The Probability of Electron Tunneling

The key requirement for a SET step in the photocatalytic process seems to be the surface complexation of the substrate, according to an exponential dependence of the probability of electronic tunneling from the distance between the two redox centers [66]. However, as was pointed out in the preceding section on the key role of back reactions, the presence of a SET mechanism could be a disadvantage from an applicative point of view. If the formed SET intermediate (e.g., a radical cation) strongly adsorbs and/or does not transform irreversibly [e.g., by loss of CO from a carboxylic acid or fast reaction with other species (e.g., superoxide or oxygen)], it can act as a recombination center, lowering the overall photon efficiency of the photocatalytic process. 

Within the approximation of the effective mass, consideration of the field created by the condensed media is confined to substitution of the real electron mass by the effective mass. Precise calculation of the effective mass is equivalent to solution of the Schrodinger equation with the consideration of the field created by the medium, and, consequently, as noted before, is hardly possible. Thus, as far as the problem of electron tunneling is concerned, the effective mass must be considered as a phenomenological parameter. In the case of tunneling with the energy I of the order of 1-5 eV, the field created by the medium apparently increases considerably the probability of electron tunneling, and the effective mass of electron can be noticeably lower than the real mass. 

To calculate the probability of electron tunneling using eqn. (18) it is necessary to know the wave functions F(q) describing the nuclear motion. At room temperature and below, it is possible to consider only the harmonic vibrations of the atoms near the equilibrium positions. Such vibrations are present even in liquids for comparatively long periods of time during which... 

Along with the harmonic approximation, the Franck-Condon approximation is usually also used in calculating the probability of electron tunneling. In accordance with the latter approximation, the exchange matrix element V(q) in eqn. (20) is factorized outside the integral sign... 

Fig. 7. The model calculation of the dependence of the probability of electron tunneling on the distance between the donor and the acceptor [27].

It has been shown in this chapter that the probability of electron tunneling from a donor to an acceptor in condensed media depends on a large number of factors. At present the theory gives quite definite ideas about the scale of influence of the various factors on the probability of tunneling. 

The probability of electron tunneling depends on the mutual orientation of the donor and the acceptor. Unfortunately, the precise character of this dependence for a given reaction is usually unknown. Therefore, analyzing the influence of the angular dependence of the tunneling probability on the overall kinetics of the reaction, it is reasonable to confine oneself to considering the simplest expression... 

DETERMINATION OF PARAMETERS CHARACTERIZING THE DEPENDENCE OF THE PROBABILITY OF ELECTRON TUNNELING ON THE DISTANCE BETWEEN THE DONOR AND ACCEPTOR... 

In Chap. 3, it was shown that the activation energy of electron tunneling reactions diminishes with decrease in temperature. Let us consider the temperature interval in which the activation energy can be considered to be approximately constant. Then, the probability of electron tunneling can be represented as... 

It will be recalled that the kinetic model on the basis of which these equations have been derived implies the presence of a strong dependence of the probability of electron tunneling on distance. Hence one can assume the... 

According to the theory outlined in Chap. 3, the probability of electron tunneling must essentially depend on the motion of atom nuclei in reacting particles. This effect has been experimentally observed [31] for reaction (14). The rate of this reaction has been found to change drastically on deuterating naphthalene. 

Using the data on the tunneling distances obtained from the EPR spectra, the values found from the reaction kinetics for the parameters ae and ve, that characterize the probability of electron tunneling between Fsf - and Vs -centres in MgO are ae = 2.2 A and ve — 104s ... 

From the results of the investigations discussed in this and in previous chapters it follows that tunneling recombination reactions proceed relatively slowly. Thus, at the typical values of the parameters ve = 10ir s 1 and ae = lA in the expression for the probability of electron tunneling, the characteristic tunneling times to distances of R = 10-20 A amount to 10 6-102s. These times are far longer than the characteristic time of diffusional approach of particles separated by the same distances in water solutions ... 

Further, using eqn. (3) from Chap. 5 for the dependence of the probability of electron tunneling from P to Q, one can easily calculate ae 1.4 A and... 

It is well known that excitation of molecules to higher electronic states leads to a decrease in their ionization potential [35]. Therefore it could be expected that excitation of donors will increase the probability of electron tunneling from these donors to acceptors so that at sufficiently high acceptor concentrations (that is at short enough distances between donor and acceptor particles) the sub-barrier electron transfer (tunneling) from a donor to acceptor will compete with the over-barrier transfer. 

The dependence of the probability of electron tunneling in PET reactions on the distance R between the reagents can be well discribed by a simple exponential law W = veexp(—2R/ae). This law is suggested by the theory and allows us to describe quantitatively all the experimental data available so far. 

Theoretical calculations of the dependence of the probability of electron tunneling on mutual orientation of donor and acceptor fragments in bridge molecules were presented in Ref. [304], With the use of a cyclic polyene model to construct wave functions of porphyrin fragments, a number of analytical expressions were obtained for orientation dependence of probabilities of tunneling between these fragments. 

The change of electronic conductivity G(r) over diameter of such two-sphere model composition as element in a system of contacting particles is shown in a Figure 10.6b. The transfer of electron across this composition consists of three stages electron tunneling over the interspace — Rq is replaced by the M/SC conductivity across a particle with subsequent electron tunneling over the further interspace R — Rq. The probability of electron tunneling falls down exponentially with increase in distance from the surface of particle. 

In structures of this kind, the probability of electron tunneling depends on the particle size, height and width of tunneling barriers, and ambient... 

When a metal such as Na is dissolved in a molten salt such as NaCl, it is found that 1 mol% of the metal gives rise to significant electronic conductance. Utilize the equation k = FNu, where IV is the number of moles ofNa per cubic centimeter and u is themobility (0.4 cm V s" ). What would be the average distance apart of the Na atoms On the basis of this distance and an approximation square-well model, calculate the probability of electron tunneling between a K atom and a... 

The net interfacial electron flux is the result of many discrete tunneling events between electron levels in phases A and B. Usually, electron tunneling is an elastic process it occurs between an occupied and an empty electron level of the same energy. Hence, the probability of electron tunneling depends on the distribution of the electron levels at both sides of the interface, and on their occupancy. The electronic structure of solids is thus of primary importance for the kinetics of interfacial electron transfer. In analogy, electrochemical electron transfer can be regarded as the result of discrete tunneling events between electron levels in a solid (the elec-... 

The Sommerfeld model is a simple quantum-mechanical model which takes the Pauli principle into account. It is sufficient for developing a model for the probability of electron tunneling events. It will therefore be discussed in some detail in the next section. A more detailed discussion can be found in Ref. [11]. 

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