the separation constant and use it to solve the boundary value problem for the quasi-radial equation with respect to the unknown phase shift and eigenfunction using the KANTBP 5M program. The results of the benchmark calculations agree with the reference calculations performed by programs implementing alternative methods in the FORTRAN language, with the required accuracy. © Pleiades Publishing, Ltd. 2025.