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REVIEW 3 major objections 4 minor 50 references

Post-CCSD(T) corrections in the S66 noncovalent interactions benchmark

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read CCSD(T) overbinds pi-stacked complexes because the usual cancellation between higher-order triples and connected quadruples fails, by about 0.04-0.10 kcal/mol in the S66 benchmark.

desk verdict Valuable new post-CCSD(T) dataset for S66; pi-stack overbinding direction is likely right but its magnitude rests on a DZP basis and needs a larger-basis check. read the letter →

arxiv 2411.12004 v3 pith:JP74BECO submitted 2024-11-18 physics.chem-ph

Emmanouil Semidalas , A. Daniel Boese , Jan M. L. Martin This is my paper · ORCID
classification physics.chem-ph
keywords noncovalentinteractionsS66benchmarkcoupledclustertheoryCCSD(T)post-CCSD(T)correctionspi-stackinghigher-ordertriplesconnectedquadruples
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

Using the S66 benchmark of biomolecular dimers, the paper computes full CCSDT and CCSDT(Q) corrections in a polarized double-zeta basis to test the assumption that CCSD(T) is essentially exact for noncovalent interactions. It finds that for hydrogen bonds, pure London complexes, and mixed-influence systems, the usual error cancellation holds: repulsive higher-order triples ($T_3-(T)$) are offset by attractive connected quadruples $(Q)$. For $\pi$-stacking complexes this cancellation starts to fail, leaving CCSD(T) overbound: the average CCSDT(Q)-CCSD(T) correction for the $\pi$-stack subset is -0.043 kcal/mol, with the fully converged parallel-displaced dimers at about -0.085 and -0.099 kcal/mol and model estimates for the remaining stacks reaching near -0.11 kcal/mol. The paper also shows that CCSD(T)$_\Lambda$ fixes the $\pi$-stack problem but overbinds London complexes, and that a two-parameter formula estimates the full CCSDT(Q)-CCSD(T) difference to 0.01 kcal/mol RMS at no steeper than $O(N^7)$ cost.

What carries the argument

The load-bearing objects are the fifth-order perturbation terms of Eq. (7): the connected-quadruples self-interaction $E^{(5)}_{QQ}$, the triples-quadruples coupling $E^{(5)}_{TQ}$, and the triples-triples interaction $E^{(5)}_{TT}$; together they make up the CCSDT(Q)-CCSD(T) difference through fifth order. The paper identifies $E^{(5)}_{QQ}$ as almost universally attractive and $E^{(5)}_{TQ}$ as antibonding, with the triples-triples term distinguishing $\pi$-stacks (antibonding) from London complexes (slightly bonding). The second key object is the two-parameter estimator of Eq. (12), $\Delta E[\mathrm{postCCSD(T)}] \approx \Delta E(T)[\alpha/(1-\beta\, \Delta E(T)/\Delta E_{\mathrm{corr}}[\mathrm{CCSD}]) - 1] + a_3(\Delta E[\mathrm{CCSDT-3}] - \Delta E[\mathrm{CCSD(T)}])$, with $a_3=1$; it converts the cheaply accessible $(T)$ energy and CCSDT-3 correction into a prediction of the full CCSDT(Q) correction. CCSDT-3 appears because it captures the lion's share of the higher-order triples effect at $O(n^3_{\rm occ}N^4_{\rm virt})$ scaling, and the geometric-series form of the first term mimics the convergence of the correlation-energy series.

What would settle it

Compute CCSDT(Q)-CCSD(T) for one of the parallel-displaced aromatic stacks, such as the benzene dimer, in a triple-zeta basis set; if the correction is substantially smaller in magnitude than -0.05 kcal/mol, the small-basis overbinding estimate does not survive basis-set extrapolation. Alternatively, run fixed-node diffusion Monte Carlo for systems 24-29 with stochastic error below 0.02 kcal/mol and compare with the CCSD(T)/CBS values.

Watch

Extended reading notes

Core claim

The paper's central claim is that CCSD(T), long treated as the gold standard for noncovalent interactions, is not uniformly reliable across the S66 dataset: for $\pi$-stacked aromatic dimers the cancellation between the (usually repulsive) higher-order triples correction $T_3-(T)$ and the (attractive) connected quadruples correction $(Q)$ breaks down. Higher-order triples become strongly repulsive for these stacks, and while $(Q)$ is also largest there, it does not fully compensate, yielding net CCSDT(Q)-CCSD(T) corrections of about -0.04 kcal/mol on average for the $\pi$-stack subset, with fully converged parallel-displaced dimers at -0.085 and -0.099 kcal/mol and model estimates for the remaining stacks reaching about -0.109 kcal/mol. The authors show by a fifth-order analysis, $E[\mathrm{CCSDT(Q)}]-E[\mathrm{CCSD(T)}] = E^{(5)}_{QQ} + 2E^{(5)}_{TQ} + E^{(5)}_{TT} + O(\lambda^6)$, that the residual is driven by the triples-triples and triples-quadruples interaction terms. They further demonstrate that CCSD(T)$_\Lambda$ moves the error from $\pi$-stacks to London complexes, and that a two-parameter model based on $(T)$, the CCSD correlation energy, and the CCSDT-3 correction reproduces the post-CCSD(T) difference with 0.01 kcal/mol RMS, allowing cheap $O(N^7)$ estimates of what full CCSDT(Q) would give.

Load-bearing premise

The overbinding estimate rests on the assumption that post-CCSD(T) corrections computed in the small cc-pVDZ(d,s) basis set represent the complete basis set limit, even though the model's $\beta$ parameter changes from -0.273 to -1.009 when the basis is enlarged.

Editorial extensions

If this is right

  • For aromatic $\pi$-stacking complexes in S66, CCSD(T)/CBS binding energies are overbound by roughly 0.04-0.10 kcal/mol, so reference tables built on CCSD(T) for such systems carry this bias.
  • CCSD(T)$_\Lambda$ removes most of the $\pi$-stack overbinding but overshoots for pure London complexes, so neither CCSD(T) nor CCSD(T)$_\Lambda$ alone matches CCSDT(Q) across all interaction categories.
  • The two-parameter model of Eq. (12) predicts CCSDT(Q)-CCSD(T) to 0.01 kcal/mol RMS for S66 and brackets the naphthalene dimer value, giving a practical $O(N^7)$ proxy for the full correction.
  • CCSDT-3 captures most of the higher-order triples effect and is a reliable stand-in for full CCSDT when estimating post-CCSD(T) corrections.
  • The gap between CCSD(T) and CCSDT(Q) for $\pi$-stacks is expected to widen for larger aromatic stacks.

Reading between the lines

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

  • If the overbinding grows with stack size as the naphthalene dimer check suggests, then CCSD(T)-based reference values for larger aromatic $\pi$-stacks, such as DNA base-pair steps or organic semiconductor dimers, may carry a systematic bias that is currently unaccounted for in benchmark tables.
  • The strong basis-set dependence of $\beta$ in Eq. (12) implies that the formula should be re-fitted rather than used with S66 parameters when applied to thermochemistry or to complexes with different monomer polarizabilities.
  • A natural testable extension is to apply Eq. (12) to a new set of $\pi$-stacked dimers with experimental binding energies; agreement within the claimed 0.01 kcal/mol RMS would support using it as a correction scheme for cheaper methods like DFT or MP2.
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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 reports CCSDT and CCSDT(Q) calculations (where feasible) for the S66 noncovalent interaction benchmark in the cc-pVDZ(d,s) basis, and analyzes post-CCSD(T) contributions decomposed by interaction category. The central finding is that for hydrogen-bonded, pure London, and mixed-influence complexes, the error cancellation between higher-order triples (T3-(T)) and connected quadruples (Q) makes CCSD(T) close to CCSDT(Q), while for pi-stacked systems this cancellation breaks down and CCSD(T) overbinds. The authors also propose simple fitted formulas, Eqs. (12) and (13), that reproduce CCSDT(Q)-CCSD(T) differences to about 0.01 kcal/mol RMS on the training set, and they compare with CCSD(T)_Lambda, FN-DMC results, a naphthalene dimer check, and a W4-11 thermochemical application.

Significance. If the double-zeta results are representative of the basis-set limit, the paper provides valuable direct evidence about the breakdown of error cancellation in CCSD(T) for pi-stacked complexes, a question of current interest in benchmarking noncovalent interactions. The calculations are state-of-the-art in scope, the authors are transparent about missing data, and the naphthalene dimer and S22 checks are genuine out-of-sample tests. The proposed low-scaling estimators are useful in practice, and the W4-11 semiquantitative transferability test is a positive feature. The significance is conditional, however, on two points: the basis-set transferability of the post-CCSD(T) corrections, and the distinction between actual CCSDT(Q) data and model estimates in the category-level statistics.

major comments (3)
  1. [Section 3 (Tables 2 and 4), with Table 3] The central quantitative claim that CCSD(T) overbinds pi-stacks by about 0.04-0.10 kcal/mol rests entirely on post-CCSD(T) corrections computed in the cc-pVDZ(d,s) basis. The only larger-basis comparison, the naphthalene dimer check, uses cc-pVDZ(d,p), which is still a double-zeta basis. Table 3 shows that the ratio Delta_E(T)/Delta_E_corr[CCSD] underlying Eq. (12) is strongly basis-set dependent, with beta changing from -0.273 at cc-pVDZ(d,s) to -1.009 at haVTZ(f,d). This does not by itself prove that the actual T3-(T) and (Q) interaction-energy corrections are basis-dependent, but it removes any default expectation that DZP post-CCSD(T) corrections are converged. Please provide at least a few larger-basis CCSDT(Q) or CCSDT(Q)-level calculations for representative pi-stacks, or state clearly that the quantitative overbinding range is a DZP result and that the robust conclusion is the sign, not the magnitude.
  2. [Section 3, Eq. (12) and Table 3] The headline RMSD of about 0.01 kcal/mol for Eq. (12) is an in-sample fit: the parameters alpha, beta, and a3 are optimized on S66 and then evaluated on the same S66 data. The out-of-sample evidence is considerably weaker: the S22 subset gives RMSD 0.021 kcal/mol, which the authors attribute to the formic acid dimer outlier, and the naphthalene dimer estimates (-0.150 and -0.201 kcal/mol for Eq. (12) and Eq. (13), respectively) bracket the actual value (-0.160 kcal/mol) with a spread comparable to the effect itself. I therefore cannot parse the abstract claim that the model 'predicts CCSDT(Q)-CCSD(T) differences to 0.01 kcal/mol RMS' as a predictive statement. Please state explicitly that the 0.01-kcal/mol figure is the training-set error, and give the out-of-sample RMS excluding the identified outlier.
  3. [Section 3 (Table 4) and Table 2] It appears that actual CCSDT(Q)-CCSD(T) values are available for only two pi-stacking systems, 24 and 25, as seen in the 'actual' column of Table 4, while the entries for systems 26-29 and 47-49 are model estimates shown in parentheses. Yet Table 2 reports category averages, such as a pi-stack average of -0.043 kcal/mol for CCSDT(Q)-CCSD(T), without indicating which entries are actual and which are estimated. The statement in the conclusions that 'one observes nontrivial discrepancies between CCSD(T) and CCSDT(Q)' for small aromatic pi-stacks is therefore based on a very small number of direct calculations augmented by the fitted models. Please specify the number of actual CCSDT(Q) data per category and, if estimates are used, provide separate statistics for actual and estimated entries.
minor comments (4)
  1. [Section 3, Table 4] The meaning of parentheses in Table 4 (model estimates) is explained only in the caption; please state this explicitly in the table itself or in the text, and mark the 'actual' column entries that are genuine CCSDT(Q) data.
  2. [Abstract and Highlights] The abstract and highlights say the present results 'corroborate' FN-DMC claims, but the FN-DMC uncertainties in Table 4 are large, and the note added in revision reports FN-DMC S66 hydrogen-bond discrepancies that are hard to rationalize. Consider softening the wording to 'consistent in sign with FN-DMC for pi-stacks' to avoid overstating the external validation.
  3. [Section 2, basis set notation] The notation 'cc-pVDZ(p,s)' appears once in Section 3 in the context of scaling Delta_E[(Q)]; please define it in Section 2 alongside cc-pVDZ(d,s) and cc-pVDZ(d,p).
  4. [Section 3, Eq. (3)] Eq. (3) is attributed to Grueneis et al. [15], but the parameters a1 and a2 are fitted in this work; please clarify explicitly which parameters are from Ref. [15] and which are refitted here.

Circularity Check

1 steps flagged · score 2.0 of 10

Physical pi-stack overbinding result is a direct calculation; the only mild circularity is the abstract's in-sample 'prediction' RMSD for the fitted two-parameter model.

  1. fitted input called prediction [Abstract; Section 3, Eqs. (10)-(12) and Table 3]
    "A fairly simple two-parameter model predicts CCSDT(Q)--CCSD(T) differences to 0.01 kcal mol−1 RMS, requiring no calculations that scale more steeply than O(N^7)."

    The 0.01 kcal/mol RMS is the training error of Eq. (12) on the S66 data used to fit alpha, beta, and a3. The paper states that Eq. (12) yields RMSD = 0.011 kcal/mol for fitted parameters and then, after fixing a3 = 1, RMSD remains 0.011 with alpha = 1.139, beta = -0.273; it later refers to 'S66 fitted parameters' when applying the model to W4-11. Thus the headline predictive accuracy is a fit residual, not an out-of-sample prediction: the parameters minimize the error on the same dataset whose RMSD is quoted. The reduction is by construction.

full rationale

The paper's central physical finding - that for pi-stacking complexes the CCSD(T) error cancellation between higher-order triples and connected quadruples breaks down and CCSD(T) overbinds - is a direct consequence of the CCSDT and CCSDT(Q) calculations reported in Tables 1 and 2, not of any fitted model. No load-bearing self-citation chain is present: earlier work by the same authors (W4-17, S66x8, the BSSE study, naphthalene dimer) is used as background or as an external validation target, not as an unverified uniqueness or existence theorem. No ansatz is smuggled in via citation; the adopted Grueneis-type expressions are explicitly credited to external work. The only circular element is a mild one: the abstract's 'predicts ... to 0.01 kcal/mol RMS' describes the training error of a model fitted to the same S66 data, so that particular number reduces to the fit by construction. The paper is transparent about the fitting, and it reports out-of-sample transferability checks on S22, W4-11, and naphthalene dimer that give the model independent support. The basis-set sensitivity of beta shown in Table 3 is a correctness risk concerning extrapolation of DZP post-CCSD(T) corrections, not a circularity of the derivation. Overall, no significant circularity compromises the paper's main physical conclusion.

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

The physical benchmark results are direct calculations; the empirical estimators carry the bulk of fitted parameters. Key free parameters are the coefficients of Eqs. (3), (5), (9), (10), (11), (12), and (13), all optimized against S66 data. The headline 0.01 kcal/mol RMSD is an in-sample fit; out-of-sample checks on S22, naphthalene dimer, and W4-11 are reported and show degraded but still useful accuracy (e.g., S22 RMSD 0.021 kcal/mol after excluding formic acid dimer).

free parameters (5)
  • Eq. (12) alpha, fitted to S66 (a3 fixed to 1) = 1.139
    Optimized to minimize RMSD of CCSDT(Q)-CCSD(T) predictions on the S66 dataset used for fitting; the same dataset yields the headline 0.011 kcal/mol RMSD.
  • Eq. (12) beta, fitted to S66 (a3 fixed to 1) = -0.273
    Basis-dependent; ranges from -0.273 (cc-pVDZ(d,s)) to -1.009 (haVTZ(f,d)) across Table 3, showing strong basis sensitivity.
  • Eq. (3) parameters a1, a2 = a1=0.9417, a2=0.1442
    Fitted to S66; perform poorly for T3-(T) with RMSD 0.035 kcal/mol, as disclosed in the text.
  • Eq. (10) scale factor b1 = 0.825 (0.872 after deleting outliers 21 and 22)
    Scaling of the CCSDT-3-CCSD(T) difference to estimate (Q); deleting two outliers improves RMSD from 0.011 to 0.0064 kcal/mol.
  • Eq. (13) coefficients = 1.085, -1.515, 0.200
    Three-parameter linear model for the post-CCSD(T) correction; system 22 is treated as an outlier in the fit.
assumptions (5)
  • standard math Coupled-cluster perturbation theory: CCSDT(Q)-CCSD(T) decomposes into three fifth-order terms (Eq. (7))
    Standard many-body perturbation theory result, cited to Bomble/Stanton and Cremer's analysis.
  • domain assumption The S66 reference geometries (MP2/cc-pVTZ BSSE-corrected, from BEGDB) are adequate for computing De corrections
    Standard benchmark practice; the paper uses them without reoptimization, symmetrizing a few systems (Section 2).
  • domain assumption BSSE contributions to post-CCSD(T) corrections are negligible
    Assumed based on the authors' separate study Ref. [40]; no BSSE correction is applied to Table 1.
  • domain assumption Fixed-node diffusion Monte Carlo is a valid reference for the pi-stacking discrepancy
    Used in Table 4 to contextualize the overbinding; the paper notes the FN-DMC uncertainties are large, so this is a soft reference.
  • ad hoc to paper The ratio Delta_E2 / Delta_Ecorr,CCSD captures the quasiperturbative triples behavior (Grueneis model)
    Eqs. (3)-(4) and (9)/(12) adopt this heuristic denominator to model T3-(T) and (Q); it is an empirical choice rather than a derived law.

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Cite this review

Pith. "Pith review of Post-CCSD(T) corrections in the S66 noncovalent interactions benchmark." pith.science (2026). https://pith.science/paper/JP74BECO

@misc{pith2026241112004,
  author       = {Pith},
  title        = {Pith review of: Post-CCSD(T) corrections in the S66 noncovalent interactions benchmark},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JP74BECO}},
  note         = {Machine review of arXiv:2411.12004}
}
abstract

For noncovalent interactions, it is generally assumed that CCSD(T) is nearly the exact solution within the 1-particle basis set. For the S66 noncovalent interactions benchmark, we present for the majority of species CCSDT and CCSDT(Q) corrections with a polarized double-zeta basis set. For hydrogen bonds, pure London complexes, and mixed-influence complexes, CCSD(T) benefits from error cancellation between (usually repulsive) higher-order triples, $T_3 - (T)$, and (almost universally attractive) connected quadruples, (Q). For $\pi$-stacking complexes, this cancellation starts breaking down and CCSD(T) overbinds; CCSD(T)$_\Lambda$ corrects the problem at the expense of London complexes. A fairly simple two-parameter model predicts CCSDT(Q)--CCSD(T) differences to 0.01 kcal/mol RMS, requiring no calculations that scale more steeply than $O(N^7)$.

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Pith tools

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