In the present work, we describe a more accurate and efficient variant of the chain-of-spheres algorithm (COSX) for exchange matrix computations. Higher accuracy for the numerical integration is obtained with new grids that were developed using global optimization techniques. With our new default grids, the average absolute energy errors are much lower than 0.1 kcal/mol, which is desirable to achieve “chemical accuracy.” Although the size of the new grids is increased by roughly a factor of 2.5, the excellent efficiency of the original COSX implementation is still further improved in most cases. The evaluation of the analytic electrostatic potential integrals was significantly accelerated by a new implementation of rolled-out versions of the Dupuis–Rys–King and Head-Gordon–Pople algorithms. Compared to our earlier implementation, a twofold speedup is obtained for the frequently used triple-ζ basis sets, while up to a 16-fold speedup is observed for quadruple-ζ basis sets. These large gains are a consequence of both the more efficient integral evaluation and the intermediate exchange matrix computation in a partially contracted basis when generally contracted shells occur. With our new RIJCOSX implementation, we facilitate accurate self-consistent field (SCF) binding energy calculations on a large supra-molecular complex composed of 320 atoms. The binding-energy errors with respect to the fully analytic results are well below 0.1 kcal/mol for the cc-pV(T/Q)Z basis sets and even smaller than for RIJ with fully analytic exchange. At the same time, our RIJCOSX SCF calculation even with the cc-pVQZ basis and the finest grid is 21 times faster than the fully analytic calculation.

中文翻译：

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一种用于交换算法的改进的球体链

在目前的工作中，我们描述了用于交换矩阵计算的球链算法（COSX）的更准确和有效的变体。使用全局优化技术开发的新网格可以获得更高的数值积分精度。使用我们新的默认网格，平均绝对能量误差远低于 0.1 kcal/mol，这是实现“化学精度”所需要的。虽然新网格的大小增加了大约 2.5 倍，但在大多数情况下，原始 COSX 实现的卓越效率仍然得到进一步提高。Dupuis-Rys-King 和 Head-Gordon-Pople 算法的推出版本的新实现显着加速了解析静电势积分的评估。与我们之前的实施相比，ζ基组，而对于四重ζ基组观察到高达 16 倍的加速。这些巨大的收益是更有效的积分评估和部分收缩基础中的中间交换矩阵计算的结果当一般收缩的壳发生时。通过我们新的 RIJCOSX 实施，我们促进了对由 320 个原子组成的大型超分子复合物的准确自洽场 (SCF) 结合能计算。对于 cc-pV(T/Q)Z 基组，相对于完全解析结果的结合能误差远低于 0.1 kcal/mol，甚至小于具有完全解析交换的 RIJ。同时，我们的 RIJCOSX SCF 计算即使使用 cc-pVQZ 基础和最精细的网格也比完全解析计算快 21 倍。