Tunneling matrix element theory

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

The operator U shifts the qj oscillator coordinate to its equilibrium through the distance QoCj/cOj, the sign depending on the state of the TLS. All the coupling now is put into the term proportional to the tunneling matrix element and the small parameter of the theory is zIq rather... 

The STM postulated tunneling matrix element distribution P(A) oc 1 /A implies a weakly (logarithmically) time-dependent heat capacity. This was pointed out early on by Anderson et al. [8], while the first specific estimate appeared soon afterwards [93]. The heat capacity did indeed turn out time dependent however, its experimental measures are indirect, and so a detailed comparison with theory is difficult. Reviews on the subject can be found in Nittke et al. [99] and Pohl [95]. Here we discuss the A distribution dictated by the present theory, in the semiclassical limit, and evaluate the resulting time dependence of the specific heat. While this limit is adequate at long times, quantum effects are important at short times (this concerns the heat condictivity as well). The latter are discussed in Section VA. 

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. 

As shown in Fig. 7.2, at a shorter proton-proton separation (/ <16 a.u. or <8 A), the electron in the Is state in the vicinity of one proton has an appreciable probability of tunneling to the Is state in the vicinity of another proton. The tunneling matrix element can be evaluated using the perturbation theory we presented in Chapter 2. A schematic of this problem is shown in Fig. 7.3. By defining a pair of one-center potentials, Ul and Ur, we define the right-hand-.side states and the left-hand-side states. Because the potential Ul is different from the potential of a free proton, Uro, the wavefunction i ii, and the energy level Eo are different from the Is state of a free hydrogen atom. (The same is true for Ur and We will come back to the effect of such a distortion later in this section. 

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... 

A straightforward goal of theories describing ET in proteins is to predict the value of the tunneling matrix element Tda- This discussion goes a step further, to define what it is about the structure of the protein that determines this value. 

This kind of semi-phenomenological theory is unsatisfactory from at least two points of view (1) No theory exists for the tunnel matrix elements that would take the complex quasiparticle structure on both sides in account and (2) the order parameter near the physical barrier behaves inhomogenously so that - if the corresponding regions are not formally included into the barrier - only the semiclassical theory of superconductivity appears appropriate (Ashauer et al. 1986). Furthermore, for reliable estimates of the size of tunnel currents, the orientation of interfaces relative to the crystal axes (Geshkenbein and Larkin 1986) and the influence of inhomogeneity on the spin-orbit coupling (Fenton 1985) may have to be taken properly into account. 

The adiabatic approximation is one of the keystones on which the theory of electron tunneling is based (see Sect. 2). In particular, the matrix element for the transition between the initial and the final electron states contain the adiabatic wave functions of the donor and acceptor. Adiabatic approximation is known [25] to have a very high degree of accuracy. Because of this the non-adiabatic effects have been neglected until recently in the theory of electron tunneling without detailed analysis of whether this can actually be done. In the present section we shall try to fill in this blank and to discuss to what extent the non-adiabatic effects can influence the process of electron tunneling. 

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 description of the theory of electron tunneling transfer is logically accomplished in this chapter - it describes the methods of calculation of the electron matrix element, whereas the methods of calculation and the form of the vibration part of the transition probability was represented in Chapter 2. Besides, in Chapter 3 the procedure of the calculation of the rate constant of tunneling transfer in the conditions of the violation of Born-Oppenheimer s approach is examined. The basic results of this chapter may be formulated as follows. 

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