{"id":"b8b4c465-3b9d-4235-a611-f4046f224684","arxiv_id":"2608.08095","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A neural-network Hamiltonian enables true on-the-fly nonadiabatic molecular dynamics in solids, replacing the ground-state classical-path approximation with excited-state forces and couplings.","lead":"This paper introduces a machine-learning framework that performs on-the-fly nonadiabatic molecular dynamics in solids, using a neural-network Hamiltonian to compute excited-state forces and couplings at low cost. It demonstrates the method on a MoS2/WS2 heterostructure, photoinduced ferroelectric switching in bilayer h-BN, and polaron formation in TiO2, including simulations with hybrid-functional accuracy.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Delta-SCF frozen-Hamiltonian forces (Eq. 2) are benchmarked only on silicon at 0.39% excitation; the TiO2 hole-polaron application is exactly the self-trapping regime where the neglected self-consistent response dominates.","rationale":"The reader's strongest claim is that on-the-fly N2AMD delivers excited-state forces and NACVs at ab initio accuracy, making CPA-free NAMD in solids possible. The load-bearing assumption is the delta-SCF force approximation of Eq. 2, and the reader correctly identified that it is validated only on silicon. My stress-test sharpens this concern: the approximation is not merely unvalidated in the applications; it is formally a frozen-orbital band-energy force, and the TiO2 hole-polaron application is precisely a regime where the neglected self-consistent response is the physical mechanism being simulated. A self-trapped polaron is stabilized by the mutual feedback between localized charge density and lattice distortion; a force computed from the ground-state Hamiltonian's orbital gradient cannot fully capture that feedback, especially at the hybrid-functional level where exact exchange is the source of the localization. The paper itself flags the approximation as the primary one, which is honest, but it does not supply the revalidation needed to support the headline applications. The h-BN ferroelectric switching may be less sensitive because the excitation is delocalized, but the polaron result is at risk. This does not invalidate the methodological framework or the silicon benchmarks, which are strong and include real speedups, transferability tests, and a Zenodo code release. It does mean the conditional verdict is appropriate: the central claim should be accepted only after the delta-SCF force approximation is revalidated in the regimes where the paper applies it, or after the claims are softened accordingly. No change to the reader's verdict is needed; my analysis reinforces it.","tokens_in":10371,"tokens_out":6938,"duration_ms":85882,"concrete_test":"Run constrained-occupation self-consistent delta-SCF (density and Hamiltonian relaxed under the excited occupations) on configurations sampled from the TiO2 polaron formation trajectory and along the h-BN AB-BA sliding path, using the same PBE/HSE settings as in the paper, and compare per-atom forces to Eq. 2. If the force MAE exceeds roughly 50 meV/Å, or if the frozen-orbital PES along the polaron distortion coordinate lacks the self-trapped minimum found self-consistently, the applications are not supported by the present validation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. 2 replaces the true excited-state force by the ground-state force plus derivatives of frozen Kohn-Sham eigenvalues. This is a non-self-consistent, Harris-type approximation. The paper's statement that \"double-counting energies vanish upon differentiation\" is not generally correct: the Hartree and XC double-counting terms depend on the density, which changes under excitation. It happens to be a good approximation in the single silicon benchmark (7.6 meV/Å MAE, 0.39% valence excitation), where the excitation is delocalized and the density response is weak. The two headline applications operate in a different regime: h-BN at 0.027 e/u.c. and especially the HSE hole polaron in anatase TiO2. Polaron formation is a self-trapping process in which the localized hole density and the lattice distortion stabilize each other through the self-consistent response of the Hamiltonian, including the exact-exchange component that is the entire reason for using HSE in this application. Eq. 2 computes the force from the gradient of the ground-state VBM orbital energy, so it cannot generate the self-consistent Fock-exchange feedback that creates the gap state. If this error is large, the claimed \"capturing real-time polaron formation\" rests on forces that are not ab initio. The paper explicitly acknowledges the approximation but does not revalidate it for either application, nor does it report force-error statistics along the trajectories.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces on-the-fly N2AMD, a machine-learning framework for nonadiabatic molecular dynamics in periodic solids. It combines an E(3)-equivariant neural-network Hamiltonian (HamGNN) with a ground-state ML potential (Allegro) to compute excited-state forces through a delta-SCF frozen-orbital approximation (Eq. 2) and nonadiabatic coupling vectors through automatic differentiation of the learned Hamiltonian (Eq. 4). This allows fewest-switches surface hopping without the Classical Path Approximation. The authors benchmark the approach on bulk silicon (Hamiltonian MAE 0.045 meV, excited-state force MAE 7.6 meV/Å, smoothed NACV MAE 0.52 meV/Å), demonstrate transferability to twisted MoS2 bilayers and Ga_xAl_1-xAs alloys, and report three applications: hole relaxation in MoS2/WS2, photoinduced ferroelectric switching in bilayer h-BN, and hole polaron formation in anatase TiO2 at the HSE level.","tokens_in":10800,"tokens_out":5991,"duration_ms":62802,"significance":"If the underlying approximations hold in the application regimes, this is a substantial advance: it removes the Classical Path Approximation for solid-state NAMD, restores the back-reaction of excited carriers on the lattice, and makes hybrid-functional NAMD tractable. The paper also ships code and data on Zenodo, which is a strong reproducibility feature. The reported 163x speedup per ionic step for a 96-atom silicon cell is significant. However, the central excited-state force approximation is validated only in a narrow regime, and the headline applications reach beyond that validation; the quantitative strength of the paper therefore depends on additional benchmarks that are currently absent.","major_comments":[{"comment":"The delta-SCF frozen-Hamiltonian force approximation in Eq. (2) is benchmarked only on bulk silicon at 0.39% valence excitation (Fig. 2a), yet it is used without revalidation for the h-BN ferroelectric switching at 0.027 e/u.c. and for the HSE hole polaron in anatase TiO2. In the polaron application, self-trapping is driven by the self-consistent response of the Hamiltonian, including the Fock-exchange component, to the localized hole; a frozen ground-state Hamiltonian gradient cannot, by construction, generate the self-consistent feedback that creates the gap state. The paper acknowledges the approximation but reports no force-error statistics along the trajectories of either application, so the central applications rest on an unverified approximation.","section":"Eq. (2) / Benchmarks"},{"comment":"The statement that \"double-counting energies vanish upon differentiation, as they do not explicitly depend on atomic positions\" is not generally correct: the Hartree and XC double-counting terms are density functionals, and the density depends on R through the orbitals. The correct statement is that the frozen-orbital delta-SCF approximation neglects these derivatives; presenting them as vanishing by construction misstates the derivation and could mislead readers about the accuracy of the force formula.","section":"Eq. (2) and surrounding derivation"},{"comment":"The NACV benchmark reports only the magnitude of the smoothed couplings d^s_ij = |(epsilon_i - epsilon_j) d_ij|, not the full NACV vectors d_ij that enter the FSSH hopping probability and the velocity rescaling described after Eq. (4). Because the direction of the NACV controls the momentum adjustment at each hop, a scalar MAE on smoothed magnitudes does not establish that the learned Hamiltonian yields accurate vectorial couplings; a vector benchmark, or at least a report of the directional error, is needed to support the on-the-fly FSSH claim.","section":"Fig. 2d / Benchmarks"},{"comment":"The TiO2 polaron demonstration claims hybrid-functional accuracy, but the manuscript provides no accuracy metrics for the HSE-trained neural Hamiltonian or for HSE delta-SCF forces. Training-data MAE, force MAE, and NACV MAE should be reported for the HSE model, as done for silicon, before the \"hybrid-functional level\" claim can be assessed quantitatively.","section":"Application 2 / Fig. 4"}],"minor_comments":[{"comment":"The phrase \"migrate to issues\" should be \"mitigate issues\".","section":"Benchmarks"},{"comment":"The phrase \"combing these key quantities\" should be \"combining these key quantities\".","section":"Workflow description"},{"comment":"The scaling comparison would be clearer if N were defined (number of atoms versus basis functions) and if the DFT scaling curve were labeled explicitly.","section":"Fig. 2b"},{"comment":"The line break after the equality sign leaves the summation bracket unbalanced in the printed text; please reformat the equation.","section":"Eq. (4)"},{"comment":"The sentence \"Here 0.39% of the valence electrons excited\" ends without a period.","section":"Benchmarks"},{"comment":"The acronym N2AMD is not expanded at first use in the main text; consider defining it explicitly as \"Neural network NAMD\".","section":"Introduction"},{"comment":"Reference [30] appears to be an unpublished manuscript; please provide its publication status or a journal reference if available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is promising and the methodology is well aligned with the journal's scope, but the main-text validation base is narrow relative to the scope of the claims. The HSE polaron application in particular should be supported by explicit HSE benchmarks of the Hamiltonian, forces, and NACVs; I would recommend requiring this in revision. The availability of code and data on Zenodo is a strong point and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick verdict: the paper's central claim is genuinely new—solid-state on-the-fly FSSH with excited-state forces and NACVs from a neural Hamiltonian, no Classical Path Approximation—and the silicon benchmarks support it. The worry you'll want to keep in mind is narrow but real: the delta-SCF frozen-Hamiltonian force formula (Eq. 2) is the load-bearing piece, and it is validated on one system only.\n\nThe good parts first. The architecture is clean: HamGNN predicts H(R), automatic differentiation gives gradient H, and the Allegro potential supplies the ground-state PES. The silicon numbers are strong: Hamiltonian MAE 0.045 meV, excited-state force MAE 7.6 meV/A against self-consistent delta-SCF, NACV MAE 0.52 meV/A. The 163x speedup over DFT at 96 atoms is realistic for this kind of ML workflow, and the transferability tests on twisted MoS2 and solid-solution compositions are a nice touch. The authors explicitly ship code and data on Zenodo, and they state the key approximation in plain language, which is more than many papers do.\n\nNow the soft spot. Eq. 2 replaces the true excited-state force by the ground-state force plus derivatives of frozen ground-state eigenvalue differences. The paper says the double-counting terms vanish upon differentiation because they do not explicitly depend on atomic positions. That is too quick: the density used in the double-counting functional is built from occupied orbitals with modified occupations, so it does depend on R. The neglect of self-consistent relaxation is fine for a dilute delocalized excitation in silicon, which is exactly the benchmark shown. It is much less obviously fine for the bilayer h-BN at 0.027 e/u.c. and especially for the HSE hole polaron in anatase TiO2, where the localized hole and the lattice distortion are stabilized by the self-consistent Fock-exchange response. If Eq. 2 cannot generate that feedback, the claimed real-time polaron formation may rest on forces that are not ab initio. The authors do not revalidate the approximation on either application system, and the dynamics plots have no error bars or ensemble statistics.\n\nNone of this undermines the core methodological claim that on-the-fly NAMD with an ML Hamiltonian is practical. But the TiO2 showcase needs closer scrutiny or a clear caveat before it can be taken as evidence. I'd send this to referees, not desk reject, and I'd ask for either a careful derivation of the double-counting term in the frozen-orbital limit or a few force-error benchmarks on the actual application systems.","headline":"Convincing proof that ML Hamiltonians can make on-the-fly NAMD in solids practical, but the delta-SCF force approximation is load-bearing and only validated on silicon.","tokens_in":11249,"tokens_out":3267,"would_cite":true,"duration_ms":33467,"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 machine-learned Hamiltonian makes on-the-fly fewest-switches surface hopping in solids practical, removing the Classical Path Approximation.","keywords":["nonadiabatic molecular dynamics","surface hopping","machine learning Hamiltonian","E(3)-equivariant neural network","excited-state forces","nonadiabatic coupling vectors","delta-SCF","polaron dynamics"],"falsifier":"Take the bilayer h-BN AB-BA sliding pathway at 0.027 e/u.c. and the HSE hole polaron in anatase TiO2, and recompute forces with fully self-consistent delta-SCF at the geometries visited by the NAMD trajectories; if the forces differ from Eq. (2) by more than a few tens of meV/Å, or if the h-BN barrier is no longer eliminated, the central claim fails.","tokens_in":10178,"feed_emoji":"⚛️","tokens_out":11112,"duration_ms":96988,"temperature":0.7,"pith_summary":"The paper introduces on-the-fly N$^2$AMD and claims it removes the main bottleneck in nonadiabatic molecular dynamics (NAMD) for periodic solids: the need to compute excited-state forces and nonadiabatic coupling vectors at every time step. Where the standard Classical Path Approximation fixes nuclei on a ground-state surface and misses the back-reaction of excited carriers, the new workflow uses an E(3)-equivariant neural network to predict the electronic Hamiltonian and obtains all required derivatives by automatic differentiation. This makes true fewest-switches surface hopping in solids practical at ab initio accuracy and, because the Hamiltonian can be trained on hybrid-functional data, at hybrid-functional accuracy. If the claim holds, the method corrects order-of-magnitude errors in carrier relaxation, enables simulation of photoinduced ferroelectric switching, and captures polaron formation in real time.","feed_headline":"Neural-net Hamiltonians bring true on-the-fly NAMD to solids","feed_subtitle":"Excited carriers reshape the lattice as they move, enabling ferroelectric switching and polaron formation.","key_machinery":"The load-bearing object is the differentiable machine-learned Hamiltonian. An E(3)-equivariant graph neural network (HamGNN) outputs the Hamiltonian matrix $H_{\\mu\\nu}$ and overlap matrix $S_{\\mu\\nu}$ from atomic positions, so $\\nabla_R H_{\\mu\\nu}$ is obtained by automatic differentiation and the overlap gradients $\\langle\\nabla_R\\phi_\\mu|\\phi_\\nu\\rangle$, $\\langle\\phi_\\mu|\\nabla_R\\phi_\\nu\\rangle$ by two-center integrals. The excited-state force is the delta-SCF expression $F \\approx F_{gs} + \\sum_{i\\in VB}(1-f_i)\\nabla_R\\epsilon_i - \\sum_{a\\in CB}f_a\\nabla_R\\epsilon_a$, and the nonadiabatic coupling vector is $d_{mn} = (\\epsilon_m-\\epsilon_n)^{-1}\\sum_{\\mu\\nu}c^*_{m\\mu}c_{n\\nu}[-\\nabla_R H_{\\mu\\nu} + \\epsilon_n\\langle\\nabla_R\\phi_\\mu|\\phi_\\nu\\rangle + \\epsilon_m\\langle\\phi_\\mu|\\nabla_R\\phi_\\nu\\rangle]$ for $m\\neq n$. Combining these with the ground-state force from the Allegro potential closes the feedback loop: the electronic state directs the nuclei, and the resulting nuclear motion feeds back into the Hamiltonian at the next time step.","core_discovery":"The paper's central claim is that a single learned Hamiltonian—an E(3)-equivariant neural network mapping atomic configurations to Hamiltonian and overlap matrices in a numerical atomic orbital basis—supplies every ingredient on-the-fly surface hopping needs: excited-state potential energy surfaces through the delta-SCF force expression of Eq. (2), nonadiabatic coupling vectors through the gradient formula of Eq. (4), and ground-state forces through a separate equivariant interatomic potential. On bulk silicon, the learned excited-state force agrees with self-consistent delta-SCF benchmarks to a mean absolute error of 7.6 meV/Å, while the ground-state approximation differs by 110.4 meV/Å, and the workflow runs a 96-atom cell 163 times faster than DFT. The method is then applied to three regimes the Classical Path Approximation cannot reach: carrier relaxation in a MoS2/WS2 heterostructure, where the relaxation rate is corrected by an order of magnitude; photoinduced ferroelectric switching in bilayer h-BN, with 82% of trajectories reversing polarization within 1.5 ps; and real-time hole polaron formation in anatase TiO2 at the hybrid-functional HSE level.","pith_inferences":["The same differentiable-Hamiltonian pipeline could be trained against self-consistent delta-SCF, GW, or BSE data, which would harden the force approximation for strongly correlated systems or high excitation densities without changing the workflow.","Because the network returns the full Hamiltonian rather than a scalar energy, the framework could be extended to spin-orbit coupling, Berry phases, or time-dependent electron propagation within the same trajectory machinery.","The demonstrated transferability suggests a cheap validation protocol: before production NAMD in a new material, compute self-consistent delta-SCF forces on a handful of representative geometries to confirm Eq. (2) holds there."],"forward_implications":["Nuclei in solid-state NAMD can evolve on excited-state potential energy surfaces, so the lattice responds to the instantaneous electronic state instead of a precomputed phonon background.","Hybrid-functional NAMD becomes feasible, as shown by the HSE simulation of hole polaron formation in anatase TiO2, removing self-interaction errors that PBE-based NAMD cannot capture.","Carrier relaxation kinetics computed with the Classical Path Approximation may be systematically revised; in the MoS2/WS2 heterostructure the correction changes the dynamics by an order of magnitude.","Photoinduced phase transitions such as sliding ferroelectric switching in bilayer h-BN become directly simulable, including hot-carrier cooling that constrained Born-Oppenheimer MD misses.","Because the learned-Hamiltonian workflow scales as $O(N^2)$ while DFT scales more steeply, the speedup over ab initio NAMD grows with system size."],"supporting_citations":[{"why":"Supplies the fewest-switches surface hopping algorithm that the on-the-fly workflow implements.","marker":"[8]"},{"why":"Defines the Classical Path Approximation (PYXAID) that the paper replaces and provides the baseline for comparison.","marker":"[12]"},{"why":"Prior workflow that introduced the HamGNN-plus-Allegro combination and is reoriented here from CPA to on-the-fly dynamics.","marker":"[22]"},{"why":"Introduces the delta-SCF method whose non-self-consistent force expression underlies Eq. (2).","marker":"[29]"},{"why":"Provides the local-orbital formula for nonadiabatic coupling vectors used in Eq. (4).","marker":"[40]"},{"why":"The E(3)-equivariant Hamiltonian neural network (HamGNN) that predicts the Hamiltonian matrix.","marker":"[41]"},{"why":"Allegro, the equivariant ground-state interatomic potential that supplies the ground-state force.","marker":"[42]"},{"why":"Phase-correction scheme applied to NACs in both methods to ensure the comparison is not artificially accelerated.","marker":"[43]"},{"why":"The screened hybrid functional HSE used for the TiO2 polaron application.","marker":"[54]"}],"fun_headline_variants":["ML Hamiltonians unlock real-time NAMD in solids","Learn Hamiltonians, simulate excited-state dynamics on the fly","One learned Hamiltonian drives on-the-fly NAMD for solids","Neural-net Hamiltonians: real-time excited-state NAMD for solids","Machine-learned Hamiltonians power on-the-fly excited-state dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the non-self-consistent delta-SCF force formula, benchmarked only on silicon at 0.39% valence excitation, remains accurate for bilayer h-BN at 0.027 e/u.c. and for the HSE hole polaron in TiO2; if self-consistent relaxation of the Hamiltonian under the excited density matters there, the simulated switching and polaron dynamics rest on incorrect forces.","fun_headline_variants_meta":{"raw":{"variants":["ML Hamiltonians unlock real-time NAMD in solids","Learn Hamiltonians, simulate excited-state dynamics on the fly","One learned Hamiltonian drives on-the-fly NAMD for solids","Neural-net Hamiltonians: real-time excited-state NAMD for solids","Machine-learned Hamiltonians power on-the-fly excited-state dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001172,"raw_usage":{"total_tokens":4918,"prompt_tokens":1085,"completion_tokens":3833,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":3747}},"tokens_in":701,"tokens_out":3833,"duration_ms":27922,"temperature":1.0,"reasoning_tokens":3747,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:25:18.612383+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the bilayer h-BN AB-BA sliding pathway at 0.027 e/u.c. and the HSE hole polaron in anatase TiO2, and recompute forces with fully self-consistent delta-SCF at the geometries visited by the NAMD trajectories; if the forces differ from Eq. (2) by more than a few tens of meV/Å, or if the h-BN barrier is no longer eliminated, the central claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fewest-switches surface hopping algorithm that the on-the-fly workflow implements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Classical Path Approximation (PYXAID) that the paper replaces and provides the baseline for comparison."},{"cited_title":"Zhang, Y","cited_arxiv_id":null,"evidence_quote":"Prior workflow that introduced the HamGNN-plus-Allegro combination and is reoriented here from CPA to on-the-fly dynamics."},{"cited_title":"Gavnholt, Delta self-consistent field method to obtain potential energy surfaces of excited molecules on surfaces, Phys","cited_arxiv_id":null,"evidence_quote":"Introduces the delta-SCF method whose non-self-consistent force expression underlies Eq. (2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the local-orbital formula for nonadiabatic coupling vectors used in Eq. (4)."},{"cited_title":"Zhong, H","cited_arxiv_id":null,"evidence_quote":"The E(3)-equivariant Hamiltonian neural network (HamGNN) that predicts the Hamiltonian matrix."},{"cited_title":"Musaelian, S","cited_arxiv_id":null,"evidence_quote":"Allegro, the equivariant ground-state interatomic potential that supplies the ground-state force."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Phase-correction scheme applied to NACs in both methods to ensure the comparison is not artificially accelerated."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The screened hybrid functional HSE used for the TiO2 polaron application."}],"review_version":1}