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Manuscript Title: Relativistic central-field Green's functions for the RATIP package
Authors: Peter Koval, Stephan Fritzsche
Program title: Xgreens
Catalogue identifier: ADWM_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 172(2005)187
Programming language: ANSI standard Fortran 90/95.
Computer: PC Pentium II, III, IV, Athlon.
Operating system: SuSE Linux 8.2, SuSE Linux 9.0.
RAM: On a standard grid (400 nodes), one central-field Green's function requires about 50 kBytes in RAM while approximately 3 MBytes are needed if saved as two-dimensional array on some external disc space.
Word size: Real variables of double- and quad-precision are used.
Keywords: Central-field Green's function, confluent hypergeometric function, Coulomb Green's function, Kummer function, multi-configuration Dirac-Fock, regular and irregular solutions.
PACS: 32.10, 32.80, 32.80Fb, 32.80Wr.
Classification: 2.7.

Nature of problem:
In atomic perturbation theory, Green's functions may help carry out the summation over the complete spectrum of atom and ions, including the (summation over the) bound states as well as an integration over the continuum [1]. Analytically, however, these functions are known only for free electrons (V(r) ≡ 0) and for electrons in a pure Coulomb field (V(r) = -Z/r). For all other choices of the potential, in contrast, the Green's functions must be determined numerically.

Solution method:
Relativistic Green's functions are generated for an arbitrary central-field potential V(r) = -Z(r)/r by using a piecewise linear approximation of the effective nuclear charge function Z(r) on some grid ri (i = 1, ..., N): Zi(r) = Z0i + Z1ir. Then, following McGuire's algorithm [2], the radial Green's functions are constructed from the (two) linear-independent solutions of the homogeneous equation [3]. In the computation of these radial functions, the Kummer and Tricomi functions [4] are used extensively.

Restrictions:
The main restrictions of the program concern the shape of the effective nuclear charge Z(r) = -rV(r), i.e. the choice of the potential, and the allowed energies. Apart from obeying the proper boundary conditions for a point-like nucleus, namely, Z(r → 0) = Znuc > 0 and Z(r → ∞) = Znuc - Nelectrons ≥ 0, the first derivative of the charge function Z(r) must be smaller than the (absolute value of the) energy of the Green's function, δZ(r)/δr < |E|.

Unusual features:
Xgreens has been designed as a part of the RATIP package [5] for the calculation of relativistic atomic transition and ionization properties. In a short dialog at the beginning of the execution, the user can specify the choice of the potential as well as the energies and the symmetries of the radial Green's functions to be calculated. Apart from central-field Green's functions, of course, the Coulomb Green's function [6] can also be computed by selecting a constant nuclear charge Z(r) = Zeff. In order to test the generated Green's functions, moreover, we compare the two lowest bound-state orbitals which are calculated from the Green's functions with those as generated separately for the given potential. Like the other components of the RATIP package, Xgreens makes careful use of the Fortran 90/95 standard.

Running time:
2 minutes on a 450 MHz Pentium III processor.

References:
[1] R.A. Swainson and G. W. F. Drake, J. Phys.A 24(1991)95.
[2] E.J. McGuire, Phys. Rev. A 23(1981)186.
[3] P.Morse and H.Feshbach, Methods of Theoretical Physics McGraw-Hill, New York 1953, (Part.1, p. 825).
[4] J. Spanier and B. Keith, An Atlas of Functions Springer, New York, 1987.
[5] S. Fritzsche, J. Elec. Spec. Rel. Phen. 114-116 (2001) 1155.
[6] P. Koval and S. Fritzsche, Comput. Phys. Commun. 152 (2003) 191.