Tunnels, one-dimensional

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. 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

As stated by inequality (2.81) (see also section 4.2 and fig. 30), when the tunneling mass grows, the tunneling regime tends to be adiabatic, and the extremal trajectory approaches the MEP. The transition can be thought of as a one-dimensional tunneling in the vibrationally adiabatic barrier (1.10), and an estimate of and can be obtained on substitution of the parameters of this barrier in the one-dimensional formulae (2.6) and (2.7). The rate constant falls into the interval available for measurements if, as the mass m is increased, the barrier parameters are decreased so that the quantity d(Vom/mn) remains approximately invariant. 

A peculiar shape-selective behavior has been observed in a Cu-based MOF containing one-dimensional tunnels with narrow necks at regular distances of... 

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]. 

The use of framework structures to minimize AH for alkali-ion electrolytes has been demonstrated to provide a means of opening up the bottlenecks to cation motion in a number of oxides (Goodenough, Hong and Kafalas, 1976). Framework structures may provide one-dimensional tunnels as in hollandite, two-dimensional transport in planes as in the )S-aluminas, or three-dimensional transport as in NASICON and LISICON. Since one-dimensional tunnels are readily blocked, the two-and three-dimensional conductors are the more interesting. 

Three-dimensional structure, with one-dimensional tunnels... 

It should be noted, however, tliat even tlie best one-dimensional tunneling estimate is still likely to underestimate the full tunneling contribution, since tunneling may occur through dimensions of the PES other than die reaction coordinate. Multi-dimensional tunneling approximations are sufficiently complex, however, tliat tliey will not be further discussed here. 

In nearly all cases, the values of kc and Tc derived from the experimental curves k(T) are not in agreement with one-dimensional tunneling calculations that utilize crystallographic and spectroscopic data to define the tunneling distance d. Furthermore, in stark contrast to the... 

The zero mode, which is associated with the longitudinal fluctuations, is now included in (4.10), and when wt>0, the determinants in (4.11) do not suffer from the zero-mode problem. The value klD is simply the rate of tunneling (3.67) in the dynamical one-dimensional barrier V%s) along the instanton trajectory. As for Bt, it incorporates the effect of transverse vibrations around the instanton trajectory. To calculate (4.10), one may employ the apparatus of Chapter 3 designed for one-dimensional tunneling. In particular, now it is possible to make use of (3.69) together with (3.66), which gives... 

Earlier, a similar instanton analysis for a PES with two transition states was performed by Ivlev and Ovchinnikov [1987], in connection with tunneling in Josephson junctions. In the language of stability parameters introduced in Section 4.1, the appearance of two-dimensional tunneling paths is signaled by vanishing of the stability parameter. As follows from (4.24), the one-dimensional tunneling path formally becomes infinitely... 

The barrier height is about 15 kcal/mol by analogy to the similar gas-phase reaction H2C + CH4- 2CH3, which was calculated by Bausch-licher et al. [1976], The C-C distance calculated in the same work is found to be equal to 2.65 A. Like reaction (6.41) any attempt to reconcile the experimental dependence k(T) with a model of one-dimensional tunneling in the barrier of indicated height leads to a tunneling distance d that is far shorter than could be reasonably rationalized based on this C-C distance. 

It should be noted that relation (2.51) is valid within the sudden approximation. However, the relaxation of heavy particle impurities typically involves motion that is slow compared with vibrations of the host lattice (i.e., the tunneling takes place in the adiabatic limit). The net effect of the adiabatic approximation is to renormalize the effective moment of inertia of the particle. This approach was used, for example, to describe vacancy diffusion in light metals. The evolution of the rate constant from Arrhenius behavior to the low-temperature plateau was described within the framework of one-dimensional tunneling of a... 

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