Linearization approximations and Liouville quantum–classical dynamics

We show that linearization methods, commonly used to approximate the evolution of the density operator in mixed quantum–classical systems, can be justified when a small parameter, the ratio of masses of the quantum subsystem and bath, is introduced. The same parameter enters in the derivation of the quantum–classical Liouville equation. Although its original derivation followed from a different formalism, here we show that the basis-free form of the quantum–classical Liouville equation for the density operator can also be obtained by linearization of the exact time evolution of this operator. These results show the equivalence among various quantum–classical schemes.