Continuum solutions to the two–center Coulomb problem in prolate spheroidal coordinates

Abstract

We present a computational and theoretical framework for solving the Schrödinger equation (SE) for the two–center Coulomb problem in prolate spheroidal coordinates when the energy of the SE is positive. A general and robust computer code has been produced that calculates the separation constants, spheroidal harmonic expansion coefficients, regular quasi–radial two–center Coulomb wave functions, and two–center Coulomb phase shifts. These quantities can be calculated over a range of internuclear separations, angular momentum projections, and continuum electron momenta. A representative set of results are presented and compared with previous calculations, excellent agreement is found in many cases while significant disagreements are found in others. Program summaryProgram title:spheroidal-cwCPC Library link to program files:https://doi.org/10.17632/nf5gw7vjnh.1Code Ocean capsule:https://codeocean.com/capsule/7638744Licensing provisions: MITProgramming language: Fortran 90Nature of problem: A robust and accurate computer code for calculating Coulomb wave functions in prolate spheroidal coordinate is necessary for the calculation of cross sections for photonionization, and electron and positron scattering on charged diatomic targets.Solution method: The angular solution is expanded in a series of spherical harmonics leading to the diagonalization of a pentadiagonal matrix. The eigenvectors contain the expansion coefficients and the eigenvalues are the separation constants which, unlike the SE in spherical coordinates, are also functions of the energy. The radial two–center Coulomb wave functions are started using a power series solution and then propagated using a linear multistep method. An asymptotic expansion is used to calculate the two–center Coulomb phase shift, and to normalize the radial wave function.Additional comments including restrictions and unusual features: Two–center Coulomb radial wave functions and phase shifts can be accurately calculated for internuclear separations from 0.001 to 60.0 a0, angular momentum projection |m| from 0 to 44, momentum k from 0.01 to 100.0 a.u., and a combined nuclear charge Z+ from 1 to 12.