Tunneling frequency

Fig. 54. Universal correlation between T , and tunneling frequency A. The values of A are given in peV (1 peV 8.066 X 10 cm = 2.42 x 10 Hz). The Zeeman frequency equals 21 MHz. The points correspond to different chemical species.

As seen from this table, the WKB approximation is reasonably accurate even for very shallow potentials. At 7 = 0 the hindered rotation is a coherent tunneling process like that studied in section 2.3 for the double well. If, for instance, the system is initially prepared in one of the wells, say, with cp = 0, then the probability to find it in one of the other wells is P( jn, t) = 5sin (2Ar), while the survival probability equals 1 — sin ( Ar). The transition amplitude A t), defined as P( + t) = A t), is connected with the tunneling frequency by... 

Static dielectric measurements [8] show that all crystals in the family exhibit a very large quantum effect of isotope replacement H D on the critical temperature. This effect can be exemphfied by the fact that Tc = 122 K in KDP and Tc = 229 K in KD2PO4 or DKDP. KDP exhibits a weak first-order phase transition, whereas the first-order character of phase transition in DKDP is more pronounced. The effect of isotope replacement is also observed for the saturated (near T = 0 K) spontaneous polarization, Pg, which has the value Ps = 5.0 xC cm in KDP and Ps = 6.2 xC cm in DKDP. As can be expected for a ferroelectric phase transition, a decrease in the temperature toward Tc in the PE phase causes a critical increase in longitudinal dielectric constant (along the c-axis) in KDP and DKDP. This increase follows the Curie-Weiss law. Sc = C/(T - Ti), and an isotope effect is observed not only for the Curie-Weiss temperature, Ti Tc, but also for the Curie constant C (C = 3000 K in KDP and C = 4000 K in DKDP). Isotope effects on the quantities Tc, P, and C were successfully explained within the proton-tunneling model as a consequence of different tunneling frequencies of H and D atoms. However, this model can hardly reproduce the Curie-Weiss law for Sc-... 

However, Ichikawa et al. [12] found a linear relationship between Tc and Ro-o, which is common to several KDP-type crystals, irrespective of whether they are deuterated or not. This finding suggests that the difference in Ro-o between KDP and DKDP (J o-o = 2.50 A for KDP and Rq-q = 2.52 A for DKDP at room temperature [10]) or the geometric isotope effect, rather than the difference in tunneling frequencies, causes the isotope effect on Tc. 

By the same token, electron transfer involves transfer of a particle between electronically coupled chemical sites and can be described as a tunneling process. In that sense, every electron transfer process involves electron tunneling with a tunneling frequency given, in the classical limit, by equation (31). 

In the methyl-substituted compound (6.20), R = CH3, the rate constants kjj and are approximately the same as for R = FI, though the energy of 3K, level is at least 0.7 kcal/mol greater than that of 3Ej level. Since the transition 3Ej—is endoergic, kXY > kYX [Eisenberger et al., 1991], We emphasized earlier that the tunneling frequencies of coherent transitions are very sensitive to any asymmetry of the potential. However, in this case the rate constant for the incoherent transition is unaffected by the difference in energies of the initial and final states. 

In conclusion of this section we present a table of barrier heights and tunneling frequencies (Table 7.2). This table is based on the data collected by Clough et al. (1981a) and, in addition, includes the results from the later papers discussed above as well as the values of V3 for some isolated molecules. 

Potential Barriers and Tunneling Frequencies for Methyl Groups in Organic Molecules... 

Covalon conduction is a spin-paired two-electron charge transfer that can take place when, the anti-symmetric normal mode of vibration, o, matches , the electron pair tunneling frequency. 

Biezo, G, J. Ladik, and J. Gergely Approximate calculation of the tunnelling frequencies of the proton in the N-H---0 hydrogen bond of the nucleotide base pairs. Physics Letters 13, 317 (1964). 

V0 is the potential height, and describes the rotation of the methyl groups. Steiskal and Gutowsky (20) give the dependence of the tunneling frequency vT of the reciprocal absolute temperature as parameter for V0. Figure 5 gives these results. 

Figure 5. Comparison of the correlation frequencies calculated according Equation 1 with the averaged tunneling frequencies vr calculated by Stejskal and Gutowsky for the methyl groups assumed as quantum-mechanical rotators as a function of the temperature for PC (x) and PMST (o). The potential height in units of kcal./mole is the parameter...

Wood1261 also deduces that the barrier is high enough to prevent tunnelling which would otherwise cause splitting of v(C=0) etc. Also he finds no tunnelling frequency vr in the far IR. But see below... 

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