Dipolar time correlation functions

Multiexponential Fitting Parameters for the Dipolar Time Correlation Functions of Water Molecules around the Three a-Helices of the Protein. Corresponding Parameters for Bulk Water is also Listed for Comparison... 

Kometani K., Shimizu H. Study of the dipolar relaxation by a continued fraction representation of the time correlation function, J. Phys. Soc. Japan 30, 1036-48 (1971). 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

Fig. 4 Dipolar reorientational time correlation function, Cw(t) for bound water molecules in the micellar solution, and for bulk water molecules.

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

The mode coupling theory of molecular liquids could be a rich area of research because there are a large number of experimental results that are still unexplained. For example, there is still no fully self-consistent theory of orientational relaxation in dense dipolar liquids. Preliminary work in this area indicated that the long-time dynamics of the orientational time correlation functions can show highly non-exponential dynamics as a result of strong in-termolecular correlations [189, 190]. The formulation of this problem, however, poses formidable difficulties. First, we need to derive an expression for the wavevector-dependent orientational correlation functions C >m(k, t), which are defined as... 

A rather simple experimental teehnique involving measurement of the time-dependent fluorescence Stokes shift (TDFSS) after an initial exeitation has been applied to measure SD in a large number of liquids. TDFSS oceurs due to dipolar solvation of the excited probe and thus gives an estimate of the solvation timeseales. In an important paper, Jimenez et al. reported the results of SD of the exeited state of the dye coumarin 343 (C343) in liquid water [14]. Their result is shown in Figure 3.13. The initial part of the solvent response of water was found to be extremely fast (few tens of femtoseconds) and it constituted more than 60% of the total solvation energy relaxation. The subsequent relaxation was found to occur in the picosecond timescale. The decay of the solvation time correlation function, S t)y was fitted to a function of the following form... 

P(co) is an internal field factor and A t) is a time-correlation function which represents the fluctuations of the macroscopic dipole moment of the volume V in time in the absence of an applied electric field. Equations (44) and (45) are a consequence of applying linear-response theory (Kubo-Callen-Green) to the case of dielectric relaxation, as was first described by Glarum in connexion with dipolar liquids. For the special case of flexible polymer chains of high molecular weight having intramolecular correlations between dipoles but no intermolecular correlations between dipoles of different chains we may write... 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

The movements capable of relaxing the nuclear spin that are of interest here are related to the presence of unpaired electrons, as has been discussed in Section 3.1. They are electron spin relaxation, molecular rotation, and chemical exchange. These correlation times are indicated as rs (electronic relaxation correlation time), xr (rotational correlation time), and xm (exchange correlation time). All of them can modulate the dipolar coupling energy and therefore can cause nuclear relaxation. Each of them contributes to the decay of the correlation function. If these movements are independent of one another, then the correlation function decays according to the product... 

In multidimensional NMR studies of organic compounds, 2H, 13C and 31P are suitable probe nuclei.3,4,6 For these nuclei, the time evolution of the spin system is simple due to 7 1 and the strengths of the quadrupolar or chemical shift interactions exceed the dipole-dipole couplings so that single-particle correlation functions can be measured. On the other hand, the situation is less favorable for applications on solid-ion conductors. Here, the nuclei associated with the mobile ions often exhibit I> 1 and, hence, a complicated evolution of the spin system requires elaborate pulse sequences.197 199 Further, strong dipolar interactions often hamper straightforward analysis of the data. Nevertheless, it was shown that 6Li, 7Li and 9Be are useful to characterize ion dynamics in crystalline ion conductors by means of 2D NMR in frequency and time domain.200 204 For example, small translational diffusion coefficients D 1 O-20 m2/s became accessible in 7Li NMR stimulated-echo studies.201... 

Sturz and DoUe measured the temperature dependent dipolar spin-lattice relaxation rates and cross-correlation rates between the dipolar and the chemical-shift anisotropy relaxation mechanisms for different nuclei in toluene. They found that the reorientation about the axis in the molecular plane is approximately 2 to 3 times slower than the one perpendicular to the C-2 axis. Suchanski et al measured spin-lattice relaxation times Ti and NOE factors of chemically non-equivalent carbons in meta-fluoroanihne. The analysis showed that the correlation function describing molecular dynamics could be well described in terms of an asymmetric distribution of correlation times predicted by the Cole-Davidson model. In a comprehensive simulation study of neat formic acid Minary et al found good agreement with NMR relaxation time experiments in the liquid phase. Iwahashi et al measured self-diffusion coefficients and spin-lattice relaxation times to study the dynamical conformation of n-saturated and unsaturated fatty acids. 

If one ignores even a small CSA contribution, the error introduced in the calculations may lead to considerable error in the calculation of rotational correlation times and their variation with temperature. In order to determine the contribution of each mechanism (dipolar and CSA) in C NMR relaxation studies, a mathematical approach has been devised [13-15] which involves combining several correlation functions followed by a series of iterative steps that lead to the correct rotational correlation times. This results in the determination of the separate dipolar and CSA contributions to the overall relaxation mechanism in C NMR relaxation studies. The following section outlines the necessary steps leading to the determination of both dipolar and CSA contributions as given in Eq. (4.4-10). 

The rotational dynamics on the protein surface is basically shaped by electrostatic interactions alone and the HBs formed by water with the protein surface break the quasi-isotropic nature of the dipolar rotation that is found in the bulk. Also, for the fiilly thermalized protein, a ratio between the characteristic times of the first and the second dipole-dipole correlation function, yff = /r, of about 5 is at variance... 

It has been known for some time that the FIDs of such systems often decay in a way that is well represented for the most part by a Weibullian [60,61]. Arguments based on consideration of correlation functions suggested that the FID of high molecular weight polydimethyl siloxane in the melt should decay for the most part with a Weibullian power of between 1.25 and 1.5 [62], and the existence of residual static dipolar interactions in these systems was confirmed by the existence of the pseudo-solid echo [63]. This reference forms part of a much larger body of work on such systems by Cohen-Addad and co-workers which it is beyond the scope of this chapter to cover in any detail, but interested readers are directed to literature such as [64] and [65]. 

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