Theoretical energy levels of $1sns$ and $1snp$ states of helium-like ions

A. Surzhykov; V. A. Yerokhin

REVIEW 2 major objections 5 minor 24 references

Theoretical energy levels of $1sns$ and $1snp$ states of helium-like ions

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper provides a complete grid of energies for the 1sns and 1snp (n=3–7) states of helium-like ions from carbon to uranium, claiming agreement with earlier theory and experiment but significantly higher accuracy.

desk verdict A useful and competently executed tables paper for He-like n=3-7 levels, with a real caveat that the QED error bars for n>=3 rest on extrapolation from n=1,2. read the letter →

arxiv 1908.08940 v1 pith:LR4GB77A submitted 2019-08-16 physics.atom-ph

V. A. Yerokhin , A. Surzhykov This is my paper · ORCID

classification physics.atom-ph

keywords helium-likeionsisoelectronicsequenceenergylevelsconfiguration-interactionmethodQEDcorrectionsfine-structureintervalsx-rayspectroscopyrelativisticatomicstructure

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 reports a systematic calculation of the energies of the $1sns$ and $1snp$ states, $n=3$–$7$, of helium-like ions across the isoelectronic sequence from carbon ($Z=6$) to uranium ($Z=92$). The claim is that the relativistic configuration-interaction method, supplemented by a model QED operator for the one-loop radiative shifts, the relativistic nuclear recoil, and the frequency-dependent Breit interaction, yields energies that agree with the previous theoretical and experimental data while being significantly more accurate. The practical point is that theoretical wavelengths of helium-like ions serve as calibration references for x-ray spectra from astrophysical and laboratory plasmas, and the $n>2$ states had not been tabulated at this level of accuracy before. All energies in the main table carry uncertainty estimates, with the QED part of the error usually dominant.

What carries the argument

The machine that carries the calculation is the configuration-interaction method in its relativistic form: the wave function is a finite sum of configuration-state functions of the Dirac–Coulomb–Breit Hamiltonian, and the energy comes from solving the secular equation for that Hamiltonian matrix. Onto these Dirac–Coulomb–Breit energies the authors add the one-loop QED shift, evaluated with the model QED operator as implemented in the QEDMOD package, plus the relativistic nuclear recoil and the frequency dependence of the Breit interaction. The QED operator is the load-bearing correction, and its uncertainty is the central estimate of the paper: comparing QEDMOD with rigorous QED results for the $n=1$ and $n=2$ states yields $(10/Z)\%$ of the QED shift for $1sns$ and $(5/Z)\%$ for $1snp$ as the quoted QED error.

What would settle it

A rigorous ab initio QED calculation of the $1s3p\,^1P_1$ or $1s3s\,^3S_1$ energy of a helium-like ion with $Z\approx 20$–$30$, compared with the QEDMOD-based shift, would settle whether the quoted $(10/Z)\%$ and $(5/Z)\%$ QED uncertainties hold for $n=3$; a disagreement exceeding the quoted uncertainty would falsify the error estimate. A high-resolution x-ray measurement of one such transition with accuracy better than the quoted theory error would also settle the point directly.

Watch

Extended reading notes

Core claim

The central result is a set of calculated energy levels for the $1sns$ and $1snp$ states with $n=3$–$7$ for every helium-like ion from $Z=6$ through $Z=92$. Each level is given as an ionization energy, an energy relative to the ground state, and, for the $n^3P_J$ levels, the two fine-structure intervals; both the theoretical error and the nuclear-radius error are quoted separately when both matter. The paper argues that the calculation matches available experiment nearly everywhere and that its numerical results are one to two orders of magnitude more accurate than most experimental data, while also improving on the older Vainshtein–Safronova values for the same states.

Load-bearing premise

The load-bearing premise is that the model QED operator, benchmarked against rigorous QED only for the $n=1$ and $n=2$ states, delivers the same accuracy at $(10/Z)\%$ and $(5/Z)\%$ of the QED shift for the $n=3$–$7$ states, where no rigorous check is reported in the paper.

Editorial extensions

If this is right

The tabulated $n=3$–$7$ energies can serve as a more accurate reference grid than the 1985 Vainshtein–Safronova values, whose accuracy the paper quantifies as about 1–2 parts in $10^{-5}$ near carbon and better at higher $Z$.
Transition wavelengths built from these levels are claimed to be one to two orders of magnitude more accurate than most of the experimental data currently available.
The $n^3P_J$ fine-structure intervals, for which the QED uncertainty nearly cancels, come with especially small relative uncertainties and are directly comparable to high-resolution measurements.
The cross-check on the $n=2$ states against full-scale QED calculations supports the reliability of the method for the higher excited states treated here.

Reading between the lines

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

If the uncertainty extrapolation from $n=1,2$ to $n=3$–$7$ holds, the same method should extend to $n>7$ and to $1snl$ states with $l>1$, where no systematic tabulation of comparable accuracy currently exists.
Because the QED error is the dominant one and the estimate rests on benchmarks at only two principal quantum numbers, a single rigorous QED calculation of one $n=3$ state would be a high-value check of the whole table.
The fine-structure intervals may be the most robust products for plasma diagnostics, since the main theoretical uncertainty cancels in the difference between the $J$ levels.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Short version: this is a data paper, and a good one. It computes 1sns and 1snp energies for n=3-7, Z=6-92, using CI plus the QEDMOD screened-QED operator, recoil, Breit frequency dependence, and uncertainty estimates. The previous standard reference, Vainshtein and Safronova 1985, stopped at n<=5 and Z<=42 and was less accurate. The n=2 cross-check against full QED, the comparison with old theory, and the experimental comparison tables are all useful. The numbers are genuinely new and will likely be used for X-ray calibration and plasma diagnostics.

I mostly agree with the reader's assessment. The central claim is that the results are significantly more accurate than previous theory, and that claim leans on the quoted uncertainties. The dominant uncertainty is the QED contribution. The authors estimate QEDMOD's error as (10/Z)% for 1sns and (5/Z)% for 1snp by comparing against rigorous QED for n=1 and n=2 only. The stress-test note is correct: no rigorous QED benchmark for n=3-7 appears in the paper, and the experimental comparisons reach only Z=26 with errors far larger than the claimed theory uncertainties. If QEDMOD's error grows with n, the Table III error bars are too small. That is a real caveat. It is not a deal-breaker. For these states the QED shift is plausibly dominated by the inner 1s electron, so an n-dependent error in the outer-electron piece would need to be large to spoil the 1/Z scaling. Still, the paper would be stronger if it stated this extrapolation more explicitly or ran one n=3 benchmark against a rigorous QED calculation.

Minor soft spots: several fine-structure intervals in Table III are quoted without explicit uncertainties, even though the text says all theoretical energies come with uncertainty estimates. Also, the data appear only as printed tables; machine-readable tables would make the paper much easier to use. The supplementary compilation of n=1 and n=2 energies is a helpful service.

The math looks sound: CI basis convergence is cross-checked, the n=2 comparison catches systematic problems, and the experimental comparisons are consistent within quoted errors. There is no circularity: the excited-state energies come from a new diagonalization, and the compiled ground-state energies are external input used only to form transition energies.

Who this is for: atomic physicists and X-ray astronomers who need wavelengths with error bars. It deserves a serious referee. The main requests should be an explicit treatment of the n>=3 QED uncertainty and error bars on every fine-structure interval. I would send it out.

Referee Report

2 major / 5 minor

Summary. This paper reports configuration-interaction calculations of the 1sns (n=3–7) and 1snp (n=3–7) states of helium-like ions from Z=6 to Z=92. The Dirac–Coulomb–Breit Hamiltonian is diagonalized in a finite CSF basis, and the resulting energies are supplemented with model-operator (QEDMOD) one-loop QED shifts, relativistic nuclear recoil, and the frequency-dependent Breit correction. For each state the paper supplies ionization energies, energies relative to the ground state, and, for 3P levels, fine-structure intervals, together with uncertainty estimates. The results are compared with the 1985 1/Z-expansion calculations of Vainshtein and Safronova and with available experimental transition energies, and the paper concludes that the new theoretical energies are 'significantly more accurate' than previous results.

Significance. If the quoted uncertainties are reliable, this is a useful data set for x-ray plasma spectroscopy and for calibration of experimental spectra of helium-like ions, filling a gap left by the most accurate ab initio treatments, which have focused on n=1 and n=2 states. The paper is transparent about the method: the CI part is checked by basis-set convergence, the n=2 case is compared with full QED calculations, and the compiled n=1,2 energies are cross-checked against independent ab initio results. The main value is the systematic tabulation for n=3–7 over a wide Z range, with an honest attempt to attach uncertainties to every energy. The central risk is that the dominant QED uncertainty is estimated by an extrapolation from n=1,2 to n=3–7 without direct benchmarks, so the strength of the headline accuracy claim depends on an assumption that the paper does not test.

major comments (2)

[Section I and Conclusion] The QED uncertainty estimate is load-bearing but rests on an unverified extrapolation. The paper states that the uncertainty of the QEDMOD results is estimated as (10/Z)% for 1sns states and (5/Z)% for 1snp states by comparing QEDMOD with rigorous QED calculations for the n=1 and n=2 states, and that QED uncertainty is usually the dominant source of the total error. No benchmark for n=3–7 is presented, and the comparisons in Tables I and II do not test the QED contribution at the claimed level: Table I is an old 1/Z-expansion calculation with limited accuracy, and Table II has experimental errors much larger than the quoted theoretical uncertainties. If the QEDMOD error does not scale as n^{-3} relative to the QED shift, or if the model potential is less accurate for a weakly bound outer electron, the quoted uncertainties in Table III will be too small and the central claim of being 'significantly more accurate' than previous theory and experiment would be weakened. I recommend that the authors either provide additional validation for n>=3 (for example, a few selected n=3 or n=4 states compared with full two-electron QED calculations, if available) or explicitly soften the accuracy claim and enlarge the QED uncertainty for n>=3 accordingly.
[Table III and Abstract] The abstract states that 'All theoretical energies are supplied with uncertainty estimates', but Table III lists many fine-structure intervals without any explicit uncertainty. Examples include the Z=6 4 3P intervals (−0.000002 and 0.000154), the Z=6 5 3P intervals (−0.000001 and 0.000078), and the Z=7 7 3P intervals (0.000007 and 0.000061). The text says that for fine-structure intervals the QED corrections nearly cancel, so the QED uncertainty is assumed negligible, but this does not address the DCB uncertainty, which should still contribute. If the missing uncertainties are intended to mean that the error is below the last quoted digit, this convention should be stated explicitly; otherwise the table is inconsistent with the abstract's claim. Please provide uncertainties for all entries or add a clear statement of the rounding/uncertainty convention.

minor comments (5)

[Table I] The 'Difference' column is not formatted with consistent decimal places (e.g., 0.0021 versus 0.0001); aligning the number of digits would improve readability.
[Table III caption] The symbol Efs is used in the table header but is not defined in the caption; please define it as the fine-structure interval in Rydbergs.
[Introduction] There is a typo in the sentence 'their relative accuracy ... was estimated to be not worse that 10−4'; 'that' should be 'than'.
[Table II] For Z=26, the 4 1P row lists two experimental values (609.72(1) and 609.69(2) from two references); indicating which reference corresponds to which value in the table header or a footnote would avoid ambiguity.
[Section II] The statement that the results of Ref. [8] 'are accurate typically to 1-2 parts in 10−5' is not fully consistent with Table I, where the 3 1S state at Z=6 differs by 0.0021 Ry, about 8×10−5 of the level energy; a more careful wording would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tabulated energies come from a new CI computation, with QEDMOD used as a calibrated model; the cited n=1,2 QED data are a validation set, not a construction input.

full rationale

The excited-state energies in Table III are obtained by solving the Dirac-Coulomb-Breit configuration-interaction eigenvalue problem (Eq. 2) and then adding recoil, frequency-dependent Breit, and QED corrections; none of the tabulated energies is taken from, or algebraically equivalent to, the cited n=1,2 QED calculations. QEDMOD is used as a model operator for one-loop QED shifts, and the paper calibrates its uncertainty by comparing QEDMOD with rigorous QED results for n=1,2, then propagates that uncertainty to n>=3. This is a calibration and extrapolation step, not a circular derivation: the n>=3 outputs are novel CI eigenvalues and the model is not fitted to those target outputs. The compiled ground-state energies used to form excitation energies are explicitly assembled from independent ab-initio QED literature, including calculations by non-overlapping groups, so using them as a reference is standard practice rather than a circular construction of the target result. Tables I and II provide external comparisons with earlier theory and experiment that are not built into the inputs. The only substantive concern is whether the (10/Z)% and (5/Z)% QEDMOD uncertainty remains valid for n>=3, where no rigorous benchmark is presented in this paper; that is a correctness or robustness risk about an extrapolated error estimate, not a circularity of the energy derivation itself. The self-citations that occur are normal references to the authors' own prior methodology and benchmark data, and they are not load-bearing in the sense that would make the predicted energies equivalent to their inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The calculation has no free parameters fitted to the target energies. Nuclear radii and masses are taken from Refs. [15,16], the QEDMOD operator is calibrated on n=1,2 QED results, and ground-state energies are compiled from prior ab initio QED calculations. The two QED uncertainty scaling rules are hand-chosen estimates. No new entities are introduced.

free parameters (2)

QED uncertainty scaling for 1sns states = (10/Z)%
Estimated by comparing QEDMOD results with rigorous QED for n=1,2 states; assumed to hold for n=3-7 (Section I).
QED uncertainty scaling for 1snp states = (5/Z)%
Estimated by the same n=1,2 comparison; assumed to hold for n=3-7 (Section I).

assumptions (4)

domain assumption Dirac-Coulomb-Breit Hamiltonian is an adequate starting point for these states
Used in Eq. (2) as the basis for the CI calculation; standard in relativistic atomic structure.
domain assumption QEDMOD model operator reproduces one-loop QED shifts for n=3-7 states to the quoted accuracy
Section I: uncertainty of QEDMOD estimated from comparison to rigorous QED for n=1,2 states, extrapolated to higher n.
domain assumption Nuclear recoil and frequency-dependent Breit corrections from Ref. [10] are complete at the stated accuracy
Section I: corrections added as described in Ref. [10], not re-derived.
domain assumption Ground-state (1s)2 energies compiled from Refs. [5,6,14,22-25] are accurate
Section II: used to convert total energies to excitation energies relative to ground state.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Theoretical energy levels of $1sns$ and $1snp$ states of helium-like ions." pith.science (2026). https://pith.science/paper/LR4GB77A

@misc{pith2026190808940,
  author       = {Pith},
  title        = {Pith review of: Theoretical energy levels of $1sns$ and $1snp$ states of helium-like ions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LR4GB77A}},
  note         = {Machine review of arXiv:1908.08940}
}

abstract

Energy levels of the $1sns$ and $1snp$ states of ions along the helium isoelectronic sequence from carbon to uranium are calculated, with $n=3$-$7$. The computation is performed within the relativistic configuration-interaction method, including the relativistic nuclear recoil effect, the leading QED effects, and the frequency dependence of the Breit interaction. All theoretical energies are supplied with uncertainty estimates.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

K. P. Dere, G. D. Zanna, P. R. Young, E. Landi, and R. S. Sutherland, Astrophys. J. Suppl. Ser. 241 , 22 (2019)

work page 2019
[2]

R. K. Smith and N. S. Brickhouse, in Advances In Atomic, Molecular, and Optical Physics , 63 , 271, Academic Press, 2014

work page 2014
[3]

u hn , P. Micke , R. Steinbr \

J. R. Stierhof , M. A. Leutenegger , S. K \"u hn , P. Micke , R. Steinbr \"u gge , C. Shah , N. Hell , M. Bissinger , M. Hisch , R. Ballhausen , M. Lang , C. Gr \"a fe , S. Wipf , R. Cumbee , G. Betancourt-Martinez , S. Park , V. Yerokhin , A. Surzhykov , M. Chung , J. Wilms , G. V. Brown , J. Crespo L \'o pez-Urrutia , and S. Bernitt , in AAS/High Energy...

work page 2019
[4]

G. W. F. Drake, Can. J. Phys. 66 , 586 (1988)

work page 1988
[5]

A. N. Artemyev, V. M. Shabaev, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. A 71 , 062104 (2005)

work page 2005
[6]

V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81 , 022507 (2010)

work page 2010
[7]

A. V. Malyshev, Y. S. Kozhedub, D. A. Glazov, I. I. Tupitsyn, and V. M. Shabaev, Phys. Rev. A 99 , 010501 (2019)

work page 2019
[8]

L. A. Vainshtein and U. I. Safronova, Phys. Scr. 31 , 519 (1985)

work page 1985

Show all 24 references

[9]

V. A. Yerokhin, A. Surzhykov, and A. M\"uller, Phys. Rev. A 96 , 042505 (2017) [(E) Phys. Rev. A 96 , 069901 (2017)]

2017
[10]

V. A. Yerokhin and A. Surzhykov, J. Phys. Chem. Ref. Dat. 47 , 023105 (2018)

2018
[11]

V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88 , 012513 (2013)

2013
[12]

Shabaev, I

V. Shabaev, I. Tupitsyn, and V. Yerokhin, Comput. Phys. Commun. 189 , 175 (2015)

2015
[13]

Shabaev, I

V. Shabaev, I. Tupitsyn, and V. Yerokhin, Comput. Phys. Commun. 223 , 69 (2018)

2018
[14]

V. A. Yerokhin and V. M. Shabaev, J. Phys. Chem. Ref. Data 44 , 033103 (2015)

2015
[15]

Angeli and K

I. Angeli and K. Marinova, At. Dat. Nucl. Dat. Tabl. 99 , 69 (2013)

2013
[16]

M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer, Chin. Phys. C 36 , 1603 (2012)

2012
[17]

Engstr\"om, P

L. Engstr\"om, P. Bengtsson, C. Jup\'en, and M. Westerlind, J. Phys. B 25 , 2459 (1992)

1992
[18]

Engstr\"om and U

L. Engstr\"om and U. Litz\'en, J. Phys. B 28 , 2565 (1995)

1995
[19]

Beiersdorfer, M

P. Beiersdorfer, M. Bitter, S. von Goeler, and K. W. Hill, Phys. Rev. A 40 , 150 (1989)

1989
[20]

J. F. Seely and U. Feldman, Phys. Rev. Lett. 54 , 1016 (1985)

1985
[21]

Indelicato, O

P. Indelicato, O. Gorceix, M. Tavernier, J. P. Briand, J. P. Desclaux, R. Marrus, and M. Prior, Z. Phys. D 2 , 149 (1986)

1986
[22]

Czarnecki and R

A. Czarnecki and R. Szafron, Phys. Rev. A 94 , 060501 (2016)

2016
[23]

V. A. Yerokhin, Phys. Rev. A 97 , 052509 (2018)

2018
[24]

A. V. Malyshev, R. V. Popov, V. M. Shabaev, and N. A. Zubova, J. Phys. B 51 , 085001 (2018)

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

K. P. Dere, G. D. Zanna, P. R. Young, E. Landi, and R. S. Sutherland, Astrophys. J. Suppl. Ser. 241 , 22 (2019)

work page 2019

[2] [2]

R. K. Smith and N. S. Brickhouse, in Advances In Atomic, Molecular, and Optical Physics , 63 , 271, Academic Press, 2014

work page 2014

[3] [3]

u hn , P. Micke , R. Steinbr \

J. R. Stierhof , M. A. Leutenegger , S. K \"u hn , P. Micke , R. Steinbr \"u gge , C. Shah , N. Hell , M. Bissinger , M. Hisch , R. Ballhausen , M. Lang , C. Gr \"a fe , S. Wipf , R. Cumbee , G. Betancourt-Martinez , S. Park , V. Yerokhin , A. Surzhykov , M. Chung , J. Wilms , G. V. Brown , J. Crespo L \'o pez-Urrutia , and S. Bernitt , in AAS/High Energy...

work page 2019

[4] [4]

G. W. F. Drake, Can. J. Phys. 66 , 586 (1988)

work page 1988

[5] [5]

A. N. Artemyev, V. M. Shabaev, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. A 71 , 062104 (2005)

work page 2005

[6] [6]

V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81 , 022507 (2010)

work page 2010

[7] [7]

A. V. Malyshev, Y. S. Kozhedub, D. A. Glazov, I. I. Tupitsyn, and V. M. Shabaev, Phys. Rev. A 99 , 010501 (2019)

work page 2019

[8] [8]

L. A. Vainshtein and U. I. Safronova, Phys. Scr. 31 , 519 (1985)

work page 1985