Autocorrelation function reorientation

Steele W. A. Molecular reorientation in liquids. II. Angular autocorrelation functions, J. Chem. Phys. 38, 2411-18 (1963). 

NMR 13C spin-lattice relaxation times are sensitive to the reorientational dynamics of 13C-1H vectors. The motion of the attached proton(s) causes fluctuations in the magnetic field at the 13C nuclei, which results in decay of their magnetization. Although the time scale for the experimentally measured decay of the magnetization of a 13C nucleus in a polymer melt is typically on the order of seconds, the corresponding decay of the 13C-1H vector autocorrelation function is on the order of nanoseconds, and, hence, is amenable to simulation. 

Formally, S2 represents a decrease in the autocorrelation function caused by the motion S2=0 corresponds to completely unrestricted motion of a bond (N-H in this case), while S2=0 is expected if the bond reorientations are frozen. It was shown recently that the order parameter may be related to the statistical mechanical properties of a protein molecule [33-35] hence, changes in the NMR-derived order parameters can indicate localized contributions to overall molecular entropy. 

Dynamics of Reorientation. The approach of Impey, Madden and McDonald is used to consider the dynamics of reorientation for water molecules. The time autocorrelation function of the second Legrendre polynomial P2 of the angle subtended by the intramolecular H-H bond vector at time t with respect to its position at time t = 0 is calculated ... 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

Microwave spectroscopy and dielectric relaxation studies probe the autocorrelation function of the total electrical polarization of the system and thereby also provide information about molecular reorientation. This information is difficult to interpret. 

The dipole autocorrelation function, , defined previously. The full-time dependence of this function for liquid carbon monoxide has been successfully determined experimentally from Fourier inversion of infrared band shapes.2,15 In fact, this was one of the reasons this system was studied. This function has also been successfully evaluated in terms of models of the molecular reorientation process.58 s memory function, KD(t), is defined by... 

Power spectra of the autocorrelation functions of the linear and angular velocities parallel and perpendicular to the C3 symmetrical axes have also been examined by Neusy et al. (32). In the rotator phase, there is good agreement with the Raman data (36). The calculated characteristic time (r4) for reorientation of the C3 axes from one [111] direction to another and also the reorientation time (r3) for rotation of molecules around the C3 axes were similar... 

Of special interest are the data for the magic analyzer orientation —30° suppressing the isotropic scattering, measured in the same experimental runs and for identical conditions as the data of Fig. 6a. The results are presented in Fig. 6b. The ordinate scale refers to the same units as in Fig. 6a, while the time scale of the abscissa is stretched. The measured transient consists of two features a pronounced signal overshoot around to = 0 due to nonresonant scattering and an exponential tail, obviously representing the anisotropic contribution. The data represent novel evidence for the exponential time dependence of the reorientational autocorrelation function or [see Equation (3)] ... 

This function is a measure of the reorientation of the component of the velocity vector parallel to the surface It is calculated by choosing a molecule and following its motion as a function of time for a specified time period, averaging the velocity autocorrelation function (the dot product of the 2D vector velocity at the time t, with the velocity at a later time in the... 

The relationship between the structure of a polymer chain and it dynamics has long been a focus for work in polymer science. It is on the local level that the dynamics of a polymer chain are most directly linked to the monomer structure. The techniques of time-resolved optical spectroscopy provide a uniquely detailed picture of local segmental motions. This is accomplished through the direct observation of the time dependence of the orientation autocorrelation function of a bond in the polymer chain. Optical techniques include fluorescence anisotropy decay experiments (J ) and transient absorption measurements(7 ). A common feature of these methods is the use of polymer chains with chromophore labels attached. The transition dipole of the attached chromophore defines the vector whose reorientation is observed in the experiment. A common labeling scheme is to bond the chromophore into the polymer chain such that the transition dipole is rigidly affixed either para 1 lei (1-7) or perpendicular(8,9) to the chain backbone. 

Other orientational correlation coefficients can be calculated in the same way as the correlation coefficients that we have discussed already. Thus, the reorientational correlation coefficient of a single rigid molecule indicates the degree to which the orientation of a molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelation function is the rotational equivalent of the velocity correlation function ... 

The Dejean-Laupretre-Monnerie (DLM) orientation autocorrelation function is based on the above description. It takes into account independent damped conformational jumps, described by the Hall-Helfand autocorrelation function, and librations of the internuclear vectors represented, as proposed by Howarth [17] (see chapter 4) by a random anisotropic fast reorientation of the CH vector inside a cone of half-angle and axis the rest position of the internuclear vector. The resulting orientation autocorrelation function can be written as... 

The detailed analysis of carbon-13 spin-lattice relaxation times of a number of polymers either in solution or in bulk at temperatures well above the glass-transition temperature has led to a general picture involving several types of motions. The segmental reorientation can be interpreted in terms of correlated conformational jumps which induce a damped orientation diffusion along the chain. It is satisfactorily described by the well-known autocorrelation functions derived from models of conformational jumps in polymer chains [4,5] which have proven to be very powerful in representing fluor-... 

Latanowicz investigated the concepts of correlation time, autocorrelation function and spectral densities in the context of the tunnelling jumps through the potential barrier, superimposed on another type of motion. Two specific case were discussed hindered methyl rotation combined with isotropic overall reorientation and jumps within a double-well hydrogen bond potential in connection with librations of the whole molecule. 

Vector autocorrelation functions are used to characterize the reorientation of some vector defined in the molecular frame. These correlation functions often have the general form ... 

Tensor autocorrelation functions are measured in some experiments. A pertinent example is a coupled relaxation CNMR experiment [16, 33-38]. In this case, Cartesian tensors fixed in the molecular frame can be used to express the results of the relaxation experiments. In a depolarized light scattering experiment, reorientation the polarizability tensor determines the observed relaxation. 

Mobility in this region is dominated by short-time motion, typically < 2 ps. After that time, all correlation of molecular motion is lost due to frequent collisions with the cavity walls. The center-of-mass velocity autocorrelation function of the penetrant exhibits typical liquid-like behavior with a negative region due to velocity reversal when the penetrant hits the cavity wall [59]. This picture has recently been confirmed by Pant and Boyd [62] who monitored reversals in the penetrant s travelling direction when it hits the cavity walls. The details of the velocity autocorrelation function are not very sensitive to the force-field parameters used. On the other hand, the orientational correlation function of diatomic penetrants showed residuals of a gas-like behavior. Reorientation of the molecular axis does not have the signature of rotational diffusion, but rather shows some amount of free rotation with rotational correlation times of the order of a few tenths of a picosecond, although dependent in value on the Lennard-Jones radii of the penetrant s atoms. 

For the dipolar relaxation mechanism, the spin lattice relaxation time is sensitive to the reorientation dynamics of the CH bond vectors. Different orientations of the CH bond result in slightly different magnetic fields at the carbon nucleus and the modulation of this field allows the spin flips to occur. When we define ecn as the unit vector along a CH bond, the second Legendre polynomial of its autocorrelation function is given by... 

A strong retardation of the rotational motion of water is usually observed near surfaces of biomolecules. The rotational motion of water is usually analyzed through the reorientational dynamics of its electrical dipole defined as the vector pointing from the water oxygen to the middle point of the two hydrogen atoms. Two first- and second-rank autocorrelation functions Ti and Ti are the time average of the Legendre polynomials Pn(cos(6>)) ... 

---