Most probable electron level

The frontier electron level of adsorbed particles splits itself into an occupied level (donor level) in a reduced state (reductant, RED) and a vacant level (acceptor level) in an oxidized state (oxidant, OX), because the reduced and oxidized particles differ from each other both in their respective adsorption energies on the interface of metal electrodes and in their respective interaction energies with molecules of adsorbed water. The most probable electron levels, gred and eqx, of the adsorbed reductant and oxidant particles are separated from each other by a magnitude equivalent to the reorganization energy 2 >. ki in the same way as occurs with hydrated redox particles described in Sec. 2.10. 

As is shown in Eqns. 2-48 and 2-49, the probability density W(e) of electron energy states in the reductant or oxidant particles is represented as a normal distribution function (Gaussian distribution) centered at the most probable electron level (See Fig. 2-39.) as expressed in Eqns. 8-10 and 8-11 ... 

Fig. 8-29. Energy diagram for the most probable electron level, eck, of oxidant particles and the conduction band edge level, Bq, in a cathodic redox electron transfer via the conduction band cathodic current is maximum when cqx equals e. ...

Complexation therefore raises the standard Fermi level of redox electrons ep(KEDcs)> provided that the affinity of ligand coordination is greater with the oxidant particle than with the reductant particle (- dGox > - dG c) whereas, the complexation lowers the standard Fermi level of redox electrons Ef(redox)> provided that the affinity of ligand coordination is smaller with the oxidant particle than with the reductant particle (- dGox < - dG o)- With a shift of the standard Fermi level of redox electrons due to complexation, the most probable electron levels esED and cox of the redox particles are also shifted in the same direction. 

As shown in Fig. 8-34, when the most probable electron level of the reductant particle is higher in the ligand-coordinated state cred(chydrated state ereix i). ibe transfer of anodic electrons occurs at higher energy levels (at less anodic potentials) with the ligand-coordinated reductant particle than with the simply hydrated reductant particle. In such a case the complexation of redox particles will accelerate the anodic transfer of redox electrons. 

FIGURE 22.2 Electron energy levels for a standard pair of hydrated redox particles and for an intrinsic semiconductor ered = the most probable electron level of oxidant, eox = the most probable electron level of reductant, 8p(redox) = standard Fermi level of redox electrons, 8p = Fermi level of an intrinsic semiconductor, v = valence band edge level, and c = conduction band edge level. 

Radiative transitions may be considered as vertical transitions and may therefore be explained in terms of the Franck-Condon principle. The intensity of any vibrational fine structure associated with such transitions will, therefore, be related to the overlap between the square of the wavefunctions of the vibronic levels of the excited state and ground state. This overlap is maximised for the most probable electronic transition (the most intense band in the fluorescence spectrum). Figure... 

Fig. 2-36. Electron energy levels in hydrated oxidant Fe and reduc-tantFe AG = energy to organize hydrate structures dGj t = energy required for dehydrated redox ions to donate or accept gaseous electrons ep.2> o = most probable electron donor level of Fe Spe +.A = most probable electron acceptor level of Fe Hj05,2.,p,j = hydrated structures cgn) = standard gaseous electron level (s 0).

Pig. 2-37. Redox reaction cycle FeJ5 - Fejj + ei iD, - FeJ in aqueous solution solid arrow=adiabatic electron transfer, dotted arrow = hydrate structure reorganization X = reorganization energy ered.d = most probable donor level eox.a = most probable acceptor level. 

The most probable donor level, ered, the most probable acceptor level, eox, and the standard Fermi level, e redox) of redox electrons are characteristic of individual redox particles but the Fermi level, e m dox), of redox electrons depends on the concentration ratio of the reductant to the oxidant, which fact is similar to the Fermi level of extrinsic semiconductors depending on the concentration ratio of the donor to the acceptor. 

For redox reactions due to cathodic transfer of electrons via the conduction band, the cathodic current is expected to be maximum when the most probable vacant level eqx of the oxidant particle is in the same level as the conduction band edge e this cathodic current gradually decreases with increasing separation of eox firom ej. as shown in Fig. 8-29. The same conclusion may also be drown fh>m Eqn. 8-61. 

Figure 47. Idealized energy levels and most probable electronic absorptions, ffabs, of one monomeric and two dimeric dye structures. The arrows represent the directions of the localized transition dipole moments. The transition in the face-to-face aggregate is representative of that in an H-aggregate and the transition in the end-to-end structure is representative of that in a J-aggregate.

In a nickel-containing enzyme various groups of atoms in the enzyme form a complex with the metal, which was found to be in the +2 oxidation state and to have no unpaired electrons. What is the most probable geometry of the Ni2+ complex (a) octahedral (b) tetrahedral (c) square planar (see Exercise 16.96) Justify your answer by drawing the orbital energy-level diagram of the ion. 

Fig. 8. Scattering the transition state from the surface. Measured vibrational distribution of NO resulting from scattering of laser-prepared NO(v = 15) from Au (111) at incidence = 5 kJ mol-1. Only a small fraction of the laser-prepared population of v = 15 remains in the initial vibrational state. The most probable scattered vibrational level is more than 150 kJ mol-1 lower in energy than the initial state. Vibrational states below v = 5 could not be detected due to background problems. These experiments provide direct evidence that the remarkable coupling of vibrational motion to metallic electrons postulated by Luntz et al. can in fact occur. (See Refs. 44 and 59.)...

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. 

Fig. S-S8. Electron levels of dehydrated redox particles, H ld + bh /h = H,d, adsorbed on an interface of metal electrodes D = state density (electron level density) 6 = adsorption coverage shVi - most probable vacant electron level of adsorbed protons (oxidants) eH(d = most probable occupied electron level of adsorbed hydrogen atoms (reductants) RO.d = adsorbed redox particles.

As discussed in Sec. 8.3.5, a redox reaction current due to electron or hole transfer depends not only on the concentration of interfadal electrons or holes at the electrode but also on the state density of the redox electrons or redox holes in the range of energy where the electron transfer takes place. Hence, it is important in the kinetics of electron or hole transfer to realize the level of the band edge Cc or Ev of the electrode relative to the most probable level cred or cox of redox electrons or redox holes in the hydrated redox particles. 

Figure 3.6 shows the various relationships between the energy levels of solids and liquids. In electrolytes three energy levels exist, Ep, redox, Eox and Ered- The energy levels of a redox couple in an electrolyte is controlled by the ionization energy of the reduced species Ered, and the electron affinity of the oxidized species Eox in solution in their most probable state of solvation due to varying interaction with the surrounding electrolyte, a considerable... 

The second and perhaps most probable explanation is damping and broadening of the resonance, due to size dependent, single electron 5d- 6p,6s interband transitions. Their explanation is that the discrete level structure of the Au 55 cluster acts as an effective decay channel. In reducing the plasmon lifetime, it would also strongly increase the bandwidth of the resonance, washing out the resonance peak. 

The basic theory of the kinetics of charge-transfer reactions is that the electron transfer is most probable when the energy levels of the initial and final states of the system coincide [5] following the Franck-Condon principle. Thus, the efficiency of the redox reaction processes is primarily controlled by the energy overlap between the quantum states in the energy bands of the semiconductor and the donor and acceptor levels of the reactants in the electrolyte (Fig. 1). In the ideal case, the anodic current density is given by the... 

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