Environmental Engineering Reference
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oscillators, which interact linearly with the reactant. The corresponding Hamiltonian is
written in the form
2 X
n
h v n ( p n þ q n ) n a X
n
H osc ¼ 1
h v n g n q n
(2 : 2)
where n labels the oscillator modes, which have frequencies v n , dimensionless
momenta p n , and coordinates q n ; g n denote the coupling constants. In many cases,
it is not only the solvent that is reorganized during electron transfer, but also inner
sphere modes of the reactants. In a first approximation, these can also be included
in the harmonic oscillator bath. However, often the frequencies of such modes also
change during the reaction; for a systematic treatment of this effect, we refer to the lit-
erature [Schmickler, 1976; Schmickler and Koper, 1999]. To a good approximation,
the solvent coordinates can be considered as classical, and we shall do this here.
Inner sphere modes are typically in the quantum range of frequencies. Such quantum
effect are an interesting topic in their own right [Schmickler, 1996; Kuznetsov, 1995],
but they cannot be considered here. We shall assume all harmonic oscillator modes to
be classical. The coupling of the electron transfer to the solvent can then be character-
ized by a single energy of reorganization defined as l ¼ P n 2 h v n g n .
The total Hamiltonian is the sum of the two terms: H ¼ H e þ 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
V k ¼ 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 2.2 Adiabatic and non-adiabatic electron transfer (schematic). The splitting at the
intersection has been exaggerated.
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