where dt is the time-step size. Note that other hopping probability schemes, such as the Landau-Zener and global flux 66 formulas can also be utilized here. To satisfy energy conservation, the nuclear velocities after hopping are rescaled along the direction of the corresponding NACV. If the kinetic energy is not sufficient for a hopping event, no action is taken and the trajectory continues to move on the original active PES, which is often denoted as frustrated hopping. The velocity rescaling and frustrated hopping procedures are kept in the original version of CTSH and our new method here, whereas based on different theoretical foundations, the energy conservation constraint can be generalized to the trajectory ensemble level. Since CTSH trajectories are evolved only with adiabatic forces, the error accumulation disappears, and thus the numerical stability is expected to be improved compared to CTMQC. From the perspective of SH, the quantum momentum term introduces quantum decoherence into the EOM of the electronic wave function. Thereby, CTSH provides a good platform to develop methods with higher performance if the potential deficiencies can be revealed. It has been known that CTSH does not fully guarantee the consistency between electronic population and the fraction of trajectories on a given PES. In the following sections, we will show that additional corrections can be introduced to solve this problem.