Pair distance histograms

The values were obtained from pair-distance histograms similar to the one shown in Figure 19.12. The numbers provide an impression of the scatter aroimd the average values given in the bottom line. 

Calculating the pair distribution function in a simulation is straightforward all we need to do is count the number of atom pairs separated by a distance in the range from r to r -I- Sr, and then normalize it. Usually, g r) is normalized by the number of pairs Nig(r,r + Sr) that would be observed in an ideal gas of the same density, so that, in the limit of large distances, r oo, where correlations disappear, g r) 1. A typical FORTRAN code to calculate g(r) from 0 to rmax with the resolution del tar is given below. The separation distance histogram is calculated for nbins = rmax/ deltar bins, for a model of N particles in a box with sides boxx. 

A radial distribution function can be determined by setting up a histogram for various distances and then looking at all pairs of molecules to construct the diagram. Diffusion coefficients can be obtained by measuring the net distances... 

Figure 19b displays the 2D histogram of the experimentally obtained conductance of N4 plotted vs distance [63]. The distance scale z is normalized with respect to z = 0 at G = 0.7 G0, to a common point. The chosen procedure is justified, because of the steep decay of the tunneling current after breaking of the last atomic contact. The histogram counts the occurrence of [log(G/Go), z ] pairs in a 2D field. Figure 19b exhibits the features of gold quantum contacts at G > Go, and a second cloud-like pattern in [10 5 10 4 G0, 0 0.5 nm]. We attribute the latter to the formation of single-molecule junctions of only one type. The center of the cloud is located at G = 3.5 4.5 x 10 5 Go, close to the peak position in the ID histogram (Fig. 19a). The extension of the cloud along the distance scale is around 0.5 nm, close to the typical length of the plateaus (the inset of Fig. 19a).

Two-particle correlation functions. A statistical approach to quantify the mixing of a heterogeneity that is represented by many particles is to calculate the distances between each pair of particles and compute the cumulative histogram H(r) which is the number of particle pairs that have a distance of less than r. The slope of log(H) versus log(r) within a particular range of r indicates the spatial dimension of the particle distribution. For example, if in a two-dimensional box calculation... 

The influence of spatial localization of two normal modes computed for cytochrome c on the frequency difference of these vibrational modes is shown in Fig. 16. We locate the largest component of each normal mode, a, and calculate the probability, P(Aco), that for another mode, (3, whose largest component lies a certain distance away from the largest component of mode a, the difference in frequency between them is Aco = coa — cop. We consider only localized modes, a, whose frequency, coa, falls between 1000 cm 1 and 2000 cm 1. Probabilities are calculated for Aco in intervals of 20 cm-1 to Aco = 600 cm-1. The solid-line histogram is an average over all pairs of modes of cytochrome c, regardless of the distance between the largest components of modes a and p. We... 

After having generated the graphs of peaks for all protein chains individually, we restored them together in their crystalline conformations. We then performed a statistical study by collecting all inter-distances between the amino acids (aa), labeled as backbone (BB) or side-chain (SD) peak. As each aa can be paired with 20 others, this data collection yielded the constmction of 20 tables of inter-distance distributions per aa (ALA-ALA, ALA-ARG etc.), with the additional distinction between SD-SD, SD-BB, and BB-BB peaks. For each aa -aay pair, the values were normalized to have a maximum value of 1.0 in each table. As an illustration, the histograms obtained for CYS-CYS and ARG-GLU appear in Figure 14-2. For the... 

Figure 14-2. Histogram of normalized inter-distance distributions between side-chain peaks for CYS-CYS and ARG-GLU amino-acid pairs...

The IR library containing both closely related, and quite different spectra of 3339 organic compounds have been used. The range of 4000-416 cm was converted into 896 bands of 4 cm", and augmented to the nearest power of 2, that is to 1024 wavelengths. The histogram of the RMS pair-wise distances for the whole set of 3339 spectra is shown (Fig. 3). To illustrate in another way spectra variation, the variance spectrum of the studied library is presented in Fig. 4. 

Fig. 3 The histogram of the RMS pair-wise distances for the whole set of 3339 spectra.

For each molecule, the distances between all generalized atom-type pairs are counted and the histograms of the counts represent a simplified but exhaustive pharmacophore fingerprint [26]. 

A natural goal of simulation would be the computation of the relative probabilities of these various states. A more elementary task is to compute the radial distribution which gives the distribution of distance between atom pairs observed. The radial density function may be approximated from a histogram of all pan-distances observed in a long simulation. (There are 21 at each step, so the amount of data is helpfully increased, reducing the sampling error .) This distribution is displayed in Fig. 3.5. The peaks of the radial distribution function are correlated with the various interatomic distances that appear in the cluster configurations shown in Fig. 3.4. 

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