Velocity Rescaling in Surface Hopping Based on Atomic Contributions to Electronic Transitions

Surface hopping is a widely used method for simulating nonadiabatic dynamics, in which nuclear motion follows classical trajectories and electronic transitions occur stochastically. To ensure energy conservation during these transitions, atomic velocities must be adjusted. Traditional velocity rescaling methods either apply a uniform adjustment to atomic velocities, which can lead to size-consistency issues, or rely on nonadiabatic coupling vectors, which are computationally expensive and may not always be available. Here, we introduce two novel velocity rescaling methods that incorporate atomic contributions to electronic transitions, derived from the one-electron transition density matrix or the density difference between states for a given transition. The first method, <i>excitation-weighted velocity rescaling</i>, redistributes kinetic energy among atoms proportionally to their contributions to the electronic transition. This is achieved through a weighted scaling factor, computed from the population analysis of the one-electron transition density matrix or the density difference of the two states involved in the transition. The second method, <i>excitation-thresholded velocity rescaling</i>, adjusts the velocities only of atoms whose contributions exceed a predefined threshold, preventing unnecessary energy redistribution to atoms with minimal involvement in the excitation. We validate these approaches through excited-state dynamics simulations of fulvene and 1<i>H</i>-1,2,3-triazole. Our results show that excitation-weighted velocity rescaling closely reproduces the adjustments based on nonadiabatic coupling vectors for both fulvene and 1<i>H</i>-1,2,3-triazole.