Δ-Machine learning for coupled potential energy surfaces: application to the three-state coupled Cl₂O⁺ and four-state...

The Δ-Machine learning (Δ-ML) approach has emerged as a powerful tool for constructing molecular diabatic potential energy matrices (DPEMs), which are essential for nonadiabatic dynamics simulations. However, successful cases are merely limited to the two-state coupled reaction systems, and the application of the Δ-ML approach to the multi-state coupled systems has rarely been discussed. In this work, we utilize a neural network (NN) based Δ-ML approach to successfully construct highly accurate DPEMs of multi-state coupled molecular systems, namely, the three-state coupled Cl₂O⁺ and four-state coupled ArNO⁺ systems, which are responsible for describing the A and B bands of the photoelectron spectra of Cl₂O and the charge transfer reaction Ar⁺ + NO(X²Π) → Ar + NO⁺(a³Σ⁺), respectively. The NN-based Δ-ML approach is demonstrated to efficiently build up these two DPEMs with high fidelity, as evidenced by about 80% of the high-level calculation costs saved. Further quantum dynamical calculations show that the A band of the photoelectron spectrum of Cl₂O computed on the new DPEM is in excellent agreement with that obtained using the DPEM constructed with a direct ML approach, validating the high accuracy of the Δ-ML approach for constructing a multi-state coupled DPEM. In addition, the spin-orbit couplings are approximately considered for the four-state DPEM of ArNO⁺, which is suitable for further studying nonadiabatic dynamics for the Ar⁺ spin-orbit state selected charge transfer reactions.