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arXiv · 2503.00977

Direct Summation of the Madelung Constant Using Axial Multipoles

Joven V. Calara·Jan D. Miller
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Abstract

We present an absolutely convergent real-space summation method for electrostatic potentials in ionic lattices. The method constructs the lattice from translated axial multipole units whose low-order moments are systematically eliminated, producing a prescribed rapid far-field decay. For the RU-13 construction, the leading contribution decays as r^(-13). In three-dimensional NaCl, this gives the Madelung constant to 13 decimal places using summation radii of only a few tens of nearest-neighbor spacings, and spherical and cubical growth geometries converge to the same limiting value. The same real-space construction is applied to nonpolar surface and edge geometries, off-lattice points, and interstitial positions. The Madelung constant of bulk CsCl was also determined with the same accuracy and convergence speed. The method is validated in higher dimensions (d<=6 in practical calculations).

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Joven V. Calara, Jan D. Miller. 2025-03-02. Direct Summation of the Madelung Constant Using Axial Multipoles. https://arxiv.org/abs/2503.00977

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