Machine Learning Nonadiabatic Dynamics: Eliminating Phase Freedom of Nonadiabatic Couplings with the State-Intraction State-Averaged Spin-Restricted Ensemble-Referenced Kohn-Sham Approach

TL;DR

The paper tackles phase freedom and singularities in nonadiabatic couplings during excited-state dynamics by formulating a phaseless training target $Δ^2$ derived from the off-diagonal diabatic Hamiltonian within the SSR(2,2) state-interaction SA-REKS framework. It trains three independent equivariant MLPs to predict $E^{PPS}$, $E^{OSS}$, and $Δ^2$, enabling phase-free, smooth diabatic predictions that reproduce ab initio ESMD for PSB3. The approach yields accurate MECI geometries, branching-plane vectors, and NACVs, with dynamics in strong agreement with reference SSR(2,2) simulations and a substantial speed-up (~100×) for nonadiabatic MD. This phaseless ML-ESMD framework advances scalable, multi-state molecular dynamics by mitigating phase-related discontinuities, paving the way for universal ML potentials across complex chemical spaces.

Abstract

Excited-state molecular dynamics (ESMD) simulations near conical intersections (CIs) pose significant challenges when using machine learning potentials (MLPs). Although MLPs have gained recognition for their integration into mixed quantum-classical (MQC) methods, such as trajectory surface hopping (TSH), and their capacity to model correlated electron-nuclear dynamics efficiently, difficulties persist in managing nonadiabatic dynamics. Specifically, singularities at CIs and double-valued coupling elements result in discontinuities that disrupt the smoothness of predictive functions. Partial solutions have been provided by learning diabatic Hamiltonians with phaseless loss functions to these challenges. However, a definitive method for addressing the discontinuities caused by CIs and double-valued coupling elements has yet to be developed. Here, we introduce the phaseless coupling term, $Δ^2$, derived from the square of the off-diagonal elements of the diabatic Hamiltonian in the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS, briefly SSR)(2,2) formalism. This approach improves the stability and accuracy of the MLP model by addressing the issues arising from CI singularities and double-valued coupling functions. We apply this method to the penta-2,4-dieniminium cation (PSB3), demonstrating its effectiveness in improving MLP training for ML-based nonadiabatic dynamics. Our results show that the $Δ^2$ based ML-ESMD method can reproduce ab initio ESMD simulations, underscoring its potential and efficiency for broader applications, particularly in large-scale and long-timescale ESMD simulations.

Figures (5)

• Figure 1: Overall schematic for this work. (a) Architecture of the NequIP-BAM model. (b) Global pooling block: ReLU activation enables learning of the phaseless coupling term $\Delta^2$(bottom), while the identity function shows discontinuities (top). (c) Nonadiabatic molecular dynamics schematic
• Figure 2: Comparison of diabatic Hamiltonian elements (in eV) between predictions from the NequIP-BAM model and reference SSR(2,2) values: (a) $E^{PPS}$ (b) $E^{OSS}$ (c) $\Delta^2$.
• Figure 3: MECI structure and its branching plane vectors: (a) Comparison of MECI structures predicted by the NequIP-BAM model (blue) and calculated by SSR(2,2)/$\omega$PBEh/6-31G* (red). (b) difference gradient vector ($\Vec{g}$) and (a) coupling derivative vector ($\Vec{h}$). The red and blue arrows denote vectors calculated by SSR(2,2)/$\omega$PBEh/6-31G* and predicted by the NequIP-BAM model respectively.
• Figure 4: Scatter plots of NACV predictions from the NequIP-BAM model compared to the reference SSR(2,2) values: Adiabatic energy difference < 0.5 eV (blue) and adiabatic energy differences $\geq$ 0.5 eV (red).
• Figure 5: Analysis of the dynamics of photoisomerization of PSB3: (a) Dihedral angle of the central C=C bond of PSB3 over time in individual trajectories. Black trajectories represent ML-based results, while red trajectories represent reference SSR-based dynamics. (b) Average electronic population with the corresponding time. The blue and red line represents ML and reference SSR population evolution, respectively. The solid line represents the BO population and the dashed line represents the averaged running state.