A Descriptor Is All You Need: Accurate Machine Learning of Nonadiabatic Coupling Vectors

TL;DR

This work tackles the challenge of learning nonadiabatic couplings (NACs) for accurate surface hopping by designing NAC-specific descriptors and a robust phase-correction framework. Using kernel ridge regression, NAC components are learned in a rotated local frame and phased iteratively, with gradient-difference descriptors—especially \Delta\nabla E—proving most informative. When integrated into fully ML-driven FSSH for fulvene (with MS-ANI energies/gradients), the approach achieves $R^2$ near 1, dramatically reduces computational cost (≈434× speedup), and enables large trajectory ensembles with reduced uncertainty. The combination of descriptor design, phase correction, and ML-NAC integration in MLatom offers a scalable path to high-accuracy nonadiabatic dynamics without on-the-fly quantum chemistry computations.

Abstract

Nonadiabatic couplings (NACs) play a crucial role in modeling photochemical and photophysical processes with methods such as the widely used fewest-switches surface hopping (FSSH). There is therefore a strong incentive to machine learn NACs for accelerating simulations. However, this is challenging due to NACs' vectorial, double-valued character and the singularity near a conical intersection seam. For the first time, we design NAC-specific descriptors based on our domain expertise and show that they allow learning NACs with never-before-reported accuracy of $R^2$ exceeding 0.99. The key to success is also our new ML phase-correction procedure. We demonstrate the efficiency and robustness of our approach on a prototypical example of fully ML-driven FSSH simulations of fulvene targeting the SA-2-CASSCF(6,6) electronic structure level. This ML-FSSH dynamics leads to an accurate description of $S_1$ decay while reducing error bars by allowing the execution of a large ensemble of trajectories. Our implementations are available in open-source MLatom.

Figures (6)

• Figure 1: Schematic workflow of our ML-driven FSSH approach. We take the data set from Ref. Martyka2025, which was generated through active learning with an MS-ANI model for energies and energy gradients employed in propagating Landau--Zener surface hopping dynamics and gapMD. Here, we apply phase correction to the data set to learn NACs, which are learned using kernel ridge regression. We show that incorporating gradient differences into the descriptor is essential for achieving high accuracy of ML NACs even without explicitly considering them within active learning. The ML models enable fast and precise predictions of all quantities required for FSSH, allowing for the execution of large ensembles of trajectories and reducing the statistical error in the final excited state populations.
• Figure 2: Phase-correcting procedure of NAC vectors introduced in this work. In the first step, couplings are rotated into a selected reference frame and then scaled by the energy gap. An iterative process is employed where a KRR model is retrained in each iteration. Initially, absolute NAC values are used to determine hyperparameters fixed within the scheme to enhance the procedure. The training uses a 5-fold cross-validation method, with NACs predicted for the remaining part of the data set. Mean squared error is calculated between predictions and positive or negative reference values. If the MSE corresponding to negative reference values is lower than that of the positive ones, the sign of the corresponding NAC in the training set is changed. The iterative procedure continues until there are no sign flips or the process has converged within a predefined patience threshold.
• Figure 3: Descriptor benchmark showing RMSE values for all combinations of investigated descriptors. Relative to equilibrium (RE), energy difference (dE, $\Delta E$), gradient difference (ddgrad, $\Delta\nabla E$), absolute value of gradient difference (adgrad, $|\Delta\nabla E|$), and Frobenius norm of gradient difference (ndgrad, $||\Delta\nabla E||_F$). The error evaluated on three test sets is significantly decreased if ddgrad descriptor is included.
• Figure 4: Phase correction of a 5950-point data set using KRR model with RE and $\Delta\nabla E$ descriptor. Phase correction has been performed iteratively with 5-fold cross-validation. In each iteration, KRR is trained with pre-defined hyperparameters. The number of sign flips (red) decreases steadily, while the average correlation across folds (light green line) increases, indicating improved phase consistency. KRR models retrained with hyperparameter optimization on intermediate data sets are evaluated on three test sets (magenta, orange, and purple lines) and surpass a correlation of 0.9 within 12 iterations and converge above 0.99 in the case of the crucial test set 2, demonstrating the effectiveness of the phase correction procedure.
• Figure 5: Excited-state population of fulvene computed using FSSH and LZSH schemes. (a) The hybrid scheme that combines CASSCF for energies and gradients with machine-learned NACs accurately reproduces the pure CASSCF population, demonstrating the quality of the ML fit. This scheme is denoted "QM($E$, $\nabla E$) + ML-NAC(ddgrad)". The population predicted by the MS-ANI and ML-NAC models (denoted "MS-ANI + ML-NAC(descriptor)") strongly depends on the choice of descriptor. When using the Relative to Equilibrium (RE) descriptor, the population is not correctly captured, whereas the $\Delta\nabla E$ (ddgrad) descriptor yields excellent agreement with the reference pure CASSCF population. (b) The importance of NACs in NAMD simulations is demonstrated by comparing the Fewest Switches Surface Hopping (FSSH) approach with the Landau--Zener Surface Hopping (LZSH) method, which requires only energies and gradients. A key advantage of ML-driven NAMD is the ability to simulate a large ensemble of trajectories (1000 ML vs. 200 QM), reducing the confidence intervals' width. LZSH population data were taken from Ref. Martyka2025. Reference trajectories were computed at the SA-2-CASSCF(6,6)/6-31G(d) level of theory. In FSSH simulations, state coefficients were corrected for decoherence using the SDM method, and upon hopping, velocities were rescaled along the direction of the NACs.