Model tunnelling

The QFH potential approximately captures two key quantum effects. When an atom is near a potential minimum, the curvature is positive and thus so is the QFH correction this models the zero-point effect. On the other hand, near potential maxima the curvature is negative, and the QFH potential models tunneling. 

ELECTRICAL RESISTANCE, HALL EFFECT, DRUDE MODEL, TUNNELING, AND THE LANDAUER FORMULA... 

For the prediction of water head changes induced by FEBEX tunnel boring, it was impossible to simulate a transient evolution due to changes in geometry with HM3D. We chose to model tunnel boring in 4 excavation phases and then simulate the steady state corresponding to the end of each phase. 

Lowndes I.S., Yang Z.Y. Jobling S., et al, 2006. A Parametric Analysis of a Tunnel Climatic Prediction and Planning Model, Tunneling and Underground Space Technology 21 520-532. 

Sun, IP, Zhao, Z.Y., Zhang, Y, 2011. Determination of three dimensional hydraulic conductivities using a combined analytical/neural network model. Tunnelling and Underground Space Technology 2011, 26(2) 310-319. 

Howarth, D.F., Adamson, W.R., Bemdt, J.R., 1989. Correlation of model tunnel boring and drilling machine performances with rock properties. Int. J. Rock Mech. Sci. 23, 171-175. 

The pathway model makes a number of key predictions, including (a) a substantial role for hydrogen bond mediation of tunnelling, (b) a difference in mediation characteristics as a function of secondary and tertiary stmcture, (c) an intrinsically nonexponential decay of rate witlr distance, and (d) patlrway specific Trot and cold spots for electron transfer. These predictions have been tested extensively. The most systematic and critical tests are provided witlr mtlrenium-modified proteins, where a syntlretic ET active group cair be attached to the protein aird tire rate of ET via a specific medium stmcture cair be probed (figure C3.2.5). 

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. 

A more detailed treatment has been given by Gurfein and his associates who chose as their pore model a cylinder with walls only one molecule thick. A few years later, Everett and Fowl extended the range of models to include not only a slit-shaped pore with walls one molecule thick, but also a cylinder tunnelled from an infinite slab of solid and a slit formed from parallel slabs of solid. 

Computations have shown that in the quantum region it is possible to have various most probable transition paths (ranging from the classical minimum energy path (MEP) to the straight-line one-dimensional tunneling of early models), depending on the PES geometry. 

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

While the goal of the previous models is to carry out analytical calculations and gain insight into the physical picture, the multidimensional calculations are expected to give a quantitative description of concrete chemical systems. However at present we are just at the beginning of this process, and only a few examples of numerical multidimensional computations, mostly on rather idealized PES, have been performed so far. Nonetheless these pioneering studies have established a number of novel features of tunneling reactions, which do not show up in the effectively one-dimensional models. 

In this section we shall consider in some detail the mechanism of coherence breakdown due to the bath, in order to clarify the physical assumptions which underlie the concept of rate constant at low temperatures. The particular tunneling model we choose is the two-level system (TLS) with the Hamiltonian... 

The simplest scheme that accounts for the destruction of phase coherence is the so-called stochastic interruption model [Nikitin and Korst 1965 Simonius 1978 Silbey and Harris 1989]. Suppose the process of free tunneling is interrupted by a sequence of collisions separated by time periods vo = to do After each collision the system forgets its initial phase, i.e., the off-diagonal matrix elements of the density matrix p go to zero, resulting in the density matrix p ... 

This simple gas-phase model confirms that the rate constant is proportional to the square of the tunneling matrix element divided by some characteristic bath frequency. Now, in order to put more concretness into this model and make it more realistic, we specify the total (TLS and bath) Hamiltonian... 

Of particular interest is the model of a bath as a set of harmonic oscillators qj with frequencies cOj, which are linearly coupled to the tunneling coordinate... 

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. 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

For this reason these vibrations influence tunneling in an entirely different way. For a model in which the reactant and product valleys are represented by paraboloids with frequencies coq and transition probability has been found to be [Benderskii et al. 1991a, b]... 

This model of classical cooperative transitions has been speculated about as an alternative to tunneling. No confirmation for such a scheme exists now. The possibility of representing the experimental form of k(T) as... 

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