{"id":"294059fe-8cab-40af-8980-817a051a801d","arxiv_id":"2509.06460","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Partial stochastic-RI CC2 computes excited-state gradients and derivative couplings with quartic scaling and manageable noise, extending CC2 to larger molecules.","lead":"This paper implements stochastic approximations to the CC2 quantum chemistry method for computing excited-state energy gradients and derivative couplings, reducing formal cost from fifth to fourth power in system size. It offers a path to excited-state dynamics simulations of molecules too large for conventional CC2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The derivative-coupling implementation is validated only against CIS/CCS and an in-house hybrid, not an independent CC2 code; if the RI-CC2 derivative-coupling reference has a bug, the reported sRI-CC2 derivative couplings inherit it.","rationale":"Good-faith reading: this is primarily an implementation/methods paper, and the gradient part is credible. Partial sRI-CC2 reproduces RI-CC2 gradients to within ~0.008 a.u. on the tested molecules, the olefin-chain errors do not grow with system size, and the fitted scaling (~O(N^3.94)) matches the designed quartic scaling. The stochastic-noise analysis in Sec. III.C also makes the Ns choices transparent. This is why I would not reject. The derivative-coupling portion is where the practical claim—'essential ingredients for large-scale nonadiabatic dynamics'—is least secure. The validation chain has a single endpoint: an unverified deterministic RI-CC2 derivative-coupling code. The authors explicitly state that 'additional tests are still needed to validate its accuracy' (Sec. III.B), an internal admission that the benchmark is incomplete. My concern is not that CC2 differs from CIS (that is physically expected), but that the sRI-CC2 error tables would be misleading if the deterministic reference contains an implementation error. The proposed independent-code or finite-difference test settles this specifically. The 'hundreds or even thousands of electrons' claim is extrapolated from much smaller test molecules, but that affects the strength of the abstract's rhetoric more than the core correctness of the method; the measured scaling itself is the relevant evidence. Overall, the reader's CONDITIONAL verdict is appropriate: the method is plausible and well supported on gradients, but the derivative-coupling benchmark needs an independent confirmation before the claim can be fully accepted.","tokens_in":19231,"tokens_out":5102,"duration_ms":64024,"concrete_test":"For H2O at a geometry where F21 is nonzero, compute the RI-CC2 derivative-coupling vector with an independent CC2 implementation—either an external code such as eT or CFOUR once their CC2 derivative-coupling modules are available, or by numerical differentiation of CC2 overlap matrix elements <L_m|R_n(x)> in a separate code using the same basis, geometry, and state pair. Compare the per-component values and the vector norm with Table V. If the independent F21 differs from the authors' RI-CC2 result by more than ~0.05 bohr^-1 (rather than agreeing within expected method tolerance), the sRI-CC2 derivative-coupling benchmark is corrupted; if it matches, the attribution of the CIS discrepancy to CC2 physics rather than an implementation error is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes excited-state derivative couplings at the sRI-CC2 level. To show that partial sRI-CC2 reproduces RI-CC2, the deterministic RI-CC2 derivative-coupling implementation must itself be correct. In Sec. III.B the authors state that independent CC2 implementations in CFOUR/eT are 'in developmental versions and inaccessible'; the only benchmarks are against CIS (Q-Chem) and the authors' own CCS code. For H2O F21, RI-CC2 differs from CIS by ~0.63 a.u. (Delta_bar_abs = 0.1912), and this discrepancy is attributed to differences in excited-state amplitudes/energies. A 'CCS->CC2' hybrid reproduces CIS, but that hybrid still relies on the same CC2 derivative-coupling machinery and does not exercise the CC2-specific amplitude response, so it cannot fully validate the implementation. Tables VII and VIII then compare sRI-CC2 only to this same RI-CC2 reference, meaning any systematic error in the deterministic derivative-coupling code is inherited by the stochastic results. This is load-bearing because the sRI-CC2 error analysis cannot distinguish stochastic error from a systematic reference error. The gradient portion does not share this weakness: the RI-CC2 gradient is checked against TURBOMOLE (Sec. III.A) and is well supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents two stochastic resolution-of-identity (sRI) variants for computing CC2 excited-state analytical gradients and derivative couplings: a 'complete sRI' approach that stochastically samples both Coulomb and exchange terms (formal O(N^3) scaling) and a 'partial sRI' approach that applies sRI only to exchange terms (formal O(N^4) scaling). The theory section adapts the standard CC2 Lagrangian formalism, introduces the Laplace-transform/sRI contraction machinery, and gives explicit working equations and scaling tables. Numerical benchmarks for gradients on H2, H2O, HF, LiH, LiF, NH3, benzene, furan, pyrrole, pyridine, and a series of olefin chains show that partial sRI-CC2 with Ns=100 reproduces the authors' deterministic RI-CC2 gradients within about 0.008 hartree/bohr maximum error, with measured scaling close to O(N^4). Derivative couplings are tested on H2O and several small molecules; partial sRI-CC2 again matches the in-house RI-CC2 reference within stochastic deviation, but the RI-CC2 derivative-coupling implementation itself is validated only against CIS/CCS, not against an independent CC2 code. The paper concludes that partial sRI-CC2 is a practical alternative for systems with hundreds or even thousands of electrons.","tokens_in":19590,"tokens_out":8409,"duration_ms":101385,"significance":"If the derivative-coupling implementation were independently validated, this paper would be a worthwhile contribution to the stochastic quantum-chemistry literature: the partial sRI scheme offers a plausible route to CC2 excited-state gradients and derivative couplings at reduced scaling, and the gradient accuracy data are encouraging. The strengths are the careful comparison of the complete and partial sRI variants, the size-dependence study on olefin chains, and the explicit scaling analysis. The weak point is the derivative-coupling validation: the only reference for the new sRI derivative couplings is the authors' own RI-CC2 code, and that code is not cross-checked against an independent CC2 derivative-coupling implementation. The practical-scale claim in the abstract also goes beyond what is demonstrated, since the largest systems in the manuscript are C10H12 (72 electrons).","major_comments":[{"comment":"The derivative-coupling reference is not independent. Sec. III.B states that the CCSD and CC2 implementations in CFOUR/eT are 'in developmental versions and inaccessible', so the RI-CC2 derivative-coupling implementation is benchmarked only against CIS/CCS. For H2O F21, RI-CC2 deviates from CIS by Delta_bar_abs = 0.1912 (Table VI); the 'CCS->CC2' hybrid recovers CIS but still uses the same CC2 derivative-coupling equations (Eqs. 38-41) and does not exercise the CC2-specific amplitude response. Tables VII-VIII therefore validate the sRI algorithm against the authors' own deterministic code, not against an external CC2 reference. Any systematic error in the RI-CC2 derivative-coupling code is inherited by the reported sRI-CC2 derivative couplings. This is load-bearing for the derivative-coupling claim. Please provide an independent CC2 benchmark when feasible, or at least a numerical finite","section":"Sec. III.B, Tables V-VIII; Eq. (38)-(41)"},{"comment":"The practical-scale claim overreaches the data. The largest systems computed in this manuscript are C10H12 (72 electrons) for both gradients and derivative couplings. The crossover with RI-CC2 is predicted by fitting at about 400 electrons, and CPU-time data stop near x=200. No calculation in this paper reaches 'hundreds or even thousands of electrons.' The quartic-scaling fits (O(N^3.94) and O(N^3.63)) support the asymptotic advantage, but the abstract's claim that partial sRI-CC2 'can handle' such systems needs either a demonstration on a system beyond the crossover or qualification as a projected capability.","section":"Abstract and Sec. IV; Figs. 2, 4"},{"comment":"The conclusion that Ns=100 is sufficient and that stochastic error does not increase with system size is drawn from olefin chains (Table IV, Fig. 1), but the broader benchmark tables show substantial system-to-system variation at fixed Ns. For derivative couplings, pyrrole and pyridine have S.D. Delta_max of 0.2995 and 0.2984 (Table VIII), compared to 0.0777 for LiH or 0.0283 for benzene. The H2O-based Ns selection (Figs. 5-6) therefore does not imply that Ns=100 is universally sufficient. Either report Ns-dependent results for the problematic cases or qualify the size-independence claim to the tested molecule classes.","section":"Sec. III.A and III.C; Tables III, VIII"}],"minor_comments":[{"comment":"Typo: 'hatree/bohr' should be 'hartree/bohr'.","section":"Table III caption"},{"comment":"The sentence 'The sRI-CC2 variant is deemed accurate if the standard deviation ... exceeds the systematic error' appears to express the opposite of the intended accuracy criterion. Clarify: presumably the deterministic RI-CC2 result should lie within the stochastic error bar rather than being exceeded by it.","section":"Sec. III.A"},{"comment":"Define the horizontal axis (presumably the number of electrons) and the meaning of the crossover coordinate x=400 in the captions or text.","section":"Figs. 2 and 4"},{"comment":"Clarify whether the averaged quantity \\bar F_mn = (F_mn - F_nm)/2 is used consistently in Tables V and VII, and how phase conventions are handled for the non-Hermitian CC2 coupling.","section":"Eq. (61)"},{"comment":"The symbol \\bar\\gamma is used both for the reduced density matrix in Eq. (27) and for the Lagrange multiplier in Eq. (33); rename one to avoid confusion.","section":"Eqs. (27) and (33)"},{"comment":"The row labeled 'CCS->CC2' in Tables V and VI is not a standard method; specify precisely which amplitudes and which equations are used in that hybrid calculation.","section":"Sec. III.B"},{"comment":"Reference [88] appears unrelated to the body of the paper and may be a typographical artifact; please check its relevance or remove it.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The gradient portion is the strongest part of the paper and could become acceptable with modest revisions. The derivative-coupling validation gap is the main barrier; if the authors can obtain even one independent CC2 derivative-coupling benchmark, or a convincing finite-difference consistency check, the paper would be substantially strengthened. I would also ask them to temper the 'hundreds or even thousands of electrons' claim unless a calculation beyond the crossover is shown. No concerns about novelty or scope for a physical-chemistry methods journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the excited-state gradient part is in good shape; the derivative-coupling part is the weak joint. The central engineering choice, applying sRI only to exchange and keeping Coulomb deterministic, is sensible, and the gradient data support it. I would send this to a referee, but I would ask for an independent derivative-coupling benchmark or an explicit limitation in the text.\n\nWhat's new: first sRI-CC2 implementation of excited-state analytical gradients and derivative couplings, including the partial exchange-only variant. It is an incremental step in the authors' own sRI-CC2 series, but a real one: the expensive exchange terms are reorganized to scale as O(N^4) instead of O(N^5), and the timing data fit that. Gradient accuracy is well documented. Max errors against RI-CC2 stay below 0.008 hartree/bohr across several molecules, and the RI-CC2 gradient itself is checked against TURBOMOLE, so the reference is not only in-house. The statistical error stays roughly flat with system size in the polyene series, which is the main practical claim.\n\nThe soft spots are where the stress-test concern lands. For derivative couplings, the RI-CC2 reference implementation is validated only against CIS and the authors' own CCS code. The paper states directly that CC2 derivative couplings in CFOUR and eT were in developmental versions and inaccessible. The 'CCS->CC2' hybrid reproduces CIS, but as the paper itself notes, it does not exercise the CC2-specific amplitude response. So a systematic error in the in-house RI-CC2 derivative-coupling code would be inherited by the sRI-CC2 results. The F21 discrepancy with CIS is large, and the attribution to excited-state amplitude differences is plausible but not independently verified. This is not fatal; the derivative-coupling code shares machinery with the better-tested gradient code. But it is the difference between 'validated' and 'consistent with an unbenchmarked reference.'\n\nMinor: the 'hundreds or even thousands of electrons' claim is extrapolated; the crossover is around 400 electrons. That is a reasonable projection, but it should be labeled as one. The complete sRI variant with Ns = 50000 is too noisy for practical use, but the partial method is the actual proposal, so this is only a mild concern.\n\nWho it's for: people working on low-scaling CC2 or nonadiabatic dynamics on large systems, and it deserves a serious referee. Recommendation: accept conditionally, with an external or at least independent derivative-coupling check, or a clearly stated limitation.","headline":"Useful but uneven: the gradient implementation is solid, and the derivative-coupling validation needs an independent reference before it carries the weight the abstract puts on it.","tokens_in":20034,"tokens_out":2461,"would_cite":true,"duration_ms":31868,"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":"This paper claims that a partial stochastic resolution of identity (sRI) applied to exchange-only integrals reduces CC2 excited-state gradients and derivative couplings from fifth-order to fourth-order scaling, making calculations on molecu","keywords":["CC2","excited-state gradients","derivative couplings","stochastic resolution of identity","scaling reduction","Laplace transform","exchange integrals","nonadiabatic dynamics"],"falsifier":"Compare the reported RI-CC2 water F21 derivative-coupling vector against an independent, released CC2 or similarity-constrained CC2 implementation; if the RI-CC2 value differs by more than the claimed sRI error, the stochastic benchmark baseline is wrong. The paper's own hybrid CCS-to-CC2 calculation reproduced the CIS result for F21 while the full CC2 did not, so resolving whether the CC2 amplitude or energy causes that discrepancy would settle whether the large F21 difference is in the method or the implementation.","tokens_in":19147,"feed_emoji":"🧪","tokens_out":6846,"duration_ms":72582,"temperature":0.7,"pith_summary":"Excited-state gradients and derivative couplings are what make CC2 useful for simulating nonadiabatic dynamics, but their steep computational scaling confines CC2 to small molecules. This paper claims that a stochastic resolution of identity (sRI), applied selectively to exchange-type integrals, reduces the cost of both CC2 excited-state gradients and derivative couplings to fourth-order scaling with a small prefactor. At a fixed number of stochastic orbitals (Ns=100), the reported errors relative to conventional RI-CC2 stay below about 0.008 hartree/bohr for gradients and remain stable as molecular size grows. A fully stochastic cubic-scaling variant exists but requires so many stochastic orbitals (Ns≈50000) that it is not practical for the tested systems. If these scaling and error behaviors hold, CC2 excited-state gradients and derivative couplings become feasible for molecules with hundreds of electrons, opening a route to large-scale nonadiabatic dynamics.","feed_headline":"Partial stochastic RI drops CC2 excited-state gradient cost to N^4","feed_subtitle":"The quartic-scaling variant also computes derivative couplings, pushing CC2 dynamics toward hundreds of electrons.","key_machinery":"The central object is the stochastic resolution of identity (sRI): random ±1 vectors {θξ} are inserted into the standard RI factorization of four-index electron repulsion integrals, replacing the auxiliary-basis index with a stochastic index whose size Ns is independent of system size. Contracted with Laplace-transformed orbital-energy denominators, the exchange part of the CC2 double-amplitude contractions becomes a product of two-index random tensors, cutting the formal cost from O(N^5) to O(N^4) in the partial variant (exchange-only sampling) and to O(N^3) in the complete variant. The partial-sRI scheme is the load-bearing design choice: it keeps the Coulomb terms deterministic so stochas","core_discovery":"The central claim is that the partial sRI-CC2 method computes excited-state analytical gradients and derivative couplings at O(N^4) scaling, down from the O(N^5) of conventional RI-CC2, by applying stochastic resolution of identity only to exchange terms in the double-amplitude contractions while keeping Coulomb terms deterministic. Empirical timings on (all-E)-olefin chains give scaling exponents of 3.94 for gradients and 3.63 for derivative couplings, versus 4.78 and 4.41 for RI-CC2, with the crossover near 400 electrons. With Ns=100 stochastic orbitals, maximum gradient errors against RI-CC2 are below 0.008 hartree/bohr and the statistical error does not grow with system size, so a fixed","pith_inferences":["The selective-sRI design principle—stochasticity only in exchange contractions—could transfer to other correlated methods such as CCSD or ADC gradients, which have similarly exchange-dominated bottlenecks.","The cubic complete-sRI route might overtake the quartic partial route only for systems substantially larger than the tested ~200-electron range; the paper's error data does not establish where that crossover lies.","A direct testable extension is to apply the partial-sRI derivative-coupling code at a near-degenerate pair of states, since conical intersections are the stated target and the noise behavior there is not yet benchmarked."],"forward_implications":["CC2 excited-state gradients and derivative couplings become affordable for molecules with hundreds of electrons, with the empirical crossover from RI-CC2 occurring around 400 electrons.","A fixed number of stochastic orbitals (Ns=100) suffices regardless of system size, so the stochastic error does not grow with molecular size.","The complete-sRI variant, though cubic in scaling, remains impractical in the tested size range because it demands roughly 50,000 stochastic orbitals to control noise.","The sRI-CC2 machinery is positioned as a step toward large-scale nonadiabatic dynamics, with the stated next step being similarity-constrained CC2 for conical intersections."],"supporting_citations":[{"why":"Introduces the stochastic resolution of identity formalism used throughout this work.","marker":"[34]"},{"why":"Establishes the sRI-CC2 ground-state and triplet excitation energy implementation that this work extends.","marker":"[42]"},{"why":"Provides the prior sRI-CC2 excited-state property framework being extended to gradients and derivative couplings.","marker":"[43]"},{"why":"Shows the earlier sRI-CC2 oscillator-strength and ground-state gradient implementation, motivating the selective-sRI choice for gradient-like properties.","marker":"[44]"},{"why":"Supplies the RI-CC2 analytic excited-state gradient formulation on which the Lagrangian and density-matrix derivations are based.","marker":"[45]"},{"why":"Supplies the coupled-cluster derivative-coupling formalism that the present implementation adapts.","marker":"[52]"},{"why":"Provides the Lagrange-multiplier route for CC2 excited-state first-order properties used to derive the gradient and derivative-coupling equations.","marker":"[79]"}],"fun_headline_variants":["Partial sRI-CC2 cuts excited-state gradients and couplings to N^4","N^4 CC2 excited-state gradients and couplings via partial sRI","Partial stochastic RI makes CC2 gradients and couplings N^4","Quartic scaling for CC2 excited-state gradients and couplings via sRI","Partial sRI-CC2: N^4 gradients and derivative couplings"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The derivative-coupling reference itself is not validated against an independent CC2 code, so the sRI-CC2 comparisons inherit any error in the RI-CC2 derivative-coupling implementation.","fun_headline_variants_meta":{"raw":{"variants":["Partial sRI-CC2 cuts excited-state gradients and couplings to N^4","N^4 CC2 excited-state gradients and couplings via partial sRI","Partial stochastic RI makes CC2 gradients and couplings N^4","Quartic scaling for CC2 excited-state gradients and couplings via sRI","Partial sRI-CC2: N^4 gradients and derivative couplings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00111,"raw_usage":{"total_tokens":4454,"prompt_tokens":727,"completion_tokens":3727,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":3631}},"tokens_in":471,"tokens_out":3727,"duration_ms":25343,"temperature":1.0,"reasoning_tokens":3631,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:34:16.102004+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the reported RI-CC2 water F21 derivative-coupling vector against an independent, released CC2 or similarity-constrained CC2 implementation; if the RI-CC2 value differs by more than the claimed sRI error, the stochastic benchmark baseline is wrong. The paper's own hybrid CCS-to-CC2 calculation reproduced the CIS result for F21 while the full CC2 did not, so resolving whether the CC2 amplitude or energy causes that discrepancy would settle whether the large F21 difference is in the method or the implementation.","supporting_citations":[{"cited_title":"Stochastic formulation of the resolution of identity: Application to second order møller–plesset perturbation theory,","cited_arxiv_id":null,"evidence_quote":"Introduces the stochastic resolution of identity formalism used throughout this work."},{"cited_title":"Stochastic resolution of identity to cc2 for large systems: Ground state and triplet excitation energy calculations,","cited_arxiv_id":null,"evidence_quote":"Establishes the sRI-CC2 ground-state and triplet excitation energy implementation that this work extends."},{"cited_title":"Stochastic resolution of identity to cc2 for large systems: excited state properties,","cited_arxiv_id":null,"evidence_quote":"Provides the prior sRI-CC2 excited-state property framework being extended to gradients and derivative couplings."},{"cited_title":"Stochastic resolution of identity to cc2 for large systems: Oscillator strength and ground state gradient calculations,","cited_arxiv_id":null,"evidence_quote":"Shows the earlier sRI-CC2 oscillator-strength and ground-state gradient implementation, motivating the selective-sRI choice for gradient-like properties."},{"cited_title":"Analytic gradients for excited states in the coupled-cluster model cc2 employing the resolution-of-the-identity approx- imation,","cited_arxiv_id":null,"evidence_quote":"Supplies the RI-CC2 analytic excited-state gradient formulation on which the Lagrangian and density-matrix derivations are based."},{"cited_title":"Communication: Non-adiabatic derivative coupling elements for the coupled cluster singles and doubles model,","cited_arxiv_id":null,"evidence_quote":"Supplies the coupled-cluster derivative-coupling formalism that the present implementation adapts."},{"cited_title":"Transition moments and excited-state first-order properties in the coupled-cluster model cc2 using the resolution-of-the- identity approximation,","cited_arxiv_id":null,"evidence_quote":"Provides the Lagrange-multiplier route for CC2 excited-state first-order properties used to derive the gradient and derivative-coupling equations."}],"review_version":1}