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Manuscript Title: A general program for computing matrix elements in atomic structure with non-orthogonal orbitals.
Authors: O. Zatsarinny
Program title: ZAP_NO
Catalogue identifier: ADDV_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 98(1996)235
Programming language: Fortran.
Computer: 386/486-BASED PCS.
Operating system: MS-DOS.
RAM: 229K words
Peripherals: disc.
Keywords: Matrix elements, Non-orthogonality, LS-coupling, Vector coupling coefficients, Complex atoms, Wave function, Bound state, Slater integrals, Multipole transitions.
Classification: 2.1.

Nature of problem:
In many atomic processes involving inner atomic shells, the relaxation of electron orbitals plays the important role. The relaxation can be most efficiently included into consideration by using the non-orthogonal orbitals for the initial and final state. This requires the evaluating of the matrix elements of various operators with respect to the non- orthogonal one-particle orbitals. Initially, these matrix elements can be expressed as weighted sums of relevant radial integrals, possibly multiplied by overlap integrals. The program computes all the arising coefficients of the radial integrals and the corresponding overlap factors.

Solution method:
At first, the configuration wave functions are expanded over the Slater determinants [1], using the combination of the vector coupling and fractional parentage methods. Then the coefficients of the radial integrals and their overlap factors are obtained from integration over all spin and angular coordinates for the separate Slater functions that is much more simple task. Intergration over the radial coordinates is defined either as unity, zero, or as expressin for radial integrals. Due to the additional task of finding the determinant expansion, the method used is more laborious than those based on the Racah techniques [2,3] and widely used now for calculating the matrix elements with orthogonal orbitals. On the other hand, the present technique admits a simple extension to the case of non-orthogonal orbitals by the most general way. Besides, considerable reduction of calculations has been achieved by using the tables of vector coupling coefficients for the individual subshells.

Restrictions:
Any number of s, p, or d electrons are allowed in a shell, but no more than two electrons or two holes in any shell of higher orbital angular momentum. In principle, the program admits the simple extension to the case of any f-subshell, but this requires the use of very large auxiliary file (about 2 Mbytes) for the corresponding vector coupling coefficients. LS-coupling is used, but its extension to jj-coupling is straightforward. The following parameters have also been adopted in the present program: maximum number of electrons is 80, maximum number of different subshells in one configuration is 20, and up to 20 configurations may be considered simultaneously in each run. These parameters may be augmented by changing dimension statements, if there is enough memory in the computer.

Running time:
The typical running time is 0.01 to 10 seconds on a 486-based PC for a single matrix element, and depends crucially on the complexity of configurations involved, primarily, on the size of determinantal expansion and the number of electrons.

References:
[1] E.U. Condon and G.H. Shortley, The theory of atomic spectra (Cambridge Univ. Press, 1935).
[2] U. Fano, Phys. Rev. A 140 (1965) 67.
[3] A. Hibbert and C. Froese Fischer, Comput. Phys. Commun. 64 (1991) 417.