{"id":"490e17ac-5486-4452-8abe-63fd24610261","arxiv_id":"2501.10907","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"VM TIDFT, a penalty-based direct minimization method, optimizes excited-state orbitals without variational collapse and gives accurate energies for many excitations, although several validation claims are overextended.","lead":"This paper presents an optimization method, VM TIDFT, that computes excited electronic states in density functional theory by gradually enforcing orthogonality with a penalty function. If it works as claimed, it gives chemists a simpler route to charge-transfer and double-electron excitations, where standard time-dependent DFT struggles.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For non-minimum excited states, the energies in Tables 2 and 3 are minima of a penalized loss, not of the bare DFT energy, and the paper does not demonstrate that the penalty gradient is exactly the constraint force; the 'accurate energies' claim for these states is therefore unsupported.","rationale":"The reader's weakest assumption identifies the same point I consider load-bearing: for states that are not minima of the bare DFT energy, the converged VM TIDFT solution is a stationary point of the penalized loss, not of E, and the physical meaning of that point is asserted rather than demonstrated. The method itself is coherent: the interstate determinant penalty is a reasonable way to build an unconstrained loss, the adjugate-based gradient in Eq. (10) is a genuine technical contribution, and the CT curves in Figs. 3 and 4 provide independent evidence that the penalty, when converged, enforces meaningful orthogonality. But the helium double-excitation and high-gradient molecular entries are the cases that most distinguish the method from ordinary ΔSCF, and those are precisely where the reported energies depend on the penalty balance. I do not think the concern warrants rejection, because a null-space projection test could settle it in the authors' favor, and the CT results give independent support. I also agree with the reader that the reporting issues (helium excluded from the MAD, 2s2p described as close) should be fixed regardless of the outcome.","tokens_in":17543,"tokens_out":10759,"duration_ms":130301,"concrete_test":"Recompute the high-gradient entries in Tables 2 and 3 with an exact null-space projection of the DFT energy gradient at the converged VM TIDFT orbitals: form the Jacobian A of the orthogonality constraints (derivatives of the state-overlap matrix elements with respect to the optimized MO coefficients) and evaluate ||P_T ∇E|| with P_T = I − A(AᵀA)⁻¹Aᵀ. If this projected gradient norm is not below roughly 10⁻⁵ Ha while the full energy gradient remains large, the point is not even a constrained stationary point of E, and the reported excitation energy is a penalty artifact. As a complementary check, rerun the same states with CP increased by factors of 10² and E0 reduced to 10⁻⁶ and 10⁻⁸; if the excitation energy and the projected gradient norm do not plateau, the high-gradient numbers are not converged DFT predictions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For several reported excited states, the converged point is not a stationary point of the DFT energy E: Table 2 lists RMS energy gradients of 7.9×10⁻² Ha for He 1s²→1s2s, 4.3×10⁻¹ Ha for He 1s²→2s2, and 3.9×10⁻¹ Ha for He 1s²→2s2p, and Table 3 lists N-phenylpyrrole, NH₃···BF₃, and pyrrole-pyrazine with similarly elevated gradients. At those points the first-order condition for the penalized loss is ∇E = −∇Ω_P, so the reported energy is the value of E at a point where the DFT functional itself is not stationary. The paper invokes Perdew-Levy extrema (Ref. 75) to argue that such points correspond to physically meaningful excited states, but that argument requires the penalty gradient to be exactly the constraint force of a well-defined orthogonality constraint, and it requires the constraint to be satisfied in the limit. With finite CP and a finite allowed DFO (default E0 = 10⁻³), this equivalence is not shown. The reporting pattern reinforces the concern: the He atom is excluded from the all-system MAD despite a 5.4 eV error, and the He 1s²→2s2p result (65.79 eV versus 60.15 eV, a 7.5 eV error) is described as close to experiment. These are exactly the cases where the energy is a property of the penalty balance rather than of the DFT functional alone, so the central accuracy claim for non-minimum states is currently unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces variable-metric time-independent DFT (VM TIDFT), an orbital-optimized DFT method for excited states that avoids variational collapse by penalizing non-orthogonality between the target state and previously optimized states. The loss functional (Eq. 1) combines the state energy with an intrastate log-determinant penalty and an interstate log-determinant penalty (Eqs. 2-3). Analytical gradients for the interstate penalty are derived (Eq. 10), and a preconditioned conjugate-gradient optimizer is used (Eq. 13, Table 1). The method is implemented in CP2K and tested on atoms and molecules, including charge-transfer curves for H2 and HeH+ and double excitations in He. The paper claims accurate excitation energies for well-behaved, charge-transfer, and double-electron excitations, with MADs of 1.02 eV (first six systems) and 0.88 eV (all listed systems with experimental values).","tokens_in":17841,"tokens_out":11475,"duration_ms":108002,"significance":"If the central accuracy claims hold, VM TIDFT is a valuable addition to the OODFT toolbox: it provides closed-form gradients, permits direct unconstrained optimization with standard algorithms, is straightforwardly extensible to spin-purified ROKS, and reproduces the correct long-range behavior of CT excitations where TDDFT fails. The analytical gradient derivation, the SVD-based adjugate algorithm for ill-conditioned overlaps, and the convergence tests across many systems are concrete strengths. The paper also deposits coordinates, which aids reproducibility. However, the significance is tempered by the incomplete support for states that are not minima of the DFT energy functional and by reporting inconsistencies that affect the claimed accuracy.","major_comments":[{"comment":"The text reporting MAD = 0.88 eV for 'all test systems with available experimental excitation energies' is only correct if the helium row is excluded from the average; with the error of 5.41 eV for helium included, the MAD over the 16 listed systems with experimental values is about 1.16 eV. The table lists helium with an experimental value and no exclusion is stated, so the reported MAD is misleading and should be corrected or explicitly justified.","section":"§Accuracy, Table 3"},{"comment":"The claim that the VM TIDFT energies for the He 1s2→2s2 and 1s2→2s12p1 transitions are 'close to the experimental values' is not supported by the numbers: the reported values of 67.63 eV (QZV3P) and 65.79 eV (aug-cc-pV5Z) differ from the experimental 60.15 eV by 5.6–7.5 eV, which is not close. This overstatement should be removed, and the large error for this high-gradient state should be acknowledged and analyzed.","section":"Double-electron excitations"},{"comment":"For states that are not stationary points of the DFT energy E (He in Table 2; N-phenylpyrrole, NH3...BF3, and pyrrole-pyrazine in Table 3), the converged loss minimum satisfies ∇E = −∇ΩP, not ∇E = 0. The paper invokes Perdew–Levy (Ref. 75) to assert that such points are physically meaningful, but it does not demonstrate that the penalty gradient approximates the exact constraint force or that the finite-DFO solutions approach the orthogonality-constrained OCDFT solution in the limit E0→0, CP→∞. A numerical demonstration (e.g., comparing VM TIDFT with OCDFT for He, or showing convergence of the reported energy as E0 is tightened) is needed to support the accuracy claim for non-minimum states.","section":"Penalty strength adjustment / Results (penalty effect)"},{"comment":"The abstract states that the method 'guarantees convergence of the excited-state optimization.' As the Conclusions note, the optimization finds a stationary point in the basin of the initial guess and does not guarantee the lowest-energy excitation; moreover, convergence of the loss does not guarantee convergence to a physical excited state for non-minimum states. Please soften or qualify this claim.","section":"Abstract / Conclusions"}],"minor_comments":[{"comment":"The 'Molecules' column contains a typo: 'formadehyde' should be 'formaldehyde'.","section":"Table 3"},{"comment":"Please identify which He transition is the 1s→2s curve and which is the 1s2→2s2 curve, as the text refers to both without a clear mapping in the figure.","section":"Fig. 1 caption"},{"comment":"Table 1 lists the default DFO threshold E0 as 10−3, but the text following Fig. 1 suggests E0 = 10−4 is acceptable; please reconcile these values.","section":"Table 1 / Fig. 1 discussion"},{"comment":"The text contains a typo: 'prportional' should be 'proportional', and 'a large number of excited state' should be 'a large number of excited states'.","section":"Computational cost"}],"recommendation":"major_revision","confidential_remarks":"The exclusion of helium from the headline MAD without even a footnote is a serious reporting issue that the editor should require the authors to fix, as it directly affects the paper's central accuracy claim. The overstatement of agreement for the He 2s2p transition is similarly concerning. If the authors can supply the requested convergence analysis for high-gradient states, the method would be a solid contribution; otherwise the accuracy claims for non-minimum states remain unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real algorithmic contribution and worth referee time, but the accuracy claims run a bit ahead of the evidence, especially for states that are not minima of the bare DFT energy.\n\nThe new piece is the interstate log-determinant penalty that lets molecular orbital coefficients serve as free variables for excited-state optimization. That is genuinely new relative to the group's earlier variable-metric work, and it is well executed: the analytical gradient through the adjugate, the SVD-based stable evaluation for ill-conditioned overlaps, and the preconditioner all make sense. The CP2K implementation and deposited structures help reproducibility. The CT curves for H2 and HeH+ look physically right, and the method handles double excitations, which TDDFT cannot. For well-behaved states, the residuals are tiny and the energies are straightforwardly meaningful.\n\nThe soft spots are the ones you flagged. The all-system MAD of 0.88 eV evidently drops the He atom, where the error is 5.4 eV, without saying why; that is misleading. The claim that the 1s2->2s2p double excitation is \"close to experiment\" when it is 65.79 eV versus 60.15 eV (PBE/aug-cc-pV5Z) is not accurate; the QZV3P number 67.63 eV is also off. And for the high-gradient states, the reported energies are minima of the penalized loss, not of E itself. The Perdew-Levy citation is relevant, but the paper does not show that the penalty gradient equals the constraint force of a clean orthogonality constraint at finite CP and E0. For He 1s->2s and 1s2->2s2, Fig. 1 gives some evidence of convergence with stricter DFO, but not enough to establish that all high-gradient entries are physically meaningful rather than penalty artifacts. The \"guarantees convergence\" phrasing in the abstract is too strong; the conclusions correctly walk it back to basin-of-attraction language.\n\nNone of this is fatal. The low-gradient results, the CT asymptotic behavior, and the reproducibility are solid enough to merit serious review. The fix is straightforward: report the MAD with and without He, soften the double-excitation wording, and add a penalty-strength dependence test for the non-minimum states. This is a paper for people working on excited-state OO-DFT; they will get something useful from it even while pushing back on the validation.","headline":"A genuinely new unconstrained excited-state optimization scheme with closed-form gradients, but its accuracy claims outrun the evidence on non-minimum states and one misleading MAD.","tokens_in":18430,"tokens_out":2963,"would_cite":true,"duration_ms":29602,"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":"A continuous log-det penalty converts excited-state DFT into a guaranteed-convergent unconstrained optimization, preventing variational collapse.","keywords":["excited states","density functional theory","orbital optimization","variational collapse","orthogonality constraint","charge-transfer excitations","double-electron excitations","nonorthogonal orbitals"],"falsifier":"Take a converged VM TIDFT solution for a state with a large residual energy gradient (e.g., He $1s^2\\rightarrow 1s2s$, gradient $7.9\\times10^{-2}$ Ha), then set the interstate penalty strength $C_P$ to zero and re-optimize the orbitals; if the state collapses to a lower state or its energy shifts by more than the expected DFT error, the reported excited-state energy depends on the penalty rather than being an intrinsic property of the DFT functional.","tokens_in":17263,"feed_emoji":"⚛️","tokens_out":5885,"duration_ms":55560,"temperature":0.7,"pith_summary":"This paper introduces variable-metric time-independent DFT (VM TIDFT), a method for computing excited electronic states by direct variational optimization. Its key move is to allow the excited states to be nonorthogonal during the optimization and to enforce orthogonality gradually with a continuous log-determinant penalty function, which converts the excited-state optimization into an unconstrained minimization over molecular orbital coefficients. Because the gradient of the resulting loss functional has a closed analytical form, standard unconstrained optimizers such as preconditioned conjugate gradient can be used with a guarantee of convergence, eliminating the variational collapse that plagues other excited-state DFT methods. The paper reports accurate excitation energies for well-behaved states and, unlike TDDFT, for charge-transfer and double-electron excitations. If the method holds up, it offers a simple, robust route to excited-state electronic structure at DFT cost.","feed_headline":"Log-det penalty stops excited-state DFT collapse","feed_subtitle":"Variable-metric TIDFT handles charge-transfer and double excitations that TDDFT misses.","key_machinery":"The central object is the interstate penalty $\\Omega^P = -C_P \\sum_{\\tau} \\ln \\det(\\Phi_\\tau \\Phi_{\\tau d}^{-1})$, built from the matrix $\\Phi_\\tau$ whose entries are the determinants of overlap matrices between occupied orbitals of different states. This penalty is zero for orthogonal states, becomes positively infinite when any two states become linearly dependent, and its strength $C_P$ is multiplied by a factor $f_{\\mathrm{upd}}$ in an outer loop until the deviation from orthogonality falls below a threshold. Its analytical gradient contains the adjugate of the state-overlap matrix, which is evaluated through the singular value decomposition so that ill-conditioned, nearly orthogonal overlaps remain numerically stable. Combined with the intrastate gradient from prior VM SCF work [54], this makes the orbital coefficients free variables of a well-behaved unconstrained minimization, which is what guarantees convergence without collapse.","core_discovery":"The central claim is that the variational collapse of excited states in orbital-optimized DFT can be prevented by a simple continuous penalty that drives the states to orthogonality without ever imposing it as a hard constraint. The loss for state $I$ is $\\Omega = \\bar{E}_I + \\Omega^P$, where the intrastate term $\\bar{E}_I$ contains the energy and a determinant-based penalty for linear independence of orbitals, and the interstate term $\\Omega^P = -C_P \\sum_{\\tau=\\alpha,\\beta} \\ln \\det(\\Phi_\\tau \\Phi_{\\tau d}^{-1})$ is zero for orthogonal states and positively infinite for linearly dependent states. With this penalty, molecular orbital coefficients become independent variables; the gradient has closed-form expressions, with the interstate gradient involving the adjugate of the MO overlap matrix computed stably via SVD, so any unconstrained optimization algorithm can be used. The paper shows with a preconditioned conjugate gradient optimizer that the procedure converges robustly, producing accurate energies for ordinary excitations and, unlike TDDFT, for charge-transfer and double-electron excitations. Importantly, the fully optimized states are stationary points of the loss, not necessarily minima of the DFT energy; the paper argues, citing the Perdew–Levy extrema work [75], that such stationary points still correspond to physically meaningful excited states.","pith_inferences":["The penalty strategy is not tied to DFT: any variational method that suffers from collapse to lower solutions, such as multiconfigurational SCF or nonorthogonal configuration interaction, could adopt the same log-determinant penalization to stabilize excited-state searches.","For states with large residual energy gradients (e.g., Table 2, He $1s^2\\rightarrow 1s2s$ with gradient $7.9\\times10^{-2}$ Ha), the reported energies depend on the assumption that the stationary point of the penalized loss is a physical state; a direct test is to turn off the penalty at the converged point and see whether the state remains stationary or collapses.","The outer-loop penalty schedule (doubling $C_P$ until DFO $< 10^{-3}$) could be replaced by a dynamic update inside the optimizer, potentially avoiding the sharp gradient spikes observed when the penalty is abruptly strengthened.","Because the interstate cost grows as $O(S^2 B^2 N)$, applications to dense excited-state manifolds with hundreds of states would need an alternative scaling or a sparsity approximation for the state-overlap matrices."],"forward_implications":["VM TIDFT reproduces the long-range Coulomb asymptote of charge-transfer excitation energies in H$_2$ and HeH$^+$, where TDDFT underestimates the tail.","Double-electron excitations, which TDDFT cannot describe, are obtained directly, e.g., He $1s^2 \\rightarrow 2s^2$ at 57.68 eV versus 57.84 eV experimental.","Because the gradient is analytic and closed-form, the method can in principle use quasi-Newton and trust-region optimizers in addition to the preconditioned conjugate gradient used here, with the same guarantee against collapse.","The analytic gradient makes excited-state atomic forces straightforward, facilitating computation of emission spectra and photophysical and photochemical dynamics.","The fully optimized states are stationary points of the loss; for systems where the energy gradient is nonzero, the penalty exactly compensates it, so the states are kept from collapsing even though they are not DFT energy minima."],"supporting_citations":[{"why":"Defines orthogonality-constrained DFT, the hard-constraint approach VM TIDFT replaces with a continuous penalty, and provides baseline OCDFT energies for comparison.","marker":"[32]"},{"why":"Supplies the intrastate energy gradient and the preconditioner that VM TIDFT reuses for the single-state term.","marker":"[54]"},{"why":"Supports the claim that stationary points of the DFT energy functional, not only minima, can represent physical excited states.","marker":"[75]"},{"why":"Documents the unconstrained optimization algorithms (conjugate gradient, quasi-Newton, trust region) whose convergence guarantees VM TIDFT inherits with its analytic gradient.","marker":"[57]"},{"why":"Provides the stable SVD-based computation of the adjugate matrix needed for the interstate gradient when state-overlap matrices are ill-conditioned.","marker":"[59]"},{"why":"Supplies the high-level wavefunction benchmark (MP2/CCSD) excitation energies used to judge VM TIDFT accuracy.","marker":"[79]"}],"fun_headline_variants":["Log-det penalty halts excited-state DFT collapse","Nonorthogonal states prevent DFT collapse","Unconstrained optimization solves excited-state DFT","Variable-metric DFT tackles tough excitations","DFT excited states without variational collapse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a stationary point of the penalized loss functional is still a physically meaningful excited state even when it is not a minimum of the DFT energy; if that premise fails, the reported energies for high-gradient states would be artifacts of the penalty rather than true DFT predictions.","fun_headline_variants_meta":{"raw":{"variants":["Log-det penalty halts excited-state DFT collapse","Nonorthogonal states prevent DFT collapse","Unconstrained optimization solves excited-state DFT","Variable-metric DFT tackles tough excitations","DFT excited states without variational collapse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000805,"raw_usage":{"total_tokens":3570,"prompt_tokens":1014,"completion_tokens":2556,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":2492}},"tokens_in":630,"tokens_out":2556,"duration_ms":19740,"temperature":1.0,"reasoning_tokens":2492,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:51:01.983480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a converged VM TIDFT solution for a state with a large residual energy gradient (e.g., He $1s^2\\rightarrow 1s2s$, gradient $7.9\\times10^{-2}$ Ha), then set the interstate penalty strength $C_P$ to zero and re-optimize the orbitals; if the state collapses to a lower state or its energy shifts by more than the expected DFT error, the reported excited-state energy depends on the penalty rather than being an intrinsic property of the DFT functional.","supporting_citations":[],"review_version":1}