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Manuscript Title: A general program for computing angular integrals of the Breit-Pauli Hamiltonian with non-orthogonal orbitals.
Authors: O. Zatsarinny, C.F. Fischer
Program title: BREIT_NO
Catalogue identifier: ADLA_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 124(2000)247
Programming language: Fortran.
Computer: Pentium-based PCs.
Operating system: Windows 97.
RAM: 2M words
Peripherals: disc.
Keywords: Atomic physics, Structure, Matrix elements, Non-orthogonality, Configuration, Interaction, Lsj-coupling, Relativistic shift, Spin-orbit interaction, Fine-structure, Splitting, Complex atoms, Functions wave, Bound states.
Classification: 2.1.

Nature of problem:
In many atomic processes involving inner atomic shells, the relaxation of electron orbitals plays an important role. The relaxation can be most naturally taken into consideration by using non-orthogonal orbitals for the initial and final state. In atomic structure calculations, the rate of convergence of the many-configuration expansion to an acceptable accuracy may be much faster if the orbitals associated with different configurations and terms are not necessarily required to be orthogonal. This requires the evaluation of 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 radial integrals, possibly multiplied by overlap integrals. The program computes all coefficients of the radial integrals and the corresponding overlap factors for matrix elements of the Breit-Pauli Hamiltonian, with any amount of non-orthogonality between the involved orbitals.

Solution method:
At first, the configuration wave functions are expanded over the Slater determinants [1], using a combination of vector coupling and fractional parentage methods [2]. Then the coefficients of the radial integrals and corresponding overlap factors are obtained from integration over all spin and angular coordinates for the separate determinants [3]. Due to the additional task of finding the determinant expansion, the method used is more laborious than those based on the Fano-Racah technique [4] and those widely used now for calculating the matrix elements with orthogonal orbitals [5,6]. On the other hand, the present technique admits a simple extension to the case of non-orthogonal orbitals in the most general way. Besides, a considerable reduction of computation can be achieved by reusing the previously obtained data allowed by the present non-orthogonal technique.

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. Only the shells outside the set of closed shells (core) common to all configurations must be specified. The core orbitals are assumed to be orthogonal to all others, and the interaction with the core is included through a redefinition of the radial integrals. A maximum 8 shells (in addition to the core) are allowed, with the maximum number of electrons equal to 80. These restrictions can be eliminated by changing the corresponding parameters and some format statements.

Running time:
The typical running time is 0.001 to 10 seconds for a single matrix element on a 200 Mhz Pentium-based PC, and depends crucially on complexity of the involved configurations, primarily, on the size of determinant expansions and the number of electrons.

[1] E.U. Condon and G.H. Shortley, The theory of atomic spectra (Cambridge Univ. Press, 1935).
[2] O. Zatsarinny, Comput. Phys. Commun. 98 (1996) 235.
[3] P.-O. Lowdin, Phys. Rev. 97 (1955) 1474.
[4] U. Fano, Phys. Rev. A 140 (1965) 67.
[5] A. Hibbert and C. Froese Fischer, Comput. Phys. Commun. 64 (1991) 417.
[6] A. Hibbert, R. Glass and C. Froese Fischer, Comput. Phys. Commun. 64 (1991) 455.