Radial Distribution Functions RDFs

D MoRSE desaiptor, radial distribution function (RDF code), WHIM descriptors, GETAWAY descriptors,... 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

The compounds were described by a set of 32 radial distribution function (RDF) code values [27] representing the 3D structure of a molecule and eight additional descriptors. The 3D coordinates were obtained using the 3D structure generator GORINA [33]. 

The radial distribution Function (RDF) of an ensemble of N atoms can be interpreted as the probability distribution to find an atom in a spherical volume of... 

A combination of physicochemical, topological, and geometric information is used to encode the environment of a proton, The geometric information is based on (local) proton radial distribution function (RDF) descriptors and characterizes the 3D environment of the proton. Counterpropagation neural networks established the relationship between protons and their h NMR chemical shifts (for details of neural networks, see Section 9,5). Four different types of protons were... 

The strncturcs in the database arc encoded using the radial distribution function (RDF) as a descriptor (cf Section 8,4,4). 

Because the correlation of atomic positions decreases as r — co, = 1. The function 47T p (, the radial distribution function (RDF), may also be... 

The case A = 2 is of greatest interest. Since the force is central, it is not necessary to use rj and ri as variables. The single variable r 2 is sufficient since the position of the center of mass is irrelevant. Thus, we have the radial distribution function (RDF), g r 12). 

The desired average is simply obtained by a time average of the given property. For example, one of the interesting properties of bulk solvents is the radial distribution function (rdf), which expresses the probability of finding a given atom type around a reference atom by... 

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

Figure 9.24 A radial distribution function (RDF) for a DRP of monospheres (right) and a scheme for its evaluation (left), Nmeenl4nR2 is an equivalent of IA(R), where A/mean is the mean number of spheres in the intervals of 0.2R. The solid curve illustrates the data obtained from neutron diffraction in liquid argon 1,2 are the experimental data by Scott [128] and Bernal [127] obtained for the models of steel spheres (cited in [127]).

The distribution of the number of full contacts in RP was studied experimentally in Ref. [106], It was shown that the radial distribution function (RDF see Figure 9.24) corresponds to a normal distribution. Table 9.6 illustrates the results of that study analyzed on the basis of Equation 9.63 and Equation 9.64 (see Section 9.7.2). 

The radial-distribution function (RDF) specifies the density of atoms or electrons as a function of the radial distance from any reference atom or electron in the system. It can be applied to both crystalline and amorphous materials, and is especially effective for amorphous material. 

Figure 5. Radial distribution function(RDF) of whole Xe molecules and clusters for w = 0.90 and 1.00 nm pores at 75.5 kPa. Solid and dotted lines denote RDFs of the whole Xe molecules and clusters.

In order to extend the analytical equations to a fractal lattice, we will need the radial distribution function rdf(r) of the Sierpinski gasket, rdf(r) dr being the average number of sites with distance between r and r + dr from a given site. For fractal lattices one has... 

Fig. 6.6. Radial distribution functions rdf(r) for the two types of the Sierpinski gaskets a and b (dots) and ideal rdfs (solid curves) with d = 1.58 and 7 = 3.65 (a) or 7 = 5.2 (b) (see equation (6.1.30)). Note, that both axes are logarithmic.

Schlesinger and Marton (15) studied the nucleation and growth of electrolessly deposited thin nickel (Ni-P) films. These studies were later extended and complemented by the studies performed by Cortijo and Schlesinger (19, 20) on radial distribution functions (RDFs). RDF curves were derived from electron diffraction data obtained from similar types of films as well as electrolessly deposited copper ones. Those studies, taken together, have elucidated the process of crystallization in the electroless deposition of thin metal films. 

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