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 summary

Program Title: Reduce3j

CPC Library link to program files: https://doi.org/10.17632/9zmx4b5sdd.1

Licensing provisions: BSD 3-clause

Programming language: Python 3.6.5

Nature 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.

中文翻译：

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一个简化Wigner 3求和的程序$Ĵ$-符号

本文介绍的程序旨在简化涉及大量威格纳产品的公式 $3Ĵ$-符号累加了各种磁性量子数。我们程序中使用的算法基于最初由Yutsis，Levinson和Vanagas开发的图形技术，后来又被其他一些技术修改。我们程序的输出表示为的乘积的加权和$3Ĵ$-， $6Ĵ$- 和 $9Ĵ$-符号，并嵌入适当的相位因子。与以前的文献相比，本程序在某种意义上更加灵活，它不仅可以处理诸如角动量耦合系数之类的闭合图，而且还可以处理开放图。我们的程序对于使用相关的电子-电子坐标进行的高精度原子和分子结构计算特别有用。

计划摘要

节目名称： Reduce3j

CPC库链接到程序文件： https : //doi.org/10.17632/9zmx4b5sdd.1

许可条款： BSD 3条款

编程语言： Python 3.6.5

问题的性质：在原子和分子结构计算中，经常遇到包含大量维格纳乘积的表达式$3Ĵ$-符号累加了各种磁性量子数。希望简化这些表达式以使最终结果紧凑，这是本文的目标。

解决方法：该程序使用量子力学角动量理论的图形技术，该技术最初是由Yutsis，Levinson和Vanagas开发的，后来由Brink和Satchler进行了修改。该程序以Python语言实现，其软件包“ Networkx”得到广泛使用。开发了友好的图形输入和输出界面。输入可以是重新耦合系数，也可以是包含以下乘积的任意表达式：$3Ĵ$-符号累加了各种磁性量子数。还开发了战略设计的优化方法来帮助实现我们的目标。我们的程序大大扩展了现有程序的实用范围，因此它现在可以满足高精度原子和分子结构计算的大多数要求。

包括限制和异常功能在内的其他注释： Reduce3j可以有效地处理任何打开或关闭的图形，只要该图形具有多个自由线，并且将来自内部线的所有磁角动量量子数求和即可。