Exploiting underlying crystal lattice for efficient computation of Coulomb matrix elements in multi-million atoms nanostructures

Atomistic modeling of nanostructures often leads to computationally challenging problems involving millions of atoms and tens of thousands of Coulomb matrix elements. In our previous work, we presented a practical solution to this problem, where quasi-linear efficiency, both in time and memory, was obtained by utilizing the fast Fourier transform. Here, we present an updated version of our highly-parallelized computer program, named Coulombo-Lattice, that eliminates the necessity of introducing an auxiliary basis set for the wave-function transfer to the computational grid. Here, we instead exploit the properties of the underlying crystal lattice and run calculations on a regular three-dimensional grid superimposed on the original, lower-symmetry lattice. Due to removal of spurious interactions from other supercells, the resulting Coulomb matrix elements are, up to numerical precision, identical to those obtained by the direct summation O(N^2) method, yet our code maintains O(N log N) scaling. We illustrate our approach by calculations involving up to 1.7 million integrals, and number of atoms reaching up to 2.8 million, for the problem of dopant charging energy for a single phosphorus dopant embedded in a silicon lattice. Next, to emphasize the broad applicability of our code, we show the results for mixed zinc-blend/wurtzite lattice systems, also known as crystal phase quantum dots.

纳米结构的原子级建模往往会遭遇计算难度极大的问题，此类问题涉及数百万个原子与数万个库仑矩阵元（Coulomb matrix elements）。在既往研究工作中，我们曾提出针对该问题的实用解决方案，通过借助快速傅里叶变换（Fast Fourier Transform, FFT）实现了时间与内存层面的准线性效率。本文我们推出高度并行化计算机程序Coulombo-Lattice的更新版本，该程序无需引入辅助基组（auxiliary basis set）即可完成波函数向计算网格（computational grid）的传递。取而代之的是，我们利用了底层晶体晶格（crystal lattice）的固有特性，在叠加于原始低对称晶格的规则三维网格上开展计算。由于消除了来自其他超胞（supercells）的虚假相互作用，所得库仑矩阵元在数值精度范围内与直接求和O(N²)方法得到的结果完全一致，但我们的程序仍保持O(N log N)的时间复杂度缩放特性。我们针对嵌入硅晶格（silicon lattice）中的单个磷掺杂剂（phosphorus dopant）的掺杂电荷能（dopant charging energy）问题，开展了涉及多达170万个积分、原子数最高达280万的计算，以此验证了所提方法的有效性。为进一步凸显本程序的广泛适用性，我们还展示了混合闪锌矿（zinc-blend）/纤锌矿（wurtzite）晶格系统（即晶体相量子点（crystal phase quantum dots））的计算结果。