Barrier, Eckart

The physical significance of quantum-mechanical tunneling was recognized very early In the development of wave mechanics, and there are many examples of physical phenomena In which tunneling Is Important. Here Is a very Incomplete list of examples, chosen principally on the basis of historical Interest  

The cold emission of electrons from a metal cathode at a hlgji negative voltage ( ). 

The effect of the double minimum in the potential energy for IIH3 on vibrational energy levels (4). 

The possibility of tunneling in a chemical reaction Involving motion of a proton or a hydrogen atom, which seems to have been first recognized by E. F. Bell ( ). 

Now that It has been made evident that tunneling is predicted by quantum mechanics, and that there are a number of physical manifestations of It, what about the particular area of chemical reactions  

---

Fig. 4. Variationally determined effective parabolic barrier frequency co ff for the Eckart barrier in units of 2n/hfi [Voth et al. 1989b], The dotted line is the high-temperature limit co = co. ...

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

Based on this physical view of the reaction dynamics, a very broad class of models can be constructed that yield qualitatively similar oscillations of the reaction probabilities. As shown in Fig. 40(b), a model based on Eckart barriers and constant non-adiabatic coupling to mimic H + D2, yields out-of-phase oscillations in Pr(0,0 — 0,j E) analogous to those observed in the full quantum scattering calculation. Note, however, that if the recoupling in the exit-channel is omitted (as shown in Fig. 40(b) with dashed lines) then oscillations disappear and Pr exhibits simple steps at the QBS energies. As the occurrence of the oscillation is quite insensitive to the details of the model, the interference of pathways through the network of QBS seems to provide a robust mechanism for the oscillating reaction probabilities. 

Fig. 6.2 (a) Bell (parabolic) and Eckart barriers, both widely used in approximate TST calculations of quantum mechanical tunneling, (b) Transmission probability (Bell tunneling) as a function of energy for two values of the reduced barrier width, a... 

A second widely used approximation uses the more smoothly shaped Eckart barrier (Fig. 6.1), which for a symmetric barrier may be expressed as V = V sech2(x) = V [2/(ex + e x)]2 where x = jts/a with s a variable dimension proportional to the displacement along MEP, and a a characteristic length. Like the Bell barrier the Eckart potential is amenable to exact solution. The solutions are similar and tunnel corrections can be substantial. In both the Bell and Eckart cases one is implicitly assuming separability of the reaction coordinate (MEP) from all other modes over the total extent of the barrier, and this assumption will carry through to more sophisticated approaches. 

QTST was applied to symmetric and asymmetric Eckart barriers in Ref 266. Variational QTST was tested on the asymmetric Eckart barrier in Ref 267. QTST is derived by rewriting the potential as a sum of a parabolic barrier term and a nonlinearity, as in Eq. 22. Therefore, it is a leading term for an expansion of the... 

Poliak and Eckhardt have shown that the QTST expression for the rate (Eq. 52) may be analyzed within a semiclassical context. The result is though not very good at very low temperatures, it does not reduce to the low temperature ImF result. The most recent and best resultthus far is the recent theory of Ankerhold and Grabert," who study in detail the semiclassical limit of the time evolution of the density matrix and extract from it the semiclassical rate. Application to the symmetric one dimensional Eckart barrier gives very good results. It remains to be seen how their theory works for asymmetric and dissipative systems. 

Figure 2.2. Variationally determined effective parabolic barrier frequency to n for the Eckart barrier in units 2-n7(fy6).

As mentioned before, the parabolic shape of the barrier can be assumed only for sufficiently small x. In reality, the potential should be finite at x— °°. As an illustration consider the Eckart barrier ... 

Figure 3.2 The pole structure of the S-matrix for a parabola (x) and an Eckart barrier ( ).

An analytical solution to the Schrodinger equation can be obtained for the so-called Eckart barrier given by [9,10],... 

Fig. 6.4.2 Tunneling probabilities for an Eckart barrier. The barrier height is 40 kJ/mol and the magnitudes of the imaginary frequency associated with the reaction coordinate are 1511 cm-1 (solid line), corresponding to the reaction H + H2, and 1511/ /2 cm-1 (dashed line), corresponding to the reaction D +D2. The step function is the transmission probability according to classical mechanics.

The correction factor due to quantum tunneling for an Eckart barrier cannot be evaluated analytically An approximate expression can, however, be derived. 

Table 6.1 Tunneling corrections as a function of temperature according to Eq. (6.40). kh is the correction factor for H + H2, where the magnitude of the imaginary frequency associated with the reaction coordinate is 1511 cm-1, kd is the correction factor for D + D2, where the magnitude of the imaginary frequency is 1511/cm-1. The tunneling probabilities are calculated for an Eckart barrier. The barrier height is Ec = 40 kJ/mol.

Figure 13 Initial probability of chemisorption, S0, vs. normal energy, En = i(cos2 0), for C2H6 and C2D6 on Ir(l 1 0)-(l x 2). Solid and dashed lines show model of tunneling through an Eckart barrier potential for C2H6 and C2D6, respectively. Data adapted from Verhoef et al. [55].

The other is of the singularities of the complex classical trajectories [19], which is peculiar to time-continuous systems and plays an important role to understand tunnehng phenomena of barrier potentials [25]. First, we briefly explain the role of singularities of classical trajectory by taking the static Eckart barrier as a simple example. 

We would like to return to the oscillating Eckart barrier. With simple intuitive consideration, we can predict what happens, if a periodically perturbation is applied to the system. In the following argument, we assume that the perturbation is sufficiently slow, namely the low-frequency limit, in which the fringed tunneling is typically observed. 

The main significance of the works [8] was in revealing the existence, irrespective of the barrier shape, of the finite low-temperature limit of the rate constant K(0). Even for Eckart barrier V x)= V /ch (2x/d), having an infinite width at = 0, the tunneling probability remains finite due to the existence of zero-point vibrations. 

In the H-atom abstraction by the methyl radical in various matrices, experimental K T) relationships can be satisfactorily described by relation (5) for Gauss and Eckart barriers at V values from 0.48 to 0.65 eV and d values from 1.7 to 2.3 A [11]. The d values correspond to the equilibrium inter-molecular distances. However, at such distances the interaction between the reagents is weak, and there are no reasons for such sharp V decrease compared with the C-H bond energy in the matrix molecules. Calculated... 

More detailed calculations of these effects were given later by Christov and Conway, who calculated proton tunneling probabilities through an Eckart barrier, the height of which was varied with potential. This gave a Tafel relation, as shown in Fig. 13, for proton transfer at a cathode for the case of complete tunneling control. In practice, both classical and nonclassical transfer occur in parallel " to relative extents dependent on temperature. 

---