Tunneling matrix element, electron-transfer

Stuchebrukhov AA (1996) Tunneling currents in electron transfer reaction in proteins. II. Calculation of electronic superexchange matrix element and tunneling currents using nonorthogonal basis sets. J Chem Phys 105(24) 10819-10829... 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

Bardeen showed that to derive the tunneling matrix element, which represents the amplitude of electron transfer between the sample and tip, explicit expressions for the wavefunctions of the tip and sample were... 

Balabin lA, Onuchic JN. Connection between simple models and quantum chemical models for electron-transfer tunneling matrix element calculations A Dyson s equations-based approach. J. Phys. Chem. 1996 100 11573-11580. 

In the early 1990s, in their studies of proton transfer in solution using Marcus rate theory Equation (5), Hynes and coworkers16 17 noticed the following limitation. If Q is the tunneling distance, it can be shown that the tunneling matrix element that appears in Equation (5) has the form A e- e. For typical electron transfer reactions... 

Effects of nuclear dynamics on electron tunneling in redox proteins have been an important question for the biological electron transfer community. While it has been understood how nuclear dynamics controls the Pranck-Condon factor, little was known until now about how the dynamics affects the tunneling matrix element. Our results show that, when tunneling is dominated by a single pathway tube, dynamical effects are small and Pathways level calculations provide reasonable results. The situation changes when several pathway tub are important and destructive interference exists among them. In this case dynamic amplification becomes important,... 

In this formula, V is the electron matrix element for electron tunneling transition, l is the distance between the centres of the D and A particles, a is the width of the charge transfer band, and EmSLX is the position of the maximum of this band. Emax = Eu — EA + A, where (ED — EA) is the difference of the redox potentials of the donor and the acceptor and A is the energy spent on the excitation of the vibrational degrees of freedom. 

It was examined in this chapter that in the standard situation the system is in equilibrium in the initial state therefore, the result of averaging out the populations of the oscillator levels in this state is their replacement with their equilibrium values (see the determination (19)). But very frequently the electron photo-transfer is studied where the initial state is the excited state of donor. The electron matrix element Vif increases exponentially with the growth of the tunneling electron energy (see Chapter 3). So, it is possible that the transfer probability in unit of time becomes bigger than the inverse time of the vibration relaxation in the donor excited state. Then, the transfer occurs from the electron s excited state with the non-relaxed or partly relaxed vibration populations. The high temperature rate constant of the... 

Some important problems of the theory of multi-phonon electron transition were not touched upon in this chapter. These are, first, the calculation of the expression for the electron matrix element at the tunneling transfer, second, the influence of medium on the electron matrix element, and, finally, the investigation of the applicability of Born-Oppenheimer approach in the electron tunneling transfer. These issues will be considered in the next chapter. 

So, the expression for the matrix element for the electron tunneling transfer in a crystal medium has the former form (see Eq. (18)), and the crystal influence is reduced to the strong change in the tunneling exponent (compare expressions (26) and (16)) and, besides, to some changes in the preexponential factor (compare expressions (19) and (27)). 

The consideration of the reactions of the electron tunneling transfer was until now based on Born-Oppenheimer s adiabatic approach (see Section 2 of Chapter 2) that was used for the description of the wave functions of the initial and final states. The electron tunneling interaction V results in the non-adiabatic transition between these states, if the matrix element Vtf... 

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