Tunneling theory

Figure 12-5. Kcprcscmauun of Uie calculated injcciiou curretu on a 111 j vs scale. Tlic dashed line indicates tile slopes predicted by Fowler Nordheiin tunneling theory lor A=0.8eV assuming that the effective mass equals the free electron mass.

The only (to the best of our knowledge) theoretical treatment of hydrogen transfer by tunnelling to explicitly recognise the role of protein dynamics, and relate this in turn to the observed kinetic isotope effect, was described by Bruno and Bialek. This approach has been termed vibration-ally enhanced ground state tunnelling theory. A key feature of this theory... 

The activation energy of most of the eh reactions, 3.5+0.5 Kcal/mole, is much less than the hydration energy of the electron, -40 Kcal/mole. There are other barriers against reaction, such as repulsion by electrons in molecules. This can only be an accident in the classical mechanism, but not in electron tunneling theory as long as the reaction is exothermic overall. 

Even Anderson et al. [39] pointed out that an important consequence of the tunnelling model was the (logarithmic) dependence of the measured specific heat on the time needed for the measurement of c. The latter phenomenon was due to the large energy spread and relaxation time of TLS. In 1978, Black [45], by a critic revision of the tunnelling theory, has been able to explain the time dependence of the low-temperature specific heat. 

Equation (12.6) is in the shape predicted by the tunnelling theory for the amorphous materials [38,39] and 8 of eq. (12.7) is within the range of values obtained for other disordered solids [40]. 

Kelley and co-workers [70, 71] measured the dynamics of the excited-state intramolecular proton transfer in 3-hydroxyflavone and a series of its derivatives as a function of solvent (Scheme 2.9). The energy changes associated with the processes examined are of the order of 3 kcal/mol or less. The model they employed in the analysis of the reaction dynamics was based upon a tunneling reaction path. Interestingly, they find little or no deuterium kinetic isotope effect, which would appear to be inconsistent with tunneling theories. For 3-hydroxy-flavone, they suggest the lack of an isotope effect is due to a very large... 

Fig. 1.20. The Bardeen approach to tunneling theory. Instead of solving the Schrddinger equation for the coupled system, a, Bardeen (1960) makes clever use of perturbation theory. Starting with two free subsystems, b and c, the tunneling current is calculated through the overlap of the wavefunctions of free systems using the Fermi golden rule.

Fig. 2.6. Quantum transmission through a thin potential harrier. From the semi-classical point of view, the transmission through a high barrier, tunneling, is qualitatively different from that of a low barrier, ballistic transport. Nevertheless, for a thin barrier, here W = 3 A, the logarithm of the exact quantum mechanical transmission coefficient (solid curve) is nearly linear to the barrier height from 4 eV above the energy level to 2 eV below the energy level. As long as the barrier is thin, there is no qualitative difference between tunneling and ballistic transport. Also shown (dashed and dotted curves) is how both the semiclassical method (WKB) and Bardeen s tunneling theory become inaccurate for low barriers.

As the semiclassical tunneling theory and Bardeen s original approach become inaccurate for potential barriers close to or lower than the energy level, the validity range of the MBA is much wider. In this subsection, the accuracy of the MBA is tested against an exactly soluble case, that is, the one-dimensional transmission through a. square barrier of thickness W=2 A (see Fig. 2.9). 

Fig. 2.10. Different approximate methods for the square harrier problem. (Parameters used W= 2 A = 4 eV Utf= 16 eV.) The original Bardeen theory breaks down when the barrier top comes close to the energy level. The modified Bardeen tunneling theory is accurate with separation surfaces either centered L = lV/2) or off-centered L = VV73). By approximating the distortion of wavefunctions using Green s functions, the error in the entire region is only a few percent.

As we have discussed in Chapter 2, a direct consequence of the Bardeen tunneling theory (or the extension of it) is the reciprocity principle If the electronic state of the tip and the sample state under observation are interchanged, the image should be the same. An alternative wording of the same... 

Feuchtwang, T. E. (1979). Tunneling theory without the transfer Hamiltonian formalism V. A theory of inelastic electron tunneling spectroscopy. Phy.s. Rev. B 20, 430-455, and references therein. 

If an electron rather than an atom tunnels, the transition probability for this particle is much larger in view of its smaller mass therefore the tunnel effect may account for the observed rates of such chemical reactions (22) in which an electron transition can be considered an essential step. Detailed calculations based on the tunneling theory have been... 

Fig. 6.17 Tunnelling and saddle point ionization in Li. (a) Experimental map of the energy levels of Li m = 1 states in a static field. The horizontal peaks arise from ions collected after laser excitation. Energy is measured relative to the one-electron ionization limit. Disappearance of a level with increasing field indicates that the ionization rates exceed 3 x 105 s 1. The dotted line is the classical ionization limit given by Eqs. (6.35) and (6.36). One state has been emphasized by shading, (b) Energy levels for H (n = 18-20, m = 1) according to fourth order perturbation theory. Levels from nearby terms are omitted for clarity. Symbols used to denote the ionization rate are defined in the key. The tick mark indicates the field where the ionization rate equals the spontaneous radiative rate, (c) Experimental map as in (a) except that the collection method is sensitive only to states whose ionization rate exceeds 3 x 105 s-1. At high fields, the levels broaden into the continuum in agreement with tunnelling theory for H (from ref. 32).

In this chapter we expand on the problem of one-dimensional motion in a potential V(x). Although it is a textbook example, we use here the less traditional Feynman path integral formalism, the advantage of which is a possibility of straightforward extension to many dimensions. In the following sections on tunneling theories we shall use dimensionless units, in which h = 1, kB = 1 and the particle has unit mass. 

This chapter is devoted to tunneling effects observed in vibration-rotation spectra of isolated molecules and dimers. The relative simplicity of these systems permits one to treat them in terms of multidimensional PES s and even to construct these PES s by using the spectroscopic data. Modern experimental techniques permit the study of these simple systems at superlow temperatures where tunneling prevails over thermal activation. The presence of large-amplitude anharmonic motions in these systems, associated with weak (e.g., van der Waals) forces, requires the full power of quantitative multidimensional tunneling theory. 

First, as the donor binding energy decreases from Fe1 to Zn1, the electronic mixing, expressed in the dependence of rate on distance k exp-(oR) should change. In tl simplest barrier tunneling theory a s (IP. - IP. . ) For this theory, then,... 

C. Simulating Photosystem Operation with Tunneling Theory. 90... 

An appreciation of the basic parameters of electron tunneling theory and a survey of the values of these parameters in natural systems allows us to grasp the natural engineering of electron transfer proteins, what elements of their design are important for function and which are not, and how they fail under the influence of disease and mutation. Furthermore, this understanding also provides us with blueprint for the design of novel electron transfer proteins to exploit natural redox chemistry in desirable, simplified de novo synthetic proteins (Robertson et al., 1994). 

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