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

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, $\Delta^2$, derived from the square of the off-diagonal elements of the diabatic Hamiltonian in the 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 $\Delta^2$ 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.