Cross-correlation function definition

The limit of this expression gives us the familiar definition for the cross-correlation function with the limits of integration redefined for a distribution about zero. 

Some basic definitions are necessary. The definition of a cross-correlation function (CCF) of two non-zero average power signals, x(t) and y(t), is ... 

The fluorescence intensity trajectories of the donor (/d(f)) and acceptor (/a(t)) give autocorrelation times (Fig. 24.2b) indistinguishable from fitting an exponential decay to the autocorrelation functions, (A/d (0) A/d (t)) and (A/a (0) A/a (t)), where A/d(t) is /d(t) — (Id), (Id) is the mean intensity of the overall trajectory of a donor, and A/a(t) has the same definition for an intensity trajectory of an acceptor. In contrast, the cross-correlation function between the donor and acceptor trajectories, (A/d (0) A/d (t)), is anticorrelated with the same decay time (Fig. 24.2b) which supports our assignment of anticorrelated fluctuations of the fluorescence intensities of the donor and acceptor to the spFRET process. 

The strength of the bath coupling to each system variable is described by the coupling constants / and, because they enter at second order, the rate constant for the dissipation process arising from each term in Eq. (38) will be proportional to f I- The only important properties of the F t) are their autocorrelation and cross-correlation functions, (FJfi)F t)) and F (0)Fi,(t)), which enter the definition of the Redfield tensor in Eq. (18). These, like the classical correlation functions discussed earlier, do not satisfy the detailed-balance relation and must be corrected in the same way. It is convenient, but not necessary, that the variables be chosen to be independent, so that the cross-correlation functions vanish. 

These symmetries allow us to make some definite statements about various correlation functions. For example, consider the two properties A (q) and A q ) corresponding to wave numbers q and q. It is clear from the foregoing that the cross correlation function is... 

The solubility behaviour of an unknown compound will serve to classify it into one of the three main divisions, namely, acidic, basic or neutral. This information, supplemented by elemental analysis if deemed necessary, and as noted above cross-correlated with spectroscopic inferences, forms the basis for the subsequent systematic search to identify definitively the functional group or groups present. It cannot be too clearly emphasised that inexperience in spectroscopic interpretation can lead to erroneous conclusions of structure. The value of chemical tests is that they reduce the chance of this happening, furthermore they are frequently easily and quickly performed and provide experience in accurate and reliable observation and reporting. 

The above definitions are particularly suitable for investigations in computer simulations. They can be applied also to a fictitious layer in the bulk liquid, except that in the latter case decay can happen by molecules crossing across the region -that is, by penetrating the sphere not allowed in the case of protein. In the case of the protein hydration layer, these survival correlation functions decay slowly for the hydration layer. 

Bond orientational correlation functions and the associated correlation times for polyethylene have been evaluated by Bahar et al. [10]. The former are determined from the definitions given by Eqs. (8) and (9) where nii in Eq. (9) is replaced by the bond vector Ij. Cross-correlation times for orientations of bonds i and k are obtained as the area under the corresponding OACF decay curves according to Eq. (3). The analysis has also been extended to the domain of normal modes by adopting the transformation... 

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