Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway.
J Comput Chem. 2013 Jun 30;34(17):1486-96. doi: 10.1002/jcc.23284. Epub 2013 Apr 3.
An analysis of Dunlap's robust fitting approach reveals that the resulting two-electron integral matrix is not manifestly positive semidefinite when local fitting domains or non-Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four-center two-electron integrals based on the resolution-of-the-identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair-atomic resolution-of-the-identity (PARI) approach, atomic-orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree-Fock and Kohn-Sham calculations, the indefinite integral matrix causes nonconvergence in the self-consistent-field iterations. In these cases, the two-electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb-metric RI method. The speed-up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple-zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky-decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations.
对 Dunlap 的鲁棒拟合方法的分析表明，当使用局部拟合域或非库仑拟合度量时，得到的双电子积分矩阵不是显式半正定的。我们提出了一种基于完全分辨近似（RI）的高度局部的四中心双电子积分评估方法，并将其应用于构建 Fock 矩阵的库仑和交换贡献。在这种对原子分辨的完全分辨近似（PARI）方法中，原子轨道（AO）乘积在以每个乘积相关的两个原子为中心的辅助函数中展开。数值测试表明，在所有 Hartree-Fock 和 Kohn-Sham 计算的 1%或更少的情况下，不定积分矩阵会导致自洽场迭代不收敛。在这些情况下，总能量的双电子贡献变为负值，这意味着电子相互作用实际上是吸引力，总能量远低于使用精确积分得到的能量。然而，在我们的绝大多数测试案例中，不定性不会干扰收敛性。总能量精度与标准库仑度量 RI 方法相当。与传统算法相比，速度提高与 RI 方法的库仑贡献相似；对于三 ζ 质量基组，交换贡献的加速因子高达 8 倍。通过引入跨越整个 AO 乘积空间的局部辅助基函数，可以在 PARI 中恢复正定积分矩阵，这可以通过使用 Cholesky 分解技术来实现。然而，局部完成会将算法的速度降低到与传统计算相当或低于传统计算的水平。
