Tunneling double-well potential

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. 

We hope to have convinced the reader by now that the tunneling centers in glasses are complicated objects that would have to be described using an enormously big Hilbert space, currently beyond our computational capacity. This multilevel character can be anticipated coming from the low-temperature perspective in Lubchenko and Wolynes [4]. Indeed, if a defect has at least two alternative states between which it can tunnel, this system is at least as complex as a double-well potential—clearly a multilevel system, reducing to a TLS at the lowest temperatures. Deviations from a simple two-level behavior have been seen directly in single-molecule experiments [105]. In order to predict the energies at which this multilevel behavior would be exhibited, we first estimate the domain wall mass. Obviously, the total mass of all the atoms in the droplet... 

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... 

The rate of tautomerization can be related to a simple model describing the transformation of the BPS to IS by considering the motion of the proton in a double-well potential, with the system either in the N—H or the NH+ configuration of the two moieties. In this model, the rate is given by the tunneling expression ... 

As we have seen, the role of phonons in two-dimensional tunneling can be elucidated by considering linearly and symmetrically coupled double-well potentials. In both cases the bending of the reaction path is caused by coupling to a vibration. The pure effect of the vibration-induced squeezing of the reactive channel (without bending) may be conventionally studied using the potential... 

Methyl-substituted malonaldehyde (a-methyl-/3-hydroxyacrolein) provides an opportunity to study the role of asymmetry of the potential profile in the proton exchange. In the initial and final states, one of the C-H bonds of the methyl group is in the molecular plane and directed toward the proton position. The double well potential becomes symmetric only due to methyl group rotation over tt/6, when the C-H bond lies in the plane perpendicular to the molecular one. As a result, proton tunneling occurs in combination with CH3 hindered rotation and the... 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

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.

The WKB approximation was applied to the symmetric double-well potential so many times that it makes it difficult to provide a comprehensive list of references. In the context of the present paper, the most important references are the famous text of Landau and Lifshits [27], where tunneling in a symmetric double-well is given as a sample problem, and the paper of Auerbach and Kivelson [11] where the symmetric double-well potential is considered as one of the model cases. 

To show how the junction rule works, consider the above example of tunneling in the double-well potential. In this case we have two nodes connected by just one tunneling path. Let the starting position of the system be in the left well with the ground-state wave function P1 = C il> (r) (Q) are assumed to be normalized, and C is the amplitude in the left well, so that I C I2 is the probability to find the system in this well. The corresponding tail of the WKB ground-state wave function under the barrier should decrease with Q exponentially,... 

Hund, one of the pioneers in quantum mechanics, had a fundamental question of relation between the molecular chirality and optical activity [78]. He proposed that all chiral molecules in a double well potential are energetically inequivalent due to a mixed parity state between symmetric and antisymmetric forms. If the quantum tunnelling barrier is sufficiently small, such chiral molecules oscillate between one enantiomer and the other enantiomer with time through spatial inversion and exist in a superposed structure, as exemplified in Figs. 19 and 24. Hund s theory may be responsible for dynamic helicity, dynamic racemization, and epimerization. 

Dolinsek et al.7 have reported the existence of quantum tunnelling in these systems. This has been observed from the temperature-independent behaviour of 8 Rb and 2H NMR SLR times at low temperatures (1.6Quantum tunnelling in these systems refers to the tunnelling of protons/deuterons in the double-well potential. In a 31P NMR relaxation time measurements by Chen et al.30 in D-RADP systems, have found an additional T1 minimum at low temperatures apart from the BPP T1 minimum at high temperatures. The low temperature minimum has been attributed to the extreme slowing down of the O-D—O intra-bond motion which is unique to the glassy phase. 

Figure 52. Double logarithmic plot of the dielectric loss data (tan 8) of several type A glass formers as a function of temperature. Below 3 K the tunneling regime with the tunneling plateau is recognized the temperature range 3 < T < 30 K corresponds to thermally activated dynamics in double-well potentials at higher temperatures, nearly constant loss is found in the corresponding spectra. (Adapted from Ref. 50.)...

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