MTA-BME Lendület Quantum Chemistry Research Group, Department of Physical Chemistry and Materials Science, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary.
J Chem Phys. 2018 Sep 28;149(12):124101. doi: 10.1063/1.5049529.
The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.
已对三中心电子排斥积分 (ERI) 的收缩球谐型高斯函数的几何导数的计算进行了优化。我们比较了基于 Obara-Saika、McMurchie-Davidson、Gill-Head-Gordon-Pople 和 Rys 多项式算法的各种方法，这些方法都使用了笛卡尔、Hermite 和混合高斯积分。后一种 ERI 同时包含了 Hermite 和笛卡尔高斯函数，它们结合了这两种基函数的优势。此外，还讨论了 ERI 导数的预筛选，并提出了一种有效的一阶 Cauchy-Schwarz 边界的近似方法。根据估计的运算次数，通过自动代码生成实现了最有前途的方案，并对其相对性能进行了评估。我们分析了同时计算壳层三重态的所有导数与每次仅计算一个自由度的导数的优势，发现前者在应用适当的预筛选时，对于三重 ζ 质量基，提供了接近一个数量级的加速。在这些情况下，使用笛卡尔高斯函数的 Obara-Saika 方法被证明是最佳方法，但当每次仅需要一个自由度的导数时，混合高斯 Obara-Saika 和 Gill-Head-Gordon-Pople 算法被预测为性能最佳的方法。
