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arXiv:2609.39176

Computing electron overlap integrals for Gausslet orbitals on cubic lattices

Xianrui Yin, Fereshteh Ghasempour, Christian B. Mendl

physics.chem-phphysics.comp-phquant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026
v1 · Sep 30, 2026v2 · Oct 5, 2026
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Abstract

Gausslet orbitals on a cubic lattice, introduced in [Steven R. White, J. Chem. Phys. 147, 244102 (2017)], represent a localized, smooth, and systematically refineable basis set featuring a "diagonal" approximability of the electron repulsion integral tensor. The present work develops efficient algorithms for evaluating overlap integrals required for electronic-structure simulations using these Gausslets, specifically the kinetic, nuclear, and electron repulsion integrals (both with and without the diagonal approximation). Computational efficiency improvements rest on the exploitation of translation, permutation, and octahedral symmetries, a reordering and precomputation of nested sums, as well as an early truncation of small coefficients. Our algorithms reduce the number of electron repulsion integrals on a $5 \times 5 \times 5$ grid from $125^4 = 244140625$ to $324275$ due to symmetries, and achieve a wall-clock runtime for evaluating the remaining integrals with a truncation tolerance of $10^{-5}$ in under 2 seconds on a laptop computer. We apply the developed methodology to compute the ground state of the hydrogen atom and molecule as a demonstration.

Comments
14 pages, 6 figures

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Gausslet orbitals on a cubic lattice, introduced in [Steven R. White, J. Chem. Phys. 147, 244102 (2017)], represent a localized, smooth, and systematically refineable basis set featuring a "diagonal" approximability of the electron repulsion integral tensor. The present work develops efficient algorithms for evaluating overlap integrals required for electronic-structure simulations using these Gausslets, specifically the kinetic, nuclear, and electron repulsion integrals (both with and without the diagonal approximation). Computational efficiency improvements rest on the exploitation of translation, permutation, and octahedral symmetries, a reordering and precomputation of nested sums, as well as an early truncation of small coefficients. Our algorithms reduce the number of electron repulsion integrals on a 5×5×55 \times 5 \times 55×5×5 grid from 1254=244140625125^4 = 2441406251254=244140625 to 324275324275324275 due to symmetries, and achieve a wall-clock runtime for evaluating the remaining integrals with a truncation tolerance of 10−510^{-5}10−5 in under 2 seconds on a laptop computer. We apply the developed methodology to compute the ground state of the hydrogen atom and molecule as a demonstration.
Slater-type orbitals (STOs) provide the physically correct description of atomic wavefunctions but have been largely replaced by Gaussian-type orbitals in computational chemistry due to the lack of closed-form multi-center integrals. We present a systematic study of amplitude encoding of STOs on quantum computers using matrix product states (MPS). For one-dimensional orbital functions of the form pd(x)e−ζxp_d(x) e^{-ζx}pd​(x)e−ζx, we derive analytical MPS constructions with constant bond dimension χ=d+1χ= d + 1χ=d+1, requiring O(n)O(n)O(n) classical and quantum resources for nnn-qubit registers with no grid sampling. We demonstrate a complete one-electron integral pipeline -- overlap, kinetic energy, and nuclear attraction -- in one dimension, validating the overlap and kinetic energy on IBM Heron processors at 5~qubits with 0.67\% hardware-induced error using Zero-Noise Extrapolation. In three dimensions, we compute multi-center overlap integrals between 1s and 2s orbitals in Cartesian coordinates with 0.02\% discretization error at 18~qubits. A systematic entanglement analysis reveals that the MPS bond dimension of three-dimensional STOs in Cartesian coordinates saturates with increasing grid resolution -- reaching $\sim$138 for the hydrogen 1s orbital at 12~qubits per coordinate -- establishing bounded encoding complexity rather than the exponential scaling initially expected. The SVD truncation threshold provides a practical resource parameter, reducing the bond dimension to 39 at threshold 10−610^{-6}10−6 with negligible accuracy loss. These results map the entanglement landscape for amplitude encoding of atomic orbitals and establish MPS-based state preparation as a viable path toward exact STO basis sets on quantum computers.
Gausslets are one of the few basis constructions for electronic structure that combine locality, orthonormality, variable resolution, and an accurate diagonal approximation for the electron-electron interaction, but the original construction is tied to one dimension. Radial gausslets extended this idea to atoms while leaving the angular degrees of freedom in spherical harmonics, so the atomic interaction remained only partially diagonal in the combined basis. Here, we introduce generalized gausslets on the sphere and combine them shell by shell with radial gausslets to form an atom-centered basis in which the electron-electron interaction takes a two-index integral-diagonal form. The angular basis starts from localized spherical Gaussians and uses injection to make a low-ℓ spherical-harmonic subspace exact. Tests of the kinetic spectrum, low-ℓ Coulomb matrix elements, spherium, first-row Hartree-Fock calculations, and He exact diagonalization show systematic convergence with increasing angular resolution. We also develop density matrix renormalization group (DMRG) methods for this basis, including compact matrix-product operators, correlated small-space starting states, Givens-rotation transfers between nearby angular sizes, and embedded sampled variance extrapolation. We show that this combination of ingredients can be used to solve the Be atom, with extrapolations in the number of angular functions but with fixed radial resolution, to within about 0.1 mH of the complete-basis-set limit exact energy. This shows that DMRG calculations of first-row atoms that include both static and accurate dynamic correlation on the same footing are feasible.
We introduce highly local basis sets for electronic structure which are very efficient for correlation calculations near the complete basis set limit. Our approach is based on gausslets, recently introduced waveletlike smooth orthogonal functions. We adapt the gausslets to particular systems using one-dimensional (1D) coordinate transformations, putting more basis functions near the nuclei, while maintaining orthogonality. Three-dimensional basis functions are composed out of products of the 1D functions in an efficient way called multislicing. We demonstrate these bases with both Hartree-Fock and density matrix renormalization group calculations on hydrogen chain systems. With both methods, we can go to higher accuracy in the complete basis set limit than is practical for conventional Gaussian basis sets, with errors near 0.1 mH per atom.
This work involves studying and developing new algorithms for molecular numerical integration used for density functional theory and new algorithms for Hartree-Fock method. -- New insight about molecular numerical integration is presented through a detailed study of the performance of some of the well known grids in addition to our implementation of the most recently developed MultiExp grid. A comprehensive study of numerical integration was conducted by evaluating several molecular properties: number of electrons, dipole moment, potential energy, and Coulomb repulsion energy using fifteen grids including a large benchmark grid. The standard grid (SG-1) and a slightly modified version of the Treutler and Alhrichs (TA) grid performed reasonably well. The MultiExp grid, which is more efficient, was studied as well and found to be less accurate. -- Studying large molecules using Hartree-Fock method is a challenge both in terms of CPU time and memory requirements. However, there is a high demand to perform quantum chemical calculations for large molecules. Projection from a smaller basis set to a larger basis set was studied in detail. It was found that projection from the STO-3G basis set to the 6-31G basis set performed well. Projection was used to develop a new version of a divide and conquer algorithm. Our divide and conquer algorithm was used to calculate the protonation energy for a series of peptides. Algorithms to skip calculating two-electron integrals of zero or negligible values are presented in addition to an algorithm to generate a better initial guess.
Abstract All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much of the foundation of modern correlated quantum chemistry calculations—on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block-diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O ( N 4.5 ) in molecular orbital representations to O ( N 2.6 ) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1–2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.
Methods for electronic structure based on Gaussian and molecular orbital discretizations offer a well established, compact representation that forms much of the foundation of correlated quantum chemistry calculations on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N4.5)O(N^{4.5})O(N4.5) in molecular orbital representations to O(N2.6)O(N^{2.6})O(N2.6) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1-2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.