How to correct Ehrenfest nonadiabatic dynamics in open quantum systems: Ehrenfest plus random force (E + σ) dynamics.

The article focuses on improving the accuracy of Ehrenfest dynamics (ED), a computationally efficient method for simulating mixed quantum–classical dynamics in open quantum systems, by incorporating a random force term, resulting in the E + σ method. This approach constructs random forces under both Markovian and non-Markovian scenarios to restore detailed balance, addressing ED's known limitations in capturing statistical fluctuations between electronic and nuclear degrees of freedom. Using the Anderson–Holstein model as a test case, the study compares E + σ with established methods such as electronic friction (EF) and surface hopping (SH), demonstrating that the non-Markovian variant (NM-ED) achieves reliable accuracy across weak to strong system–bath coupling regimes and under non-equilibrium conditions. The work also presents a general numerical scheme to generate non-Markovian random forces for complex systems, highlighting that updating the random force correlation function at larger time intervals can reduce computational cost without sacrificing accuracy.