arxiv: 2605.04517 · v1 · submitted 2026-05-06 · ⚛️ physics.chem-ph · cond-mat.str-el· physics.atom-ph· physics.comp-ph

Recognition: unknown

Angular Gausslets

Steven R. White

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:05 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.str-elphysics.atom-phphysics.comp-ph

keywords angular gaussletsradial gaussletsatomic basis setsDMRGelectron correlationberyllium atomdiagonal Coulomb integralsangular extrapolation

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The pith

Angular gausslets paired with radial ones form an atomic basis where the electron-electron interaction becomes exactly diagonal in two indices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs generalized gausslets on the sphere that start from localized spherical Gaussians and use an injection step to keep low angular-momentum spherical harmonics exact. These angular functions are then combined shell by shell with existing radial gausslets to produce a full atom-centered basis. In the resulting representation the two-electron repulsion integrals collapse to a simple two-index diagonal form instead of the usual four-index tensors. The construction is tested for convergence on kinetic spectra, Coulomb elements, spherium, Hartree-Fock, and exact diagonalization of helium before being paired with new DMRG tools. The combination is applied to the beryllium atom, where angular extrapolation at fixed radial resolution reaches energies within 0.1 millihartree of the complete-basis-set limit.

Core claim

Generalized gausslets on the sphere are built from localized spherical Gaussians with an injection procedure that enforces exactness in a low-ℓ spherical-harmonic subspace. When tensor-producted shell by shell with radial gausslets the combined basis renders the electron-electron Coulomb interaction diagonal in its two indices. Systematic tests confirm convergence of the kinetic spectrum, low-ℓ Coulomb matrix elements, spherium energies, first-row Hartree-Fock results, and helium exact diagonalization. New DMRG machinery, including compact matrix-product operators, correlated initial states, Givens transfers, and embedded sampled variance extrapolation, is developed for this basis. The full

What carries the argument

The angular gausslet: a localized spherical function obtained by injecting an exact low-ℓ spherical-harmonic subspace into a basis of spherical Gaussians, which when combined with radial gausslets diagonalizes the two-electron interaction to two-index integrals.

If this is right

First-row atoms become treatable by DMRG with both static and dynamic correlation obtained on the same footing and with controlled extrapolation.
The two-index diagonal form of the repulsion integrals reduces the cost of integral storage and contraction inside the DMRG sweep.
Angular resolution can be increased independently of radial resolution, allowing separate extrapolation in each direction.
The same shell-by-shell construction supplies a route to systematic improvement of Hartree-Fock and correlated atomic calculations without changing the radial grid.
DMRG methods developed for this basis, such as Givens-rotation transfers and embedded sampled variance extrapolation, become available for any system using the same angular-radial separation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same angular-radial separation and diagonal interaction could be placed at multiple nuclear centers to treat molecules, although the paper tests only single atoms.
Because the interaction is already diagonal, other many-body solvers that rely on two-body operators may become cheaper without further approximation.
Holding radial resolution fixed while refining only the angular part may lower the total computational effort compared with uniform refinement in both variables.
The extrapolation procedure demonstrated for beryllium could be applied to other first-row atoms or ions to test whether 0.1 millihartree accuracy remains reachable.

Load-bearing premise

Increasing the number of angular functions while keeping radial resolution fixed will drive the total energy to the complete basis set limit without leaving appreciable residual error from angular incompleteness or DMRG truncation.

What would settle it

A beryllium calculation at substantially higher angular resolution but the same radial functions that shifts the extrapolated energy by more than 0.1 millihartree would show that the claimed convergence to the complete basis set limit has not occurred.

Figures

Figures reproduced from arXiv: 2605.04517 by Steven R. White.

**Figure 1.** Figure 1: FIG. 1. Centered orthonormalized Gaussian functions on the view at source ↗

**Figure 3.** Figure 3: FIG. 3. Effect of injection for the sweet-spot case view at source ↗

**Figure 5.** Figure 5: FIG. 5. Kinetic-energy eigenvalues for the angular basis with view at source ↗

**Figure 6.** Figure 6: FIG. 6. Total energy errors in Hartrees for the spherium system view at source ↗

**Figure 8.** Figure 8: FIG. 8. Ground-state energy error for the He atom from two view at source ↗

**Figure 9.** Figure 9: FIG. 9. Transfer diagnostic view at source ↗

**Figure 11.** Figure 11: FIG. 11. Final target-size extrapolation for Be. The corrected view at source ↗

read the original abstract

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-$\ell$ spherical-harmonic subspace exact. Tests of the kinetic spectrum, low-$\ell$ Coulomb matrix elements, spherium, first-row Hartree--Fock calculations, and He exact diagonalization show systematic convergence with increasing angular resolution. We also develop DMRG methods for this basis, including compact MPOs, correlated small-space starting states, Givens-rotation transfers between nearby angular sizes, and embedded sampled variance extrapolation (ESVE). 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 which include both static and accurate dynamic correlation on the same footing are feasible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

White's angular gausslets give a workable way to make the full atomic electron repulsion diagonal in a mixed radial-angular basis, but the Be atom claim of 0.1 mH CBS accuracy needs a radial-resolution check to hold up.

read the letter

The new piece is the spherical gausslet construction: localized spherical Gaussians plus an injection step that keeps the low-ℓ spherical-harmonic subspace exact. When these are combined shell-by-shell with the existing radial gausslets, the two-electron integrals become diagonal in the combined index. That is the main technical advance over the earlier radial-only work, and it is cleanly described in the abstract and the tests that follow from it. The paper also shows the expected systematic improvement in kinetic spectra, low-ℓ Coulomb elements, spherium, and first-row Hartree-Fock as angular resolution increases, plus some practical DMRG machinery (compact MPOs, Givens transfers, ESVE extrapolation) that lets them run the Be calculation at all. Those parts look solid and worth having in the literature for anyone who wants a local, orthonormal, variable-resolution atomic basis with cheap interactions. The Be result is the part that needs a closer look. They extrapolate only in the number of angular functions while holding the radial gausslet resolution fixed and report agreement to roughly 0.1 mH with the exact CBS energy. That agreement is only convincing if the radial incompleteness error is already well below 0.1 mH at the chosen resolution; angular extrapolation cannot cancel radial error. The abstract and the stress-test note do not mention a radial-resolution scan or a comparison at higher radial cutoff, so it is possible the quoted accuracy partly reflects a lucky radial choice rather than a true demonstration that the basis reaches the CBS limit. If the full manuscript contains that scan, the claim strengthens; if not, it is a gap that a referee would flag. This paper is aimed at people who already use DMRG or local bases for small atoms and molecules and who care about treating static and dynamic correlation on equal footing. A reader working on first-row benchmark calculations or on new atomic basis constructions will get the most out of it. The construction itself is novel enough and the numerical evidence is systematic enough that it deserves a serious referee rather than a desk reject, even if the Be extrapolation needs one extra figure to be fully convincing.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces generalized angular Gausslets constructed from localized spherical Gaussians with injection to preserve exact low-ℓ spherical-harmonic subspaces. These are combined shell-by-shell with radial Gausslets to produce an atom-centered basis in which the electron-electron interaction takes a two-index integral-diagonal form. Specialized DMRG methods are developed, including compact MPOs, correlated starting states, Givens-rotation transfers, and embedded sampled variance extrapolation (ESVE). Systematic convergence is demonstrated on the kinetic spectrum, low-ℓ Coulomb matrix elements, spherium, first-row Hartree-Fock, and He exact diagonalization. The central numerical result is a Be-atom DMRG calculation with extrapolation in angular resolution at fixed radial resolution, reported to reach within ~0.1 mH of the exact complete-basis-set limit energy.

Significance. If the Be result is confirmed to be free of significant radial incompleteness, the work would demonstrate a viable route to accurate DMRG calculations for first-row atoms that treat static and dynamic correlation on equal footing within a single basis framework. Strengths include the locality/orthonormality/diagonal-interaction properties of the basis, the systematic convergence shown across multiple tests, and the development of tailored DMRG infrastructure (compact MPOs, ESVE). The fixed-radial-resolution extrapolation, however, requires additional verification before the 0.1 mH claim can be regarded as a general demonstration of reaching the CBS limit.

major comments (1)

[Be-atom results (abstract and corresponding results section)] The central claim (abstract and Be-atom results section) that angular extrapolation at one fixed radial resolution yields an energy within 0.1 mH of the exact CBS limit is load-bearing for the paper's numerical demonstration. No radial-resolution scan or comparison at a higher radial resolution is reported; angular extrapolation cannot remove radial-basis incompleteness error. Without such a check, the reported agreement could partly reflect a radial error that happens to be small at the chosen resolution rather than a true CBS result.

minor comments (2)

Quantitative error bars, explicit basis-set parameters (radial and angular cutoffs), and full data tables are absent from the abstract and the high-level description of the tests; their inclusion would allow independent assessment of the reported systematic convergence.
[DMRG methods section] The description of the embedded sampled variance extrapolation (ESVE) and the Givens-rotation transfers between angular sizes would benefit from additional algorithmic detail or pseudocode to aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript's contributions and for the detailed comment on the Be-atom results. We respond to this comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Be-atom results (abstract and corresponding results section)] The central claim (abstract and Be-atom results section) that angular extrapolation at one fixed radial resolution yields an energy within 0.1 mH of the exact CBS limit is load-bearing for the paper's numerical demonstration. No radial-resolution scan or comparison at a higher radial resolution is reported; angular extrapolation cannot remove radial-basis incompleteness error. Without such a check, the reported agreement could partly reflect a radial error that happens to be small at the chosen resolution rather than a true CBS result.

Authors: We concur that angular extrapolation does not address potential radial incompleteness and that no radial scan is presented for the Be DMRG calculations. The radial gausslet basis parameters were chosen to achieve high accuracy as validated through the systematic tests on smaller systems and the kinetic energy spectrum. Nevertheless, we recognize that this leaves the radial error unquantified for Be specifically. In the revised manuscript, we will add a paragraph in the results section discussing the expected radial error based on the basis construction and the He benchmark, and we will modify the abstract to read 'to within about 0.1 mH of the complete basis set limit exact energy at the chosen radial resolution'. We believe this provides an honest qualification of the result while preserving the demonstration of the method's capabilities. revision: yes

Circularity Check

0 steps flagged

No significant circularity in basis construction or numerical demonstration

full rationale

The paper constructs angular gausslets from localized spherical Gaussians with injection to enforce low-l exactness, then combines shell-by-shell with radial gausslets to obtain a two-index diagonal electron-electron interaction form. All reported results (kinetic spectrum, Coulomb elements, spherium, HF, He ED, and Be DMRG) are obtained from explicit numerical computations that converge systematically with angular resolution. The Be-atom claim is an independent DMRG calculation plus standard angular extrapolation compared against the known external CBS limit; it does not reduce by construction to any fitted parameter or prior self-citation. No self-definitional loops, fitted-input predictions, or load-bearing ansatzes imported via citation appear in the derivation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; specific free parameters, axioms, or invented entities cannot be identified without the full manuscript. The basis construction itself introduces resolution and localization parameters whose values are not stated.

pith-pipeline@v0.9.0 · 5565 in / 1123 out tokens · 17240 ms · 2026-05-08T17:05:06.300656+00:00 · methodology

discussion (0)

Reference graph

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We will refer to this procedure as sampled variance extrap- olation (SVE)

Although this trimming introduces a bias, that bias decreases smoothly as the state approaches an eigenstate, so it can be absorbed into the extrapolation. We will refer to this procedure as sampled variance extrap- olation (SVE). In the Be calculations below, we explored both c = 0.9 and c = 0.85 and examined the statistical errors in the variance surrog...

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DMRG convergence in an angular basis Before detailing the DMRG lifting and extrapolation techniques, we address simple DMRG convergence with the bond dimension. The basis sequences were all based on identical fairly fine and extensive radial gausslet bases with s = 0.2 and Rmax = 30, with 41 radial functions. We first consider the largest system in our se...

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Wavefunction transfer Our first sequence had angular sizes of Ns = 10, 15, 24, 32, 48. Subsequently, we added Ns = 36, trans- ferring the starting MPS for the DMRG from the Ns = 32 system. The reason for considering both 32 and 36, nearby points, was that 32 in the HF studies was exceptionally accurate, unlike the other points, so that it might not be use...
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Variance extrapolations: SVE and ESVE How close can we come to the complete basis set fully correlated Be energy ( Eex = −14.6673561335 Ha[33]), us- ing large basis sets and extrapolation in the angular basis size, but no cusp corrections? Because relatively small bond dimension was needed for highly accurate DMRG, ordinary SVE to improve the ground state...
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Angular-basis diagnostics For each final angular orbital ϕa(Ω), we assign a cen- ter ˆna by the prototype Gaussian carrying the largest Gaussian-part coefficient, j(a) = arg max j |Dja|,ˆn a ≡ˆnj(a),(A1) where Dja are the Gaussian-part coefficients of the con- tracted form in Eq. (25), specialized here to the angular basis on the sphere. The diagnostics u...
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