High-precision ab initio calculations of the spectrum of Lr$^{+}$

A. Borschevsky; E. Eliav; E. V. Kahl; J. C. Berengut; M. Laatiaoui

REVIEW 3 major objections 4 minor 31 references

High-precision ab initio calculations of the spectrum of Lr$^{+}$

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper predicts the first systematic, high-precision spectrum of Lr+, identifying two strong ground-state transitions at 31540 and 47295 cm^-1 with uncertainties of at least 389 cm^-1.

desk verdict First real Lr+ spectrum calculation, with a useful slate of atomic properties for the planned experiment—but the uncertainty bars are not as conservative as claimed, and one benchmark level sits well outside the quoted 389 cm−1 floor. read the letter →

arxiv 1908.04578 v1 pith:MB472OII submitted 2019-08-13 physics.atom-ph

E. V. Kahl , J. C. Berengut , M. Laatiaoui , E. Eliav , A. Borschevsky This is my paper · ORCID

classification physics.atom-ph

keywords lawrenciumsuperheavyelementsatomicspectroscopyFockspacecoupledclusterCI+MBPTenergylevelsEinsteincoefficientslaser

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

Singly ionized lawrencium (Lr+), with 103 protons, is made only one atom at a time, and no experimental spectrum exists. This paper tries to close that gap by computing the low-lying spectrum from first principles with two independent relativistic many-body methods and using the lighter homolog lutetium (Lu+) as a benchmark. It predicts that two electric-dipole transitions from the ground state are strong enough for laser searches: the $7s7p\,{}^3P_1$ state at $31540\,\mathrm{cm}^{-1}$ and the $7s7p\,{}^1P_1$ state at $47295\,\mathrm{cm}^{-1}$, with recommended uncertainties of at least $389\,\mathrm{cm}^{-1}$. If right, the theory shrinks the experimental search window from an unknown spectrum to a few specific wavelengths, and it provides lifetimes and branching ratios needed to plan the measurement.

What carries the argument

The central device is homologue benchmarking with two independent relativistic methods: Fock space coupled cluster (FSCC), a multireference approach that builds excited states by adding electrons to a closed-shell reference, and configuration interaction plus many-body perturbation theory (CI+MBPT), which combines a CI expansion with second-order core-valence corrections. Both start from the projected Dirac-Coulomb-Breit Hamiltonian and include QED corrections through separate operators. Agreement between the two methods, validated by reproducing measured Lu+ levels and transition rates, is used to set the Lr+ error bars and to justify transferring the demonstrated accuracy to the heavier ion.

What would settle it

A laser scan of the predicted $7s7p\,{}^3P_1$ transition near $31540\,\mathrm{cm}^{-1}$ that finds no resonance within the stated uncertainty band, or finds the $7s7p\,{}^1P_1$ line at $47295\,\mathrm{cm}^{-1}$ displaced by more than the recommended uncertainty, would refute the assumption that Lu+ accuracy transfers to Lr+.

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Extended reading notes

Core claim

The paper claims to provide the first large-scale systematic calculation of the Lr+ spectrum. Both methods place the ground state at $7s^2\,{}^1S_0$ and agree on the ordering of the low-lying $6d7s$ and $7s7p$ levels; the recommended energies are the mean of the FSCC and CI+MBPT results, with uncertainty at least $389\,\mathrm{cm}^{-1}$ or the inter-method difference. For the two transitions proposed for experiment, the paper gives $7s7p\,{}^3P_1$ at $31540 \pm 389\,\mathrm{cm}^{-1}$ (lifetime 14.5 ns, branching ratio 0.90 to ground) and $7s7p\,{}^1P_1$ at $47295 \pm 1048\,\mathrm{cm}^{-1}$ (lifetime 1.1 ns, branching ratio 0.96). It also finds that the $6d7s\,{}^3D_1$ level decays only by a suppressed M1 transition and has a lifetime of about 25 days. The accuracy argument rests on Lu+: average theory-experiment differences of $-263\,(348)\,\mathrm{cm}^{-1}$ for FSCC and $16\,(389)\,\mathrm{cm}^{-1}$ for CI+MBPT over the eight lowest relevant levels.

Load-bearing premise

The entire error budget for Lr+ rests on the assumption that the accuracy demonstrated for Lu+ transfers to Lr+ unchanged, since no Lr+ experimental data exists to test it.

Editorial extensions

If this is right

Experimenters can begin a laser search by scanning a window around $31540\,\mathrm{cm}^{-1}$ for the $7s7p\,{}^3P_1$ state, the strongest accessible ground-state transition with the smallest recommended uncertainty.
The $7s7p\,{}^1P_1$ line at $47295\,\mathrm{cm}^{-1}$, with a branching ratio of 0.96 and a lifetime near 1 ns, provides a second, independent resonance to confirm the spectrum.
The predicted lifetimes and branching ratios allow estimates of detector sensitivity and required beam time for an experiment producing about one ion per second.
The very long predicted lifetime of the $6d7s\,{}^3D_1$ level (about 25 days) means population can accumulate in a metastable state, which any excitation scheme must take into account or could exploit.

Reading between the lines

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

The paper does not spell this out, but the same dual-method benchmarking could be used to set search windows for the next even heavier ions, where no homologous experimental anchor exists.
A precise remeasurement of the Lu+ $3P_2$ g-factor would be a cheap indirect test of the wavefunction-quality assumption: the paper predicts 1.50 against a tabulated 1.66 and notes the assignment may be erroneous.
The metastable $6d7s\,{}^3D_1$ state, with its 25-day lifetime, could in principle serve as an optical-clock or trapping state for Lr+ if the ion can be held long enough; this goes beyond the paper's experimental guidance.

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Signed reviews

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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

Colleague,

You should know this paper is genuinely new: it gives the first large-scale ab initio spectrum of Lr+, including energies, g-factors, Einstein A coefficients, branching ratios, and lifetimes, with no experimental input for Lr+ itself. The validation on Lu+ is an honest attempt to gauge accuracy, and the two independent methods (FSCC and CI+MBPT) mostly bracket the known Lu+ levels. The convergence tests for basis size and MBPT partial waves are reported, not just waved at. For a superheavy ion where the planned optical search needs a narrow window, this is exactly the kind of predictive work that moves the field forward.

The main soft spot is the uncertainty prescription. The paper calls the 389 cm−1 floor “conservative,” but the Lu+ data in their own Table I contradict that. For 5d6s 1D2, both methods give ~17884 cm−1 while experiment is 17332 cm−1—a true error of ~550 cm−1, well outside the ±389 interval. The method difference is only 17 cm−1 there, so the max(method difference, 389) rule cannot catch the shared systematic bias. That matters for Lr+: the headline line at 31540 cm−1 could easily be off by more than the quoted ±389, and an experimental scan planned around that width might miss it. This does not invalidate the central prediction—the two methods agree and the Lu+ benchmark is broadly within a few hundred cm−1—but it means the stated uncertainties are not rigorous bounds, just rough estimates. The paper should say so plainly and maybe widen the search guidance to a few thousand cm−1.

Minor issues: the A-coefficient uncertainties are not quantified (10–30% is quoted, but that is from Lu+), and the g-factor of the Lu+ 3P2 level sits far from experiment, which the authors flag as a possible misassignment in the data. Those are side notes, not fatal. The citation pattern is clean: prior Lr+ work is limited to ionization potentials or a misidentified ground state, and the authors correctly situate themselves. No circularity or fabrication concerns.

Net: this paper deserves a serious referee. It is a solid, useful contribution for the atomic-physics and superheavy-element community. My advice to the editor: send it out, but ask the authors to revise the uncertainty discussion—either remove the word “conservative” or re-derive the error from the full Lu+ benchmark distribution, including the 1D2 outlier. The predictions themselves should stand.

— [Your name]

Referee Report

3 major / 4 minor

Summary. The paper reports relativistic Fock-space coupled-cluster (FSCC) and CI+MBPT calculations of the low-lying excitation spectrum, g-factors, transition rates, branching ratios, and lifetimes of Lr+ and its lighter homologue Lu+. The authors benchmark both methods against experimental Lu+ energy levels and A coefficients, then use the Lu+ comparison to assign uncertainties to their recommended Lr+ energies, which are taken as the mean of the two methods with an uncertainty of max(method difference, 389 cm^-1). The main output is a set of recommended Lr+ excitation energies, of which the two strongest ground-state transitions are identified as 7s7p 3P1 at 31540 cm^-1 and 7s7p 1P1 at 47295 cm^-1, meant to guide the planned optical spectroscopy of Lr+. No experimental Lr+ data are used in the calculations.

Significance. If the predictions are reliable, this is the first systematic theoretical spectrum of Lr+ and will directly inform an ongoing experimental search, making the paper valuable to both atomic theory and superheavy-element spectroscopy. The study combines two independent state-of-the-art methods, presents explicit basis-convergence and MBPT partial-wave-convergence tests, and does not fit any Lr+ experimental data, so the recommended energies are genuinely ab initio. The Lu+ comparison is a useful validation of both methods and of the predicted A coefficients. However, the paper's central uncertainty prescription is not supported by its own benchmark, as one Lu+ level shows an error of about 550 cm^-1 while the quoted uncertainty floor is 389 cm^-1; because the experimental search window will be set by these quoted errors, this issue is load-bearing for the paper's main application.

major comments (3)

[Section III, Table I] The uncertainty prescription described in Section III is not conservative by the paper's own Lu+ benchmark. For Lu+ 5d6s 1D2, CI+MBPT gives 17892 cm^-1 and FSCC gives 17875 cm^-1, while the experimental value is 17332 cm^-1. The recommended mean is therefore 17884 cm^-1, an error of about 552 cm^-1, yet the formula max(method difference, 389 cm^-1) assigns this level an uncertainty of only 389 cm^-1. Because 389 cm^-1 is the standard deviation of the CI+MBPT residuals, it is expected that some individual residuals exceed it; calling that value 'conservative' is contradicted by an in-sample case. This matters directly for the primary Lr+ search line at 31540 cm^-1, whose quoted ±389 cm^-1 is set by the same floor and would not cover a shared bias of the magnitude seen in Lu+.
[Section III] Using the difference between the two calculated energies as an error estimate is not sensitive to systematic errors common to both methods. The Lu+ 5d6s 1D2 case shows that the two methods can agree to 17 cm^-1 while both being about 550 cm^-1 from experiment, so a small method difference cannot be taken as evidence of small total error. Both calculations start from the same projected Dirac-Coulomb-Breit Hamiltonian and use similar QED-model corrections, so shared systematic errors are plausible. I ask the authors to replace or supplement the max(method-difference, standard-deviation) rule with a bound that covers the observed Lu+ residuals, for example the maximum residual or an expanded uncertainty, and to discuss which physical effects could produce a common bias in Lr+ but not in Lu+.
[Section III] The statement 'We expect similar accuracy for the calculated transition energies of the heavier homologue of Lu+, Lr+' is the only justification for transferring the Lu+ error estimate to Lr+. The paper should make this transferability argument more concrete, for example by comparing the Lr+ energies with the independent prior calculations of Dzuba et al. and Cao and Dolg, or by estimating the sensitivity of the 7s7p levels to higher-order correlation, higher partial waves, and the QED-model uncertainty. Without such a check, the quoted uncertainties for Lr+ rest on an untested assumption; this is not a fatal flaw, but it is load-bearing for the experimental search-window application.

minor comments (4)

[Table V] In the first two rows of Table V, the lower level is labeled '7s2 2 S0' but the ground state is 7s2 1S0; the superscript should be 1.
[Section II B] The phrase 'Land` e g-factors' contains a formatting artifact; it should read 'Landé g-factors'.
[Section II A, Eq. (4)] The definition of the even-tempered exponents in Eq. (4) is hard to read because the value of gamma is split across lines; please format it as a single number.
[Abstract] The hyphen in 'branching-ratios' should be removed for consistency with the main text and standard usage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lr+ predictions are produced by two independent ab initio methods and are not fitted to any Lr+ experimental data.

full rationale

The central output of the paper, the recommended Lr+ excitation energies, is computed as the unweighted mean of FSCC and CI+MBPT results obtained from the projected Dirac-Coulomb-Breit Hamiltonian. No Lr+ experimental level is used to determine or adjust any parameter in either calculation, so the predicted energies are not equivalent by construction to the inputs. The Lu+ comparison in Table I uses independent published experimental data (refs. [29,30]) as an external benchmark, and the uncertainty prescription (max of method difference and the 389 cm-1 standard deviation of CI+MBPT Lu+ errors) is a calibration on that homologue, not a fit to the target. The self-citations (TRAFS-3C, ambit, and earlier FSCC/CI+MBPT papers) are references to the methods and codes themselves; their validity is supported by the external Lu+ agreement, so they are not load-bearing self-citations. The statement 'we expect similar accuracy' for Lr+ is an explicit transferability assumption and a limitation, not a circular step. Even if the uncertainty floor is questioned (e.g., the Lu+ 5d6s 1D2 level has a true error around 550 cm-1 at the quoted mean), that is a correctness or calibration risk, not a circularity, because the prediction does not reduce to the Lu+ data used to set the error bar.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No parameter is fitted to Lr+ or Lu+ experimental data. All listed items are convergence or model choices. No new physical entities are introduced.

free parameters (4)

Virtual orbital energy cutoff = 200 a.u.
Orbitals with energies above 200 a.u. are omitted from correlation; a truncation chosen by hand with no explicit convergence curve shown.
FSCC model space sizes = 13s11p9d8f6g5h (Lu+), 14s12p10d9f6g5h (Lr+)
Maximum model spaces used in coupled cluster; convergence with model space size is stated as verified but no table is shown.
CI and emu CI truncations = single excitations to 16 spdfg; single and double excitations to 12 spdfg; Ndominant threshold
Basis and dominant-configuration thresholds in CI+MBPT; convergence checks are given in Section II.
MBPT partial-wave limit = orbitals up to 35 spdfghi (l <= 6)
Second-order MBPT basis truncation; adding l >= 7 changes energies by less than 50 cm-1.

assumptions (5)

domain assumption The projected Dirac-Coulomb-Breit Hamiltonian, correct to second order in the fine-structure constant, is an adequate starting point for Lr+ and Lu+.
Used throughout, Eq (1); neglects higher-order QED and non-QED effects beyond the model operators added in Section II.
domain assumption Lu+ is a valid lighter homologue whose calculation accuracy transfers to Lr+.
Section III: 'We expect similar accuracy for the calculated transition energies of the heavier homologue of Lu+, Lr+.' This is the load-bearing premise for the Lr+ error budget.
domain assumption The ground state of Lr+ is 7s2 1S0.
FSCC starts from closed-shell Lr3+ and adds two electrons; CI builds on 7s2 in a V(N-1) potential. The previous Hartree-Fock identification as 6d2 is dismissed in Section I.
domain assumption Model-space, basis, and MBPT truncations are converged to the claimed accuracy.
Section II reports convergence tests: increasing the dominant-configuration set changes energies by less than 0.01%, basis beyond 16spdfg by about 1%, and l >= 7 MBPT orbitals by less than 50 cm-1.
domain assumption The model Lamb shift operator and radiative potential adequately describe QED effects for these ions.
Section II applies QEDMOD/MLSO for FSCC and the Flambaum-Ginges radiative potential for CI+MBPT; QED corrections lower energies by 100-400 cm-1 and are not directly benchmarked.

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Pith. "Pith review of High-precision ab initio calculations of the spectrum of Lr$^{+}$." pith.science (2026). https://pith.science/paper/MB472OII

@misc{pith2026190804578,
  author       = {Pith},
  title        = {Pith review of: High-precision ab initio calculations of the spectrum of Lr$^+$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MB472OII}},
  note         = {Machine review of arXiv:1908.04578}
}

abstract

The planned measurement of optical resonances in singly-ionised lawrencium (Z = 103) requires accurate theoretical predictions to narrow the search window. We present high-precision, ab initio calculations of the electronic spectra of Lr$^+$ and its lighter homologue lutetium (Z = 71). We have employed the state-of-the-art relativistic Fock space coupled cluster approach and the AMBiT CI+MBPT code to calculate atomic energy levels, g-factors, and transition amplitudes and branching-ratios. Our calculations are in close agreement with experimentally measured energy levels and transition strengths for the homologue Lu$^+$ , and are well-converged for Lr$^+$ , where we expect a similar level of accuracy. These results present the first large-scale, systematic calculations of Lr$^+$ and will serve to guide future experimental studies of this ion.

Figures

Figures reproduced from arXiv: 1908.04578 by the authors.

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