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Applied Sciences

Efficient GPU Implementation of the McMurchie–Davidson Method for Shell-Based ERI Computations

Haruto Fujii, Yasuaki Ito, Nobuya Yokogawa, Kanta Suzuki, Satoki Tsuji, Koji Nakano, Victor Parque, Akihiko Kasagi

Published Feb 27, 2025• 6 citations
Publisher

Abstract

Quantum chemistry offers the formal machinery to derive molecular and physical properties arising from (sub)atomic interactions. However, as molecules of practical interest are largely polyatomic, contemporary approximation schemes such as the Hartree–Fock scheme are computationally expensive due to the large number of electron repulsion integrals (ERIs). Central to the Hartree–Fock method is the efficient computation of ERIs over Gaussian functions (GTO-ERIs). Here, the well-known McMurchie–Davidson method (MD) offers an elegant formalism by incrementally extending Hermite Gaussian functions and auxiliary tabulated functions. Although the MD method offers a high degree of versatility to acceleration schemes through Graphics Processing Units (GPUs), the current GPU implementations limit the practical use of supported values of the azimuthal quantum number. In this paper, we propose a generalized framework capable of computing GTO-ERIs for arbitrary azimuthal quantum numbers, provided that the intermediate terms of the MD method can be stored. Our approach benefits from extending the MD recurrence relations through shells, batches, and triple-buffering of the shared memory, and ordering similar ERIs, thus enabling the effective parallelization and use of GPU resources. Furthermore, our approach proposes four GPU implementation schemes considering the suitable mappings between Gaussian basis and CUDA blocks and threads. Our computational experiments involving the GTO-ERI computations of molecules of interest on an NVIDIA A100 Tensor Core GPU (NVIDIA, Santa Clara, CA, USA) have revealed the merits of the proposed acceleration schemes in terms of computation time, including up to a 72× improvement over our previous GPU implementation and up to a 4500× speedup compared to a naive CPU implementation, highlighting the effectiveness of our method in accelerating ERI computations for both monatomic and polyatomic molecules. Our work has the potential to explore new parallelization schemes of distinct and complex computation paths involved in ERI computation.

DOI
10.3390/app15052572

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Quantum chemistry offers the formal machinery to derive molecular and physical properties arising from (sub)atomic interactions. However, as molecules of practical interest are largely polyatomic, contemporary approximation schemes such as the Hartree–Fock scheme are computationally expensive due to the large number of electron repulsion integrals (ERIs). Central to the Hartree–Fock method is the efficient computation of ERIs over Gaussian functions (GTO-ERIs). Here, the well-known McMurchie–Davidson method (MD) offers an elegant formalism by incrementally extending Hermite Gaussian functions and auxiliary tabulated functions. Although the MD method offers a high degree of versatility to acceleration schemes through Graphics Processing Units (GPUs), the current GPU implementations limit the practical use of supported values of the azimuthal quantum number. In this paper, we propose a generalized framework capable of computing GTO-ERIs for arbitrary azimuthal quantum numbers, provided that the intermediate terms of the MD method can be stored. Our approach benefits from extending the MD recurrence relations through shells, batches, and triple-buffering of the shared memory, and ordering similar ERIs, thus enabling the effective parallelization and use of GPU resources. Furthermore, our approach proposes four GPU implementation schemes considering the suitable mappings between Gaussian basis and CUDA blocks and threads. Our computational experiments involving the GTO-ERI computations of molecules of interest on an NVIDIA A100 Tensor Core GPU (NVIDIA, Santa Clara, CA, USA) have revealed the merits of the proposed acceleration schemes in terms of computation time, including up to a 72× improvement over our previous GPU implementation and up to a 4500× speedup compared to a naive CPU implementation, highlighting the effectiveness of our method in accelerating ERI computations for both monatomic and polyatomic molecules. Our work has the potential to explore new parallelization schemes of distinct and complex computation paths involved in ERI computation.
The two-electron repulsion integral (ERI) is a fundamental component in quantum chemistry, involving the calculation of integrals over four basis functions. Quantum chemistry computations require O(M4) 4 ERI calculations for M basis functions, making ERI evaluations a significant bottleneck in these computations. In this paper, we propose an efficient GPU-based parallelization of the McMurchie-Davidson algorithm, which recursively computes ERI. Our method focuses on optimizing the computations by leveraging the shell parameter, which characterizes the shape of the basis functions. The results demonstrate that the shell-based optimization achieves a performance improvement of up to 15% compared to the nonoptimized approach.
ABSTRACT In Quantum chemical computation, numerical schemes such as the Hartree–Fock (HF) and density functional theory (DFT) are widely used to solve the Schrödinger equation numerically, to realize experiment‐free prediction and analysis of key molecular properties such as structure and energy. Computing one‐electron integrals, such as kinetic energy integrals and nuclear attraction integrals, is essential in both HF and DFT to characterize the molecular electronic states. However, as molecules of practical interest grow in size and angular momentum, computing one‐electron orbitals becomes computationally expensive in most cases. Although computing kinetic energy integrals on CPUs is straightforward, bottlenecks in CPU‐GPU data transfer have often been overlooked. In this study, we propose an efficient method to compute both the kinetic‐energy and nuclear‐attractive integrals on GPUs. First, we explicitly and symbolically expand recurrence relations based on the Obara–Saika and McMurchie–Davidson methods to eliminate redundant operations, thus improving computational efficiency. Second, we implemented a hybrid method that selects the best/fastest of both methods depending on the integration task. Third, we achieved further speedups by using CUDA streams to parallelize the execution of multiple kernels and efficiently utilize multiprocessor resources on the GPU. Computational experiments using NVIDIA A100 GPUs and Intel Xeon Gold 6338 CPU on relevant molecules of interest demonstrated the superiority of our one‐electron integral GPU implementations, achieving a speedup of 20.2 times over PySCF, and a speedup of 132.6 times over GPU4PySCF.
Computational quantum chemistry aims to derive properties of a molecule, such as its energy, without the need for actual experimental work. This is done by taking the types and coordinates of atoms within the molecule as input. In this study, we accelerate the computation of the two-electron repulsion integrals that mainly dominate the computing time. Specifically, we focus on the McMurchie-Davidson algorithm, where the two-electron repulsion integrals are obtained through recurrence computation. The idea of the proposed GPU implementation is to expand the recurrence relations and efficiently compute them using multiple threads of the GPU. Our proposed method resulted in an acceleration of up to 3.5 times compared to GPU implementations using recursive functions, and up to 17 times compared to CPU implementations similar to those using the GPU.
In this article, several optimization methods of two-electron repulsion integral calculations on a graphic processing unit (GPU) are presented. These methods are based on the investigations of the method presented by McMurchie and Davidson (MD). A new Boys function evaluation method for the GPU calculation is introduced. The series summation, the error function, and the finite sum formula method are combined; thus, good performance on the GPU can be achieved. By taking some theoretical study of the McMurchie-Davidson recurrence relations, three major optimization approaches are derived from the deduction of the general term formula for the Hermite expansion coefficient. The three approaches contain a new form of the Hermite expansion coefficients with corresponding recurrence relations, which is more efficient for one-electron integrals and [ss|∗∗] or [∗∗|ss] type two-electron integrals. In addition, a simple yet efficient new recurrence formula for the coefficient evaluation is derived, which is more efficient both in float operations and memory operations than its original one. In average, the new recurrence relation can save 26% float operations and 37% memory operations. Finally, a common sub-expression elimination (CSE) method is implemented. This CSE method is directly generated from some equalities we discovered from the general term formula other than by computer algebra system software. This optimized method achieved up to 3.09 speedups compared to the original MD method on the GPU and up to 92.75 speedups compared to the GAMESS calculation on the central processing unit.
High Resolution Image Download MS PowerPoint Slide We introduce the first version of GPU4PySCF, a module that provides GPU acceleration of methods in PySCF . As a core functionality, this provides a GPU implementation of two-electron repulsion integrals (ERIs) for contracted basis sets comprising up to g functions using the Rys quadrature. As an illustration of how this can accelerate a quantum chemistry workflow, we describe how to use the ERIs efficiently in the integral-direct Hartree–Fock build and nuclear gradient construction. Benchmark calculations show a significant speedup of 2 orders of magnitude with respect to the multithreaded CPU Hartree–Fock code of PySCF and the performance comparable to other open-source GPU-accelerated quantum chemical packages, including GAMESS and QUICK, on a single NVIDIA A100 GPU.
ABSTRACT In quantum chemistry, fundamental molecular orbital methods such as the Hartree–Fock theory are pragmatically constrained by the computational complexity behind two‐electron repulsion integrals (ERIs) encoding electron Coulomb interactions. Although the computational cost of ERIs roughly scales with the fourth power of the atom count in the target molecule, it can be reduced by screening techniques based on the Schwarz inequality. However, the Schwarz screening involves evaluating the necessity of computation for individual ERIs based on their upper bounds, which results in substantial overhead consuming most of the total computation time, especially for large‐sized molecules. To tackle this bottleneck, we propose a new dynamic screening algorithm and its efficient GPU implementation, which drastically reduces the number of upper bound evaluations. Our parallel algorithm can discard numerically insignificant ERIs without even evaluating their upper bounds by dynamic scheduling of thread blocks, allowing to traverse only the significant ERIs. The merits and performance of our GPU implementation has been evaluated through computational experiments using an NVIDIA A100/H200 GPU with a relevant set of low and medium molecular‐weight compounds. As a result, our dynamic screening reduces the Schwarz upper bound evaluations by up to and achieves a speedup of up to compared to examining the necessity of computation for all ERIs. Our work has the potential to further enhance the performance frontiers of GPU‐based large‐scale molecular quantum chemistry calculations.