Dynamic Screening of Two-Electron Repulsion Integrals in GPU Parallelization

Two-electron repulsion integrals (ERIs) are among the most foundational quantities for numerically solving the Schrödinger equation, accounting for the Coulomb interactions between electrons in a molecule. Calculating ERIs is a computationally intensive task of determining integral values for every combination of four atomic orbitals, which dominates the execution time of fundamental frameworks in quantum chemistry such as the Hartree-Fock method. It is known that the numerous ERI calculations can be significantly reduced using the Schwarz inequality. However, the Schwarz screening requires evaluating the upper bound for every integral value, leading to the time complexity comparable to the entire ERI calculations themselves. This paper proposes a dynamic screening algorithm that minimizes the number of the upper bound evaluations using the asynchronous parallelism of GPUs. Our parallel algorithm performed by dynamically scheduled CUDA blocks can discard most ERI calculations without even evaluating their upper bounds. Furthermore, by screening ERIs at the level of the finest integral units, primitive integrals, we reduce more integral calculations over coarse-grained screening. Experimental results for various molecules using an NVIDIA A100 GPU demonstrate that the proposed dynamic screening achieves up to an 18.0-fold speedup compared to the non-screening ERI computation while keeping the energy error below 1.0 × 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−7</sup> hartree.