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arXiv:2607.00102

A differential derivation of the Obara-Saika relation for Gaussian electron repulsion integrals

Charles C. Forgy, David A. Mazziotti

physics.chem-phquant-phSubmitted Jun 30, 2026• Announced Jul 2, 2026
arXivHTMLPDF

Abstract

The Obara-Saika (OS) method is one of the most widely used techniques in quantum chemistry for evaluating electron repulsion integrals (ERIs) via a set of recurrence relations that build higher angular momentum integrals from lower-order ones. The original derivation by Obara and Saika proceeded by directly relating integrals of differing angular momentum. In this work, we present a compact novel derivation of the OS vertical recurrence relation based solely on differential relations between Gaussian basis functions, expanding on a method suggested in earlier work. By explicitly deriving the required derivative expressions we identify all non-zero primitive terms contributing to the full ERI to develop a hierarchical formulation of the OS recursion relations. This approach has pedagogical value as a rigorous and self-contained derivation. Additionally, the resulting organization exposes independent primitive derivative quantities and may be useful for code generation and parallel implementations on modern GPU architectures.

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The Obara-Saika (OS) method is one of the most widely used techniques in quantum chemistry for evaluating electron repulsion integrals (ERIs) via a set of recurrence relations that build higher angular momentum integrals from lower-order ones. The original derivation by Obara and Saika proceeded by directly relating integrals of differing angular momentum. In this work, we present a compact novel derivation of the OS vertical recurrence relation based solely on differential relations between Gaussian basis functions, expanding on a method suggested in earlier work. By explicitly deriving the required derivative expressions we identify all non-zero primitive terms contributing to the full ERI to develop a hierarchical formulation of the OS recursion relations. This approach has pedagogical value as a rigorous and self-contained derivation. Additionally, the resulting organization exposes independent primitive derivative quantities and may be useful for code generation and parallel implementations on modern GPU architectures.
Abstract—Evaluation of electron repulsion integrals (ERIs) takes considerable time in modern quantum chemistry applications and also presents a certain difficulty to be efficiently computed on GPUs. Here, we describe a novel methodology for generating high-arithmetic-density kernels for ERI evaluation of d and higher angular momentum functions, as well as highlight challenges associated with the efficient use of GPUs for ERIs. Employing this approach allows to perform the ERI evaluations substantially faster than just using traditional CPU codes. Index Terms—GPUs; quantum chemistry; gaussian integrals; d-functions I.
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.
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.
Electron-repulsion integrals describe the Coulomb interaction between charge distributions built from orbital basis functions. Most integral algorithms generate these quantities through Cartesian Gaussian functions, whose angular shapes are written as powers of xxx, yyy, and zzz, and then transform the result to spherical functions. This route is effective, but from ddd shells onward the Cartesian representation contains more functions than the spherical space required by the calculation. Direct real or complex solid-harmonic engines work in that target space from the beginning. They therefore produce a smaller final Coulomb tensor while preserving the ordering, phase, and magnetic-quantum-number labels that describe its angular structure. Following this structure beyond integral evaluation reveals direct connections to the algorithms that use the tensor. Simple analytical counts quantify tensor size, angular blocks, radial Slater--Condon parameters, and pair-space work. These quantities guide low-rank factorization, local Hamiltonian construction, quantum simulation, and transformations to spinor or effective-model bases. In this way, solid-harmonic integral engines provide a direct bridge between efficient integral generation and structured many-electron computation.