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arxiv: 2605.18488 · v1 · pith:2LWFKU54new · submitted 2026-05-18 · ⚛️ physics.comp-ph · physics.atom-ph

SPARC-atomSFE: Spectral finite-element package for atomic structure calculations in density functional theory

Qihao Cheng , Shubhang Krishnakant Trivedi , Phanish Suryanarayana This is my paper

Pith reviewed 2026-05-20 02:13 UTC · model grok-4.3

classification ⚛️ physics.comp-ph physics.atom-ph
keywords atomic structure calculationsdensity functional theoryspectral finite elementKohn-Sham equationsexchange-correlation functionalspseudopotentialshybrid functionalsrandom phase approximation
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The pith

The SPARC-atomSFE package performs atomic Kohn-Sham DFT calculations to 1 micro-Hartree accuracy across local to nonlocal functionals using spectral finite elements.

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

The paper introduces SPARC-atomSFE as a new computational package for atomic structure calculations inside Kohn-Sham density functional theory. It discretizes the radial equations on an adaptive grid with high-order Lagrange polynomial basis functions placed at Legendre-Gauss-Lobatto nodes and integrates with Gauss-Legendre quadrature. The implementation covers all-electron and norm-conserving pseudopotential treatments for a hierarchy of exchange-correlation approximations that includes hybrids and the random phase approximation. Validation shows that total energies and other quantities match literature references to within 1 micro-Hartree once the reported grid and polynomial parameters are used. Readers interested in reliable atomic benchmarks would care because these calculations test functionals that later appear in molecular and solid-state work.

Core claim

SPARC-atomSFE solves the atomic Kohn-Sham equations by means of a spectral finite-element discretization on an adaptive radial grid. Element nodes follow the Legendre-Gauss-Lobatto distribution, the basis consists of high-order C0-continuous Lagrange polynomials, and integrals are evaluated by Gauss-Legendre quadrature. The code supports both all-electron and pseudopotential calculations and implements local, semilocal, hybrid, and random-phase-approximation functionals, the latter via either the generalized Kohn-Sham or optimized-effective-potential route. Systematic convergence studies identify the parameters needed for target accuracy, and representative calculations confirm agreement to

What carries the argument

Adaptive spectral finite-element discretization of the radial Kohn-Sham problem using high-order Lagrange polynomials on Legendre-Gauss-Lobatto nodes together with Gauss-Legendre quadrature.

If this is right

  • Both generalized Kohn-Sham and optimized-effective-potential routes become available for eigenvalue-dependent functionals such as hybrids and RPA.
  • Users can reach any target accuracy by systematically refining the radial grid or raising the polynomial degree.
  • All-electron and norm-conserving pseudopotential results are obtained inside the same code and discretization framework.
  • Atomic energies serve as clean benchmarks for developing or validating new exchange-correlation approximations.

Where Pith is reading between the lines

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

  • The adaptive radial mesh could be generalized to multi-center molecular geometries by allowing element refinement around each nucleus.
  • Direct timing comparisons against established atomic codes that use finite differences or B-splines would reveal any efficiency gains from the spectral-element approach.
  • The OEP implementation for nonlocal functionals opens a route to other advanced many-body corrections that depend on orbital eigenvalues.

Load-bearing premise

The chosen adaptive grid, polynomial degree, and quadrature rules remain free of significant numerical artifacts for the tested atoms and functionals once the reported convergence parameters are used.

What would settle it

A total-energy calculation for the neon atom in the LDA functional that deviates from the accepted literature value by more than 1 micro-Hartree when the package is run at the convergence settings stated in the paper.

Figures

Figures reproduced from arXiv: 2605.18488 by Phanish Suryanarayana, Qihao Cheng, Shubhang Krishnakant Trivedi.

Figure 1
Figure 1. Figure 1: Illustration of the spectral finite-element framework employed in SPARC-atomSFE. Left: domain of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of eigenvalue errors for the radial Schrödinger equation corresponding to [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of the total energy and occupied eigenvalues with respect to domain size [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of SPARC-atomSFE all-electron Kohn-Sham DFT results with literature for various exchange-correlation [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the total energy and occupied eigenvalues with respect to domain size [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of SPARC-atomSFE all-electron Kohn-Sham DFT results with literature for various exchange-correlation [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
read the original abstract

We present SPARC-atomSFE, a spectral finite-element package for accurate and efficient atomic structure calculations within the framework of Kohn-Sham density functional theory. The package supports both all-electron and norm conserving pseudopotential calculations across a comprehensive hierarchy of exchange-correlation approximations, spanning local, semilocal, and nonlocal functionals. The latter includes hybrid functionals and the many-body random phase approximation, for which we implement both the generalized Kohn-Sham approach and the optimized effective potential (OEP) method, with OEP necessary for eigenvalue-dependent functionals. Spatial discretization is based on an adaptive grid with element nodes distributed according to the Legendre--Gauss--Lobatto scheme, high-order $C^{0}$-continuous Lagrange polynomial basis functions, and Gauss--Legendre quadrature for numerical integration. We present systematic convergence studies and identify the computational parameters required to achieve target accuracies. We validate the accuracy of SPARC-atomSFE through representative calculations spanning the various exchange-correlations approximations, obtaining results that generally agree with values in the literature to within $1~\mu\text{Ha}$ or better.

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.

Referee Report

1 major / 4 minor

Summary. The manuscript presents SPARC-atomSFE, a spectral finite-element code for Kohn-Sham DFT atomic calculations. It supports all-electron and norm-conserving pseudopotential treatments across local, semilocal, hybrid, and RPA functionals, implementing both generalized Kohn-Sham and optimized effective potential (OEP) approaches for eigenvalue-dependent cases. Discretization relies on an adaptive radial grid with Legendre-Gauss-Lobatto nodes, C0 Lagrange polynomials of high order, and Gauss-Legendre quadrature. Systematic convergence studies are used to select parameters for target accuracy, and the code is validated by direct comparison to literature total energies and eigenvalues, with reported agreement generally within 1 μHa or better.

Significance. If the reported numerical accuracies are confirmed, the package supplies a useful, reproducible platform for high-precision atomic reference data, especially for advanced nonlocal functionals that require OEP. The combination of adaptive high-order elements with broad functional coverage and explicit convergence diagnostics adds value for method benchmarking and functional testing in atomic DFT.

major comments (1)
  1. [Convergence studies and validation sections] The central accuracy claim (agreement to 1 μHa) rests on the assertion that the chosen adaptive-grid, polynomial degree, and quadrature settings are fully converged. The manuscript should include explicit refinement tables or figures (e.g., energy change upon increasing polynomial degree by 2 or halving the adaptive mesh spacing) for the most demanding cases: all-electron runs near the nuclear cusp and OEP calculations with eigenvalue-dependent functionals. Without such data, it is difficult to confirm that discretization error lies well below the target threshold for every functional class shown.
minor comments (4)
  1. [Abstract] Abstract, line 3: 'exchange-correlations approximations' should read 'exchange-correlation approximations'.
  2. [Discretization section] The notation for the radial adaptive grid and the mapping from reference element to physical element should be stated once in a single equation block rather than repeated in prose.
  3. [Figures] Figure captions for the convergence plots would benefit from explicit mention of the atom, functional, and quantity (total energy or eigenvalue) being plotted, together with the final parameter set used for the validation table.
  4. [Results] A short paragraph comparing wall-time or iteration counts against at least one existing atomic DFT code (e.g., a finite-difference or Gaussian-basis implementation) would help readers assess the practical efficiency of the spectral-element approach.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the constructive recommendation for minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Convergence studies and validation sections] The central accuracy claim (agreement to 1 μHa) rests on the assertion that the chosen adaptive-grid, polynomial degree, and quadrature settings are fully converged. The manuscript should include explicit refinement tables or figures (e.g., energy change upon increasing polynomial degree by 2 or halving the adaptive mesh spacing) for the most demanding cases: all-electron runs near the nuclear cusp and OEP calculations with eigenvalue-dependent functionals. Without such data, it is difficult to confirm that discretization error lies well below the target threshold for every functional class shown.

    Authors: We agree that additional explicit refinement data would further strengthen the convergence evidence presented in the manuscript. While systematic convergence studies were performed to identify parameters achieving the target accuracy, we will incorporate new tables and figures in the revised manuscript. These will report energy changes upon increasing the polynomial degree by 2 and upon halving the adaptive mesh spacing, specifically for the most demanding cases of all-electron calculations near the nuclear cusp and OEP calculations involving eigenvalue-dependent functionals. This will explicitly demonstrate that discretization errors remain well below the 1 μHa threshold across the functional classes considered. revision: yes

Circularity Check

0 steps flagged

No circularity: SPARC-atomSFE is a software implementation validated against independent external literature

full rationale

The manuscript presents a spectral finite-element code for atomic Kohn-Sham DFT calculations. It adopts standard discretization (adaptive radial grid, Legendre-Gauss-Lobatto nodes, C0 Lagrange polynomials, Gauss-Legendre quadrature) and reports systematic convergence studies followed by direct numerical comparison of total energies and eigenvalues to previously published reference values across local, semilocal, hybrid, and RPA functionals. No result is obtained by fitting a parameter to a subset of the same data and then relabeling it a prediction; no central premise rests on a self-citation chain whose own justification is internal to the present work; and the validation step is explicitly benchmarked against external literature rather than closed under the code's own outputs. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard validity of Kohn-Sham DFT for atoms and on the numerical convergence of the chosen finite-element discretization; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption Kohn-Sham density functional theory provides a sufficiently accurate framework for the atomic systems and properties considered.
    Invoked throughout the description of all-electron and pseudopotential calculations.

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