Double-well

Figure Al.1.4. Wavefimctions for the four lowest states of the double-well oseillator. The ground-state wavefiinetion is at the bottom and the others are ordered from bottom to top in tenns of inereasing energy.

Figure Al.1.5. Ground state wavefimetion of the double-well oseillator, as obtained in a variational ealeulation usmg eight basis funetions eentred at the origin. Note the spurious oseillatory behaviour near the origin and the loeation of the peak maxima, both of whieh are well inside the potential minima.

The decrease in reactivity with increasing temperature is due to the fact that many low-energy ion-molecule reactions proceed tln-ough a double-well potential with the following mechanism [82] ... 

Gillam M J 1987 Quantum-classical crossover of the transition rate in the damped double well J. Phys. C Solid State Phys. 20 3621... 

Figure B3.4.10. Schematic figure of a ID double-well potential surface. The reaction probabilities exliibit peaks whenever the collision energy matches the energy of the resonances, which are here the quasi-bound states in the well (with their energy indicated). Note that the peaks become wider for the higher energy resonances—the high-energy resonance here is less bound and Teaks more toward the asymptote than do the low-energy ones.

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

Fig. 2. The fluctuating difference between the proton potential at the product side relative to that at the reactant side (the difference between the two wells in a double-well proton potential). Whenever this difference is close to zero, tunneling conditions are favourable.

Figure 1 Double well potential for a generic conformational transition showing the regions of reactant and product states separated by the transition state surface.

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. 

Fig. 10. Energy levels in symmetric double well. Arrows indicate interdoublet transitions induced by vibrations.

Coleman s method can be applied to finding the ground state tunneling splitting in a symmetric double well [Vainshtein et al. 1982], for some... 

An example of a numerically calculated trajectory in a symmetric double well is presented in fig. 29 for the Hamiltonian... 

Leggett et al. [1987] have set forth a rigorous scheme that reduces a symmetric (or nearly symmetric) double well, coupled linearly to phonons, to the spin-boson problem, if the temperature is low enough. However, in the case of nonlinear coupling (which is necessary to introduce in order to describe the promoting vibrations), no such scheme is known, and the use of the spin-boson Hamiltonian together with (3.67) relies rather on intuition, and is not always Justifiable. 

Fig. 33. Three-dimensional instanton trajeetories of a partiele in a symmetrie double well, interaeting with symmetrieally and antisymmetrieally eoupled vibrations with eoordinates and frequeneies q, to, and ru, respeetively. The curves are 1, ru, ru, P ojq (MEP) 2. to, (u, < (Oq (sudden approximation) 3. ru, < cOq, oj, P ojo 4. to, > (Oq, < (Oq.

The contour plot is given in fig. 43. As remarked by Miller [1983], the existence of more than one transition states and, therefore, the bifurcation of the reaction path, 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 this PES has been carried out by Benderskii et al. [1991b]. The... 

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... 

Besides the deviation mentioned above, the main problem with the dynamical information from the MF approximation is that it contains only one positive frequency and so the resulting real-time correlations cannot be damped or describe localizations on one side of the double well due to interference effects, as one expects for real materials. Thus we expect that the frequency distribution is not singly peaked but has a broad distribution, perhaps with several maxima instead of a single peak at an average mean field frequency. In order to study the shape of the frequency distribution we analyze the imaginary-time correlations in more detail. 

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... 

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