How to correct Ehrenfest nonadiabatic dynamics in open quantum systems: Ehrenfest plus random force (E$+$$\sigma$) dynamics

One key challenge in the study of nonadiabatic dynamics in open quantum systems is to balance computational efficiency and accuracy. Although Ehrenfest dynamics (ED) is computationally efficient and well-suited for large complex systems, ED often yields inaccurate results. To address these limitations, we improve the accuracy of the traditional ED by adding a random force (E$+$$\sigma$). In this work, the construction of random forces is considered in Markovian and non-Markovian scenarios, and we ensure the dynamics satisfy the detailed balance in both scenarios. By comparing our E$+$$\sigma$ with existing methods such as the electronic friction model and surface hopping, we furthermore validate its reliability. In addition, E$+$$\sigma$ model still retains the high efficiency of the ED and does not incur much additional computation. We believe that this method provides an alternative to accurately describe mixed quantum-classical dynamics in open quantum systems, particularly for large complex systems.