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The Journal of Chemical Physics

Optimizing two-electron repulsion integral calculations with McMurchie–Davidson method on graphic processing unit

Yingqi Tian, Bingbing Suo, Yingjin Ma, Zhong Jin

Published Jul 20, 2021• 21 citations
Publisher

Abstract

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.

DOI
10.1063/5.0052105

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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.
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.
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.
Modern videogames place increasing demands on the computational and graphical hardware, leading to novel architectures that have great potential in the context of high performance computing and molecular simulation. We demonstrate that Graphical Processing Units (GPUs) can be used very efficiently to calculate two-electron repulsion integrals over Gaussian basis functions [Formula: see text] the first step in most quantum chemistry calculations. A benchmark test performed for the evaluation of approximately 10(6) (ss|ss) integrals over contracted s-orbitals showed that a naïve algorithm implemented on the GPU achieves up to 130-fold speedup over a traditional CPU implementation on an AMD Opteron. Subsequent calculations of the Coulomb operator for a 256-atom DNA strand show that the GPU advantage is maintained for basis sets including higher angular momentum functions.
High Resolution Image Download MS PowerPoint Slide We report an implementation of the McMurchie–Davidson evaluation scheme for 1- and 2-particle Gaussian AO integrals designed for processors with Single Instruction Multiple Data (SIMD) instruction sets. Like in our recent MD implementation for graphical processing units (GPUs) [ Asadchev, A.; Valeev, E. F. . J. Chem. Phys. 2024, 160, 244109 .], variable-sized batches of shellsets of integrals are evaluated at a time. By optimizing for the floating point instruction throughput rather than minimizing the number of operations, this approach achieves up to 50% of the theoretical hardware peak FP64 performance for many common SIMD-equipped platforms (AVX2, AVX512, NEON), which translates to speedups of up to 30 over the state-of-the-art one-shellset-at-a-time implementation of Obara–Saika-type schemes in Libint for a variety of primitive and contracted integrals. As with our previous work, we rely on the standard C++ programming language─such as the std::simd standard library feature to be included in the 2026 ISO C++ standard─without any explicit code generation to keep the code base small and portable. The implementation is part of the open source LibintX library freely available at https://github.com/ValeevGroup/libintx .
In this article, we present an effective approach to calculate quantum chemical two-electron integrals over basis sets consisting of Gaussian-type basis functions on graphical processing unit (GPU). Our framework generates several different variants called routes to the same integral problem with different integral algorithms (McMurchie-Davidson, Head-Gordon-Pople, and Rys) and precision. Each route is benchmarked on more GPU architectures, and with this data, a model is fitted to select the best available route for an integral task given a GPU architecture. Moreover, this approach supports the computation of high angular momentum orbitals up to g effectively on GPU, tested up to cc-pVQZ-sized basis sets. Rigorous analysis is shown regarding the effectiveness of our method. Molecule simulations with several basis sets are measured using NVIDIA GTX 1080 Ti, NVIDIA P100, and NVIDIA V100 cards.