As described in Paper I 1 , the PLDM approach evolves the reduced density matrix of the system by propagating the forward and backward quantum coefficients (mapping variables), and coupled to the classical equation of motion for the nuclear DOF. Although the original PLDM approach was developed based on the partially linearized path-integral formalism using the MMST mapping representation, it can be recast using forward and backward expansion coefficients c a and ca (see discussions in Paper I 1 ), evolving according to