Spectral function density

McCain D 0 and Markley J L 1986 Rotational spectral density functions for aqueous sucrose ... 

The energy spectral density function (or power spectrum) P f) is given by the absolute square of P f) ... 

We can consider EMS to be a direct probe for the energy-momentum spectral density function... 

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. 

The relevant contribute of relaxation measurements on the use of NMR spectroscopy in studying interactions can be argued by considering the relationship between relaxation rates and spectral density function being the latter related to the correlation time, which accounts for the molecular motion. Therefore, spin-lattice and spin-spin can be used to probe interactions between, in principle, every species bearing an active NMR nucleus. 

With analogy to electric circuits, a transfer function of the antenna can be calculated and the response of the antenna to an incoming wave obtained. The output signal is usually expressed as antenna cross-section. It is defined as the ratio between the total energy absorbed by the antenna and the incident spectral density function of the incident wave. In the case of Nautilus antenna (2300 kg, 3 x 0.6 m) the cross-section is of the order of 10 25m2 Hz. 

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 ... 

The case of parallel orientations (e J. Oz) differs radically from the previously considered one, since the frequency dependence of the spectral density function is specified by the fractional power law 109... 

Up to this point only overall motion of the molecule has been considered, but often there is internal motion, in addition to overall molecular tumbling, which needs to be considered to obtain a correct expression for the spectral density function. Here we apply the model-free approach to treat internal motion where the unique information is specified by a generalized order parameter S, which is a measure of the spatial restriction of internal motion, and the effective correlation time re, which is a measure of the rate of internal motion [7, 8], The model-free approach only holds if internal motion is an order of magnitude (<0.3 ns) faster than overall reorientation and can therefore be separated from overall molecular tumbling. The spectral density has the following simple expression in the model-free formalism ... 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

Two spectral density functions determine the relaxation rates in [Eqs. (2) and (3)]. 

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. 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

It is assumed that the noise voltage n(t) is the result of a real stationary process (Davenport and Root, 1958) with zero mean. Because it can be shown that the spectral density function S(f) is the Fourier transform of the autocorrelation function of the noise, it follows that the rms noise is given by... 

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