{"id":"0eb459c2-e2a7-48af-90b9-3e12e0189a51","arxiv_id":"2508.21453","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new variant of Hartree-Fock, Convex HF, projects out the unstable orbital rotation near a conical intersection and reintroduces it in a final diagonalization, giving continuous ground- and excited-state surfaces.","lead":"Convex Hartree-Fock is a modified mean-field method that stays stable exactly where ordinary Hartree-Fock fails: at molecular geometries where the ground and first excited states touch. This yields smooth energy surfaces for photochemical simulations at low computational cost.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Continuity of the projected Hessian eigenvector r1 is assumed, not established; any eigenvector swap between nearby geometries reintroduces the discontinuities CVX-HF is designed to remove.","rationale":"The method is plausible: removing the unstable Hessian direction from the SCF optimization and reintroducing it in a final small diagonalization is a clever way to avoid the HF instability at conical intersections. The numerical examples are visually convincing and the SI provides some supporting extensivity and multi-projection data. However, the entire construction relies on the selected eigenvector r1 being a continuous function of nuclear coordinates. This is not a mathematical consequence of the equations; it depends on the behavior of the Hessian spectrum and on the iterative solver tracking a consistent branch. The paper's own Methods section acknowledges the need for a diabatization step to keep SAD guesses consistent between nearby geometries, which is an implicit admission that continuity is not automatic. The reader identified this same weakest assumption, and the proposed test would settle it directly. No other issue appears more load-bearing: the Brillouin-type vanishings used in the Hamiltonian are justified by the projected-gradient condition, and the SI proof for replacing the metric matrix is acceptable apart from a zero-energy edge case. The central claim is therefore conditional on the continuity test, exactly as the reader concluded; no change of verdict is needed.","tokens_in":16030,"tokens_out":6971,"duration_ms":82737,"concrete_test":"Run the ammonia 2D scan on a grid twice as dense. At each converged point, save r1 and the CVX-HF reference determinant; compute |<r1(R+δ)|r1(R)>| and the determinant overlap for adjacent points. Also restart 5 geometries near the CI from a perturbed SAD guess and from the previous κ (the 'diabatized' restart) and compare the final S0-S1 gap and r1. If the overlap drops below 0.9 within one grid step, or the restart changes the energy by more than the convergence threshold, the continuity assumption is violated and the surfaces may have hidden kinks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the lowest Hessian eigenvector r1 defined in Eq. (15), and therefore the projector P = I - r1 r1^T used in Eq. (14), varies smoothly with geometry. This is the linchpin: the final Hamiltonian (Eq. 16) is built from r1, and any discontinuity in r1 translates directly into a discontinuity in S0/S1. The paper never tests this. The algorithm (step 4) determines r1 by Davidson diagonalization at each SCF iteration; step 8 uses a 'diabatized' SAD restart when continuing along a surface, an ad hoc procedure not proven to track the same eigenvector. If the two lowest Hessian eigenvalues cross, or the Davidson solver converges to a different eigenvector character at adjacent grid points, the converged κ jumps. Then the CVX-HF reference, the matrix elements in Eq. (16), and the resulting eigenstates are discontinuous. The reported scans use coarse grids and visually smooth surfaces, so a kink smaller than the grid spacing would go unnoticed. Without an overlap analysis of r1 between neighboring geometries, or a restart-sensitivity check, the continuity of the PES is not verified. The SI extensivity tables do not address this.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a modified Hartree–Fock method, 'Convex Hartree–Fock' (CVX-HF), designed to describe ground-state conical intersections at mean-field cost. The HF optimization is constrained to a subspace in which the Hessian is made positive definite by removing the component of the gradient along the lowest Hessian eigenvector r1 (Eqs. (14)–(15)). The ground and excited states are then obtained by diagonalizing the Hamiltonian in the basis {|HF>, |R1>, |~nu>} (Eq. (16)). The method is tested on three systems: ammonia, 2,4-cyclohexadien-1-ylamine, and the HBDI- GFP chromophore. In each case, TDA-TDHF shows discontinuities, negative excitation energies, or non-converged regions, while CVX-HF yields visually smooth S0/S1 surfaces with the intersection appearing as a single point. The paper claims that the framework introduces the coupling elements needed to capture conical intersections and the geometric phase effect.","tokens_in":16289,"tokens_out":6384,"duration_ms":67849,"significance":"If the claims are correct, CVX-HF would provide a practically useful, mean-field-cost route to correct S0/S1 conical intersection topology, with potential applications in nonadiabatic dynamics. The central construction is not circular: the coupling <HF|H|R1> is the residual gradient along the projected mode, determined by the self-consistent equations, and the method is not fitted to known intersections. The paper includes useful numerical experiments on three distinct systems, and the SI provides geometries, g/h vectors, and size-intensivity tests. However, the two strongest claims—continuity of the surfaces and capture of the geometric phase—are not quantitatively established. The continuity of the selected Hessian eigenvector r1 is assumed but never tested, and the geometric phase is asserted without being computed. These are load-bearing gaps that should be addressed before the central claims can be accepted.","major_comments":[{"comment":"The central continuity claim is not verified. The final Hamiltonian in Eq. (16) is built from r1, the lowest Hessian eigenvector, and the projector P = I - r1 r1^T used in Eq. (14). Algorithm 1 obtains r1 by Davidson diagonalization at each SCF iteration (step 4), and step 8 uses a 'diabatization' of SAD guesses that is never defined. Nothing in the manuscript shows that r1 varies smoothly with nuclear coordinates or that the Davidson solver tracks the same eigenvector across adjacent geometries. If the two lowest Hessian eigenvalues cross, or if the solver converges to a different eigenvector, P, the CVX-HF reference, the matrix elements in Eq. (16), and hence S0/S1 become discontinuous—reintroducing the very problem the method is designed to solve. The reported scans are coarse, so kinks smaller than the grid spacing are invisible; the SI extensivity tables do not address this. Please","section":"Algorithm 1, step 4 and Eq. (15)"},{"comment":"The assertion that the CVX-HF eigenstates 'correctly capture conical intersections and the geometric phase effect' is not supported by any computation in the manuscript. The paper reports only energy surfaces. The geometric phase is a phase change of the electronic wave function along a closed loop around a conical intersection; no Berry phase, line integral, or sign change of the adiabatic wave function is computed. At minimum, compute the loop integral of the derivative coupling or the sign change of the wave function (for example, using the overlap of |R1> or the full-space eigenstate along a closed path encircling the intersection). If such a calculation is not possible, the claim should be tempered to 'correct conical intersection topology at the mean-field level.'","section":"Discussion, first paragraph"},{"comment":"The claim that the CVX-HF intersection is 'limited to a single point' is based on visual inspection of the colormap in Fig. 1(b). No quantitative analysis of the energy gap is given: no fit of E1-E0 to a linear function of the branching coordinates, no g/h vectors for the CVX-HF intersection, and no loop test to distinguish a true conical intersection from a very narrowly avoided crossing. The SI provides the CVX-HF CI geometry for 2,4-cyclohexadien-1-ylamine (Table S8) but not a characterization of the branching space. Please quantify the degeneracy (e.g., show that E1-E0 scales linearly with displacement in two directions away from the CI) or weaken the claim.","section":"Fig. 1 and Applications, ammonia"}],"minor_comments":[{"comment":"The expression 'n excited' is undefined; it should be 'n_proj' or 'n_projected'. Also, step 8 refers to a 'diabatization step of the new C0 coefficients' that is not described anywhere; please define it or provide a reference.","section":"Algorithm 1, step 4"},{"comment":"The third component of SFS x - x is written as '-r1,mu r1,nu x_nu', but this should be a sum over nu (or the index nu should be contracted). The notation currently mixes mu and nu inconsistently.","section":"SI S2, Eq. (20)"},{"comment":"The caption uses 'CVX-HF = TDA-TDHF' and 'CVX-HF = TDA-TDHF 2'. This could be misread as equality of methods; clarify that it refers to the orbitals being visually indistinguishable.","section":"Fig. 2 caption"},{"comment":"The caption of Fig. 4 says 'The TDHF (a) and CVX-HF (b)', while the text refers to 'TDA-TDHF' throughout. Please make the method labeling consistent.","section":"Fig. 4 caption and text"}],"recommendation":"major_revision","confidential_remarks":"The continuity of r1 is the main technical risk. If the authors can supply overlap analyses and a geometric-phase computation, the paper could become a strong contribution. The current manuscript is within the scope of a specialized electronic-structure journal, but the geometric-phase claim is overstated relative to the evidence. Also note that the implementation is in a private development version of eT, so reproducibility is limited; this is common but worth flagging in the revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Interesting paper. The core idea is genuinely new: instead of fighting the near-zero Hessian eigenvalue at a ground-state conical intersection, the authors project it out of the SCF optimization and then put it back in a tiny final Hamiltonian diagonalization. That's a neat trick, and it works on the three test systems—ammonia, 2,4-cyclohexadien-1-ylamine, and HBDI−—where TDA-TDHF either gives negative excitation energies or fails to converge. The SI comparisons to CCSD/FCI for ammonia strengthen the case. The method is not fitted to the intersections; the coupling element is the residual gradient along the projected mode, determined self-consistently. That's a real contribution.\n\nThe soft spots are real but not disqualifying. The Discussion claims the framework 'captures conical intersections and the geometric phase effect.' The surfaces are shown, but the geometric phase is never computed. That's an overclaim and should be retracted or tested. More importantly, the whole construction assumes the lowest Hessian eigenvector r1 varies smoothly with geometry. The paper doesn't test this. The algorithm re-diagonalizes the Hessian at every SCF iteration, and the restart procedure uses a diabatized SAD guess, but there is no overlap analysis of r1 between neighboring points. If the Davidson solver ever swaps to a different eigenvector, or if the Hessian eigenvalues cross, the projector P and the final Hamiltonian jump, and the surfaces would be discontinuous. The reported grids are coarse enough to hide a small kink. This is the one assumption that could sink the method's central promise, and it is left unexamined.\n\nI'd also note no code or data are shipped, which is common for a theory paper but makes independent verification slower.\n\nNone of this is fatal. The idea is simple, the derivations are clean, the extensivity check with He atoms is a nice touch, and the authors are appropriately modest about the HF-level accuracy. With a stability analysis of r1 and a softened geometric-phase claim, this would be a solid paper in its niche. It deserves referee time now.","headline":"A genuinely new SCF-based route to continuous ground-state conical intersections, held back by an untested continuity assumption and an unsupported geometric-phase claim.","tokens_in":16778,"tokens_out":3931,"would_cite":true,"duration_ms":39460,"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":"Convex Hartree-Fock keeps ground and excited surfaces continuous at conical intersections by optimizing in a subspace and reintroducing the removed direction at the end.","keywords":["Convex Hartree-Fock","conical intersections","ground-state degeneracy","Hessian eigenvector projection","Tamm-Dancoff approximation","potential energy surfaces","geometric phase","nonadiabatic dynamics"],"falsifier":"Compute CVX-HF energies along a closed loop around the ammonia conical intersection and also along a path where the two lowest Hessian eigenvalues cross; if the overlap of r1 with its value at the previous geometry drops sharply and the S0/S1 gap jumps, the smoothness assumption fails. Alternatively, compare the E1-E0 map against FCI on a fine grid near the intersection: any seam or discontinuity beyond the single point would refute the claim.","tokens_in":15875,"feed_emoji":"🧪","tokens_out":7037,"duration_ms":68940,"temperature":0.7,"pith_summary":"Conical intersections between ground and excited states are where molecules switch electronic state efficiently, but standard Hartree-Fock and time-dependent Hartree-Fock break down there: the lowest excitation energy collapses toward zero, the Hartree-Fock Hessian develops a zero-curvature direction, and potential energy surfaces become discontinuous or fail to converge. This paper introduces Convex Hartree-Fock (CVX-HF), which optimizes the reference determinant in a subspace with that soft direction projected out, making the ground-state optimization convex, and then reintroduces the removed direction as an explicit configuration in a small final Hamiltonian diagonalization. The claim is that this two-step procedure supplies the missing coupling elements, so the resulting S0 and S1 surfaces are continuous and the intersection has the correct single-point topology, as demonstrated for ammonia, 2,4-cyclohexadien-1-ylamine, and the GFP chromophore HBDI-. If correct, the method offers a mean-field-cost route to ground-state conical intersections for photochemistry and nonadiabatic dynamics, where multiconfigurational methods are often prohibitively expensive.","feed_headline":"Convex Hartree-Fock captures conical intersections at mean-field cost","feed_subtitle":"Removing the soft Hessian direction during optimization and restoring it later keeps S0/S1 surfaces continuous.","key_machinery":"The central object is the Hessian eigenvector r1 associated with the smallest curvature direction of the Hartree-Fock energy, together with the full-space Hamiltonian matrix built in the basis {|HF>, |R1>, |~nu>}. r1 is the orbital-rotation direction along which the Hessian eigenvalue vanishes as a conical intersection is approached; removing its projection from the gradient and from the orbital rotation parameter makes the optimization convex (Eqs. 14-15). The reintroduced state |R1> is the single excitation generated by r1 acting on the reference, and |~nu> are the orthogonal single excitations. Diagonalizing this small Hamiltonian reintroduces the coupling between ground and excited state","core_discovery":"The central claim is that ground-state conical intersections, normally inaccessible to single-reference mean-field methods, can be captured by a modified Hartree-Fock optimization followed by a Hamiltonian diagonalization. Near a degeneracy, the lowest eigenvalue of the TDHF response problem and the corresponding Hessian eigenvalue both approach zero; the first-order gradient along that eigenvector r1 therefore has almost no steering effect, which triggers bifurcations and discontinuous surfaces. CVX-HF solves the projected stationarity condition together with the Hessian eigenvector equation for r1, so the energy is convex in the subspace orthogonal to r1. The removed direction is then used","pith_inferences":["The continuity argument implicitly relies on r1 varying smoothly with nuclear coordinates. A natural stress test is to follow a path where the two lowest Hessian eigenvalues cross: if r1 and r2 exchange character, the projected subspace can jump and the CVX-HF surfaces may develop a small seam that the current examples do not probe.","Because the final diagonalization is essentially a two-state-plus-bath model built from a mean-field reference, CVX-HF could be viewed as a systematic way to generate diabatic-like states from a single determinant, connecting to diabatization procedures beyond conical intersections.","The method's cost profile, one convex Hartree-Fock optimization plus a small eigenvalue problem, suggests it could enable on-the-fly nonadiabatic dynamics for medium-sized chromophores where multireference methods are currently too expensive, though this computational extrapolation is not tested in the paper.","For intersections involving more than two states, or seams of higher dimension, the single-vector projection may be insufficient; the paper notes multiple projections are possible, but their robustness along extended seams is not benchmarked."],"forward_implications":["CVX-HF gives continuous S0/S1 potential energy surfaces in regions where TDA-TDHF shows negative excitation energies, unconverged Hartree-Fock solutions, or discontinuous surfaces, at essentially Hartree-Fock cost.","The converged CVX-HF orbitals can be used as a reference for correlated methods such as coupled cluster or perturbation theory, preventing Hartree-Fock-level discontinuities from propagating into more accurate models.","The same orbital-rotation parameterization can be applied within Kohn-Sham DFT, so TDDFT could inherit the same correction to conical-intersection topology.","Additional Hessian eigenvectors can be projected and the final space can be enlarged with extra configurations, including double excitations, which covers multichromophore systems and cases where double excitations matter.","Excitation energies remain size-intensive when non-interacting molecules or solvent atoms are added, which is important for modeling solvated chromophores."],"supporting_citations":[{"why":"Establishes that zero-energy eigenvectors of the TDHF response problem are also zero-energy eigenvectors of the Hartree-Fock Hessian, the link that turns the vanishing excitation energy into the direction removed by CVX-HF.","marker":"[25]"},{"why":"Supplies the ammonia S0/S1 conical intersection test case and the CIC-TDA method with correct intersection dimensionality against which CVX-HF is compared.","marker":"[13]"},{"why":"Documents the failures of single-reference and coupled-cluster methods at ground-state conical intersections due to the geometric phase, motivating the need for a modified mean-field reference.","marker":"[9]"},{"why":"Provides the generalized coupled cluster framework for ground and excited state intersections that CVX-HF is benchmarked against and whose orbitals CVX-HF could seed.","marker":"[28]"},{"why":"The superposition-of-atomic-densities initial guess used to start the CVX-HF optimization in every calculation.","marker":"[21, 22]"},{"why":"Defines the Tamm-Dancoff approximation used as the benchmark method against which CVX-HF results are compared.","marker":"[24]"},{"why":"Analyzes the stability of the TDHF eigenvalue problem, underlying the instability analysis that leads to the projection strategy.","marker":"[23]"}],"fun_headline_variants":["Convex HF finds ground-state conical intersections cheaply","Mean-field theory now sees ground-state crossings","Tailored HF optimization keeps S0/S1 surfaces continuous","Removing a soft Hessian direction restores degeneracy detection","Convex HF: ground-state conical intersections without multireference"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The method assumes that the selected Hessian eigenvector r1 tracks smoothly with nuclear geometry, so the projected subspace and converged reference are continuous; if r1 changes abruptly between neighboring geometries, the surfaces could regain discontinuities.","fun_headline_variants_meta":{"raw":{"variants":["Convex HF finds ground-state conical intersections cheaply","Mean-field theory now sees ground-state crossings","Tailored HF optimization keeps S0/S1 surfaces continuous","Removing a soft Hessian direction restores degeneracy detection","Convex HF: ground-state conical intersections without multireference"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1073,"prompt_tokens":673,"completion_tokens":400,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":321}},"tokens_in":417,"tokens_out":400,"duration_ms":4508,"temperature":1.0,"reasoning_tokens":321,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:19:27.329067+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute CVX-HF energies along a closed loop around the ammonia conical intersection and also along a path where the two lowest Hessian eigenvalues cross; if the overlap of r1 with its value at the previous geometry drops sharply and the S0/S1 gap jumps, the smoothness assumption fails. Alternatively, compare the E1-E0 map against FCI on a fine grid near the intersection: any seam or discontinuity beyond the single point would refute the claim.","supporting_citations":[{"cited_title":", author Bulik, I","cited_arxiv_id":null,"evidence_quote":"Establishes that zero-energy eigenvectors of the TDHF response problem are also zero-energy eigenvectors of the Hartree-Fock Hessian, the link that turns the vanishing excitation energy into the direction removed by CVX-HF."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ammonia S0/S1 conical intersection test case and the CIC-TDA method with correct intersection dimensionality against which CVX-HF is compared."},{"cited_title":"Understanding failures in electronic structure methods arising from the geometric phase effect","cited_arxiv_id":"2411.08209","evidence_quote":"Documents the failures of single-reference and coupled-cluster methods at ground-state conical intersections due to the geometric phase, motivating the need for a modified mean-field reference."},{"cited_title":", author Kj nstad, E","cited_arxiv_id":null,"evidence_quote":"Provides the generalized coupled cluster framework for ground and excited state intersections that CVX-HF is benchmarked against and whose orbitals CVX-HF could seed."},{"cited_title":"& author Head-Gordon, M","cited_arxiv_id":null,"evidence_quote":"Defines the Tamm-Dancoff approximation used as the benchmark method against which CVX-HF results are compared."},{"cited_title":"& author Simons, J","cited_arxiv_id":null,"evidence_quote":"Analyzes the stability of the TDHF eigenvalue problem, underlying the instability analysis that leads to the projection strategy."}],"review_version":1}