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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
TL;DR
Accurate excited-state simulation requires handling multiple coupled energy surfaces, intersections, and related electronic properties without the prohibitive cost of high-level quantum chemistry. LUSH learns a smooth latent Hamiltonian and transforms additional latent operators into the adiabatic frame. Models achieve chemical accuracy for QM9 ground-state energies and reproduce key excited-state landscapes and spectra for thymine and azobenzene.
Problem
Excited-state machine-learning potentials must represent multiple coupled states and conical intersections, while existing approaches do not unify energetics with transition properties and nonadiabatic couplings.
Method
LUSH encodes arbitrary molecular geometries into fixed-size latent representations, learns an implicit-basis Hamiltonian, diagonalizes it for adiabatic states, and rotates other operators into that basis.
Results
LUSH approaches chemical accuracy for QM9 ground-state energies and reproduces the relevant ground- and excited-state landscapes, transitions, and absorption spectra of thymine and azobenzene.
Takeaways & Limitations
The framework provides a unified description of electronic states, conical intersections, nonadiabatic couplings, and transition operators within a latent Hamiltonian representation.
Abstract
from arXiv · showhide
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 is difficult to model because accurate electronic-structure methods are costly, while naive multi-state machine-learning potentials are not smooth at state intersections. LUSH addresses this by learning a smooth latent Hamiltonian whose diagonalization recovers adiabatic states and related properties.
- High-level ab initio methods provide accurate excited-state descriptions but impose dramatic computational costs that limit photochemical simulations.
- Excited-state landscapes comprise tightly coupled functions for multiple state energies, unlike ground-state potentials that typically represent one energy surface.
- Naive multi-state MLIPs fail near conical intersections because adiabatic eigenstates are non-smooth where states intersect.
- Finite Hamiltonian submatrices offer a practical representation because photochemistry usually involves a small number of low-lying electronic states.
- LUSH learns a general smooth latent Hamiltonian, diagonalizes it for adiabatic energies and conical intersections, and rotates additional operators into the adiabatic frame.
A neural network for electronic-state Hamiltonians
LUSH converts molecular geometries into fixed-size single- and pair-state representations, then generates latent operators including a Hamiltonian that is diagonalized and used to transform other properties. Its training combines energy, excitation-energy, and oscillator-strength objectives while addressing conical-intersection boundary effects and derivative properties.
- Network architecture: An SE(3)-invariant MPNN encodes atom-local geometry, and cross attention maps variable-size molecular graphs into fixed-length latent tokens refined by self attention.
- Network architecture: Pair lifting combines refined single-state tokens into an explicit state-pair tensor, which is projected and globally refined before operator generation.
- Electronic (transition) properties from unified operator representations: Parallel operator heads produce latent matrices; the symmetrized Hamiltonian is diagonalized for energies and eigenvectors, while other operators are rotated into the same adiabatic basis.
- Electronic (transition) properties from unified operator representations: Latent operator matrices represent operators in an implicit electronic-state basis, whose diagonal and off-diagonal adiabatic elements encode state expectations and transition properties.
- Electronic (transition) properties from unified operator representations: Hamiltonian diagonalization lets the network learn smooth matrix elements instead of derivative discontinuities, while NACs follow from the eigenvalue problem and Hellmann–Feynman relation.
- Training procedure and loss definition: The loss combines energy MSE with smooth-L1 losses for excitation energies and oscillator strengths, and all three terms supervise predicted state energies.
- Training procedure and loss definition: At intersections at the spectrum boundary, the highest retained state can become discontinuous and contaminate other states, motivating adaptive state weighting and an unsupervised buffer dimension.
Results
LUSH reproduces ground- and excited-state behavior across thymine and azobenzene, including energies, oscillator strengths, conical intersections, nonadiabatic couplings, and critical geometries. Its limitations are most evident for twisted azobenzene structures because the training data underrepresent cis-like geometries.
- Ground-state benchmark: 48 hours on one NVIDIA Tesla V100 brought QM9 validation energies near chemical accuracy (~1 kcal/mol).The 5.95-million-parameter model used the lowest Hamiltonian eigenvalue for total-energy prediction.
- Thymine: Thymine predictions closely match TDDFT excitation energies and oscillator strengths, including state tracking through an avoided crossing.The oscillator strengths for the two transitions flip consistently with the changing state character.
- Thymine: A learned thymine conical intersection lies near the ab initio seam, with a TDDFT state gap of 3.1 ⋅108$ eV.The predicted zero-gap geometry was found despite the training dataset containing no corresponding conical intersections.
- Thymine: LUSH produces a conical intersection in the thymine branching plane and a vortex-like NAC field associated with Berry phase accumulation.The branching-plane energy surfaces show the conical topology, while NAC projections reflect the gap variation along the two lifting coordinates.
- Azobenzene: For azobenzene, LUSH reproduces reference absorption spectra and largely matches potential-energy surfaces across trans and cis systems.Agreement is reported for relative peak positions, relative intensity, and scans relaxed according to the reference excited-state surfaces.
- Azobenzene: LUSH accurately reproduces azobenzene minima, while MECIs deviate more strongly and twisted geometries can reach relative-energy errors of -1.8 eV.MECI RMSD values reach 0.473 Å, and the poor twisted-geometry description is attributed to few cis-like training geometries.
Conclusion and Outlook
LUSH constructs a latent electronic-state Hamiltonian whose diagonalization yields adiabatic properties, while extending to transition operators and accurately describing ground, excited, and intersecting states. The authors identify future extensions toward larger-scale nonadiabatic dynamics and broader intersection coverage.
- LUSH predicts ground and excited electronic states by constructing and diagonalizing a latent electronic-state Hamiltonian.
- The framework extends beyond energies and derivative properties to latent operators such as transition dipoles evaluated in the adiabatic frame.
- Models reproduce ground- and excited-state landscapes, transitions, absorption spectra, conical intersections, and non-adiabatic coupling topography for thymine and azobenzene.
- The authors propose extending excited-state calculations with NACs toward nonadiabatic dynamics at longer times and larger length scales.
- LUSH combines quantum-chemical mathematical structure with deep-network representation learning without imposing strictly adiabatic or diabatic constraints.
Methods and Computational Details
The methods construct latent state-pair and operator representations with neural-network components, then train against state-resolved electronic properties. Azobenzene data combine thermal sampling and excited-state dynamics using a specified electronic-structure reference.
- The pair representation is initialized by a pair lift, dimensional projection, residual network, and AlphaFold3-inspired pairformer block.
- Parallel heads project refined state-pair features into latent Hamiltonian and transition-dipole operator representations.
- LUSH removes absolute Hamiltonian scaling by making the predicted matrix traceless, then restores geometry-dependent diagonal offsets from a learned state representation.
- The loss includes mean-square error in atomization energies across states and training samples, using state-resolved reference energies.
- Excitation-energy and oscillator-strength losses use separate weighting scales, with ε = 10^8 eV and ε = 10^8 a.u., respectively.
- Azobenzene geometries are optimized with a Levenberg–Marquardt-type quasi-Newton scheme in delocalized internal coordinates, while MECIs minimize energy-gap and average-energy objectives.
- Azobenzene training geometries combine 423 K Wigner samples around cis and trans minima with nonadiabatic molecular-dynamics configurations.
Supplementary information
The supplementary information provides additional database, atomic-reference-energy, and secondary-results details for QeMFi and azobenzene.
- Supplementary analyses cover the QeMFi and azobenzene databases, atomic reference energies, and secondary results in Figures S1–S6 and Table S1.
Supplementary Materials for
The supplied supplementary-materials passage identifies the paper and its authors but provides no substantive supplementary content.
- The paper is titled “Latent unified smooth Hamiltonians for excited state chemistry.”
- The listed authors are David Juergens, Martin Stöhr, Andreas E. Hillers-Bendtsen, O. Jonathan Fajen, and Todd J. Martı́nez.
Supplementary Figures
The supplementary figures assess LUSH training and behavior across QM9, QeMFi, thymine, and azobenzene, while supplementary tables report azobenzene properties and dataset-specific atomic reference energies.
- QM9: 0.059 eV (~1.36 kcal/mol) validation set U−U_ref mean absolute error is approached by the ~7.05 million-parameter QM9 model.The QM9 models differ in hidden dimension, with d=256 and d=128 configurations.
- QeMFi: 135,000 QeMFi def2-TZVP samples from nine molecules are split randomly into 90% training and 10% test sets for monitoring total loss, energies, gaps, and oscillator strengths.
- Thymine: The thymine crossing-tracking figure compares LUSH with TDDFT along held-out geometries and shows predicted values closely tracking the reference, including an S3/S4 crossing.
- Thymine: 6.53e-03 eV is the smallest S1/S2 energy gap among 15,000 thymine def2-TZVP calculations in QeMFi.
- Azobenzene: The azobenzene supplementary materials compare S0, S1, and S2 potential-energy surfaces and report vertical excitation energies and oscillator strengths for trans and cis forms.
- Reference energies: Atomic reference energies are dataset-specific: Hartree-Fock/cc-pv5z for QM9, a linear fit for QeMFi, and BHLYP/def2-SVP calculations for azobenzene.