Minimum-distance distribution function

However, the RDF(CM) is not appropriate when the solute is an elongated molecule, as it was discussed before, for the case of (3-carotene in several solvents [47] and benzophenone in water [50], In these cases of elongated solutes, an appropriate function is the minimum-distance distribution function (MDDF), where the histogram used to calculate the distribution function is not the distance between the CMs of solute-solvent, but the minimum distance between them. The MDDF is defined as... 

Figure 7-5. The RDF between the center of mass (dashed line) and the minimum-distance distribution function (solid line) between the benzophenone (shown in the inset) and the water molecules...

Fig. 6 Minimum distance distribution function (MDDF) for the syn and anti mesityl oxide isomers with polarized (a) and non-polarized (b) atomic charges...

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

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). 

Radial distribution functions are characterized by the distance of closest approach d, the position and height h of the main peak, the position r " of the minimum following the main peaks, and the coordination numbers CN, defined by... 

The use of the periodic boundary conditions in the two directions perpendicular to the interface normal (X and Y) implies that the system has infinite extent in these directions. To make the computational cost reasonable, one must truncate the number of interactions that each molecule experiences. The simplest possible technique is to include, for each molecule i, the interaction with all the other molecules that are within a sphere of radius which is smaller than half the shortest box axis. One selects, from among the infinite possible images of each molecule, the one that is the closest to the molecule i under consideration. This is called the minimum image convention, and more details about its implementation can be found elsewhere [2]. To arrive at the correct bulk properties, any ensemble average calculated by this technique must be corrected for the contribution of the interactions beyond the cutoff distance. The fixed analytical corrections are calculated by assuming some simple form of the statistical mechanics distribution function for distances greater then R. ... 

Fig. 5.4 Generic radial distribution functions gMO for the metal ion in water for cases of (a) strong coordination and (b) weak coordination. Tmo is the optimum metal ion-oxygen distance in the solvated species. The coordination number is estimated by integrating the distribution function out to the minimum at rg. (From reference 8, with permission.)...

The radial distribution function, g(r), can be determined experimentally from X-ray diffraction patterns. Liquids scatter X-rays so that the scattered X-ray intensity is a function of angle, which shows broad maximum peaks, in contrast to the sharp maximum peaks obtained from solids. Then, g(r) can be extracted from these diffuse diffraction patterns. In Equation (273) there is an enhanced probability due to g(r) > 1 for the first shell around the specified molecule at r = o, and a minimum probability, g(r) < 1 between the first and the second shells at r = 1.5cr. Other maximum probabilities are seen at r = 2(7, r = 3 o, and so on. Since there is a lack of long-range order in liquids, g(r) approaches 1, as r approaches infinity. For a liquid that obeys the Lennard-Jones attraction-repulsion equation (Equation (97) in Section 2.7.3), a maximum value of g(r) = 3 is found for a distance of r = <7. If r < cr, then g(r) rapidly goes to zero, as a result of intermolecular Pauli repulsion. 

There is one parameter in Bjerrum s treatment which so far remains arbitrary. The upper limit of the integral in eqn. 5.3.6, d, was fixed by Bjerrum aXd = I = (ze) /2eyfcr because at this distance the distribution function eqn. 5.3.5 presents a minimum. In this case... 

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