Autocorrelation function corresponding spectral density

In an "analog optical mixing experiment one measures either the time autocorrelation function of the photomultiplier output current or its corresponding spectral density. In the former case, the photomultiplier output is analyzed by an autocorrelator and in the latter, by a spectrum analyzer. 

The spectral density J(w) which we have already mentioned is the frequency spectrum corresponding to the autocorrelation function G(t), i.e., it is the Fourier transform of it, so that... 

The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

In nematic systems the two uncoupled fluctuation modes are purely relaxational with Lorenzian spectral densities. In general, the autocorrelation function of the scattered field includes two exponentials corresponding to decay times of the two fluctuation modes. 

Initial conditions for the wavepacket calculations correspond to the Franck-Condon geometry, and the effective-mode expansion is defined with respect to this reference geometry. Figure 15.3 shows the time-dependent diabatic S2 populations and autocorrelation functions C(f) = (V (0) vf(f)) generated from the successive spectral density approximants J (o), M = 1,...,3. All orders agree over the shortest time scale ( 5 fs), and the orders M = 2,3 are found to be very close over the complete observation interval. The M = 3 result is virtually indistinguishable from the result obtained for the reference spectral density and can be considered converged. 

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