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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.

Restrictions:
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.