{"id":"d5e9bc5a-c94d-4924-806c-9dc945b9a072","arxiv_id":"2504.18516","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"First relativistic coupled-cluster calculations through quadruple excitations give electric field gradients that yield Q(7Li) = -0.0386 b, supporting the newer smaller value, and Q(27Al) = 0.1466 b, matching the recommended value.","lead":"Scientists built a new way to compute electron properties in molecules with very high accuracy, including relativistic effects and electron correlations beyond the usual doubles approximation. They used it to measure the shapes of lithium and aluminum nuclei, finding that lithium's nucleus may be rounder than the currently accepted value.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reported Q values rest on composite T/Q corrections (Eqs. 59-61) computed in tiny active spaces and bases with no convergence test; if those corrections shift when the active space or basis is enlarged, the central conclusions are not yet supported.","rationale":"The Reader's weakest assumption correctly identifies the composite T/Q correction as the load-bearing element. The implementation itself has plausible independent support: the MRCC comparison for LiH dipole moments (Table IV) checks the generated lambda equations and density matrices against an established code, and the reported agreement to roughly 1e-8 is real evidence. The concern is not about code correctness but about the scientific protocol: the high-order corrections are computed in a regime where no convergence evidence is given, and the final Q values carry no uncertainty estimate. The 45 Eh vs 4.5 Eh cutoff discrepancy is a secondary reproducibility issue. I therefore agree with the Reader's CONDITIONAL verdict rather than moving it: the technical advance is credible, but the quantitative conclusions should not be taken as definitive until the transferability of Eqs. (59)-(61) is tested or bounded. The proposed LiF calculation directly probes the weakest point with one high-order EFG computation in an expanded basis and active space; a shift beyond about 20% of the reported triples correction would exceed what the current error budget permits.","tokens_in":27456,"tokens_out":10903,"duration_ms":110494,"concrete_test":"For LiF, recompute the CCSDT correction T = q_CCSDT - q_CCSD of Eq. (59) in three ways: (a) dyall.v3z with the reported (3,68) active space; (b) dyall.v4z with the same (3,68) active space; (c) dyall.v3z with an all-electron active space covering a comparable virtual range. If T changes by more than 20% of its reported value 0.000279 Eh/a0^2 between (a) and either (b) or (c), the composite scheme of Eq. (61) has a basis/active-space transferability error larger than the experimental error budget, and the central Q(7Li) claim should be treated as conditional pending a converged correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical conclusions, Q(7Li) = -0.0386 b and Q(27Al) = 0.1466 b, are obtained through the composite scheme of Eqs. (59)-(61): a large-basis CCSD EFG plus triple and quadruple corrections evaluated in dyall.v3z with the severely truncated active spaces of Table III (e.g., LiF and AlF both use (3,68) spinors). The EFG is an r^-3-weighted property, so corrections involving core-valence correlation and tight basis functions near the nucleus can be sizable. No convergence test of T or Q with respect to active space or basis is reported; the text only says that a smaller basis with all virtuals gives corrections of the same order of magnitude, which is far coarser than the 0.0072% / 0.0089% error budget set in Sec. IV.1. The independent MRCC check validates a nonrelativistic LiH dipole in a 6-31G basis; it does not validate this active-space truncation for EFGs. The unresolved 45 Eh vs 4.5 Eh virtual-cutoff discrepancy in Sec. IV also weakens reproducibility. If either correction changes by more than about 10-20% when the basis or active space is enlarged, the agreement for 27Al and the revision of 7Li are not quantitatively established. No uncertainty is attached to the final Q values.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports the first implementation of expectation values (analytic one-electron properties) for CCSDT and CCSDTQ within the DIRAC program package, using the tenpi code generator for the lambda equations and one-body density matrices. The implementation is validated against MRCC for the nonrelativistic LiH dipole moment. The main application is the computation of electric field gradients at Li in LiH, LiF, LiCl and at Al in AlH, AlF, AlCl, AlBr, followed by extraction of the nuclear quadrupole moments Q(7Li) and Q(27Al) via the experimental NQCC relationship and a linear regression. A composite scheme (Eqs. 59-61) combines large-basis CCSD EFGs with triple and quadruple corrections computed in smaller bases and active spaces. The authors obtain Q(7Li) = -0.0386 b, smaller in magnitude than the currently recommended value, and Q(27Al) = 0.1466 b, in agreement with the recommended value.","tokens_in":27821,"tokens_out":10021,"duration_ms":91393,"significance":"If the reported numerical results are sustainable, the paper makes two useful contributions: it demonstrates a working and validated route to high-order relativistic coupled-cluster expectation values, and it provides an independent molecular determination of Q(7Li) and Q(27Al) that may inform the ongoing debate about the 7Li quadrupole moment. The cross-validation of the tenpi-generated CCSDT/CCSDTQ expectation values against MRCC, the systematic treatment of rovibrational averaging, and the attention to basis-set convergence at the SCF/CCSD level are concrete strengths. However, the central quantitative claims rely on a composite approximation whose convergence is not demonstrated, and the reported final Q values carry no uncertainty. The paper is therefore a promising contribution whose current numerical conclusions should be treated as provisional pending the additional tests and clarifications described below.","major_comments":[{"comment":"The triple and quadruple corrections to the EFG are computed in the dyall.v3z basis with active spaces of (3,68) spinors at CCSDT and (3,18) spinors at CCSDTQ, freezing the Li 1s and Al core electrons. Since the EFG is an r^-3-weighted property, core-valence correlation in the higher-order corrections can be significant. No convergence test of the T and Q corrections with respect to active space or basis is reported; the statement that a smaller basis with all virtuals gives corrections of the same order of magnitude does not bound the error at the 0.0072%/0.0089% level set in Sec. IV.1. Please provide convergence data or conservative error estimates for T and Q, because the central Q(7Li) and Q(27Al) values depend directly on these additivity assumptions.","section":"IV.1, Eqs. (59)-(61), Table III"},{"comment":"The text states that Pernpointner and Visscher excluded orbitals above 4.5 Eh (AlF, AlCl) and 4.4 Eh (AlBr), but in the next paragraph it is stated that for AlF the EFG stabilizes once a threshold of 45 Eh is reached. This is a tenfold discrepancy that must be clarified, since the large-basis CCSD reference value and hence the final Q values depend on the actual cutoff. The cutoffs used for AlCl and AlBr are also not explicitly specified. Please correct the numbers and document the cutoff for each system.","section":"IV, virtual-orbital cutoff discussion"},{"comment":"The Q columns for LiH are inconsistent with Eq. (1). For LiH with NQCC = 0.346750 MHz and q(HF) = -0.039726 Eh/a0^2, Eq. (1) gives Q(HF) = -0.037148 b, as listed. Adding the tabulated CCSD EFG correction of +0.001020 gives q = -0.038706 and Q = -0.038128 b, so the CCSD Q increment should be about -0.00098 b, not -0.002578 b; correspondingly, the triple increment should be about +0.00005 b, not +0.001549 b. Please correct the table or explain the convention used for the Q columns; as written, the individual LiH contributions do not follow from the EFG values via Eq. (1).","section":"V.B, Table V"},{"comment":"The final quadrupole moments are obtained by a slope fit of NQCC versus EFG, but no uncertainty, goodness-of-fit, or weighted-regression details are reported. This matters for both central claims: for 27Al, the per-molecule final values in Table VII range from 0.145716 to 0.148547 b, a spread larger than the experimental uncertainty of the recommended value (0.1466 +/- 0.0010 b), and for 7Li the proposed revision of Q is a small effect. Please report the slope, its uncertainty, and the fit statistics so that the agreement with the recommended 27Al value and the deviation for 7Li can be properly assessed.","section":"V.C, Figs. 4-5 and Table VII"}],"minor_comments":[{"comment":"The headers state that Q is given in mb, but the values in the text and abstract are clearly in barns (about -0.0386 b for 7Li and 0.1466 b for 27Al). Please correct the units consistently.","section":"Tables V and VII"},{"comment":"There are several typographical and grammatical errors, including 'inclusion fo the triple excitations', 'Pernpointner and Visscher showed that excluding orbitals above 4.5 Eh ... A contribution to such unstability is as follows', and 'the two approaches are entirely equivalent' followed by redundant references. A careful proofread is recommended.","section":"Introduction and Sec. IV"},{"comment":"The convergence-threshold column lists different threshold values for DIRAC and MRCC runs, but the criterion used for the thresholds is not defined. Stating the convergence criterion for the amplitudes and multipliers would make the validation easier to interpret.","section":"Table IV"},{"comment":"The data availability statement says the repository will be made open after acceptance. Making the input files and output data available during review would allow the composite-scheme convergence issue to be checked by the referees.","section":"Ref. [124]"}],"recommendation":"major_revision","confidential_remarks":"This is a potentially valuable implementation paper, and the MRCC cross-check for LiH gives confidence in the code itself. The main risk is that the final Q values are overinterpreted: the composite T/Q corrections are computed with very small active spaces and no convergence test, and the reported numbers lack uncertainty estimates. I would ask the authors to add convergence checks for the T and Q corrections, state the virtual-orbital cutoff clearly, correct the LiH entries in Table V, and provide uncertainty estimates for the fitted Q values before the numerical claims are accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine implementation advance, and the 7Li result is likely right, but the paper currently overstates the certainty of its headline Q values. The composite treatment of triple and quadruple corrections has no convergence test, no error estimate, and an unresolved cutoff inconsistency.\n\nWhat is actually new and good: the authors extend the tenpi code generator to lambda equations and density matrices for CCSDT and CCSDTQ, giving the first relativistic molecular expectation values at these excitation levels. The cross-validation against MRCC for the LiH dipole at CCSD, CCSDT, and CCSDTQ is clean, with agreement at the 10^-8 level. The equivalent-line factor 1/n! extension and the OpMin integer-prefactor fix are real engineering contributions. The EFG basis-set tuning against a 0.0072%/0.0089% target is thoughtful, and the 7Li value matching Guan's atomic determination through a completely different route is a meaningful data point.\n\nThe soft spots are serious but localized. The triple and quadruple corrections in Eqs. (59)-(61) are computed in dyall.v3z with tiny active spaces—CCSDT (3,68), CCSDTQ (3,18)—freezing Li 1s and the Al core. The EFG is r^-3-weighted, so core-valence correlation and tight basis functions matter. The statement that a smaller basis with all virtuals gives corrections of the same order of magnitude is not a substitute for a convergence test. If either correction shifts by 10-20% in a larger basis, the stated 0.0072%/0.0089% budget is gone. The identically zero quadruple corrections for Li in Table V are suspicious. The 45 Eh versus 4.5 Eh virtual-cutoff discrepancy in Sec. IV is unexplained and hurts reproducibility, and the data repository is still not available. None of this undermines the implementation validation, but it does mean the quantitative Q values are not yet established at the claimed accuracy.\n\nWho this is for: anyone working on heavy-element molecular properties, nuclear quadrupole moments, or automated CC code generation. The implementation and the 7Li comparison deserve serious referee time.\n\nRecommendation: send to peer review, but require major revision: add convergence tests for T/Q corrections with respect to active space and basis, resolve or at least explain the 45/4.5 Eh discrepancy, attach uncertainty estimates to the final Q values or soften the claims, and make the repository available. With those changes this becomes a solid, citable paper.","headline":"A real implementation advance—relativistic CCSDT/CCSDTQ expectation values in DIRAC—with solid cross-validation, but the final Q values rest on an untested composite approximation and need a major revision before the accuracy claims hold.","tokens_in":28306,"tokens_out":1706,"would_cite":true,"duration_ms":20060,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"First implementations of CCSDT and CCSDTQ expectation values in a relativistic molecular program yield Q(27Al) = 0.1466 b, matching the recommended value, and Q(7Li) = -0.0386 b, a revision of the accepted lithium quadrupole moment.","keywords":["coupled cluster","CCSDT","CCSDTQ","analytic derivatives","electric field gradient","nuclear quadrupole moment","relativistic quantum chemistry","code generation"],"falsifier":"Recalculate the full-triples correction to the lithium electric field gradient for LiF in a larger (quadruple-zeta) basis with the same correlated electrons: if the correction shifts the final gradient by more than the paper's own 0.0072% relative-accuracy target, the composite transfer is not converged and the reported $Q({}^{7}\\mathrm{Li})$ does not meet the claimed precision. An independent arbitration would be a direct determination of the $^{7}\\mathrm{Li}$ quadrupole moment from atomic hyperfine structure alone, free of molecular electric-field-gradient theory, pushed beyond the current experimental uncertainty.","tokens_in":27240,"feed_emoji":"⚛️","tokens_out":35994,"duration_ms":288118,"temperature":0.7,"pith_summary":"The paper reports the first implementation of molecular expectation values at the CCSDT and CCSDTQ levels — coupled-cluster wave functions that include all triple (and, for CCSDTQ, all quadruple) excitations — in a relativistic framework, with the equations generated automatically by the tenpi code-writing toolchain and embedded in the DIRAC program package. The authors use the new capability to compute electric field gradients at the lithium and aluminum nuclei of a series of small molecules, from which they extract nuclear electric quadrupole moments, a measure of the shape of the nuclear charge distribution. They obtain $Q({}^{27}\\mathrm{Al}) = 0.1466$ b, in agreement with the recommended value, which they take as validation of the method. For $^{7}\\mathrm{Li}$ they obtain $-0.0386$ b, smaller in magnitude than the long-standing recommended $-0.0406$ b and identical to a recent value from precision spectroscopy of $^{7}\\mathrm{Li}^{+}$; they conclude that the accepted lithium value should be revisited.","feed_headline":"Revises lithium-7 quadrupole moment: new calculation says -0.0386 b","feed_subtitle":"A new high-order relativistic quantum method confirms the aluminum value and backs the disputed lithium one.","key_machinery":"The argument runs on three mechanisms. First, the Lagrangian formulation of coupled-cluster properties: the energy is made stationary with respect to both cluster amplitudes and Lagrange multipliers, the multipliers obey a single perturbation-independent linear system (the Lambda equations), and first-order properties follow from the trace of the symmetrized one-body density matrix with the perturbation integrals — no response equation per field strength is needed. Second, the tenpi code generator, which encodes every diagram as a 13-integer string and automatically derives, factorizes, and optimizes the amplitude, Lambda, and density-matrix diagrams; the new implementation extends the symmetry factor for equivalent lines from the $1/2$ rule for pairs to the general $1/n!$ rule for $n$ equivalent lines, which first becomes necessary at triple and quadruple excitations. Third, a composite scheme for the applications: the reference electric field gradient is computed at CCSD in a large property-optimized basis, the triple and quadruple excitation corrections are computed in a smaller triple-zeta basis with only two or three correlated electrons and at most 78 spinors, and rovibrational averaging is applied through a perturbative expansion of the property along the bond coordinate.","core_discovery":"The paper's central claim is that analytic first derivatives of the energy — hence expectation values of one-electron operators — can be generated and implemented automatically for high-order coupled-cluster models in a relativistic setting, and that the resulting CCSDT/CCSDTQ code is accurate enough to extract nuclear quadrupole moments at the level of the experimental uncertainties of the measured coupling constants. The code is validated in two ways: non-relativistic dipole moments of LiH agree with an established independent program to eight significant digits at every excitation level, and the relativistic treatment of the aluminum series reproduces the recommended $Q({}^{27}\\mathrm{Al}) = 0.1466$ b. The lithium application then makes a stronger claim: full iterative triples give a triple-excitation correction to the electric field gradient roughly an order of magnitude smaller than the perturbative T(CCSD) treatment used in the earlier molecular determination, so the resulting $Q({}^{7}\\mathrm{Li}) = -0.0386$ b agrees with the precision-spectroscopy value rather than with the widely used $-0.0406$ b. The authors therefore assert that the accepted $^{7}\\mathrm{Li}$ quadrupole moment should be re-examined.","pith_inferences":["The composite scheme's transferability is directly testable: recompute the triple correction for LiF in a larger basis with the same active space; if the correction moves by more than the paper's relative-accuracy target, the reported $Q({}^{7}\\mathrm{Li})$ would need to be re-derived. The paper reports no such convergence test.","Because lithium is the lightest nucleus in the set, its revised quadrupole moment isolates electron correlation as the decisive factor — relativistic corrections are minuscule at $Z=3$. Read this way, the lithium result is primarily a statement about correlation treatment, while the aluminum result is a joint statement about relativity and correlation.","The generated density matrices are property-agnostic, so the same machinery should deliver other one-electron properties at CCSDT/CCSDTQ level with no extra formalism. One caveat the paper itself supplies is that its basis construction deliberately omitted tight s functions because s orbitals barely contribute to electric field gradients, so properties that sample s density at the nucleus would ne"],"forward_implications":["The accepted molecular value $Q({}^{7}\\mathrm{Li}) = -0.0406$ b is called into question; the paper's $-0.0386$ b matches an independent precision-spectroscopy determination.","Comparing full iterative triples with the perturbative T(CCSD) scheme, the paper argues that the earlier estimate of the triple-excitation correction to the lithium electric field gradient is too high by roughly an order of magnitude.","Quadruple excitations change the lithium electric field gradients by less than $10^{-6}\\ E_h/(e a_0^2)$ and can be neglected there, but they shift the aluminum gradients by up to $4\\times10^{-4}\\ E_h/(e a_0^2)$ in AlH, so CCSDTQ matters at the stated accuracy for the lightest aluminum compounds.","The linear-regression extraction of $Q({}^{27}\\mathrm{Al}) = 0.1466$ b over the aluminum series agrees with the recommended value, which the authors read as confirmation that the composite protocol of large-basis CCSD plus small-basis triple and quadruple corrections is sound."],"supporting_citations":[{"why":"Supplies the general analytic first-derivative formalism for arbitrary coupled-cluster models on which the automatic Lambda and density-matrix generation is based.","marker":"[35]"},{"why":"Provides the relativistic coupled-cluster module and its hand-coded CCSD expectation value, against which the autogenerated CCSD code is validated.","marker":"[55]"},{"why":"Prior relativistic study showing that triple excitations affect the aluminum electric field gradient, and the source of the comparison value Q(27Al) = 0.1460 b.","marker":"[56]"},{"why":"Precedent for extracting quadrupole moments by combining large-basis CCSD with smaller-basis triple and quadruple corrections.","marker":"[57]"},{"why":"The open-source code generator that automatically produces, factorizes, and optimizes the amplitude, Lambda, and density-matrix equations implemented here.","marker":"[71]"},{"why":"An established independent program used to validate the non-relativistic CCSD, CCSDT, and CCSDTQ dipole moments at every excitation level.","marker":"[75]"},{"why":"The earlier molecular determination of Q(7Li) = -0.0406 b from LiH, LiF, and LiCl that the paper challenges.","marker":"[129]"},{"why":"The combined fully numerical and basis-set LiH electric field gradient benchmark that established the prior lithium quadrupole-moment reference.","marker":"[146]"},{"why":"The recent precision-spectroscopy determination of Q(7Li) = -0.0386(5) b that the paper's lithium value matches.","marker":"[150]"},{"why":"The molecular determination cited as the source of the recommended Q(27Al) = 0.1466 b that the paper reproduces.","marker":"[151]"}],"fun_headline_variants":["High-order relativistic calculation trims lithium-7 quadrupole moment","CCSDT/CCSDTQ expectation values shrink lithium-7 quadrupole moment","Lithium-7 quadrupole moment re-examined: high-order relativistic result -0.0386 b","New method: electric field gradients at CCSDTQ yield Li-7 quadrupole moment -0.0386 b"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the triple- and quadruple-excitation corrections to the electric field gradient, computed in a small triple-zeta basis with only two or three correlated electrons and at most 78 spinors, remain valid when added to a value computed in a much larger basis with far more correlated electrons; the paper gives no error estimate or convergence study for this transfer.","fun_headline_variants_meta":{"raw":{"variants":["High-order relativistic calculation trims lithium-7 quadrupole moment","CCSDT/CCSDTQ expectation values shrink lithium-7 quadrupole moment","Lithium-7 quadrupole moment re-examined: high-order relativistic result -0.0386 b","New method: electric field gradients at CCSDTQ yield Li-7 quadrupole moment -0.0386 b"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001373,"raw_usage":{"total_tokens":5596,"prompt_tokens":1005,"completion_tokens":4591,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":4490}},"tokens_in":621,"tokens_out":4591,"duration_ms":33552,"temperature":1.0,"reasoning_tokens":4490,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:14:54.581502+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recalculate the full-triples correction to the lithium electric field gradient for LiF in a larger (quadruple-zeta) basis with the same correlated electrons: if the correction shifts the final gradient by more than the paper's own 0.0072% relative-accuracy target, the composite transfer is not converged and the reported $Q({}^{7}\\mathrm{Li})$ does not meet the claimed precision. An independent arbitration would be a direct determination of the $^{7}\\mathrm{Li}$ quadrupole moment from atomic hyperfine structure alone, free of molecular electric-field-gradient theory, pushed beyond the current experimental uncertainty.","supporting_citations":[{"cited_title":"Pernpointner \\ and\\ author L","cited_arxiv_id":null,"evidence_quote":"Prior relativistic study showing that triple excitations affect the aluminum electric field gradient, and the source of the comparison value Q(27Al) = 0.1460 b."},{"cited_title":"Stopkowicz \\ and\\ author J","cited_arxiv_id":null,"evidence_quote":"Precedent for extracting quadrupole moments by combining large-basis CCSD with smaller-basis triple and quadruple corrections."},{"cited_title":"Urban \\ and\\ author A","cited_arxiv_id":null,"evidence_quote":"The earlier molecular determination of Q(7Li) = -0.0406 b from LiH, LiF, and LiCl that the paper challenges."},{"cited_title":"Sundholm , author P","cited_arxiv_id":null,"evidence_quote":"The combined fully numerical and basis-set LiH electric field gradient benchmark that established the prior lithium quadrupole-moment reference."},{"cited_title":"Kellö \\ and\\ author A","cited_arxiv_id":null,"evidence_quote":"The molecular determination cited as the source of the recommended Q(27Al) = 0.1466 b that the paper reproduces."}],"review_version":1}