A program for simplifying summation of Wigner 3[formula omitted]-symbols

Abstract

The program presented in this paper aims to simplify formulas involving products of a large amount of Wigner 3j-symbols summed over various magnetic quantum numbers. The algorithm used in our program is based on the graphical techniques originally developed by Yutsis, Levinson, and Vanagas, and later revised by some others. The output of our program is expressed as a weighted sum of products of 3j-, 6j- and 9j-symbols, with appropriate phase factors embedded. Compared with the previous literature, the present program is more flexible in a sense that it can deal with not only closed diagrams, such as the angular momentum recoupling coefficients, but also open diagrams. Our program is particularly useful to high precision atomic and molecular structure calculations using correlated electron–electron coordinates. Program summaryProgram Title: Reduce3jCPC Library link to program files:https://doi.org/10.17632/9zmx4b5sdd.1Licensing provisions: BSD 3-clauseProgramming language: Python 3.6.5Nature of problem: In atomic and molecular structure calculations, one often encounters expressions containing products of a large amount of Wigner 3j-symbols summed over various magnetic quantum numbers. It is desirable to simplify these expressions to make the final results compact, which is the goal of this paper.Solution method: The program uses the graphical techniques of quantum mechanical angular momentum theory, originally developed by Yutsis, Levinson, and Vanagas, and later modified by Brink and Satchler. The program is implemented in Python language, with its package “Networkx” extensively used. A friendly graphical input and output interface is developed. The input can be either a recoupling coefficient or an arbitrary expression containing products of the 3j-symbols summed over various magnetic quantum numbers. A strategically designed optimization method is also developed to help achieve our goal. Our program has significantly expanded the scope of usefulness of the existing programs so that it can now meet most of the demands from high precision atomic and molecular structure calculations.Additional comments including restrictions and unusual features: Reduce3j can effectively handle any open or closed graph, as long as the graph has more than one free line, and all the magnetic angular momentum quantum numbers from internal lines are summed over.