Programs in Physics & Physical Chemistry
|[Licence| Download | New Version Template] adrj_v1_0.tar.gz(335 Kbytes)|
|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.|
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.
The radial components of the Coulomb wave and Green's functions are implemented in position space, following the representation of Swainson and Drake . For the computation of these functions, however, use is made of Kummer's functions of the first and second kind  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.
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.
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.
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.
|||H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, (Kluwer Academic Publishers, 1977).|
|||R.A. Swainson and G.W.F. Drake, J. Phys. A 24 (1991) 95.|
|||M. Abramowitz and I.A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York 1965).|
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