Computer Physics Communications Program Library

Programs in Physics & Physical Chemistry

PROGRAM SUMMARY

Manuscript Title: Algebraic tools for dealing with the atomic shell model. I. Wavefunctions and integrals for hydrogen-like ions.
Authors: Andrey Surzhykov, Peter Koval, Stephan Fritzsche
Program title: DIRAC
Catalogue identifier: ADUQ
Journal reference: Comput. Phys. Commun. 165(2005)139
Programming language: Maple 8 and 9.
Computer: All computers with a license for the computer algebra package Maple [1].
Keywords: Analytical solution, Coulomb-Green's function, Coulomb problem, Dirac equation, energy level, expectation value, hydrogen-like ion, hydrogenic wavefunction, matrix element, radial integral, special functions.
PACS: 01.50.Lc, 03.65.-w, 03.65.Pm, o3.65.Sq, 31.10.+z, 31.15.-p.
Classification: 2.1, 2.7.
Nature of problem: Analytical solutions of the hydrogen atom are widely used in very different fields of physics [2,3]. Despite the rather simple structure of the hydrogen-like ions, however, the underlying 'mathematics' is not always that easy to deal with. Apart from the well-known level structure of these ions as obtained from either the Schrodinger or Dirac equation, namely, a great deal of other properties are often needed. These properties are related to the interaction of bound electron(s) with external particles and fields and, hence, require to evaluate transition amplitudes, including wavefunctions and (transition) operators of quite dfferent complexity. Although various special functions, such as the Laguerre polynomials, spherical harmonics, Whittaker functions, or the hypergeometric functions of various kinds can be used in most cases in order to express these amplitudes in a concise form, their derivation is time consuming and prone to error. In addition to their complexity, moreover, there exist a large number of mathematical relations among these functions which are difficult to remember in detail and which have often hampered quantitative studies in the past.
Solution method: A set of Maple procedures is developed which provides both the nonrelativistic and relativistic (analytical) solutions of the `hydrogen atom model` and which facilitates the symbolic evaluation of various transition amplitudes.
Restrictions: Over the past decades, a large number of representations have been worked out for the hydrogenic wave and Green's functions, using different variables and coordinates [2]. From these, the position-space representation in spherical coordinates is certainly of most practical interest and has been used as the basis of the present implementation. No attempt has been made by us so far to provide the wave and Green's functions also in momentum space, for which the relativistic momentum functions would have to be constructed numerically. Although the DIRAC program supports both symbolic and numerical computations, the latter are based on MAPLE's standard software floating-point algorithms and on the (attempted) precision as defined by the global Digits variable. Although the default number, Digits = 10, appears sufficient for many computations, it often leads to a rather dramatic loss in the accuracy of the relativistic wave functions and integrals, mainly owing to MAPLE's imprecise internal evaluation of the corresponding special functions. Therefore, in order to avoid such computational difficulties, the Digits variable is set to 20 whenever the DIRAC program is (re-)loaded.
Unusual features: The DIRAC program has been designed for interactive work which, apart from the standard solutions and integrals of the hydrogen atom, also support the use of (approximate) semirelativistic wave functions for both, the bound- and continuum states of the electron. To provide a fast and accurate access to a number of radial integrals which arise frequently in applications, the analytical expressions for these integrals have been implemented for the one-particle operators r^k, e^-σr, d^m/dr^m, jL^(kr) as well as for the (so-called) two-particle Slater's integrals which are needed to describe the Coulomb repulsion among the electrons. Further procedures of the DIRAC program concern, for instance, the conversion of the physical results between different unit systems or for different sets of quantum numbers. A brief description of all procedures, as available in the present version of the DIRAC program, is given in the user manual Dirac-commands.pdf which is distributed together with the code.
Running time: Although the program replies promptly to most requests, the running time also depends on the particular task.
References:
[1]	Maple is a registered trademark of Waterloo Maple Inc.
[2]	H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, Berlin 1957.
[3]	J. Eichler and W. Meyerhof, Relativistic Atomic Collisions, Academic Press, New York, 1995