Nonadiabatic Molecular Dynamics on Real-Time Excited-State Surfaces via Machine Learning Hamiltonians.

Simulating the coupled, nonequilibrium dynamics of electrons and nuclei is a central challenge in chemistry, physics, and materials science, governing phenomena from photocatalysis to quantum information. The primary bottleneck has been the lack of a general, accurate, and efficient method for modeling the complete excited-state landscape: the potential energy surfaces, forces, and nonadiabatic couplings for multiple electronic states. While machine learning has revolutionized ground-state simulations and shown promise for excited states in molecules, a unified framework that solves the complete multistate problem for general condensed matter systems has remained elusive. Here, we introduce on-the-fly neural network nonadiabatic molecular dynamics (NAMD), a machine learning framework that makes on-the-fly NAMD in solids a reality. By employing an equivariant neural network to predict the system Hamiltonian, the framework delivers excited-state energies, forces, and nonadiabatic coupling vectors at a fraction of the cost of ab initio calculations. Crucially, it allows simulations with hybrid functional accuracy, a level of approach previously inaccessible for NAMD. We showcase its capabilities with three topical examples: correcting order-of-magnitude errors in carrier dynamics predicted by conventional procedure in a MoS_{2}/WS_{2} heterostructure, simulating previously inaccessible photoinduced ferroelectric switching, and capturing real-time polaron formation in TiO_{2} at the hybrid-functional level. On-the-fly neural network NAMD moves beyond the limitations of equilibrium theory, establishing a new paradigm for the predictive, first-principles design of materials operating far from equilibrium.