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Manuscript Title: The two-electron atomic systems. S-states.
Authors: Evgeny Z. Liverts, Nir Barnea
Program title: TwoElAtom-S
Catalogue identifier: AEFK_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 181(2010)206
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
Keywords: energies, wave functions, helium-like ions, matrix, determinant.
PACS: 31.15.A-, 31.15.ac, 31.15.ae, 31.15.-p.
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)