{"id":"236e4b4b-ecb3-4e0c-915c-37f8393c323e","arxiv_id":"2608.08784","paper_version":1,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors build a self-consistent double-hybrid density functional by inserting a one-body MP2 correlation potential into the generalized Kohn-Sham Hamiltonian, and report improved potential energy curves, self-interaction errors, and charged noncovalent interactions.","lead":"Electrons in molecules feel each other through correlated motion. This paper puts MP2-level correlation directly into the orbital equations of density functional theory, creating a self-consistent double-hybrid method, and tests it on several chemistry benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing gap is that Eq. (8) treats vOBMP2 as the orbital derivative of EOBMP2_c, but neither EOBMP2_c nor the total OBDH energy is defined or differentiated anywhere; without that identity Eq. (9) is not variational.","rationale":"I read the paper as proposing a new self-consistent double-hybrid scheme whose core theoretical step is the placement of the OBMP2 one-body potential inside the GKS Hamiltonian. For that placement to be more than a heuristic Fock operator, the potential must be the orbital derivative of the correlation energy that appears in Eq. (7). The preprint does not provide this derivation, does not define EOBMP2_c, and does not state the total OBDH energy. The numerical results, including H2 and LiH dissociation, He2+ and Be2 PECs, the SIE4x4 MAD of 17.9 kcal/mol, and the embedding convergence, are coherent and suggestive, but they all assume the energy whose curves are plotted is well-defined. If vOBMP2 is not a functional derivative, the method reduces to a non-variational self-consistent MP2-like Fock approximation and the claimed variational interpretation collapses. I do not assert the identity is false; I assert it is load-bearing and unproven. The paper cites Refs. 44-45 for vOBMP2 but does not reproduce the derivation or the energy expression, and no machine-checked proof, code, or raw data are provided. The two fitted parameters and the absence of baseline double-hybrid comparisons are secondary concerns; the missing derivative identity is the single most load-bearing issue because it determines whether the method is what the title and abstract claim. A direct analytic or numerical check of δE_c/δφ versus vOBMP2 would settle the matter. Until then, rejection with possibility of conditional acceptance after the derivation is appropriate.","tokens_in":13280,"tokens_out":5705,"duration_ms":59710,"concrete_test":"Implement the OBMP2 correlation energy EOBMP2_c from the amplitudes of Eq. (6) as defined in Ref. 44, and at the converged OBDH solution for H2/cc-pVDZ at a stretched bond length compute the orbital-gradient vector ∂E_c/∂κ_ai by finite difference over orbital-rotation parameters; compare numerically with ⟨a|vOBMP2|i⟩. If they agree to ~1e-6, the derivative identity holds and the variational claim is supported. As a second check, write down the total OBDH energy expression including the αc EOBMP2_c term and verify that the SCF stationary point is a minimum along the H2 dissociation coordinate and that the resulting curve matches Fig. 2a; if no energy expression can be written, the reported PEC is undefined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, that SCF convergence of Eq. (9) yields orbitals optimized in the presence of MP2 correlation, rests on the unstated identity δEOBMP2_c[{φ}]/δφ_i*(r) = vOBMP2 φ_i(r), where vOBMP2 is the one-body operator imported from Refs. 44-45. The paper never writes EOBMP2_c explicitly, never gives the total OBDH energy expression whose dissociation curves appear in Fig. 2, and never derives the stationarity condition. Eq. (7) defines S symbolically with an EOBMP2_c term, and Eq. (8) is said to follow from 'the GKS procedure,' but GKS only justifies an effective Hamiltonian after the orbital derivative of the model functional is known. In OBMP2, v is obtained through canonical transformation plus cumulant reduction; its relation to a differentiable energy functional is nontrivial and cannot be assumed merely because v is one-body. If the identity is false, the converged orbitals are not the minimum of the stated S, the method is a non-variational self-consistent Fock approximation, and the energies plotted for OBDH and sub-OBDH are not defined as functionals of the orbitals. The paper's own emphasis that no response equations or orbital gradients are used highlights exactly why this derivative relation is non-obvious. Without the derivation, Eq. (9) is an assertion, not a variational GKS equation, so the improvements in dissociation, SIE, and Be2 binding are not attributable to a well-defined self-consistent double-hybrid energy.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a self-consistent double-hybrid density functional, OBDH, in which the one-body OBMP2 correlation potential is inserted directly into a generalized Kohn-Sham effective Hamiltonian. A computationally reduced variant, sub-OBDH, combines OBDH with projector-based embedding and concentric localization truncation of the virtual space. The authors benchmark OBDH on diatomic potential energy curves (H2, LiH, He2+, Be2), on the SIE4x4 self-interaction-error dataset, and sub-OBDH on embedding convergence, torsional profiles of ethanol/1-butanol, and noncovalent interactions in charged Mg2+ complexes. The central claim is that SCF convergence of Eq. (9) variationally optimizes orbitals in the presence of MP2-level correlation without the optimized effective potential or response equations.","tokens_in":13541,"tokens_out":7612,"duration_ms":85586,"significance":"If the variational foundation were established, OBDH would be a practically important step: a self-consistent double-hybrid at second-order cost that avoids the OEP, with promising embedding behavior and competitive errors on charged noncovalent interactions. The numerical results are suggestive, especially the stable sub-OBDH errors below about 1.1 kcal/mol on the Mg2+ clusters. However, the theoretical core of the paper is currently incomplete: the model functional as written does not generate the stated effective Hamiltonian, and the key identification of vOBMP2 with an orbital derivative of a correlation energy is asserted rather than derived. The benchmarks are therefore not yet attributable to a well-defined self-consistent double-hybrid energy functional. The manuscript also does not include the raw data or a complete specification of the energy expression, which limits reproducibility.","major_comments":[{"comment":"The central theoretical step is not demonstrated: Eq. (8) treats vOBMP2 as if it were the orbital derivative of EOBMP2_c, but EOBMP2_c[{φ}] is never defined and its functional derivative is never computed. In the GKS framework, the one-body operator in the orbital equation is δS/δφ_i*(r); for a correlation functional built from MP2 amplitudes, that derivative contains response terms of the amplitudes and orbital eigenvalues. The text instead imports vOBMP2 from Refs. 44-45 and explicitly states that no response equations are used. The identity δEOBMP2_c/δφ_i* = vOBMP2 φ_i is therefore an additional postulate, not a consequence of the GKS procedure. Without this identity, Eq. (9) is not the stationarity condition of the functional S in Eq. (7), and the variational claims in the abstract and conclusion are unsupported. The authors must define EOBMP2_c, state the total OBDH energy expression, and derive the orbital derivative or prove that vOBMP2 is exactly that derivative.","section":"Theory, Eqs. (7)-(9)"},{"comment":"Equation (7) is inconsistent with Eq. (8). The text says S includes the Hartree energy, but Eq. (7) as written contains only Ts, αx EHF_x, and αc EOBMP2_c, with no Hartree term and no semilocal exchange-correlation term. Equation (8), in contrast, contains V_H(r) and V_DFA_xc(r) with the scalings (1-αx) and (1-αc). Consequently, Eq. (8) is not obtained by orbital differentiation of Eq. (7) as written, and the claim that the effective Hamiltonian is systematically derived from this S cannot be checked. The model functional must be written completely, including the Hartree energy and the semilocal exchange-correlation energy, before the GKS orbital equation can be derived.","section":"Theory, Eq. (7) and Eq. (8)"},{"comment":"The potential energy curves labeled OBDH in Fig. 2 and the SIE4x4 reaction energies in Table I are not tied to any stated total energy expression. Because neither EOBMP2_c nor the total OBDH energy is defined, the reader cannot reproduce the plotted curves or verify that they correspond to a converged solution of Eq. (9). The same issue affects the dissociation energies: without an energy expression, the claim that fully self-consistent OBDH 'describes dissociation properly' is not a well-defined statement about a functional. A complete specification of the energy expression used in every reported number is required.","section":"Results, Fig. 2 and Table I"}],"minor_comments":[{"comment":"The fitting procedure for αx and αc should be fully specified: which IP subset of GMTKN55 was used, how many molecules, what basis set, and what reference values. The current caption is too terse for reproducibility.","section":"Results, Fig. 1"},{"comment":"Figure 3 is referenced in the text, but the plot is not present in the version under review; the raw data are said to be in the Supporting Information, which is also not included. Please provide both.","section":"Results, Fig. 3"},{"comment":"The axis labels in Fig. 4 appear corrupted in the manuscript text (e.g., strings of digits without clear tick labels), likely a typesetting problem; please regenerate the figure.","section":"Results, Fig. 4"},{"comment":"For Be2, the paper states that the OBDH curve is closer to experiment than B3LYP or CCSD(T), but no quantitative well depth or equilibrium distance is reported; please include these values.","section":"Results, Be2 discussion"},{"comment":"The phrase 'OBDH consistently outperforms standard DFT' is broader than the evidence: the paper reports only PECs, SIE4x4, embedding convergence, and a small set of charged noncovalent complexes, with no thermochemistry or kinetics benchmarks.","section":"Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the missing derivation of the functional derivative and the total energy. If the authors can supply a rigorous derivation of δEOBMP2_c/δφ_i* and show that it equals vOBMP2 (or correct the method to a non-variational statement), the paper could be a valuable contribution. The other inconsistencies, especially Eq. (7) vs. Eq. (8), must also be fixed. I found no evidence of misconduct; the concern is purely about the validity of the central theoretical claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look. The idea is to put the OBMP2 one-body potential directly into a GKS effective Hamiltonian and call the resulting SCF process a self-consistent double-hybrid. That is genuinely new: no OEP, no perturbative orbital relaxation. The numerical figures are encouraging—the H2/LiH dissociation curves behave, SIE4x4 MAD is clearly lower than PBE/PBE0/B3LYP, and the sub-OBDH embedding converges fast. Give credit where due.\n\nBut the central theoretical step is missing. The paper defines S with an EOBMP2_c term, then asserts the GKS Hamiltonian is ... with vOBMP2. To be a variational GKS scheme you need to show that vOBMP2 is (or yields) the functional derivative of EOBMP2_c with respect to the orbitals. The paper doesn't write EOBMP2_c explicitly, doesn't write the total OBDH energy, and doesn't derive the stationarity condition. The text says the operator is imported from prior work, and claims no response equations are needed. That is precisely why the derivative relation is not obvious. If it's false, the method is a non-variational Fock-like approximation and the plotted 'energies' are not well-defined. This is a load-bearing gap, exactly as the stress-test note says.\n\nSmaller issues: the two mixing parameters are fitted to an IP subset of GMTKN55, which is fine but should be labeled as a training set. No code or raw data are provided. And the benchmarks compare to standard DFT and MP2 but not to existing double-hybrids like B2PLYP or oo-PBE-QIDH, so the practical claim about double-hybrid accuracy is undersupported.\n\nThis is not a desk-reject. It is a credible new combination with at least some encouraging evidence, and the missing derivation is probably available to the authors, since OBMP2 is their own prior work. As a referee I'd want the derivation and the total energy expression before accepting. Send it to peer review, and if the authors can supply those missing pieces, the paper could become a solid contribution.","headline":"Plausible and useful new combination, but the central variational step is asserted rather than derived; the paper needs that derivation before the results can be trusted.","tokens_in":14169,"tokens_out":2583,"would_cite":false,"duration_ms":29018,"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":"Self-consistent double-hybrid DFT fixes MP2 dissociation failure.","keywords":["double-hybrid density functional theory","one-body second-order Møller-Plesset perturbation theory","generalized Kohn-Sham","self-consistent orbitals","projector-based embedding","concentric localization","self-interaction error","potential energy curves"],"falsifier":"Compute the OBDH total energy at the converged solution and at orbitals rotated by a small unitary transformation; if the energy changes to first order under the rotation, the assumed functional-derivative relation is false and OBDH collapses to a non-variational self-consistent Fock approximation. A numerical finite-difference derivative of the OBMP2 correlation energy against orbital rotations would give the same verdict directly.","tokens_in":12955,"feed_emoji":"⚛️","tokens_out":7732,"duration_ms":67270,"temperature":0.7,"pith_summary":"The paper introduces OBDH, a double-hybrid density functional that makes MP2-level correlation self-consistent by placing the one-body second-order Møller-Plesset (OBMP2) correlation potential $\\hat{v}^{\\mathrm{OBMP2}}$ directly inside the generalized Kohn-Sham Hamiltonian. Conventional double hybrids compute a perturbative MP2 correction on top of fixed DFT orbitals, so the orbitals are not optimized for the final energy expression. OBDH instead iterates the effective Hamiltonian until convergence, giving orbitals variationally relaxed in the presence of both exact exchange and MP2 correlation without an optimized effective potential or response equations. The authors benchmark OBDH and its embedding-based variant sub-OBDH on diatomic potential energy curves, self-interaction errors, torsional barriers, and charged non-covalent complexes, reporting that OBDH removes the unphysical dissociation divergence of MP2 and non-iterative double hybrids and reduces the SIE4x4 mean absolute deviation to 17.9 kcal/mol. If the method holds up, it offers a practical route to self-consistent double hybrids at MP2 cost.","feed_headline":"Self-consistent double-hybrid DFT fixes MP2 dissociation failure","feed_subtitle":"OBDH optimizes orbitals with MP2 correlation, beating PBE and B3LYP on SIE and bond-breaking curves.","key_machinery":"The central object is the OBMP2 one-body correlation potential $\\hat{v}^{\\mathrm{OBMP2}}$, a correlated Fock-like operator built from a unitary canonical transformation of the molecular Hamiltonian followed by a cumulant approximation that keeps only one-body terms. In OBDH it enters the GKS Hamiltonian with mixing coefficients $\\alpha_x$ and $\\alpha_c$, alongside semilocal exchange-correlation, Hartree, and exact-exchange terms. The same operator structure lets the correlation part be embedded through projector-based embedding and concentric localization in sub-OBDH, where SPADE partitions the occupied space and CL truncation compresses the virtual space.","core_discovery":"The central claim is that inserting the OBMP2 one-body correlation operator $\\hat{v}^{\\mathrm{OBMP2}}$ into the generalized Kohn-Sham effective Hamiltonian yields a self-consistent double-hybrid method, avoiding the optimized effective potential, response equations, and perturbative orbital relaxation. The SCF equation $\\hat{H}_{\\mathrm{eff}}^{\\mathrm{OBDH}} \\phi_i = \\varepsilon_i \\phi_i$ is solved with amplitudes and the correlation potential updated from the current orbitals and eigenvalues at each cycle, so the converged orbitals are 'optimized in the presence of both exact exchange and MP2-level dynamic correlation.' The paper reports accurate potential energy curves for H$_2$, LiH, He$_2^+$, and Be$_2$, a SIE4x4 MAD of 17.9 kcal/mol that beats PBE, PBE0, and B3LYP, and faster convergence of the embedded sub-OBDH variant with respect to localization truncation than MP2-in-DFT.","pith_inferences":["If a standalone total-energy expression for OBDH is provided and shown to be stationary at the SCF solution, the method would support analytic gradients and response properties, which standard double hybrids lack.","The parameter choice $\\alpha_x=0.5$, $\\alpha_c=0.4$ is fitted to ionization potentials in GMTKN55; broad thermochemical benchmarks would test whether the improvements transfer to barriers, thermochemistry, and non-covalent neutral complexes.","A practical test of the functional-derivative assumption would be numerical differentiation of $E_c^{\\mathrm{OBMP2}}$ with respect to orbital rotations and comparison with $\\hat{v}^{\\mathrm{OBMP2}}$; the paper currently does not report this check.","The charged NCI results hint at applicability to ion-binding sites in enzymes, but the present evidence covers only small magnesium clusters, so extension to open-shell transition-metal centers remains untested."],"forward_implications":["If OBDH is variational as claimed, self-consistent double hybrids no longer require optimized effective potentials or perturbative orbital relaxation steps, simplifying the methodology.","The correct dissociation behavior of H$_2$ and LiH indicates that self-consistency removes the long-range divergence of MP2 and non-iterative double hybrids.","The SIE4x4 MAD of 17.9 kcal/mol, compared with 38.0 for PBE, 28.6 for B3LYP, and 25.5 for PBE0, implies that OBDH substantially reduces delocalization error in charged radicals.","The Be$_2$ curve approaching experiment without an unphysical barrier suggests OBDH captures long-range dispersion in weakly bound systems better than B3LYP and MP2.","Sub-OBDH's insensitivity to concentric localization shell truncation implies the OBMP2 correlation is well localized, favoring embedding-based scaling to larger systems."],"supporting_citations":[{"why":"Supplies the working expression for the OBMP2 one-body correlation potential entering Eq. (8).","marker":"[44]"},{"why":"Extends OBMP2 to a self-consistent treatment, grounding the iterative update of the correlated potential.","marker":"[45]"},{"why":"Formalizes the generalized Kohn-Sham framework that justifies including nonlocal orbital-dependent operators in the effective Hamiltonian.","marker":"[13]"},{"why":"Provides the GKS procedure used to derive the OBDH effective Hamiltonian and SCF equations.","marker":"[56]"},{"why":"Introduces projector-based embedding, the foundation of the sub-OBDH cost reduction.","marker":"[57]"},{"why":"Supplies the concentric localization truncation that compresses the virtual space in sub-OBDH.","marker":"[59]"},{"why":"Provides the SPADE orbital partition used to define active and environment subsystems in sub-OBDH.","marker":"[66]"},{"why":"Supplies the GMTKN55 IP training subset and the SIE4x4 dataset with reference values used for benchmarks.","marker":"[67]"},{"why":"Provides the LNO-CCSD(T) reference and comparison methods for charged noncovalent complexes.","marker":"[72]"}],"fun_headline_variants":["Self-consistent double-hybrid DFT skips OEP and orbital relaxation","New double-hybrid DFT embeds MP2 directly in SCF","Double-hybrid DFT gets self-consistent without OEP or perturbative fixes","OBDH: DFT with self-consistent MP2 correlation beats PBE, B3LYP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the OBMP2 one-body potential is exactly the derivative of the OBMP2 correlation energy with respect to the orbitals, so the self-consistent solution truly minimizes the stated energy functional; the paper does not prove this derivative relation.","fun_headline_variants_meta":{"raw":{"variants":["Self-consistent double-hybrid DFT skips OEP and orbital relaxation","New double-hybrid DFT embeds MP2 directly in SCF","Double-hybrid DFT gets self-consistent without OEP or perturbative fixes","OBDH: DFT with self-consistent MP2 correlation beats PBE, B3LYP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":3049,"prompt_tokens":976,"completion_tokens":2073,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":1985}},"tokens_in":592,"tokens_out":2073,"duration_ms":15703,"temperature":1.0,"reasoning_tokens":1985,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:25:03.055655+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the OBDH total energy at the converged solution and at orbitals rotated by a small unitary transformation; if the energy changes to first order under the rotation, the assumed functional-derivative relation is false and OBDH collapses to a non-variational self-consistent Fock approximation. A numerical finite-difference derivative of the OBMP2 correlation energy against orbital rotations would give the same verdict directly.","supporting_citations":[],"review_version":1}