Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA.
J Chem Phys. 2013 May 21;138(19):194107. doi: 10.1063/1.4802773.
We investigate the application of molecular quadratures obtained from either standard Becke-type grids or discrete variable representation (DVR) techniques to the recently developed least-squares tensor hypercontraction (LS-THC) representation of the electron repulsion integral (ERI) tensor. LS-THC uses least-squares fitting to renormalize a two-sided pseudospectral decomposition of the ERI, over a physical-space quadrature grid. While this procedure is technically applicable with any choice of grid, the best efficiency is obtained when the quadrature is tuned to accurately reproduce the overlap metric for quadratic products of the primary orbital basis. Properly selected Becke DFT grids can roughly attain this property. Additionally, we provide algorithms for adopting the DVR techniques of the dynamics community to produce two different classes of grids which approximately attain this property. The simplest algorithm is radial discrete variable representation (R-DVR), which diagonalizes the finite auxiliary-basis representation of the radial coordinate for each atom, and then combines Lebedev-Laikov spherical quadratures and Becke atomic partitioning to produce the full molecular quadrature grid. The other algorithm is full discrete variable representation (F-DVR), which uses approximate simultaneous diagonalization of the finite auxiliary-basis representation of the full position operator to produce non-direct-product quadrature grids. The qualitative features of all three grid classes are discussed, and then the relative efficiencies of these grids are compared in the context of LS-THC-DF-MP2. Coarse Becke grids are found to give essentially the same accuracy and efficiency as R-DVR grids; however, the latter are built from explicit knowledge of the basis set and may guide future development of atom-centered grids. F-DVR is found to provide reasonable accuracy with markedly fewer points than either Becke or R-DVR schemes.
我们研究了从标准 Becke 型网格或离散变量表示（DVR）技术获得的分子求积在最近开发的电子排斥积分（ERI）张量最小二乘张量超收缩（LS-THC）表示中的应用。LS-THC 使用最小二乘拟合对 ERI 的双侧面伪谱分解进行正则化，该分解跨越物理空间求积网格。虽然此过程在技术上适用于任何网格选择，但当求积被调谐以准确再现主轨道基二次乘积的重叠度量时，效率最高。适当选择的 Becke DFT 网格可以大致达到此属性。此外，我们提供了采用动力学社区的 DVR 技术来生成两个不同类别的网格的算法，这些网格大致达到此属性。最简单的算法是径向离散变量表示（R-DVR），它对角化每个原子的径向坐标的有限辅助基表示，然后组合 Lebedev-Laikov 球面求积和 Becke 原子分区以产生全分子求积网格。另一个算法是全离散变量表示（F-DVR），它使用全位置算子的有限辅助基表示的近似同时对角化来产生非直接乘积求积网格。讨论了所有三类网格的定性特征，然后在 LS-THC-DF-MP2 中比较了这些网格的相对效率。粗 Becke 网格的精度和效率与 R-DVR 网格基本相同；然而，后者是从基组的显式知识构建的，并且可能指导未来原子中心网格的发展。F-DVR 被发现具有合理的准确性，并且点数明显少于 Becke 或 R-DVR 方案。
