Asymmetric Double Well Potential

Figure 6.8. Fluorescence excitation spectra of matrix isolated 9-deuteroxyphenalenone (lower) and methyl-9-deuteroxyphenalenone (upper) as examples of nearly symmetric and asymmetric double well potentials for hydrogen transfer, shown on the left. The suppression of hot band 01 is shown in a separate spectrum at 3.5 K. Due to asymmetry of the potential, the wave functions are linear combinations of the left and right well function with different amplitudes if/t = a2 + b2, ij/a = -b2 + a2, b/a = 0.22 and 0.80 in the ground and excited states of the methyl derivative. (From Barbara et al. [1989].)...

It is interesting to note that, for malonic acid (which is structurally related to DMMA), the activation energy measured from XH NMR Tx measurements [170] is 5.6 kj mol-1, which is significantly lower than in DMMA and is assigned to proton jumps between the two minima of an asymmetric double well potential. This emphasises the importance of the effect of the crystal packing on the asymmetry of the potential function, which defines the mechanism of the proton dynamics in carboxylic acid dimers. 

At temperatures say, below 70 K further relaxation processes occur. There, the frequency dependence of "(v) changes from a negative slope to a positive one. In this temperature range, thermally activated dynamics in asymmetric double-well potentials (ADWP) and, below 10 K, tunneling phenomena were discussed relating to the so-called low-temperature anomalies of glasses.27,30,31... 

Figure 30. A schematic presentation of the symmetric and asymmetric double-well potentials.

Genzel et al. (1983) and Kremer et al. (1984) reported picosecond relaxations in proteins, including lyophilized hemoglobin and lysozyme, that were described in terms of processes occurring in asymmetric double-well potentials, likely the NH OC hydrogen bridges of the... 

Figure 3. Asymmetric double-well potential v q). The minima are located at qo the energy of the localized states P) and a) are, respectively, A/2.

The work on the formic acid dimer focused on the double-well potential for a highly symmetric system. An attempt to locate a double-well potential for a less symmetric system was made by Zielinski and Poirier (1984). They studied the formamide dimer and isolated a possible structure for the transition state for a double-proton transfer along the reaction path to the formimidic acid dimer (a dimer of the enol form of formamide) using the 3-21G basis set. The proposed transition state is only slightly less stable than the formimidic acid dimer. In other words, a very asymmetric double-well potential was found with a very shallow well on the formimidic acid dimer side of the reaction. It will be interesting to see the shape of the function for a double-proton transfer between formamide and amidine, which would more closely mimic the double-proton transfer that may be possible for the A-T pair. 

In this section, we shall demonstrate how to evaluate the linear response of an assembly of noninteracting polar Brownian particles in a uniaxial potential —Zf(n e)2 (K is the anisotropy constant, e = p/p, and n is the unit vector in the direction of the easy axis) with a superimposed strong dc electric field Fo. In order to retain axial symmetry, we suppose that the field Fo and axis of the uniaxial anisotropy potential n are directed along the Z axis of the laboratory coordinate system. The total potential is an asymmetric double-well potential, which can be written in a dimensionless form as... 

Figure 7.07. Schematic illustration of an NSR process due to a transition 1- 2 in an asymmetric double-well potential (ADWP) characterizing the local low-energy configurations in glassy solids (after Kanert et al., 1994).

The larger kinetic H/D isotope effects in the parent radical can be explained in terms of the higher symmetry of the parent radical as compared to the di-fert-butyl radical. In the latter, the methyl groups on both sides of the ring are not ordered, leading to effective asymmetric double well potentials of the H-transfer. These examples show how subtle structural effects can lead to very different H-transfer properties. 

Fig. 3.3. Ground state potential and asymmetric double-well potential associated with the phenomenon of exciton self-trapping, as a function of the coordinate rj that undergoes a strong displacement upon self-trapping. F is the bottom of the free-exciton band, in which the lattice is not distorted (77 = 0), S denotes the lowest self-trapped exciton state, and U is the barrier height. The luminescence from the self-trapped state is red-shifted relative to the free-exciton luminescence. Upon photoexcitation of the system, two pathways towards the self-trapped state occur. The first possibility is that the created excitons first relax towards the bottom of the free-exciton well, after which they may further relax to the self-trapped state through tunneling or a thermoactivated process. This pathway is indicated by the filled arrows. The second possibility is that high-energy (hot) excitons relax directly to the self-trapped state, as indicated by the open arrow. Reprinted with permission from Knoester et al. (47). Copyright Elsevier (2003).

Fig. 3.3 Ground state potential and asymmetric double-well potential associated with the phenomenon of exciton selftrapping 73...

This section considers a single asymmetric double-well potential. At low temperatures a quantum mechanical description is necessary, and only the lowest energy eigenstates will be relevant. If the energy asymmetry of the wells is not too great then it will be sufficient to describe the problem in terms of a two-state basis, where the two states are localized in each of the two wells of the potential. These two states compose the TLS. 

In early interpretations, static distributions of asymmetric double-well potentials (ADWP) were assumed to exist for the locally mobile ions, providing a wide range of relaxation times. This model, which was based on ideas of PoUak and Pike [32], has often been used to fit experimental data [33, 34]. 

The probability of proton tunnelling across the barrier of a double-well potential strongly depends on its symmetry [74, 75]. Figure 9.6 shows a symmetrical and an asymmetrical double-well potentials, their energy levels and eigenfunctions. In the case of the symmetrical PES profile (degenerate systems). 

Fig. 9.6a-c. Schematic representation of the tunnelling mechanism in a symmetrical (malonalde-hyde) and an asymmetrical (substituents Rj R2) potential [74,75] a energy levels of the system b wave functions of the corresponding states c time dependence of the function of state ij/o (x, t) that describes the proton motion in a symmetrical and an asymmetrical double-well potential. (Adopted from Refs. [74,75] with permission from the American Chemical Society)... 

ENERGY LEVELS IN A DOUBLE WELL POTENTIAL 2.6.1 Asymmetric Double Well Potential... 

Here we consider the energy levels or the energy shifts in an asymmetric double well potential such as that shown in Figure 2.6. The diagrammatic technique mentioned in Section 2.4 can be applied. The corresponding diagram is shown in Figure 2.7. 

Figure 6 Asymmetric double-well potential U x) with y = 0.9 (see Eqs. [24,25]).

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