Tunneling matrix element, electron-transfer electronic coupling

Several curves shown in Fig. 6 correspond to different transition states of electron transfer reaction. The transition state is the resonance between donor and acceptor electronic states. Such a resonance is achieved in the course of thermal fluctuations of the real system. The fluctuations of the polar medium cause the shifts of the potentials of the donor and acceptor complexes. Since fluctuations are random, the two off resonant levels can move to resonance by moving one level up, or, by moving another level down, or moving one level up and another, simultaneously, down, until their energies match. All such cas can be reali d in reality, so the transition state is not uniquely defined. The question is how different these possible transition states in a real system can be It is clear that in all cases the position of the pair of resonating states with respect to other states in the system will be different. Different will be the barrier that electron tunnels through, and therefore different will be the coupling matrix element for each individual transition state. 

The first quantum-mechanical consideration of ET is due to Levich and Dogonadze [7]. According to their theory, the ET system consists of two electronic states, that is, electron donor and acceptor, and the two states are coupled by the electron exchange matrix element, V, determined in the simplest case by the overlap between the electronic wave functions localized on different redox sites. Electron transfer occurs by quantum mechanical tunneling but this tunneling requires suitable bath fluctuations that bring reactant and product energy levels into resonance. In other words, ET has... 

To be more precise, we assume that there is no coupling in the sense that the matrix element of the perturbation between states of the two systems is very small compared with current bond energies. Then, in addition to the polarization effect just illustrated, and provided that element is not exactly zero, an effect associated with the possibility of electron tunneling can take place charge transfer. 

The magnitude of the electronic matrix coupling element can influence the kinetics of electron transfer in two ways. First, for weak coupling (small H ), the frequency of electron tunneling within the transition state (and hence tne preexponential component of the rate) is given by [2] ... 

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