Radial distribution function definition

When the system contains more than one component it is important to be able to explore the distribution of the different components both locally and at long range. One way in which this can be achieved is to evaluate the distribution function for the different species. For example in a binary mixture of components A and B there are four radial distribution functions, g (r), g (r), g (r) and g (r) which are independent under certain conditions. More importantly they would, with the usual definition, be concentration dependent even in the absence of correlations between the particles. It is convenient to remove this concentration dependence by normalising the distribution function via the concentrations of the components [26]. Thus the radial distribution function of g (r) which gives the probability of finding a molecule of type B given one of type A at the origin is obtained from... 

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). 

A number of simulation [114,115,131-133] and experimental [116,134] studies have explored the structure of ILs. The radial distribution functions obtained from these studies are similar to those of high temperature fused salts, in that they show definite association between oppositely charged ions. Close examination... 

Using the pair-wise additivity of U(R), it is possible to integrate Eq. (18) over the equilibrium configurations of (N — 2) particles. If one then uses the definition of the radial distribution function, an expression for E in terms of g( r) and u(r) is obtained, and it is referred to as the energy equation... 

These diagrams indicate the limit of the hydration shell in the gas-phase ion as the first minimum in the radial distribution function. It is well pronounced for K, which has 8 molecules as the calculated coordination number on the cluster curiously, the sharpness of the definition for Na" is less atAl= 6 (and sometimes 7). The influence... 

Figure 1.10 Geometrical quantities in defining the proximal radial distribution function gp j(r) of Eq. (1.14). The surface proximal to the outermost carbon (carbon /), with area fi, (r) r, permits definition of the mean oxygen density in the surface volume element, conditional on the chain configuration p gprox ( )-...

Figure 7.7 The radial distribution function ko( ) for oxygen atoms about a K+ ion in liquid water. See Rempe et al. (2004). The dashed curve is the contribution to Ko(r) from the nearest four oxygen atoms, and the dashed-dot curve is the contribution from the 5 and 6 nearest oxygen atoms. Notice the lack of definition obtained from the Ko(r) solely because a minimum separating a from a 2 mean hydration shell is indistinct.

Having defined the different Interactions occurlng In [3.6.1], we now need to specify the probability of finding an Ion a at some position r. The one-particle (singlet) density p fr jls defined In sec. I.3.9d as the number of particles per volume at position r. Now we apply the definition to Ions. The radial distribution function g (r)and the ion-wall total correlation function h (r) follow from (1.3.9.22 and 23] as... 

This definition simplifes considerably in the case of an isotropic liquid for which the single particle density is a constant (i.e. p(r) = N/V), resulting in the radial distribution function g(r). As said above, these definitions will form a part of our later thinking. 

In this section, we illustrate the general features of the radial distribution function (RDF), g(R), for a system of simple spherical particles. From the definitions (2.31) and (2.39) (applied to spherical particles), we get... 

By introducing an approximation into Eq. IV. 18, we can gain insight into the formula for the heat of transport. It suffices to use the assumption of regular solution theory that the radial distribution function is independent of composition.4 Under this assumption we may differentiate Eq. IV. 19 (using the definition of partial molecular enthalpy) to obtain expressions for ha/ in terms of thermodynamic quantities. Substituting these into Eq. IV. 18 yields... 

The radial distribution function, g(r), is the ratio of the density distribution of a type of site / at a given distance from a given type of site i, to the average density of site / in the system. A more technical definition, which allows the binning of this function into discrete intervals of r, is ... 

The situation is far more complicated for non-spherical or more complex solvent molecules. In the first place, the very concept of a hard-core diameter is not a well-defined quantity. For water, for instance, one may conveniently choose the effective diameter of the water molecule as the location of the first peak of the radial distribution function g R) for pure water. If we adopt this definition, we find that there exists a small positive temperature dependence of the molecular diameter of water. The rationale for this behavior is quite simple. In liquid water at room temperature, most of the water molecules are engaged in hydrogen bonds. The optimal distance for a hydrogen bond is about 2.76 A, well within the effective hard-core diameter assigned to a water molecule, about 2.8 A. As the temperature increases, one should consider at least two competing effects. On the one hand, we have the kinetic energy effect that was described above, which tends to decrease the effective diameters of the free water molecules. On the other hand, hydrogen-bonded pairs are broken as we increase the temperature hence. 

Perhaps some of the most important information on the mode of molecular packing of water in the liquid state is contained in the radial distribution function, which, in principle, can be obtained by processing X-ray or neutron scattering data. There are, however, several difficulties in extracting the proper information from the experimental data. First, it should be kept in mind that the full orientation-dependent pair correlation function cannot be obtained from such an experiment. Instead, only information on the spatial pair correlation function is accessible. We recall the definition of this function,... 

Figure 15.5. The simplest geometrical criterion for the H-bond controls either the Rqo or i Ho distances, and the respective radial distribution functions are used to assess the allowable length of the H-bond. Commonly the position of the first minimum is assumed. More complex geometric definitions include additional terms describing the deviation of the H-bond from linearity.

As we can see from Fig. 2, the TB and REBO relaxations give similar results for the radial distribution function. The relaxation of the RMC models results in an intensity increase along with a decrease in width of the first peak. These results arises from the different bond length constraints used in RMC and the REBO or TB potential. In RMC, it is assumed that two atoms can have a minimum distance of approach of 1.2 A and that two atoms are considered bonded if the distance between them is less than 1.6 A. This definition is obtained from the... 

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