Temperature dependence of tunneling

The way this has been expected to Influence an isotopic rate constant ratio has been discussed frequently in the literature (14), and Is illustrated In Fig. 5. lUnetlc-isotope-effect data can be expressed in the form 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

To answer this. Stern and I (22) tued the same model systems previously Investigated by Schneider and Stern, but Includ the tunneling correction In the Isotope effect. Values of v were those calculated In their work barrier heights V were not needed In their work, and we have simply assumed values of 1, 5, 10, 20, and 30 kcal/mole for Vg. A symmetrical Eckart potential was i ised. For reasons discussed in our paper (22), two sets of calculations were made one with Vg = Vg and one with Vg = Vg + 

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Classical Free Energy and Temperature Dependence of Tunneling... 

Fig, 22. Temperature dependence of tunneling DOS for the optimum doped Bi2212 sample (7 ... 

This formula, aside from the prefactor, is simply a one-dimensional Gamov factor for tunneling in the barrier shown in fig. 12. The temperature dependence of k, being Arrhenius at high temperatures, levels off to near the cross-over temperature which, for A = 0, is equal to ... 

Fig. 53. The temperature dependence of 7", in tiglic acid. Methyl tunnel splitting (ueV)...

It is noteworthy that the above rule connects two quite different values, because the temperature dependence of is governed by the rate constant of incoherent processes, while A characterizes coherent tunneling. In actual fact, A is not measured directly, but it is calculated from the barrier height, extracted from the Arrhenius dependence k T). This dependence should level off to a low-temperature plateau at 7 < This non-Arrhenius behavior of has actually been observed by Punnkinen [1980] in methane crystals (see fig. 1). A similar dependence, also depicted in fig. 1, has been observed by Geoffroy et al. [1979] for the radical... 

Based on C-H versus C-D zero point vibrational differences, the authors estimated maximum classical kinetic isotope effects of 17, 53, and 260 for h/ d at -30, -100, and -150°C, respectively. In contrast, ratios of 80,1400, and 13,000 were measured experimentally at those temperatures. Based on the temperature dependence of the atom transfers, the difference in activation energies for H- versus D-abstraction was found to be significantly greater than the theoretical difference of 1.3kcal/mol. These results clearly reflected the smaller tunneling probability of the heavier deuterium atom. 

Product analysis by NMR indicated an isotope effect at 118°C of = 2.14, corrected for numbers of H versus D. On lowering the temperature to -12°C, however, it was found that the isotope effect increased to 3.25. Referring to earlier experimental results on the C-H shift in methylchlorocarbene, " the authors cited the normal temperature dependence of the isotope effect as evidence against tunneling in 64. In retrospect, however, as noted above, theoretical support for an atypical inverse temperature dependence in methylchlorocarbene has been refuted. Hence, the involvement of tunneling in 62/64 at ambient temperatures is still an open question. 

On the other hand, the low-conductance values (L) give a poor linear correlation of the molecular length with an approximate decay constant fiN 0.45 0.09, distinctively different from the H and M sequences. The estimated value of fiN(L) is rather close to results reported by Cui [28] and Haiss [243]. Haiss et al. [244] found a pronounced temperature dependence of these L values, which scales logarithmically with 7 1 in the temperature range 293-353 K, indicating a transport mechanism different from a simple tunneling model. 

The second issue of interest is the temperature dependence of elastic tunneling spectroscopy. Because the bands are intrinsically wide, spectra measure at 5 K are similar in line shape to those measured near 300 K. In any case, the integrated normalized intensities,... 

The photolyses of diazirines 9a and 9b were similarly studied in Ar matrices at 10-34.5 K 59 Eq. 10. Benzylchlorocarbene (10a) and its ct,a-d2 analogue (10b) were observed by UV or IR (10b) spectroscopy, and their decay to styrenes 11 and 12 could be monitored. Tunneling in these 1,2-H(D) shifts was indicated by (a) much higher rates of carbene decay at 10 K than could be anticipated from extrapolation of the 298 K LFP kinetic data, (b) a kinetic isotope effect (KIE) for the 1,2-H(D) shifts estimated at 2000, and (c) little temperature dependence of the rate at low temperature.59 Accepting that QMT is important in the very low temperature H shifts of carbenes 10 and 18, the obvious question becomes is QMT important at higher or even ambient temperatures ... 

The second method of detecting tunnelling relies on the fact that the primary hydrogen KIE shows an anomalous temperature dependence when significant tunnelling takes place. In the absence of tunnelling, the temperature dependence of the rate constant should follow the Arrhenius equation (42)... 

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