S-states of helium-like ions

Abstract

A simple Mathematica (version 7) code for computing S-state energies and wave functions of two-electron (helium-like) ions is presented. The elegant technique derived from the classical papers of Pekeris is applied. The basis functions are composed of the Laguerre functions. The method is based on the perimetric coordinates and specific properties of the Laguerre polynomials. Direct solution of the generalized eigenvalues and eigenvectors problem is used, distinct from the Pekeris works. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of the results and computation times depend on the basis size. The ground state and the lowest triplet state energies can be computed with a precision of 12 and 14 significant figures, respectively. The accuracy of the higher excited states calculations is slightly worse. The resultant wave functions have a simple analytical form, that enables calculation of expectation values for arbitrary physical operators without any difficulties. Only three natural parameters are required in the input. The new version of Mathematica code takes into account the fact that the negative hydrogen ion has only one bound state. New version program summary Program title: TwoElAtomSL(SH) Catalogue identifier: AEHY_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHY_v2_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 27 998 No. of bytes in distributed program, including test data, etc.: 286 543 Distribution format: tar.gz Programming language: Mathematica 7.0 and 8.0 Computer: Any PC with a Mathematica installation Operating system: Any which supports Mathematica; tested under Microsoft Windows XP and Linux SUSE 11.0 RAM: ⩾ 10 9 bytes Classification: 2.1, 2.2, 2.7, 2.9 Catalogue identifier of previous version: AEHY_v1_0 Journal reference of previous version: Comput. Phys. Comm. 182 (2011) 1790 Does the new version supersede the previous version?: Yes Nature of problem: The Schrödinger equation for atoms (ions) with more than one electron has not been solved analytically. Approximate methods must be applied in order to obtain the wave functions or other physical attributes from quantum mechanical calculations. Solution method: The S-wave function is expanded into a triple set of basis functions which are composed of the exponentials combined with the Laguerre polynomials in the perimetric coordinates. Using specific properties of the Laguerre polynomials, solution of the two-electron Schrödinger equation reduces to solving the generalized eigenvalues and eigenvector problem for the proper Hamiltonian. The unknown exponential parameter is determined by means of minimization of the corresponding eigenvalue (energy). Reasons for new version: The need to take into account the fact that the negative hydrogen ion ( Z = 1 ) has only one bound (ground) state. Summary of revisions: Minor amendments were made in Cell 2 and Cell 5 of both TwoElAtomSH and TwoElAtomSL programs. Restrictions: Firstly, the too large length of expansion (basis size) takes too much computation time and operative memory giving no perceptible improvement in accuracy. Secondly, the number of shells Ω in the wave function expansion enables one to calculate the excited nS-states up to n = Ω + 1 inclusive. Running time: 2–60 minutes (depends on basis size and computer speed).