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Manuscript Title: Relativistic wave and Green's functions for hydrogen-like ions.
Authors: P. Koval, S. Fritzsche
Program title: GREENS
Catalogue identifier: ADRJ_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 152(2003)191
Programming language: C++.
Computer: PC Pentium III, PC Athlon.
Operating system: Linux 6.1+, SuSe Linux 7.3, SuSe Linux 8.0, Windows 98.
RAM: 300K words
Word size: 8
Keywords: Confluent hypergeometric function, Coulomb-Green's function, Hydrogenic wave function, Kummer function, Nonrelativistic, Relativistic, Two-photon ionization cross section, Whittaker function, Atomic physics.
Classification: 2.7.

Nature of problem:
In order to describe and understand the behaviour of hydrogen-like ions, one often needs the Coulomb wave and Green's functions for the evaluation of matrix elements. But although these functions have been known analytically for a long time and within different representations [1,2], not so many implementations exist and allow for a simple access to these functions. In practice, moreover, the application of the Coulomb functions is sometimes hampered due to numerical instabilities.

Solution method:
The radial components of the Coulomb wave and Green's functions are implemented in position space, following the representation of Swainson and Drake [2]. For the computation of these functions, however, use is made of Kummer's functions of the first and second kind [3] which were implemented for a wide range of arguments. In addition, in order to support the integration over the Coulomb functions, an adaptive Gauss-Legendre quadrature has also been implemented within one and two dimensions.

Restrictions:
As known for the hydrogen atom, the Coulomb wave and Green's functions exhibit a rapid oscillation in their radial structure if either the principal quantum number or the (free-electron) energy increase. In the implementation of these wave functions, therefore, the bound-state functions have been tested properly only up to the principal quantum number n ~ 20, while the free-electron waves were tested for the angular momentum quantum numbers K <= 7 and for all energies in the range 0 ... 10|E1s|. In the computation of the two-photon ionization cross sections sigma2, moreover, only the long-wavelength approximation (e**iK.R ~ 1) is considered both, within the nonrelativistic and relativistic framework.

Unusual features:
Acces to the wave and Green's functions is given simply by means of the GREENS library which provides a set of C++ procedures. Apart from these Coulomb functions, however, GREENS also supports the computation of several special functions from mathematical physics (see section 2.4) as well as of two-photon ionization cross sections in long-wavelength approximation, i.e. for a very first application of the atomic Green's functions. Moreover, to facilitate the integration over the radial functions, an adaptive Gauss-Legendre quadrature has been also incorporated into the GREENS library.

Running time:
Time requirements critically depends on the quantum numbers and energies of the functions as well as on the requested accuracy in the case of a numerical integration. One value of the relativistic two-photon ionization cross section takes less or about one minute on a Pentium III 550 MHz processor.

References:
[1] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, (Kluwer Academic Publishers, 1977).
[2] R.A. Swainson and G.W.F. Drake, J. Phys. A 24 (1991) 95.
[3] M. Abramowitz and I.A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York 1965).