Spectral densities rotational motions

The small step rotational diffusion model has been employed to extract rotational diffusion constants Dy and D from the measured deuterium spectral densities in liquid crystals [7.25, 7.27, 7.46, 7.49 - 7.53]. Both the single exponential correlation functions [Eq. (7.54)] and the multiexponential correlation functions [Eq. (7.60)] have been used to interpret spectral densities of motion. However, most deuterons in liquid crystal molecules are located in positions where they are rather insensitive to motion about the short molecular axis. Thus, there is a large uncertainty in determining D or Tq (t o) because of Dy > D and the rather small geometric factor [doo( )] for most deuterons in liquid crystal molecules. For 5CB, it is necessary to fix [7.52] the value of D using the known activation... 

Finally, restricted 7-motion within an apex angle 0o [Eq. (7.80)] has been employed to interpret [7.11] spectral densities of aromatic deuterons in the nematic and smectic A phases of 50.7-d4. Assuming that a-, and 7-motion are completely uncorrelated, and neglecting the -motion [i.e., set / (0) = /3 t)], the spectral densities were evaluated by describing the a-motion by a simplified model of uniaxial free rotational diffusion about the director and the 7-motion by the reduced correlated functions given in Eq. (7.80). It remains to be examined whether the anisotropic viscosity model in conjunction with restricted 7-motion [i.e., Eq. (7.82)] would be better in interpreting spectral densities of motion in various smectic phases. Also, there is still no convincing NMR evidence for biased 7-motion in nematic or smectic A phases. 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

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. 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

Now the expectation (mean) value of any physical observable (A(t)) = Yv Ap(t) can be calculated using Eq. (22) for the auto-correlation case (/ = /). For instance, A can be one of the relaxation observables for a spin system. Thus, the relaxation rate can be written as a linear combination of irreducible spectral densities and the coefficients of expansion are obtained by evaluating the double commutators for a specific spin-lattice interaction X in the auto-correlation case. In working out Gm x) [e.g., Eq. (21)], one can use successive transformations from the PAS to the (X, Y, Z) frame, and the closure property of the rotation group to rewrite D2mG(Qp ) so as to include the effects of local segmental, molecular, and/or collective motions for molecules in LC. The calculated irreducible spectral densities contain, therefore, all the frequency and orientational information pertaining to the studied molecular system. 

Notice the presence of a spectral density at zero frequency in i 2 arising from bz t)bz Q)dt (which evidently does not require to switch to the rotating frame). This zero frequency spectral density will be systematically encountered in transverse relaxation rates and, in the case of slow motions, explains... 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

If this observation corresponds to the true situation in solution then the internal motions are an order of magnitude faster than the rotational correlation time. Under such circumstances, the spectral density function used in these calculations is incorrect. This aspect requires further investigation, particularly once the data from dynamics calculations specifically including water become available. 

The lattice models provide useful interpretations of spin relaxation in dissolved polymers and rubbery or amorphous bulk polymers. Very large data bases are required to distinguish the interpretive ability of lattice models from other models, but as yet no important distinction between the lattice models is apparent. In solution, the spectral density at several frequencies can be determined by observing both carbon-13 and proton relaxation processes. However, all the frequencies are rather hl unless T2 data are also included which then involves the prospect of systematic errors. It should be mentioned that only effective rotational motions of either very local or very long range nature are required to account for solution observations. The local... 

For small or medium-sized molecules undergoing rotational motion in the extreme narrowing limit, the spectral density is field-independent and the n.0.e. 

This result suggests, if it is assumed that a C-H heteronuclear dipolar relaxation mechanism is operative, that methyl protons dominate the relaxation behavior of these carbons over much of the temperature range studied despite the 1/r dependence of the mechanism. The shorter T] for the CH as compared to the CH2 then arises from the shorter C-H distances. Apparently, the contributions to spectral density in the MHz region of the frequency spectrum due to backbone motions is minor relative to the sidegroup motion. The T p data for the CH and CH2 carbons also give an indication of methyl group rotational frequencies. 

Some systems cannot be well described by translational motion, so instead they require a model based on rotational diffusion. The commonly used model is one where the nuclei and radical form a bound complex, then this complex rotates to modulate dipolar coupling.74 Here, the overall correlation time consists of the rotational correlation time of the solvent complex, xT, and the exchange rate of molecules in and out of the complex, tm, where 1 /tc 1 /tr I 1 /tm. The form of this spectral density function is simpler4,25 ... 

The fluctuations are often caused by atomic motion e.g. Brownian motion in liquids, ionic hopping, molecular rotations, librations and atomic vibrations. These motions are often complex and it is the range of frequencies that are present in the motions that determine relaxation. The spectral density function describes the relative intensities of different frequencies in the motions and can be used to calculate relaxation rates. 

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