Autocorrelation function approximations

Figure 1.20 (A) Site-specific R relaxation rates of sRii from Anabaena sp. PCC7120 (ASR) determined at 12 kHz spinlock power and (B) motional correlation times estimated using single exponential autocorrelation function approximation. Repr/nfed from [250], Copyright 2014 American Chemical Society.

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

This function is a continuous analogue of the frequencies derived from the quasiharmonic approximation. Information about the intensities can be obtained by using the dipole moment autocorrelation function in place of the velocity. The advantage of using MD to build up information about the vibrational modes of the polymer is that the approach incorporates an averaging over many vibrational states of a complex molecule, which may be changing conformation... 

On the other hand, the undamped autocorrelation function (17) we have obtained within the standard approach avoiding the adiabatic approximation must lead after Fourier transform to spectral densities involving very puzzling Dirac delta peaks given by... 

Note that this last expression is nothing but the closed form [90] of the autocorrelation function obtained (as an infinite sum) in quantum representation III by Boulil et al.[87] in their initial quantum approach of indirect damping. Although the small approximation involved in the quantum representation III and avoided in the quantum representation II, both autocorrelation functions are of the same form and lead to the same spectral densities (as discussed later). 

It has been pointed out over the years that the simple exponential function of the form where / is travel time from the source, appears to approximate the Lagrangian velocity autocorrelation function R t) rather well (Neumann, 1978 Tennekes, 1979). If R(t) = exp(-l/r), then the mean square particle displacement is given by (Taylor, 1921)... 

This result is interesting, since it gives the slip length as a function of parameters that can be measured experimentally or a priori, for simple systems in a linear approximation. The bulk shear viscosity can be approximated from the literature, and the monolayer density can be determined from optical techniques. To a first approximation, for rigidly adsorbed layers, the sliptime is related to the autocorrelation function of random momentum fluctuations in the film, given by [40]... 

The PRBS is to be preferred to other random inputs with approximately impulse-shaped autocorrelation functions for the following reasons ... 

The first relevant quantity required to obtain the rates is the autocorrelation function which are shown in Fig.8 for the ground vibrational level of the two excited electronic states. The two cases present a very similar behavior. Simply, for the A case its decay seems much faster. What is notorious is the large difference between the EP halfwidths as a function of the energy for the two electronic states, of approximately 2-3 orders of magnitude, as shown in Fig.9. This is explain by the norm of the initial wavepackets, which is much smaller for the B state, because its well is at larger R and shorter r, where the non-adiabatic couplings are much smaller. 

Fig. 50. Velocity autocorrelation function for argon at 90 K. O, From the molecular dynamics calculation of Rahman [451] ——, from the Langevin approximation, exp — mDtfkuT D is taken as 2.72 X 10 9 m2 s"1.

Thus the irreversible relaxation of the average magnetization is (in linear approximation) determined by the autocorrelation function of the equi-... 

This is the general formula (in the linear noise approximation) for the autocorrelation function of the fluctuations in a stable stationary state. Hence it is possible to write down the fluctuation spectrum in an arbitrary system without solving any specific equations. This fact is the basis of the customary noise theory. 

Thus we have derived the Kramers equation (VIII.7.4) as an approximation for short tc. It becomes exact in the white noise limit (3.12). The coefficient of the fluctuation term is the integrated autocorrelation function of the fluctuating force, in agreement with (IX.3.5) and (IX.3.6).110... 

As a consequence of the single chain approximation the conformation statistics of any two chains are independent of each other. Correspondingly, the mean autocorrelation function is given by... 

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... 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

---