VPA: Computer program for the computation of the phase shift in atom–atom potential scattering using the Variable Phase Approach

Abstract

A computer code for the computation of the phase shift in atom–atom and electron–atom potential scattering is presented. The phase shift is the central quantity required for the calculation of all types of scattering cross sections. The program uses the Variable Phase Approach (VPA). This is the only exact method for the direct calculation of the scattering phase shift. All other methods are based on examining the large distance behavior of the exact solution of the Schrödinger equation. Such methods yield the phase shift only modulo π. The absolute value of the phase shift and its variation with scattering energy is, however, needed for a full understanding of the scattering process, such as for instance in the study of shape resonances and Glory oscillations. The VPA has been sparingly used owing to the instability of the underlying equations and the consequent difficulty of writing computer code to solve them. We present a computer code for the efficient implementation of the VPA method for atom–atom scattering problems over a wide range of scattering energies. The code works for potentials which are singular and for those that are non-singular at the origin. An example of the implementation of the code is given for both an interaction potential with an attractive well and for a purely repulsive potential. Program summaryProgram title: VPA:Variable Phase ApproachCPC Library link to program files: https://dx.doi.org/10.17632/zh248d2362.1Code Ocean capsule: https://dx.doi.org/10.24433/CO.8146850.v1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: To compute the quantum mechanical phase shift in an atom–atom or an electron–atom potential scattering problem. The interaction potential, dependent on a single radial coordinate, is defined by the user.Solution method: The Variable Phase Approach is used. This is the only available method for the direct calculation of the phase shift. The initial internuclear separation for the integration is chosen with the help of the analytical solution for a hard sphere core for all orbital angular momentum quantum numbers L of interest. The first-order nonlinear equation is integrated numerically with a modified LSODA19,20 program. The analytic formula for calculation of the Jacobian of the equation is also provided.Additional comments including restrictions and unusual features: Computer codes for the atom–atom interaction potential, POTENT1, and for its derivative with respect to the inter nuclear separation, DPOTENT1, must be supplied by a user. The calculation of the spherical Bessel and Neumann functions required is performed explicitly for large l angular momentum quantum numbers where the standard upward iterative generation of the functions becomes unstable.