Accelerating Correlated Wave Function Calculations with Hierarchical Matrix Compression of the Two-Electron Integrals

TL;DR

This paper introduces a hierarchical matrix compression technique for electron repulsion integrals in quantum chemistry, significantly accelerating correlated wave function calculations by reducing computational complexity.

Contribution

The authors develop a hierarchical $ ext{H}^2$ matrix approach for ERI tensors, enabling faster MP2 calculations with asymptotic complexity better than $ ext{O}(N^2)$.

Findings

Achieves $ ext{O}(N^2 ext{log} N)$ time and space complexity for MP2 energy calculations.

Numerical tests confirm asymptotic complexity improvements for linear alkanes and water clusters.

Demonstrates practical efficiency gains in quantum chemical computations.

Abstract

Leveraging matrix sparsity has proven a fruitful strategy for accelerating quantum chemical calculations. Here we present the hierarchical SOS-MP2 algorithm, which uses hierarchical matrix ($\mathcal{H}^{2}$) 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 $\mathcal{H}^{2}$ 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 $\mathcal{O}(N^{2}\log N)$ and a space complexity of $\mathcal{O}(N^{2}\log N)$, where $N$ denotes the number of basis functions. Numerical tests show asymptotic time and space complexities better than $\mathcal{O}(N^{2})$ for both linear alkanes and three-dimensional water clusters.