Calculation of the matrix elements of the Coulomb interaction involving relativistic hydrogenic wave functions

• Published in: Computer physics communications Vol. 212; pp. 283 - 284
• Main Author: Sarkadi, L.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-03-2017
• Subjects: Coulomb interaction; Dirac–Coulomb functions; Ionization; Matrix elements; Multipole series expansion; Relativistic hydrogenic wave functions; Dirac–Coulomb functions; Multipole series expansion; Ionization; Relativistic hydrogenic wave functions; Coulomb interaction; Matrix elements
• ISSN: 0010-4655; 1879-2944
• DOI: 10.1016/j.cpc.2016.10.019

The program MTRDCOUL [1] calculates the matrix elements of the Coulomb interaction between a charged particle and an atomic electron, ∫ψf∗(r)∣R−r∣−1ψi(r)dr. Bound-free transitions are considered, and relativistic hydrogenic wave functions are used. In this revised version a bug discovered in the F3Y CPC Program Library subprogram [2] is fixed.
Program Title: MTRDCOUL
Program Files doi: “http://dx.doi.org/10.17632/4cmts2c49b.1 ”
Licensing provisions: GNU GPL v3
Programming language: Fortran 77
Journal reference of previous version: Comput. Phys. Commun. 141 (2001) 73
Does the new version supersede the previous version?: Yes
Reasons for the new version: In some applications MTRDCOUL led to unexpected results that were traced back to the erroneous execution of the subprogram F3Y [2]. For example, in some cases F3Y yielded completely different values for the inputs (l,m1,l,m2,l′,m′) and (l,m2,l,m1,l′,m′), while, for symmetry reason, one expects equal results. In the new version this error in F3Y was corrected.
Summary of revisions: In the line AAQQ0036 of the original F3Y program [2] the incorrect assignment A2=L1−M1−N2 was replaced by A2=L1−M1−N1. The corrected F3Y was tested by comparing its results with those obtained by a program written for the integral of the product of three spherical harmonics using the 369j program [3]. An agreement within 10−14 was found between the two calculations for all possible arguments of the function belonging to the values of li up to 10.
Nature of problem: The theoretical description of the excitation and ionization of atoms by charged particle impact often requires the knowledge of the matrix elements of the Coulomb interaction. Considering that the program can easily be extended to the calculations of matrix elements between wave functions other than the hydrogenic ones, it may find a broad application including the treatment of the electron–electron correlation problems.
Solution method: The algorithm is based on the multipole series expansion of the Coulomb potential.
Additional comments including Restrictions and Unusual features: The matrix elements are calculated with the following restrictions. The initial bound states are limited to 1s1/2, 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2. The quantum number l in the final state has a maximum value of 10.
Acknowledgments
This work was supported by the National Scientific Research Foundation (OTKA, Grant No. K109440). [1]L. Lugosi and L. Sarkadi, Comput. Phys. Commun. 141 (2001) 73.[2]A. Liberato de Brito, Comput. Phys. Commun. 25 (1982) 81.[3]L. Wei, Comput. Phys. Commun. 120 (1999) 222; Erratum: 182 (2011) 1199.Appendix
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