Energy surface for electron transfer

Figure 6. Diabatic potential energy surfaces for electron transfer reactions in the system AL/B.

Fig. 8. Typical profiles of the potential-energy surfaces for electron-transfer in the isergonic region (for 2 = 1.8 eV). The dotted lines represent the diabatic (non-interacting) states (HDa = 0). Adiabatic states (solid lines) are presented with the values of the electronic coupling element Hda...

Figure 13 Potential energy surfaces for electron transfer reactions. Hamionic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Namre Publishing Group, www.nature.com)...

Figure 11.5 (a) Potential energy surface for electron transfer from a reactant state (S -CB) to a... 

Q (r — fb). In this case, and for transfer of one electron, A(R ) = A(R ) is the difference between the electrostatic potentials at the A and B centers that is easily evaluated in numerical simulations. An example of such result, the free energy surfaces for electron transfer within the Fe i /Fe redox pair, is shown in Fig. 16.5. The resulting curves are fitted very well by identical shifted parabolas. Results of such numerical simulations indicate that the origin of the parabolic form of these free energy curves is more fundamental than what is implied by continuum linear dielectric theory. 

Transient Response. The current analysis performed above demonstrates that H+ ion adsorption at the photoanode/electrolyte interface decreases the electronic energy barrier for electron transfer from the bulk conduction band to the electrode surface. 

Organic liquids with low vapor pressure and liquefied rare gases are amenable to electron injection from the vapor phase (Sato et al., 1956 Sommer, 1964 Watson and Clancy, 1965). Usually, a hot filament is used as the electron source. The electrons are drawn to the liquid surface by an electric field. The principle of the method is depicted in Figure 4a. Depending on the energy barrier for electron transfer at the... 

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

FIGURE 34.8 Free-energy surfaces for the dissociative electron transfer reaction (a) for the solvent polarization (b) along the coordinate r of the molecnlar chemical bond. corresponds to stable molecule in oxidized form. U" is the decay potential for the rednced foim. AFj and AF are the partial free energies of the transition determining mntnal arrangement of the two sets of the free-energy surfaces. 

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... 

Figure 2.9 Model potential energy surface for combined electron and proton transfer. is the solvent coordinate for electron transfer and Q2 that for proton transfer. (See color insert.)...

Figure 2.10 Potential energy surface for combined electron and proton transfer.

Fig. 2 The molecular structure of I—III and the representation of the potential energy surfaces for adiabatic to nonadiabatic electron transfer reactions...

Electrode reactions are inner-sphere reactions because they involve adsorption on electrode surfaces. The electrode can act as an electron source (cathode) or an electron sink (anode). A complete electrochemical cell consists of two electrode reactions. Reactants are oxidized at the anode and reduced at the cathode. Each individual reaction is called a half cell reaction. The driving force for electron transfer across an electrochemical cell is the Gibbs free energy difference between the two half cell reactions. The Gibbs free energy difference is defined below in terms of electrode potential,... 

The classical Morse curve model of intramolecular dissociative electron transfer, leading to equations (3.23) to (3.27), involves the following free energy surfaces for the reactant (Grx-) and product (Gr +x ) systems, respectively ... 

---