Distance space

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

The value of embodies the conformation-independent 3D arrangement of the atoms of the ligands of a chirality center in distance space and thus cannot distinguish between enantiomers. This distinction is introduced by the descriptor S , , . 

Topological descriptors and 3D descriptors calculated in distance space", such as 3D autocorrelation, surface autocorrelation, and radial distribution function... 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

The procedure of DG calculations can be subdivided in three separated steps [119-121]. At first, holonomic matrices (see below for explanahon) with pairwise distance upper and lower limits are generated from the topology of the molecule of interest. These limits can be further restrained by NOE-derived distance information which are obtained from NMR experiments. In a second step, random distances within the upper and lower limit are selected and are stored in a metric matrix. This operation is called metrization. Finally, all distances are converted into a complex geometry by mathematical operations. Hereby, the matrix-based distance space is projected into a Gartesian coordinate space (embedding). 

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

In this context, time, speed and distances (space) are important parameters to be kept in mind, leading to the so-called "spatiotemporal" or "proximity theory". This theory states that "the reaction rate (speed) is a sensitive inverse function of distance and time". 

Typical spotted arrays would have 100 to 150-micron (pm) diameter features while photolithographically prepared in situ arrays may have features on the order of 2 to 20 pm. The separation between elements is usually measured in terms of a center-to-center distance, spacing, or pitch. Thus, for a printed array, two adjacent spots in the array, e.g., each at 100-pm spot diameter, might have a center-to-center distance of 150 pm, or the spots would be separated by 50 pm from their edges. The number of spots per square centimeter usually defines the spot density. As an example, an array manufactured by Affymetrix (Santa Clara, CA) at >280,000 elements per... 

With a Fourier transformation of (k) in the distance space, one obtains a separation of the contribution of the various coordination shells. This Fourier transform yields the structural parameters Rj, Nj and ah and thus the near range order of the specimen with respect to the absorbing atoms. The EXAFS analysis for the different absorber atoms within the material yields their specific near range order. Thus, one may get the structure seen form several kinds of absorbing atoms. EXAFS does not require highly crystalline materials. It is a suitable method to study disordered, or even amorphous, structures. The a values provide quantitative information about the thermal and structural disorder. 

Since PathFinder works in path/distance space, the frame of reference for every molecule is internal and, therefore, no pairwise alignment is necessary when molecules are compared. PathFinder at the same time incorporates information on both overall shape (long distances) and local topology (shorter distances). 

Figure 4. (a). Raw XAS data for the Fe k-edge (7112 eV) of an Fe foil incorperating the XANES and EXAFS regions with the cubic spline funtion (dotted line) representing the background to be subtracted, (b). Normalized EXAFS data that has been transformed to k-space using Eq. 9. And (c). Fourier Transformed EXAFS spectrum in distance space. 

The XRD patterns of zirconium sulfate pillared clays obtained after 90 hours of intercalation with different zirconium acetate concentrations using 0.5 as sulfate to Zr ratio and the same clay concentration as used earlier are presented in Fig. 5. The diffraction data show the appearance of two first order reflections. The first one is at 23.4 A for the lowest zirconium concentration and appears as a shoulder at the same distance for 0.05 mol/L concentration. The second reflection is observed at approximately 12.3 A for the lowest concentration and at 13.7 A for 0.1 mol/L zirconium acetate. The first one results from the intercalation of sulfated zirconium species. Those species are more voluminous than the non sulfated one which gives a distance spacing at only 19.6 A. The better intercalation of sulfated zirconium species at low Zr concentration is probably due to the slow progress of polycondensation reactions. This process reduces the number of different zirconium species and gives a better cristallinity of the solid. Table 2 summarizes the textural properties of samples prepared with different zirconium concentrations. The decrease of the surface area with the decrease of the Zr concentration is probably due to the increase of the sodium clay layers by comparison with the intercalated layers. The microporous volume increases when the Zr concentration decreases. The higher microporosity is due to the important basal distance of this sample. 

The 2D RDF shows a distance axis and a property axis showing the partial atomic charge distribution. Since the probability-weight fnnction p is omitted, the fnnction simply splits into distance space and property space. The distance axis is equivalent to a one-dimensional RDF, whereas the property axis shows the charge distribution at a certain distance. The two intense peaks represent the C-H and C-H ... 

Smoothing Parameter (B) is an exponential factor that defines the width of the Gaussian distribution around a peak in the distance space of an RDF descriptor. It can be interpreted as a temperature factor that describes the movement of atoms within a molecule. 

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