Frequency, effective

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

In addition, the frequency cooo, as well as the tunneUng distance can also be extracted from the same empirical data. Thus all the information needed to construct a PES is available. Of course, this PES is a rather crude approximation, since all the skeleton vibrations are replaced by a single mode with effective frequency cooo and coupling parameter C. From the experimental data it is known that the strong hydrogen bond (roo < 2.6 A) is usually typical of intramolecular hydrogen transfer. 

Like displacement probes, velocity transducers have an effective frequency range of about 10 to 1000 Hz. They should not be used to monitor frequencies above or below this range. 

These effective frequencies are chosen such that contributes only for optical processes with at least three non-zero frequency arguments, while u 4 is only non-zero if all four frequency arguments are non-zero. During the implementation it was also found that this choice for the effective frequencies leads to the most compact expressions for the coefficients Ak,i,m in terms of D n,m,l). Using a similar notation as in Eq. (73), 7 (o o Wi,o 2i 3) can up to sixth order in the frequencies be expanded... 

We note that 02 4 vanishs if two of the four the frequency arguments Wq, wi, u>2, W3 become zero. For the case that three frequency arguments are equal (third harmonic generation) both mixed-symmetry effective frequencies w 2 vanish. The coefficients Amsi Bms, etc. 

Figure 16. The effective frequency 0(x) for two transversal excitations in HO2. Solid line shows the results of numerical solution of Eqs. (79) and (80). Dotted line represents the frequency in the adiabatic approximation, y = 1[3] corresponds to the mode (0,0,1) [(1,0,0)] (see Fig. 14). Taken from Ref. [32].

The present approach has been applied to the experiment done by Nelsen et ah, [112], which is a measurement of the intramolecular electron transfer of 2,7-dinitronaphthalene in three kinds of solvents. Since the solvent dynamics effect is supposed to be unimportant in these cases, we can use the present theory within the effective ID model approach. The basic parameters are taken from the above reference except for the effective frequency. The results are shown in Fig. 26, which shows an excellent agreement with the experiment. The electronic couphng is quite strong and the perturbative treatment cannot work. The effective frequencies used are 1200, 950, and 800 cm for CH3CN, dimethylformamide (DMF), and PrCN [113]. 

For simple outer-sphere electron transfer reactions, the effective frequency co is determined by the properties of the slow polarization of the medium. For a liquid like water, where the temporal relaxation of the slow polarization as a response to the external field is single exponential, tfie effective frequency is equal to... 

Equation (9) was obtained using the assumption that the vibrational subsystem is in the state of thermal equilibrium corresponding to the initial electron state. The expression for the effective frequency a>eff has the form5... 

Thus the effective frequency with which activated complexes are transformed into reaction products is kBT/h. At a temperature of 300 °K, this group has a value of 6 x 1012 sec-1, which is comparable in magnitude to normal molecular vibration frequencies. 

Now, comparing 7Sf and /Sf ex, we observe also some splitting modifications. We may attribute them to some additional changes in the frequency gap. Indeed, as shown in Section V.B, the full treatment of the Fermi coupling mechanism leads to a displacement of the potential of the fast mode (both in energy and position) that does not appear within the exchange approximation. The fast mode then involves an effective frequency that differs from oo0, which leads to an effective gap which differs from (oo0 — 2oog). 

Moreover, because the driven term depends on A, a strengthening of the Fermi coupling implies an increase of the effective frequency gap between the coupled levels, and therefore an increase of the intensity gap, that is at the opposite of the behavior observed within the exchange approximation. This... 

The effective frequencies that characterize solvent response can be characterized more quantitatively from several points of view, including generalized Langevin theory [367-372], Brownian oscillators [373, 374], and instantaneous normal modes [375],... 

Here, L total is the depth of the etched hole per pulse and is assumed to be the sum of photochemical and photothermal contributions, Tphoto and Thermal, respectively 0Ceff is the effective photon absorption coefficient of the medium and can vary with laser emission characteristics, e g., photon density Fis the incident laser fluence Fth is the medium s threshold fluence A and F are the effective frequency factor with units of pm/pulse and the effective activation energy with units of J/cm2, respectively, for the zeroth-order thermal rate constant F0, comparable in magnitude to Fth, is important only at low fluences.64 Equation (5) is obtained after assuming that the polymer temperature T in the laser-exposed region of mass mp and the thermal rate constant k are given, respectively, as... 

The effective activation energy and effective frequency factor, calculated. In the above expression Kj, K2, and standfor the rate constant for the different steps of the reaction ... 

Evaluate the effective frequency of the product flow vibrator and the oscillating rotor at the different possible adjustments develop history and date. 

In this chapter, we first present (Section 10b) some qualitative aspects of MPE processes based on their visualization via the configuration coordinate model. We next discuss the problem more quantitatively (Section 10c). It is also to be noted that although the configuration coordinate model (i.e., visualization in terms of a single effective frequency) follows from the quantitative treatment only in certain approximations, in practice most of the literature does use such approximations in one form or another. Finally (Section lOd), we review the literature. 

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