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Manuscript Title: Accurate numerical solution of the radial Schrodinger and Dirac wave equations.
Authors: F. Salvat, J.M. Fernandez-Varea, W. Williamson Jr
Program title: RADIAL
Catalogue identifier: ADBP_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 90(1995)151
Programming language: Fortran.
Computer: IBM 80486/66.
Operating system: MS-DOS Rel. 5.0.
RAM: 280K words
Word size: 8
Keywords: Schrodinger equation, Dirac equation, Central fields, Bound states, Eigenvalues, Free states phase shifts, General purpose, Differential equations.
Classification: 4.3.

Nature of problem:
This subroutine package provides numerical solutions of the Schrodinger and Dirac radial equations for central fields such that nu(r) equivalent r V(r) is finite for all r and tends to constant values when r -> 0 and r -> infinity. Normalized radial functions, eigenvalues for bound states and phase shifts for free states are calculated to a prescribed accuracy, which is specified by the input parameter epsilon.

Solution method:
The potential function nu(r) is represented by the natural cubic spline that interpolates a table introduced by the user. The radial functions are evaluated, for the cubic spline field, by means of their exact power series expansions. Free-state radial functions are normalized by matching the outward numerical solution with the "outer" solution, express as a linear combination of the regular and irregular Coulomb functions.

RADIAL may be unable to solve equations for states with energies that are too close to zero. In the case of weakly bound states, it is necessary to use a radial grid that is dense enough to separate consecutive zeros of the radial functions. The limitation for free states stems from difficulties in computing accurate Coulomb functions for r less than the Coulomb turning point; in practice, these types of difficulties are found only for very small energies, of the order of 0.01 atomic units or less.

Unusual features:
The spline-power series solution method permits a direct control of truncation errors. The input parameter epsilon governs the accuracy of power series summations; with epsilon = 10**-n, results with (n-1)- decimal-place accuracy are obtained. When using optimum accuracy (i.e. epsilon=10**-15 with double-precision arithmetic) truncation errors are effectively eliminated. Radial functions are evaluated at the points of a grid arbitrarily selected by the user, which may be different from the grid where the potential function is tabulated.

Running time:
The running time largely depends on the energy and the desired accuracy. The evaluation of the ground state of a screened Coulomb field with epsilon=10**-13 takes less than 5 seconds.