Double well potential tunneling splitting

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

Needless to say, tunneling is one of the most famous quantum mechanical effects. Theory of multidimensional tunneling, however, has not yet been completed. As is well known, in chemical dynamics there are the following three kinds of problems (1) energy splitting due to tunneling in symmetric double-well potential, (2) predissociation of metastable state through... 

Fig. 1. Symmetric double-well potential U-(Q) for a pseudo-JT molecule with two nondegenerate electronic terms coupled to one low-symmetry mode [equation (9)]. The curve corresponds to strong coupling case with k = 4 and a relatively large energy gap, A = 12 (both in units of hcS). The dashed line represents the twofold degenerate ground-state energy level subject to a tunneling splitting.

Fig. 2 (a) Two inde[>endent single-well potential curves for the vibrating OH groups, (b) The double-well potential curve for the hydrogen-bonded OH group (-OH 0= vs. =0- HO-). The solid curve indicates a symmetric combination of the two individual proton wave-functions and the broken curve is for an antisymmetric combination. The tunnelling splitting (Ao) is also shown. 

It is worthwhile to note that the charged Bose gas trapped in the double well potential Ua of Eq. 50 behaves as an inverted Josephson junction (N-S-S-N). The super-current, which accompanies the matter wave coherence, is induced between the degenerate resonance states of the adjacent wells at the frequency of the tunnel splitting A response time, as is typical of tunnel junctions (whose frequency cutoff is much smaller than the vibrational frequency even for nano junctions). The coherent oscillations of the Josephson current can be observed by virtue of their slow frequency A V which is robustly controlled by the bias voltage. 

If on the other hand AFJ is much smaller than Ipv, the eigenstates 1), 12) essentially coincide with the handed states L), /2). In that case the absolute value of the difference E2 — 1] between the eigenstates is approximately 2 V v - We can estimate the size of the tunneling splitting for typical chiral molecules for instance within the Wentzel-Kramers-Brillouin approximation. If we assume a quartic double well potential with a barrier height of 200 kJ mol a barrier Avidth of 200 pm and a tunneling mass... 

Fig. 3. Three-spin magnetic polaron which is regarded as the EPR active center in the CuOj plane. The Jahn-Teller distorted polaron has two degenerated configurations as indicated by the dashed lines. The inset shows the corresponding double-well potential with the excited vibronic state (dashed line) and the ground state split by tunneling (sohd line). From Kochelaev et al. (1997).

In the first part of this section, the instanton theory [2] is explained by taking the motion of a particle of mass m in one-dimensional potential V x). Tunneling splitting in a symmetric double well potential and decay of metastable state by tunneling through a potential barrier are employed as examples. In the second subsection, it is shown that the results can be reproduced by the WKB method with slight modification. 

In this subsection, we show that the same results as those obtained by the instanton theory in the previous subsection can be derived by the WKB theory with a slight modification [43,46]. We consider the tunneling splitting in a symmetric double well potential and as usual we use the asymptotic WKB wave function localized in one of the wells—say, left-side well ... 

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