Fourier Transform of Time-Correlation

In a theoretical treatment, it is necessary to make approximations in the derivation of the spectral densities (Appendix A.2 - equation (A7)), that is, the Fourier transforms of time correlation functions of perturbations used to express the nuclear spin relaxation times. These theories have been tested against experiments and their limitations have been examined under varying conditions. The advantage of MD simulations to evaluate the theoretical models is the realism of the description and that many approximations in the theoretical model can be tested separately. Because of the conceptual differences between theories and the arbitrariness in their parameterization, it is often not possible discriminate between... 

The above equations provide two alternative routes for calculating kinetic coefficients from simulations of a system at equilibrium. Averages in the above equations are ensemble averages, hence the results are ensemble-sensitive. The time correlation functions contain more information than just the kinetic coefficients. The Fourier transforms of time correlation functions can be related to experimental spectra. Nuclear magnetic resonance (NMR) measures the time correlation functions of magnetization, which is related to the reorientation of particular bonds in the polymer molecule inelastic neutron scattering experiments measure the time correlation functions of the atom positions infrared and Raman scattering spectroscopies measure the time correlation function of dipole moments and polarizabilities of the molecules. 

The connection between the rate constants W and molecular parameters is a complex subject that can only be outlined here. Basically, the dipole-dipole interaction is a through-space effect in which one nuclear magnetic dipole interacts with the local field created by a second nucleus. The value of W depends upon the component of this field that fluctuates at the frequency of the transition, that is at 0, (o and 2Larmor frequency. Working out the algebra shows that the cross-relaxation rates are proportional to spectral densities, which are Fourier transforms of time correlation functions that describe molecular motions ... 

---