We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches.
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 characterize the entanglement structure of multi-electron atoms in the angular momentum eigenbasis—the basis whose orbitals are products of hydrogenic radial functions and spherical harmonics, labeled by quantum numbers
(n,l,m). Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant
κ=−1/16) contributes
10–
39 of the correlation energy but less than
0.2 of the entanglement entropy; the off-diagonal electron repulsion contributes
100 of the entanglement. Entanglement entropy scales as
S∼Z−2.56 across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from
42 to
99 density in a step-function transition at the identity. The nonzero ERI count is
Z-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—
1s→2s→2p—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to
≈2×10−3 by
Z=4 and below
10−3 for
Z≥5, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is
R-independent, with core/bond entropy ratios of
50× at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the
O(Q2.5) Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
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.
Coulomb integrals, i.e., matrix elements of bare or screened Coulomb interaction between one-electron orbitals, are fundamental objects in many approaches developed to tackle the challenging problem of calculating the electronic structure of strongly correlated materials. In this paper, Coulomb integrals are analyzed by considering both the point-group symmetry of the site occupied by the atom in the crystal or molecule and the permutation symmetries of the orbitals in the integrals. In particular, the case where one-electron orbitals form the basis of a general (i.e., a real, complex, or pseudoreal) irreducible representation is considered. Explicit formulas are provided to calculate all integrals of the interaction tensor in terms of a minimum set of independent ones. The effect of a symmetry breaking is also investigated by describing Coulomb integrals of a group in terms of those of one of its subgroups. We developed the specific example of
O(3) as the larger group which can therefore be used to quantify the deviation of a specific system from the spherical symmetry. Possible applications of the presented framework include the calculation of solid-state and molecular spectroscopies via multiplet techniques, dynamical mean-field theory, or the
GW approximation.