The Weak-Coupling Case

In general the temperature dependence of the nonradiative processes is reasonably well understood. However, the magnitude of the nonradiative rate is not, and can also not be calculated with any accuracy except for the weak-coupling case. The rea.son 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 fiom parabolic behaviour in the configurational coordinate diagram (anhar-monicily) may influence the nonradiative rate with many powers of ten. However, it will be clear that the offset between the two parabolas (AR) is a very important parameter for the nonradiative transition rate. This rate will increase dramatically if AR becomes larger. 

We consider first the weak-coupling case (S 0), and subsequentfy the intermediate- and strong-coupling cases (S 0). 

Nonradiative transitions in the weak-coupling approximation are probably the best understood nonradiative processes. The experimental data relate mainly to the rare-eaith ions, as far as their sharp-line transitions are considered (i.e. intra-4/ -configuration transitions). The topic has been discussed in books and review papers [1,2,44]- We summarize as follows  

For transitions between levels of a 4/ configuration the temperature dependence of the nonradiative rate is given by 

W(0) is laige for low p, i.e. for small AH or high vibrational frequencies. Further 

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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]... 

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]. 

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),... 

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. 

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. 

As a rule, the weak coupling case is observed in solution, the intermediate coupling case in crystalline solids and oriented membranes, and the strong coupling case within conjugated macromolecules. In order to quantify the energy transfer, we will consider a molecular complex composed of two molecules a and b. The Hamiltonian of this system is given by... 

In the weak coupling case, where the excitation energy is localized, a different treatment is needed. A generalized formula for the excitation energy transfer of weakly coupled systems was derived by Forster. He wrote equations describing the rate of a transition from a localized excited state, ipa M, to another localized excited state, VWi- When the distance between the two molecules, Rab, is not too short, the resonance integral can be approximated by the interaction energy between the transition dipole moments, Pa and Pb,... 

In polyatomic molecules belonging to the weak-coupling case (glyoxal, propynal) the density of rovibronic triplet levels /, J ), which may be collisionally coupled to the initially excited state, is already so high... 

It formed the basis of the first solid state laser in I960. This emission consists of two sharp lines (the so-called R lines) in the far red (see Fig. 3.18). Since it is a line, it must be due to the transition -> A2 (Fig. 2.9) generally speaking the emission of transition metal ions originates from the lowest excited state. The life time of the excited state amounts to some ms, because the parity selection rule as welt as the spin selection rule apply. The emission line is followed by some weak vibronic transitions obviously this emission transition belongs to the weak-coupling case. 

In Figure 4.7, we show the line shape and Q for the weak, slow modulation limit, case 2. The line shape [Fig. 4.7(a)] is a Lorentzian with a width T, to a good approximation, thus the features of the line shape do not depend on the properties of the coupling of the SM to an environment such as V and R. On the other hand, Q [Fig. 4.7(h)], shows a richer behavior. Recalling Eq. (4.67) for T l/R in Eq. (4.66), it exhibits doublet peaks separated by tP(F) with the dip located at = 0, and its magnitude is proportional to l/R. We will later show that this kind of a doublet and a dip in (2 is a generic feature of the weak coupling case, found not only in the slow but also in the fast modulation case considered in Section VI.A.3. 

By inserting into Eq. (213) the appropriate form of pit) from Eq. (211), we obtain the expressions for S(r) and L(r) corresponding to the coherent and incoherent excitation conditions. We may deduce, in this way, the general form of the decay in typical cases, summarized in Table 1. These are represented schematically in Figs. 8-10. The weak-coupling case is approximated... 

Fig. 8. Time dependence of (a) S and (b) L in the weak-coupling case incoherent (broken line) and coherent (solid line) excitation.

Under an incoherent excitation, the overall decay is described as a sum of exponentials in the weak-coupling, as well as in the strong-coupling case. S and L decrease monotonically with time, but not necessarily with the same rate. The decay time of levels with high s content is more rapid. The main difference between both cases is the initial value of the S(to)/L to) ratio. In the weak-coupling case, we have on the average 1 and... 

The strong-coupling case is probably an exception (in methylene and formaldehyde the Si-T, interaction corresponds clearly to the weak-coupling case), but it occurs at least in three triatomic molecules NO2, SO2, and CS2, where an anomalously long fluorescence decay was observed for the first time in the classical work of Douglas (1966). We limit our discussion to the most extensively studied and yet not completely elucidated case of the visible transition in NO2 [(an exhaustive review is given by Hsu et al. (1978)]. 

This behavior corresponds clearly to the weak-coupling case. The (symmetry-allowed) mixing takes place between a few vibronic levels, and under a selective excitation (and in absence of collisional relaxation), only the resonance emission occurs. 

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