The two-electron atomic systems. S-states

Evgeny Z. Liverts*, Nir Barnea

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


A simple Mathematica program for computing the S-state energies and wave functions of two-electron (helium-like) atoms (ions) is presented. The well-known method of projecting the Schrödinger equation onto the finite subspace of basis functions was applied. The basis functions are composed of the exponentials combined with integer powers of the simplest perimetric coordinates. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of results and computation time depend on the basis size. The precise energy values of 7-8 significant figures along with the corresponding wave functions can be computed on a single processor within a few minutes. The resultant wave functions have a simple analytical form consisting of elementary functions, that enables one to calculate the expectation values of arbitrary physical operators without any difficulties. Program summary: Program title: TwoElAtom-S. Catalogue identifier: AEFK_v1_0. Program summary URL: Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland. Licensing provisions: Standard CPC licence, No. of lines in distributed program, including test data, etc.: 10 185. No. of bytes in distributed program, including test data, etc.: 495 164. Distribution format: tar.gz. Programming language: Mathematica 6.0; 7.0. Computer: Any PC. Operating system: Any which supports Mathematica; tested under Microsoft Windows XP and Linux SUSE 11.0. RAM:≥ 109 bytes. Classification: 2.1, 2.2, 2.7, 2.9. 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 basis set in three perimetric coordinates. Method of projecting the two-electron Schrödinger equation (for atoms/ions) onto a subspace of the basis functions enables one to obtain the set of homogeneous linear equations F . C = 0 for the coefficients C of the above expansion. The roots of equation det (F) = 0 yield the bound energies. Restrictions: First, the too large length of expansion (basis size) takes the too large computation time giving no perceptible improvement in accuracy. Second, the order of polynomial Ω (input parameter) in the wave function expansion enables one to calculate the excited nS-states up to n = Ω + 1 inclusive. Additional comments: The CPC Program Library includes "A program to calculate the eigenfunctions of the random phase approximation for two electron systems" (AAJD). It should be emphasized that this fortran code realizes a very rough approximation describing only the averaged electron density of the two electron systems. It does not characterize the properties of the individual electrons and has a number of input parameters including the Roothaan orbitals. Running time: ∼10 minutes (depends on basis size and computer speed).

Original languageAmerican English
Pages (from-to)206-212
Number of pages7
JournalComputer Physics Communications
Issue number1
StatePublished - Jan 2010


  • Determinant
  • Excited energies
  • Ground state energy
  • Helium-like ions
  • Matrix
  • Wave functions


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