Gaussian dependence

Table II presents the vadues of v, the rate constant for the electron transfer reaction with the donor and acceptor in contact, calculated by deconvolution of the fluorescence decay curves for a number of excited porphyrin-cOkyl halide systems. It appears that the rate parauneter depends strongly on the calculated exothermicity for these reactions. Parauneter i/ contadns information about the Framck-Condon factor of the electron-tramsfer reaction, which is in itself dependent on the reaction exothermicity and reorgauiization energy (22.23). Whether the rate constauit for the electron-transfer reactions depends on the exothermicity in the manner predicted by theory, that is with a simple Gaussian dependence (22), cannot be ainswered at present because of the uncertainties in the energetics of the particular reactions studied here.

Thus, classical Marcus theory predicts an electron transfer rate that has a Gaussian dependence on the free energy of the reaction (Marcus, 1956 Marcus and Sutin, 1985). 

FIGURE 3. Marcus theory predicts a Gaussian dependence of the electron transfer rate ket on free energy AG, which appears as a parabolic dependence on a log plot. The maximum rate is found when the driving force matches the reorganization energy, AG = fiX. 

The majority of peroxy radical UV line shapes show a Gaussian dependence of absorption cross section on the logarithm of the wavelength. As... 

We see that there are two causes for the shift of the emission to higher energies with Increasing temperature the first, mentioned above. Is the frequency effect (if As > 0), while the second Is the deviation from a Gaussian dependence of emission intensity on frequency. Even without the frequency effect the maximum shifts to the blue with Increasing temperature. 

Figure 8.19 shows the deformation resistance at 353 K, 6.5 K above Tg, albeit at a 50-fold-increased strain rate that brings the initial response close to Tg. The behavior is nearly completely rubbery in form, as is shown clearly when the strain is considered in terms of its Gaussian dependence on the principal extension ratio, X, as is shown in the plot of Fig. 8.20 when the stress is plotted against g X) =1 — 1 /I, without any dilatancy consideration in the deformation resistance that becomes inoperative above Tg. The dependence of a on g X) is linear with the exception of the region near where g X) 0, for which there is a vestigial, very minor, plastic-like behavior because of the 50-fold increase in strain rate. 

A typical basis function is a hxed linear combination of simpler, primitive functions. Such a composite function is termed a contracted basis function. Each primitive basis function is centered at an atomic nucleus and has a Gaussian dependence on distance from that nucleus. Except for s-functions, it also has a Cartesian factor to describe its angular dependence. Eor example, a p primitive function looks like xexp(—(r ), where ( (zeta) is the exponent of the Gaussian function. A p contracted function is a hxed linear combination of two or more primitive functions with different exponents. 

As an illustration of the depletion technique, the solid curve in Fig. 5.16A shows an excitation function with a Gaussian dependence on position,... 

We have seen that if the interactions of a quantum system with its surroundings fluctuate rapidly, the decay of coherence in an ensemble of such systems can be described by a relaxation matrix of microscopic first-order rate cmistants. But if the energies of the basis states vary statically from system to system, coherence decays with a Gaussian dependence on time (Box 10.4 Figs. 10.2 and 10.8). We now seek a more general expressirm for pure dephasing to connect the domains in which the fluctuations of the surroundings are either very slow or very fast. 

The transverse fields of the fundamental modes have a Gaussian dependence on R, as is clear from Tables 14-1 and 14-2. By substituting Fq info Table 13-2, page 292, we obtain simple expressions for quantities of interest, such as the normalization N. The fraction of power f within 0 1 increases with increasing V and decreases with... 

An exception is the infinite parabolic profile of Section 14-4. The fundamental-mode fields of Table 14-2, page 307, have the simple Gaussian dependence exp( — F ) and other modal properties have very elementary forms, from which their physical behavior is immediately apparent. To these facts we add the observation that the fundamental-mode intensity pattern - and hence the field distribution - for step and clad power-law profiles... 

The basis of the Gaussian approximation for circular fibers is the observation that the fundamental-mode field distribution on an arbitrary profile fiber is approximately Gaussian. Coupled with the fact that the same field on an ihfinite parabolic-profile fiber is exactly Gaussian, the approximation fits the field of the arbitrary profile fiber to the field of an infinite parabolic-profile fiber. The optimum fit is found by the variational procedure described in Section 15-1. Now in Chapter 16, we showed that the fundamental-mode field distribution on an elliptical fiber with an infinite parabolic profile has a Gaussian dependence on both spatial variables in the cross-section. Accordingly, we fit the field of such a profile to the unknown field of the noncircular fiber of arbitrary profile by a similar variational procedure, as we show below [1, 2],... 

Consider, for example, a circular fiber which has a perturbation with the Gaussian dependence... 

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