Reorganization Energy X

We begin with the solvent reorganization energy, Xsoiv The average position and orientation of a solvent molecule depend on the local charges. Still, it is fair to hope that may be taken into account via the dielectric constant by application of the Bom approximation. In the case of metal complexes, we may assume that they are spherical. In the case of t-systems, this is not, of course, a reasonable approximation and improved expressions have now been derived. 

The dielectric constant depends on phase, temperature, and pressure. For water, the relative dielectric constant is e, = 80 at T = 293 K (20°C). At the freezing point T = 273 K (20° lower), we have e, = 88, and at 40°C (20° higher) we have e, = 72. The dependence on tanperature is thus linear at these temperatures. 

We assume that the two metal complexes are not necessarily identical and let the radii be aj and aj and the distance between them R, counted from the centers. We assume that the charge of the whole complex is increased by one unit. The Bom energy for polarization energy around a charge Z with radius a, may be written as 

In Equation 10.30, e is the relative dielectric constant (=8,). There are two contributions to the dielectric constant from the solvent the polarization due to electronic polarizability when the water molecules are fixed and the polarization caused by the change of geometry and orientation. The first part is an electronic part, which should not be counted since it does not depend on the nuclei and is already (in principle) included in the calculation of the electronic wave function. 

As the system moves from -Qo to Qo, the energy of the upper PES is lowered since the solvent geometry readjusts in the move. Uj is thus a positive contribution to the positive reorganization energy. 

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In the high temperature limit where all the nuclear motions coupled to the process can be described classically, the nuclear factor is expressed in terms of only two parameters the driving force of the reaction AG°, and the whole reorganization energy X (expressions (13) and (14)). Detailed calculations carried out in the case of cytochrome c have demonstrated that AG° is a complex quantity, which depends not only on the electronic properties of the redox centers but also on those of the protein and of the surrounding solvent [100]. Usually, AG can be evaluated from measurements of redox potentials and of eventual interaction energies between the different parts of the systems (Appendix). 

The thermally averaged transition probability above is expressed in a form appropriate for harmonic reactant and product potential energy surfaces, in terms of the harmonic frequency os and reorganization energy X [40, 41]. It is Eq. 4 which provides us with the required connection between interaction... 

Figure 4. Schematic diagram to show the reorganization energy X for nonisotopic reactions for harmonic free energy profiles. This figure shows a normal region activation barrier when-AG° < an activationless situation when -AC =. l.and an inverted region activation barrier when-AG° > A for the harmonic potential inii andGfin represent the initial (reactant) and the final (product) system free energy, respectively.

The localized electron level of hydrated particles in aqueous solutions, different from that of particles in solids, does not remain constant but it fluctuates in the range of reorganization energy, X, because of the thermal (rotational and vibrational) motion of coordinated water molecules in the hydration structure. The electron levels cox,a and esmo are the most probable levels of oxidants and reductants, respectively. 

The most probable levels ered and eox of the reductant and oxidant may be derived from the standard Fermi level, ES)ii Ty>Y. and the reorganization energy, X. As an example. Fig. 2-42 shows the standard Fermi level e e.3,/r,j ) (= - 5.27 eV) with the most probable levels er,2 d hydrated Fe and Fe ions,... 

Figure 8-33 shows the shift of the level of redox electrons that results from the complexation of redox particles. For simple cases, the reorganization energy, X, of redox particles may be assumed nearly the same in the complex redox particles (coordinated with ligands) as in the simply hydrated redox particles (coordinated with water molecules). 

The rates of electron-transfer reactions can be well predicted provided that the electron transfer is a type of adiabatic outer-sphere reaction and the free-energy change of electron transfer and the reorganization energy (X) associated with the electron transfer are known [1-7]. This means that electron-transfer reactions can be designed quantitatively based on the redox potentials and the reorganization energies of molecules involved in the electron-transfer reactions. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

Thus, in order to understand biological electron transfer in a theoretical context, we wish to characterize the parameters which control the prefactor, A, through the distance dependence (the "a" parameter) and characterize the reorganization energy, X. 

Here V is the matrix element which describes the coupling of the electronic states of reactants and products, S is known as the electron-vibration coupling constant which is equal to the inner-shell reorganization energy X- expressed in units of vibrational quanta,... 

Figure 4.2 illustrates the parabolic free-energy surfaces as a function of reaction coordinate. In particular, three different kinetic regimes are shown in accordance with the classical Marcus theory. The reorganization energy, X, represents the change in free energy upon transformation of the equilibrium conformation of the reactants to the equilibrium conformation of the products when no electron is... 

Figure 3.5. Transition from contact to remote transfer in a polar solvent with increasing contact reorganization energy X(a) — A I /-o. Above the energy scheme of electron transfer at different exergonicity. Below the r dependence of the corresponding rates (thick lines) in comparison to an exponential decrease of the normalized tunneling rate V(r) (thin line).

Figure 2 Plots of log vs. AGg-j- calculated by using theory. Solid line calculated by using semiclassical theory [Eq. (4), see text] and dashed line calculated by using quantum theory [Eq. (5)]. Rate maximum occurs when the total reorganization energy (X) is equal to the driving force (e.g., when X = IAGCTI).

Application of Hush theory to the observed IPCT bands yielded information about the relationship between optical and thermal ET in these systems. The redox potentials of both the metal dithiolene donors and the viologen acceptors can be systematically varied, which, in turn, tunes the thermodynamic driving force for electron transfer. The researchers found that the IPCT band energy increases linearly with more positive free energy AG for ET, and that the reorganization energy (x) remains constant with variation in the metal or cation redox potentials (66, 67). 

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