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Manuscript Title: Z-expansion of matrix elements of one-electron operators for many- electron atoms.
Authors: M.N. Lewis
Catalogue identifier: AACH_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 18(1979)109
Programming language: Fortran.
Computer: CDC 6400.
Operating system: SCOPE.
RAM: 50K words
Word size: 60
Peripherals: magnetic tape.
Keywords: Atomic physics, Theory, Z-expansion, Perturbation theory, One-electron operators, Single particle Substitutions sums, Hydrogenic, Continuum singularity Integration, Structure.
Classification: 2.1.

Nature of problem:
Program SSTR (Substitution Sums for Transitions), computes the generalized single-particle substitution sums used in obtaining the coefficient of the second term in the Z-expansion of the matrix elements of one-electron operators in many-electron atoms. The calculations correspond in complexity to those needed for the contributions E'2 in the second-order energy, but not to those needed for E''2. The program is generalized to use one-electron operators corresponding to r**kk, for brevity referred to as a 'transition', from a state with quantum numbers n1l1 to state n2l2 in the presence of the n0l0 core electron.

Solution method:
Each generalized single-substitution sum is a sum of terms of the form R**k(alpha,beta,gamma,nl) T**kk(delta,nl) / E(alpha+beta-gamma-nl), with delta = alpha or beta or gamma. The substituted wave functions P(nl) are members of a complete set of hydrogenic wave functions and the generalized sums are found, as in program SPSS, by the summation over discrete n from n = l+1 to n = next-1, by extrapolation from n = next to infinity, and by integration over the continuum energy.

For high principal quantum numbers of the fixed wave functions it may be necessary to use double precision for parts of the calculations. For restrictions on the scaling factor SX in subroutine PRTX.

Unusual features:
Program SSTR selects the generalized sums needed for the 'transition' and can punch results on cards. SSTR could be adapted to obtain perturbation theory summations for problems with other definite hydrogenic integrals in the numerator of the expression to be summed.

Running time:
Transitions 3p-3s and 3d-3p with a 1s core electron take a total time of 3.7 s for 7 to 10 significant figures, or 2.3 s for 4 to 7 figures.