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 [Pg.2981]

J. Ulstrup, J. Jortner, Effect of intramolecular quantum modes on free energy relations for electron transfer reactions, J. Chem. Phys., 1975, 63, 4358-4368. [Pg.267]

The situation is simplest if the frequency ft), of the quantum mode is so high that its quantum of energy is much higher than the thermal energy, ficoi / > kT, so that it is not excited in the initial state. In this case, we only have to consider transitions from the ground state 0 to the nth excited state. Let [Pg.580]

When the quantum of energy is of the same order of magnitude as the thermal energy, fico kT, we have to consider transitions from excited initial states m to final states n, and perform a thermal average over the initial states of the quantum modes [Pg.581]

We next consider the case in which the electron-transfer terms are small so that the first-order perturbation theory can be applied and assuming that all the phonon modes are classical - quantum modes will be considered below. In this case, the coordinates of the oscillators can be treated as external parameters. It is then convenient to include the electron-phonon coupling - the last term in Eq. (12) - into the electronic energy of the reactant, and rewrite the first and last term as [Pg.579]

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore [Pg.266]

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