RADIAL: A Fortran subroutine package for the solution of the radial Schrödinger and Dirac wave equations

Abstract

The Fortran subroutine package RADIAL for the numerical solution of the Schrödinger and Dirac wave equations of electrons in central potentials V(r) is described. The considered potentials are such that the function V(r) ≡ rV (r) is finite for all r and tends to constant values when r → 0 and r → ∞. This includes finite-range potentials as well as combinations of Coulomb and finite-range potentials. The function V(r) used in the numerical calculation is the natural cubic spline that interpolates a table of values provided by the user. The radial wave equations are solved by using piecewise exact power series expansions of the radial functions, which are summed up to the prescribed accuracy so that truncation errors can be completely avoided. The RADIAL subroutines compute radial wave functions, eigenvalues for bound states and phase shifts for free states. Specific subroutines are also provided for computing the radial functions and phase shifts for free states of complex optical potentials having a finite-range absorptive imaginary part. The solution subroutines are accompanied by example main programs, as well as with specific programs that perform calculations relevant in atomic, nuclear, and radiation physics (the self-consistent solution of the Dirac-Hartee-Fock-Slater equations for neutral atoms and positive ions, and the calculation of cross sections for elastic scattering of high-energy electrons and positrons by atoms and of nucleons by nuclei). The distribution package includes a detailed manual with a description of the basic physics and the mathematical formulas implemented in the subroutines.