arxiv: 2603.22646 · v2 · submitted 2026-03-23 · ⚛️ physics.chem-ph · cond-mat.str-el· physics.atom-ph· physics.comp-ph

Recognition: no theorem link

Radial Gausslets

Steven R. White

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:00 UTC · model grok-4.3

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

keywords radial gaussletsgausslet basiselectronic structureatomic calculationsHartree-Fockexact diagonalizationdiagonal interactionsradial basis sets

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The pith

Generalizing gausslets to radial coordinates creates a compact basis for atomic calculations with diagonal electron interactions.

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

Gausslets simplify electronic structure calculations by allowing diagonal, two-index terms for electron-electron interactions, unlike standard four-index terms. Their original construction was limited to Cartesian coordinates due to their one-dimensional nature. This work generalizes the gausslet to the radial coordinate in spherical symmetry for atoms. The resulting radial gausslets form a compact set that maintains the diagonal property and shows accuracy in Hartree-Fock and exact diagonalization calculations on atomic systems. A sympathetic reader would care because this could reduce computational cost for atomic and molecular problems by shrinking the basis size and simplifying interactions.

Core claim

The paper establishes that radial gausslets, obtained by generalizing the 1D gausslet construction to the radial coordinate, provide a very compact radial basis with a relatively modest number of functions while preserving diagonal electron-electron interaction terms. This is demonstrated through applications to Hartree-Fock and exact diagonalization on atomic systems.

What carries the argument

Radial gausslets: a radial generalization of gausslets that maintains compactness and the diagonal property of interactions.

If this is right

The basis uses a modest number of functions for accurate atomic calculations.
Electron-electron interactions are diagonal, reducing computational complexity.
Accuracy is illustrated for Hartree-Fock and exact diagonalization on atoms.
This approach leverages spherical symmetry for efficiency in atomic systems.

Where Pith is reading between the lines

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

Such a basis might extend to molecular calculations by combining with angular parts.
Diagonal interactions could enable new Monte Carlo or tensor network methods for electrons.
The compactness may allow larger atomic systems to be treated exactly.
Testing on more atoms could reveal if the generalization holds for heavier elements.

Load-bearing premise

The 1D gausslet construction can be directly generalized to the radial coordinate while preserving compactness and the diagonal interaction property without significant errors.

What would settle it

Performing Hartree-Fock calculations on the helium atom and finding that the energy error does not decrease as expected with the number of radial gausslets or requires far more functions than a standard basis would indicate the generalization does not work as claimed.

Figures

Figures reproduced from arXiv: 2603.22646 by Steven R. White.

**Figure 2.** Figure 2: FIG. 2. Radial gausslets, formed from the boundary gausslets [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Error in the energy of the hydrogen atom versus [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Error in the RHF energy of the He atom energy [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Error in the exact diagonalization (full-CI) energy [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original abstract

Gausslets are one of the few examples of basis sets for electronic structure which allow for two-index/diagonal electron-electron interaction terms. A weakness of gausslets is that, because of their 1D origin, they have been tied to Cartesian coordinates. Here we generalize the gausslet construction for the radial coordinate in three dimensions for atomic basis sets. These radial gausslets make a very compact radial basis with a relatively modest number of functions, with diagonal interaction terms. We illustrate the accuracy of this construction with Hartree--Fock and exact diagonalization on atomic systems.

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 radial gausslets give a compact atomic basis that keeps two-electron integrals diagonal, but the numerical support is still light.

read the letter

The core news is that gausslets can be extended to the radial coordinate while preserving the diagonal property of the electron-electron interactions. That is the main advance: a basis that stays compact for atoms and avoids the usual dense two-body integrals that slow down Hartree-Fock and configuration interaction. The construction is presented as a direct generalization of the original 1D Cartesian version, and the paper walks through how the radial functions are built and then used in atomic calculations. It shows results from Hartree-Fock and exact diagonalization on a few atoms, which at least confirms the method runs and produces reasonable energies. That is useful concrete work, and the diagonal-interaction feature is rare enough that even a modest improvement in radial bases is worth noting. The stress-test worry about the r² measure and the Coulomb kernel introducing off-diagonal terms does not appear to be a problem in the actual construction; the paper maintains the diagonal property through the way the functions are defined and integrated. The main limitation is the lack of hard numbers. The abstract and description give no basis sizes, no error tables, and no side-by-side comparisons against standard radial bases such as Slater-type orbitals or optimized Gaussians. Without those, it is difficult to judge how compact the basis really is or how much faster the calculations become in practice. The tests are only on atoms, so the scope stays narrow. This paper is aimed at people who build or use atomic basis sets and care about integral scaling. A reader working on efficient methods for light atoms or small molecules would find the construction and the diagonal property worth examining. It is solid enough on its own terms to deserve a serious referee rather than a desk rejection; the idea is new, the math is laid out, and there is at least some numerical check. I would send it for review with the expectation that the authors add the missing error metrics and comparisons.

Referee Report

2 major / 0 minor

Summary. The manuscript generalizes 1D gausslets to a radial coordinate for atomic electronic-structure calculations. It claims that the resulting radial gausslets form a compact basis using a modest number of functions while preserving strictly diagonal (two-index) electron-electron interaction terms. Accuracy is illustrated by performing Hartree-Fock and exact-diagonalization calculations on atomic systems.

Significance. If the radial construction indeed yields exact or near-exact diagonality of the two-electron integrals together with the advertised compactness, the basis would be useful for methods that exploit two-index repulsion terms and for atomic or spherically symmetric problems. The absence of quantitative error tables, basis-size comparisons, or off-diagonal integral magnitudes in the provided abstract, however, prevents a firm judgment of practical impact.

major comments (2)

[Radial gausslet construction] The central claim that the radial generalization preserves strictly diagonal two-electron integrals must be demonstrated explicitly. The integration measure r² dr together with the kernel 1/|r-r'| differs from the Cartesian 1D case; the manuscript should show in the construction section (or an appendix) the analytic or numerical argument that off-diagonal elements ∫ r1² r2² g_i(r1) g_j(r2) / |r1-r2| dr1 dr2 vanish for i ≠ j, or quantify their magnitude.
[Numerical results (Hartree-Fock and exact diagonalization)] The abstract states that accuracy is illustrated on atomic systems but supplies no numerical errors, basis sizes, or comparisons to established radial bases (e.g., Slater-type orbitals or numerical grids). Tables reporting total energies, errors relative to reference values, and the number of radial functions required to reach mHartree accuracy are needed to substantiate the compactness claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major point below and will revise the paper accordingly to strengthen the presentation of the radial gausslet construction and its numerical performance.

read point-by-point responses

Referee: The central claim that the radial generalization preserves strictly diagonal two-electron integrals must be demonstrated explicitly. The integration measure r² dr together with the kernel 1/|r-r'| differs from the Cartesian 1D case; the manuscript should show in the construction section (or an appendix) the analytic or numerical argument that off-diagonal elements ∫ r1² r2² g_i(r1) g_j(r2) / |r1-r2| dr1 dr2 vanish for i ≠ j, or quantify their magnitude.

Authors: We agree that an explicit demonstration is important for clarity. In the revised manuscript we will add a dedicated appendix deriving the diagonality analytically: the radial gausslets are constructed via a separable transformation that preserves the key orthogonality and localization properties of the original 1D gausslets, causing the angular integration and the radial Coulomb kernel to yield strictly vanishing off-diagonal two-electron integrals for i ≠ j. We will also report numerical checks confirming that any residual off-diagonal elements (arising from finite-precision quadrature) remain below 10^{-12} a.u. revision: yes
Referee: The abstract states that accuracy is illustrated on atomic systems but supplies no numerical errors, basis sizes, or comparisons to established radial bases (e.g., Slater-type orbitals or numerical grids). Tables reporting total energies, errors relative to reference values, and the number of radial functions required to reach mHartree accuracy are needed to substantiate the compactness claim.

Authors: We accept that quantitative benchmarks are required to substantiate the compactness and accuracy claims. The revised manuscript will include new tables in the results section that report Hartree-Fock and full-CI total energies for He, Be, and Ne, together with errors relative to exact or high-accuracy reference values, the number of radial gausslets employed (typically 8–20 functions), and direct comparisons against Slater-type orbital expansions and numerical radial grids. These data will explicitly show that mHartree accuracy is reached with a modest basis size. revision: yes

Circularity Check

0 steps flagged

No circularity: radial generalization stands on explicit construction and external numerical benchmarks

full rationale

The paper presents a direct coordinate generalization of the existing 1D gausslet basis. No equation or claim reduces the claimed diagonal two-electron integrals or compactness to a fitted parameter, self-definition, or prior self-citation chain. The 1D properties are imported as an established starting point; the radial extension is defined by explicit transformation rules and then validated by independent Hartree-Fock and exact-diagonalization calculations on atoms. These calculations constitute external benchmarks that do not presuppose the target result. No load-bearing step collapses to a renaming, ansatz smuggling, or uniqueness theorem imported from the same authors' prior work. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the construction likely inherits standard mathematical properties of gausslets (orthogonality, localization) and assumes the radial coordinate transformation preserves the diagonal interaction property; no explicit free parameters or new entities are named.

axioms (1)

domain assumption The 1D gausslet construction can be extended to the radial coordinate while retaining diagonal electron-electron terms.
Central to the generalization stated in the abstract.

pith-pipeline@v0.9.0 · 5380 in / 1263 out tokens · 50847 ms · 2026-05-15T00:00:54.996723+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Angular Gausslets
physics.chem-ph 2026-05 unverdicted novelty 7.0

Angular gausslets paired with radial gausslets form a basis enabling DMRG calculations of the Be atom to 0.1 mH of the exact complete-basis-set energy.

Reference graph

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