Separation distance vector, between

The autocorrelation A(Af) is equal to zero for a random distribution of the two elements. It is positive if species separated by Af tend to be of equal type and negative in case of a preference for different atoms separated by Af. If Af is the vector between nearest neighbours (NN), positive A(Af) means demixing tendency, negative A(Af) a tendency towards ordering. The maximum positive value that A(Af) can approach is +1 (in case of complete phase separation where one never finds an unlike atom in a distance Af), whereas the minimum value is given by... 

Let us start with a few definitions. A lattice plane of a given 3D BL contains at least three noncollinear lattice points and this plane forms a 2D BL. A family of lattice planes of a 3D BL is a set of parallel equally-spaced lattice planes separated by the minimum distance d between planes and this set contains all the points of the BL. The resolution of a given 3D BL into a family of lattice planes is not unique, but for any family of lattice planes of a direct BL, there are vectors of the reciprocal lattice that are perpendicular to the direct lattice planes. Inversely, for any reciprocal lattice vector G, there is a family of planes of the direct lattice normal to G and separated by a distance d, where 2jt/d is the length of the shortest reciprocal lattice vector parallel to G. A proof of these two assertions can be found in Ashcroft and Mermin [1]. 

Second, we should consider the effect of the separation distance between molecules on the interaction forces. It was found that these fall off exponentially as the distance between atoms increases. The explanation for this behavior can be found in classical Newton mechanics according to Newton s laws, a force, F, is a push or pull exerted on a body it is a vector quantity with magnitude (newton, N = kgms 2, in the SI system) and direction. The work, W, done by the force acting on a body is given as... 

The energy expressions in MM2 and MM3 contain terms for both electrostatic and hydrogen bond energies. It is notable that the MMx series of force fields employs bond dipoles in contrast to the use of partial atomic charges, to compute the electrostatic interaction energies. The classical electrostatic interaction energy between two dipole vectors and i.j and H2 separated by a distance vector Rj2 may be expressed as follows... 

Electrostatic repulsions in nearly salt-free solutions create sufficient order for the appearance of a well-defined peak in the low angle scattered intensity (107-114). In a plot of scattered intensity versus scattering vector q, Figure 8 displays how these peaks vary with c (114). Analogous peaks do not occur for flexible neutral polymers. When polyelectrolyte chains are dilute (but for c not too far below c ), the angular position of the peaks is associated with the average distance d between physically separated chains d is measured and predicted to be proportional to c (73,107,115). The spacing arises from the electrostatically... 

P(x) is the radial distribution function (RDF) as introduced in Section I.I.2.I.4. It is the product of electronic densities attached to two points of the real space separated by vector x, with x being the distance between a pair of atoms in a set of atoms arranged disorderly. P(x) is thus a probability it is a generalization of the Patterson function of a crystal [10]. It corresponds to the Fourier transform of I(s). Reciprocally to Equation (1.8),... 

The original 2D autocorrelation analysis calculates a vector based on the distances between all atoms of a structure and any property of these atoms.205>206 For each pair of atoms, the distance between the atoms (number of bonds between them) and the product of the properties is noted. Each element of the autocorrelation vertor is the sum of these products for one particular distance. A separate autocorrelation vector is calculated for each property of interest—typically, volume, electronegativity, hydrogen bonding character, hydrophobicity. As a final step a principal components analysis reduces the number of variables to consider. 

A very different approach to the use of property information is described by Sadowski et al. Some molecular property, such as the molecular electrostatic potential, is calculated at each of the points on a molecular surface, and the distance calculated between each pair of these points. The range of possible inter-point distances is binned so that each distance is associated with a specific distance range. If two surface points are separated by a distance that lies within the r-th distance range, then those points make a contribution to the r-th bin that is equal to the product of the property values at those two points. Thus, if the possible distances are divided into n bins, each molecule is represented by an n-element vector and then the similarity between a pair of molecules is calculated by coniparing their corresponding vectors, a procedure that is clearly related to the topological sequences approach described in Section 2.2. 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

The permanent dipole moment p (Eq. 1, vector entities are in bold type) between two equal but opposite charges, separated by a distance r, is defined as the product of the charge q and the distance r ... 

The magnetostatic potential about an isolated pair of spheres with dipole moments of M, and M2 will now be considered. M, and M2 are vectors. The magnetostatic potential between Mx and M2 separated by distance R is given by Eq. 36. 

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