Gaussian function oxidation

Figure 10. VEH-INDO/SCI calculated absorption spectra of sexithiophene in the neutral (solid line), singly oxidized (dotted line), and doubly oxidized (dashed line) states. The spectra are simulated by means of Gaussian functions whose full width at half maximum is set to 0.2 eV.

Table 5.3 Contributions of -orbitals to the total electron density at the iron nucleus (in a.u. ) as a function of oxidation state and configuration. Calculations were done with the spin-averaged Hartree-Fock method and a large uncontracted Gaussian basis set. (17 1 Ip 5d If)... 

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 ... 

However, the Lorentzian form of the dipolar broadening function, which has the advantage of mathematical simplicity, is not suitable for an interpretation in terms of second moments it is replaced with a Gaussian dipolar function S(oa, AG), where the parameters AG correspond to the appropriate fractions of the square root of the intra-group rigid lattice second moments. With appropriate values for AG, calculated and experimental line shapes I(oo) are found to be in a good agreement for cross-linked polyethylene oxide) swollen in chloroform 1U). 

In this study of the 3d metal oxides (ScO - ZnO) the popular hybrid functional B3LYP was used. The Gaussian 94 [20] and Gaussian 98 [21] programs were used. In these, the B3LYP exchange-correlation functional is of the form [22] ... 

The gaussian types of functions (Eqs. 6.37 and 6.38) yield the distribution of electronic levels of the redox system. Wred describes the fluctuation and therefore the distribution of the occupied electronic level and the distribution of the empty level. The density of electronic states is proportional to the concentration of the reduced (Cred) and oxidized species (cq ) of the redox system. Accordingly, the total distribution is given by ... 

In the present author s view, the Gaussian distribution function based on the solvent fluctuation model, which is developed for a simple redox couple, is used too often even when the basic assumption is not valid. For example, this type of distribution function is often drawn for the hydrogen evolution reaction where the oxidized state is H+ and reduced state is H2.105 Certainly the nature of the solvation is completely different between H+ and H2. Moreover, when one considers the kinetics of the hydrogen evolution reaction, one should consider not the energy level of H+/H2 but that of H+/H(a) as Gurney did. 

Recently Nakabayashi et a/.106 107 questioned the validity of the Gaussian distribution function based on the fact that the rate of a highly exothermic electron transfer reaction did not agree with the prediction based on this distribution function and proposed an alternate function (vibronic model), which extends to a smaller energy for the oxidized form and to a larger energy for the reduced form (Fig. 19). 

As was explained in Section 2.9.10, the reduced and oxidized ions of a redox couple interact with the solvent dipoles by ion-dipole interaction. This influences the energy of the electronic states. The fluctuation of the solvent molecules around the ion with only a statistical equilibrium solvation leads to a distribution of the electron energies around a central value of Gaussian form. Two energy distribution functions describe the energy distribution, one for the reduced ions (the occupied states) and the other for the oxidized ions (the unoccupied states). This was shown in Figure 2.33. The development of two different distribution functions is based on stable oxidation states. In each state the ion-dipole interaction can achieve a quasi equilibrium distribution. 

Considerations of interfacial electron transfer require knowledge of the relative positions of the participating energy levels in the two (semiconductor and solution) phases. Models for redox energy levels in solution have been exhaustively treated elsewhere [27, 28]. Besides the Fermi level of the redox system (Eq. 6), the thermal fluctuation model [27, 28] leads to a Gaussian distribution of the energy levels for the occupied (reduced species) and the empty (oxidized species) states, respectively, as illustrated in Fig. 5(a). The distribution functions for the states are given by... 

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