Probability distribution, distance

By including characteristic atomic properties, A. of atoms i andj, the RDF code can be used in different tasks to fit the requirements of the information to be represented. The exponential term contains the distance r j between the atoms i andj and the smoothing parameter fl, which defines the probability distribution of the individual distances. The function g(r) was calculated at a number of discrete points with defined intervals. 

Fw is the force on particle i due to particle j, and is assumed to depend only upon the distance between the particles the prime on the summation means that the term j — i is omitted. The i -particle probability distribution function is defined as follows ... 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

Many investigators have used the Ogston model and its fundamental idea as a basis for their models. Most recently, Johansson and Elvingson [182] obtained the probability distribution g(r) for spaces in a random suspension of fibers i.e., the probability that a randomly chosen point in a network of fibers is found at a radial distance r to the fiber of closest approach. For a cylindrical cell (CC) model, which consists of an infinite cylindrical cell, containing solvent and polymer, with the polymer represented as a rod centered in the cell, they obtained g(r) for one cylindrical cell as... 

The angular-dependent adiabatic potential energy curves of these complexes obtained by averaging over the intermolecular distance coordinate at each orientation and the corresponding probability distributions for the bound intermolecular vibrational levels calculated by McCoy and co-workers provide valuable insights into the geometries of the complexes associated with the observed transitions. The He - - IC1(X, v" = 0) and He + 1C1(B, v = 3) adiabatic potentials are shown in Fig. 3 [39]. The abscissa represents the angle, 9,... 

The least-square solution itself does not depend on the probability distribution of y it is simply a minimum-distance estimate. Later in this Chapter, it will be shown, however, that its sampling properties are most easily described when the measurements are normally distributed. 

The deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

With this equation in place we now may evaluate the statistical average for any distance k= h-j along the chain, where h and j denote the position of chain segments. Fourier transforming r n ) (Eq. 3.3) we obtain a probability distribution in Fourier space ... 

Fig. 4. Backfolding in dendrimers as predicted by analytical theory [12]. Free end probability distribution function of the radial distance for generations 2-7. All data has been calculated assuming a realistic excluded volume parameter of the segments of the dendrimer (see [12] for further details). Reproduced with permission from [12]...

At the beginning the interatomic distances were determined by measuring the positions of the maxima and minima on the interference pattern. Soon, however, a more direct method was proposed by Pauling and Brockway for determining the interatomic distances. They obtained a so-called radial distribution by Fourier-transforming the estimated intensity data. The radial distribution is related to the probability distribution of interatomic distances. The position of maximum on the radial distribution gives the interatomic distance, while its halfwidth provides information on the associated vibrational amplitude. 

Like the Coulombic forces, the van der Waals interactions decrease less rapidly with increasing distance than the repulsive forces. They include interactions that arise from the dipole moments induced by nearby charges and permanent dipoles, as well as interactions between instantaneous dipole moments, referred to as dispersion forces (Israelachvili 1992). Instantaneous dipole moments can be thought of as arising from the motions of the electrons. Even though the electron probability distribution of a spherical atom has its center of gravity at the nuclear position, at any very short instance the electron positions will generally not be centered on the nucleus. 

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