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PEACE: Covariant learning of nonadiabatic manifolds with parity-resolved Hamiltonians

Rongzhi Gao, Shuguang Chen, Yang Zhou, GuanHua Chen, Ziyang Hu, ChiYung Yam

arXiv:2610.09576v1physics.chem-phcs.LG

TL;DR

Nonadiabatic molecular dynamics needs learned representations that respect electronic symmetry and consistently relate energies to interstate couplings. PEACE combines a parity-equivariant latent Hamiltonian with a learned electronic connection, and reproduces alkene population dynamics while extending to spin–orbit-coupled simulations.

  • Problem

    Learning nonadiabatic couplings is challenging because they depend on electronic-state phases, can diverge as energy gaps close, and may show geometric-phase behavior.

    Source: Introduction · p. 1

  • Method

    PEACE combines a parity-constrained latent Hamiltonian with a learned electronic connection and uses a common covariant derivative to obtain forces and energy-gap-weighted couplings.

    Source: Introduction · p. 1

  • Results

    Across three alkene benchmarks, PEACE achieved SOTA overall performance, closely matched reference population decay times, and extended to CH2S spin–orbit-coupled dynamics.

    Source: Results · p. 4

  • Takeaways & Limitations

    The ablations show complementary contributions from opposite-parity Hamiltonian blocks and the electronic connection, while low training loss alone does not establish physical fidelity.

    Source: Discussion · p. 4

Abstract

from arXiv · show

Nonadiabatic molecular dynamics provides mechanistic insight into light-driven processes and informs the design of molecules and materials for solar energy conversion, photocatalysis and photo switching. Accurately describing these processes requires a representation that respects electronic symmetry and consistently relates energies to interstate couplings. Here we introduce PEACE, which combines a parity-equivariant latent Hamiltonian with a learned electronic connection. Controlled ablations reveal the complementary roles of symmetry-allowed state mixing and electronic-frame variation in reproducing crossing structures and relaxation dynamics. PEACE closely reproduces excited-state population dynamics from first-principles simulations, while its extension to spin-orbit coupling enables simulations of intersystem crossing. These results demonstrate that a more complete incorporation of the underlying physics into learned electronic representations leads to more accurate predictions of nonadiabatic dynamics.

Introduction

PEACE combines a parity-constrained latent Hamiltonian with a learned electronic connection to represent energies and nonadiabatic couplings consistently. This addresses challenges in learning couplings and captures both symmetry-dependent state mixing and electronic-frame variation.

  • Nonadiabatic couplings can depend on electronic-state phases, diverge near closing energy gaps and show double-valued geometric-phase behavior.Energy-gap weighting removes the explicit inverse-gap factor but generally leaves a non-conservative coupling field.Source: Introduction · p. 1
  • Existing flexible coupling representations can predict energies and couplings through separate readouts without explicitly connecting them through a shared electronic Hamiltonian.SchNarc’s scalar-gradient formulation also imposes a conservative-field approximation and can suppress symmetry-allowed out-of-plane components at planar geometries.Source: Introduction · p. 1
  • Hamiltonian elements must respect electronic-state symmetry, while derivative couplings reflect both eigenvector rotation and changes in the underlying electronic basis.A strictly diabatic basis is not generally available for a finite electronic subspace of polyatomic molecules, so residual couplings can influence dynamics.Source: Introduction · p. 1
  • PEACE combines a latent Hamiltonian and learned electronic connection, using explicit parity constraints and a common covariant derivative for energies, forces and nonadiabatic couplings.Source: Introduction · p. 1

Results

PEACE combines a parity-equivariant latent Hamiltonian with a learned electronic connection to represent energies, forces and nonadiabatic couplings consistently. Ablations, alkene benchmarks and a spin–orbit extension demonstrate its performance across crossing structures, dynamics and coupling predictions.

The architecture of PEACE

  • PEACE represents reflection symmetry with parity-labelled Hamiltonian blocks and combines Hamiltonian eigenvector rotation with a learned electronic connection to derive forces and smoothed nonadiabatic couplings.The model jointly trains on energies, forces and SNACs; the connection preserves conservative forces while expanding the coupling representation.Source: Results · p. 2

Ablation of odd-parity and electronic connection

  • Removing the connection increased SNAC MAE from 0.117 to 0.410 eV Å-1 and degraded coupling directions, although the localized S1/S2 gap minimum remained.The simulations also showed slower S1 depletion; the explicit connection improves derivative-coupling representation beyond Hamiltonian eigenvector variation.Source: Results · p. 3, Figure 2 · p. 3
  • Without opposite-parity Hamiltonian blocks, test SNAC MAE rose to 0.257 eV Å-1 and the localized gap minimum became an extended low-gap valley, with distorted couplings and persistent excited-state populations.The connection contributes to derivative couplings, whereas the Hamiltonian determines the energy-gap landscape and opposite-parity blocks enable symmetry-allowed state mixing.Source: Results · pp. 3–4, Figure 2 · p. 3
  • Removing both components raised SNAC MAE to 0.687 eV Å-1 and strongly suppressed S2 relaxation despite an energy MAE of 0.033 eV; full PEACE reproduced the S2-to-S1-to-S0 population sequence.The ablations show that low training loss and small energy errors alone do not establish physical fidelity.Source: Figure 2 · p. 3, Results · p. 4

Benchmarking alkene photo dynamics

  • Across C2H4, C3H6 and C4H8, PEACE achieved the best overall benchmark performance, with energy MAEs of 0.008–0.022 eV and force MAEs of 0.036–0.054 eV Å-1.Fitted S1 decay times differed from reference values by at most 1.9 fs across the three alkenes.Source: Results · p. 4
  • Under the tested hardware configurations, PEACE inference took 7.66–12.44 ms per geometry versus 6.77–13.39 s for the CASSCF workflow, a hardware-specific wall-clock ratio of about 1,000.PEACE used one NVIDIA H100 GPU; the CASSCF workflow used six CPU workers on an Intel Xeon Platinum 8480C system.Source: Results · p. 4

Spin–orbit couplings and intersystem crossing

  • On CH2S, PEACE predicted complex spin–orbit matrix elements with 0.265 cm-1 MAE and 0.397 cm-1 RMSE, and enabled 3-ps surface hopping with 95.4% S1 and 2.8% T1 populations.The extension retains separate singlet and triplet Hamiltonian and connection representations and adds an equivariant spin–orbit readout.Source: Results · p. 4

Discussion

The controlled ablations show that symmetry-allowed Hamiltonian mixing and electronic-frame variation contribute complementarily to crossing structure and coupling predictions. Future extensions target condensed environments, active learning and optional state-overlap supervision.

Discussion

  • Controlled ablations show that opposite-parity Hamiltonian blocks and the electronic connection make complementary contributions to crossing structure and coupling predictions, so physical fidelity requires more than low training loss.The discussion recommends assessing local crossing behaviour and ensemble relaxation alongside static prediction errors.Source: Discussion · p. 4
  • Extending PEACE to condensed environments will require suitable descriptions of environmental and intermolecular interactions; active learning and optional state-overlap supervision are also proposed.The proposed extensions aim to preserve efficient inference while broadening the photochemical and photophysical processes accessible to PEACE.Source: Discussion · p. 4

Methods

PEACE represents electronic states with a parity-resolved latent Hamiltonian and a learned electronic connection. The methods use these components to derive energies, forces and couplings, extend the representation to spin–orbit coupling, and evaluate it in trained models and surface-hopping simulations.

Parity-resolved and covariant representations

  • PEACE predicts a real symmetric latent Hamiltonian and real antisymmetric electronic connection, then uses a common covariant derivative for forces and smoothed nonadiabatic couplings.The Hamiltonian is diagonalized for adiabatic energies; the connection represents changes in the latent electronic frame.Source: Methods · pp. 4–5

Nonadiabatic molecular dynamics

  • The ensemble-population comparison covers C2H4, C3H6 and C4H8 over 100 fs and CH2S over 3 ps, with PEACE and reference curves shown.The alkene reference ensembles contain 200 trajectories each, while the PEACE ensembles contain 919, 906 and 920 trajectories, respectively.Source: Figure 3 · p. 5

Parity-resolved and covariant representations

  • Opposite-parity Hamiltonian elements vanish at reflection-invariant geometries but can develop away from symmetry, allowing latent states of opposite parity to mix.The first-order term captures local coupling onset, while the model retains nonlinear geometric dependence at asymmetric geometries.Source: Methods · p. 6

Spin-orbit couplings

  • The spin–orbit extension assembles singlet and triplet Hamiltonians with symmetry-appropriate SOC vectors into a complex Hermitian Hamiltonian.Singlet–singlet and same-root triplet SOC blocks are zero in this representation; reverse SOC blocks are Hermitian conjugates.Source: Methods · p. 6

Reference data and training

  • Training fits energies, forces, smoothed nonadiabatic couplings and energy gaps, with auxiliary parity-sector and signed-gap supervision for confidently labelled symmetric configurations.The SNAC loss minimizes over one consistent set of electronic-state signs for all state pairs within each geometry.Source: Methods · pp. 6–7

Nonadiabatic molecular dynamics

  • Surface-hopping ensembles compare PEACE with reference trajectories for alkenes and CH2NH2+, while CH2S trajectories include both nonadiabatic and spin–orbit couplings.Alkene and CH2NH2+ trajectories run for 100 fs; CH2S trajectories run for 3 ps, with a 0.5 fs nuclear step and 25 electronic substeps.Source: Methods · p. 7

Reference data and comparison models

The study assembled molecule-specific reference datasets and retrained comparison models under matched training settings. The datasets span three alkenes and CH2NH2+, with an external CH2NH2+ geometry set also described.

Reference datasets

  • The alkene datasets contain 7,500, 7,500 and 15,000 geometries for C2H4, C3H6 and C4H8, respectively, at SA(3)-CASSCF(2,2)/cc-pVDZ.The alkene reference calculations follow the electronic-structure level used in the SPaiNN benchmark.Source: Reference data and comparison models · p. 9
  • The CH2NH2+ ablation dataset contains 4,000 MR-CISD/aug-cc-pVDZ geometries, and its external set contains 770 geometries from the SchNarc benchmark.Training, validation and test counts are given in Supplementary Table 2.Source: Reference data and comparison models · p. 9

Comparison models

  • Alkene and CH2NH2+ comparison models were retrained on the same datasets, splits, batch size and epoch count as PEACE.The CH2S reference labels were recomputed at CASSCF(6,5)/def2-SVP, and its spin–orbit couplings used atomic mean-field spin–orbit integrals.Source: Reference data and comparison models · p. 9

Timing and numerical symmetry tests

The timing benchmark compares PEACE with CASSCF under specified hardware and workflow conditions, so its reported ratios are configuration-specific. Numerical tests separately assess transformation consistency under rotations, reflections, and translations.

Timing and numerical symmetry tests

  • Timing ratios are specific to the tested CPU/GPU configurations and workflows, which include energy, force, and coupling evaluation plus interface and data-transfer work.The benchmark used 4,000 configurations from 20 complete trajectories per alkene; PEACE inference ran at batch size one after warm-up, with three repeats.Source: Timing and numerical symmetry tests · p. 9, Figure 14 · p. 16
  • Learning curves compare PEACE with connection and opposite-parity-block ablations using validation loss and RMSEs for energies, forces, and SNACs.Models used 2,500 training and 250 validation geometries for 1,000 epochs, with stars marking validation-selected checkpoints.Source: Figure 1 · p. 9
  • Supplementary Fig. 15 reports FP64 numerical symmetry-consistency tests evaluating relative errors in energies, forces, and smoothed nonadiabatic couplings across 32 molecular geometries subjected to rotations, reflections, and translations.Source: Figure 15 · p. 16
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