{"id":"ee7c0d5d-fee2-4b90-a578-9055027c514d","arxiv_id":"2411.16599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"ppRPA with hybrid functionals containing 10-20% exact exchange predicts double excitation energies with about 0.35-0.40 eV mean absolute error, comparable to CCSDT and CASPT2.","lead":"This paper benchmarks the particle-particle random phase approximation (ppRPA) for computing double excitation energies in 21 molecules and two diamond defects. It finds that ppRPA with hybrid density functionals containing about 10-20% exact exchange reaches average errors near 0.35-0.40 eV, similar to costly wavefunction methods but at much lower cost.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table I MAEs are computed on different per-functional subsets because of convergence N/A entries, so the 0.35–0.40 eV accuracy claim and the CCSDT/CASPT2 comparison rest on an incomplete and non-random dataset.","rationale":"The paper's central quantitative claim is the MAE table. The table is the only evidence for the 0.35–0.40 eV accuracy figure, and it is built from a per-functional set of converged calculations. A reader cannot tell whether the differences between functionals reflect method quality or simply which calculations failed. For example, the largest errors in the converged columns appear for exactly the kind of multireference cases that fail to converge in other columns, so the missing data are likely informative. The TBE question raised in the reader's weakest-assumption is real but less decisive: Ref. 13 is a peer-reviewed benchmark with high-level references, and any bias in the TBEs would affect ppRPA and the WFT methods similarly. The subset inconsistency, by contrast, directly invalidates the quantitative comparison as presented. The fix is straightforward: report MAEs on a common subset and give the WFT numbers for the same states. This is a request for additional analysis rather than a demonstration that the method fails, so the conditional verdict is unchanged.","tokens_in":114,"tokens_out":7641,"duration_ms":352427,"concrete_test":"Recompute all MAE and MSE values using only the subset of states for which every functional of interest has a converged value, and preferably the subset common to all eight functionals. In addition, take the CCSDT, CASPT2, CC3, and SA-CASSCF energies for those same states from Ref. 13 (or recompute them) and report the MAEs side-by-side with ppRPA on that common subset. If the best ppRPA MAE on the common subset rises above about 0.5 eV, or if CC3 or SA-CASSCF outperform ppRPA on the same states, the abstract claim of parity with CCSDT and CASPT2 should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Each MAE in Table I is over a different subset of the 26 states, because many (N−2)-electron SCF calculations fail to converge (N/A entries). This matters because the missing states are not random. For cyclopentadienethione, the three functionals that do converge all give errors near −1.06 eV (M06-2X: 4.269 vs TBE 5.329; CAM-B3LYP: 4.310; ωB97X-D: 4.206), yet B3LYP has N/A for this state. TPSSh, which yields the lowest MAE (0.350 eV), has N/A for at least acrolein, benzoquinone, cyclopentadienethione, cyclopentadienone (both states), diazete, and pyrazine (both states). If those states have similarly large errors for TPSSh, its MAE would increase materially. The subsequent comparison with WFT methods quotes global MAEs from Ref. 13 (SA-CASSCF 0.48 eV, CC3 0.56 eV) without restricting them to the same subset, and the claimed parity with CCSDT/CASPT2 is not documented with any numerical table. The headline conclusion that ppRPA with 10–20% exact exchange achieves 0.35–0.40 eV accuracy therefore depends on an apples-to-oranges dataset.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript benchmarks the particle-particle random phase approximation (ppRPA) for computing vertical double excitation energies in 21 molecular systems and two point defects in diamond, using eight density functional approximations as starting points. The authors report mean absolute errors (MAEs) of about 0.35–0.40 eV for functionals with 10–20% exact exchange, claim accuracy comparable to high-level wave function methods such as CCSDT and CASPT2, demonstrate a ΔSCF-based ppRPA calculation for the acrolein double excitation, and apply ppRPA to the NV− and VC defects in bulk periodic systems. The central claim is that ppRPA is an accurate and efficient alternative for double excitations in both molecular and periodic systems.","tokens_in":15719,"tokens_out":3901,"duration_ms":38298,"significance":"If the accuracy claims hold, the paper would provide a practical low-scaling method for double excitations that is competitive with much more expensive wave function methods. The work also extends ppRPA to a new starting-point construction (ΔSCF) and to periodic defect systems, which are of genuine interest in materials chemistry. A strength is that the benchmark uses the well-established theoretical best estimates (TBEs) from Ref. 13 and does not fit any parameters to target energies. However, the central accuracy comparison is weakened by missing data and by an incomplete documentation of the wave function reference values, so the paper's quantitative conclusions are not fully supported as written.","major_comments":[{"comment":"The MAEs and MSEs in Table I are computed over different subsets of the 26 molecular states because many (N−2)-electron SCF calculations failed to converge (N/A entries). For example, TPSSh, which gives the lowest MAE (0.350 eV), is missing at least acrolein, benzoquinone, cyclopentadienethione, both cyclopentadienone states, diazete, and both pyrazine states. If these missing states have large errors analogous to those seen for other functionals (e.g., cyclopentadienethione errors near −1 eV), the reported MAE could increase substantially. The authors should report the number of states included for each functional, recompute MAEs on the common subset of states for all functionals, and state explicitly how the comparison with reference wave function methods is affected by the differing subsets.","section":"Table I and Section IV A"},{"comment":"The text states that ppRPA 'provides similar accuracy as CCSDT and CASPT2' and that ppRPA is more accurate than SA-CASSCF and CC3, but the only numerical MAE values given for comparison are 0.48 eV (SA-CASSCF) and 0.56 eV (CC3), quoted globally from Ref. 13. No MAE values for CCSDT or CASPT2 are provided, and the global values from Ref. 13 are not restricted to the states for which ppRPA results are available. The claim of parity with CCSDT/CASPT2 therefore cannot be verified. The authors should tabulate the MAEs of CCSDT, CASPT2, and the other WFT methods on the same subset of states used for the ppRPA MAEs, and, if possible, provide per-state errors to enable a rigorous comparison.","section":"Section IV A, comparison with WFT methods"},{"comment":"The defect excitation energies are obtained using a two-point supercell-size linear extrapolation E(1/N) = E∞ + a/N, but the manuscript reports only the extrapolated values without error bars or the individual supercell data. Since the experimental ranges are narrow (e.g., 1.76–1.85 eV for NV−), the spread of 1.67–2.09 eV across functionals is material to the claim of small starting-point dependence. The authors should provide the raw supercell results, the fitted values, and an estimate of the extrapolation uncertainty, or at least include the two-point data in the Supporting Information.","section":"Section IV B and Table III"}],"minor_comments":[{"comment":"The statement that ppRPA with 10–20% exact exchange 'can provide more accurate double excitation energies compared to GGAs and meta-GGAs' is not strongly supported by Table I: the MAE for PBE (0.378 eV) is close to that of B3LYP (0.393 eV) and TPSSh (0.361 eV), and the differences may be within statistical noise given the differing subsets. The authors should soften this statement or support it with a statistical analysis.","section":"Section IV A, paragraph on functional trends"},{"comment":"The discussion of the acrolein ΔSCF calculation could be clearer. Figure 1 shows qualitative energy levels, but the text does not explain how the maximum overlap method (MOM) was used to converge the (N−2)-electron state, nor how the resulting ppRPA calculation differs from the standard ground-state-based calculation. A brief description of the MOM procedure and the convergence criteria would help readers reproduce the result.","section":"Section IV A and Figure 1"},{"comment":"There are minor typographical errors, including 'results form ppRPA' (should be 'from') in Section IV A, and 'nitrosomethan' in Table I (should be 'nitrosomethane'). The abbreviation 'DFA' is used without definition; it should be defined at first use (e.g., density functional approximation).","section":"Throughout"},{"comment":"In Table III, the column header 'dominant configuration in 1A1' applies to the ground state for VC and to the excited state for NV−; this is confusing because for NV− the 1A1 is the excited state of interest, while for VC the 1A1 is the ground state. The authors should clarify the row/state labels to avoid ambiguity.","section":"Table III"}],"recommendation":"major_revision","confidential_remarks":"The paper builds on a substantial body of work from the same groups (ppRPA developments; Refs. 60–65, 72–76, 79–81, 87), and the manuscript is somewhat self-referential in its claims of novelty. This is not by itself disqualifying, but the benchmark's incompleteness and the missing WFT reference numbers are the key technical concerns. If the authors can provide a complete dataset (or a well-defined common subset) and a rigorous comparison with CCSDT/CASPT2 on that subset, the paper would be a solid contribution. Otherwise, the headline accuracy claim remains unverified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what the title says: it benchmarks ppRPA on a 21-molecule double-excitation set across eight functionals, reports MAEs around 0.35–0.40 eV for the best ones, and adds two genuinely new things—the first ppRPA run from a Delta-SCF excited (N−2)-electron determinant, and applications to periodic point defects. The benchmark against the Kossoski TBEs is careful, the NTO analysis is a nice check on the character of the states, and the defect calculations extend the method to the bulk. None of it is circular: no target energies are used to fit parameters, and the method itself is grounded in prior ppRPA theory. That part deserves credit.\n\nThe soft spot is real and it is in the headline claim. Table I has a lot of N/A entries because many (N−2) SCF calculations failed to converge. The MAEs are computed over different subsets for each functional, and the missing states are not random. TPSSh, which gives the lowest MAE (0.350 eV), has N/A for acrolein, benzoquinone, cyclopentadienethione, cyclopentadienone (both states), diazete, and pyrazine (both states). If those states are systematically harder and have larger errors, the ordering among functionals and the absolute MAE could shift materially. The comparison to WFT is also under-documented: the paper quotes SA-CASSCF (0.48 eV) and CC3 (0.56 eV) from Ref. 13 but does not restrict those to the same subset, and the claimed parity with CCSDT/CASPT2 is asserted without a numerical table. For a reader, the 0.35–0.40 eV number is the take-home, so it needs to rest on comparable data.\n\nThe defect section is a useful demonstration but sketchy: no active-space sizes, no supercell sizes, no error bars on the two-point extrapolation. That limits how much weight one can put on the defect numbers, though the errors around 0.1–0.2 eV for B3LYP/HSE03 on the VC 1E state are plausible.\n\nWho is this for? People working on double excitations in photochemistry or defect physics who want a cheaper alternative to CASPT2/CCSDT. It is a solid within-subfield contribution, not a paradigm shift, and the central idea—that ppRPA with modest exact exchange gets double excitations right—is supported in spirit even if the exact MAE is fragile.\n\nRecommendation: send it to peer review. The missing-data problem is fixable with a proper common-subset analysis or a discussion of why the N/A states fail, and the WFT comparison needs its own table. The paper is worth the referee time.","headline":"A useful, honest benchmark of ppRPA for double excitations, with a real but fixable apples-to-oranges problem in the headline MAEs.","tokens_in":16313,"tokens_out":1404,"would_cite":true,"duration_ms":14768,"reading_group":"yes","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 shows that ppRPA with about 10–20% exact exchange predicts double excitation energies with roughly 0.4 eV mean absolute error, comparable to CCSDT and CASPT2, at much lower cost.","keywords":["particle-particle random phase approximation","double excitations","excitation energies","density functional theory","exact exchange","point defects","delta-SCF","benchmark"],"falsifier":"Recompute the 21-molecule benchmark against full configuration interaction or EOM-CC4 reference energies for a subset of the molecules; if ppRPA's mean absolute error with TPSSh or B3LYP rises above about 0.5 eV while the reported CCSDT and CASPT2 errors stay at their benchmark values, the claim of comparable accuracy would be refuted.","tokens_in":15231,"feed_emoji":"⚛️","tokens_out":10087,"duration_ms":85691,"temperature":0.7,"pith_summary":"The paper argues that double excitation energies, which standard time-dependent DFT misses and high-level wave function methods compute only at steep cost, can be obtained cheaply from the particle-particle random phase approximation (ppRPA). Across a 21-molecule double-excitation benchmark, ppRPA built on Kohn-Sham orbitals from functionals with about 10–20% exact exchange reaches mean absolute errors near 0.35–0.40 eV, an accuracy the paper places alongside CCSDT and CASPT2. It also shows ppRPA describing double-excitation-character states in periodic point defects, and introduces a $\\Delta$SCF-based starting point that lets the $(N-2)$-electron reference be an excited determinant when orbital ordering shifts, as in acrolein. If these results hold, ppRPA becomes a practical alternative for studying dark, doubly excited states in molecules and solids.","feed_headline":"ppRPA matches wave-function accuracy for double excitations","feed_subtitle":"Cheap DFT-based method reaches CCSDT/CASPT2-level errors on 21 molecules and diamond defects.","key_machinery":"The machinery is the pairing-matrix fluctuation propagator $K_{pqrs}(\\omega)$, the dynamic fluctuation of $\\langle\\Psi_0^N|\\hat a_p \\hat a_q|\\Psi_0^N\\rangle$, which obeys a Dyson-like equation $K = K^0 + K^0 V K$ and becomes a generalized eigenvalue problem with matrix blocks $A_{ab,cd}=\\delta_{ac}\\delta_{bd}(\\epsilon_a+\\epsilon_b)+\\langle ab||cd\\rangle$ and analogous $B$ and $C$ blocks. The eigenvalues are two-electron addition or removal energies of the $(N\\pm2)$-electron system, and the difference between two such eigenvalues is the $N$-electron double excitation energy. This object carries the argument because it treats the two added or removed electrons in a subspace-configuration-interaction manner while the remaining electrons are described by DFT; active-space truncation and density fitting keep the computational cost at $O(N_{\\rm act}^4)$.","core_discovery":"The central discovery is that the ppRPA eigenvalue problem, solved on top of an $(N-2)$-electron Kohn-Sham reference, yields vertical double excitation energies of the $N$-electron system whose average error can be as low as 0.350 eV (TPSSh) and around 0.39–0.43 eV for B3LYP and HSE03, close to the reported errors of CCSDT and CASPT2 and better than SA-CASSCF and CC3 on the same benchmark. The accuracy holds for both genuine and partial double excitations, and for the two periodic defects tested, NV$^-$ and VC in diamond, ppRPA reproduces the measured $1A_1$ and $1E$ energies with errors much smaller than TD-DFT. The paper also demonstrates that when removing two electrons changes the orbital order, the $(N-2)$-electron calculation can be seeded from a $\\Delta$SCF excited determinant, extending ppRPA to references that are not ground states.","pith_inferences":["An implication the paper leaves implicit is that the 10–20% exact-exchange window might transfer to other density-based two-particle methods, so the optimal fraction could be tuned without re-benchmarking 21 molecules; testing ppRPA on a second benchmark set would reveal whether the window is universal.","A testable extension is to start from $\\Delta$SCF $(N-2)$-electron states beyond the acrolein example, for instance orbitals from a different occupied manifold, which would probe whether the method's accuracy survives when the reference determinant is not the ground state.","A caution implied by the benchmark design is that the 21-molecule and two-defect sample is small, so the reported mean absolute error is a small-sample estimate; a broader test set including larger conjugated systems would show whether the 0.35–0.40 eV floor persists.","If the accuracy claim holds, ppRPA becomes a practical screening tool for double-excitation spectroscopy in extended $\\pi$-systems and defect qubits where CASPT2-scale costs are prohibitive; this is a forward-looking consequence the paper points to but does not itself demonstrate."],"forward_implications":["ppRPA with hybrid functionals in the 10–20% exact-exchange range gives double-excitation mean absolute errors of about 0.35–0.40 eV on the 21-molecule benchmark, close to CCSDT and CASPT2 rather than to SA-CASSCF and CC3.","The method also produces double-excitation energies for bulk point defects, including the $1A_1$ state of NV$^-$ and the $1E$ state of VC in diamond, with errors around 0.1–0.5 eV, smaller than TD-DFT's roughly 1 eV error.","Starting the $(N-2)$-electron SCF from a non-ground-state determinant, justified by $\\Delta$SCF, lets ppRPA handle cases where simple removal from the HOMO would misalign orbitals, as in the acrolein $1A'$ state when using B3LYP.","Active-space truncation and density fitting keep the cost at $O(N_{\\rm act}^4)$ with iterative diagonalization, making the approach feasible for systems where CCSDT and CASPT2 are prohibitively expensive.","ppRPA treats genuine and partial double excitations with comparable accuracy, reflecting its two-electron-in-CI treatment of the particle-particle channel."],"supporting_citations":[{"why":"It supplies the theoretical best estimates and the classification of genuine vs partial double excitations that Table I benchmarks against.","marker":"Ref. 13"},{"why":"It introduces the pairing-matrix fluctuation formalism from which the ppRPA eigenvalue equations used here are derived.","marker":"Ref. 60"},{"why":"It demonstrated ppRPA double excitation energies for small molecules with roughly 0.5 eV errors, the baseline this work improves and extends.","marker":"Ref. 62"},{"why":"It supplies the fast ppRPA point-defect methodology and the two-point supercell extrapolation used for NV$^-$ and VC.","marker":"Ref. 63"},{"why":"It provides the point-defect geometries, the correlated excited-state ppRPA treatment, and the natural-transition-orbital analysis used in this work.","marker":"Ref. 64"},{"why":"It supplies the iterative diagonalization algorithm and benchmark that make full-space ppRPA calculations feasible for the 21 molecules.","marker":"Ref. 79"},{"why":"It provides the active-space truncation with $O(N_{\\rm act}^4)$ scaling used for the periodic defect calculations.","marker":"Ref. 81"},{"why":"It gives the theoretical foundation for $\\Delta$SCF that justifies starting ppRPA from an excited $(N-2)$-electron determinant in the acrolein case.","marker":"Ref. 87"}],"fun_headline_variants":["ppRPA matches CCSDT and CASPT2 accuracy for double excitations","Cheap ppRPA predicts double excitations accurately in solids and molecules","Double excitations: ppRPA is a fast alternative to high-level methods","ppRPA from ΔSCF references yields accurate double excitation energies","ppRPA: accurate and efficient for double excitations in bulk and molecules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The accuracy claims for molecules rest on the assumption that the reference double-excitation energies from the benchmark of Ref. 13 are the true values; if those references are systematically wrong, the reported mean absolute errors, and the comparison to CCSDT and CASPT2, lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["ppRPA matches CCSDT and CASPT2 accuracy for double excitations","Cheap ppRPA predicts double excitations accurately in solids and molecules","Double excitations: ppRPA is a fast alternative to high-level methods","ppRPA from ΔSCF references yields accurate double excitation energies","ppRPA: accurate and efficient for double excitations in bulk and molecules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000379,"raw_usage":{"total_tokens":2002,"prompt_tokens":923,"completion_tokens":1079,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":991}},"tokens_in":539,"tokens_out":1079,"duration_ms":9855,"temperature":1.0,"reasoning_tokens":991,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:56:02.308365+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the 21-molecule benchmark against full configuration interaction or EOM-CC4 reference energies for a subset of the molecules; if ppRPA's mean absolute error with TPSSh or B3LYP rises above about 0.5 eV while the reported CCSDT and CASPT2 errors stay at their benchmark values, the claim of comparable accuracy would be refuted.","supporting_citations":[],"review_version":1}