Single mode approximation

Fig. 14. Dependence of the quantity (fito/gy l V )k where k is the rate constant for spin conversion on halkgT as calculated from the single-mode approximation Eq. (79) solid lines) and the full expression Eq. (77) for the value q = 0.3 of the solvent reorganization parameter dashed lines). Data are given for p = — 2.0, 0, -I- 2.0 and for the value of the coupling parameter S = 15. According to Ref. [117]...

In an (average) single mode approximation, the Franck-Condon factor, eqs 3-5, can be simplified and it takes the well known form (2, 4, 7-9)... 

Figure 2. Theoretical prediction for the temperature dependence of the electron transfer rate for activated and for activationless processes. Solid lines are calculated for a continuum of vibrational modes dotted lines represent the single-mode approximation (6, 8). Upper curve AE, —2000 cm 1 P, 20 and S, 20. Lower curves AE, —800 cm"1 P, 8 and S, 20.

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

Thus, one can see that the single-mode approximation allows us to describe linear viscoelastic behaviour, while the characteristic quantities are the same quantities that were derived in Chapter 6. To consider non-linear effects, one must refer to equations (9.52) and (9.53) and retain the dependence of the relaxation equations on the anisotropy tensor. 

In the time domain, the single-mode approximation, equation Eq. (191), is equivalent to assuming that the relaxation function Cy(t) as determined by the exact equation, Eq. (182) (which in general comprises an infinite number of Mittag-Leffler functions), may be approximated by one Mittag-Leffler function only, namely... 

The latter expression clearly indicates that % decreases with increases in F. In addition, using the above expressions for k and we write the total free energy density for this ID single mode approximation as ... 

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