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Manuscript Title: HYDMATEL: a code to calculate matrix elements for hydrogen-like atoms.
Authors: M.L. Sanchez, A. Lopez Pineiro
Program title: HYDMATEL
Catalogue identifier: ACLO_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 75(1993)185
Programming language: Fortran.
Computer: UNISYS 5000/55.
Operating system: UNIX.
RAM: 1800K words
Word size: 32
Keywords: Atomic physics, Structure, Matrix elements, R-dependent operator, Hydrogenic wavefunctions.
Classification: 2.1.

Nature of problem:
Many problems in atomic physics need the calculation of matrix elements for r-dependent operators. The calculation of these elements could be carried out by resolving the integral analytically, but when n is much greater than l, Gordon's formula or other analytically expanded formulae are not only hard to use, but have no numerical validity for large n due to successive cancellations between positive and negative terms. Several alternative methods have therefore been developed: asymptotic expansions, relationships between the Laguerre polynomials, algebraic methods such as the joint use of the hypervirial theorem and the sum rule, hypervirial and Hellmann-Feynman theorems, group theory with second quantization formalism, and, finally, the use of ladder operators. None of them, however, has solved the most complicated and general case which is that in which the states involved differ in the principal quantum number.

Solution method:
Using a simple procedure, based on the joint use of the hypervirial theorem and of certain ladder operators, we have obtained recurrence relations that we have implemented in the program HYDMATEL. These expressions compute recursively matrix elements of the operators r**k, e**-sr, d**m/dr**m, or any combination of them.

Running time:
Running times depend on the principal quantum numbers involved and the power of the differential operator, m, but not on k and s. As an example, for n=10, calculation of all matrix elements < n'l'|d**m/dr**m|n l > requires 3.2 seconds, when m=0, and 4.5 seconds when m=5; for n=50, m=0 requires 14.8 seconds and 52.0 seconds when m=5. By contrast, 169.9 seconds and more than 24 hours are required for the m=0 matrix elements with n=10 and n=50 respectively, resolving the integral analytically.