Hellmann-Feynman Theorem Time-Dependent

From (16) we straightforwardly obtain the time-dependent Hellmann-Feynman theorem... 

The above expansion is known as the Kubo relation [14,15] and was developed within the framework of propagator methods. We will look at this alternative approach in section 2.4. When we insert the Kubo relation on the right hand side of the time-dependent Hellmann-Feynman theorem (19) and do the time averaging we obtain... 

The MF force conserves the total quantum-classical energy (2) as established by the time-dependent version of the Hellmann-Feynman theorem [12,72]. For a given wave function of the quantum subsystem the MF force defines a unique classical trajectory. This feature of the MF method is both its major advantage and disadvantage. MF gives an optimal classical description of the exact wave packet that remains localized over the time of experiment. On the other hand, MF fails to capture the splitting of the wave packet that occurs with asymptotically distinct reaction channels, such that quantum states of the solute subsystem are correlated with diverging solvent evolutions. 

From the variational condition (3.9), (t) r satisfies a generalization of the time-dependent Hellmann-Feynman theorem, and if we consider in the Hamiltonian (3.2) an external perturbation bV(co) with amplitude s and periodicity T = In/co, we obtain... 

The time-dependent stationary conditions (3.9) and the corresponding Hellmann-Feynman theorem (3.11) are the basic equations for the determination of the response functions of the molecular solutes in the presence of periodic external perturbations. 

Within the variational time-dependent approach of Sect. 3.1.1, the molecular response functions (3.13) are determined by expanding the time-dependent wave-function >/ (t) > and the time-averaged free-energy functional (3.10) in orders of the perturbation, and by imposing that the variational condition (3.9) is satisfied at the various order. The response functions are then identified by means of the Hellmann-Feynman theorem (3.11), as terms of the expansion of the quasi-free-energy. 

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