Tunnelling Eckart barrier

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. 

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].

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. 

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. 

The potential energy surface for the proton motion is not of a simple shape in our model calculation. Hence, the Eckart barrier formula is used for proton tunneling by adjusting the two variables in it [41]. Thus, the barrier for the proton transfer was fitted to the following Eckart formula ... 

Finally, to calculate the current, the entropy of the activation is added to the activation energy, and then the relevant part for the proton tunneling is evaluated numerically using the Eckart barrier formalism for EH+ tunn(E) ... 

Figure 7.29 Tunneling probability for the indicated Eckart barrier with three different critical vibrational frequencies. The latter are related to the curvature at the top of the barrier. Note that the transmission coefficient is equal to 0.5 at the saddle point.

An example of tunneling in which both the normal and the deuterated species were measured is in HCl or DCl loss from the ethyl chloride ion (Booze et al., 1991). Ab initio molecular orbital calculations (Morrow and Baer, 1988) demonstrated that the HCl loss channel proceeds via a substantial barrier of 1 eV (23 kcal/mol). Yet, the onset for HCl loss occurs at about 0.3 eV. Furthermore, the reaction is very slow (k = 10 sec ). These facts all point to a reaction that proceeds via tunneling. The ab initio calculations provide the vibrational frequencies for both the ion and the transition state, which is located at a saddle point. In addition, they furnish the activation energy as well as the curvature of the barrier (the imaginary frequency). Thus, all the information required in the utilization of the Eckart barrier and equation (7.65) is given. 

Shavitt, The Tunnel Effect Correction to the Rates of Reactions with Parabolic and Eckart Barriers, University of Wisconsin Theoretical Chemistry Laboratory Report WlS-AEC-23 (1959). 

ABSTRACT. The results of multireference singles and doubles Cl calculations of potential energy surfaces for hydrogen atom addition to O2, N2, and NO and recombination of OH -h O are discussed. The errors due to the use of externally contracted Cl and due to the neglect of correlation of O 2s and N 2s electrons are analyzed. Similarities and differences between the surfaces for the addition reactions are discussed. The calculated HN2 addition surface is used in a simple dynamical treatment (one-dimensional tunneling through an Eckart barrier) to estimate the lifetime of the HN2 species. The OH -h O recombination potential is found to exhibit complex features which require that electrostatic forces (dipole-quadrupole) and chemical forces be treated consistently. 

The similarity of scISM and DMBE reaction paths suggests that their tunnelling corrections should be similar. The simplest realistic tunnelling correction is that of the Eckart barrier. This barrier can be fitted to the scISM reaction path using its asymptotic limits, the barrier height AV j and the curvature of this path at the classical transition state. This latter... 

Thus, sclSM rate constants, given by the procedure described above, are not as easily accessible as their classical counterparts. In contrast, the tunnelling corrections now extend the accuracy of the calculated rates to lower temperatures, and, with the ZPE corrections, provide reasonable estimates of the KlEs. Figure 6.19 iUustrates the results obtained for the H+H2 and H+D2 systems with the ZPE corrections described in Appendix IV and the tunnelling corrections of an Eckart barrier fitted to the classical path. This figure also presents... 

Figure 6.19 Comparison between the rate constants of the reactions H+Hj and H+Dj, in units of mol dm sec , calculated by the scISM with Eckart-barrier tunnelling corrections and by the VTST with least-action tunnelling corrections (dotted lines) with the experimental rates.

I. Shavitt, The tunnel effect correction to the rates of reactions with parabolic and Eckart barriers. University of Wisconsin Theoretical Chemistry Laboratory Report WIS-AEC-23, Madison,... 

The tunnelling correction P is the transmission probability through the potential barrier averaged over all possible crossing points and potential energies . An asymmetrical banier of the Eckart type l is assumed in the present model. 

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