Intensity correlation

Generalized Two-Dimensional (2D) correlation analysis is a powerful tool applicable to data obtained from a very broad range of measurements, such as chromatography or infrared spectroscopy. Relationships among systematic variations in infrared spectra are obtained as a function of spectroscopic frequencies. In this paper, the variation is induced by the introduction of small doses of CO in the catalytic cell, inducing a pressure change and a modification of adsorbed CO concentration. The correlation intensities are displayed in the form of 2D maps, usually referred to as 2D correlation spectra. 2D correlation analysis can help us to solve the complexity of the spectra... 

D correlation analysis is a powerful tool applicable to the examination of data obtained from infrared spectroscopy. The correlation intensities, displayed in the form of 2D maps, allow us to correlate the shift induced by CO adsorption and acidity of sites in dealuminated zeolites. Results are in accordance with previous results, obtained using only IR measurements, proving the validity of this technique. New correlations allowed the assignment of very complex groups of bands, and 2D correlation revealed itself as a great help for understanding acidity in dealuminated zeolites. 2D correlation has allowed us to validate the model obtained by NMR. 

At the simplest level, when there are relatively few conformations to be considered, each of them can be generated and the correlation intensities can be used as a guide to choosing between these possibilities. The relevant distances are readily available from MOMEC and/or HyperChem. 

Intensities of through-space correlations between atoms in multidimensional NMR are related to the separation (d) between those atoms by a d term. The intensities are usually standardized with respect to a correlation with a known atom-atom separation such as a geminal H H contact and then the correlation intensities can be converted to distances. These distances can then be included as constraints in the energy minimization process, leading to a geometry with the required atom-atom separations. However, it is important to realize that factors other than atom-atom separation can influence the intensity of the correlation, and therefore, it is preferable to include the distance information based on these correlations as soft restraints rather than constraints. 

The computation of asynchronous 2D correlation intensity is somewhat more complicated. Two approaches can be used (i) using the Hilbert transform and (ii) a direct procedure, obtaining similar results (for a detailed discussion on the asynchronous calculation, see the Further Reading section)... 

Figure 10.2 Bidimensional correlation maps corresponding to changes in intensity of a band composed of one (bottom) or two peaks (top). Synchronous maps are located at the left and asynchromous to the right. The x- and y-axes correspond to the numbers of the point on the artificial curve and the z-axis is the correlational intensity. Roughly they represent a protein amide I band in a DjO buffer. Negative peaks are shaded...

The ansatz has a global extensive part, exp(r" ) n 2) based on a closed shell n 2 reference vacuum and a local correlation intensive part, whose wave function is lij/) = exp(T" ) n 2), where is the Cl-like... 

The real and imaginary components, 3>(t i, V2) and F(vi, V2), of the cross-correlation function X(t) are referred to, respectively, as the synchronous and asynchronous correlation intensity. These quantities are related to the in-phase and quadrature spectra of dynamic dichroism by... 

The two-dimensional nature of our correlation analysis arises from the fact that the correlation intensities are obtained by comparing the time dependence of... 

Figure 1-10. A schematic contour diagram of a synchronous 2D IR correlation spectrum 31. Shaded areas represent negative correlation intensity.

The correlation intensity at the diagonal position of a synchronous 2D spectrum (Figure 1-10) corresponds to the autocorrelation function of perturbation-induced... 

Figure 1-14. A synchronous 2D IR spectrum of a thin film of atactic polystyrene in the CH-stretching vibration region at room temperature. Regular one-dimensional spectra of the same system are provided at the top and left of the 2D spectrum for reference. Shaded areas represent negative correlation intensity.

Now, one can define the complex 2D correlation intensity between 0 and y v2, t). 2D correlation is nothing but a quantitative comparison of spectral intensity variations observed at two different spectral variables over some finite observation internal between Tmin and T t T = Tmax - Tmm). Equation 3.11 yields the complex 2D correlation intensity. 

They are respectively referred to as the synchronous and asynchronous 2D infrared spectra. The synchronous spectrum characterizes the degree of coherence between the dynamic fluctuations of signals measured at two wavenumbers, and the correlation intensity becomes significant only if the reorientation rates of dipole transition moments are similar to each other. The asynchronous spectrum, however, characterizes the independent, uncoordinated out-of-phase fluctuations of the signals. Hence the asynchronous correlation intensity becomes non-vanishing only if the signals vary at difierent rates. 

The synchronous and asynchronous (i.e. quadrature) correlation intensities, V2) and (vi,V2), of the dynamic spectrum are given by... 

The synchronous correlation intensity, (vi, V2), characterizes the degree of coherence between two signals that are measured simultaneously and is maximized when the variations of the two dynamic infrared dichroism signals are totally in phase with each other and minimized when the two signals are out of phase. Conversely, the asynchronous correlation intensity (vi, V2) characterizes the degree of coherence between two signals that are measured at two different instants that are separated in time by a correlation time x/2co. Thus, the maximum value of... 

Figure 21.1 Schematic illustrations of (a) synchronous and (b) asynchronous 2D correlation spectra. White and gray areas in the contour maps represent positive and negative correlation intensities, respectively.

The explicit analytical expressions given by Equations (FI 6) and (F24), obtained for the synchronous and asynchronous spectrum, are well suited for the efficient machine computation of correlation intensities from discretely sampled and digitized spectral data. If a discretely sampled dynamic spectrum y(v -, t,) with the total of n points of wavenumber Vj is obtained for m times at each point of time tj, with a constant time increment, that is, t,+i - t, i = At, the integrations in Equations (FI6) and (F24) ean be replaced with summations. 

To extraa more information from the spearal data, 2D-COS can be employed. Basically, this analysis method ae-ates a pair of synchronous (vj,v2) and asynchronous F(vi,V2) 2D correlation spectra, where the spectral variables vi and V2 are wavenumbers. The synchronous 2D correlation intensity (vi,V2) represents the overall similarity or coincidental changes between two separate intensity variations measured at different spectral variables during variation of the external perturbation. The as3mchronous 2D correlation intensity 1 (vi,V2) may be regarded as a measure of dissimilarity or more strictly speaking, out-of-phase charaaer of the spectral intensity variations. 

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