Clifford边界条件：Madelung常数的简单直接和计算
Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants
摘要：
我们提出了一个简单的直接和方法的有效评估格和周期固体。它包括两个主要原则：i）创建一个具有Clifford torus拓扑结构的超胞，Clifford torus是一个平坦的，有限的和无边界的流形; ii）通过将Clifford torus嵌入空间中的欧氏距离定义为Clifford torus上两点之间的距离的重正化。我们的方法不需要任何积分变换，也不需要对电荷进行重整化。我们说明了我们的方法，将其应用于计算的马德隆常数的离子晶体。我们表明，朝着无限大的系统的收敛是单调的，这使得一个简单的外推的马德隆常数。我们能够以惊人的精度恢复马德隆常数，并且计算成本几乎可以忽略不计，即，在笔记本电脑上几秒钟。
英文摘要：
We propose a simple direct-sum method for the efficient evaluation of lattice sums in periodic solids. It consists of two main principles: i) the creation of a supercell that has the topology of a Clifford torus, which is a flat, finite and border-less manifold; ii) the renormalization of the distance between two points on the Clifford torus by defining it as the Euclidean distance in the embedding space of the Clifford torus. Our approach does not require any integral transformations nor any renormalization of the charges. We illustrate our approach by applying it to the calculation of the Madelung constants of ionic crystals. We show that the convergence towards the system of infinite size is monotonic, which allows for a straightforward extrapolation of the Madelung constant. We are able to recover the Madelung constants with a remarkable accuracy, and at an almost negligible computational cost, i.e., a few seconds on a laptop computer.
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