Strong weak coupling limit

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

In the previous section we have dealt with a simple, but nevertheless physically rich, model describing the interaction of an electronic level with some specific vibrational mode confined to the quantum dot. We have seen how to apply in this case the Keldysh non-equilibrium techniques described in Section III within the self-consistent Born and Migdal approximations. The latter are however appropriate for the weak coupling limit to the vibrational degrees of freedom. In the opposite case of strong coupling, different techniques must be applied. For equilibrium problems, unitary transformations combined with variational approaches can be used, in non-equilibrium only recently some attempts were made to deal with the problem. [139]... 

In Equation 6.88, K0 is the equilibrium constant for the formation of the collision complex, (V) 2 is the electronic coupling, and F is the Franck-Condon factor. In contrast to the radiationless relaxation, the energy transfer process cannot be rationalized only in the limit of the strong and weak coupling limits shown in Figure 6.16. 

However, in the preceding two decades, there have been many experimental discoveries, beside high-Tc superconductivity, evidencing that we do not have yet the proper theoretical skills and tools to deal well with strongly correlated electron systems. For instance, heavy-fermions, fractional quantum Hall effect, ladder materials, and very specially high-Tc superconductivity seem not accessible from the weak coupling limit. 

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. 

When the donor and acceptor are sufficiently close, as in an ion pair or in covalently linked complexes, electron transfer can be promoted by the absorption of light. An absorption band corresponding to the light induced electron transfer is usnally called a charge transfer (CT) absorption band . The molecnlar parameters that determine the CT absorptivity, bandwidth, and band shape are the same molecular parameters that determine the magnitude of the electron-transfer rate constant.In the weak-coupling limit, the absorptivity of the CT absorption band is small (much less than 10 cm ) in the strong-... 

The evolution of a t) is most complicated if /j2 is of similar magnitude to A 1, 2 strong coupling regime). However, in the weak coupling limit, where... 

In the weak coupling limit the density operator of the spin system does not evolve, that is, the polarization of the first spin is invariant aU) = A2 (see Fig. lA). In the intermediate case of strong coupling (Fig. IB), the density operator evolves in an oscillatory fashion. Starting from cr(0) = /j, terms Ujyl2x hxhy and U xhx + are created periodically, as... 

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).

In the weak-coupling limit becomes much larger than the length of the unit cell a for rrani -polyacetylene) and the Peierls gap has a strong effect... 

The form of the X part of the ABX spectrum cannot be deduced from this simple analysis. In general it contains 6 lines, rather than the four which would be expected in the weak coupling limit. The two extra lines are combination lines which become observable when strong coupling is present. 

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