Inner-sphere mode

There is a small complication in that the frequency to is different for the reduced and oxidized states so that one has to take an average frequency. Marcus has suggested taking u>av = 2woxu>red/(wox + a>reci)-When several inner-sphere modes are reorganized, one simply sums over the various contributions. The matter becomes complicated if the complex is severely distorted during the reaction, and the two states have different normal coordinates. While the theory can be suitably modified to account for this case, the mathematics are cumbersome. 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

Let us consider the reorganization of an inner-sphere mode. Typically the modes have such high frequencies (hui > kT) that we can assume them to be in their ground state before the reaction.1 Therefore thermal averaging is not required, and Eq. (19.25) simplifies to ... 

If in addition one inner-sphere mode of frequency oj, with Tuo 3> kT, is reorganized, the total rate constant can be written as a sum over partial rates ... 

Both the total rate kox and the partial rates are shown in Fig. 19.3 as a function of the energy change Ae e/ — e. As might be expected, the transitions to excited inner-sphere modes become important only... 

According to a recent model (13) nuclear tunneling factors for the inner-sphere modes can be defined by... 

Figure 2. Initial ( (/a) and final ( J/b) state potential-energy contours for the complete (two-mode) active space the abscissa refers to the inner-sphere mode and the ordinate governs the low-frequency active solvent mode. The difference in frequencies leads to a curved reaction path. Equilibrium coordinate values for the reactant ( j/A) and product ( J/b) states are labeled qA and qB, respectively. For the case of qin, qB° - qA° = Aqin°, as given by Eq. 16.

Considering that -O-H may be a weaker complex than -O-M, formation of the latter would be relatively independent of pH. The latter complex would involve a strong bond (e.g., chemisorption). The same explanation applies to anion adsorption. For example, phosphate (P04) adsorption by oxides may take place in an outer- or inner-sphere mode of the monodentate or bidentate type (Fig. 4.7). 

Rate formulations that treat the inner-sphere mode(s) quantum mechanically and the outer sphere modes classically are used rather widely. The rate expression for a single harmonic quantum mode is... 

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