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REVIEW 4 major objections 5 minor 31 references

Helium-like ions $(Z,e,e)$ in Lagrange Mesh method, interpolating the highly-accurate energy spectra vs. Z

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that one nine-parameter rational function reproduces the lowest S-state energies of helium-like ions for all relevant $Z$, with the Lagrange mesh method supplying 14–15 digit benchmarks.

desk verdict The excited-state LMM tables are a plausible new dataset, but the 10-14 s.d. interpolation claim is unsupported: it is validated only in-sample and the paper contains internal coefficient mismatches. read the letter →

arxiv 2608.10695 v1 pith:6WO6ZA5U submitted 2026-08-11 quant-ph math-phmath.MPphysics.atom-ph

H Olivares-Pilon , JC Lopez-Vieyra , AV Turbiner This is my paper · ORCID
classification quant-phmath-phmath.MPphysics.atom-ph
keywords helium-likeionsLagrangemeshmethodtwo-pointPadéapproximantsecondcriticalchargePuiseuxexpansion1/Zenergyinterpolation
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

This paper tries to establish that the non-relativistic energies of the lowest S-states of helium-like ions — two electrons around a nucleus of charge $Z$ — are given for every physically relevant $Z$ by a single nine-parameter rational function of $\lambda = (Z - Z_B)^{1/2}$, where $Z_B$ is the second critical charge at which the state ceases to exist. To supply this claim with data, it uses the Lagrange mesh method, which it argues reaches 14–15 significant digits for $Z \leq 50$ with modest computing effort, including $Z = 20$–$50$ where no accurate references existed before. The picture rests on matching the large-$Z$ $1/Z$ expansion to the Puiseux expansion in half-integer powers around $Z_B$, a branch-point structure previously confirmed for the ground state and here extended to the first excited singlet and two lowest triplet S-states. If correct, the result replaces dozens of separate high-precision calculations with one closed-form energy curve that is exact as $Z \to \infty$.

What carries the argument

The machinery is a generalized two-point Padé approximant, equation (13): the ratio of a degree-9 polynomial to a degree-5 polynomial in $\lambda = \sqrt{Z - Z_B}$. The substitution $\lambda^2 = Z - Z_B$ converts the Puiseux expansion at the second critical charge into a Taylor series at $\lambda = 0$ and the $1/Z$ expansion into a Laurent series at infinity, so the rational function interpolates between the two asymptotic regimes. Four constraints (14) impose the exact leading terms: the constant $E_B$ and the vanishing of the $\lambda$ term for the singlets at $Z_B$, and the exact $\epsilon_0 Z^2$ and $\epsilon_1 Z$ coefficients at large $Z$. The remaining nine free parameters are fitted to the Lagrange-mesh energies, and the poles of the approximant are reported to lie away from the physical $\lambda \geq 0$ semi-axis, avoiding spurious wiggles in the energy.

What would settle it

Compute the energy of, say, the $1^1S$ state at $Z = 15$ or $Z = 35$ with an independent high-precision variational method to 12 decimal digits and compare with equation (13); disagreement beyond the claimed 10–14 significant digits would falsify the interpolation ansatz. Alternatively, numerically continue the fitted Padé function to complex $\lambda$ and test whether the nearest singularity to the physical axis is exactly the asserted branch point $Z_B$, and whether increasing the degree $N$ changes the predicted energies by more than the claimed accuracy.

Watch

Extended reading notes

Core claim

The central claim is that equation (13), the two-point Padé approximant $E(\lambda) = P_9(\lambda)/Q_5(\lambda)$ in $\lambda = (Z-Z_B)^{1/2}$, reproduces the energies of the $1^1S$, $2^1S$, $2^3S$, and $3^3S$ states to 10–14 significant digits over the entire physically relevant range of $Z$. The interpolant is constrained so that, at $Z = Z_B$, it yields the constant term $E_B$ and, for the singlet states, sets the half-integer term $(Z-Z_B)^{1/2}$ to zero; at large $Z$, it reproduces exactly the leading coefficients $\epsilon_0$ and $\epsilon_1$ of the $1/Z$ expansion, so the error vanishes as $Z \to \infty$. The remaining nine parameters are fitted to the Lagrange-mesh data of Tables I–IV, and the paper reports at least 12 s.d. agreement for the ground state, 9–12 s.d. for $2^1S$, at least 13 s.d. for $2^3S$, and all 14 s.d. for $3^3S$ across $Z \in [1,50]$. The paper also extends the second-critical-charge picture: the two singlet states share $Z_B \approx 0.904854$ and are analytically connected through a square-root branch point, as are the two triplet states at $Z_B \approx 0.8799$.

Load-bearing premise

The load-bearing premise is that the true energy of each state traces out the exact smooth nine-parameter curve used for the fit, with no extra singularities nearer than the assumed critical charge; the curve is only checked against the same data points that produced it.

Editorial extensions

If this is right

  • The energies of the $1^1S$, $2^1S$, $2^3S$, and $3^3S$ states of any helium-like ion in the non-relativistic static approximation can be obtained from one closed-form rational function to 10–14 significant digits, with the error going to zero as $Z \to \infty$.
  • The Lagrange mesh method with a $50 \times 50 \times 40$ lattice gives 14–15 significant digits for these four states at $Z \leq 50$, filling the previously unknown $Z = 20$–$50$ region with first-time accurate results.
  • The second-critical-charge picture extends beyond the ground state: the two singlet states share one branch point and the two triplet states share another, linking the states through level crossings in the analytic continuation.
  • The same two-point Padé interpolation scheme is proposed as a general tool for any excited state of the helium-like sequence and, in principle, for other few-electron atomic systems, provided the critical charge and expansion coefficients are known.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the interpolant is a closed-form rational function, derivatives of the energy with respect to $Z$ are available analytically; the paper does not pursue this, but it could directly yield isotope shifts, polarizabilities, or other $Z$-derivative observables.
  • The paper fixes $Z_B$ from earlier fits rather than letting it float in the nine-parameter fit; fitting $Z_B$ as a free parameter would test whether the branch-point value is independently determined by the Lagrange-mesh data at the claimed accuracy.
  • The claimed generality to arbitrary excited states is demonstrated on only four S-states; a direct next test is to apply the same form to a higher Rydberg state or to a P-state, where the branch-point structure is not yet established.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper reports Lagrange-mesh method (LMM) calculations for four low-lying S-states of helium-like ions, 1^1S, 2^1S, 2^3S, and 3^3S, for Z up to 50, and claims 14-15 significant digits for this numerical method. It then constructs a generalized two-point Padé approximant in the variable lambda = sqrt(Z - Z_B), matching the large-Z 1/Z expansion and the small-|Z - Z_B| Puiseux expansion around a second critical charge Z_B. For each state the approximant is stated to have nine free parameters fitted to LMM data, and the paper claims 10-14 significant digits for any physically relevant Z, with absolute accuracy as Z -> infinity. Explicit parameters and comparison tables are provided for all four states.

Significance. If the accuracy claims hold, the paper would be a useful contribution: it provides an explicit, cheap interpolation formula that could replace separate high-precision variational calculations for each Z, and it offers LMM data for Z = 20-50 where the authors state no accurate comparison results exist. The tables are explicit, the method is reproducible, and the LMM results for Z <= 10 agree with independent high-precision references to roughly the claimed number of digits. However, the central interpolation claim is currently validated only in-sample, and there are internal inconsistencies between the text and the parameter tables that must be resolved before the claimed accuracy can be accepted.

major comments (4)
  1. [§IV.A, Tables I-IV] The central claim that the nine-parameter Padé approximant (13)/(16) reproduces 10-14 s.d. for 'any physically relevant Z' is not independently established. Section IV.A states that the parameters are fixed by fitting the LMM-based data in Table I, and Tables I-IV contain exactly the Z values shown in the accuracy comparison. Thus the agreement is an in-sample goodness-of-fit, not a predictive test. No held-out Z values (for example Z = 15, 25, 35, 45) and no cross-validation are reported, and for Z > 10 there is no independent high-precision data set used for comparison. Please provide an out-of-sample test or at least a residual/convergence analysis showing that the fitted form does not merely interpolate its training points.
  2. [§IV.B, Table V; §IV.C-D, Table VI] The text's claim that the interpolant reproduces the Puiseux coefficients is contradicted by the tables. Section IV.B says the coefficient p1 = -1.12349 is correctly reproduced for 2^1S, but Table V lists p1 = -0.843 for 2^1S. Similarly, Sections IV.C and IV.D say q1 = ±0.011065 and p1 = -0.884233 from Eq. (8) are reproduced, but Table VI lists q1 = -0.0210 and -0.009 and p1 = -0.8091 and -0.8736 for the two triplet states. This discrepancy is load-bearing because the Puiseux matching is the stated theoretical basis of the interpolation. Please clarify which values are outputs of the unconstrained fit and which are imposed constraints, and correct the text or the tables.
  3. [§II, Eq. (8), ref. [18]] The triplet-state input is not independently verifiable. The values Z_B = 0.8799, E_B = -0.384708, and the Puiseux coefficients in Eq. (8) are said to come from the unpublished work in progress [18], and they are obtained by fitting the same LMM data that is later used to assess the interpolation. The paper provides no derivation or external evidence for these quantities. Since the small-Z behavior of the Padé approximant depends on them, this is a load-bearing unverified input. Please include the supporting analysis or a reference where it is published.
  4. [§I, Tables II-IV (Z = 20-50)] The LMM accuracy claim of 14-15 s.d. for Z = 20-50 is not supported by any external comparison. The tables mark these rows as 'first calculation' with no accurate reference values, and the paper reports no convergence study with respect to lattice size for these Z values. The interpolation accuracy statement inherits this uncertainty, because the Padé parameters are fitted to these LMM numbers. Please add lattice-convergence evidence for large Z or explicitly qualify the accuracy claim as an internal LMM estimate.
minor comments (5)
  1. [Abstract and Conclusions] The phrase 'any physically-relevant nuclear charge Z' overstates the tested range. The tables cover Z = 1 to 50 plus the asymptotic limit; the accuracy at intermediate or larger Z is asserted, not demonstrated. Please qualify the statement to the computed range or provide additional test points.
  2. [Eqs. (11)-(13)] The parameter-count discussion is confusing: Eq. (11) says the total number of free parameters is 2N+5, then says fixing a0 = E_B reduces it to 2N+4, while Eq. (13) is later said to depend on 9 free parameters with four constraints. Please reconcile these counts in a way that matches the explicit form of Eq. (13).
  3. [§IV.B, Table II] The text says 12 s.d. are reproduced for Z <= 10 and then the accuracy decreases to 9 s.d. at Z = 40, but for Z = 40 the difference between the second and third columns is about 4.7 x 10^-8, which is roughly 10 significant digits in relative terms. The mixing of 'significant digits' and 'decimal digits' should be made consistent.
  4. [Footnote [24]] The paper relies on the ground-state Puiseux expansion, yet footnote [24] states that this expansion was never published in complete form. Please provide the complete expansion in an appendix or point to a published source that contains it.
  5. [Figure 1 caption] The caption describes the 2^3S state as a red dashed line and the 2^1S state as a gray dashed line, but the plot legend described in the caption may be difficult to distinguish. Please verify the line styles and colors so the figure is unambiguous.

Circularity Check

3 steps flagged · score 6.0 of 10

The claimed 10–14 s.d. accuracy of the Padé interpolation is supported only by in-sample fits to the same LMM energies used to determine its nine parameters, and the triplet critical charge is imported from an unpublished self-citation.

  1. fitted input called prediction [Sec. IV, paragraph following Eq. (12); Sec. IV.A]
    "The remaining nine free parameters in gPade(9/5)(λ)3,4 ... are found by fitting the available numerical data for the ground state, see [9]. Data for the excited states from Tables II-IV, obtained via the LMM, can be fitted with the intention of reproducing 13-14 s.d. in energies correctly. ... A comparison of the numerical LMM data against the two-point Padé approximant ones (see Table I) shows that the Padé approximant reproduces at least 12 s.d. in the energy for all Z ∈ [1, 50]."

    The nine free parameters of (13) are fitted to the very LMM energy columns that Tables I–IV then use as evidence of accuracy. With nine parameters and roughly thirteen fitted Z values per state, reproducing the fitted nodes is an in-sample goodness-of-fit, not a predictive test. No held-out Z-values (e.g., 15, 25, 35, 45) and no range outside [1,50] are checked, so the abstract's 'accuracy of 10-14 s.d. for any physically-relevant Z' is not independently established; it is a statement about the fitted data set.

  2. self citation load bearing [Sec. II, before Eq. (8)]
    "In [18] it is shown that by making a detailed analysis of the exponential decay rate of the accurate variational trial wavefunctions for various Z at large distances the second critical charges for the first two spin-triplet states 2^3S and 3^3S can be found: with high accuracy these critical charges coincide! They are equal to Z_B(2^3S) = Z_B(3^3S) = 0.8799."

    The entire Puiseux expansion (8) for the triplet states, and hence the branch point around which the two-point Padé interpolation for 2^3S and 3^3S is built, is anchored to Z_B = 0.8799 solely by reference [18]. Reference [18] is listed as 'work in progress' by two of the three present authors, so the load-bearing input is an unpublished self-citation. If this critical charge were incorrect, the claimed asymptotic and half-integer-power matching for the triplet states would lose its foundation; no independent calculation or external benchmark is supplied.

1 more flagged steps
  1. fitted input called prediction [Sec. II after Eq. (6); Sec. IV.A]
    "Coefficients {p, q} are found by making an interpolation (6) with accuracy 3-5 s.d. of the thirteen energies at Z = 0.904854, 0.91, 0.91028..., 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00, 2.00, respectively, calculated using the LMM, see [9] [24]. ... The next terms in the expansions with coefficients p1 = −1.12349 and ǫ2 = −0.15766643, respectively, are reproduced accurately, as shown in Table V."

    The Puiseux coefficients {p,q} are themselves fit outputs from the same LMM energies, and the Padé parameters are also fitted to the same data. Thus the 'reproduction' of p1 and q1 by the Padé formula is a comparison between two fits of the same numbers, not an independent confirmation of the branch-point expansion. Moreover, the claimed p1 = −1.12349 disagrees with Table V, which lists p1 = −1.118 for 1^1S and −0.843 for 2^1S; likewise Eq. (8) gives q1 = ±0.011065 for triplets while Table VI lists −0.0210 and −0.009, so the asserted reproduction is not even numerically consistent.

full rationale

The paper is not circular at the Hamiltonian level: the Lagrange-mesh method is a genuine numerical solver, and the exact 1/Z coefficients ǫ0 and ǫ1 are independent known inputs that the Padé form matches by construction. For Z = 1–10 the LMM energies also agree with external variational results, giving the underlying numerical data some independent support. The circularity lies in the central validation of the interpolation claim: all nine free parameters are fitted to the LMM data in Tables I–IV, and those same tables are then presented as demonstrating 12–14 s.d. accuracy, with no held-out Z values. The triplet critical charge, the foundation of the Puiseux expansion for 2^3S and 3^3S, is imported from an unpublished same-author manuscript [18], and the Puiseux coefficients used as constraints are themselves fits to the same LMM data. These features make the global '10–14 s.d. for any Z' claim an in-sample fit plus self-consistency check rather than an independent prediction, so the partial-circularity score is 6 rather than higher, because the LMM data have external anchors at low Z and the 1/Z matching coefficients provide genuinely independent constraints.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The interpolation rests on a mix of standard quantum-mechanical setup, an unproven analyticity assumption about the critical charge, and many parameters fitted to the paper's own numerical data. The nine Pade coefficients per state are fitted to LMM energies; ZB and EB come from earlier fits by the same group, and for triplets from an unpublished work in progress. No new physical entities are introduced.

free parameters (8)
  • ZB_singlet = 0.9048539992
    Second critical charge for 1^1S and 2^1S, taken from earlier fits by the authors [16,17]; not fitted in this paper but an input on which the Pade variable lambda depends.
  • ZB_triplet = 0.87989
    Second critical charge for 2^3S and 3^3S, taken from unpublished work in progress [18]; only 4-5 s.d. are quoted.
  • EB_singlet = -0.407932489
    Energy at ZB for the singlet states, from prior author fits; built into the Pade numerator as a0.
  • EB_triplet = -0.3847081
    Energy at ZB for the triplet states, from prior author fits [18].
  • Nine Pade coefficients for 1^1S (a2,a3,a4,a5,a9,b1,b2,b3,b4) = -5.6981404503284, -17.989697949178, -18.992880737496, -47.232296202503, 23.798877891853, 6.2140006981088…
    Fitted to LMM energies in Table I; values from Table V, left column.
  • Nine Pade coefficients for 2^1S (a2,a3,a4,a5,a9,b1,b2,b3,b4) = 6.322467305554, -26.598886009815, 20.011169176394, -43.386785729570, 13.165328519353, 15.027759860647…
    Fitted to LMM energies in Table II; values from Table V, right column.
  • Nine Pade coefficients for 2^3S (a1,a2,a3,a4,a5,a6,a9,b1,b2,b3,b4) = 0.1346410928570494, -2.9863549938707585, -1.1579741129463628, -5.585962716446034, -8.312824729944447…
    Fitted to LMM energies in Table III; values from Table VI, left column, using the relaxed E_{1,2} form.
  • Nine Pade coefficients for 3^3S (a1,a2,a3,a4,a5,a6,a9,b1,b2,b3,b4) = 0.7062083507800702, -2.743811129460274, 1.591050751980635, -4.449017775986878, -2.606064850185295, -1.697461065068710…
    Fitted to LMM energies in Table IV; values from Table VI, right column, using the relaxed E_{1,2} form.
assumptions (6)
  • domain assumption Non-relativistic Born-Oppenheimer Hamiltonian with infinite nuclear mass and point-like Coulomb interactions describes the computed energies.
    Eq. (1) defines the model. The authors themselves note in the Conclusions that QED and relativistic corrections limit the non-relativistic theory to about 4 s.d., so the 14-15 s.d. numbers are for this model, not physical helium.
  • ad hoc to paper For each state there exists a second critical charge ZB where the energy has a Puiseux expansion with only integer and half-integer powers.
    Section II asserts this on physics grounds and checks it numerically for some states; it is not proven from the Hamiltonian.
  • domain assumption The 1/Z perturbation series has finite radius of convergence whose boundary is set by the ZB branch point.
    Kato's theorem is cited in Section III for finite radius, but the identification of the boundary with ZB is a conjecture.
  • ad hoc to paper The generalized two-point Pade approximant in Eq. (13) is an adequate global interpolating form.
    Section IV states the form was chosen after several attempts; no derivation shows this rational function class captures the true analytic structure.
  • ad hoc to paper For triplet states, ZB=0.8799 and the Puiseux coefficients in Eq. (8) are correct.
    These values are taken from unpublished work in progress [18] and from fits to LMM energies in Tables III-IV; there is no independent published derivation.
  • domain assumption The LMM 50x50x40 lattice gives 14-15 s.d. accuracy for all four states and all Z in [1,50].
    Section I states this but the paper reports no convergence study for the new states; it relies on prior work [9,11].

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Pith. "Pith review of Helium-like ions $(Z,e,e)$ in Lagrange Mesh method, interpolating the highly-accurate energy spectra vs. Z." pith.science (2026). https://pith.science/paper/6WO6ZA5U

@misc{pith2026260810695,
  author       = {Pith},
  title        = {Pith review of: Helium-like ions $(Z,e,e)$ in Lagrange Mesh method, interpolating the highly-accurate energy spectra vs. Z},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WO6ZA5U}},
  note         = {Machine review of arXiv:2608.10695}
}
abstract

Two alternative approaches for studying Helium-like atomic ions in non-relativistic quantum mechanics are proposed: (I) a numerical approach, based on the Lagrange-mesh method which can easily reach up to 14-15 significant digits in the energy spectrum for any nuclear charge $Z$ with modest CPU time in single processor mode and (II) a highly-accurate, few-parametric interpolation formula for the energies {\it vs.} $Z$. The interpolation formula of general nature is proposed, it can be applied to the energies of any excited state of the helium-like sequence. It is based on matching the $1/Z$-expansion at large $Z$ and the Puiseux expansion with integer and half-integer powers around the so-called second critical charge $Z_B$, introduced by F and D Stillinger (1969, 1974), confirmed by the present authors in 2019 for the ground state $1^1 S$, then revisited here, and extended to the excited states in the present work. For example for the first two spin-singlet $1^1 S$, $2^1 S$ and the first two spin-triplet $2^3 S$, $3^3 S$ states this interpolation formula with nine free parameters can reach an accuracy of 10-14 significant digits (s.d.) in the energies for any physically-relevant nuclear charge $Z$, giving absolute accuracy at large $Z$. Many results are obtained for the first time.

Figures

Figures reproduced from arXiv: 2608.10695 by the authors.

Figure 1
Figure 1. FIG. 1. Energy of the helium-isoelectronic sequence as a fun [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Works this paper leans on

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    However, they can be useful in s earch for a new physics and also in mathematics studies of three-body Coulomb systems

    Needless to say that such accuracies are well beyond of t oday’s physics reach, both experi- mentally and theoretically. However, they can be useful in s earch for a new physics and also in mathematics studies of three-body Coulomb systems. 22

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    This expansion was never published in complete form

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    It has to be remembered that the coefficient in front of λ0 is already fixed equal to EB, hence, effectively, n0 is reduced by one

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    It implies that, in particular, some coefficients in fron t of odd powers of λ are constrained to vanish

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    Equivalently, in front of λ4 and λ2 23

Pith tools

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