Reorganization energy vibrations

Fig. 11. (a) Diagram of energy levels for a polyatomic molecule. Optical transition occurs from the ground state Ag to the excited electronic state Ai. Aj, are the vibrational sublevels of the optically forbidden electronic state A2. Arrows indicate vibrational relaxation (VR) in the states Ai and Aj, and radiationless transition (RLT). (b) Crossing of the terms Ai and Aj. Reorganization energy E, is indicated. 

The change in the inner-sphere structure of the reacting partners usually leads to a decrease in the transition probability. If the intramolecular degrees of freedom behave classically, their reorganization results in an increase in the activation barrier. In the simplest case where the intramolecular vibrations are described as harmonic oscillators with unchanged frequencies, this leads to an increase in the reorganization energy ... 

In order to simplify the expression for G, one has to employ a sufficiently simple model for the vibrational modes of the system. In the present case, the solvent contribution to the rate constant is expressed by a single parameter E, the solvent reorganization energy. In addition, frequency changes between the initial and final states are neglected and it is assumed that only a single internal mode with frequency co and with the displacement Ar is contributing to G. Thus the expression for G reduces to [124] ... 

In the conversion case, the solvent reorganization energy is very small and thus the one-mode expression [124] for the vibrational overlap factor G is generally adequate such that ... 

Here (x)t and (x)f denote the mean values of the relative coordinate x over the states of the proton in the first and second potential wells, respectively. Equation (107) shows that the inertia effects lead to a decrease of the activation factor in the transition probability due to an increase of the reorganization energy. The greater the mass, m of the tunneling particle and the frequency of the vibrations of the atom, w0, the greater is this effect. The above result corresponds to the conclusion drawn in Ref. 66. 

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. 

In the cases treated in the present paper, we do not have a reorganization energy because, for example as shown in Figures 5 and 10, the two diabatic states between which electron transfer occurs (e.g., the SS a and excited-Rydberg states) cross so close (i.e., within the zero-point vibrational motion of the SS bond) to the minimum on the Rydberg-state surface as to render A essentially zero. In more traditional electron-transfer events, A contains contributions from the... 

There is no such simple expression for the internal reorganization energy Aj, which must be given as a sum of terms related to the vibrational states of the reactants and products... 

Let us now consider the system with an arbitrary spectrum of normal vibrations. In this case normal vibrations should be first divided into classical < T) and quantum (cok > T) vibrations. If the reorganization energy of the classical vibrations exceeds the reaction exothermicity then, neglecting the excitation and absorption of phonons with the frequencies k > T, in the same way as when deriving eqn. (41), i.e. taking into account, for the quantum degrees of freedom, only the transitions (0 - 0), we obtain for the probability of tunneling the expression... 

In the case of local vibrations the shift of the reduced normal coordinates upon transition from the initial to the final state can be of the order of unity or more and even considerably more than unity. Essential changes of vibrational frequencies are also possible. A different picture presents itself in the case of the delocalized vibrations. The number of the delocalized vibrations is of the order of magnitude N, where N is the number of atoms in a crystal. If A is the characteristic shift of the equilibrium position for the delocalized vibrations upon the transition from the initial to the final diabatic term, then the full change of the energy of the delocalized vibrations upon this transition has a value of the order of No A2. This quantity is comparable with the reorganization energy, and thus, along with Er, is of the order of 0.05... 

---