Linear response theory quantum form

The quantum mechanical forms of the correlation function expressions for transport coefficients are well known and may be derived by invoking linear response theory [64] or the Mori-Zwanzig projection operator formalism [66,67], However, we would like to evaluate transport properties for quantum-classical systems. We thus take the quantum mechanical expression for a transport coefficient as a starting point and then consider a limit where the dynamics is approximated by quantum-classical dynamics [68-70], The advantage of this approach is that the full quantum equilibrium structure can be retained. 

Interaction between quantum systems and classical flelds is not problematic. It is the basis of almost all forms of optical spectroscopy where the transition dipole operator of the system interacts with the electric and magnetic flelds of light. It is a necessary ingredient of linear response theory, and also of the Redfleld relaxation mechanism. The starting point for all these examples is the quantum Liouville equation... 

This formula can be derived either from classical considerations or by using quantum mechanical perturbation theory. In either case it is clear that only linear polarizabilities are included that is, the induced moments described by Eq. (29) are a linear response to the fields and field gradients of the neighbouring molecules. For a complete description we should also allow for contributions to the induced moments that depend on quadratic and higher powers of the electric fields. For a uniform field, we should use, instead of Eq. (24), the form... 

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