Mixed Quantum–Classical Dynamics with Machine Learning-Based Potentials via Wigner Sampling

Abstract

Machine learning-based approaches for surface hopping (SH) offer the prospect of SH simulations with ab initio accuracy, but with a computational cost more similar to classical molecular dynamics simulations. However, such approaches in the adiabatic basis are difficult due to the need to fit a machine learning model to reproduce the nonadiabatic coupling, which rapidly changes in the vicinity of a conical intersection. Previous approaches have typically dealt with this difficulty by either computing the hopping probabilities using methods that do not require the explicit nonadiabatic coupling or by employing adaptive sampling. In this study, we introduce a new approach using a simple modification of Wigner sampling to generate appropriate training data. Test SH simulations on the two-state spin-boson Hamiltonian system show that Wigner sampling with an appropriately selected data set can reduce the size of the training data set by up to a factor of 7.5 per degree of freedom compared to previously linear sampling-based approaches.

Molecular excited states through a machine learning lens

“…One has to ensure that the ML training set includes nuclear configurations featuring large coupling magnitudes, 29 which can be obtained through a careful selection of training points. These points can be handpicked, 29 or selected by using dense sampling of all points, 38 or sampled in specific regions, 139 or included through active learning. 41,[43][44][45] The coupling vectors tend to infinity at zero energy gaps, therefore it is also helpful to train the model on these vectors multiplied by the energy gap.…”

Excited-state dynamics with machine learning

“…This poses a severe problem for learning them [3] as, on the one hand, the training set may not contain enough (if any at all) points with substantial NAC values and, on the other hand, such a narrow function is intrinsically difficult to learn with ML. Possible solutions are including points with substantial NAC values into the training set [60,69,75] and switching to reference calculations in the region of small gaps, where the probability of large NAC values is larger [69]. An additional simple solution is to multiply values of NACs by the gap 𝐸 M − 𝐸 B to generate easier to learn values [73][74] (Figure 4).…”

“…Optimal coverage here means that the most representative configurations during excited-state processes should be included, enabling a good description of points around conical intersections. To obtain a satisfactory training set, many strategies can be used: sampling from MD trajectories [68]; sampling from conformational space [69] (e.g., by farthest-point sampling [106]); including enough points near conical intersections through handpicking [60] or sampling around conical intersections [75]; excluding problematic data points in critical regions [74].…”

MLatom Software Ecosystem for Surface Hopping Dynamics in Python with Quantum Mechanical and Machine Learning Methods

“…These packages have interfaces to the third-party electronic structure software needed to calculate the energies and forces for the electronic states involved with the quantum mechanical (QM) methods. The progress in machine learning (ML), particularly in the context of surface-hopping dynamics ,,− (see also reviews − ), shows the potential of substituting slow QM with fast ML models for evaluating forces and energies. This potential is underutilized for the LZBL approximation, although there is a growing interest in using ML to accelerate LZBL surface hopping dynamics.…”