Fast convergent series are presented for lattice sums associated with the simple cubic, face-centered cubic, body-centered cubic, and hexagonal close-packed structures for interactions described by an inverse power expansion in terms of the distances between the lattice points, such as the extended Lennard-Jones potential. These lattice sums belong to a class of slowly convergent series, and their exact evaluation is related to the well-known number-theoretical problem of finding the number of representations of an integer as a sum of three squares. We review and analyze this field in some detail and use various techniques such as the decomposition of the Epstein zeta function introduced by Terras or the van der Hoff–Benson expansion to evaluate lattice sums in three dimensions to computer precision.

中文翻译：

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立方和六方密排结构的快速收敛晶格和的分析方法

对于与简单立方、面心立方、体心立方和六方密排结构相关的晶格和，提出了快速收敛级数，这些结构的相互作用由晶格点之间的距离的逆幂展开描述，例如扩展的伦纳德-琼斯势。这些格和属于一类缓慢收敛的级数，它们的准确评估与著名的数论问题有关，即寻找一个整数表示为三个平方和的次数。我们详细地回顾和分析了这个领域，并使用各种技术，例如 Terras 引入的 Epstein zeta 函数的分解或 van der Hoff-Benson 展开来评估三个维度的晶格和，以达到计算机精度。