In this work, we use the TSH method to perform the on-the-fly non-adiabatic dynamics for the realistic molecular system coupled to the cavity. Here, we briefly describe the TSH dynamics for the coupled moleculecavity hybrid system, whereas the details can be found in our previous work. We use the fourth-order Runge-Kutta method to integrate the propagation of the quantum subsystem with the Lindblad dynamics, through the EOMs presented in Eq. 27a and Eq. 27b. Specifically, for each nuclear time step, we propagate the quantum subsystem using Eq. 27a and then propagate using Eq. 27b. The classical subsystem (nuclear DOF) is propagated using Newton's EOM with the velocity Verlet algorithm. In the TSH dynamics, the nuclear force comes from only one specific polariton state |E I (R(t))⟩ (eigenstate of V , see Eq. 17) as follows