Radial distribution function defined

The right-hand side of this equation is the first term of the density expansion of the radial distribution function which is valid only in the limit of low concentrations (p 0). For arbitrary concentrations, the radial distribution function defines the potential of the mean force w(r) by a similar equation to Eq. (47), where u(r) is substituted by w(r). In general, the potential of the mean force does not coincide with u(r) except in the limit of low concentrations. Therefore, a more general method to determine the effective pair potential should be developed. 

Figure 12.5. The radial distribution function for crystalline and amorphous Si. The curves show the quantity G(r) = g(r)/(47tr dr), that is, the radial distribution function defined in Eq.(12.19) divided by the volume of the elementary spherical shell (47rr dr) at each value of r. The thick solid line corresponds to a model of the amorphous solid, the thin shaded lines to the crystalline solid. The atomic positions in the crystalline solid are randomized with an amplitude of 0.02 A, so that G(r) has finite peaks rather than 5-functions at the various neighbor distances the peaks corresponding to the first, second and third neighbor distances are evident, centered at r = 2.35, 3.84 and 4.50 A, respectively. The values of G(r) for the crystal have been divided by a factor of 10 to bring them on the same scale as the values for the amorphous model. In the results for the amorphous model, the first neighbor peak is clear (centered also at 2.35 A), but the second neighbor peak has been considerably broadened and there is no discernible third neighbor peak.

Many of the equilibrium properties of such systems can be obtained through the two-body reduced coordinate distribution function and the radial distribution function, defined in Eqs. (27.6-5) and (27.6-7). There are a number of theories that are used to calculate approximate radial distribution functions for liquids, using classical statistical mechanics. Some of the theories involve approximate integral equations. Others are perturbation theories similar to quantum mechanical perturbation theory (see Section 19.3). These theories take a hard-sphere fluid or other fluid with purely repulsive forces as a zero-order system and consider the attractive part of the forces to be a perturbation. ... 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

Show thar, if the radial distribution function is defined as P = r2R2, then the expression for P for an s-orbital is... 

The radial distribution function for isobutane (Fig. 1, curve C) shows well-defined peaks at 1.13,... 

For high densities, g cannot be set equal to one, and the collision function becomes much more complex and so is not given here. It turns out, however, that instead of using the full radial distribution function, it is sufficient to use the value at contact r — R, so that we define a new function ... 

The most widely used measure of structure in fluids is the pair correlation function (1-6) (or radial distribution function) gij(r). It is defined so that... 

In amorphous solids there is a considerable disorder and it is impossible to give a description of their structure comparable to that applicable to crystals. In a crystal indeed the identification of all the atoms in the unit cell, at least in principle, is possible with a precise determination of their coordinates. For a glass, only a statistical description may be obtained to this end different experimental techniques are useful and often complementary to each other. Especially important are the methods based on diffraction experiments only these will be briefly mentioned here. The diffraction pattern of an amorphous alloy does not show sharp diffraction peaks as for crystalline materials but only a few broadened peaks. Much more limited information can thus be extracted and only a statistical description of the structure may be obtained. The so-called radial distribution function is defined as ... 

For visualization purposes we have made plots of pair distribution functions, defining the electron-nuclear radial probability distribution function D(ri) by the formula... 

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). 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

The average local electrostatic potential V(r)/p(r), introduced by Pohtzer [57], led Sen and coworkers [58] to conjecture that the global maximum in V(r)/p(r) defines the location of the core-valence separation in ground-state atoms. Using this criterion, one finds N values [Eq. (3.1)] of 2.065 and 2.112 e for carbon and neon, respectively, and 10.073 e for argon, which are reasonable estimates in light of what we know about the electronic shell structure. Politzer [57] also made the significant observation that V(r)/p(r) has a maximum any time the radial distribution function D(r) = Avr pir) is found to have a minimum. 

The moments of Y are obtained from similar expressions simply by changing the signs of all g /, g f and g f that appear in Eqs. 6.78 through 6.80, so that we need not repeat those expressions here. We note that the reduced mass is m, B is short for Bj.cJ, and the Vv, Vv> are the vibrational averages of the interaction potential. Superscripted Roman numericals I. .. IV mean the first. .. fourth derivatives with respect to R. The radial distribution functions g = g(R) depend on the interaction potentials, Vv, Vv>, and are thus subscripted like the potentials the low-density limit of the distribution function will be sufficient for our purposes. The functions g and g M are defined in Eq. 6.23. The notation f f R)d2R stands for 4n /0°° / (R) R2 dR as usual. 

Approximate evaluations of the radial distribution function in dense systems are being obtained as solutions to integral equations derived from firsl principles under well-defined approximations. 

The form factor is defined in terms of the radial distribution function... 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

The structure of liquids can be analyzed by the calculated radial distribution function (RDF), which defines the solvation shells. In Fig. 16.1, the calculated RDF of the liquid Aris shown, and in Table 16.1, the structure is compared with the experimental results. Four solvation shells are well defined. The spherical integration of these peaks defines the coordination number, or the number of atoms in each solvation shell. The first shell that starts at 3.20A has a maximum at 3.75A, and ends at 5.35 A, has an average of 13 Ar atoms. Therefore, in the first solvation shell, there is a reference Ar atom surrounded by other neighboring 13 Ar atoms. All the maxima of the RDF, shown in Table 16.1, are in good agreement with the experimental results obtained by Eisenstein and Gingrich [29], using X-ray diffraction in the liquid Ar in the same condition of temperature and pressure. The calculated... 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

Take the position of one atom as origin and denote the atomic density, in a given direction, at a distance r from the centre by the number of atoms between r and dr, p(r). The volume of the corresponding spherical shell is 47rr2dr and the number of atoms in the shell, 47rr2p(r)dr, defines the radial distribution function. The intensity of scattered radiation from the sample is given by the Fourier transform... 

III. 1.2. The radial distribution function. The radial distribution function is defined by the expression (28)... 

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