Radial distribution function orientation

For molecules the radial distribution function can be extended with orientational degrees of freedom to characterize the angular distribution. 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

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. 

The structure of the adsorbed ion coordination shell is determined by the competition between the water-ion and the metal-ion interactions, and by the constraints imposed on the water by the metal surface. This structure can be characterized by water-ion radial distribution functions and water-ion orientational probability distribution functions. Much is known about this structure from X-ray and neutron scattering measurements performed in bulk solutions, and these are generally in agreement with computer simulations. The goal of molecular dynamics simulations of ions at the metal/water interface has been to examine to what degree the structure of the ion solvation shell is modified at the interface. 

A model-free approach to analysis of DEER data in the absence of orientation selection was proposed based on shell factorization.22 The decay curves are simulated as the products of orientationally averaged thin shells of interacting electrons. The dipolar time-evolution data can be separated into a linear contribution and a non-linear contribution from background. The linear contribution can be converted to a radial distribution function for spin-spin interaction. 

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

Fig. 6. Radial distribution functions C-C, C-CI. and Cl-Cl in CC14 at 280, 380, and 430 K and 1.0 GPa pressure showing the changes from an ordered to an orientationally ordered phase. Inset shows a plot of cos 9 against temperature. Here 9 is the smallest angle between the (a + 7) direction and the four Cl-Cl bonds. The dashed line is drawn to guide the eye. (From Yashonath and Rao (19).)...

Conformer sequence probabilities Radial distribution functions Scattering functions Orientation correlation functions Mechanical properties Distribution of free volume... 

Data available from a computer simulation is not similarly limited, as a complete description of the system, the positions and orientations of all its molecules, is immediately available therefore the full molecular pair distribution function should, in principle, be obtainable. Still the practical considerations of accumulating and presenting this 6-dimensional function have made it virtually inaccessible, despite its obvious importance. Computer simulation studies of molecular liquids and solutions have then traditionally relied almost exclusively upon radial distribution functions to provide structural information. 

The Patterson function is a map that indicates all the possible relationships (vectors) between atoms in a crystal structure. It was introduced by A. Lindo Patterson " in 1934, inspired by earlier work on radial distribution functions in liquids and powders. In crystals the directionality as well as the lengths of vectors between atoms (atomic distances) can be deduced. By contrast, in liquids and powders the geometric information that can be obtained is limited to interatomic distances, because in these the molecules are randomly oriented. While the use of the Patterson function revolutionized the determination of crystal structures of small molecules in the 1930s to 1950s, direct methods are now the most widely used methods for obtaining structures of small organic molecules. The Patterson function, however, continues to play an essential part in the determination of crystal structures of inorganic compounds and macromolecules. It is also very useful when the structure of a small molecule proves difficult to solve by direct methods. 

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