Perturbation theory electron-transfer reactions

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

Returning to equation (38), in the limit that ve vn, Ke = 1 and vet = vn. Electron transfer reactions that fall into this domain where the probability of electron transfer is unity in the intersection region have been called adiabatic by Marcus. Reactions for which Kei < 1, have been called non-adiabatic . In the limit that ve 2vn and e = vjvn, the pre-exponential term for electron transfer is given by vet = ve. This was the limit assumed in the quantum mechanical treatment using time dependent perturbation theory. 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

For photo-induced electron transfer (ET) reactions [53], there exist three cases depending on their mechanism (1) non-adiabatic, diabatic, or weak coupling case, (2) adiabatic, or strong coupling case, and (3) charge transfer complex case. This section shall focuses on case (1) to which perturbation theory can be applied. 

Quantum mechanical approaches for describing electron transfer processes were first applied by Levich [4] and Dogonadze, and later also in conjunction with Kuznetsov [5]. They assumed the overlap of the electronic orbitals of the two reactants to be so weak that perturbation theory, briefly introduced in the previous section, could be used to calculate the transfer rate for reactions in homogeneous solutions or at electrodes. The polar solvent was here described by using the continuum theory. The most important step is the calculation of the Hamiltonians of the system. In general terms the latter are given for an electron transfer between two ions in solution by... 

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