Compressing the Four-Index Two-Electron Repulsion Integral Matrix using the Resolution-of-the-Identity Approximation Combined with the Rank Factorization Approximation

The four-index two-electron repulsion integral (4-2ERI) matrix is compressed using the resolution-of-the-identity (RI) approximation combined with the rank factorization approximation (RFA). The 4-2ERI is first approximated by the RI product. Then, the singular value decomposition (SVD) approximation is used to eliminate low-weighted singular vectors. The SVD RI approximation maintains the canonical form of the RI approximation and introduces a tunable compression factor. The characteristics of the SVD RI approximation along with the stochastic RI and natural auxiliary function approximation were numerically examined by applying these methods to the closed-shell second-order Moller-Plesset perturbation theory (MP2). The results show that, while the SVD RI approximation yields large errors for absolute properties (e.g., the correlation energy), it provides accurate relative properties (potential energy surface, binding energy) of the applied ab initio method (e.g., RHF, MP2).