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Finding the ultra-narrow $^3\!P_2 \rightarrow \, ^3\!P_0$ electric quadrupole transition in Ni$^{12+}$ ion for an optical clock

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

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Pith's one-line read The energy of the ultra-narrow 3P2 → 3P0 electric quadrupole transition in Ni12+ is predicted at 20081(10) cm^-1 and measured at 20078.984(10) cm^-1, an agreement of 2 cm^-1.

desk verdict First optical excitation of the Ni12+ clock transition, with a 10 cm^-1 ab initio prediction that landed 2 cm^-1 from the measured value; a strong letter with one modest caveat about the full-CI residual estimate. read the letter →

arxiv 2502.05386 v1 pith:PO242C34 submitted 2025-02-08 physics.atom-ph

Charles Cheung , Sergey G. Porsev , Dmytro Filin , Marianna S. Safronova , Malte Wehrheim , Lukas J. Spie{ss} , Shuying Chen , Alexander Wilzewski
show 2 more authors
José R. Crespo López-Urrutia Piet O. Schmidt
This is my paper · ORCID
classification physics.atom-ph
keywords Ni12+highlychargedionopticalclockelectricquadrupoletransitionconfigurationinteractioncoupled-clusteratomicphysicsfrequencymetrology
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 predicts the energy of the strongly forbidden 3s2 3p4 3P2 → 3s2 3p4 3P0 electric quadrupole transition in the Ni12+ ion, a candidate for a highly stable optical clock with a natural linewidth of only 8 mHz. Using two independent theoretical methods — a hybrid approach combining configuration interaction with coupled-cluster and a pure 16-electron CI calculation — the authors obtain 20081(10) $cm^{-1}$, an uncertainty of 0.05%. Working from this prediction, they located the transition experimentally in six hours, measuring 20078.984(10) $cm^{-1}$, only 2 $cm^{-1}$ away. The close agreement for a 16-electron system is presented as evidence that full CI calculations can guide searches for clock transitions in other highly charged ions.

What carries the argument

The load-bearing tool is a pure CI calculation for all 16 electrons of the Ni12+ ground configuration, broken into additive contributions from excitations of the outer six electrons, inner ten electrons, extra reference configurations, and an estimated residual full-CI term. The residual is estimated through a linear relation between a configuration's relative weight and its energy contribution, extrapolated from computed configurations. A hybrid CI+all-order method with a coupled-cluster effective Hamiltonian provides an independent cross-check, and a neural-network-based configuration selection reduced memory and time needs.

What would settle it

If another independent computation of the 3P0 energy, or a future more precise measurement, deviates from 20081 $cm^{-1}$ by more than 10 $cm^{-1}$, the claimed uncertainty and the extrapolation of the weight-energy relation would be falsified. A direct check would be an independent CI calculation with a different code or basis that does not use the same linear extrapolation.

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Extended reading notes

Core claim

The central claim is that a fully converged 16-electron configuration-interaction calculation can predict the energy of the forbidden 3P2 → 3P0 clock transition in Ni12+ with an uncertainty of only 10 $cm^{-1}$, sufficient to locate the transition with rapid laser scans. The prediction, 20081(10) $cm^{-1}$, differs from the measured value 20078.984(10) $cm^{-1}$ by 2 $cm^{-1}$. The authors argue this validates their method for other complex atomic systems and enables a Ni12+ optical clock.

Load-bearing premise

The estimate of the residual full-CI contribution assumes that the linear weight–energy relation found for computed configurations holds for the far more numerous configurations that cannot be computed directly, with the 3P0 correction being small (-3 $cm^{-1}$) but not independently verified.

Editorial extensions

If this is right

  • The Ni12+ clock transition is now accessible for quantum-logic spectroscopy and the development of a high-precision optical clock.
  • The demonstrated accuracy of the 16-electron CI method supports its application to other highly charged ions where clock transitions have not yet been located.
  • The separation of CI into additive contributions and the weight-energy estimate extends the practical size of CI calculations beyond the current limits of exact diagonalization.

Reading between the lines

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

  • The neural-network configuration-selection algorithm, developed for this calculation, could be applied to other large CI problems where the full expansion is intractable.
  • The measured 2 cm^-1 discrepancy suggests the remaining uncertainty may be dominated by the estimated full-CI residual or QED corrections; a systematic study of similar transitions could refine these estimates.
  • The same fast-scanning search strategy, combined with precise predictions, could be used to find other forbidden transitions in HCIs that were previously considered too weak to search for.
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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

3 major / 4 minor

Summary. The paper predicts the energy of the strongly forbidden 3s2 3p4 3P2 → 3s2 3p4 3P0 electric quadrupole transition in Ni12+ as 20081(10) cm−1, using two independent many-body approaches: a hybrid CI+all-order method and a pure 16-electron CI calculation. The prediction was then used to search for and find the transition experimentally at 20078.984(10) cm−1, only 2 cm−1 from the theoretical value. The authors claim that this level of agreement, with a stated 0.05% uncertainty for a 16-electron system, is unprecedented and qualifies their method for future atomic-structure calculations.

Significance. If the uncertainty estimate holds, this is a landmark result: it demonstrates that a full 16-electron configuration-interaction calculation can predict a state energy to about 0.05% accuracy, and the experimental confirmation in a few hours is a powerful validation. The paper also contributes an open-source parallel CI package, a neural-network-based configuration selection tool, and a two-method cross-check that agree to 5 cm−1 for the target level. The main weakness is the unquantified extrapolation used to estimate the residual full-CI contribution, which is central to the claimed 10 cm−1 uncertainty budget.

major comments (3)
  1. [Theory: pure CI method (Table II, column 'Estimate full CI')] The estimate of the contribution of all omitted configurations, labeled 'Estimate full CI', is obtained from a linear relation between configuration weight and energy contribution, but no uncertainty is assigned to this extrapolation. The 10 cm−1 uncertainty budget appears to rely on the difference between the two methods and the agreement for the 3P1 level, and does not explicitly include the error in this extrapolation. The 1S0 level in Table II shows a discrepancy of −66 cm−1 from experiment, with an 'Estimate full CI' correction of only −24 cm−1, demonstrating that the method's accuracy is strongly level-dependent. Therefore, the small −3 cm−1 correction for 3P0 does not by itself guarantee that the extrapolation error is small. Please provide a quantitative bound for the extrapolation error, for example by testing the linear relation on additional levels or by varying the slope within a plausible range, or explicitly state that the 10 cm−1 budget excludes this source and justify why it is negligible.
  2. [Theory: pure CI method, paragraph on 'Estimate full CI'] The description 'a relative weight of 0.01% leading to a correction of about −11 cm−1' is ambiguous: it is unclear whether this refers to the contribution of a single configuration or to the integrated contribution of all configurations with weights below a threshold. The manuscript does not provide the actual fitting parameters, the range of weights over which the linear relation was validated, or a plot of the data supporting the 'linear dependence (similar for all four levels)' claim. Without this information, the 'Estimate full CI' column in Table II is not reproducible and its reliability cannot be independently assessed.
  3. [Conclusion] The conclusion states that the work demonstrates 'full convergence of a 16-electron CI computation', but this overstates the status of the calculation: the largest part of the configuration space is not computed directly but estimated via the extrapolated linear relation. A more precise wording would be 'convergence with respect to explicitly included configurations and an estimated residual contribution', which would be consistent with the actual procedure described in the text.
minor comments (4)
  1. [Throughout] Several formulas are typeset incorrectly in the text as provided, e.g., '3s23p4' instead of '3s^2 3p^4' and 'Ni12+' instead of 'Ni$^{12+}$'. These should be corrected in the final version.
  2. [References] Reference [12] is not a standard citation but a footnote explaining the treatment of k and l partial waves. It should be renumbered as a footnote or integrated into the main text.
  3. [Table I and Table II] The column groups such as '17 spd fg', 'l >6', 'Extra conf.', and 'Estimate full CI' are not fully defined in the captions. Please add a sentence explaining the meaning of the grouped columns, e.g., that '17 spd fg' refers to orbitals with n up to 17 and partial waves s, p, d, f, g, and that the contributions are incremental from the previous column.
  4. [Figure 1] The caption 'Illustration of the basis set upscale' is vague. The figure should describe what the blocks represent (e.g., principal quantum number ranges and partial waves) and how it relates to the convergence procedure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 3P2–3P0 energy is computed ab initio and the experimental value is used only as a posteriori validation, not as an input.

full rationale

The paper's central prediction (20081(10) cm^-1 for the 3P2–3P0 E2 transition) is obtained from two independent many-body treatments, CI+all-order and 16-electron CI. No parameter is fitted to the 3P0 energy or to the measured clock transition. The experimental 3P1, 1D2, and 1S0 levels are used as benchmarks in Tables I and II, not as constraints in the calculation. The self-citations to the CI package pCI [13,14], the CI+all-order formalism [10], the recurrent basis [11,18], and the neural-network selection [19] are method citations; they do not smuggle the predicted transition energy into the calculation. The 'Estimate full CI' procedure in the pure CI section is an empirical extrapolation of a weight–energy relation tested on computed configurations; it is explicitly acknowledged as an estimate ('impossible to compute directly'), is applied uniformly to the low-lying levels, and contributes only -3 cm^-1 to the 3P0 energy. Its reliability concerns the uncertainty budget, but it is not circular because the correction is not constructed from the target 3P0 energy. The agreement with the subsequently measured value (20078.984(10) cm^-1) is an external check, not an input. Thus no step in the derivation reduces by construction to its own input.

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

The central value 20081 cm^-1 is obtained without fitting any parameter to the target transition. The calculation does depend on the assumed completeness of the CI model, the empirical full-CI extrapolation, and the accuracy of QED corrections. These are the main assumptions that would affect the claimed 10 cm^-1 uncertainty.

free parameters (1)
  • Full-CI weight-to-energy conversion factor = -11 cm^-1 per 0.01% relative weight
    Empirically derived from the computed contributions of 200, 2000, and 9500 reference configurations; used to estimate the contribution of all remaining configurations. Not fitted to experimental data.
assumptions (4)
  • domain assumption The CI model space with single and double excitations from the chosen reference configurations is sufficient to converge the low-lying state energies to a few cm^-1.
    The paper demonstrates convergence in n and partial waves, but completeness of the reference set is assumed.
  • ad hoc to paper The relationship between configuration relative weight and energy contribution is linear and can be extrapolated to all omitted configurations.
    This linear relation is found empirically and used to estimate the full-CI correction. It is not derived from first principles.
  • domain assumption QED corrections computed in Ref. [15] are accurate to 10% or better for these levels.
    The paper relies on the accuracy of QED corrections from prior work; in Table II, QED contributes +33 cm^-1 to the 3P0 energy.
  • domain assumption The frozen-core approximation in the CI+all-order method, with core-valence correlations treated by coupled-cluster, is valid for this system.
    The hybrid approach separates the 10 core electrons and 6 valence electrons; the validity of this separation is standard in atomic structure theory.

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Cite this review

Pith. "Pith review of Finding the ultra-narrow $^3\!P_2 \rightarrow \, ^3\!P_0$ electric quadrupole transition in Ni$^{12+}$ ion for an optical clock." pith.science (2026). https://pith.science/paper/PO242C34

@misc{pith2026250205386,
  author       = {Pith},
  title        = {Pith review of: Finding the ultra-narrow $^3\!P_2 \rightarrow \, ^3\!P_0$ electric quadrupole transition in Ni$^12+$ ion for an optical clock},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO242C34}},
  note         = {Machine review of arXiv:2502.05386}
}
abstract

The Ni$^{12+}$ ion features an electronic transition with a natural width of only 8 mHz, allowing for a highly stable optical clock. We predict that the energy of this strongly forbidden $3s^2 3p^4\, ^3\!P_2 \rightarrow 3s^2 3p^4 \, ^3\!P_0$ electric quadrupole transition is 20081(10) cm$^{-1}$. For this, we use both a hybrid approach combining configuration interaction (CI) with coupled-cluster (CC) method and a pure CI calculation for the complete 16-electron system, ensuring convergence. The resulting very small theoretical uncertainty of only 0.05\% allowed us to find the transition experimentally in a few hours, yielding an energy of 20078.984(10) cm$^{-1}$. This level of agreement for a 16-electron system is unprecedented and qualifies our method for future calculations of many other complex atomic systems. While paving the way for a high-precision optical clock based on Ni$^{12+}$, our theory and code development will also enable better predictions for other highly charged ions and other complex atomic systems.

Figures

Figures reproduced from arXiv: 2502.05386 by the authors.

Figure 1
Figure 1. FIG. 1: Illustration of the basis set upscale. 17 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Measurement of the motional excitation after [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Coulomb Crystallization of Highly Charged Ni^12+ Ions in a Linear Paul Trap

    physics.atom-ph 2025-04 conditional novelty 6.0 of 10

    Ni12+ ions were sympathetically cooled with laser-cooled Be+ in a room-temperature Paul trap to form a two-species Coulomb crystal at the 100 mK level.

Reference graph

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