Bond-vector distribution function

Chains of the usual length between junctions in a rubber network consist of several hundred skeletal bonds. The distribution function W(r) for the vector r connecting the ends of a chain of this length is satisfactorily approximated by the Gaussian function i.e.,... 

We also examined the fold statistics in this Ciooo system. The distribution of the inter-stem vectors connecting stems linked by the loops, and their radial distribution function again indicated that about 60-70% of the folds are short loops connecting the nearest or the second and third nearest stems, though the crystallization did not complete. The presence of local order in the under cooled melt in the present Ciooo system is also examined through the same local order P(r) parameter, the degree of bond orientation as a function of position r, but again we did not detect any appreciable order in the undercooled melt. 

This function is the integrally normalized probability for each water molecule being oriented such that it makes an angle B between its OH bond vectors and the vector from the water oxygen to the carbon atom. This function is calculated for those molecules within 4.9 A of the carbon atom (nearest neighbors), as this distance marks the first minimum in the pair distribution function for that atom. The curve in Figure 10 is typical for hydrophobic hydration (22). 

Figure 10. Distribution of orientations for water molecules adjacent to the exocyclic methylene carbon atom C6 as calculated from a molecular dynamics simulation of a-D-glucopyranose in aqueous solution. The function plots the frequency of occurrence of an angle between the water OH bond vectors and the vector from the carbon atom to the water oxygen atom. A value of cos( ) of 1.0 corresponds to an OH bond vector pointing directly away from the carbon atom. (Reproduced from Ref. 32. Copyright 1989 American Chemical Society.)...

A method is developed for calculating even moments of the end-to-end distance r of polymeric chains, on the basis of the RIS approximation for rotations about skeletal bonds. Expressions are obtained in a form which is applicable in principle to arbitrary k, but practical applications are limited by a tremendous increase in the order of the matrices to be treated, with increasing k. An application is made to the PE chain by using the familiar three-state model. Approximate values of the distribution function Wn (r) of the end-to-end vector r, Wn (0), and , are calculated from these even moments. 

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

The flexible helix modeled here is best described by the entire array of conformations it can assume. A comprehensive picture of this array is provided by the three-dimensional spatial probability density function Wn(r) of all possible end-to-end vectors (25, 35). This function is equal to the probability per unit volume in space that the flexible chain terminates at vector position relative to the chain origin 0,as reference. An approximate picture of this distribution function is provided by the three flexible single-stranded B-DNA chains of 128 residues in Figure 5(a). The conformations of these molecules are chosen at random by Monte Carlo methods (35, 36) from the conformations accessible to the duplex model. The three molecules are drawn in a common coordinate system defined by the initial virtual bond of each strand. For clarity, the sugar and base moieties are omitted and the segments are represented by the virtual bonds connecting successive phosphorus atoms. 

Here the orientational dependence is expressed in terms of scalar products between the unit intersite vector R and unit bond vectors Zj and Z2, defined to point from the center of molecules 1 and 2, respectively, toward the atoms whose interactions are being calculated (see Table 1). The combination of anisotropy of these two sets of symmetry-related S functions shifts the repulsive wall outward at the end of the molecule and close to the intramolecular bond, allowing closer approach in the plane through the atom perpendicular to the intramolecular bond. Thus this picture of the effective shape of the N2 molecule is compatible with its charge distribution. 

The theory of AIM allows one to study the concept of chemical bond and the bond strength in terms of electron density distribution function [6, 193]. It exploits the topological features of electron density and thereby a definition of chemical bonding through bond path and bond critical point (BCP). A BCP (it is a point at which gradient vector vanishes, Vp(r) = 0) is found between the... 

Several structural properties are typically monitored to characterize water in these systems. Some of these are the density distribution of water molecules with respect to the surface, surface area per water molecule, the root-mean-square displacements from the optimal surface positions for corrugated surfaces, angular distributions of the dipolar and O—H bond vectors with respect to the surface normal, and moments of the angular distributions. Most of these characterize the water structure with reference to the metal surface. In contrast, the radial distribution function, number of nearest neighbors per molecule, and number of hydrogen bonds per molecule are used to characterize water-water interactions. 

Fig. 19.8. A proposed secondary strucmre of Fe403(0H)4(N03)2 I.5H2O shown as a projection on the plane of the paper. The vectors a and h are the probable Fe-Fe interactions corresponding to the two peaks at 5.6 and 6.4 A in the radial distribution function. The NOj ions are thought to be hydrogen bonded to the HjO molecules on the surface. (Brady et ai, 1968.)...

The probability distribution of bond vectors follows a Gaussian function. 

The OPs developed by Santiso and Trout [65] are extracted from a generalized pair distribution function. All OPs used in this stody were based on [dmim ] no particular OPs were defined for [Cl ]. In Fig. 1 two ion pairs are shown, where the absolute orientation of each cation is given by the vectors qi and q2, which are normal to the imidazolium ring of each cation. The distance OP provides quantification for the various center of mass (COM) distances between the cations. The bond orientation OP measures the orientation of bonds joining the center-of-mass of the cations, while the relative orientation OP measures the orientation of one cation with respect to another one [65,67]. All the OPs are defined per cation and per peak... 

The distribution function just derived is an expression for the probability that an end-to-end vector terminates in the volume element centered about the coordinates x,y,z (at the end of the vector ). Figure 1.31 is drawn for a random flight of 10,000 steps, each step being 0.25 nm, almost double the length of a chain segment of polyethylene. Because of bond restrictions that will be discussed later in this section. 

Since the bond vectors r are independent of each other, the distribution function for the polymer conformation is written as... 

The freely jointed chain model has an exact analytical solution for the distribution function of the end-to-end vector. The probability that the chain of n bonds has the end-to-end vector r is... 

A polymer chain changes its conformation by thermal motion. The probability of finding a particular conformation of the chain in the heat reservoir of the absolute temperature T is given by the canonical distribution function. If one end xo of a chain is fixed at the origin of the coordinates (Figure 1.4), and the other end x is fixed at the position vector R, the end-to-end vector R is given by the sum of all bond vectors... 

Because the energy of orientation measured from the reference direction parallel to the end vector is /I R// = /a cos 0, the orientational distribution function of the bond vector is proportional to exp[/acos0, / B7"]. Because the tension is related to the end-to-end distance by (1.28b), the orientational distribution under a fixed R is given by the probability... 

As is usual in statistical mechanics, (2.8) contains more information than is really needed. Experimental quantities are obtainable from reduced distributions, integrals of (2.8) over a subset of the bond vectors or of (2.2) over a subset of the segment positions. Consider the end-to-end distribution function... 

Whilst some of these cases have been treated previously, Eichinger s method allows the calculation not only of statistical averages, but also of the distribution function itself. For polymer molecules with real bond angles and restricted bond rotation, the rotational isomeric state (RIS) model has proved powerful. Flory has summarized some of the most important results of the treatment, and Mark has considered the applications, particularly to bulk polymers and networks, pioneered by his group. A recent paper uses the RIS model to calculate the distribution function of the end to-end vector distribution for short polymer chains. 

MD methods allow us to follow the coordinates / , and velocities Vj of all atoms throughout the simulations, and insight into the local order can be found from the distributions of the bond (0 ) and dihedral angles iviju)- The pair distribution function (PDF) g r) is a spherically averaged distribution of interatomic vectors. 

---