Inverted regime

Early studies showed tliat tire rates of ET are limited by solvation rates for certain barrierless electron transfer reactions. However, more recent studies showed tliat electron-transfer rates can far exceed tire rates of diffusional solvation, which indicate critical roles for intramolecular (high frequency) vibrational mode couplings and inertial solvation. The interiDlay between inter- and intramolecular degrees of freedom is particularly significant in tire Marcus inverted regime [45] (figure C3.2.12)). 

The inverted regime (—AG > X) for strongly exergonic processes leads to a decrease of the ET rate. 

Table 9.4 represents the calculated AG values for the charge separation and charge recombination processes. Hereby, the charge recombination falls into the inverted regime of the Marcus parabola. With these values in hand, it was possible to place the different possible reaction pathways in a state diagram (Fig. 9.25). 

Figure 14 Ti (p, T) vs. temperature for the solvent ethane at fixed density (the critical density) and the theoretically calculated curve. The scaling factor, frequency co, and the hard sphere diameters are the same as those used in the fit of the 34°C density-dependent data, i.e., there are no free parameters in this calculation. Notice the presence of an inverted regime, i.e., a range of temperatures for which the lifetime increases with temperature, contrary to expected behavior. The lifetime peaks at 375 K before decreasing with temperature. Remarkably, the theory captures this phenomenon, though it overestimates the drop in the lifetime with temperature after 375 K.

A more dramatic prediction is the existence of the inverted regime in the dependence of kb a on the driving force Eab = Ea — Eb. Equation (16.59) shows that the rate increases as Eab grows from zero, however beyond Eab = E r this dependence is inverted, and the rate decreases with further increase in this force. Reflection on the geometrical origin of this observation shows it to be related to the way by which the crossing point between two shifted parabolas changes when one of these parabolas move vertically with respect to the other, see Fig. 16.3(a). The verification of this prediction, pictorially displayed in Fig. 16.3(b) has provided a dramatic evidence in support of this picture. ... 

It is of interest to note another characteristic of the inverted regime The anticorrelation between the nuclear shift A. and the activation energy E a as seen from Eq. (16.49) and from the fact that E scales like A. In this limit of small nuclear... 

The mclusion of high-frequency molecular modes in the spirit of Eq. (16.62) is critical for the quantitative aspects of this behavior, in particular in the inverted regime (see Fig. 13 in the Review by Bixon and Jortner cited at the end of this chapter). 

In this system electron transfer is believed to be in the inverted regime. The transfer rate is seen to correlate with the solvent dielectric relaxation when this relaxation is fast enough, but decouples from it in a slow solvent, in this case triacetine. 

Figure 9.20. Population of the reactant state as a function of time normalized with respect to its stationary value in the normal (AG = 0) and the inverted (AG = — 2 ,) regimes calculated on the basis of a stochastic Markovian model of reversible ET assisted by a fast vibrational mode with E = ) = 9k T, and KXi/k T= 10. In the normal regime, the reaction is always activated and well described by a single exponential. Dashed line corresponds to the long-time limit in the inverted regime, described by /c j. (Reproduced from [309] with permission. Copyright (2001) by the American Institute of Physics.)...

ET rate via electronic coupling for a multi-dimensional system in the Marcus inverted regime, (a) pEa—6.7, (b) pSa = 10.0, and (c) pEa = 20.0. Ea represents the minimum energy on the seam surface. Solid line present result dashed line the results predicted from the LZ formula dotted line results from perturbation theory. 

As a numerical example, we investigate ET in the Marcus inverted regime to reveal the solvent effect. The potentials in the fast q) and slow (x) coordinates are modeled by two shifted harmonic oscillators, although the approaches can be straightforwardly applied to anharmonic systems. The parameters. 

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