Atomic multipole moments associated with a spherical volume fully residing within a topological atom (i.e., the β sphere) can be obtained analytically. Such an integration is thus free of quadrature grids. A general formula for an arbitrary rank spherical harmonic multipole moment is derived, for an electron density comprising Gaussian primitives of arbitrary angular momentum. The closed expressions derived here are also sufficient to calculate the electrostatic potential, the two types of kinetic energy, as well as the potential energy between atoms. Some integrals have not been solved explicitly before but through recursion and substitution are broken down to more elementary listed integrals. The proposed method is based on a central formula that shifts Gaussian primitives from one center to another, which can be derived from the well‐known plane‐wave expansion (or Rayleigh equation). © 2018 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.

中文翻译：

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对拓扑原子中 β 球体 3D 体积的性质进行完全分析整合

与完全位于拓扑原子（即 β 球体）内的球形体积相关的原子多极矩可以通过分析获得。因此，这种集成没有正交网格。对于包含任意角动量的高斯基元的电子密度，推导出任意秩球谐多极矩的一般公式。这里导出的闭合表达式也足以计算静电势，两种动能，以及原子之间的势能。一些积分以前没有明确解决，但通过递归和替换被分解为更基本的列出积分。所提出的方法基于将高斯基元从一个中心移动到另一个中心的中心公式，这可以从众所周知的平面波展开（或瑞利方程）推导出来。© 2018 作者。计算化学杂志由 Wiley Periodicals, Inc. 出版。