Radial distribution function liquid

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

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... 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Fig. 6.2 Radial distribution function determined from a lOOps molecular dynamics simulation of liquid argon at a temperature of 100K and a density of 1.396gcm. ...

FIGURE 2.2 Radial distribution functions for (a) a hard sphere fluid, (A) a real gas, (c) a liquid, (li) a crystal. 

A very important aspect of both these methods is the means to obtain radial distribution functions. Radial distribution functions are the best description of liquid structure at the molecular level. This is because they reflect the statistical nature of liquids. Radial distribution functions also provide the interface between these simulations and statistical mechanics. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

Figure 4.1-11 The EXAFS data and pseudo-radial distribution functions of Co(ll) in (a) basic and (b) acidic chloroaluminate ionic liquid. Reproduced from reference 46 with permission.

Figure 9.24 A radial distribution function (RDF) for a DRP of monospheres (right) and a scheme for its evaluation (left), Nmeenl4nR2 is an equivalent of IA(R), where A/mean is the mean number of spheres in the intervals of 0.2R. The solid curve illustrates the data obtained from neutron diffraction in liquid argon 1,2 are the experimental data by Scott [128] and Bernal [127] obtained for the models of steel spheres (cited in [127]).

Fig. 55. Radial distribution function for liquid water synthesized as a mixture of ice I, ice II and ice III (from Ref. 75>)...

The ordering of the anions in bmimX ionic liquids has also been suggested by our recent large-angle x-ray scattering experiment on liquid bmimi [23]. Figure 13 shows a differential radial distribution function obtained for liquid bmimi at room temperature. Clear peaks in the radial distribution curve are... 

Figure 13. Differential radial distribution function of liquid bmimi by large-angle x-ray scattering [23].

The second category of methods uses a more general approach, based on fundamental concepts in statistical mechanics of the liquid state. As mentioned above, the Hwang and Freed theory (138) and the work of Ayant et al. (139) allow for the presence of intermolecular forces by including in the formulation the radial distribution function, g(r), of the nuclear spins with respect to the electron spins. The radial distribution function is related to the effective interaction potential, V(r), or the potential of mean force, W(r), between the spin-carrying particles through the relation (138,139) ... 

Once the structures of liquid and amorphous InP have been generated using MD simulations, various properties of these materials can be explored. To give just one example, the structure of disordered materials is often characterized using the radial distribution function ... 

Figure 9.4 Radial distribution functions for liquid (left) and amorphous (right) InP. In each case, distribution functions for In In, P P, In P atom pairs and for all atoms are shown. The zeros for the P P, In P, and total distribution functions are displaced vertically by 1, 2, and 3 units, respectively. (Reprinted by permission from the source cited in Fig. 9.2.)...

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