Features of the Radial Distribution Function

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 (2.32) and (2.40) (applied to spherical particles), we get 

This general relation will be used to extract information on the behavior of g(R) for some simple systems. A more useful expression, which we shall 

The RDF for an ideal gas can be obtained directly from definition (2.49). Putting Uji = 0 for all configurations, the integrations become trivial and we get 

As we expect, g(R) is practically unity for any value of R. This is an obvious reflection of the basic property of an ideal gas, i.e., absence of correlation follows from absence of interaction. The term N is typical of constant-volume systems. At the thermodynamic limit N oo, V- oo NfV = const, this term may, for most purposes, be dropped. Of course, in order to get the correct normalization of g(R), one should use the exact relation (2.54), which yields 

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For colloidal liquids, Eqs. (19-21) refer to the excess energy [second term of the right-hand side of Eq. (19)], the osmotic pressure and osmotic compressibility, respectively. They show one of the important features of the radial distribution function g(r), namely, that this quantity bridges the (structural) properties of the system at the mesoscopic scale with its macroscopic (thermodynamic) properties. 

SOME GENERAL FEATURES OF THE RADIAL DISTRIBUTION FUNCTION 35... 

Some general features of the radial distribution function... 

This is valid for any Ra. However, since in general we do not know the various radial distribution functions, this relation is not useful. Before transforming (8.6) into a more useful and computable form, we recall the following two characteristic features of the radial distribution functions gis(R). 

The latter should be compared with the value of 4.5 A in the experimental curve of g R).] This result indicates that as one couples the HB part of the pair potential, the tendency to form a tetrahedral geometry increases, and therefore the characteristic peak at about 4.5 A is developed. Other features of the radial distribution function obtained from these computations are not in agreement with the experimental results. For instance, the average coordination number obtained for the case 7 0.3 was... 

These two features of the radial distribution function lead to the following conclusion The basic geometry around a single molecule in... 

All of the features of the radial distribution function were simulated by the model particles. There is a sharp first peak at R = R/a = 0.975, corresponding to = 2.75 A (with a = 2.82 A). The second peak occurs at jR = 1.69, corresponding to = 4.76 A, which is a little above the experimental value of about R = 4.6 A. Also, the average coordination number, computed up to the first minimum (following the first maximum), is 5.5, which is somewhat higher than the experimental value of about 4.4. Figure 6.32 also shows the running coordination number, i.e., the function... 

With reference to the minima of the radial distribution function D r), SCF analyses [61] using the near-Hartree-Fock wavefunctions of dementi [64] indicate that the numbers of electrons found in the inner shell extending up to the minimum of D r) amount to = 2.054 e (Be), 2.131 (C), 2.186 (O), 2.199 (F) and 2.205 electron (Ne). The results of Smith et al. [65] bearing on the boundaries in position space that enclose the exact number given by the Aufbau principle support the idea of physical shells compatible with that principle. The maxima of D r), on the other hand, also appear to be topological features indicative of shells, their positions correlate well with the shell radii from the Bohr-Schrodinger theory of an atom... 

Figure 9. A schematic illustration of the origin of structural features in the radial distribution function. Atoms are shown as lying on sharply defined rings for simplicity. Broadening is incorporated in g r).

This structural feature has affected almost all subsequent attempts to theorize about the properties of liquid water and aqueous solutions. Bernal and Fowler were also the first to attempt a theoretical calculation of the radial distribution function of water. Such an undertaking could not have been feasible without a detailed characterization of the structure of the various components presumed to exist in the liquid. 

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. 

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