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Extension of the Active-Orbital-Based and Adaptive CC($P$;$Q$) Approaches to Excited Electronic States: Application to Potential Cuts of Water

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Pith's one-line read This paper shows that active-orbital and adaptive CC(P;Q) methods reproduce CCSDT/EOMCCSDT ground- and excited-state water potential cuts along O–H bond breaking.

desk verdict A solid, well-benchmarked extension of CC(P;Q) to excited states; the accuracy on water holds, and the open question is transferability, not correctness. read the letter →

arxiv 2411.11245 v2 pith:ADXO4FBC submitted 2024-11-18 physics.chem-ph physics.comp-ph

Karthik Gururangan , Jun Shen , Piotr Piecuch This is my paper · ORCID
classification physics.chem-phphysics.comp-ph
keywords coupled-clustertheoryequation-of-motionCC(PQ)methodologyCC(t3)adaptiveselectionoftriplesexcitedstatespotentialenergysurfacewaterdissociation
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 is a first test of two cost-saving coupled-cluster strategies—active-orbital-based CC(t;3) and adaptive CC(P;Q)—on excited electronic states. The test case is water breaking one O–H bond, for which full CCSDT/EOMCCSDT potential cuts are known to be nearly exact. The paper shows that both methods reproduce those CCSDT/EOMCCSDT curves for 12 singlet and triplet states with mean unsigned errors of about 0.2–1.0 millihartree, while putting only a small fraction of the triple excitations into the iterative calculation. A sympathetic reader should care because excited-state potential surfaces at this accuracy normally cost far more, and the adaptive variant needs no user-chosen active orbitals.

What carries the argument

The object that carries the argument is the CC(P;Q) energy correction $\delta_\mu(P;Q)=\sum_{|\Phi_K\rangle\in\mathcal H^{(Q)}} \ell_{\mu,K}(P) M_{\mu,K}(P)$, in which the Hilbert space is split into an iterative P space and a noniterative Q space. $M_{\mu,K}(P)$ are the generalized moments of the CC/EOMCC equations projected on Q-space determinants, and $\ell_{\mu,K}(P)$ are coefficients built from the left bra states with Epstein–Nesbet denominators. CC(t;3) chooses the P-space triples by active orbitals (here the three highest occupied and two lowest unoccupied RHF orbitals), while adaptive CC(P;Q) selects the leading 1–2% of triples by the size of their $\ell_{\mu,K}M_{\mu,K}$ contributions, with no active-space input; the leftover Q-space triples enter through $\delta_\mu(P;Q)$. The method's power is that a tiny, selected subset of triples is treated iteratively and the rest perturbatively, so the expensive $N^8$ triple step of full CCSDT/EOMCCSDT is avoided.

What would settle it

Run the adaptive CC(P;Q)[%T=2] protocol on a different stretched-bond system, such as ozone or N2, using a basis larger than TZ, and compare the resulting ground- and excited-state potentials against CCSDT/EOMCCSDT or full CI; if the mean unsigned error exceeds roughly 1–2 mH, or if more than 2% of triples is needed to remove the qualitative failures of CR-EOMCC(2,3), the selection assumption is falsified for that case.

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Extended reading notes

Core claim

The central claim is that the CC(P;Q) moment-expansion hierarchy, in both its active-orbital CC(t;3) form and its black-box adaptive form, can converge ground- and excited-state potential cuts of water along the O–H dissociation coordinate to the parent CCSDT/EOMCCSDT results. Relative to CCSDT/EOMCCSDT, CC(t;3) has MUEs between 0.166 and 0.951 mH over 12 states, and adaptive CC(P;Q) with only 2% of the triples in the P space has MUEs between 0.197 and 0.993 mH and NPEs between 0.133 and 1.572 mH. The improvement is most dramatic for the 2 3A″ and 3 3A″ states, where the earlier CR-EOMCC(2,3) corrections produce errors of 13 and 42 mH in NPE and a spurious bump in the 3 3A″ curve; CC(t;3) removes the bump and reduces these NPEs to 0.255 and 0.608 mH. The paper also establishes that CCSDT/EOMCCSDT itself is an excellent benchmark here, with MUEs below 1.6 mH relative to full CI for all 12 states. This is the first use of these two CC(P;Q) variants for excited electronic states.

Load-bearing premise

The weakest link is the assumption that a very small set of triple excitations, chosen by active orbitals or by moment-size ranking, captures almost all of the correlation that a full triples treatment would add; this has been demonstrated for one molecule, one basis set, and one active-space choice.

Editorial extensions

If this is right

  • Excited-state potential surfaces of near-EOMCCSDT quality become practical in the stretched-bond regions where EOMCCSD and CR-EOMCC(2,3) fail, because the leading triples enter the iterative wave function rather than only a correction.
  • The adaptive variant removes the need for an active-orbital choice, so the method can be applied as a black-box approximation to systems where full triples are too costly.
  • The largest CR-EOMCC(2,3) failures on water, the 2 3A″ and 3 3A″ curves, are repaired: their errors relative to EOMCCSDT drop from tens of millihartree to below one millihartree in CC(t;3) and to below one millihartree in most adaptive 2% calculations.
  • Only about 300 triply excited determinants per symmetry (2% of the total) suffice in this system to bring the MUE below about 1 mH, indicating the iterative cost can be far below the roughly 31,000 triples used by full EOMCCSDT.
  • Because CCSDT/EOMCCSDT is shown to be within about 1.6 mH of full CI for all 12 states, these low-cost approximations inherit a near-exact description of the same curves.

Reading between the lines

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

  • The 38%-vs-2% comparison implies that moment-based adaptive selection is more efficient than the active-space heuristic: the same accuracy is reached with far fewer iterative triples, so chemical-intuition active spaces may be over-inclusive.
  • The selection heuristic is not yet established for other molecules; a direct test would run the same 2% adaptive protocol on another multireference bond-breaking case, such as ozone or stretched N2, against CCSDT/EOMCCSDT or full CI.
  • If the heuristic transfers, this recipe could become a practical default for photochemical and spectroscopic studies, where many excited-state curves are needed and full CCSDT is feasible only for small molecules.
  • The remaining roughly 0.2 mH floor in the adaptive results suggests a possible refinement: using higher-rank moments or a second adaptive pass on the Q space to reduce that floor without admitting more triples into the iterative P space.
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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

0 major / 4 minor

Summary. This manuscript extends the CC(P;Q) hierarchy to excited states by combining active-orbital-based CC(t;3) and adaptive CC(P;Q) with EOMCC, and benchmarks the resulting methods on the ground and 11 excited singlet/triplet A′/A′′ states of water along the O–H dissociation coordinate using the TZ basis. The parent CCSDT/EOMCCSDT potentials are first validated against full CI in Table 1 (MUEs 0.28–1.53 mH, NPEs 0.57–5.50 mH). The approximate methods are then compared with these parents in Tables 2 and 3, showing that CC(t;3) reproduces CCSDT/EOMCCSDT with MUEs below 1 mH and NPEs below 0.61 mH, while adaptive CC(P;Q) using 2% of triples in the iterative P spaces gives MUEs of 0.197–0.993 mH and NPEs of 0.133–1.572 mH, and both methods improve on CR-CC(2,3)/CR-EOMCC(2,3) in strongly stretched regions. The paper is framed as a single-molecule benchmark, with broader testing explicitly deferred to future work in Sec. 4.

Significance. The significance is incremental but real: it demonstrates that the adaptive, orbital-free variant of CC(P;Q), previously tested mainly on ground states, extends successfully to excited states, including states with strong multireference character such as 23A′′ and 33A′′ where CR-EOMCC(2,3) fails badly. Strengths of the paper include the deterministic and complete set of 12 states, the external full-CI grounding of the parent CCSDT/EOMCCSDT data, the clear reporting of MUE/NPE statistics with no post-hoc state exclusions, and the availability of total energies in Supplementary Data and of the CCpy code on GitHub. The main limitation—validation on a single molecule and one basis set—is acknowledged by the authors in Sec. 4 and does not undermine the stated water-benchmark claim.

minor comments (4)
  1. [Abstract and Sec. 3.3] The abstract and Sec. 3.3 state that the adaptive calculations offer significantly reduced computational costs, but the manuscript reports only the fraction of triples in the iterative P spaces (1–2% versus the full CCSDT/EOMCCSDT triple manifolds), not wall-clock timings or operation counts for the excited-state implementation; since the noniterative Q-space correction still spans the remaining triples, a few timing data or an explicit complexity statement would make the cost claim easier to verify.
  2. [Sec. 3.1] The adaptive algorithm parameters (1% growth rate and %T = 1 and 2% final spaces) are stated, but it is not specified whether %T refers to the final converged fraction after the growth process or to a target fraction imposed at each recursion step; a one-sentence definition would improve reproducibility.
  3. [Sec. 3.1 and Fig. 1] The manuscript does not describe how the individual EOMCC(P) roots are tracked through avoided crossings when constructing the state-specific P spaces; because the reported MUE/NPE values require consistent state labels along ROH, a sentence on root assignment would strengthen the comparisons.
  4. [Tables 2 and 3] The footnotes define the active space for CCSDt/CC(t;3) only as the three highest occupied and two lowest unoccupied RHF orbitals; specifying whether this selection is by RHF orbital energy and collecting the symmetry labels already given in Sec. 3.1 in a single place would make the tables more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No meaningful circularity: the water-benchmark claim is a direct numerical comparison against independently benchmarked CCSDT/EOMCCSDT, not a fit or self-citation tautology.

full rationale

The paper's central claims—that CC(t;3) and adaptive CC(P;Q) reproduce the CCSDT/EOMCCSDT water potential cuts—are checked directly against those parent calculations in Tables 2 and 3, and the parent is itself validated against full CI in Table 1 using full CI results from Refs. [13,16]. No parameter in the CC(P;Q) workflow is fitted to the CCSDT/EOMCCSDT target energies: the moment corrections of Eq. (4) are built from CC(P)/EOMCC(P) amplitudes and Hamiltonian matrix elements, and the adaptive selection of triples uses those same moment contributions as a convergence guide rather than any target energy. The active-orbital CC(t;3) active-space choice is explicitly user-defined and acknowledged as such; that is a transferability limitation, not a hidden input-output identity. The paper's own Sec. 4 states that testing other systems and larger basis sets is future work, which is a scope limitation rather than a circularity. The extensive citations to the authors' earlier CC(P;Q) papers describe the method's derivation, but the numerical benchmark in this paper is an independent application against an external full-CI anchor. I find no step in which a prediction is equivalent by construction to its input, so the circularity score is 0.

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

The central claim rests on the pre-existing CC(P;Q) moment-expansion formalism and on the numerical benchmarks against CCSDT/EOMCCSDT and full CI. The paper introduces no new entities. The adjustable parameters are the adaptive algorithm growth rate, the 1% and 2% triple fractions, and the CC(t;3) active space choice; none are fit to the target CCSDT/EOMCCSDT energies, but the demonstrated accuracy is contingent on them.

free parameters (4)
  • adaptive_P_space_triple_fraction_1_percent = 1% of Sz=0 triple excitations
    User-selected cutoff determining how many triply excited determinants enter the iterative CC(P)/EOMCC(P) spaces; the central accuracy claims for adaptive CC(P;Q)[%T=1] depend on it.
  • adaptive_P_space_triple_fraction_2_percent = 2% of Sz=0 triple excitations
    Second user-selected cutoff; results improve from %T=1 to %T=2, so the reported MUEs depend on this manually chosen threshold.
  • adaptive_algorithm_growth_rate = 1% growth rate
    Growth rate in the relaxed adaptive selection algorithm (Sec. 3.1), chosen by the authors; affects which triples are selected.
  • CC(t;3)_active_space = three highest occupied and two lowest unoccupied RHF orbitals
    Active orbital set defining CCSDt/EOMCCSDt P spaces; results (Tables 2 and 3) are conditional on this choice, which may not generalize to other molecules.
assumptions (4)
  • domain assumption The CCSDT/EOMCCSDT water potential cuts accurately approximate the full CI results of Refs. [13,16].
    Used as the benchmark truth in Tables 2 and 3; supported by Table 1 (MUE 0.28-1.53 mH, NPE 0.57-5.50 mH) but only for this molecule and basis.
  • standard math The moment expansion δμ(P;Q) = Σ ℓμ,K Mμ,K in Eq. (4) converges to the parent CCSDT/EOMCCSDT energies for the selected P/Q partitions.
    Invoked in Sec. 2; this is the core approximation of the CC(P;Q) framework, justified by prior work rather than proven in this paper.
  • domain assumption The TZ basis set and RHF reference with frozen 1s oxygen orbital are adequate for the water potential cuts.
    Computational details in Sec. 3.1; inherited from Refs. [13,16], not revalidated here.
  • domain assumption Geometries optimized at CCSD/cc-pVTZ from Ref. [16] define the reaction path.
    Sec. 3.1; the potential cuts depend on these geometries.

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Cite this review

Pith. "Pith review of Extension of the Active-Orbital-Based and Adaptive CC($P$;$Q$) Approaches to Excited Electronic States: Application to Potential Cuts of Water." pith.science (2026). https://pith.science/paper/ADXO4FBC

@misc{pith2026241111245,
  author       = {Pith},
  title        = {Pith review of: Extension of the Active-Orbital-Based and Adaptive CC($P$;$Q$) Approaches to Excited Electronic States: Application to Potential Cuts of Water},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADXO4FBC}},
  note         = {Machine review of arXiv:2411.11245}
}
abstract

We report the first study using active-orbital-based and adaptive CC($P$;$Q$) approaches to describe excited electronic states. These CC($P$;$Q$) methodologies are applied, alongside their completely renormalized (CR) coupled-cluster (CC) and equation-of-motion (EOM) CC counterparts, to recover the ground- and excited-state potential cuts of the water molecule along the O-H bond-breaking coordinate obtained in the parent CC/EOMCC calculations with a full treatment of singles, doubles, and triples (CCSDT/EOMCCSDT). We demonstrate that the active-orbital-based and adaptive CC($P$;$Q$) approaches closely approximate the CCSDT/EOMCCSDT data using significantly reduced computational costs while improving the CR-CC and CR-EOMCC energetics in stretched regions of the O-H bond-breaking potentials.

Figures

Figures reproduced from arXiv: 2411.11245 by the authors.

Figure 1
Figure 1. A comparison of the potential cuts of water, plotted as functions of ROH and corresponding to the H2O → H+OH dissociation channels that correlate with the X 2Π ground state and the lowest-energy 2Σ + and 2Σ − states of OH, obtained with (a) CCSD/EOMCCSD, (b) CR-CC(2,3)/CR￾EOMCC(2,3), (c) adaptive CC(P)/EOMCC(P) using 1% of triply excited determinants in the P spaces, (d) adaptive CC(P;Q) using 1% of triply excited d… view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.