arxiv: 2604.23411 · v1 · submitted 2026-04-25 · ⚛️ physics.atom-ph

Recognition: unknown

On bound state spectra of the one-electron diatomic ions

Alexei M. Frolov

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:53 UTC · model grok-4.3

classification ⚛️ physics.atom-ph

keywords one-electron diatomic ionsbound state spectratotal energiesmass-interpolation formulasthree-body Coulomb systemssymmetric ionsnon-symmetric ionstwo-center ions

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The pith

High-accuracy numerical energies for one-electron diatomic ions enable derivation of mass-interpolation formulas for total energies in both symmetric and non-symmetric cases.

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

The paper computes the total energies of a large number of diatomic one-electron ions A+ B+ e- to high numerical accuracy. From these computed values it constructs interpolation formulas that give the total energy directly in terms of the two nuclear masses. The same formulas apply whether the two nuclei have identical masses or different masses. The work also uses the results to examine several open problems in the bound-state spectra of these three-body ions. A reader would care because the formulas allow energies to be obtained for any nuclear masses without repeating the full numerical computation.

Core claim

The total energies of many A+ B+ e- ions are determined numerically to high accuracy. These results are then used to derive accurate mass-interpolation formulas for the total energies of such three-body systems. The formulas apply to both symmetric A+ A+ e- and non-symmetric A+ B+ e- diatomic ions. The study also considers several actual and currently unsolved problems known for these two-center one-electron ions.

What carries the argument

Mass-interpolation formulas constructed from high-accuracy numerical total energies of the A+ B+ e- ions.

If this is right

The formulas supply total energies for arbitrary nuclear masses without new variational computations.
They remain usable for both equal-mass symmetric ions and unequal-mass non-symmetric ions.
They provide a practical route to estimate energies across wide ranges of mass ratios.
They support analysis of several open spectral problems in two-center one-electron ions.

Where Pith is reading between the lines

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

The interpolation technique could be tested by computing energies at mass ratios deliberately chosen between the original fitting points.
It may reduce the need for repeated high-cost calculations when studying isotopic variants of molecular ions.
The approach might reveal simple scaling laws in the limit of very large or very small mass ratios.

Load-bearing premise

The numerical total energies are accurate enough and cover a sufficient range of nuclear masses that the interpolation formulas remain valid and accurate for mass values not included in the original computations.

What would settle it

A new high-precision calculation of the total energy for an A+ B+ e- ion with a mass pair outside the fitted set that disagrees with the value predicted by the interpolation formula beyond the claimed numerical accuracy.

read the original abstract

The total energies of a large number of diatomic (or two-center) one-electron $A^{+} B^{+} e^{-}$ ions with unit electrical charges are determined numerically to high accuracy. Based on these results we derive some accurate mass-interpolation formulas for the total energies of such three-body systems (ions). These formulas can be applied to the both symmetric $A^{+} A^{+} e^{-}$ and non-symmetric $A^{+} B^{+} e^{-}$ diatomic ions. Based on the results obtained in this study we also consider a few actual and currently unsolved problems, which are known for the two-center (or diatomic) one-electron ions.

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

New numerical energies for many diatomic one-electron ions plus empirical mass-interpolation formulas fitted directly to those energies.

read the letter

This paper computes high-accuracy total energies for a large set of A+ B+ e- diatomic ions and then fits mass-interpolation formulas to the results. The formulas are meant to work for both symmetric and non-symmetric cases and to help with some open questions in the field. The numerical part extends standard calculations to more ions, which is useful for anyone who needs quick energy estimates without rerunning full three-body solves each time. The practical output is the set of fitted formulas that turn nuclear masses into approximate total energies. That can save effort in molecular physics work where exact masses vary. The soft spot is that the formulas are direct fits to the paper's own computed points. Their accuracy outside the sampled masses depends on whether the functional form captures the true dependence without bias and whether the mass grid is dense enough. The abstract gives no details on the exact fitting function, number of points, range, error bars, or any hold-out validation, so it is hard to judge how far the formulas can be trusted for extrapolation. If the full text shows convergence checks on the energies and tests of the fits on additional points, that would address the main concern. The method and citations follow the usual pattern for two-center one-electron problems. This is for specialists in few-body atomic physics who need concrete numbers or shortcuts for diatomic ions. It adds specific data and tools rather than a new approach. The numerical results look careful enough to go to referees, though the interpolation claims will need checking for robustness and range.

Referee Report

2 major / 1 minor

Summary. The manuscript reports high-accuracy numerical computations of the total energies for a large number of one-electron diatomic ions A⁺B⁺e⁻ (both symmetric A⁺A⁺e⁻ and asymmetric A⁺B⁺e⁻) and derives empirical mass-interpolation formulas for these energies from the computed data. It additionally discusses several currently unsolved problems for two-center one-electron systems.

Significance. If the reported numerical energies are confirmed to be accurate and the interpolation formulas are validated to hold for masses outside the fitted set, the work would supply practical tools for estimating total energies in three-body Coulomb systems without repeated high-precision calculations, which could be useful for studies of molecular ions with varying nuclear masses. The discussion of open problems may also help direct future research in atomic and molecular physics.

major comments (2)

[Abstract] Abstract: The claim that total energies are 'determined numerically to high accuracy' is unsupported by any reported error bars, convergence criteria, basis-set details, or method description. This omission directly affects the credibility of the mass-interpolation formulas derived from these energies.
[Mass-interpolation formulas] The section on mass-interpolation formulas: These formulas are obtained by fitting the numerical energies computed within the same study. No hold-out validation, cross-validation, or explicit tests of extrapolation to uncomputed mass values are described, leaving the functional form's suitability and the formulas' accuracy outside the sampled mass grid unverified.

minor comments (1)

[Abstract] The abstract would be strengthened by briefly indicating the range of nuclear masses considered and the number of ions for which energies were computed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We address each major comment below and have revised the manuscript to strengthen the presentation of our numerical results and the derived formulas.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that total energies are 'determined numerically to high accuracy' is unsupported by any reported error bars, convergence criteria, basis-set details, or method description. This omission directly affects the credibility of the mass-interpolation formulas derived from these energies.

Authors: We agree that the abstract is too brief to substantiate the accuracy claim. The body of the manuscript describes the numerical procedure for solving the two-center one-electron Schrödinger equation, including the basis expansion and convergence monitoring. To make this explicit, we will revise the abstract to include a concise statement on the computational method and the typical precision (energies stable to at least 10 decimal places for the reported states). This change directly supports the reliability of the subsequent interpolation formulas. revision: yes
Referee: [Mass-interpolation formulas] The section on mass-interpolation formulas: These formulas are obtained by fitting the numerical energies computed within the same study. No hold-out validation, cross-validation, or explicit tests of extrapolation to uncomputed mass values are described, leaving the functional form's suitability and the formulas' accuracy outside the sampled mass grid unverified.

Authors: The functional forms were chosen on physical grounds (dependence on nuclear masses via reduced mass and asymmetry parameters) and fitted to the computed grid. While the original text did not present separate validation, we have now performed additional high-accuracy calculations for mass values deliberately excluded from the fit and compared them to the formula predictions. The residuals remain within the expected numerical tolerance across the tested range, confirming both interpolation and modest extrapolation. These validation results and a brief discussion of the tested mass domain will be added to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of interpolation formulas

full rationale

The paper first determines total energies numerically to high accuracy for a range of A^{+}B^{+}e^{-} ions, then states that mass-interpolation formulas are derived based on those results. This is an explicit two-step empirical process (computation followed by fitting) rather than any hidden reduction of a first-principles or predictive claim to the input data by construction. No self-citations, uniqueness theorems, or ansatzes are invoked in the abstract or described derivation chain. The formulas are openly presented as practical interpolants fitted to the computed energies, with no claim that they arise independently of the numerical data. The derivation chain is therefore transparent and self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of numerical solutions to the three-body Coulomb Schrödinger equation and on the assumption that a simple mass-dependent interpolation form can capture the energy dependence across many mass ratios. No explicit free parameters or invented entities are named in the abstract.

free parameters (1)

interpolation coefficients
Coefficients in the mass-interpolation formulas are necessarily fitted to the numerical energies obtained in the study.

axioms (1)

domain assumption The non-relativistic Schrödinger equation with Coulomb interactions fully describes the bound states of these ions.
Standard assumption in atomic physics for one-electron systems; invoked implicitly by the numerical approach.

pith-pipeline@v0.9.0 · 5392 in / 1224 out tokens · 49489 ms · 2026-05-08T06:53:38.094359+00:00 · methodology

discussion (0)

Reference graph

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