SIAM Journal on Applied Mathematics, Volume 81, Issue 2, Page 694-717, January 2021.
The electrostatic fields and potentials associated with neutral periodic crystals are defined by sums that are not absolutely convergent. The sums depend on the order of summation. The mean-zero periodic solution ${\underline \phi}$ of Poisson's equation provides a natural potential and electric field. This sum is inconsistent with the electric field in ferroelectric materials. We introduce summation methods based on the concept of neutral charge groups, a notion common in computational chemistry. If a charge group has moments of order $<\kappa$ vanishing, then the sum for $\partial^\alpha \phi$ defined by summing first by charge groups is absolutely convergent for $|\alpha|\ge 3-\kappa$. In the borderline case $|\alpha| = 2-\kappa$, consider charge groups coming from primitive unit cells. For a macroscopic shape $\Omega$, the unit-cell $R\,\Omega$ algorithm sums over unit cells in $R\,\Omega$ and then takes the limit $R\to \infty$. This yields an answer for $\partial^\alpha\phi$ that depends on $\Omega$ and the unit cell. The limits differ from $\partial^\alpha{\underline \phi}$ by a constant given as an easily approximable integral. For quadrupolar primitive unit cells, $\kappa=2$. The unit-cell $R\,\Omega$ algorithm gives a potential $\phi$ that differs from ${\underline \phi}$ by a constant. This constant enters additively in the energy content per unit volume. If one could grow or cut crystals respecting these unit cells, this shape-dependent energy effect could be verified experimentally. For dipolar primitive unit cells, $\kappa=1$. Given a shape $\Omega$ and unit cell, the algorithm yields a well-defined electric field that differs from the mean-zero periodic field by a constant vector. If one could grow or cut crystals respecting these dipolar unit cells, this shape-dependent ferroelectric effect [C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, New York, 2004] could be verified experimentally.

中文翻译：

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周期性晶体静电的电荷基团求和方法

SIAM应用数学杂志，第81卷，第2期，第694-717页，2021年1月。
与中性周期晶体相关的静电场和电势由并非绝对收敛的和定义。总和取决于求和的顺序。泊松方程的均值零周期解$ {\下划线\ phi} $提供了自然势和电场。该总和与铁电材料中的电场不一致。我们介绍基于中性电荷基团的概念的求和方法，中性电荷基团是计算化学中的常见概念。如果某个计费组的阶次$ <\ kapp $消失，则首先由计费组求和定义的$ \ partial ^ \ alpha \ phi $的总和对于$ | \ alpha | \ ge 3- \ kappa绝对收敛$。在临界情况下，$ | \ alpha | = 2- \ k $，请考虑来自原始单位单元的电荷组。对于宏观形状$ \ Omega $，单元格$ R \，\ Omega $算法对$ R \，\ Omega $中的单元格求和，然后将限制$ R \限制为\ infty $。这将产生$ \ partial ^ \ alpha \ phi $的答案，该答案取决于$ \ Omega $和单位单元格。限制与$ \ partial ^ \ alpha {\ underline \ phi} $的区别在于，常量为易于近似的整数。对于四极基本单元，$ \ kappa = 2 $。单元格$ R \，\ Omega $算法给出的潜在$ \ phi $与$ {\下划线\ phi} $相差一个常数。该常数累加输入每单位体积的能量含量。如果可以根据这些晶胞生长或切割晶体，则可以通过实验验证这种形状依赖性能量效应。对于偶极基本单元，$ \ kappa = 1 $。给定形状$ \ Omega $和晶胞，该算法会产生一个定义明确的电场，该电场与均值零周期场相差一个常数向量。如果一个人可以生长或切割与这些偶极子晶胞有关的晶体，则这种形状依赖的铁电效应[C。Kittel，《固态物理导论》，第8版，纽约，威利，2004年]可以通过实验进行验证。