Correlation, Fourier transform

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

For strongly structured microemulsions, g is negative, and the structure functions show a peak at nonzero wavevector q. As long as g < 2 /ca, inverse Fourier transform of S q) still reveals that the water-water correlation functions oscillate rather than decay monotonically. The lines in phase space where this oscillating behavior sets in are usually referred to as disorder lines, and those where the maximum of S q) moves away from zero as Lifshitz lines. ... 

In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

Given a real-valued sequence (Jq, Ci,. ..,(T v-i, the the correlation function R t) and Fourier Transformation may be defined in both a continuous and discrete form ... 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

The qualitative difference between low-density and high-density rotational relaxation is clearly reflected in the Fourier transform of the normalized angular momentum correlation function ... 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

It is often useful to deal with the statistics in Fourier space. The Fourier transform of the correlation is called the power spectrum... 

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

Given equilibrium quantum expectation values, we can calculate moments of the infra-red vibrational lineshape using a procedure originally outlined by Gordon.The infrared vibrational lineshape is given as the Fourier transform of the dipole moment correlation function ... 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

Two densities P(r) and p (r) are said to be indistinguishable if all their correlation functions and c are identical. As shown by Mermin and collaborators their Fourier transforms... 

Figure 7. Fourier transform of image reported in Figure 6. The (+1) and (—1) and cross-correlation zones are shown together with the zone used to obtain Figure 8. The selected zone corresponds to a spatial resolution in phase of 0.4nm.

Although the idea of generating 2D correlation spectra was introduced several decades ago in the field of NMR [1008], extension to other areas of spectroscopy has been slow. This is essentially on account of the time-scale. Characteristic times associated with typical molecular vibrations probed by IR are of the order of picoseconds, which is many orders of magnitude shorter than the relaxation times in NMR. Consequently, the standard approach used successfully in 2D NMR, i.e. multiple-pulse excitations of a system, followed by detection and subsequent double Fourier transformation of a series of free-induction decay signals [1009], is not readily applicable to conventional IR experiments. A very different experimental approach is therefore required. The approach for generation of 2D IR spectra defined by two independent wavenumbers is based on the detection of various relaxation processes, which are much slower than vibrational relaxations but are closely associated with molecular-scale phenomena. These slower relaxation processes can be studied with a conventional... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

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