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.
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.
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
5×5×5 grid from
1254=244140625 to
138187 due to symmetries, and achieve a wall-clock runtime for evaluating the remaining integrals with a truncation tolerance of
10−5 in under 1 second on a laptop computer. We apply the developed methodology to compute the ground state of the hydrogen atom and molecule as a demonstration.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
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
x,
y, and
z, and then transform the result to spherical functions. This route is effective, but from
d 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.
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.