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Manuscript Title: Numerical and symbolic calculation of the multipole matrix elements in the axial and triaxial harmonic oscillator basis.
Authors: V. Martin, L.M. Robledo
Program title: MLM
Catalogue identifier: ADEF_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 99(1996)113
Programming language: Mathematica.
Computer: PC-Compatible.
Operating system: Windows 3.1/2.2.3, OS/2 3.0, AIX v4.1.4.
RAM: 8M words
Keywords: Atomic physics, Theoretical methods, Computer algebra, Symbolic computation, Multipole matrix elements, Harmonic oscillator basis, Axial and triaxial symmetry.
Classification: 2.9, 5.

Nature of problem:
The expressions of the multipole operators Qlambda mu are calculated in the axial and triaxial harmonic oscillator basis. The selection rules for the axial case are also implemented.

Solution method:
The numerical (a number for a given, fully determined, set of quantum numbers) and symbolic (an expression for a given set of symbols representing the quantum numbers and a numeric quantity that specifies the allowed variation of the symbols) algorithms are very different. Algorithms for both kind of problems are developed in this paper. The numerical one is adequate to be implemented in a nonrecursive way as in Fortran programs. The symbolic one is used here to calculate algebraic expressions, although it can also be utilized in the same way as the numerical algorithm if the language allows for recursion. In the standard (numerical) algorithm, the integrals which express the matrix elements are substituted by a finite sum with limits depending on numerically fixed quantum numbers. These limits cannot be specified if an expression instead of a number is sought. To circumvent this problem a recursive algorithm is used. In this case the dependence of the limits of the sums can be shifted from the actual numbers to their allowed variation. The resulting algorithm is able to produce expressions in terms of the symbols representing the quantum numbers.

Running time:
The basic commands - those providing an expression for a matrix element once a lambda, mu and fixed numerical variations of the symbols representing the quantum numbers are given - take just a few seconds. Full tables generated using the selection rules command depend heavily on the multipolarity of the operator.