Spectral Density Distributions

The n-th spectral density distribution moment of a general Hamiltonian is defined as... 

Consider the relation between the correlation function and its spectral density. Slower and faster decays of the correlation function (i.e. slower and faster motions) give narrower and wider distributions of the spectral density, respectively. Figure 11 (a) shows some decay curves of the correlation function for motions with different xc values and (b) indicates the distributions of their respective spectral densities that are obtained by Fourier transform of the decay curves. Here A, B, C in (a) are the decay curves with xca> tcb> and A, B, C in (b) are the distribution of their respective spectral densities. As can be seen, the decay becomes slower and the spectral density distribution becomes narrower as the Xc increases. Assume here that xca cb i cc and the amorphous phase involves two independent motions dictated by xca and xcb whereas the crystalline-amorphous interphase involves two motions dictated by xca and Xcc-Here, xca characterizes a local molecular motion with relation to few carbon atoms in the main molecular chain, and xcb>tcc a somewhat long-ranged motion with relation to a conformational change of ca. 10-20 carbon atoms. In other words, it is assumed that somewhat long-ranged motion is different between the two phases but local motion is the same, the former is dictated by xcb or Xcc and the latter by a common relaxation time xca-... 

J.A Karwowski, Spectral density distribution moments, in Handbook of Molecular Physics and Quantum Chemistry, vol. 1 Fundamentals, ed. S. Wilson, P.F. Bernath and R. McWeeny, chapter 29, Wiley, Chichester, 2003. 

Example plot of power spectral density distribution... 

The spectral analysis of the records shown in Fig. 25.6 has been performed, and the spectral density distributions are correspondingly shown in Fig. 25.7. It can be seen that the dominant waves in those events, which have the highest energy density, are all around 21 to 22 min. 

Cole and Davidson s continuous distribution of correlation times [9] has found broad application in the interpretation of relaxation data of viscous liquids and glassy solids. The corresponding spectral density is ... 

The largest correlation times, and thus the slowest reorientational motion, were shown by the three C- Fl vectors of the aromatic ring, with values of between approximately 60 and 70 ps at 357 K, values expected for viscous liquids like ionic liquids. The activation energies are also in the typical range for viscous liquids. As can be seen from Table 4.5-1, the best fit was obtained for a combination of the Cole-Davidson with the Lipari-Szabo spectral density, with a distribution parame-... 

Wiener inverse-filter however yields, possibly, unphysical solution with negative values and ripples around sharp features (e.g. bright stars) as can be seen in Fig. 3b. Another drawback of Wiener inverse-filter is that spectral densities of noise and signal are usually unknown and must be guessed from the data. For instance, for white noise and assuming that the spectral density of object brightness distribution follows a simple parametric law, e.g. a power law, then ... 

The 327-670 GHz EPR spectra of canthaxanthin radical cation were resolved into two principal components of the g-tensor (Konovalova et al. 1999). Spectral simulations indicated this to be the result of g-anisotropy where gn=2.0032 and gi=2.0023. This type of g-tensor is consistent with the theory for polyacene rc-radical cations (Stone 1964), which states that the difference gxx gyy decreases with increasing chain length. When gxx-gyy approaches zero, the g-tensor becomes cylindrically symmetrical with gxx=gyy=g and gzz=gn. The cylindrical symmetry for the all-trans carotenoids is not surprising because these molecules are long straight chain polyenes. This also demonstrates that the symmetrical unresolved EPR line at 9 GHz is due to a carotenoid Jt-radical cation with electron density distributed throughout the whole chain of double bonds as predicted by RHF-INDO/SP molecular orbital calculations. The lack of temperature... 

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. 

To compare the time scales of the dynamics characterization produced by each model, the spectral density or correlation function can be written as a distribution of exponential correlation times. For a correlation function, (t), the general expression is CO... 

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