PROGRAM SUMMARY
Title of program:
GREENS
Catalogue identifier:
ADRJ
Ref. in CPC:
152(2003)191
Distribution format: tar gzip file
Operating system: Linux 6.1+,SuSe Linux7.3,SuSe Linux8.0,Windows98
High speed store required:
300K words
Number of bits in a word:
8
Number of lines in distributed program, including test data, etc:
29852
Keywords:
Confluent hypergeometric function, Coulomb-Green's function,
Hydrogenic wave function, Kummer function, Nonrelativistic,
Relativistic, Two-photon ionization cross section,
Whittaker function, Atomic physics.
Programming language used: C++
Computer:
PC Pentium III ,
PC Athlon .
Nature of physical 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.
Method of solution:
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.
Typical 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).