Beyond the Parabolas

The overlap A can be replaced by A when A 7ip . Equation [63] then leads to the ground-state energy E( Y ) (zero temperature for the electronic subsystem) often used to describe adiabatic heterogeneous 

The paradigm of free energy surfaces provides a very convenient and productive conceptual framework to analyze the thermodynamics and dynamics of electronic transitions in condensed phases. It, in fact, replaces the complex dynamics of a quantum subsystem interacting with a many-body 

Clearly, the MFI description does not capture all possible complicated mechanisms of ET activation in condensed phases. The general question that arises in this connection is whether we are able to formulate an extension of the mathematical MH framework that would (1) exactly derive from the system Hamiltonian, (2) comply with the fundamental linear constraint in Eq. [24], (3) give nonparabolic free energy surfaces and more flexibility to include nonlinear electronic or solvation effects, and (4) provide an unambiguous connection between the model parameters and spectroscopic observables. In the next section, we present the bilinear coupling model (Q model), which satisfies the above requirements and provides a generalization of the MH model. 

It has in fact been anticipated for many years that the CT free energy surfaces may deviate from parabolas. A part of this interest is provoked by experimental evidence from kinetics and spectroscopy. Eirst, the dependence of the activation free energy, Ff , for the forward (/ = 1 ) and backward i = 2) reactions on the equilibrium free energy gap AFq (ET energy gap law) is rarely a symmetric parabola as is suggested by the Marcus equation,Eq. [9]. Second, optical spectra are asymmetric in most cases and in some cases do not show the mirror symmetry between absorption and emission.In both types of experiments, however, the observed effect is an ill-defined mixture of the intramolecular vibrational excitations of the solute and thermal fluctuations of the solvent. The band shape analysis of optical lines does not currently allow an unambiguous separation of these two effects, and there is insufficient information about the solvent-induced free energy profiles of ET. 

Nonlinear solvation (breakdown of assumption 4 in the Introduction) has long been considered as the main possible origin of nonparabolic free 

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