Neese Frank, Colinet Pauline, DeSouza Bernardo, Helmich-Paris Benjamin, Wennmohs Frank, Becker Ute
Department of Molecular Theory and Spectroscopy, Max-Planck-Institut für Kohlenforschung, D-45470 Mülheim an der Ruhr, Germany.
FAccTs GmbH, Rolandstraße 67, 50667 Köln, Germany.
J Phys Chem A. 2025 Mar 13;129(10):2618-2637. doi: 10.1021/acs.jpca.4c07415. Epub 2025 Mar 3.
In this work, we describe the development of a new algorithm for the computation of Coulomb-type matrices using the well-known resolution of the identity (RI) or density fitting (DF) approximation. The method is linear-scaling with respect to system size and computationally highly efficient. For small molecules, it performs almost as well as the Split-RI-J algorithm (which might be the most efficient RI-J implementation to date), while outperforming it for larger systems with about 300 or more atoms. The method achieves linear scaling through multipole approximations and a hierarchical treatment of multipoles. However, unlike in the fast multipole method (FMM), the algorithm does not use a hierarchical boxing algorithm. Rather, close-lying objects like auxiliary basis shells and basis set shell pairs are grouped together in spheres that enclose the set of objects completely, which includes a new definition of the shell-pair extent that defines a real-space radius outside of which a given shell pair can be safely assumed to be negligible. We refer to these spheres as "bubbles" and therefore refer to the algorithm as the "Bubblepole" (BUPO) algorithm, with the acronym being RI-BUPO-J. The bubbles are constructed in a way to contain a nearly constant number of objects such that a very even workload arises. The hierarchical bubble structure adapts itself to the molecular topology and geometry. For any target object (shell pair or auxiliary shell), one might envision that the bubbles "carve" out what might be referred to as a "far-field surface". Using the default settings determined in this work, we demonstrate that the algorithm reaches submicro-Eh and even nano-Eh accuracy in the total Coulomb energy for systems as large as 700 atoms and 7000 basis functions. The largest calculations performed (the crambin protein solvated by 500 explicit water molecules in a triple-ζ basis) featured more than 2000 atoms and more than 33,000 basis functions.
在这项工作中，我们描述了一种用于计算库仑型矩阵的新算法的开发，该算法使用了著名的单位分解（RI）或密度拟合（DF）近似。该方法相对于系统大小呈线性缩放，计算效率极高。对于小分子，它的性能几乎与Split-RI-J算法（可能是迄今为止最有效的RI-J实现方式）相当，而在具有约300个或更多原子的较大系统中则表现更优。该方法通过多极近似和多极的分层处理实现线性缩放。然而，与快速多极方法（FMM）不同，该算法不使用分层装箱算法。相反，诸如辅助基壳和基组壳对之类的紧邻对象被分组在完全包围该组对象的球体中，这包括对壳对范围的新定义，该定义定义了一个实空间半径，在该半径之外可以安全地假定给定的壳对可忽略不计。我们将这些球体称为“气泡”，因此将该算法称为“气泡极”（BUPO）算法，其首字母缩写为RI-BUPO-J。气泡的构建方式是包含几乎恒定数量的对象，从而产生非常均匀的工作量。分层气泡结构会根据分子拓扑和几何形状进行自我调整。对于任何目标对象（壳对或辅助壳），可以设想气泡“雕刻”出所谓的“远场表面”。使用在这项工作中确定的默认设置，我们证明该算法在总库仑能量方面对于多达700个原子和7000个基函数的系统达到了亚微埃哈甚至纳埃哈的精度。所进行的最大计算（用三重ζ基组中的500个显式水分子溶剂化的克拉宾蛋白）包含超过2000个原子和超过33000个基函数。
