Second-order corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged energy in the HD$^+$ and H$_2^+$ ions

Jean-Philippe Karr; Vladimir I. Korobov; Zhen-Xiang Zhong

REVIEW 4 major objections 4 minor 25 references

Second-order corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged energy in the HD$^+$ and H$_2^+$ ions

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper calculates the complete set of second-order perturbative corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged ro-vibrational energies of the hydrogen molecular ions, with relative numerical uncertainty…

desk verdict First numerical results for the second-order mα^6 corrections in H2+ and HD+, worth refereeing, but the convergence claims outrun the evidence shown. read the letter →

arxiv 2506.22164 v1 pith:2QDMMEVJ submitted 2025-06-27 physics.atom-ph

Vladimir I. Korobov , Jean-Philippe Karr , Zhen-Xiang Zhong This is my paper

classification physics.atom-ph

keywords hydrogenmolecularionsmalpha^6correctionsrecoilsecond-orderperturbationtheoryspin-averagedenergiesro-vibrationaltransitionsexponentialvariationalbasisQED

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 reports the first numerical evaluation of the complete set of second-order perturbative corrections of relativistic orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged ro-vibrational energies of the hydrogen molecular ions H$_2^+$ and HD$^+$. The naive Rayleigh-Schrödinger expressions for these corrections are singular, so the authors separate off the singular parts through a regularized operator and compute the finite remainders $E'_B$, $E'_R$, $E_S^{(0)}$, and $E_S^{(1)}$ for states with $L=0$–4 and $v=0$–4. A convergence study of the numerically most difficult term, $E'_B$, indicates that all tabulated digits are reliable to better than $10^{-5}$ relative. These values supply the previously missing second-order piece of the $m\alpha^6$ correction and, when combined with first-order effective-Hamiltonian expectation values that are currently being computed, should allow theory to advance beyond the 1 part-per-trillion level for ro-vibrational transition frequencies.

What carries the argument

The load-bearing mechanism is the second-order resolvent formula $A Q(E_0-H_0)^{-1}Q B$, evaluated numerically through the intermediate state $\psi^{(1)}$ defined by $(E_0-H_0)\psi^{(1)}=(B-\langle B\rangle)\psi_0$. Because the naive matrix elements for $B=H_B$ and $B=H_{\rm ret}$ diverge, the authors use the identity $H'_B=H_B-(E_0-H_0)U-U(E_0-H_0)$ with the two-centre weight $U=c_1/r_1+c_2/r_2$ to split each singular second-order term into a finite regularized matrix element $E'_B$ or $E'_R$ plus terms that move into the first-order effective Hamiltonian. The numerical engine is the exponential variational expansion of [24], organized in the multi-layered form of [21,22]: several subsets approximate the regular part of $\psi^{(1)}$ while layers with exponents up to $10^6$ reproduce its $\ln r_a$ singular behaviour near the nuclei. For the rank-1 spin-orbit operator $H_S$, intermediate angular momenta $L'=L-1,L,L+1$ are treated separately and combined into $E_S^{(0)}$ and the spin-orbit coefficient $E_S^{(1)}$.

What would settle it

Recompute $E'_B$ for the $(L=0, v=0)$ ground state of HD$^+$ with an independent method, for example a different intermediate-state basis or a direct numerical solution of the equation for $\psi^{(1)}$, and compare with Table I's converged entry $-0.3009339$. Then repeat the same basis-enlargement test for $E'_R$, $E_S^{(0)}$, and $E_S^{(1)}$; if any value shifts by more than about $10^{-5}$ relative when $N'$ grows beyond 12000 or exponents exceed $10^6$, the paper's convergence claim is falsified.

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Extended reading notes

Core claim

The central claim is that the divergent second-order matrix elements $E_B = \langle H_B Q(E_0-H_0)^{-1}Q H_B \rangle$ and $E_R = 2\langle H_B Q(E_0-H_0)^{-1}Q H_{\rm ret} \rangle$ can be regularized by the substitution $H'_B = H_B - (E_0-H_0)U - U(E_0-H_0)$ with $U=c_1/r_1+c_2/r_2$, leaving finite second-order contributions $E'_B$ and $E'_R$ plus singular terms to be absorbed into the first-order $m\alpha^6$ effective Hamiltonian. The authors compute these regularized contributions, together with the regular spin-orbit second-order terms $E_S^{(0)}$ and $E_S^{(1)}$, by solving the three-body equation $(E_0-H_0)\psi^{(1)}=(B-\langle B\rangle)\psi_0$ with a multi-layered exponentially correlated variational basis whose largest exponents reach $10^6$. The final tables give all four contributions for a range of rotational and vibrational states of both ions; the dominant term $E'_B$ is shown by a basis-size study to converge to the displayed digits, and the same accuracy is assumed for the smaller terms. The paper presents this as the first step toward total $m\alpha^6$ and $m\alpha^6(m/M)$ energy shifts, with the corresponding first-order effective-Hamiltonian calculations in progress.

Load-bearing premise

The load-bearing premise is that the multi-layered variational basis, with sizes up to $N'=12000$ and exponents up to $10^6$, represents the logarithmically singular intermediate state $\psi^{(1)}$ accurately enough that every digit in Tables II and III is converged; the dedicated convergence study covers only the dominant term $E'_B$, and the same accuracy is assumed for the three smaller terms.

Editorial extensions

If this is right

Combined with the first-order effective-Hamiltonian expectation values and their already-regularized forms, these numbers complete the $m\alpha^6$ and $m\alpha^6(m/M)$ corrections, with an uncertainty contribution from the second-order part below $10^{-6}\,\alpha^4 E_h/h \simeq 19$ Hz.
For some ro-vibrational transitions the theoretical uncertainty, currently about $8\times10^{-12}$ relative, should improve by one to two orders of magnitude, sharpening tests of the proton-electron mass ratio and constraints on a hypothetical fifth force.
The $E_S^{(1)}$ columns give the second-order spin-orbit coefficients at order $m\alpha^6$ for each tabulated state, extending the hyperfine-structure results of [21,22] to these transitions.
The tables provide a complete numerical target for the $m\alpha^6$ second-order part: any independent method that treats the singular structure correctly must reproduce $E'_B$, $E'_R$, $E_S^{(0)}$, and $E_S^{(1)}$ for these states.

Reading between the lines

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

Editorial inference: The same $U$-regularization split should carry over to the next-order $m\alpha^7\ln\alpha$ divergences, so the second-order machinery presented here may be reusable with only the replacement of the perturbing operator.
Editorial inference: A dedicated convergence study of $E'_R$ and $E_S^{(0)}$ would settle whether the paper's global $10^{-5}$ uncertainty estimate is conservative, since the present evidence for those terms is indirect.
Editorial inference: Once the first-order expectation values are available, comparing the full $m\alpha^6$ shift with older adiabatic estimates will quantify the nonadiabatic recoil effects that the previous approach could only estimate from hydrogen-atom theory.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read on 2506.22164. The genuinely new thing is the numbers: first numerical evaluation of the second-order mα^6 and mα^6(m/M) corrections in H2+ and HD+ in the three-body formalism. The operators were derived earlier by this group, but nobody had computed them. That alone makes the paper useful for the precision-spectroscopy crowd.

What it does well: the singular-separation identity (Eq. 10) is stated cleanly, and the multi-layered exponential basis is a sensible way to handle the logarithmic singular ψ^(1). Table I shows a clear convergence pattern for E'_B, the dominant and most singular term, with the value settling around -0.300934. That is real evidence the method works.

The soft spots are real but contained. First, the anomalous Table I entry at (N=3000, N'=12000) reads -0.3019389, three orders of magnitude off from its neighbors. It has to be a typo or a genuine instability; the paper cannot stand with it unexplained. Second, the converged value from Table I (-0.300934) disagrees with Table III's -0.300935 for the same HD+ ground-state E'_B. That is a one-part-in-a-million discrepancy in a table where the text claims all digits are converged. One of the two tables is rounded or not actually converged at the stated level. Third, there is no convergence study at all for E'_R, E_S^(0), E_S^(1); the abstract's <1e-5 uncertainty is inferred from E'_B alone. That inference may hold, but it is not demonstrated.

None of this undermines the physical framework or the overall approach. The discrepancy is at the 1e-6 level, and the claimed final uncertainty is 1e-5, so the abstract's headline number might survive. But the 'all digits converged' sentence in Section IV is too strong, and the anomalous entry must be addressed.

The paper is for specialists in hydrogen-molecular-ion QED and precision tests of fundamental constants. It deserves a serious referee. I would send it to peer review with a request that the authors reconcile the two tables, correct or explain the outlier, and report convergence evidence or error bars for the smaller terms. If those are fixed, the numbers will be citable. Even as is, it is a solid stepping-stone result, honestly described as incomplete until the first-order effective-Hamiltonian terms appear.

Referee Report

4 major / 4 minor

Summary. The manuscript computes the complete set of second-order perturbative corrections of orders mα^6 and mα^6(m/M) to the spin-averaged rovibrational energies of H2+ and HD+. After separating the singular parts of the second-order Breit–Pauli contributions through the transformation of Eqs. (8)–(11), the authors evaluate the regularized terms E'_B, E'_R, E_S^(0), and E_S^(1) using variational exponential bases with multi-layered exponents. Tables II and III list results for L = 0–4 and v = 0–4 (and one v = 9 state for HD+). The authors claim that all digits shown are converged and estimate a relative numerical uncertainty below 10^-5, which they state is sufficient to support future theoretical improvements beyond the 1 ppt level for ro-vibrational transition frequencies.

Significance. If the numerical values are correct, this paper supplies a previously missing piece of the mα^6 and mα^6(m/M) corrections in the three-body formalism, moving beyond the adiabatic approximation for these orders. The separation of singularities in Eq. (10) is clearly laid out, and the decomposition of the spin-orbit contributions in Eqs. (14)–(15) is useful. The paper is the first to present numerical results for the complete set of these second-order terms, and the application to future precision spectroscopy of hydrogen molecular ions is well motivated. However, the significance currently rests on convergence evidence that is incomplete and internally inconsistent; the central numerical claim is therefore not yet established.

major comments (4)

[Table I] Table I contains an anomalous entry at N = 3000, N' = 12000: the value -0.3019389 differs from all neighboring entries (near -0.30093) by about 1e-3, which is three orders of magnitude larger than the convergence trend shown elsewhere in the table. This entry directly contradicts the statement that all digits in Tables II and III are converged. If it is a typographical error, it must be corrected; if it is real, it indicates a numerical instability in the small-N plus high-exponent regime that is exactly the regime used for the final results. The authors must resolve this entry and explain its origin.
[Section IV, Tables II and III] The convergence study in Table I covers only E'_B for the HD+ (L = 0, v = 0) ground state. There are no analogous convergence studies for E'_R, E_S^(0), or E_S^(1), nor for higher L and v states or for H2+. The abstract's claim of relative numerical uncertainty below 10^-5 and the statement that all digits are converged are therefore extrapolated from a single term and state. Please provide convergence evidence for each type of contribution, at least for representative states of both ions, or explicitly restrict the uncertainty claim to E'_B.
[Table I versus Table III] For the same quantity, the HD+ ground-state E'_B, the converged-looking value in Table I at N = 6000, N' = 12000 is -0.3009339, whereas Table III lists -0.300935. The difference of 1.1e-6 in the sixth decimal place exceeds what 'all the digits indicated in the Tables are converged' should allow. The two tables need to be made consistent, and the final value and its uncertainty should be stated with the number of digits that can actually be defended.
[Eq. (11)] The regularized recoil term E'_R is introduced by saying that Hret is 'treated in a similar way', but no explicit transformed operator or cancellation identity equivalent to Eq. (10) is shown for Hret. Since E'_R is one of the four quantities tabulated and is part of the completeness claim, the regularization step needs to be stated explicitly or referenced to a specific equation in refs. [20, 21, 23].

minor comments (4)

[Abstract] The phrase 'less then 10^-5' should read 'less than 10^-5'.
[Section III] The sentence 'the calculation in divided into the angular momentum components' appears to contain a typo; it should read 'the calculation is divided into'.
[General] Since the effective Hamiltonian and the regularization used here are taken from earlier papers by the same authors, the manuscript should state explicitly which parts of the derivation are new and which are adopted from refs. [20, 21, 23], so that the reader can assess the completeness claim without consulting all prior works.
[Abstract] The stated numerical uncertainty is attributed to the second-order terms only; consider clarifying in the abstract that the <10^-5 relative uncertainty refers to these calculated contributions and not yet to the total mα^6 correction, which also requires the first-order effective Hamiltonian terms.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found; same-author citations supply prior derivations but do not feed the numerical values back into the inputs.

full rationale

The quantities in Eqs. (4)-(7) are explicit matrix elements of the Breit-Pauli operators (3) with the nonrelativistic Green function; the regularization identity (10) is an algebraic rearrangement, and the singular terms are to be canceled by first-order effective-Hamiltonian contributions that are not computed here. No experimental energies or benchmark values are used to set constants, and no fitted parameter is renamed as a prediction. The effective Hamiltonian and regularization are taken from Refs. [20,21,23], which include the present authors, but those are prior derivations with stated assumptions and the paper's new content is the numerical evaluation, which is not determined by the citations. The convergence argument is not circular either: it compares variational basis sizes, although it is incomplete (Table I covers only E'_B for HD+ ground state) and internally inconsistent (the N=3000/N'=12000 entry -0.3019389 deviates from neighbors, and Table I's converged -0.3009339 differs at the 1e-6 level from Table III's -0.300935). That is a numerical-evidence concern, not a circularity. Score 2 reflects the presence of several same-author citations in the load-bearing theory chain; none of them reduces the claimed result to its own input.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The calculation rests on established QED Hamiltonians and a variational basis; no new physical entities are introduced. The main free parameters are the basis size and exponent ranges, which are numerical controls rather than fitted physical constants.

free parameters (1)

Variational basis sizes N and N' and exponent ranges (β, γ up to 10^6) = N=3000-6000, N'=8000-12000 per Table I
The intermediate-state basis must include large exponents to represent the ln(r) singular behavior; convergence is established by increasing these parameters, so they are numerical control parameters rather than physical constants.

assumptions (4)

domain assumption The Breit-Pauli Hamiltonian H^(4) in Eq. (2), restricted to spin-averaged terms, is the correct leading relativistic (mα^4) Hamiltonian.
Used as the perturbation operator H_B, H_ret, H_S in Eqs. (4)-(7); standard QED reduction, cited to prior work.
domain assumption The effective mα^6 and mα^6(m/M) Hamiltonian derived in refs [20,21] is complete for spin-averaged energies.
The paper computes second-order matrix elements of this Hamiltonian; if the Hamiltonian is incomplete, the numbers are incomplete.
domain assumption The counterterm U=c1/r1+c2/r2 with c_a from Eq. (9) fully removes the singular parts of E_B and E_R.
The regularization in Eqs. (8)-(11) assumes that all singularities are of the form U and that the remaining E'_B and E'_R are finite; the paper relies on refs [20,23] for this.
domain assumption The multi-layered exponential basis can represent the intermediate function ψ^(1) with sufficient accuracy at the largest basis sizes used.
Convergence is demonstrated numerically for E'_B in Table I, but the same assumption is carried over to E'_R and spin-orbit terms without a dedicated study.

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Cite this review

Pith. "Pith review of Second-order corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged energy in the HD$^+$ and H$_2^+$ ions." pith.science (2026). https://pith.science/paper/2QDMMEVJ

@misc{pith2026250622164,
  author       = {Pith},
  title        = {Pith review of: Second-order corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged energy in the HD$^+$ and H$_2^+$ ions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QDMMEVJ}},
  note         = {Machine review of arXiv:2506.22164}
}

abstract

The relativistic second-order corrections at orders $m\alpha^6$ and $m\alpha^6(m/M)$ in hydrogen molecular ions are calculated. Convergence of numerical results is studied, which allows estimating the relative numerical uncertainty to be less then $10^{-5}$. This accuracy is sufficient to enable future improvement of theoretical predictions beyond the 1 ppt (part-per-trillion) precision level for the ro-vibrational transition frequencies.

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Alighanbari, G.S

S. Alighanbari, G.S. Giri, F.L. Constantin, V.I. Korobov, and S. Schiller, Precise test of quantum electrodynamics and determination of fundamental constants with HD + ions, Nature 581, 152 (2020)

work page 2020

[2] [2]

Patra, M

S. Patra, M. Germann, J.-Ph. Karr, M. Haidar, L. Hilico, V.I. Korobov, F.M.J. Cozijn, K.S.E. Eikema, W. Ubachs, and J.C.J. Koelemeij, Proton-electron mass ratio from laser spectroscopy of HD + at the part-per-trillion level, Science 369, 1238 (2020)

work page 2020

[3] [3]

Kortunov, S

I. Kortunov, S. Alighanbari, M.G. Hansen, G.S. Giri, S. Schiller, and V.I. Korobov, Proton-electron mass ratio by high- resolution optical spectroscopy of ion ensemble in the resolved-carrier regime. Nature Phys. 17, 569 (2021)

work page 2021

[4] [4]

Alighanbari, I.V

S. Alighanbari, I.V. Kortunov, G.S. Giri, and S. Schiller, Test of charged baryon interaction with high-resolution vibrational spectroscopy of molecular hydrogen ions. Nat. Phys. 19, 1263 (2023)

work page 2023

[5] [5]

Schwegler, D

N. Schwegler, D. Holzapfel, M. Stadler, A. Mitjans, I. Sergachev, J.P. Home, and D. Kienzler, Trapping and ground-state cooling of H + 2 . Phys. Rev. Lett. 131, 133003 (2023)

work page 2023

[6] [6]

Alighanbari, M.R

S. Alighanbari, M.R. Schenkel, V.I. Korobov, and S. Schiller, High-accuracy laser spectroscopy of H + 2 and the proton- electron mass ratio, to appear in Nature

work page

[7] [7]

Karr and J.C

J.-Ph. Karr and J.C. Koelemeij, Extraction of spin-averaged rovibrational transition frequencies in HD + for the determi- nation of fundamental constants. Mol. Phys. 121, e2216081 (2023)

work page 2023

[8] [8]

Tiesinga, P.J

E. Tiesinga, P.J. Mohr, D.B. Newell, and B.N. Taylor, CODATA recommended values of the fundamental physical con- stants: 2022. Rev. Mod. Phys. 97, 025002 (2025)

work page 2025