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REVIEW 4 major objections 4 minor 157 references

Highly Accurate Expectation Values Using High-Order Relativistic Coupled Cluster Theory

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2504.18516 v1 pith:WPK4ONZ3 submitted 2025-04-25 physics.chem-ph

Gabriele Fabbro , Jan Brandejs , Trond Saue This is my paper · ORCID
classification physics.chem-ph
keywords coupledclusterCCSDTCCSDTQanalyticderivativeselectricfieldgradientnuclearquadrupolemomentrelativisticquantumchemistrycodegeneration
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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
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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

4 major / 4 minor

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.

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 (4)
  1. [IV.1, Eqs. (59)-(61), Table III] 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.
  2. [IV, virtual-orbital cutoff discussion] 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.
  3. [V.B, Table V] 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).
  4. [V.C, Figs. 4-5 and Table VII] 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.
minor comments (4)
  1. [Tables V and VII] 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.
  2. [Introduction and Sec. IV] 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.
  3. [Table IV] 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.
  4. [Ref. [124]] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the central derivation: computed EFGs enter the experimental NQCC relation as independent inputs, and the only self-citations are non-load-bearing implementation/protocol support.

full rationale

The final quadrupole moments are extracted through Eq. (1), NQCC = 234.9647 × Q × q, using experimental NQCCs (Tables I-II) as independent inputs and EFGs computed from the CC density matrix. The slope of the linear regressions in Figures 4-5 determines Q; the computed EFGs are not adjusted to reproduce the tabulated Q values, and the basis-set exponents are optimized against large-basis EFG references rather than against the experimental NQCCs. The composite correction scheme of Eqs. (59)-(61) is an acknowledged approximation: the T and Q corrections are small-basis differences added to a large-basis CCSD value, which is a stated modeling assumption, not a fitting of the final answer. The expectation-value implementation is cross-checked against the independent MRCC package for LiH dipole moments (Table IV), so the tenpi self-citations (Refs. 71-72) are implementation support rather than load-bearing evidence for the numerical conclusions. The authors' earlier projection analysis (Ref. 8) motivates the basis-set protocol but is not used to define the final Q values. No step was found in which a prediction reduces, by construction or by self-citation, to its own input. Remaining concerns about active-space truncation, the 45 Eh vs 4.5 Eh virtual-cutoff discrepancy, and the absence of explicit convergence tests for the T/Q corrections are accuracy and reproducibility issues, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard CC machinery plus a composite protocol whose additivity with severe active-space truncation is the main untested pillar. The largest uncertainty is the transferability of the tiny-active-space triple and quadruple corrections to the large-basis CCSD EFG.

free parameters (2)
  • Tight basis exponents (Li p, d; Al p, d, f) = Li p: 435.417100 a0^-2, d: 62.168460 a0^-2; Al p: 24750.350000 a0^-2, d: 1233.029836 a0^-2, f: 548.276242 a0^-2
    Chosen by even-tempered augmentation followed by line search to reproduce the EFG from a larger basis at SCF level; these are hand-tuned to the property of interest, not the final Q.
  • Virtual orbital energy cutoff for Al CCSD = 45 Eh
    Chosen by monitoring EFG stabilization in AlF and applied to AlCl and AlBr; ad hoc threshold, sensitive to inclusion of the whole p-shell.
assumptions (5)
  • standard math The similarity-transformed CC Hamiltonian expansion truncates after four commutators (Baker-Campbell-Hausdorff) for the DCG Hamiltonian with up to two-electron operators.
    Standard CC theory; the expression for H-bar in Eq. (13) is well-established.
  • domain assumption No-pair approximation: only positive-energy orbitals are retained at the correlated level, with MOs optimized at HF level.
    Standard in relativistic quantum chemistry; stated in Sec. II A.
  • ad hoc to paper Composite scheme additivity: the triple and quadruple excitation corrections computed with small active spaces in the v3z basis are transferable to the large-basis CCSD result.
    Eq. (61) sums corrections from small-basis small-active-space high-order calculations onto the large-basis CCSD value; no error bound given.
  • domain assumption s orbitals contribute negligibly to the EFG and can be omitted from basis set augmentation.
    Based on projection analysis and previous work (Ref 8); stated in Sec. IV.1.
  • ad hoc to paper Freezing Li 1s and Al core electrons in the CCSDT/CCSDTQ calculations does not significantly affect the triple and quadruple corrections to the EFG.
    Active spaces of 2-3 correlated electrons are used for higher-order corrections; core-valence correlation may matter for a core-weighted property, and this is not tested.

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Pith. "Pith review of Highly Accurate Expectation Values Using High-Order Relativistic Coupled Cluster Theory." pith.science (2026). https://pith.science/paper/WPK4ONZ3

@misc{pith2026250418516,
  author       = {Pith},
  title        = {Pith review of: Highly Accurate Expectation Values Using High-Order Relativistic Coupled Cluster Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPK4ONZ3}},
  note         = {Machine review of arXiv:2504.18516}
}
abstract

This work presents the automatic generation of analytic first derivatives of the energy for general coupled-cluster models using the \text{tenpi} toolchain. We report the first implementation of expectation values for CCSDT and CCSDTQ methods within the DIRAC program package for relativistic molecular calculations. As pivotal calculations, we focus on the electric field gradient (EFG) evaluated at the lithium nucleus in LiX (X = H, F, Cl) compounds, enabling the extraction of the nuclear electric quadrupole moment $Q({}^{7}Li)$, and at the aluminum nucleus in AlY (Y =H, F, Cl, Br) compounds, for the determination of $Q({}^{27}Al)$. These high-order methods are applied to compute corrections for triple and quadruple excitations for the EFG, a crucial quantity for determining nuclear quadrupole moments. We obtain $Q({}^{27}Al)$ = 0.1466 b, in excellent agreement with the recommended value, and $Q({}^{7}Li)$ = -0.0386 b, which is smaller than the currently recommended value, that indicates the need for further investigation.

Figures

Figures reproduced from arXiv: 2504.18516 by the authors.

Figure 1
Figure 1. FIG. 1. Diagrammatic structure of the coupled-cluster Lagrangian: the central diagram is obtained by [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Elements of the Hamiltonian contributing to disconnected diagrams in the [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Contribution to the EFG from the HF orbitals in AlF molecule. [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Linear fit of the nuclear quadrupole coupling constant (NQCC) as a function of the electric field [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Linear fit of the nuclear quadrupole coupling constant (NQCC) as a function of the electric field [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]

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