Eq. 21 can be viewed as taking the mixed quantumclassical approximation on -i ℏ [ ĤS , ρS ] in Eq. 12 then tracing out the classical DOFs R. Eq. 21 can be viewed as a special case of the recently developed density matrix hybrid method that combines MQC methods and master equation approaches, where the master equation is chosen to be Lindblad dynamics.