Efficient GPU Implementations of Three-Center Two-Electron Repulsion Integrals

In computational quantum chemistry, the computation of three-center two-electron repulsion integrals (also termed three-center ERIs) is essential for density fitting. Due to the large number of integral elements and the induced combinatorial computational complexity, the community has actively pursued the acceleration/speedup of ERI calculations to achieve pragmatic levels of efficiency. From the perspective of GPU acceleration, atomicAdd is known to incur significant memory overhead: The frequent collisions and retrials of value aggregation in global GPU memory lead to substantial performance degradation. To tackle this issue, we propose new thread mapping strategies for three-center two-electron integrals on GPUs, aiming at reducing the computational cost associated with value aggregation. Our methods are based on the idea of suitable substitutions of device-level reduction (atomicAdd) with efficient warp- and thread-level reduction, such as warp-shuffle and register accumulation. As a result, our computational experiments using an Intel Xeon Gold 6338 CPU, an NVIDIA A100 GPU, and relevant molecules of interest show the superiority against the conventional thread mapping scheme, achieving up to 2.76 speedups to compute three-center ERIs more efficiently. Moreover, compared to well-known quantum chemistry software such as PySCF and GPU4PySCF, our method achieved up to $11.90\times$ speedups over PySCF and up to $4.99\times$ speedups over GPU4PySCF. Our method has the potential to further enhance the performance, extensibility, and versatility of GPU-accelerated quantum chemical computations.