{"id":"1f158d19-7093-4c1a-abeb-ac36072583e8","arxiv_id":"2411.08207","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Keeping two electronic states at a small fixed energy gap, the tube algorithm finds approximate minimum energy conical intersections, and CCSD versions of these structures match CASSCF and SF-TDDFT reference geometries for three molecules.","lead":"This paper tests an optimization trick that locates near-degenerate molecular geometries, called ε-MECIs, without computing the difficult nonadiabatic coupling vectors normally required. It shows the trick lets coupled cluster theory, a powerful quantum chemistry method, reproduce ground-state intersection structures for ethylene, azobenzene, and uracil that match other established methods.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The tube algorithm's ε-MECI is not unique: for a linear cone, E_m is constant on the I_ε ellipse, so branching-plane coordinates are initial-guess dependent; reported CCSD-vs-CASSCF geometry agreement may not test convergence to the true MECI.","rationale":"The reader's weakest_assumption identified the right broad premise: the ε-MECI at finite ε must be shown to approximate the true MECI. My analysis sharpens this into a mechanism: in the linear-cone limit, the upper-state energy is exactly constant on the ellipse I_ε ∩ branching plane, so the geometry is underdetermined along that ellipse. This is a stronger, mechanism-level version of the same concern and explains why the finite-ε comparison with CASSCF geometries is not by itself a valid convergence test. The paper's own convergence data (SI Table 2) show stability of selected internal coordinates as ε is lowered from 0.27 to 0.07 eV, which is useful but does not probe the flat direction; the proposed branching-plane displacement test would do so. If the test shows multiple converged geometries with equal energy, the claim that CCSD quantitatively reproduces the CASSCF MECI structure is underdetermined, though the seam-coordinate part of the geometry may still be accurate. No fatal inconsistency is apparent, so the manuscript remains conditionally acceptable pending this validation; the verdict should stay CONDITIONAL rather than being upgraded or rejected.","tokens_in":9982,"tokens_out":11427,"duration_ms":136364,"concrete_test":"For the uracil S0/S1 6S5 ε-MECI (ε=0.07 eV), re-run the CCSD tube optimization from two starting geometries obtained by displacing the reported structure along the two branching-plane vectors (e.g., ±0.1 Å along g and h evaluated with SA-CASSCF at that geometry). If the two optimizations converge to geometries with the same seam-coordinate dihedral (C4C5C6N1) but noticeably different branching-plane coordinates (C4C5C6H5), then the reported structure is one of a continuum and the comparison with CASSCF in Fig. 5 cannot support the central claim. If all starting points converge to the same geometry, the flat-direction degeneracy is broken by higher-order terms and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the converged ε-MECI geometry, not just its energy, is a faithful proxy for the true MECI on the seam. In the linear-cone model underlying the paper's geometric picture (Eqs. (1)–(4)), the adiabatic energies in the branching plane are E_{n,m}(x,y,s) = E0(s) ∓ sqrt(g^2 x^2 + h^2 y^2). On the isosurface I_ε this square root equals ε/2, so the upper-state energy is E_m = E0(s) + ε/2. Therefore E_m is constant along the entire ellipse I_ε ∩ branching plane for each fixed seam coordinate s. The objective minimized in Eq. (5) has a continuum of stationary points: any point on the ellipse at the optimal s is a minimum, and the projected gradient cannot move the geometry along the ellipse. The converged ε-MECI therefore inherits its branching-plane coordinates from wherever the optimization first reached I_ε, not from the seam itself. For the ground-state cases the reported ε values are 0.07–0.27 eV, so this ellipse can have finite radius. Differences such as uracil oop-O C2N3C4O8 = 94° vs CASSCF 113° and C4–O8 = 1.42 vs 1.52 Å may be partly this finite-ε ambiguity rather than CCSD error. The paper's claim that CCSD 'can provide an accurate description of conical intersections with the ground state' thus rests on the unverified premise that the reported geometries are representative of the true MECI, not merely arbitrary points on the tube. The SI shows stability across ε for one starting point, but that does not test the flat direction along the ellipse.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The Letter analyzes an optimization algorithm, called the tube algorithm, for locating minimum energy conical intersections without using nonadiabatic coupling vectors. The method minimizes the energy of the upper state on the isosurface I_ε = {R : E_m(R) - E_n(R) = ε}, using the projected gradient in Eq. (5). The authors validate the algorithm on the S1/S2 intersection in uracil by comparing with the gradient projection method, and then apply it to S0/S1 intersections in ethylene, azobenzene, and uracil at the CCSD level, comparing the resulting ε-MECI geometries with CASSCF and SF-TDDFT reference structures. On this basis they conclude that CCSD, despite its convergence problems at degeneracies, can provide an accurate description of conical intersections with the ground state and may be suitable for nonadiabatic dynamics targeting ground-state relaxation.","tokens_in":10252,"tokens_out":5191,"duration_ms":56625,"significance":"If the tube algorithm is reliable, it is a useful tool because it removes the need for derivative couplings and because it sidesteps coupled-cluster convergence problems near the seam. The paper has clear strengths: the geometric picture of the tube, the open repository of optimized structures, the stepwise ε-reduction protocol, and the S1/S2 uracil comparison against an independent optimizer. However, the central claim that finite-ε CCSD tube minima represent true S0/S1 MECIs is not fully established: for the S0/S1 cases the smallest converged gaps are 0.07–0.27 eV, and the paper gives no proof or systematic ε→0 convergence study for ground-state intersections. The flat-direction degeneracy of the upper-state energy on I_ε makes the branching-plane coordinates of the converged geometry potentially path-dependent, so the agreement with CASSCF structures may be less decisive than presented.","major_comments":[{"comment":"For a locally linear conical intersection, the two adiabatic energies in the branching plane are E_{n,m} = E_0(s) ∓ sqrt(g^2 x^2 + h^2 y^2). On the isosurface I_ε the square root is fixed to ε/2, so the upper-state energy E_m = E_0(s) + ε/2 is, to leading order, independent of the branching-plane coordinates (x,y) along the ellipse. The projected gradient in Eq. (5) therefore has no restoring component along that ellipse, and the converged ε-MECI's branching-plane coordinates are determined by the path by which the optimization first reaches I_ε rather than by the true seam. With reported S0/S1 gaps of 0.07–0.27 eV this ellipse is not negligibly small, so discrepancies such as the uracil oop-O C2N3C4O8 angle (94° vs 113°, Fig. 5 and SI Table 2) may reflect this finite-ε ambiguity rather than CCSD error. A concrete test would be to restart the same ε optimization from different initial branching-plane coordinates and report the spread in the converged internal coordinates, or to project the converged ε-MECI onto the seam and compare with a direct MECI optimization.","section":"Tube algorithm, Eqs. (4)–(6)"},{"comment":"The paper states that 'for sufficiently small ε, the tube folds around the crossing seam' and that the ε-MECI gives 'an accurate approximation to the minimum energy conical intersection,' but no proof or quantitative convergence analysis is provided for the S0/S1 cases. The only direct ε→0 validation is the S1/S2 uracil case (Fig. 2), where ε is reduced to 0.0027 eV with SCCSD. For the ground-state intersections the smallest converged values are 0.07 eV (uracil 6S5), 0.14 eV (uracil oop-O), 0.20 eV (azobenzene), and 0.27 eV (ethylene) (Figs. 3–5 and SI Table 2), and the SI reports that CCSD ground-state equations do not converge for smaller energy gaps. As presented, the concluding claim that CCSD 'can provide an accurate description of conical intersections with the ground state' is therefore not established; the authors should either supply a convergence study or a proof for S0/S1, or explicitly restrict the claim to finite-ε tube minima.","section":"ε→0 limit and ground-state convergence"},{"comment":"The accuracy assessment relies on comparisons to CASSCF and SF-TDDFT structures computed with different basis sets and active spaces, for example CCSD/cc-pVDZ vs CASSCF(10/8)/6-31G* for uracil and CCSD/6-31G vs 5SA-CASSCF(6/6)/6-31G for azobenzene. This confounds the effect of the tube approximation with method and basis-set differences. A consistency check at a common basis for at least one S0/S1 case would strengthen the claim that the reported geometry differences are dominated by the electronic-structure treatment rather than by the finite-ε tube effect.","section":"Comparison with reference geometries"}],"minor_comments":[{"comment":"The phrase 'showing that it minimizes the energy on hypersurfaces that envelop the intersection seam' overstates what is demonstrated: the main text gives the projected-gradient construction and numerical evidence, but not a proof that the ε-MECI converges to the true MECI as ε→0.","section":"Abstract and Introduction"},{"comment":"A few reference formatting issues should be corrected, including 'Bernhard Schlegel, H.' (Ref. 10, should list H. Bernhard Schlegel), 'Chem. Phys. Letters' (Refs. 10 and 28, journal is Chem. Phys. Lett.), and the accented name 'MartÍnez' in Ref. 22.","section":"References"},{"comment":"The column headers in SI Table 2 interleave CCSD ε values and literature reference columns without clearly marking which ε belongs to which method; a clearer layout (e.g., subheadings 'CCSD' and 'Reference') would help readers verify which geometry is being compared.","section":"Supporting Information, Table 2"},{"comment":"The text says the three methods agree with '53° with SF-TDDFT and 52° for CCSD and CASSCF' for the C4C5C6N1 dihedral; SI Table 2 shows 52° only at ε = 0.07 eV, with 50° and 51° at larger ε, so the ε dependence of the comparison should be stated explicitly.","section":"Main text, uracil 6S5 paragraph"},{"comment":"The stepwise procedure is described as starting with CCSD at ε = 0.14 eV and switching to SCCSD for ε ≤ 0.014 eV, while Fig. 2 shows CCSD also at ε = 0.027 eV; the text and figure caption should be harmonized to avoid confusion about which method was used at each ε.","section":"Section on S1/S2 uracil"}],"recommendation":"major_revision","confidential_remarks":"The flat-direction degeneracy of the upper-state energy on I_ε is the key technical issue, and it directly affects the interpretation of the S0/S1 results. The paper is otherwise a useful contribution, and the S1/S2 validation is convincing. I would ask for a starting-point-dependence study along the branching-plane ellipse and a more guarded statement of the CCSD ground-state claim. The companion work in Ref. 27 is relevant but not essential to this revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Three things to know up front. The gradient in Eq. (5) is the same as Harabuchi et al.'s (Ref. 28), and the authors say so; credit where due. The new pieces are the geometric framing of I_ε as a tube enveloping the seam, and the first systematic demonstration that CCSD can locate S0/S1 ε-MECIs. The caveat the paper never quite faces: on the tube, the upper-state energy is flat along the branching-plane ellipse, so the branching-plane coordinates of a converged ε-MECI are inherited from wherever the optimizer first reaches I_ε, not determined by the minimization.\n\nWhat's good: the paper is honest and well executed. Structures are on Zenodo; the S1/S2 uracil study directly shows the ε-MECI converging to the gradient-projection MECI as ε decreases; the S0/S1 geometries for ethylene, azobenzene, and uracil match independent CASSCF and SF-TDDFT references well, especially the azobenzene dihedrals and uracil 6S5 puckering (52° vs 52°). For anyone wanting a size-extensive single-reference route to approximate ground-state crossing geometries, this is a practical result.\n\nSoft spots, in proportion. The flat-direction issue is real: for a linear cone, E_m is constant on each I_ε ellipse, so the objective has a continuum of stationary points. The SI's stability across ε (uracil 6S5 dihedrals shift only 1–2° from ε = 0.27 to 0.07 eV) looks reassuring but does not test that flat direction, because the stepwise procedure restarts from the previous geometry and follows one ray. A multi-start test would settle it. Second, the ground-state ε values (0.07–0.27 eV) are not small, and the ε→0 limit is shown directly only for the S1/S2 intersection. Saying CCSD 'can provide an accurate description of conical intersections with the ground state' goes a step beyond the evidence, which shows accurate near-seam geometries at finite ε. Third, minor: at these ε values the ellipse radius can be a few hundredths of an Å, the same order as some reported CCSD–CASSCF deviations; part of that scatter may be finite-ε ambiguity rather than method error.\n\nOn the stress-test: the strongest version — that the geometry agreement 'may not test convergence to the true MECI' — is mathematically right but overstated as a threat. The ambiguity vanishes as ε→0, the S1/S2 study shows the limit reached smoothly, and the close agreement with independent literature geometries would be unlikely if the branching-plane coordinates were arbitrary.\n\nWho it's for: people doing nonadiabatic dynamics or benchmarking single-reference against multiconfigurational methods at crossings. It deserves a serious referee; the right outcome is minor-to-moderate revision asking for a discussion of the flat direction, ideally a multi-start check, and a tempered conclusion.","headline":"Honest, useful tube-algorithm paper for CCSD ground-state ε-MECIs; flat-direction non-uniqueness is a real caveat the authors don't discuss, but the method and S1/S2 validation hold up.","tokens_in":10903,"tokens_out":9062,"would_cite":true,"duration_ms":85545,"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":"Minimizing the upper electronic state on a fixed small energy gap gives accurate minimum-energy conical intersections and lets coupled cluster theory describe ground-state crossings in ethylene, azobenzene, and uracil.","keywords":["minimum energy conical intersection","tube algorithm","coupled cluster singles and doubles","nonadiabatic coupling-free optimization","isosurface energy gap","similarity constrained coupled cluster","ground-state conical intersection","photochemistry"],"falsifier":"For one ground-state case, take the uracil 6S5 S0/S1 intersection and compute the $\\varepsilon$-MECI at a sequence of decreasing $\\varepsilon$ down to the coupled cluster convergence limit, while also obtaining a reference MECI with a method that can converge on the seam itself; if the tube minima do not converge to the seam minimum, for instance if the C4C5C6N1 dihedral shifts by more than a few degrees, the central approximation fails for ground states.","tokens_in":9659,"feed_emoji":"⚛️","tokens_out":9811,"duration_ms":90312,"temperature":0.7,"pith_summary":"The paper proposes and tests a way to find minimum-energy conical intersections, the geometries where two electronic states are degenerate and population can switch between surfaces, without computing nonadiabatic coupling vectors. Instead of converging onto the crossing seam itself, the algorithm minimizes the upper state's energy on the nearby hypersurface where the two states are separated by a small fixed energy difference, here called the tube. The paper argues that for small enough energy gaps this tube wraps around the seam and its minimum is an accurate stand-in for the true crossing minimum, and that the optimization never touches the degenerate point, so the numerical breakdowns that plague coupled cluster theory near conical intersections are avoided. Using this approach, the paper demonstrates that coupled cluster singles and doubles reproduces multireference and spin-flip TDDFT geometries for ground-state intersections in ethylene, azobenzene, and uracil, suggesting coupled cluster dynamics could describe nonradiative relaxation to the ground state.","feed_headline":"Tube algorithm finds crossing minima without derivative couplings","feed_subtitle":"Minimizing the upper state on a fixed small energy gap lets CCSD match multireference geometries for three molecules.","key_machinery":"The tube isosurface $I_\\varepsilon = \\{R : E_m(R) - E_n(R) = \\varepsilon\\}$ is the central object; it is an $(N-1)$-dimensional hypersurface in energy parallel to the crossing seam, and its limit as $\\varepsilon \\to 0$ is the seam itself. The work is carried by the modified gradient $G^\\varepsilon_{nm} = P^\\varepsilon_{nm} \\nabla E_m + 2(E_m - E_n - \\varepsilon) g^\\varepsilon_{nm}/\\|g^\\varepsilon_{nm}\\|$ with $P^\\varepsilon_{nm} = 1 - g_{nm} g_{nm}^T$, where $g_{nm} = \\nabla(E_n - E_m)$. The first term moves the geometry along the tube to reduce the upper-state energy; the second term drives the geometry onto the tube. Since only $g_{nm}$ is required, no nonadiabatic coupling vectors are needed.","core_discovery":"The paper's central claim is that a minimum-energy conical intersection can be approximated without ever evaluating the crossing seam or nonadiabatic coupling vectors. For two states $m$ and $n$, one fixes a small positive energy $\\varepsilon$ and minimizes the upper state's energy on the isosurface $I_\\varepsilon = \\{R : E_m(R) - E_n(R) = \\varepsilon\\}$, which for small $\\varepsilon$ forms a tube enveloping the seam and collapses onto the seam as $\\varepsilon \\to 0$. The paper shows that the appropriate modified gradient needs only the energy-difference gradient $g_{nm} = \\nabla(E_n - E_m)$, and that the converged $\\varepsilon$-MECI geometries approach the gradient-projection MECI for an excited-state crossing in uracil. For ground-state crossings, the paper claims that CCSD $\\varepsilon$-MECIs of ethylene, azobenzene, and uracil agree quantitatively with state-averaged CASSCF and spin-flip TDDFT reference structures, which is the first demonstration that coupled cluster theory can describe ground-state conical intersections.","pith_inferences":["Editorial inference: the drift of the $\\varepsilon$-MECI geometry as $\\varepsilon$ shrinks could serve as a diagnostic of how close a given electronic structure method is to a genuine degeneracy; a geometry that stabilizes as $\\varepsilon \\to 0$ signals a converged approximation, while one that keeps shifting signals an avoided crossing or a method artifact.","Editorial inference: running surface-hopping or mean-field dynamics on the $I_\\varepsilon$ tube with a fixed small $\\varepsilon$ might let coupled cluster trajectories avoid the singular region altogether while still sampling the relevant branching-plane topography; this is a testable extension the paper does not pursue.","Editorial inference: because the gradient expression applies to any pair of states whose energy gap can be differentiated, the approach could be extended to intersections between other excited states, or to state pairs in multireference methods, without derivative couplings."],"forward_implications":["Coupled cluster singles and doubles can be used, at least for the molecules tested, to locate meaningful S0/S1 conical intersection structures despite its known convergence problems at true degeneracies.","The algorithm places no special demand on the electronic structure method: any method with analytic or numerical energy gradients for the two states can use it.","Stepwise reduction of $\\varepsilon$ gives a practical route to approach the seam from above, with the converged geometry at a larger $\\varepsilon$ serving as an initial guess for a smaller $\\varepsilon$.","If the S0/S1 accuracy holds more generally, coupled cluster theory becomes a candidate for nonadiabatic dynamics simulations that describe nonradiative relaxation to the ground state.","The optimization can converge to an avoided crossing rather than a true intersection, and the paper notes that a geometric-phase loop is needed to confirm a genuine conical intersection."],"supporting_citations":[{"why":"Defines the gradient projection method that minimizes the upper state on the crossing seam, the baseline the tube algorithm is compared with throughout.","marker":"[10]"},{"why":"Proposed the shifted small-energy-gap gradient whose functional form the tube algorithm adopts; the paper's contribution is the geometric interpretation of that gradient as motion on a tube.","marker":"[28]"},{"why":"Provides the similarity constrained coupled cluster model used as the reference for the S1/S2 uracil convergence test that validates the $\\varepsilon \\to 0$ limit.","marker":"[24]"},{"why":"Supplies the analytical molecular gradients and nonadiabatic couplings implementation that makes the coupled cluster geometry optimizations possible.","marker":"[26]"},{"why":"Supplies the state-averaged CASSCF ethylene S0/S1 MECI geometry that the CCSD $\\varepsilon$-MECI is compared against.","marker":"[38]"},{"why":"Supplies the SA-CASSCF azobenzene CI-rot MECI geometry used as the reference for the CCSD structure.","marker":"[39]"},{"why":"Supplies the spin-flip TDDFT uracil 6S5 geometry that the CCSD $\\varepsilon$-MECI matches.","marker":"[40]"},{"why":"Supplies the CASSCF uracil 6S5 and oop-O geometries and the nomenclature used to name the two S0/S1 intersections.","marker":"[41]"}],"fun_headline_variants":["Crossing minima via energy tube, no coupling vectors","Fixed energy gap finds MECIs in coupled cluster theory","Enveloping the seam: CCSD locates ground-state crossings","No derivative couplings: tube method locates crossing minima","Small energy gap yields crossing minima in CCSD"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's results depend on the assumption that the finite energy gaps it actually converged to, 0.27 eV for ethylene, 0.20 eV for azobenzene, and 0.07 to 0.14 eV for uracil ground-state crossings, are small enough that the tube minimum is close to the true seam minimum; that limit is demonstrated directly for one excited-state crossing but not for the ground-state crossings.","fun_headline_variants_meta":{"raw":{"variants":["Crossing minima via energy tube, no coupling vectors","Fixed energy gap finds MECIs in coupled cluster theory","Enveloping the seam: CCSD locates ground-state crossings","No derivative couplings: tube method locates crossing minima","Small energy gap yields crossing minima in CCSD"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000271,"raw_usage":{"total_tokens":1604,"prompt_tokens":899,"completion_tokens":705,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":626}},"tokens_in":515,"tokens_out":705,"duration_ms":7305,"temperature":1.0,"reasoning_tokens":626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:52:36.457587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For one ground-state case, take the uracil 6S5 S0/S1 intersection and compute the $\\varepsilon$-MECI at a sequence of decreasing $\\varepsilon$ down to the coupled cluster convergence limit, while also obtaining a reference MECI with a method that can converge on the seam itself; if the tube minima do not converge to the seam minimum, for instance if the C4C5C6N1 dihedral shifts by more than a few degrees, the central approximation fails for ground states.","supporting_citations":[{"cited_title":"J.; Robb, M","cited_arxiv_id":null,"evidence_quote":"Defines the gradient projection method that minimizes the upper state on the crossing seam, the baseline the tube algorithm is compared with throughout."},{"cited_title":"Exploring approximate geometries of minimum energy conical intersections by TDDFT calculations","cited_arxiv_id":null,"evidence_quote":"Proposed the shifted small-energy-gap gradient whose functional form the tube algorithm adopts; the paper's contribution is the geometric interpretation of that gradient as motion on a tube."},{"cited_title":"F.; Koch, H","cited_arxiv_id":null,"evidence_quote":"Provides the similarity constrained coupled cluster model used as the reference for the S1/S2 uracil convergence test that validates the $\\varepsilon \\to 0$ limit."},{"cited_title":"F.; Angelico, S.; Koch, H","cited_arxiv_id":null,"evidence_quote":"Supplies the analytical molecular gradients and nonadiabatic couplings implementation that makes the coupled cluster geometry optimizations possible."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the state-averaged CASSCF ethylene S0/S1 MECI geometry that the CCSD $\\varepsilon$-MECI is compared against."},{"cited_title":"Probing the → * photoisomerization mechanism of cis-azobenzene by multi-state ab initio on-the-fly trajectory dynamics simulation","cited_arxiv_id":null,"evidence_quote":"Supplies the SA-CASSCF azobenzene CI-rot MECI geometry used as the reference for the CCSD structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the spin-flip TDDFT uracil 6S5 geometry that the CCSD $\\varepsilon$-MECI matches."},{"cited_title":"J.; Szymczak, J","cited_arxiv_id":null,"evidence_quote":"Supplies the CASSCF uracil 6S5 and oop-O geometries and the nomenclature used to name the two S0/S1 intersections."}],"review_version":1}