Wirz Lukas N, Reine Simen S, Pedersen Thomas Bondo
Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo , P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
J Chem Theory Comput. 2017 Oct 10;13(10):4897-4906. doi: 10.1021/acs.jctc.7b00801. Epub 2017 Sep 25.
The definiteness of the Mulliken and Dirac electron repulsion integral (ERI) matrices is examined for different classes of resolution-of-the-identity (RI) ERI approximations with particular focus on local fitting techniques. For global RI, robust local RI, and nonrobust local RI we discuss the definiteness of the approximated ERI matrices as well as the resulting bounds of Hartree, exchange, and total energies. Lower bounds of Hartree and exchange energy contributions are crucial as their absence may lead to variational instabilities, causing severe convergence problems or even convergence to a spurious state in self-consistent-field optimizations. While the global RI approximation guarantees lower bounds of Hartree and exchange energies, local RI approximations are generally unbounded. The robust local RI approximation guarantees a lower bound of the exchange energy but not of the Hartree energy. The nonrobust local RI approximation guarantees a lower bound of the Hartree energy but not of the exchange energy. These issues are demonstrated by sample calculations on carbon dioxide and benzene using the pair atomic RI approximation.
针对不同类别的单位分解（RI）电子排斥积分（ERI）近似，特别是局部拟合技术，研究了穆利肯（Mulliken）和狄拉克（Dirac）ERI矩阵的确定性。对于全局RI、稳健局部RI和非稳健局部RI，我们讨论了近似ERI矩阵的确定性以及由此产生的哈特里（Hartree）、交换能和总能量的界限。哈特里和交换能贡献的下限至关重要，因为它们的缺失可能导致变分不稳定性，在自洽场优化中引起严重的收敛问题，甚至收敛到虚假状态。虽然全局RI近似保证了哈特里和交换能的下限，但局部RI近似通常是无界的。稳健局部RI近似保证了交换能的下限，但不能保证哈特里能的下限。非稳健局部RI近似保证了哈特里能的下限，但不能保证交换能的下限。通过使用对原子RI近似对二氧化碳和苯进行的样本计算证明了这些问题。
