Legendre functions reorientation

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

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... 

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... 

Most experimental techniques do not probe reorientational motions directly but measure the time-dependence of properties which themselves are sensitive to orientation. Thus in far infrared and Raman spectroscopy one measures dipole or polarizability fluctuations with respect to particular normal modes of vibration. Although the interpretation can be complicated, in cases where the fluctuations arise predominantly from rotational motions of localized groups correlation functions of the first and second Legendre polynomials, C t) and 2(1) respectively, can be retrieved from the data. 

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>)) ... 

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