Symmetric Double Well Potential

Using the spectral function method. Miller [17,55] obtained the following expression for the tunneling splitting, which is applicable to the excited state E  

The modified WKB method can also be applied to low excited states. This is discussed for the case of a multidimensional problem in Chapter 6. 

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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... 

The single particle auto-correlation time tc in Eq. 9 can, of course, exhibit also a non-critical temperature dependence. Consider a set of independent hydrogen bonds with symmetric double well potentials and a barrier a between the wells. In this case the motion is thermally activated and tc shows an Arrhenius behaviour ... 

In previous work, the iGLE and WiGLE models have been illustrated through the use of free-particle, biased, and biased-washboard potentials. Rather than repeat these calculations, in this section we illustrate the dramatic role that the asymmetry in the nonstationary friction can play in the dynamics of the symmetric double-well potential. The specific question to be explored is whether the equilibrium position of the double-well particles is affected by the asymmetry in the nonstationary friction. 

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. 

By application of the external constant field of arbitrary strength, the symmetric double-well potential may be gradually transformed into a single-well (dipolar) one on growth of the bias field, SR maximum noticeably shifts to higher temperatures. 

Consider two enantiomorphous molecules, R (right-handed) and L (left-handed), isolated from their surroundings and from external fields but not from each other. That is, they are allowed to interact. In the quantum-mechanical treatment of this system, as two particles in a one-dimensional symmetric double-well potential, the two states are degenerate in energy and are related to wavefunctions TR and 4Y localized in the two potential wells. Superposition of these wavefunctions yields ground and first excited states F+ and P ... 

We define as modulation theory an approach to a noncanonical distribution based on the modulation of processes that with no modulation would yield canonical distributions. For instance, a double-well potential under the influence of white noise yields a Poisson distribution of the time of sojourn in the two wells [150]. In the case of a symmetric double-well potential we have... 

Clearly, Eq. (275) indicates that between state A and state B there are the intermediate states Wab and Wba- Identifying the intermediate states Wab and Wba with the state C in the Gray-Rice theory, one sees that (first-order) RIT and the Gray-Rice theory are based upon the same reaction mechanism. Specifically, for the case of a symmetric double-well potential, RIT yields the rate constant [56]... 

Isomerization Rate Constants for Symmetric Double-Well Potential Systems Defined in Table XIV... 

V(x) is assumed to be the usual symmetrical double-well potential [V(x) - 8x /2 + fix /4] the third term on the right-hand side of Eq. (77) is the coupling between the Brownian particle and the external radiation field, which is characterized through its autocorrelation function... 

Quantum effects and strong interactions with vibrating surrounding atoms complicate the detailed study of proton transfer in the hydrogen bond AH B. Owing to the small mass, quantum tunneling of the proton plays an important role at a symmetric double-well potential. 

Figure 3.1 Typical potential energy curves for strong, low-barrier hydrogen bonds. The homoconjugated HjOj (a) exhibits a (relaxed) symmetric, double-well potential as a function ofthe difference of the bridging...

Another typical example of the stochastic resonance system is the nonlinear bistable doublewell dynamic system, which describes the overdamped motion of a Brownian jjartide in a symmetric double-well potential in the presence of noise and periodic forcing as shown in... 

These are the ordinary Marcus curves for a symmetric systan. The ground state forms a symmetric double-well potential. The BO approximation does not work here because if we are at one minimum, the electron is at the donor D and if we are at the other minimum, the electron is at the acceptor A. 

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