A linear scaling hierarchical block low-rank representation of the electron repulsion integral tensor

Efficient representations of the electron repulsion integral (ERI) tensor and fast algorithms for contractions with the ERI tensor often employ a low-rank approximation of the tensor or its sub-blocks. Such representations include density fitting (DF), the continuous fast multipole method (CFMM), and, more recently, hierarchical matrices. We apply the $H2$ hierarchical matrix representation to the ERI tensor with Gaussian basis sets to rapidly calculate the Coulomb matrices in Hartree–Fock and density functional theory calculations. The execution time and storage requirements of the hierarchical matrix approach and the DF approach are compared. The hierarchical matrix approach has very modest storage requirements, allowing large calculations to be performed in memory without recomputing ERIs. We interpret the hierarchical matrix approach as a multilevel, localized DF method and also discuss the close relationship between the hierarchical matrix approaches with CFMM. Like CFMM, the hierarchical matrix approach is asymptotically linear scaling, but the latter requires severalfold less memory (or severalfold less computation, if quantities are computed dynamically) due to being able to efficiently employ low-rank approximations for far more blocks.