{"id":"a52f0e79-7455-4b28-bcc5-0ae751107146","arxiv_id":"1908.08940","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new set of theoretical energies with uncertainties is reported for 1sns and 1snp states (n=3-7) of helium-like ions across the periodic table.","lead":"This paper computes energy levels of excited states in helium-like ions, atoms with two electrons, from carbon to uranium. The resulting reference tables could sharpen the analysis of X-ray spectra from hot plasmas in space and in the laboratory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The accuracy claim depends on assuming QEDMOD's n=1,2 error bounds hold for n=3-7, with no benchmark for those states; an n-dependent model error would shrink the claimed advantage.","rationale":"The reader identified the same load-bearing weakness: QEDMOD uncertainties are extrapolated from n=1 and n=2 benchmarks to n=3-7 without direct validation, and QED dominates the error budget. My stress-test pass confirms this is the most consequential assumption. I checked the tables for internal consistency and found no arithmetic or sign errors in the sampled rows, and the relation E + Eio = ground-state ionization energy holds for the low-Z cases I verified. The paper is a competently executed data calculation, but the central 'significantly more accurate' claim is conditional on the n-extrapolation of QEDMOD accuracy. Since the reader already assigned CONDITIONAL, this critique does not change the verdict; it sharpens the condition that should be satisfied before the quoted uncertainties are used as calibration-grade values.","tokens_in":92740,"tokens_out":8558,"duration_ms":94622,"concrete_test":"Perform a rigorous two-electron QED calculation of the one-loop QED shift for the 1s3s and 1s3p states at a representative set of ions, e.g. Z=6, Z=26, Z=54, and Z=92, using the same ab initio method as in Refs. [5,6], and compare with the QEDMOD shifts used in Table III. If the relative difference exceeds (10/Z)% for any 1sns state or (5/Z)% for any 1snp state, the uncertainty estimates in Table III must be revised upward and the accuracy claim should be re-evaluated. A minimal version of this check is to run the n=3 case at Z=26 and Z=92, where QED is largest relative to the DCB energy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section I estimates the QED uncertainty as (10/Z)% for 1sns and (5/Z)% for 1snp states by comparing QEDMOD with rigorous QED calculations of the n=1 and n=2 states. The headline claim of being 'significantly more accurate' than previous theory and experiment relies on these quoted uncertainties, because QED shifts are stated to be the dominant error source, especially for heavier ions. No rigorous QED benchmarks for n=3-7 are presented, and the comparisons in Table I (Vainshtein and Safronova) and Table II (experiments) do not provide a tight test of the QED contribution at n>=3: the former is an old 1/Z-expansion calculation with limited accuracy, and the latter extends only to Z=26 with experimental errors much larger than the claimed theory uncertainties. QEDMOD is a local model potential fitted to reproduce one-loop QED shifts; its error for a weakly bound excited outer electron, where the spatial sampling of the potential and the two-electron screening differ from the n=1,2 cases, need not follow the same (10/Z)% and (5/Z)% scaling. If the QEDMOD error grows with n or with Z in a way not captured by the n=1,2 comparison, the quoted uncertainties in Table III are too small and the central accuracy claim is weakened. The paper is internally consistent, and this is not a disagreement with consensus; it is an unsupported extrapolation that is directly load-bearing for the stated uncertainty budget.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":93040,"tokens_out":5621,"duration_ms":56106,"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":[{"comment":"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.","section":"Section I and Conclusion"},{"comment":"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.","section":"Table III and Abstract"}],"minor_comments":[{"comment":"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.","section":"Table I"},{"comment":"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.","section":"Table III caption"},{"comment":"There is a typo in the sentence 'their relative accuracy ... was estimated to be not worse that 10−4'; 'that' should be 'than'.","section":"Introduction"},{"comment":"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":"Table II"},{"comment":"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.","section":"Section II"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the calculations are likely useful to the atomic-physics and plasma-spectroscopy communities. The main risk is the QEDMOD uncertainty extrapolation from n=1,2 to n=3–7, which directly supports the 'significantly more accurate' claim. If the authors cannot provide additional benchmarks for higher-n states, they should be asked to either increase the quoted QED uncertainties for n>=3 or moderate the accuracy claim. The missing uncertainties on some fine-structure intervals are a smaller but still tractable issue that should be fixed during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nI 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.\n\nMinor 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":93585,"tokens_out":3285,"would_cite":true,"duration_ms":36001,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["helium-like ions","isoelectronic sequence","energy levels","configuration-interaction method","QED corrections","fine-structure intervals","x-ray spectroscopy","relativistic atomic structure"],"falsifier":"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.","tokens_in":92527,"feed_emoji":"⚛️","tokens_out":11155,"duration_ms":90696,"temperature":0.7,"pith_summary":"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.","feed_headline":"Energy levels of helium-like ions sharpened from carbon to uranium","feed_subtitle":"Excited n=3–7 states now carry per-ion uncertainties that beat most measured x-ray lines.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Implements the model QED operator used for the one-loop QED energy shifts.","marker":"[12, 13]"},{"why":"Defines the model QED operator method on which QEDMOD is based.","marker":"[11]"},{"why":"Rigorous QED calculation of n=1 and n=2 states used to benchmark the QEDMOD uncertainty and to compile ground-state energies for heavy ions.","marker":"[5]"},{"why":"Rigorous QED results for low-Z ions used in the same benchmarking and in the n=1,2 compilation.","marker":"[6]"},{"why":"Previous 1/Z-expansion theory for n>2 states, the main comparison baseline.","marker":"[8]"},{"why":"Describes the configuration-interaction method and correction chain applied to lithium-like ions that this paper adapts.","marker":"[10]"},{"why":"Provides the hydrogenic 1s energies subtracted from the total energies to obtain ionization energies.","marker":"[14]"},{"why":"Supplies the nuclear rms radii that enter the heavy-ion uncertainty budgets.","marker":"[15]"},{"why":"Supplies the nuclear masses used in the relativistic recoil correction.","marker":"[16]"},{"why":"Independent ab initio QED calculation used to cross-check the n=1 and n=2 compilation at high Z.","marker":"[25]"}],"fun_headline_variants":["Helium-like ion levels upgraded to n=7","Precision energies for helium-like ions Z=6–92","Theoretical levels beat measurement for helium-like ions","Carbon to uranium: helium-like ion energies refined"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Helium-like ion levels upgraded to n=7","Precision energies for helium-like ions Z=6–92","Theoretical levels beat measurement for helium-like ions","Carbon to uranium: helium-like ion energies refined"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000124,"raw_usage":{"total_tokens":1009,"prompt_tokens":759,"completion_tokens":250,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":375,"completion_tokens_details":{"reasoning_tokens":187}},"tokens_in":375,"tokens_out":250,"duration_ms":2869,"temperature":1.0,"reasoning_tokens":187,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:58:12.064479+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the model QED operator method on which QEDMOD is based."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rigorous QED calculation of n=1 and n=2 states used to benchmark the QEDMOD uncertainty and to compile ground-state energies for heavy ions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rigorous QED results for low-Z ions used in the same benchmarking and in the n=1,2 compilation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous 1/Z-expansion theory for n>2 states, the main comparison baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the configuration-interaction method and correction chain applied to lithium-like ions that this paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the hydrogenic 1s energies subtracted from the total energies to obtain ionization energies."},{"cited_title":"Angeli and K","cited_arxiv_id":null,"evidence_quote":"Supplies the nuclear rms radii that enter the heavy-ion uncertainty budgets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nuclear masses used in the relativistic recoil correction."}],"review_version":1}