Generalized solvent coordinate

Time-resolved fluorescence spectroscopy of polar fluorescent probes that have a dipole moment that depends upon electronic state has recently been used extensively to study microscopic solvation dynamics of a broad range of solvents. Section II of this paper deals with the subject in detail. The basic concept is outlined in Figure 1, which shows the dependence of the nonequilibrium free energies (Fg and Fe) of solvated ground state and electronically excited probes, respecitvely, as a function of a generalized solvent coordinate. Optical excitation (vertical) of an equilibrated ground state probe produces a nonequilibrium configuration of the solvent about the excited state of the probe. Subsequent relaxation is accompanied by a time-dependent fluorescence spectral shift toward lower frequencies, which can be monitored and analyzed to quantify the dynamics of solvation via the empirical solvation dynamics function C(t), which is defined by Eq. (1). 

Fig. 1 Potential energy i s. generalized solvent coordinate q in Marcus theory.

These ideas can be put into quantitative terms within a simple, one-dimensional model in which the (free) energy of the system is plotted as a function of a generalized solvent coordinate q characterizing the solvation of the reactant. There are two such curves (see Fig. 1), one for each side of Eq. (1). Each curve attains its minimum at the equilibrium configuration for the corresponding state. If we develop each curve into a Taylor series about its minimum and keep terms up to second order, we obtain the harmonic approximation familiar through many branches of physics and chemistry in which each curve is represented by a parabola. We make the further assumption that the curvatures of both parabolas are equal - this can be... 

The main points can be explained within the one-dimensional model outlined in the preceding section in which the free energy is taken to be a function of a generalized solvent coordinate - the two-dimensional... 

We have already emphasized the important role of the solvent dynamics and structure in the kinetics of electron-transfer reactions. During the last few years, a number of classical molecular dynamic (MD) simulations have been performed to obtain free energy surfaces of the reaction [21-27]. These simulations require explicit interaction potentials between the constituents of the system the reactants, the solvent, and the electrode. Again, a generalized solvent coordinate is used to... 

The situation takes on an additional dimension if the positions of the charged or polar groups near the chromophore fluctuate rapidly with time. One way to describe the effects of such fluctuations is to write the energies of the ground and excited electronic states ( and ) as harmonic functions of a generalized solvent coordinate (X) ... 

Fluctuating interactions with the solvent thus broaden the vibronic absorption lines of the chromophore and shift them to higher energies relative to Eg. As discussed above, however, the mean energy of interaction can shift Eg either upward or downward depending on the chromophore and the solvent. We will discuss generalized solvent coordinates further in Chaps. 5 and 10. 

Fig. 8.7 Dependence of the excited-state energy surfaces on a generalized solvent coordinate in cases of strong (A) and weak (B) exciton interactions. Hu and 7/22 (dashed lines) are the excited-state energies of individual molecules, which are assumed to be identical. Eb+ and (solid lines) are the energies of the two exciton states as given by Eqs. (8.9b-8.9d). (Exciton state Fb+ is assumed arbitrarily to have a higher energy than I b- ) Th ground-state energy is off scale at the bottom. In the units of energy used for the ordinate scale, H12 is 0.4 in (A) and 0.04 in (B)...

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