Strongly 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. 

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... 

Strong electronic-vibrational interaction in the electrolyte renders the tunneling process from the semiconductor irreversible, obviating oscillations which increase residence time in the semiconductor and thus intraband thermalization therein. This irreversibility has its basic origin in the electronic particle tunneling from the semiconductor where the electron-phonon interaction is weak to a strongly-coupled state of the electrolyte where the electronic-vibrational interaction is strong. 

In the present section I discuss examples where the treatment has been extended in two different ways, either going beyond the QVC approach or including more than two strongly coupled states in the analysis. 

Considerable insight has been obtained into the properties of the IDs in Figures 4.S-4.8, where the multidimensional distribution Ii (m, n a, b) of i accepting modes is represented, respectively, for the cases of moderately strong-coupled states... 

Figure 4.1 The intramolecular distribution /) (m,n a, t>) as function of n for preselected levels m of the electronic excited state (strongly coupled states a = 12, fc = 0.05).

Figure4.5 The multidimensional distribution / (0,n a,b) (in the parallel-mode approximation) for moderately strongly coupled states a = 2, b = —0.2 and different orders of degeneracy i.

For the study of time-resolved processes such as discussed in Chapter 6, but with many strongly coupled states in the manifold ipi, a closed-form solution cannot be carried through analytically and approximate treatments are necessary. This is not the case when solving Equation 6.58 for many states, which can always be integrated by numerical methods. All require modern, high-speed computers for their execution and the development of numerical recipes to handle large determinants. Furthermore, numerical solutions may often be obtained much more easily than closed-form solutions and may be sufficiently accurate, as the physical situation warrants. On the other hand, the closed-form solution given by Equation 6.58 serves as a convenient introduction to pursue much more difficult problems when possible. 

The strong dependence of the PES on molecular orientation also leads to strong coupling between rotational states, and hence rotational excitation/de-excitation in the scattering. This has been observed experimentally for H2 scattering from Cu surfaces. Recent work has shown that for H2 the changes m rotational state occur almost exclusively when the molecular bond is extended, that is, longer than the gas-phase equilibrium value [ ]. 

B. The Study of a Real Three-State Molecular System Strongly Coupled (2,3) and (3,4) Conical Intersections... 

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). 

The contrast in knowledge is a result of the degree of complexity of materials properties elastic piezoelectric solids have perhaps the least complex behaviors, whereas ferroelectric solids have perhaps the most complex mechanical and electrical behaviors of any solid under shock compression. This complexity is further compounded by the strong coupling between electrical and mechanical states. Unfortunately, much of the work studying ferroelectrics appears to have underestimated the difficulty, and it has not been possible to carry out careful, long range, systematic efforts required to develop an improved picture. 

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