Strongly weakly coupled states

If all states of the system are strongly coupled to each other, the system dynamics can only be described by completely solving the above equations. However, it is extremely unlikely that this is the case. Rather, it is commonly found that some pairs of states are strongly coupled and other pairs of states are weakly coupled. Then we expect that the population transfers among strongly coupled states dominate the system dynamics and that it should be possible to study the n-state system dynamics in the subspace of strongly coupled states with a correction from the influence of the weakly coupled states. 

We prove our statement in two steps First, we consider the special case of a Hilbert space of three states, the two lowest of which are coupled strongly to each other but the third state is only weakly coupled to them. Then, we extend it to the case of a Hilbert space of N states where M states are strongly coupled to each other, and L = N — M) states, are only loosely coupled to these M original states (but can be stiongly coupled among themselves). 

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 ... 

The most amazing are the results for weak coupling. It appears that the gap function could have sizable values at finite temperature even if it is exactly zero at zero temperature. This possibility comes about only because of the strong influence of the neutrality condition on the ground state preference in quark matter. Because of the thermal effects, the positive electrical charge of the diquark condensate is easier to accommodate at finite temperature. We should mention that somewhat similar results for the temperature dependence of the gap were also obtained in Ref. [21] in a study of the asymmetric nuclear matter, and in Ref. [22] when number density was fixed. 

Weak coupling (U AE, Aw U As) The interaction energy is much lower than the absorption bandwidth but larger than the width of an isolated vibronic level. The electronic excitation in this case is more localized than under strong coupling. Nevertheless, the vibronic excitation is still to be considered as delocalized so that the system can be described in terms of stationary vibronic exciton states. 

The excited vibrational states can be considered as quasi-eigenstates [41]. As can be seen in the simplified scheme of Figure 2.2, these states are a result of the relatively strong coupling between a zero-order bright state (ZOBS), namely i >, with several zero-order dark states (ZODS), l > [48], that are further weakly coupled to the bath states that include a dense manifold of nearly equally coupled levels with a finite decay rate. 

Note that in this context, one often speaks of an inner-sphere mechanism if there is a strong electronic coupling between R and P in the transition state, and conversely, of an outer-sphere mechanism, if the interaction is weak (Eberson, 1987). 

As in paper [5], we start from a system Hamiltonian consisting of three (one ground g) and two nonadiabatically coupled excited (j) ) and 1 states strongly coupled to a reaction mode, which in turn is weakly coupled to a dissipative environment (see Fig. 1). The bath degrees of freedom are integrated out in the framework of Redfield theory, and the signals are calculated according to the explicit formulas derived in [6,7]. 

Viewed in the time domain, the replacement of M(a>) by M washes out the details of the time variation within Q space. For this approximation to be useful, all strongly coupled states should be included in the P space and the Q space should not include any states that couple strongly to the P space (weak coupling assumption). We now find that the population dynamics of the m levels within the P space is governed by the equations of motion... 

Figure 11.7 Potential energy diagrams for a two-component mixed valence device with (a) negligible, (b) weak and (c) strong electronic coupling. The dashed curves represent unperturbed zero-order states. The horizontal axis represents generalised nuclear coordinates, including contributions from both ligand atom positions and solvent sphere.

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