Strong-coupling limit

In this model there is a quantitative difference between RLT and electron transfer stemming from the aforementioned difference in phonon spectra. RLT is the weak-coupling case S < 1, while for electron transfer in polar media the strong-coupling limit is reached, when S > 1. In particular, in the above example of ST conversion in aromatic hydrocarbon molecules S = 0.5-1.0. 

In the strong-coupling limit at high temperatures the electron transfer rate constant is given by the Marcus formula [Marcus 1964]... 

Other studies have also been made on the dynamics around a conical intersection in a model 2D system, both for dissociative [225] and bound-state [226] problems. Comparison between surface hopping and exact calculations show reasonable agreement when the coupling between the surfaces is weak, but larger errors are found in the strong coupling limit. 

Fig. 2. Correlation diagram of the doublet states between weak (left) and strong (right) spin-orbit coupling. In the strong coupling limit the splitting pattern is determined by the pseudo-./ quantum number...

Figure 3. The zero-resonon state 0) in a two-leg ladder. It is made of all vertical rung singlets. It is the reference state in the strong coupling limit J J.

Figures 2 and 3 contain plots of the two non-zero RFs for M = A and T2, respectively, as a function of the coupling strength KT. Also included in each figure are the original analytical calculations of Bates et al. [3] (who also included corrections due to anisotropy) and the numerical results of O Brien [9] (The key is shown as an insert in each figure.) It should be emphasized that, even though the results shown cover the range KT = 0-2.5, all but the numerical results of O Brien [9] are only strictly valid in the strong coupling limit (KT larger than unity).

We use now the results of the foregoing section to discuss the electronic transport properties of our model in some limiting cases for which analytic expressions can be derived. We will discuss the mean-field approximation and the weak-coupling regime in the electron-bath interaction as well as to elaborate on the strong-coupling limit. Furthermore, the cases of ohmic (s = 1) and superohmic (s = 3) spectral densities are treated. 

When a Jahn-Teller-distorted C q molecule is placed in a solid-state environment, strain or crystal-field perturbations may play a decisive role in the selection and/or enhancement of the Jahn-Teller-distorted configuration. In an attempt to investigate the Jahn-Teller-distorted C6 0 molecules in the solid state, C60-tetraphenylphosphonium iodide has been synthesized and studied [40]. The well-known Flu(l) and Flu(2) modes were found to split into doublets at room temperature indicating a D5d or D3d and not a D2u Jahn-Teller configuration. These results are consistent with a dynamic Jahn-Teller effect in the strong coupling limit or with a static distortion stabilized by the low-symmetry perturbations. 

The strong coupling limit, Hund s case (c)161, cannot be dealt with in first order. Rather, it requires the inclusion of higher-order (at least second-order) spin-orbit coupling in the calculations. 

The theory of CDW and SDW instabilities has received much attention it differentiates between the weak-coupling limit (U intermediate-coupling limit [50,51,52], and the strong-coupling limit ( U>t) [35,53,54],... 

In equation (5), is the equilibrium constant for the outer-sphere association of the donor and acceptor, is the electronic transmission coefficient (the probability that products form once the nuclear configuration of the transition state is achieved), Vnu is the effective frequency for nuclear motion along the reaction coordinate in the neighborhood of the transition state, and the nuclear transmission coefficient nu is the classical exponential function of the activation energy. The weak-coupling limit corresponds to the limit in which Kei < 1, and for the strong-coupling limit /Cei = 1. 

In the infinitely strong coupling limit Hartmann-Hahn limit r]2l -> 0) Eq. (30) may be simplified to... 

Figure 7 Illustration of the mixing between two resonances — for the double-well potential shown in (a) — as a function of the potential parameter a (the width of the inner potential well) and its influence on the resonance energies (middle two panels) and widths r (lower two panels, plotted on logarithmic scales). Solid lines represent the narrow resonance n located in the inner well, while the dashed lines indicate the broad resonance b localized in the outer well. The weak-coupling limit is shown in (b) and (c), while the strong-coupling limit is illustrated in (d) and (e). In the example discussed in the text, Vi = 8 and V2 = 17 for (b) and (c) and V2 = 11 for (d) and (e).

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