where L [ρ(t)], L Eh [ρ(t)], and L R [ρ(t)] are defined in Eq. A6, Eq. A20, and Eq. A24, respectively. The dynamics associated with the relaxation component L R [ρ(t)] in Eq. A24 cannot be captured through deterministic dynamics. This is because the off-diagonals must be decohered independent of the diagonals, which requires the use of a random phase that cannot appear in a deterministic method governed solely by a Hamiltonian. Instead of using a modified Hamiltonian in the relaxation stage of the Ehrenfest+R method, the electronic expansion coefficients {c i,ξ } of state i and trajectory ξ at each time step are directly modified to attempt to achieve the correct Lindblad dynamics. These coefficient modifications for states |0⟩ and |1⟩ can be summarized as