This was discussed by Gray and Manopolous as the Symplectic integrator of TDSE. For the mapping approaches with an explicit zero point energy factor (such as SQC and spin-LSC ), a similar argument can also be carried out (for example, Eqs. 94-95 in Ref. 57,58). Historically, the PLDM and related mapping dynamics approaches are often solved with a simple velocity verlet algorithm or complete symplectic algorithm (that requires diagonalizing V ), which are expensive to perform when having a large N in HTC Hamiltonian.