Linear response theory classical form

The spectra s (v) will be described here in terms of a linear-response theory. We shall employ the specific form [GT, VIG] of this theory, called the ACF method, which previously was termed the dynamic method. The latter is based on the Maxwell equations and classical dynamics. A more detailed description of this method is given in Section II. Taking into attention the central role of the model suggested here, we, for the sake of completeness, give below a brief list of the main assumptions employed in our variant of the ACF method. 

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... 

The present contribution concerns an outline of the response tlieory for the multiconfigurational self-consistent field electronic structure method coupled to molecular mechanics force fields and it gives an overview of the theoretical developments presented in the work by Poulsen et al. [7, 8, 9], The multiconfigurational self-consistent field molecular mechanics (MCSCF/MM) response method has been developed to include third order molecular properties [7, 8, 9], This contribution contains a section that describes the establisment of the energy functional for the situation where a multiconfigurational self-consistent field electronic structure method is coupled to a classical molecular mechanics field. The second section provides the necessary background for forming the fundamental equations within response theory. The third and fourth sections present the linear and quadratic, respectively, response equations for the MCSCF/MM response method. The fifth 283... 

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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