Weak-coupling case

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. 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

The quantity bk has a physical meaning as the spectral density of photons emitted by the dressed excited particle, and its explicit form is rather complicated [11]. However, for weak coupling case it reduces to a simple form ... 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

The resulting expression is especially simple in the weak coupling case. In this case, the two propagators in Eq. (12) can be approximated by their first order (i.e, single hop) terms. (The zeroth order term makes no contribution of Kjf as long as i 5 f) In this weak coupling limit, the expression for Pif (t) can be expressed as ... 

The shift of the emission maximum relative to the absorption maximum, the so-called Stokes shift, is determined by the value of Qq-Qo (see Fig. 1). For the equal force constant case this Stokes shift is equal to 2Shv [2], This indicates that the Stokes shift is small for the weak-coupling case and large for the strongcoupling case. It is also clear that the value of the Stokes shift, the shape of the optical bands involved, and the strength of the (electron-vibrational) coupling are related. For a more detailed account of these models the reader is referred to the literature mentioned above [1-4]. 

The problems in case of the rare earth ions, a weak-coupling case, are completely different. Progress has only started a couple of years ago and will certainly accelerate in the coming years. 

Fig. 5. Schematic representation of various energy curves due to weak (w) and strong (s) interactions between two harmonic oscillators. Energy in arbitrary units with the following values (k/2)% = 3, E = 0 for strong coupling, U = 7, for weak coupling, U = 5. The dotted curves indicate the intermediate states for weak coupling case, [W (w)].

For photo-induced electron transfer (ET) reactions [53], there exist three cases depending on their mechanism (1) non-adiabatic, diabatic, or weak coupling case, (2) adiabatic, or strong coupling case, and (3) charge transfer complex case. This section shall focuses on case (1) to which perturbation theory can be applied. 

Figure 6.1 shows simulation of the population dynamics of the two vibronic manifolds. The populations pbv>b and pcw>cw(T) after excitation are calculated for (a) the weak coupling case and (b) the strong coupling case. Figure 6.1 clearly shows the population transfer between the two electronic states due to the creation of the vibronic coherence. 

For the weak coupling case (class II complexes), in which the distortion seen in Fig. 2 is small, Hush derived the following relationships of intervalence band properties For the symmetric system (Fig. la),... 

For weak coupling cases, Hush showed that the intensity of an intervalence transition was related to the extent of coupling between donor and acceptor. The derivation (11) begins by considering the theoretical expression for oscillator strength,... 

Up to this point it was assumed that the return from the excited state to the ground state is radiative. In other words, the quantum efficiency (q), which gives the ratio of the numbers of emitted and absorbed quanta, was assumed to be 100%. This is usually not the case. Actually there are many centers which do not luminescence at all. We will try to describe here the present situation of our knowledge of nonradiative transitions that is satisfactory only for the weak-coupling case. For detailed reviews the reader is referred to ref. 11. 

In general the temperature dependence of the nonradiative processes is reasonably well understood. However, the magnitude of the nonradiative rate is not, and cannot be calculated with any accuracy except in the weak-coupling case. The reason for this is that the temperature dependence stems from the phonon statistics which is known. However, the physical processes are not accurately known. Especially the deviation from parabolic behavior in the configurational coordinate diagram (anharmonicity) may influence the nonradiative rate with many powers of ten (11). 

IV. Nonradiative Transitions A Qualitative Approach A. The Weak-Coupling Case... 

The small upper limit of the Davydov splitting established by the low temperature piezomodulation spectra of PTS indicates that the interchain coupling of electronic transitions is negligible for that system. While the sidechains of the phenylurethane series may change this somewhat, it is unlikely that the interaction will exceed the weak coupling case. This is confirmed by the bandwidth studies of the reflection spectra where the coupling may approach the intermediate case. 

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