Simple tunneling mechanism

These results cannot be explained by a simple tunneling mechanism in which no decay of carriers are assumed before reaching electrodes [68,71]. They... 

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

First, we shall discuss reaction (5.7.1), which is more involved than simple electron transfer. While the frequency of polarization vibration of the media where electron transfer occurs lies in the range 3 x 1010 to 3 x 1011 Hz, the frequency of the vibrations of proton-containing groups in proton donors (e.g. in the oxonium ion or in the molecules of weak acids) is of the order of 3 x 1012 to 3 x 1013 Hz. Then for the transfer proper of the proton from the proton donor to the electrode the classical approximation cannot be employed without modification. This step has indeed a quantum mechanical character, but, in simple cases, proton transfer can be described in terms of concepts of reorganization of the medium and thus of the exponential relationship in Eq. (5.3.14). The quantum character of proton transfer occurring through the tunnel mechanism is expressed in terms of the... 

We should remember (1) that the activation energy of eh reactions is nearly constant at 3.5 0.5 Kcal/mole, although the rate of reaction varies by more than ten orders of magnitude and (2) that all eh reactions are exothermic. To some extent, other solvated electron reactions behave similarly. The theory of solvated electron reaction usually follows that of ETR in solution with some modifications. We will first describe these theories briefly. This will be followed by a critique by Hart and Anbar (1970), who favor a tunneling mechanism. Here we are only concerned with fe, the effect of diffusion having been eliminated by applying Eq. (6.18). Second, we only consider simple ETRs where no bonds are created or destroyed. However, the comparison of theory and experiment in this respect is appropriate, as one usually measures the rate of disappearance of es rather than the rate of formation of a product. 

On the other hand, the low-conductance values (L) give a poor linear correlation of the molecular length with an approximate decay constant fiN 0.45 0.09, distinctively different from the H and M sequences. The estimated value of fiN(L) is rather close to results reported by Cui [28] and Haiss [243]. Haiss et al. [244] found a pronounced temperature dependence of these L values, which scales logarithmically with 7 1 in the temperature range 293-353 K, indicating a transport mechanism different from a simple tunneling model. 

Electron transfer from the excited states of Fe(II) to the H30 f cation in aqueous solutions of H2S04 which results in the formation of Fe(III) and of H atoms has been studied by Korolev and Bazhin [36, 37]. The quantum yield of the formation of Fe(III) in 5.5 M H2S04 at 77 K has been found to be only two times smaller than at room temperature. Photo-oxidation of Fe(II) is also observed at 4.2 K. The actual very weak dependence of the efficiency of Fe(II) photo-oxidation on temperature points to the tunneling mechanism of this process [36, 37]. Bazhin and Korolev [38], have made a detailed theoretical analysis in terms of the theory of radiationless transitions of the mechanism of electron transfer from the excited ions Fe(II) to H30 1 in solutions. In this work a simple way is suggested for an a priori estimation of the maximum possible distance, RmSiX, of tunneling between a donor and an acceptor in solid matrices. This method is based on taking into account the dependence... 

We will consider the general features of a emission, and then we will describe them in terms of a simple quantum mechanical model. It turns out that a emission is a beautiful example of the quantum mechanical process of tunneling through a barrier that is forbidden in classical mechanics. 

The model put forward above for proton transfer by a tunneling mechanism based on naphthol/NH3 cluster data is both simple and sufficient to explain (nearly quantitatively) the observed cluster behavior. This same model can be used to explain cluster matrix isolation behavior as well (Brucker and Kelley 1987a,b, 1988, 1989a,b,c Brucker et al. 1991 Swinney and Kelley 1991). 

Regardless of the precise value of /(P BPh ) it is clear that the values obtained are much (some three orders of magnitude) smaller than expected from the rate of charge separation (2.8 ps [116]) when simple tunneling theory is applied [50,57,113,117,118]. This also seems to indicate the need for another intermediate (which need not function as a true electron acceptor but could act as a transmitting medium via a superexchange mechanism). 

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]. 

We refer to Fig. 6-6 and Section 3.2. in Chapter 8 for a discussion of the physical nature of this mechanism. The following simple tunneling current form, cf Chapter 8 catches presently the essence ... 

Theoretical models beyrmd the simple BCS mechanism are discussed to elucidate the inconsistency between the tunneling results inside the vortex core and other data... 

This type of motion, where the electron is chemically forbidden in certain regions of space, resembles the tunneling mechanism in physics, first discovered in radioactive nuclear decays. Since the term tunneling has been an accepted name for decades, there is no reason to adopt another name. However, the original tunneling model is a qualitative model. In ordinary quantum chemical calculations, there is no simple way to calculate tunneling barriers for the electron and there is also no reason to do so. Quantitative results can be obtained with the help of molecular orbital (MO) methods. The electron tunneling model is based on the overlap between the D and A wave functions and this still holds true in MO models. 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

The take-home message here is that conductivity measurements in single-molecule junctions are difficult to analyze without the support of quantum mechanical calculations that include the metal electrodes. This is very much the domain of specialists, and the simple rules discussed for analyzing elastic tunneling spectra in other junction types generally do not apply for metal-single-molecule-metal junctions. 

Complications that arise with this simple reaction are twofold. First, because of the low mass of the hydrogen atom its movement frequently exhibits non-classical behavior, in particular quantum-mechanical tunneling, which contributes significantly to the observed kinetic isotope effect, and in fact dominates at low temperature (Section 6.3). Secondly, in reaction 10.2 protium rather than deuterium transfer may occur ... 

Consider path integrals to arise from a foundation built from the solutions of two elementary (meaning simple yet profound) quantum mechanical problems, a one-dimensional tunneling problem [66], and a propagation in two or more dimensions by more than one alternative path [67]. 

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