Application of General-order Relativistic Coupled-cluster Theory to Estimate Electric-field Response Clock Properties of Ca$^+$ and Yb$^+$

B. K. Sahoo; Yanmei Yu

REVIEW 3 major objections 4 minor 40 references

Application of General-order Relativistic Coupled-cluster Theory to Estimate Electric-field Response Clock Properties of Ca$^+$ and Yb$^+$

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

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

Pith's one-line read Relativistic coupled-cluster calculations with triple excitations fix the electric-field response properties of the Yb+ optical-clock states, yielding a differential polarizability of 38(4) a.u. in accord with measurement.

desk verdict Solid Ca+ results and plausible Yb+ numbers, but the Yb+ D3/2 benchmark rests on an untested additivity that contradicts the stated protocol. read the letter →

arxiv 2502.06225 v1 pith:ISANEGJL submitted 2025-02-10 physics.atom-ph

YanMei Yu , B. K. Sahoo This is my paper · ORCID

classification physics.atom-ph

keywords relativisticcoupled-clusterdipolepolarizabilityquadrupolemomentopticalclockytterbiumioncalciumfinite-fieldmethodtripleexcitations

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 tries to establish that the electric-field response properties of the Yb+ ion's optical clock states—the differential scalar dipole polarizability, the quadrupole moment, and the quadrupole polarizability—can be computed accurately enough to serve as benchmarks for systematic-effect corrections in clocks. Using general-order relativistic coupled-cluster theory through singles, doubles, and triples (RCCSDT) in a finite-field scheme, the authors obtain a differential scalar dipole polarizability of 38(4) a.u. for the 6s 2S1/2 → 5d 2D3/2 clock transition, matching a 2018 measurement, and a quadrupole moment of 1.973(31) a.u. for the 5d 2D3/2 state, matching a 2020 measurement. The central lesson is that triple excitations, previously ignored in most RCC property calculations, are decisive for these quantities. They also validate the method on the lighter Ca+ ion, whose similar clock states allow a more complete correlation treatment.

What carries the argument

The central object is the general-order relativistic coupled-cluster expansion with single, double, and triple excitations (RCCSDT), applied through a finite-field approach: the energy of each clock state is computed under a series of static electric fields (and field gradients), and the dipole polarizability, quadrupole moment, and quadrupole polarizability are read off from the coefficients of the polynomial energy shift versus field strength. This bypasses direct property-evaluation expressions, and the authors argue it satisfies the Hellmann-Feynman theorem in the RCC framework. The strategy of first benchmarking on Ca+—whose similar 4s–3d clock transitions allow nearly complete correlation—is used to decide basis-set and triple-excitation handling before transferring the method to Yb+.

What would settle it

A full RCCSDT calculation in the 4ζ basis for the Yb+ 5d 2D3/2 state (or an equivalent large-basis treatment of triples) would settle whether the additivity assumption holds; if the computed Δαd shifts by more than the quoted 4 a.u., the recommended value is biased. Alternatively, a more precise experimental measurement of Δαd for the Yb+ clock transition with uncertainty below 2 a.u. would directly test the central value.

Watch

Extended reading notes

Core claim

The paper's central claim is that the previously discrepant theoretical and experimental values for the Yb+ E2 clock transition's electric-field response arise primarily from missing triple-excitation correlations. Computing energies under external electric fields and field gradients with RCCSDT wavefunctions, and extracting polarizabilities and quadrupole moments from polynomial fits of those energies, the authors obtain ground-state αd = 63.53(35) a.u., excited-state αd = 101(4) a.u. and αT = −72(5) a.u. for the 5d 2D3/2 state, differential Δαd = 38(4) a.u., and quadrupole moment Θ = 1.973(31) a.u. These numbers agree with the most recent experiments and support using them as benchmarks, while also explaining why older calculations and one of the two experimental groups sat systematically higher.

Load-bearing premise

The final Yb+ values rely on the assumption that triple-excitation contributions computed with a smaller (2ζ or 3ζ) basis can be added as a correction to RCCSD results from a larger (4ζ) basis; if triple corrections depend strongly on basis size, the recommended values could be biased by more than the quoted uncertainties.

Editorial extensions

If this is right

The recommended values give clock-systematics corrections with uncertainties small enough to improve the Yb+ E2 clock systematic budget.
The result resolves the conflict between the two Yb+ differential-polarizability measurements, siding with the 2018 value.
The quadrupole moment agrees with recent experiment, implying the quadrupole shift can be reduced in Yb+ E2 clock operations.
The method provides first reported values of αq for the Yb+ clock states, useful for estimating gradient-induced shifts.
The improved Ca+ values also strengthen benchmarks for the Ca+ optical clock.

Reading between the lines

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

One could test whether the same RCCSDT-plus-basis-correction strategy applies to other heavy-ion clocks such as Sr+ or Ba+, where triple excitations may be underestimated as well.
A direct extension would be to compute the E3 clock state (4f13 6s2 2F7/2) polarizabilities with this method; the paper does not attempt that transition.
The finite-field extraction of αq from third-order energy terms may be sensitive to the fitting range, so future work should report stability across field strengths.
The explanation for older RCCSD results suggests that earlier sum-over-states analyses may need to revisit triple-excitation effects in their property evaluations.

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

Good to see this one. It's a serious ab initio effort: first finite-field RCCSDT calculation of dipole and quadrupole polarizabilities and quadrupole moments for Ca+ and Yb+ clock states, and the first alpha_q values for the Yb+ 2D3/2 state. The final Yb+ D3/2 numbers land close to recent measurements, and the Ca+ proxy study is a reasonable strategy.

The soft spot is in how the Yb+ D3/2 values are assembled. Section III describes a standard composite: RCCSD at the largest basis plus the (RCCSDT−RCCSD) correction from a smaller basis. For Table III, that gives 108.36 + (103.31 − 119.71) = 91.96 a.u. What the paper actually does is take the 2ζ RCCSDT value 103.31 and add an SD-level basis shift from 3ζ to 4ζ of −2.10, getting 101.21 a.u. The two procedures differ by 9.25 a.u., far outside the quoted 4 a.u. uncertainty, and the text doesn't acknowledge the switch. The same mismatch appears in Table II, where the final ground-state polarizability is shifted by Δbasis even though the text says those corrections go into the uncertainty budget.

The bigger worry is that the additivity is unvalidated for the target states. The paper's own ground-state data show the triple correction is strongly basis dependent: ΔPT is −2.27 a.u. at 2ζ and −0.53 a.u. at 3ζ for αS_d(Yb+). For the 2D3/2 state, no RCCSDT result exists beyond 2ζ, so the large triple correction (16.4 a.u.) could change significantly with basis. The quoted ±4 a.u. uncertainty is built from incremental SD-level corrections, not from the additivity assumption, so its coverage is unknown. And the statement that a prior RCCSD calculation by the same group cannot be reproduced due to 'implementation differences' is a reproducibility flag that needs an explanation.

What I credit: the calculations are genuinely ab initio, no experimental fitting, and the experiment comparisons are retrospective validation. The work gives the clock community new numbers for alpha_q and a plausible resolution of the theory-experiment discrepancy for ΔαS_d of the E2 transition. The Ca+ results are on firmer ground because triples are included in larger bases there.

This deserves a serious referee, but not acceptance as written. The authors should either compute the 3ζ RCCSDT for the D3/2 state or justify why the 2ζ triple correction is adequate, reconcile the method description with the executed arithmetic, and explain the Ref. [35] discrepancy. As is, I'd treat the final Yb+ D3/2 values as preliminary rather than benchmarks.

Referee Report

3 major / 4 minor

Summary. The manuscript reports general-order relativistic coupled-cluster (RCC) calculations, at the RCCSD and RCCSDT levels, of static electric dipole polarizabilities, quadrupole moments, and quadrupole polarizabilities for clock-relevant states of Ca+ and Yb+. A finite-field approach is used to extract these properties from energy shifts. The authors present recommended values for Yb+ (e.g., αS_d = 101(4) a.u. for the 5d 2D_3/2 state, ΔαS_d = 38(4) a.u. for the 2S_1/2 → 2D_3/2 transition, and Θ = 1.973(31) a.u. for the 2D_3/2 state) and argue that triple excitations are decisive in bringing the calculated values into agreement with recent experiments. The Ca+ calculations are used as a lighter proxy to validate the role of triples and to guide basis-set choices for Yb+.

Significance. If correct, the recommended values would be a valuable ab initio benchmark for systematic-effect corrections in Ca+ and Yb+ optical clocks. The work uses a genuine general-order RCC implementation with triples and a finite-field method that avoids property-evaluation issues of some earlier approaches; it is also free of experimental input in the Hamiltonian. The Ca+ analysis is a useful methodological demonstration. However, the final Yb+ numbers rely on a composite-scheme assumption that is not validated and that is implemented inconsistently with the scheme stated in Sec. III. Because the main claim is to provide benchmark-quality values with reliable uncertainties, this issue must be resolved before the numbers can be adopted with confidence.

major comments (3)

[Sec. IV, Table III] The composite scheme actually used for the 2D_3/2 state differs from the scheme stated in Sec. III. Sec. III defines the final value as RCCSD(large basis) + [RCCSDT(small basis) − RCCSD(small basis)]. For Table III this would give αS_d = 108.36 + (103.31 − 119.71) = 91.96 a.u. Instead, the text combines the 2ζ RCCSDT value with the 3ζ→4ζ basis shift obtained from RCCSD: 103.31 + (108.36 − 110.46) = 101.21 a.u. The two prescriptions differ by 9.25 a.u., which is more than twice the quoted uncertainty of 4 a.u. This directly affects the recommended αS_d = 101(4) a.u. and ΔαS_d = 38(4) a.u., and therefore the claimed agreement with the experimental result of Ref. [13]. The authors must either reconcile the two schemes or justify why the second one is the correct composite estimate.
[Sec. IV, Tables II and III] The additivity assumption that triple-excitation corrections are nearly independent of basis size is contradicted by the paper's own ground-state data. For αS_d of Yb+, Table II gives ΔPT = −2.27 a.u. at 2ζ but −0.53 a.u. at 3ζ, a factor-of-four change. No RCCSDT result is reported for the 2D_3/2 state beyond 2ζ, so the triple correction in Table III is taken at the smallest basis, where basis error is largest. The uncertainty budget for the final 2D_3/2 values is built from Δbasis, Δ4d, Δvirt, and Δsaug, but it does not include the uncertainty in the additivity assumption itself. The error bars therefore have unknown coverage, and the recommended values could be biased beyond the quoted uncertainties.
[Sec. IV, Table V] For the 5d 2D_5/2 state, no RCCSDT calculation is reported. The final values are obtained from RCCSD only, with an uncertainty estimated by multiplying Δbasis by a factor of 2 'to be on safer side.' This is an ad hoc uncertainty that is not based on an actual treatment of triple excitations, and it is not justified by the Ca+ analysis, where triples contribute at the level of several percent to the 2D-state polarizabilities. As the paper recommends αS_d = 89(6) a.u., αT_d = −74(2) a.u., and Θ = 3.06(7) a.u. for this state, the reliability of these benchmark values is not established.

minor comments (4)

[Sec. IV, text near Table III] The discussion refers to 'Exp-2015' for Ref. [12], but Ref. [12] is Schneider et al. (2005); this appears to be a typographical error.
[Sec. IV, Table IV] The final Θ is quoted as 1.973(31) a.u., but adding the listed corrections (1.985 − 0.025 + 0.017 − 0.004 − 0.005) gives 1.968 a.u.; the manuscript should clarify how 1.973 is obtained.
[Sec. IV, Table V and text] The notation 'e32-CCSD' and '5D5/2' appears in the table and accompanying text; these should be 'e23-CCSD' and '2D_5/2' or '5d 2D_5/2'.
[Sec. IV, Table III] The text states that the ground-state polarizability used in the final ΔαS_d is αS_d = 65.53(35) a.u., but the recommended ground-state value in Table II and the Summary is 63.53(35) a.u.; this appears to be a typographical error.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: finite-field RCC results are self-contained; the Yb+ mixed-basis triples additivity is an accuracy caveat, not a by-construction reduction.

full rationale

The derivation is an ab initio finite-field coupled-cluster calculation: the DCG Hamiltonian (Eq. 6) contains no empirical atomic constants, and alpha_d, alpha_q, and Theta are obtained as Taylor coefficients of RCCSD/RCCSDT energies computed at several field strengths (Eqs. 14-15). No experimental value is used to set or tune the Hamiltonian, and the agreements with Refs. [13], [14], and [36] are retrospective validations rather than fitted inputs. The same-group RCC results cited for comparison (Refs. [19], [22], [35]) do not supply any premise required for the present numbers. The main substantive concern adding a 2-zeta RCCSDT triple correction to 3-zeta/4-zeta RCCSD basis shifts for the Yb+ 2D3/2 state is a potential accuracy/validity limitation, and it may not follow the composite scheme announced in Section III, but it does not make the final prediction equivalent by construction to its input; the values are still determined by the RCC energies rather than by the experimental benchmarks. The paper itself flags the 2-zeta SDT calculation as a computational compromise, which is a stated limitation rather than a disguised input. Hence no circular step can be exhibited.

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

The calculation is ab initio with no free parameters fitted to experimental data. The central values depend on the DCG Hamiltonian, the RCCSDT truncation, the finite-field truncation of the energy series, and an additivity assumption connecting small-basis triple-excitation corrections to large-basis RCCSD results. The latter is an ad hoc modeling choice specific to this paper.

free parameters (1)

Uncertainty inflation factor for 5D5/2 state = 2.0
In Table V, the uncertainty for the 4f14 5d 2D5/2 state is estimated by applying a factor of 2 to the basis set correction Delta_basis 'to be on safer side'. This is a hand-chosen multiplier; it affects only the quoted error bar, not the central values, and is not load-bearing for the main 2D3/2 results.

assumptions (4)

domain assumption The Dirac-Coulomb-Gaunt (DCG) Hamiltonian, Eq. (6), with a Gaussian nuclear charge distribution is an adequate Hamiltonian for calculating these properties at the quoted accuracy.
Used throughout; QED corrections beyond Gaunt are neglected, which is standard for these ions but an approximation.
domain assumption The energy of the atomic states can be represented by the truncated Taylor expansions in Eqs. (1) and (2); higher-order terms in the field and field-gradient are negligible for the field strengths used ([0,0.0025] a.u. and [1.5,30]x10^-6 a.u.).
Finite-field extraction relies on this; if nonlinear terms were significant, the fitted polarizabilities would be biased.
domain assumption The RCCSDT approximation for T and the corresponding R operator for excited states is sufficient; effects of quadruple and higher excitations are negligible.
The method truncates the coupled-cluster excitation operator at triples. The paper argues triples matter but does not estimate quadruple contributions.
ad hoc to paper The composite scheme is valid: the RCCSDT correction computed in a small basis (2z or 3z) can be added to RCCSD results at larger basis (4z) to obtain the final values.
This is the key extrapolation used in Tables I, II, III, and V. It is not directly validated and is the weakest structural premise.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Application of General-order Relativistic Coupled-cluster Theory to Estimate Electric-field Response Clock Properties of Ca$^+$ and Yb$^+$." pith.science (2026). https://pith.science/paper/ISANEGJL

@misc{pith2026250206225,
  author       = {Pith},
  title        = {Pith review of: Application of General-order Relativistic Coupled-cluster Theory to Estimate Electric-field Response Clock Properties of Ca$^+$ and Yb$^+$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISANEGJL}},
  note         = {Machine review of arXiv:2502.06225}
}

abstract

Accurate calculations of electric dipole polarizabilities ($\alpha_d$), quadrupole moments ($\Theta$), and quadrupole polarizabilities ($\alpha_q$) for the clock states of the singly charged calcium (Ca$^+$) and ytterbium (Yb$^+$) ions are presented using the general-order relativistic coupled-cluster (RCC) theory. Precise knowledge of these quantities is immensely useful for estimating uncertainties caused by major systematic effects such as the linear and quadratic Stark shifts and black-body radiation shifts in the optical Ca$^+$ and Yb$^+$ clocks. A finite-field approach is adopted for estimating these quantities, in which the first-order and second-order energy level shifts are analyzed by varying strengths of externally applied electric field and field-gradient. To achieve high-accuracy results in the heavier Yb$^+$ ion, we first calculate these properties in a relatively lighter clock candidate, Ca$^+$, which involves similar clock states. From these analyses, we learned that electron correlation effects arising from triple excitations in the RCC theory contribute significantly to the above properties, and are decisive factors in bringing the calculated values closer to the experimental results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 37 canonical work pages

[35]

B. K. Sahoo, B. P. Das, and D. Mukherjee, Phys. Rev. A, 79, 052511 (2009)

work page 2009
[13]

C. F. A. Baynham, E. A. Curtis, R. M. Godun, J. M. Jones, P. B. R. Nisbet-Jones, P. E. G. Baird, K. Bongs, P. Gill, arXiv:1801.10134v3, (2020)

arXiv 2020
[1]

Taylor, M

P. Taylor, M. Roberts, S. V. Gateva-Kostova, R. B. M. Clarke, G. P. Barwood, W. R. C. Rowley, and P. Gill, Phys. Rev. A 56, 2699 (1997)

work page 1997
[2]

Roberts, P

M. Roberts, P. Taylor, S. V. Gateva-Kostova, R. B. M. Clarke, W. R. C. Rowley, P. Gill, Phys. Rev. A 60, 2867 (1999)

work page 1999
[3]

S. A. Webster, R. M. Godun, S. A. King, G. Huang, B. Walton, V. Tsatourian, H. S. Margolis, S. Lea and P. Gill, IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control 57, 3 (2010)

work page 2010
[4]

C. Tamm, N. Huntemann, B. Lipphardt, V. Gerginov, N. Nemitz, M. Kazda, S. Weyers and E. Peik, Phys. Rev. A 89, 023820 (2014)

work page 2014
[5]

R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, K. Bongs, and P. Gill, Phys. Rev. Lett. 113, 210801 (2014)

2014
[6]

Huntemann, B

N. Huntemann, B. Lipphardt, Chr. Tamm, V. Gerginov, S. Weyers, and E. Peik, Phys. Rev. Lett. 113, 210802 (2014)

work page 2014

Show all 40 references

[7]

Filzinger, S

M. Filzinger, S. D¨ orscher, R. Lange, J. Klose , M. Steinel , E. Benkler , E. Peik , C. Lisdat , and N. Huntemann, Phys. Rev. Lett. 130, 253001 (2023)

2023
[8]

Lange, N

R. Lange, N. Huntemann, J. M. Rahm, C. Sanner, H. Shao, B. Lipphardt, C. Tamm, S. Weyers, and E. Peik, Phys. Rev. Lett. 126, 011102 (2021)

2021
[9]

Sanner, N

C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, Nature (London)567, 204 (2019)

2019
[10]

Toﬀul, C

A. Toﬀul, C. F. A. Baynham, E. A. Curtis, A. O. Parsons, B. I. Robertson, M. Schioppo, J. Tunesi, H. S. Margolis, R. J. Hendricks, J. Whale, R. C. Thompson, and R. M. Godun, Metrologia 61, 045001 (2024)

2024
[11]

V. V. Flambaum and V. A. Dzuba, Can. J. Phys. 87, 25 (2009)

2009
[12]

Schneider, E

T. Schneider, E. Peik, and Chr. Tamm, Phys. Rev. Lett., 94, 230801 (2005)

2005
[14]

Lange, N

R. Lange, N. Huntemann, C. Sanner, H. Shao, B. Lip- phardt , Chr. Tamm, and E. Peik, Phys. Rev. Lett. 125, 143201 (2020)

2020
[15]

Migdalek, J

J. Migdalek, J. Quant. Spectrosc. Radiat. Transf. 28, 61(1982)

1982
[16]

S. N. Lea, S. A. Webster, and G. P. Barwood, Proceed- ings of the 20th EFTF, 302, (2006)

2006
[17]

U. I. Safronova and M. S. Safronova, Phys. Rev. A 79, 022512 (2009)

2009
[18]

S. G. Porsev, M. S. Safronova, and M. G. Kozlov, Phys. Rev. A 86, 022504 (2012)

2012
[19]

A. Roy, S. De, Bindiya Arora and B. K. Sahoo, J. Phys. B: At. Mol. Opt. Phys. 50, 205201 (2017)

2017
[20]

Itano, Phys

Wayne M. Itano, Phys. Rev. A 73, 022510 (2006)

2006
[21]

K. V. P. Latha, C. Sur, R. K. Chaudhuri, B. P. Das, and D. Mukherjee, Phys. Rev. A 76, 062508 (2007)

2007
[22]

D. K. Nandy and B. K. Sahoo, Phys. Rev. A 90, 050503(R) (2014)

2014
[23]

X. T. Guo, Y. M. Yu, Y. Liu, and B. B. Suo, Chin. Phys. B 29 053101 (2020)

2020
[24]

T. Chen, L. Wu, R.-K. Zhang, Y.-B. Tang, J. Jiang, and C.-Z. Dong, Chin. Phys. B 32, 053206 (2023)

2023
[25]

B. K. Sahoo and B. P. Das, Phys. Rev. A 84, 010502(R) (2011)

2011
[26]

P. J. Blythe, S. A. Webster, K. Hosaka and P. Gill, J. Phys. B 36, 981 (2003)

2003
[27]

R. F. Bishop, Theoretica Chimica Acta 80, 95 (1991)

1991
[28]

DIRAC, a relativistic ab initio electronic structure p ro- gram, Release DIRAC22 (2022), written by H. J. Aa. Jensen, R. Bast, A. S. P. Gomes, T. Saue and L. Viss- cher, with contributions from I. A. Aucar, V. Bakken, C. Chibueze, J. Creutzberg, K. G. Dyall, S. Dubillard, U. E...

2022
[29]

T. Saue, R. Bast, A. S. P. Gomes, H. J. A. Jensen, L. Visscher, I. A. Aucar, R. Di Remigio, K. G. Dyall, E. Eliav, E. Faßhauer, T. Fleig, L. Halbert, E. D. Hedeg ˚ ard, B. Helmich-Paris, M. Iliaˇ s, C. R. Jacob, S. Knecht, J. K. Laerdahl, M. L. Vidal, M. K. Nayak, M. Olejnicza...

2020
[30]

K´ allay,, P

M. K´ allay,, P. G. Szalay, and P. R. Surj´ an, J. Chem. Phys. 117, 980 (2002)

2002
[31]

K´ allay, and J

M. K´ allay, and J. Gauss, J. Chem. Phys. 121, 9257 (2004)

2004
[32]

K´ allay, Z

Mrcc, a quantum chemical program suite written by M. K´ allay, Z. Rolik, J. Csontos, I. Ladj´ anszki, L. Szegedy, B. Lad´ oczki, and G. Samu. See also Z. Rolik, L. Szegedy, I. 10 Ladj´ anszki, B. Lad´ oczki, and M. K´ allay, J. Chem. Phys. 139, 094105 (2013), as well as: www.mrcc.hu

2013
[33]

Dyall and A.S.P

K.G. Dyall and A.S.P. Gomes, unpublished
[34]

Arora, M

B. Arora, M. S. Safronova, and C. W. Clark, Phys. Rev. A 76, 064501 (2007)

2007
[36]

Huang, H

Y. Huang, H. Guan, M. Zeng, L. Tang, and K. Gao, Phys. Rev. A, 99, 011401(R) (2019)

2019
[37]

C. E. Theodosiou, L. J. Curtis, and C. A. Nicolaides, Phys.Rev. A 52, 3677 (1995)

1995
[38]

P. S. Barklem and B. J. O’Mara, Mon. Not. R. Astron. Soc. 300, 863 (1998)

1998
[39]

Andr´ e S. P. Gomes, K. G. Dyall, L. Visscher, Theor. Chem. Acc. 127, 369 (2010)

2010
[40]

Kramida, Y

A. Kramida, Y. Ralchenko, J. Reader, and NIST ASD Team https://physics.nist.gov/asd

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [35]

B. K. Sahoo, B. P. Das, and D. Mukherjee, Phys. Rev. A, 79, 052511 (2009)

work page 2009

[2] [13]

C. F. A. Baynham, E. A. Curtis, R. M. Godun, J. M. Jones, P. B. R. Nisbet-Jones, P. E. G. Baird, K. Bongs, P. Gill, arXiv:1801.10134v3, (2020)

arXiv 2020

[3] [1]

Taylor, M

P. Taylor, M. Roberts, S. V. Gateva-Kostova, R. B. M. Clarke, G. P. Barwood, W. R. C. Rowley, and P. Gill, Phys. Rev. A 56, 2699 (1997)

work page 1997

[4] [2]

Roberts, P

M. Roberts, P. Taylor, S. V. Gateva-Kostova, R. B. M. Clarke, W. R. C. Rowley, P. Gill, Phys. Rev. A 60, 2867 (1999)

work page 1999

[5] [3]

S. A. Webster, R. M. Godun, S. A. King, G. Huang, B. Walton, V. Tsatourian, H. S. Margolis, S. Lea and P. Gill, IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control 57, 3 (2010)

work page 2010

[6] [4]

C. Tamm, N. Huntemann, B. Lipphardt, V. Gerginov, N. Nemitz, M. Kazda, S. Weyers and E. Peik, Phys. Rev. A 89, 023820 (2014)

work page 2014

[7] [5]

R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, K. Bongs, and P. Gill, Phys. Rev. Lett. 113, 210801 (2014)

2014

[8] [6]

Huntemann, B

N. Huntemann, B. Lipphardt, Chr. Tamm, V. Gerginov, S. Weyers, and E. Peik, Phys. Rev. Lett. 113, 210802 (2014)

work page 2014