The L-MFE algorithm outlined in Eq. 72 will, in principle, generate different results compared to Eq. 21 where the Lindblad dynamics are propagated deterministically in the Liouville space. This is because the L-MFE method adds random phases to state |0⟩ (for the L = |0⟩⟨1| jump operator) through T (see Eq. 69b) for each individual trajectory. Further, these random phases in c 0,ξ (t) will also influence the magnitude of populations through coupling term V 01 (R ξ )+ε 01 . Thus, L-MFE adds different phases onto ρ 01,ξ (as well as on ρ 10,ξ ) and generates different populations ρ 00,ξ and ρ 11,ξ for each individual trajectory. The L L [ρ] term in Eq. 21, on the other hand, causes deterministic changes for the density matrix elements of all trajectories. Thus, the random phases in L-MFE will influence the nuclear forces in Eq. 19 and these different nuclear forces for each trajectory further influences the nuclear motion for R, and eventually back influence the electronic dynamics through e -i ℏ ( ĤQ+ ĤQC(R))dt in Eq. 72. Future research will focus on investigating the difference between Eq. 21 and Eq. 72. Nevertheless, numerical results obtained for the model systems (Eqs. 76a-76h and Eq. 77) produce visually identical results using either Eq. 21 or Eq. 72.