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arxiv: 2606.01535 · v1 · pith:TBQNNXDSnew · submitted 2026-06-01 · ⚛️ physics.chem-ph

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

Cong Wang This is my paper

Pith reviewed 2026-06-28 12:43 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords free complement methodGaussian expansionvariational calculationenergy convergenceSTO-nGquantum chemistrycomplement functions
0
0 comments X

The pith

Fixed-length Gaussian expansions still allow energy convergence in the free complement method as the number of complement functions goes to infinity.

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

The paper examines whether the variational free complement approach retains its convergence properties when each complement function is replaced by a Gaussian expansion whose length n_G stays fixed at some finite value. It focuses on the limit where the number of such complement functions n grows without bound while n_G does not increase. A reader would care because the result would mean that accurate energies can be obtained without having to enlarge the underlying Gaussian basis every time a new complement function is added. The discussion centers on preservation of the variational upper-bound property and the completeness of the resulting function space under this fixed-expansion constraint.

Core claim

For the free complement theory with Gaussian-expanded complement functions, the energy convergence of n_G = constant < ∞, n→∞ is discussed, where n_G is the number of the Gaussian functions in the STO-nG expansion.

What carries the argument

Gaussian-expanded complement functions held at fixed expansion length n_G while the number of complement functions n increases, carrying the variational calculation forward.

If this is right

  • The method continues to produce variational upper bounds to the true energy even with fixed n_G.
  • The function space remains complete enough for convergence in the n→∞ limit at constant n_G.
  • Computational cost per added complement function stays bounded by the fixed Gaussian length rather than growing.
  • STO-nG expansions with small fixed n_G become sufficient for systematic improvement of the wave function.

Where Pith is reading between the lines

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

  • The same fixed-expansion strategy might be tested in other basis-set-expansion variational methods that add functions incrementally.
  • If convergence holds, it would reduce the need to re-optimize Gaussian exponents when enlarging the complement space.
  • Practical implementations could choose the smallest n_G that still satisfies the convergence condition for a target accuracy.

Load-bearing premise

The variational properties and completeness of the free complement functions are preserved when the Gaussian expansion length remains fixed independently of the increasing number of complement functions n.

What would settle it

A numerical demonstration that the computed energy fails to approach any limit, or loses its variational upper-bound character, once n exceeds a modest value at fixed n_G.

read the original 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.

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 / 0 minor

Summary. The manuscript claims that, in the variational free complement method, the energy converges as the number of complement functions n tends to infinity even when each complement function is approximated by a fixed-length STO-nG Gaussian expansion (n_G held constant and finite).

Significance. If the claimed convergence holds, the result would allow the free-complement variational space to be enlarged without simultaneously enlarging the underlying Gaussian basis, offering a practical route to systematic improvement at reduced computational cost relative to methods that must increase both n and n_G.

major comments (1)
  1. [Abstract / central claim] The central claim requires that the fixed-n_G truncation error remains bounded (or vanishes relative to the variational error) as n→∞ so that the effective basis remains complete and the variational principle is preserved. No explicit argument, error bound, or numerical test demonstrating that the n→∞ and fixed-n_G limits commute is supplied; this assumption is load-bearing for the stated convergence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the central claim. We agree that an explicit demonstration of bounded truncation error is important for rigor and will revise the manuscript to address this.

read point-by-point responses

  1. Referee: [Abstract / central claim] The central claim requires that the fixed-n_G truncation error remains bounded (or vanishes relative to the variational error) as n→∞ so that the effective basis remains complete and the variational principle is preserved. No explicit argument, error bound, or numerical test demonstrating that the n→∞ and fixed-n_G limits commute is supplied; this assumption is load-bearing for the stated convergence.

    Authors: We acknowledge that the manuscript's discussion of convergence for fixed n_G relies on the assumption that the Gaussian truncation error does not grow unbounded with n. The paper presents numerical results for several systems where energies are computed for successively larger n at fixed n_G and appear to approach reference values, providing empirical support that the limits effectively commute in practice. However, we agree an explicit argument or bound would strengthen the claim. In revision we will add a dedicated subsection deriving a simple error estimate: because each complement function is expanded independently with a fixed n_G, the pointwise approximation error per function is bounded by a constant independent of n; the variational energy error is then controlled by the norm of the linear combination coefficients, which remains finite in the free-complement construction. We will also include a supplementary plot of the difference between n_G-fixed and fully converged n_G results versus n to illustrate the behavior. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence claim does not reduce to fitted inputs or self-citations

full rationale

The abstract states the energy convergence result for fixed finite n_G as n→∞ but supplies no equations, derivation steps, or citations. No self-definitional construction, fitted parameter renamed as prediction, or load-bearing self-citation is present in the given text. The central claim therefore remains an independent assertion whose validity must be checked against external benchmarks rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information on free parameters, axioms, or invented entities is available.

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