周期系统的Clifford边界条件：1、2、3维立方晶体的Madelung常数
Clifford Boundary Conditions for Periodic Systems: the Madelung Constant of Cubic Crystals in 1, 2 and 3 Dimensions
摘要：
在这项工作中，我们证明了鲁棒性的实空间方法治疗的无限系统所描述的周期性边界条件，我们最近提出的[J。17，7090]。在我们的方法中，我们提取片段，即，一个超胞，然后将其拓扑修改为Clifford环面的拓扑，Clifford环面是一个平坦的、有限的、无边界的流形。然后，我们通过将两点之间的距离定义为Clifford环面的嵌入空间中的欧氏距离来重新归一化两点之间的距离。用我们的方法，我们已经能够计算出文献中的参考结果具有显着的准确性，并在一个非常低的计算工作量。在这项工作中，我们表明，我们的方法是强大的超级细胞的形状。特别是，我们表明，马德隆常数收敛到相同的值，但收敛性质是不同的。我们的方法与原子数量呈线性关系。在笔记本电脑上计算马德隆常数只需几秒钟，相对精度约为10 $^{-6}$。
英文摘要：
In this work we demonstrate the robustness of a real-space approach for the treatment of infinite systems described with periodic boundary conditions that we have recently proposed [J. Phys. Chem. Lett. 17, 7090]. In our approach we extract a fragment, i.e., a supercell, out of the infinite system, and then modifying its topology into the that of a Clifford torus which is a flat, finite and border-less manifold. We then renormalize the distance between two points by defining it as the Euclidean distance in the embedding space of the Clifford torus. With our method we have been able to calculate the reference results available in the literature with a remarkable accuracy, and at a very low computational effort. In this work we show that our approach is robust with respect to the shape of the supercell. In particular, we show that the Madelung constants converge to the same values but that the convergence properties are different. Our approach scales linearly with the number of atoms. The calculation of Madelung constants only takes a few seconds on a laptop computer for a relative precision of about 10$^{-6}$.
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