Time-scale framework

Calculations within tire framework of a reaction coordinate degrees of freedom coupled to a batli of oscillators (solvent) suggest tliat coherent oscillations in the electronic-state populations of an electron-transfer reaction in a polar solvent can be induced by subjecting tire system to a sequence of monocliromatic laser pulses on tire picosecond time scale. The ability to tailor electron transfer by such light fields is an ongoing area of interest [511 (figure C3.2.14). 

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

In solution, NMR investigations of fluorenyl complexes give rise to only seven C signals, either due to a symmetric position of the lithium cation relative to the car-banion framework or to fast exchange on the NMR time scale under the experimental... 

The developed model was applied to the EPS experiment (Fig.lb) to extract information on the water dynamics. Similar to the previous report [17], the EPS function decreases rapidly at a time scale of -0.5 ps, then raises again at -2 ps, and finally falls off to zero. The EPS functions acquired while keeping the delays tn (empty circles) and t23 (solid circles) fixed [20], are shifted along the vertical axis which is a consequence of the relatively short excited-state lifetime (700 fs). The peak in the EPS function around -2 ps is explained in the framework of our model as arising from interference between the chromophore and solvent responses. The delicate balance between phases of genuinely nonlinear and thermal contributions as the delay t12 between the two excitation pulses is increased, leads to the enhancement of the integrated signal that is measured in the EPS experiment. 

The spiro(indanedione) anion radical in Scheme 3-61 was studied via ESR and UV/visible spectroscopy (Maslak et al. 1990). The spectra clearly indicated delocalization of the unpaired spin density over the entire molecular framework. The unpaired electron undergoes simultaneous delocalization between the halves (in the ESR time scale). The observed spectra were independent of the counterion (Li+, Na+, and K+), thus excluding any ion-pairing complication. As a general inference, an unpaired electron spends its time on both half-shaped orbitals, with no geometrical changes in the molecular skeleton of this anion radical. 

Time scaling. Because the phase-vocoder (the short-time Fourier transform) gives access to the implicit sinusoidal model parameters, the ideal time-scale operation described by Eq. (7.3) can be implemented in the same framework. Synthesis time-instants / are usually set at a regular interval / +1 - / = R. From the series of synthesis time-instants / analysis time-instants / are calculated according to the desired time warping function tua = T l(t ). The short-time Fourier transform of the time-scaled signal is then ... 

The present chapter introduces the reader to singular perturbation theory as the framework for modeling and analyzing systems with multiple-time-scale dynamics, which we will make extensive use of throughout the text. 

Within the framework proposed in Kumar et al. (1998), a time-scale decomposition is initially used to derive separate representations of the slow and fast dynamics of (2.36) in the appropriate time scales and to provide some insight into the variables that should be used as part of the desired coordinate change. Specifically, by multiplying Equation (2.36) by e and considering the limit e —> 0,... 

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