{"id":"1ca19ab9-c83f-4643-abb1-df08437fc58f","arxiv_id":"2501.18135","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Plugging frozen-pair coupled-cluster amplitudes into the ISR(2) excited-state formalism yields a polynomial-scaling, Hermitian method with about 0.2 eV vertical excitation errors and robust potential-energy-surface topology.","lead":"This paper tests whether coupled-cluster methods that are good at describing static correlation in ground states can also describe excited states when plugged into the intermediate state representation (ISR). The best variant, CCDf1-ISR(2), predicts vertical excitation energies of small organic molecules within about 0.2 eV and correctly describes avoided crossings where equation-of-motion CCSD fails.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"CCDf1-ISR(2) robustness is limited by the (1+T2) reference truncation, which the authors themselves flag as over-stabilizing excited states at strong correlation.","rationale":"The reader's weakest_assumption identifies exactly the same structural issue: the ISR(2) excited states inherit a first-order reference while the ground-state energy is full CC. The authors' own Hubbard data and concluding statement confirm that this imbalance is real, so the concern is grounded in the manuscript rather than in an external objection. The paper has genuine supporting evidence: the Quest #1 benchmark is a standard test set, the N2 and formaldehyde surfaces are internally consistent, and the qualitative improvement over EOM-CCSD at the avoided crossing is plausible. However, none of those tests isolate the faithfulness of (1+T2)|Φ0> in the strongly correlated regime, which is exactly the regime advertised in the abstract. The 0.2 eV claim also overreaches for double-excitation states, where Table 1 gives 0.53 eV MAE. These are real limitations but not fatal flaws in the methodology as a proof of concept. A conditional acceptance with explicit scope qualifications is the appropriate verdict: the central construction is promising, but the robustness claim should be bounded to the regime where the first-order reference truncation is valid, and the accuracy claim should be qualified to the tested Quest #1 single-excitation set rather than small organic molecules in general. Secondary manuscript issues such as broken cross-references and the pCCD-ISR(2) exclusions do not change this assessment.","tokens_in":30087,"tokens_out":7133,"duration_ms":73481,"concrete_test":"On a small exact system where static correlation can be tuned continuously, e.g., the 4-site half-filled Hubbard chain or the symmetric stretch of H4 in a minimal basis, compute for each U/|t| (or bond length R): (i) the fidelity f = |<Φ0|(1+T2)^† e^{T2}|Φ0>|^2 / [<Φ0|(1+T2)^†(1+T2)|Φ0> <Φ0|e^{T2†} e^{T2}|Φ0>]; (ii) the CCDf1-ISR(2) excitation energy; (iii) the FCI excitation energy. If the signed error grows sharply once f drops below a threshold such as 0.9, the first-order reference truncation is the factor limiting robustness; if the error stays small even when f is poor, the concern is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"CCDf1-ISR(2) builds the ISR(2) excited-state Hamiltonian from the first-order wave function (1+T2)|Φ0> (Eq. 19) while the ground-state energy and amplitudes come from the full exponential e^{T2}|Φ0>. This mixed level of treatment is the method's core premise, and the authors explicitly state in Sec. 4.1 that it over-stabilizes excited states at large U/|t|, and in the Conclusions that 'a more rigorous approach than CC-ISR(2) should treat the ground and excited state wave function at the same level of approximation.' The consequence is that the central claim of robustness in the face of static correlation is only demonstrated in regimes where (1+T2) remains a faithful stand-in for the CC reference. The paper shows this up to U/|t|≈8 for the 10-site Hubbard chain and qualitatively for N2, but it does not test the truncation where T2 amplitudes become large, for example near avoided crossings or in the strongly correlated regions of polyenes. The Quest #1 0.21 eV MAE is computed at equilibrium geometries of mostly weakly correlated molecules, so it does not exercise the assumption. The double-excitation states in Table 1 already show a 0.53 eV MAE, which is outside the abstract's 'about 0.2 eV' claim. This is load-bearing because the method's novelty is precisely that static correlation is carried through the reference amplitudes, and the first-order truncation is the point where that transfer is least secure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a family of excited-state methods that combine 'addition-by-subtraction' coupled-cluster reference wave functions (pCCD, CCD0, CCD1, CCDf0/CCDf1) with the intermediate state representation (ISR) at second order, yielding CC-ISR(2) approaches. The central claim is that CCDf1-ISR(2) is robust against static correlation, provides enough dynamical correlation to give excitation energies accurate to about 0.2 eV for small organic molecules, and, thanks to the Hermitian ISR construction, correctly describes avoided crossings where EOM-CCSD fails. The paper benchmarks these methods on the 10-site Hubbard model, N2 dissociation, the formaldehyde PES, the Quest #1 database, and double-excitation states in linear polyenes. Overall, the work is a well-structured exploration of an interesting idea, with credible evidence that certain CC-ISR(2) variants outperform ADC(2) and EOM-CCSD in statically correlated regimes, but the abstract's accuracy claim is overstated and the reliance on a first-order (1+T2) reference for the excited-state Hamiltonian is a load-bearing approximation that deserves more scrutiny.","tokens_in":30398,"tokens_out":5544,"duration_ms":56316,"significance":"If the central claims hold, this is a valuable contribution to excited-state quantum chemistry: a polynomial-scaling, single-reference, spin-pure, Hermitian method that handles static correlation better than standard ADC(2) or EOM-CCSD, with potential utility in photodynamics and avoided-crossing topology. The paper provides reproducible benchmarks (Hubbard model, N2 PES, Quest #1, polyenes) and a transparent discussion of the method's limitations. The Quest #1 MAE of 0.21 eV for CCDf1-ISR(2) and the smooth N2 PES are concrete strengths. However, the significance is tempered by the fact that the double-excitation results are considerably less accurate (MAE 0.53 eV in Table 1) and by the authors' own admission that the first-order wave function truncation over-stabilizes excited states in strongly correlated regimes.","major_comments":[{"comment":"The abstract's claim that CCDf1-ISR(2) predicts excitation energies 'to within about 0.2 eV in small organic molecules' is contradicted by Table 1, which reports a mean absolute error of 0.53 eV for CCSDf1-ISR(2) on four alkenes, with individual errors as large as 1.45 eV (hexatriene 2^1Ag: TBE 5.09 eV vs. 6.54 eV). The 0.21 eV MAE quoted in Sec. 4.4 refers specifically to the 52 singlet excitations in the Quest #1 database at equilibrium geometries, which is a different and narrower class of states. The abstract and concluding statements should be qualified to the Quest #1 benchmark and should not imply that the cited accuracy extends to double-excitation-dominated states.","section":"Abstract and Sec. 4.5, Table 1"},{"comment":"The ISR(2) excited-state Hamiltonian is built from the first-order wave function (1+T2)|Φ0>, whereas the ground-state CC energy and amplitudes come from the full exponential e^{T2}|Φ0>. The authors themselves show in Sec. 4.1 that this imbalance over-stabilizes excited states at large U/|t| (the excited-state energies begin to decrease incorrectly beyond U/|t|~10), and in the Conclusions they state that 'a more rigorous approach than CC-ISR(2) should treat the ground and excited state wave function at the same level of approximation.' Because the paper's central claim is robustness in the face of static correlation, this truncation is load-bearing. The current evidence for robustness is limited to U/|t|≤8 in the Hubbard chain and to the equilibrium and moderately stretched regions of N2; the method is not tested in the regime where T2 amplitudes become large, such as strongly correlated polyene geometries or the avoided-crossing region of formaldehyde. The authors should either provide such tests or explicitly limit the robustness claim to the tested correlation regimes.","section":"Sec. 4.1 and Eq. (19)"},{"comment":"The presentation of the double-excitation results is more positive than the data warrant. The text states that CCSDf1-ISR(2) 'performs slightly better than ADC(2) (by about 0.1 eV) even for double excitations' and is 'on par with EOM-CCSD' for the 1Ag states, but the mean absolute error of 0.53 eV is more than twice the 0.2 eV accuracy claimed in the abstract. For the 2^1Ag states specifically, the errors are 0.93 eV (butadiene), 1.45 eV (hexatriene), and 1.27 eV (octatetraene), which are not quantitatively accurate. The conclusion that 'improving the ground-state reference can impart improvements to the predicted excitation energies' is supported only in a weak sense (a small MAE reduction relative to ADC(2)), and the text should clearly distinguish qualitative from quantitative accuracy when discussing double excitations.","section":"Sec. 4.5, Table 1"}],"minor_comments":[{"comment":"The manuscript contains many unresolved placeholder references such as 'Table ??', 'Fig. ??', and 'Figure ??' (e.g., Sec. 4.1, Sec. 4.2, Sec. 4.3). These should be resolved before the paper can be properly assessed by readers.","section":"Throughout (e.g., Sec. 4.1, 4.2, 4.3)"},{"comment":"The acronym 'FpiCCD' is introduced without definition, and the subsequent text uses 'CCDf1' instead; please clarify the relationship between these terms.","section":"Sec. 2.1"},{"comment":"The CASSCF@NEVPT2 reference is approximated by CASSCF alone at R_NN = 0.9 and 1.0 Å, but this is only mentioned in the figure caption, not in the main text; this approximation should be stated explicitly in the text.","section":"Sec. 4.2 and Fig. 2"},{"comment":"TD-DFT with ωB97X-D is used as a qualitative reference for the formaldehyde avoided crossing; the paper should note that TD-DFT is not a high-accuracy benchmark for excited-state topology and that the agreement is only qualitative.","section":"Sec. 4.3 and Fig. 3"},{"comment":"The statement that 'many physical systems fall within U/|t|≤8' is supported by citations to Hubbard-model literature, but a more quantitative argument or a diagnostic based on the size of T2 amplitudes would strengthen the claim that the tested range covers physically relevant strong correlation.","section":"Sec. 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper presents a promising idea with mostly sound benchmarking, but the abstract's accuracy claim is too broad and the central (1+T2) truncation deserves further testing or a more restrained claim. The unresolved placeholders in the manuscript are unusual and should be fixed. I think the paper is suitable for publication in a good quantum chemistry journal after major revisions that address the accuracy claim and the static-correlation robustness issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read. The genuinely new piece is testing addition-by-subtraction CCD references inside Dreuw's CC-ISR(2): pCCD, CCD0/1, CCDf0/1, with canonical and Brueckner orbitals. That sounds incremental—the ISR framework and each ground-state method already exist—but the results are not trivial. The standout is CCDf1/BCCDf1: about 0.2 eV MAE on Quest #1, smooth N2 PES without barriers, and the formaldehyde avoided-crossing topology where EOM-CCSD fails. The pCCD-ISR(2) failure is a genuinely informative negative result, and tying it to singles/reference coupling and orbital invariance is plausible and well argued.\n\nThe math and data look solid. Excitation energies are benchmarked against independent theoretical best estimates, no parameters are fitted to the targets, and the basis-set checks support the main trends. The citation pattern is honest: prior work by Dreuw, Hodecker, Bulik, Henderson, Scuseria, and Boguslawski gets proper credit.\n\nThe real limitation is one the authors themselves flag in Sec. 4.1 and the conclusions: ISR(2) is built from the first-order wave function (1+T2)|Phi0> while the ground-state energy and amplitudes come from the full exponential e^{T2}|Phi0>. This mixed level of treatment over-stabilizes excited states at strong correlation. The stress-test note lands, but as a boundary on the robustness claim, not as a fatal flaw. The paper demonstrates robustness up to U/|t| ~ 8 in the Hubbard model and across the tested PES regions; it does not claim to hold where the first-order approximation breaks down.\n\nSoft spots, in order. (1) The abstract's \"within about 0.2 eV\" overstates coverage: the double-excitation polyene benchmark has MAE 0.53 eV. The abstract should say single excitations or include the caveat. (2) The pCCD-ISR(2) Quest MAE is computed after excluding five failed degeneracies for CO and N2; defensible, but the main text should show that failure rate more prominently. (3) The manuscript is littered with broken cross-references (Table ??, Fig. ??), which makes it harder than it should be to verify claims. (4) None of these undermine the central benchmark result.\n\nWho gets value: method developers working on ISR/ADC variants and computational photochemists who need a Hermitian, polynomial-scaling alternative to EOM-CC for excited-state topology. BCCDf1-ISR(2) is a concrete, practical recommendation. This deserves a serious referee—send it out, with requested fixes on the abstract wording and the broken references.","headline":"A solid, honest benchmark paper showing CCDf1-ISR(2) is a useful Hermitian excited-state method, with an overbroad abstract claim and a real but contained limitation from the (1+T2) reference.","tokens_in":30965,"tokens_out":2979,"would_cite":true,"duration_ms":31279,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that planting static-correlation-preserving coupled cluster amplitudes into the intermediate state representation yields polynomial-cost excited-state energies accurate to about 0.2 eV and correct same-symmetry crossing…","keywords":["excited states","static correlation","intermediate state representation","coupled cluster","addition-by-subtraction CC","CCDf1-ISR(2)","conical intersections","polynomial scaling"],"falsifier":"Compute CCDf1-ISR(2) vertical excitation energies and excited-state potential energy surfaces for a transition-metal complex with known multiconfigurational excited states (for example an iron or chromium photocatalyst) and compare against a large-active-space CASPT2 or NEVPT2 benchmark. If the errors grow systematically beyond the roughly 0.2 eV seen for small organic molecules, or if same-symmetry crossings lose their correct topology, the claim of robustness in the face of static correlation would be falsified.","tokens_in":29872,"feed_emoji":"⚛️","tokens_out":7786,"duration_ms":70724,"temperature":0.7,"pith_summary":"This paper is trying to establish that the robustness of addition-by-subtraction coupled cluster methods against static correlation can be carried over from ground states to excited states by using those CC amplitudes inside the intermediate state representation (ISR). Its central result is that CCDf1-ISR(2), built from a reference that keeps singlet-paired correlation while freezing pre-computed triplet-paired amplitudes, stays stable where ordinary CC and EOM-CC methods diverge and predicts excitation energies to within about 0.2 eV on small organic molecules. The same Hermitian construction reproduces the avoided crossing between the $2^{1}$A1 and $3^{1}$A1 states of formaldehyde that EOM-CCSD gets wrong. A reader should care because this is a polynomial-scaling, black-box, spin-pure route to statically correlated excited states, an area usually reserved for factorial multireference methods.","feed_headline":"Frozen-pair coupled cluster predicts excited states to 0.2 eV","feed_subtitle":"The Hermitian ISR construction also fixes same-symmetry crossings that EOM-CCSD misses, at polynomial cost.","key_machinery":"The central object is the ISR(2) Hamiltonian matrix $M_{IJ} = \\langle \\tilde{\\Psi}_I | \\hat{H} - E_0 | \\tilde{\\Psi}_J \\rangle$, whose correlated excited states are built by applying physical excitation operators to the reference and orthogonalizing them by Gram-Schmidt; this makes the eigenvalue problem Hermitian. The paper's innovation is to supply the reference wave function $(1 + \\hat{T}_2)|\\Phi_0\\rangle$ from addition-by-subtraction CC methods instead of from MP2. In CCDf1, the triplet-paired amplitudes are solved first and then frozen while the singlet-paired channel is recoupled by solving the external CC equations to infinite order, so the reference carries the static correlation that ISR(2) then projects into the excited-state manifold.","core_discovery":"The paper claims that the quality of the reference wave function, not just the excitation manifold, decides whether second-order ISR can describe static correlation in excited states. The authors insert ground-state amplitudes from pCCD, CCD0, CCD1, CCDf0, and CCDf1 into the ISR(2) secular problem built from the first-order CC wave function $(1+\\hat{T}_2)|\\Phi_0\\rangle$, and show that CCDf1-ISR(2) smoothly dissociates N2 in the ground and $1^1\\Pi_g$ states, tracks the ten-site Hubbard model well beyond the interaction strength where EOM-CCSD and ADC(2) break down, and reproduces the formaldehyde avoided-crossing topology with mean absolute errors of 0.21 eV over 52 Quest #1 singlet excitations. The recommended variant, BCCDf1-ISR(2), uses Brueckner orbitals to remove the reference-singles coupling and gives the same accuracy with a more even error distribution.","pith_inferences":["We infer that the unbalanced treatment of the reference -- full exponential $e^{\\hat{T}_2}$ in the ground state versus first-order $(1+\\hat{T}_2)$ in ISR(2) -- is the main source of the asymmetric over-stabilization the authors observe at strong interaction strengths, and that a matched-order ISR built on the full exponential would be the natural next test.","We infer that the optimal-reference concept is portable: any ground-state method that captures static correlation in a single-determinant framework, such as orbital-optimized pair theories or regularized CC, could be substituted into the same ISR(2) machinery with minimal reimplementation.","We infer that because ISR(2) gives size-intensive oscillator strengths and correct same-symmetry crossing topology, it is a promising engine for nonadiabatic dynamics simulations, a use the authors flag but do not yet demonstrate.","We infer that the pCCD-ISR(2) failure with canonical orbitals and rescue by Brueckner orbitals implies that orbital invariance, not reference quality alone, controls whether an optimal-reference excited-state method works; future methods should be screened for orbital-rotation sensitivity."],"forward_implications":["CCDf1-ISR(2) reaches mean absolute errors of about 0.21 eV on the Quest #1 singlet excitation set, matching ADC(2) while remaining stable when the MP2 reference diverges.","The Hermitian ISR construction lets CCDf1-ISR(2) give the correct 2^1A1/3^1A1 avoided-crossing topology in formaldehyde, a case where standard EOM-CCSD predicts a spurious degeneracy.","BCCDf1-ISR(2), with Brueckner orbitals, removes outliers and is recommended for quantitative work, so the method is ready for benchmarking on photochemistry problems that need potential energy surface shapes.","CCSDf1-ISR(2) improves on ADC(2) for 1Ag states with substantial double-excitation character in polyenes, showing that a better ground-state reference also helps doubly excited states.","All of these variants scale polynomially (the CCDf1 bottleneck is O(N^6)), so the approach offers a single-reference, black-box alternative to active-space methods for statically correlated excited states."],"supporting_citations":[{"why":"Introduces the CCD-ISR(2) framework that this paper extends by swapping in addition-by-subtraction CC amplitudes.","marker":"[112]"},{"why":"Establishes that single-reference CCD0 and CCD1 can describe static correlation in ground states, the property being transplanted to excited states.","marker":"[120]"},{"why":"Presents the recoupled frozen-amplitude CCDf0/CCDf1 methods used as the CCDf1-ISR(2) reference.","marker":"[129]"},{"why":"Defines seniority-zero pCCD and notes its orbital-invariance issues that motivate Brueckner-orbital variants.","marker":"[94]"},{"why":"Supplies the Quest #1 database of theoretical best estimates used for the 0.21 eV accuracy benchmark.","marker":"[139]"},{"why":"Documents the formaldehyde same-symmetry crossing problem used to test ISR topology against EOM-CCSD.","marker":"[108]"},{"why":"Provides the intermediate state representation formalism on which the ISR(2) secular equations rest.","marker":"[110]"},{"why":"Shows earlier evidence that CC amplitudes improve ADC/ISR results, the direct predecessor of this study.","marker":"[115]"}],"fun_headline_variants":["Coupled cluster tames static correlation in excited states","Excited states at polynomial cost: new CC method hits 0.2 eV","0.2 eV accuracy for excited states with paired CC-ISR","Fixing same-symmetry crossings: CC-ISR beats EOM-CCSD","Polynomial-cost excited states: paired CC meets 0.2 eV"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes that the first-order coupled cluster wave function $(1 + \\hat{T}_2)|\\Phi_0\\rangle$ is a faithful reference for the excited states whenever the underlying ground-state amplitudes are good; the authors note in Section 4.1 that using the full exponential in the ground state but only the linearized wave function in ISR(2) over-stabilizes excited states at large interaction strength.","fun_headline_variants_meta":{"raw":{"variants":["Coupled cluster tames static correlation in excited states","Excited states at polynomial cost: new CC method hits 0.2 eV","0.2 eV accuracy for excited states with paired CC-ISR","Fixing same-symmetry crossings: CC-ISR beats EOM-CCSD","Polynomial-cost excited states: paired CC meets 0.2 eV"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000679,"raw_usage":{"total_tokens":3147,"prompt_tokens":1071,"completion_tokens":2076,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":1978}},"tokens_in":687,"tokens_out":2076,"duration_ms":13939,"temperature":1.0,"reasoning_tokens":1978,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:32:36.518044+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute CCDf1-ISR(2) vertical excitation energies and excited-state potential energy surfaces for a transition-metal complex with known multiconfigurational excited states (for example an iron or chromium photocatalyst) and compare against a large-active-space CASPT2 or NEVPT2 benchmark. If the errors grow systematically beyond the roughly 0.2 eV seen for small organic molecules, or if same-symmetry crossings lose their correct topology, the claim of robustness in the face of static correlation would be falsified.","supporting_citations":[{"cited_title":"R.;\\ \\ Dreuw, A","cited_arxiv_id":null,"evidence_quote":"Introduces the CCD-ISR(2) framework that this paper extends by swapping in addition-by-subtraction CC amplitudes."},{"cited_title":"W.;\\ \\ Henderson, T","cited_arxiv_id":null,"evidence_quote":"Establishes that single-reference CCD0 and CCD1 can describe static correlation in ground states, the property being transplanted to excited states."},{"cited_title":"A.;\\ \\ Henderson, T","cited_arxiv_id":null,"evidence_quote":"Presents the recoupled frozen-amplitude CCDf0/CCDf1 methods used as the CCDf1-ISR(2) reference."},{"cited_title":"M.;\\ \\ Scuseria, G","cited_arxiv_id":null,"evidence_quote":"Defines seniority-zero pCCD and notes its orbital-invariance issues that motivate Brueckner-orbital variants."},{"cited_title":"A mountaineering strategy to excited states: Highly accurate reference energies and benchmarks","cited_arxiv_id":null,"evidence_quote":"Supplies the Quest #1 database of theoretical best estimates used for the 0.21 eV accuracy benchmark."},{"cited_title":"Can coupled-cluster theory treat conical intersections? J","cited_arxiv_id":null,"evidence_quote":"Documents the formaldehyde same-symmetry crossing problem used to test ISR topology against EOM-CCSD."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the intermediate state representation formalism on which the ISR(2) secular equations rest."},{"cited_title":"L.;\\ \\ Rehn, D","cited_arxiv_id":null,"evidence_quote":"Shows earlier evidence that CC amplitudes improve ADC/ISR results, the direct predecessor of this study."}],"review_version":1}