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Latent unified smooth Hamiltonians for excited state chemistry

David Juergens, Martin Stöhr, Andreas E. Hillers-Bendtsen, O. Jonathan Fajen, Todd J. Martínez

arXiv:2609.01871v1physics.chem-phcs.LG

TL;DR

Photochemical modeling requires reliable excited-state descriptions, yet machine-learning interatomic potentials have not broadly extended their ground-state success to this setting. LUSH learns a latent electronic Hamiltonian, and its models reproduce ground- and excited-state landscapes and absorption spectra for thymine and azobenzene.

  • Problem

    Photochemistry needs reliable excited-state modeling, but machine-learning interatomic potentials have not broadly translated their ground-state success because excited-state landscapes couple multiple energies.

    Source: Introduction · p. 2

  • Method

    LUSH learns a smooth, implicit-basis latent electronic Hamiltonian whose diagonalization yields adiabatic energies and whose derivatives provide non-adiabatic couplings.

    Source: Introduction · p. 3

  • Results

    On thymine and azobenzene, LUSH models reproduce ground- and excited-state landscapes, key photochemical potential-surface features, and absorption spectra.

    Source: Conclusion and Outlook · p. 13

  • Takeaways & Limitations

    The authors conclude LUSH balances quantum-chemistry inductive bias with deep networks’ representation-learning capacity by encoding general multistate structure rather than restrictive constraints.

    Source: Conclusion and Outlook · p. 14

Abstract

from arXiv · show

We describe a neural network architecture and training procedure designed to model electronic ground and excited states of arbitrary molecular systems. By indirectly learning a latent, implicit basis representation of the electronic-state Hamiltonian, the model offers a unified treatment of multiple electronic states, conical intersections, and non-adiabatic couplings. The formalism can be further extended to learn consistent latent representations of additional operators such as transition dipole moments, for example. To demonstrate the general capabilities of our architecture, we train and evaluate networks on two realistic photochemical systems, thymine and azobenzene. The resulting models accurately reproduce energies and oscillator strengths for the ground- and low-lying excited states relevant to the photochemistry of these systems. We highlight the performance of the trained networks by studying critical molecular geometries, including conical intersections and excited state minima. By construction, the proposed framework also recovers the emergence of Berry phase accumulation around conical intersections. By pairing key mathematical structure from quantum chemistry with the representation learning power of transformers, the presented architecture offers a qualitatively new path toward fast and accurate ground- and excited-state simulations.

Introduction

Excited-state photochemistry requires costly electronic-structure calculations, while conventional smooth multi-state potentials struggle with conical intersections. LUSH learns a latent Hamiltonian to model excited-state energies and related electronic properties in a unified framework.

Introduction

  • High-level excited-state calculations are costly because photochemistry requires extensive dynamics and sampling, and can depend sensitively on the molecular environment.These demands make biologically and technologically relevant time and length scales difficult to reach with available methods.Source: Introduction · p. 2
  • Conventional smooth multi-state potentials struggle with conical intersections because adiabatic energies are nonsmooth there, causing artifacts such as spurious avoided crossings.The passage also describes artificially smoothed double crossings where an excited state falls below the supposed ground state.Source: Introduction · p. 2
  • A finite Hamiltonian submatrix can focus on the few low-lying states relevant to photochemistry, but common adiabatic and diabatic representations limit transferable, unified modeling.Diabatic forms lack a unique global representation and can be cumbersome to prepare; adiabatic energies have derivative discontinuities near intersections. Existing encoded-energy approaches do not unify energies with properties such as transition dipoles and nonadiabatic couplings.Source: Introduction · pp. 2–3
  • LUSH learns a general latent Hamiltonian that smoothly interpolates across configurations and yields adiabatic energies, including at conical intersections.The model computes nonadiabatic couplings from Hamiltonian derivatives using the Hellmann–Feynman relation, and can represent other operators such as transition properties in the same framework.Source: Introduction · p. 3

A neural network for electronic-state Hamiltonians

LUSH maps molecular geometries to latent Hamiltonian and operator representations, diagonalizes the Hamiltonian to obtain adiabatic states, and rotates other operators into the same basis. Its training procedure combines state-property losses with state weighting to address limitations near intersections at the spectrum boundary.

Network architecture

  • An SE(3)-invariant message-passing encoder and learned-latent cross-attention convert variable-size molecular graphs into fixed-size tokens, which are pair-lifted and refined before parallel operator heads.The Hamiltonian head produces a matrix that is symmetrized and diagonalized for state-energy predictions; the other operator heads produce latent representations.Source: Figure 1 · p. 4, Network architecture · pp. 4–5, Electronic (transition) properties… · p. 5

Electronic (transition) properties from unified operator representations

  • Diagonalizing the latent Hamiltonian yields adiabatic states, while rotating other latent operators into that eigenbasis gives state expectation values and off-diagonal transition properties.The Hamiltonian eigenvalues and eigenvectors also define non-adiabatic coupling vectors; this construction encodes conical-intersection branching planes and vortex-like coupling topology underlying Berry-phase accumulation. The coupling formula is exact when the chosen state space is geometry-independent and very accurate when its geometry dependence is weak.Source: Electronic (transition) properties… · pp. 5–6

Training procedure and loss definition

  • Training targets include electronic state energies, excitation energies, and oscillator strengths, with phase-invariant losses applied to these quantities.The energy loss is MSE, while excitation-energy and oscillator-strength absolute errors use smooth-L1 losses; the energy-derived terms also supervise predicted state energies.Source: Training procedure and loss definition · pp. 6–7
  • Because a truncated Hamiltonian becomes non-smooth at its spectral boundary, LUSH state-weights the loss and treats the highest eigenstates as buffers to protect lower-state interpolation.At least the two highest states can serve as buffers, leaving the lowest states fully accounted for; additional buffer states can help near more complex degeneracies.Source: Training procedure and loss definition · pp. 7–8

Results

LUSH is evaluated on QM9 ground-state energies and on thymine and azobenzene excited-state landscapes, including conical intersections, nonadiabatic couplings, and critical geometries.

Chemical accuracy for ground state energetics

  • On QM9, a 5.95-million-parameter LUSH model approached chemical accuracy (~1 kcal/mol) on 100 validation molecules in about 48 hours on one V100 GPU.The experiment used the lowest of eight predicted eigenvalues as the total energy; excited-state loss components were not used.Source: Chemical accuracy for ground state… · p. 8

Recovering conical intersections and NAC fields

  • LUSH found a zero-gap thymine S1/S2 intersection despite no such intersections in QeMFi, and its branching-plane surfaces show the conical shape.The predicted geometry is near the TDDFT intersection seam, with a reported TDDFT gap of 3.1 ⋅108$ eV; the branching-plane construction identifies the two coordinates that lift degeneracy.Source: Recovering conical intersections and… · p. 9, Figure 2 · p. 9
  • LUSH reproduces the NAC vortex around the thymine intersection, whose topology produces Berry-phase accumulation along paths encircling the intersection.NAC vectors are guaranteed to be singular at degeneracies; their branching-plane projections are larger along the coordinate where the energy gap varies more rapidly.Source: Recovering conical intersections and… · pp. 9–10, Figure 2 · p. 9

Excited state landscape of azobenzene

  • LUSH closely reproduces reference azobenzene absorption spectra and potential-energy scans across states, while swapping the nearly degenerate S1 and S2 states for trans-azobenzene.The scans use geometries relaxed according to FOMO-hh-TDA; S1 is dark and S2 bright in trans-azobenzene, so an additional reference state is included for matching.Source: Excited state landscape of azobenzene · p. 10, Figure 3 · p. 11
  • Twisted azobenzene geometries, including the cis minimum, are poorly described because training data contain few cis-like structures, though qualitative landscape features are captured.Agreement improves for planar-angle scans when geometries are relaxed with LUSH; the model was not trained on gradients and obtains local curvature through interpolation.Source: Excited state landscape of azobenzene · pp. 10–11
  • MECI RMSD reaches 0.473 Å and twisted-geometry ΔΔL reaches -1.8 eV, while LUSH reproduces minima and planar S1/S2 CI energetics well.The paper attributes the twisted-geometry discrepancy to limited low-energy cis-like training configurations; overall, the predicted MECI structures and energetics are in good agreement with the reference.Source: Excited state landscape of azobenzene · p. 12, Figure 4 · p. 12

Conclusion and Outlook

LUSH constructs a latent electronic-state Hamiltonian to predict ground and excited states, intersections, and related properties. Tests on QM9, thymine, and azobenzene show accurate electronic-structure predictions, while future work targets larger-scale nonadiabatic simulations.

Conclusion

  • Diagonalizing LUSH's latent Hamiltonian yields adiabatic energies and intersections, while the framework also predicts forces, non-adiabatic couplings, and latent operators such as transition dipoles.General latent operators can be evaluated in the adiabatic frame.Source: Conclusion and Outlook · pp. 12–13
  • LUSH combines fixed-count cross-attention latents, pairwise Pairformer refinement, a differentiable eigensolver stable at degeneracies, and gap-weighted properties to isolate a fixed-size low-energy Hamiltonian.The gap weighting limits the Hamiltonian submatrix without contaminating low-lying states when degeneracies occur at the spectrum boundary.Source: Conclusion and Outlook · p. 13
  • LUSH reaches chemical accuracy on QM9 ground states and reproduces thymine and azobenzene photochemical surfaces, absorption spectra, and conical-intersection NAC topography at a fraction of reference-data computational cost.The models provide a uniform description of ground and excited states and transitions, with accuracy on par with quantum-chemical reference data.Source: Conclusion and Outlook · p. 13

Outlook

  • Future work will extend nonadiabatic dynamics to longer times and larger systems, add energy-gradient and phaseless-NAC training, and explore ML/MM coupling toward biomolecular complexes.The current gradient-free-loss models already describe critical geometries well; the authors expect the extended loss to improve performance further.Source: Conclusion and Outlook · p. 13
  • Future work will test whether state-character information lets LUSH recover conical intersections missing from reference data, while its mathematical structure balances quantum-chemistry inductive bias with deep-network representation learning.The example given is intersections missed by TDDFT101.Source: Conclusion and Outlook · pp. 13–14

Methods and Computational Details

The methods refine state-pair representations into latent Hamiltonian and transition-dipole operators, train them with energy and spectroscopy losses, and evaluate azobenzene using reference calculations and geometry searches.

Additional details on LUSH Pairformer

  • The LUSH Pairformer refines a lifted state-pair representation through axial triangle multiplication and attention; its attention cost remains modest for typical photochemistry models with fewer than 10 states.The block is an in-house implementation of the AlphaFold3 pairformer [63].Source: Additional details on LUSH Pairformer · p. 14
  • Parallel heads project the refined state-pair representation into latent Hamiltonian and transition-dipole operators, with output dimensions of one and three, respectively.The three transition-dipole components represent the x, y, and z directions.Source: Additional details on LUSH Pairformer · p. 14
  • The Hamiltonian is made traceless before a learned geometry-dependent diagonal shift restores its absolute spectral offset, allowing the pairformer to focus on excitation energies.The shift is obtained from the mean of an offset-head output applied to the refined single-state representation.Source: Additional details on LUSH Pairformer · p. 14

Details of loss definition

  • The loss combines mean-square atomization-energy error across states and samples with smooth-L1 losses for excitation energies and oscillator strengths.State-resolved reference energies use isolated-atom Hartree–Fock values for QM9, fitted atom energies for QeMFi, and in-vacuo BHLYP/def2-SVP values for azobenzene.Source: Details of loss definition · pp. 14–15

Geometry optimization and conical intersection search

  • Azobenzene MECIs are searched by penalizing the energy gap while minimizing the average state energy, using relaxed gradient tolerances for MECI searches and surface scans.Geometry optimizations use a Levenberg–Marquardt-type quasi-Newton scheme with delocalized internal coordinates in geomeTRIC; the stated RMS and maximum-gradient tolerances are 1.7e-3 and 1.0e-3, respectively.Source: Geometry optimization and conical… · p. 15

Azobenzene reference calculations

  • Azobenzene training geometries come from 423 K harmonic Wigner distributions around cis and trans minima and nonadiabatic dynamics sampling excited-state landscapes.Reference calculations use FOMO-hh-TDA-BHLYP/def2-SVP [95, 87], and the ab initio multiple spawning method [104, 105]; a five-state LUSH model uses two buffer states to represent the three studied states. Calculations were performed with TeraChem [88, 89].Source: Azobenzene reference calculations · pp. 15–16

Supplementary information

The supplementary information provides additional database, atomic-reference-energy, and secondary-results analyses for QeMFi and azobenzene.

Supplementary Materials for

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Supplementary Materials for

Supplementary Figures

The supplementary material reports LUSH training and evaluation details, including a QM9 validation error, thymine state-crossing and energy-gap analyses, azobenzene training-data coverage, and dataset-specific atomic reference energies.

Supplementary Figures

  • The d=256 LUSH model reached a validation mean absolute error of 0.059 eV (~1.36 kcal/mol) for QM9 U−U_ref prediction.The model had ~7.05 million parameters; the two models differed in hidden dimension (256 versus 128).Source: Figure S1 · p. 25
  • LUSH training on the 135,000-sample, nine-molecule QeMFi def2-TZVP dataset used a 90/10 random train/test split.The supplementary figure tracks total loss, state energies, state energy gaps, and oscillator strengths during training.Source: Figure S2 · p. 26

LUSH FOMO-hh-TDA-BHLYP/def2-SVP

  • On an unseen thymine optimization trajectory, LUSH tracked TDDFT predictions and captured an S3/S4 crossing between frames 1 and 2 through oscillator-strength switching.The model was pretrained on QeMFi def2-TZVP data; the trajectory geometries were not in the training set.Source: Figure S3 · p. 27
  • The smallest S1/S2 gap among 15,000 thymine QeMFi def2-TZP calculations was 6.53e-03 eV.The supplementary histogram summarizes the S1/S2 gaps across the dataset.Source: Figure S4 · p. 27
  • The azobenzene training data contained no geometries within 0.2 Å of cis-azobenzene or the S1/S0 reactive and S2/S1 twisted conical intersections.Among geometries with CNNC dihedral angle <45°, the closest sample was 2.7 eV above cis-azobenzene.Source: Figure S6 · p. 29

Computation of atomic reference energies

  • Atomic reference energies were dataset-specific: QM9 used Hartree-Fock/cc-pv5z isolated-atom energies, QeMFi used a linear fit to molecular formula and ground-state energies, and azobenzene used ROKS BHLYP/def2-SVP single-atom calculations.For QeMFi, the linear model was fit by least squares.Source: Computation of atomic reference energies · p. 30

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