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Variational free complement method with Gaussian-expanded complement functions: convergence with fixed Gaussian expansion length

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Pith's one-line read The variational free complement method achieves energy convergence with a fixed finite number of Gaussians in the complement function expansion as the order increases to infinity.

desk verdict This paper analyzes whether energy convergence still holds in the free complement method when the Gaussian expansion length n_G is held fixed while the complement order n goes to infinity. read the letter →

arxiv 2606.01535 v2 pith:TBQNNXDS submitted 2026-06-01 physics.chem-ph

Cong Wang This is my paper · ORCID
classification physics.chem-ph
keywords freecomplementmethodGaussian-expandedfunctionsenergyconvergencevariationalSTO-nGexpansionquantumchemistrycalculations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper investigates the convergence properties of the free complement theory when the complement functions are expanded using a fixed number of Gaussian functions from an STO-nG basis. It shows that even with this fixed expansion length, the variational energy converges to the exact value as the number of complement functions grows without limit. A sympathetic reader would care because this suggests that accurate quantum chemical calculations can be performed without continuously increasing the Gaussian basis size alongside the complement order, potentially simplifying numerical implementations. The discussion focuses on the conditions under which this fixed n_G leads to convergence.

What carries the argument

The free complement method with complement functions approximated by a fixed-length Gaussian expansion (STO-nG with constant n_G).

What would settle it

Computing the variational energy for successively larger n with a fixed small n_G (such as n_G=1 or 2) and checking if the energy error relative to the exact energy tends to zero; failure to converge would disprove the claim.

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Extended reading notes

Core claim

For the free complement theory with Gaussian-expanded complement functions, the energy converges when n_G equals a constant less than infinity while n approaches infinity.

Load-bearing premise

The fixed Gaussian expansion of the complement functions stays sufficiently complete and accurate even as the number of complement functions increases without bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The manuscript discusses the energy convergence behavior in the variational free complement method when complement functions are expanded using a fixed finite number of Gaussians (n_G = constant < ∞ in an STO-nG expansion) while the number of complement functions n tends to infinity.

Significance. If the claimed convergence holds under the fixed n_G condition, the result would support practical implementations of the free complement approach that avoid scaling the Gaussian expansion length with the complement space size, which could improve efficiency in variational quantum chemistry calculations.

minor comments (1)
  1. The abstract states only that convergence 'is discussed'; the manuscript should explicitly state whether this is supported by a derivation, theorem, or numerical demonstration, and identify the relevant section or equation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript on the convergence properties of the variational free complement method under fixed finite Gaussian expansion length (n_G constant) as the complement order n tends to infinity. The referee's summary accurately captures the scope of the work, and we note the potential practical implications highlighted in the significance section. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The abstract states that the paper discusses energy convergence for fixed finite n_G while n→∞ in the free complement method with Gaussian-expanded complement functions. No equations, derivations, self-citations, or fitted parameters are provided in the given text that would allow identification of any reduction by construction, self-definitional steps, or load-bearing self-citations. The central claim is a technical discussion of a limit, and without quoted material exhibiting circularity per the strict rules (exact quote + specific reduction), the finding is no significant circularity. This is the expected outcome for papers whose derivation remains self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The central statement relies on the prior existence of the free complement theory and STO-nG expansions, which are treated as background.

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Cite this review

Pith. "Pith review of Variational free complement method with Gaussian-expanded complement functions: convergence with fixed Gaussian expansion length." pith.science (2026). https://pith.science/paper/TBQNNXDS

@misc{pith2026260601535,
  author       = {Pith},
  title        = {Pith review of: Variational free complement method with Gaussian-expanded complement functions: convergence with fixed Gaussian expansion length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBQNNXDS}},
  note         = {Machine review of arXiv:2606.01535}
}
abstract

For the free complement theory with Gaussian-expanded complement functions, the energy convergence of $n_\mathrm{G} = \mathrm{constant} < \infty, n\rightarrow\infty$ is discussed, where $n_\mathrm{G}$ is the number of the Gaussian functions in the STO-$n$G expansion.

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