Accelerating correlated wave function calculations with hierarchical matrix compression of the two-electron integrals

Leveraging matrix sparsity has proven to be a fruitful strategy for accelerating quantum chemical calculations. Here, we present the hierarchical SOS-MP2 algorithm, which uses hierarchical matrix ( H 2 ) compression of the electron repulsion integral (ERI) tensor to reduce both time and space complexity. This approach is based on the atomic orbital Laplace transform MP2 calculations, leveraging the data-sparsity of the ERI tensor and the element-wise sparsity of the energy-weighted density matrices. The H 2 representation approximates the ERI tensor in a block low-rank form, taking advantage of the inherent low-rank nature of the repulsion integrals between distant sets of atoms. The resulting algorithm enables the calculation of the Coulomb-like term of the MP2 energy with a theoretical time complexity of O ( N 2 ⁡ log ⁡ N ) and a space complexity of O ( N 2 ⁡ log ⁡ N ) , where N denotes the number of basis functions. Numerical tests show asymptotic time and space complexities better than O ( N 2 ) for both linear alkanes and three-dimensional water clusters.