Radial distributions

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)... 

In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

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. 

Flere g(r) = G(r) + 1 is called a radial distribution function, since n g(r) is the conditional probability that a particle will be found at fif there is another at tire origin. For strongly interacting systems, one can also introduce the potential of the mean force w(r) tln-ough the relation g(r) = exp(-pm(r)). Both g(r) and w(r) are also functions of temperature T and density n... 

Figure A2.3.7 The radial distribution function g r) of a Lemiard-Jones fluid representing argon at T = 0.72 and p = 0.844 detennined by computer simulations using the Lemiard-Jones potential.

Figure A2.4.1. Radial distribution fiinction g(R ) for water (dashed curve) at 4 °C and 1 atm and for liquid argon (fiill curve) at 84.25 K and 0.71 atm as functions of the reduced distance R = R/a, where a is the molecular diameter from [1].

Figure A2.4.4. Plot of the radial distribution difference fiinction Ag(r) against distance r (pm) for a 1.46 M solution of NiCl in D O. From [5].

Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

A more subtle point concerns scaling of -xwith density. Among tlie various possibilities that exist, either employing local densities obtained from numerically calculated radial distribution fiinctions [38]... 

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

Fig. 4. Radial distribution functions between the centre of a test cavity and the (jxygen atom of the surrounding water. The curves correspond to the different barrier heights for the softcore interaction illustrated in Fig. 3...

D MoRSE desaiptor, radial distribution function (RDF code), WHIM descriptors, GETAWAY descriptors,... 

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]. 

Figure 8-6. Comparison of the radial distribution function of the ctiair, boat, and twist conformations of cyclohexane (hydrogen atoms are not considered).

Topological descriptors and 3D descriptors calculated in distance space", such as 3D autocorrelation, surface autocorrelation, and radial distribution function... 

Models with 32 Radial Distribution Function Values and Eight Additional Descriptors... 

The compounds were described by a set of 32 radial distribution function (RDF) code values [27] representing the 3D structure of a molecule and eight additional descriptors. The 3D coordinates were obtained using the 3D structure generator GORINA [33]. 

The radial distribution Function (RDF) of an ensemble of N atoms can be interpreted as the probability distribution to find an atom in a spherical volume of... 

Figure 10.1-5. Predicted versus experimental solubility values of 496 compounds in the test set by a back-propagation neural network with 32 radial distribution function codes and eight additional descriptors.

A combination of physicochemical, topological, and geometric information is used to encode the environment of a proton, The geometric information is based on (local) proton radial distribution function (RDF) descriptors and characterizes the 3D environment of the proton. Counterpropagation neural networks established the relationship between protons and their h NMR chemical shifts (for details of neural networks, see Section 9,5). Four different types of protons were... 

The strncturcs in the database arc encoded using the radial distribution function (RDF) as a descriptor (cf Section 8,4,4). 

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. ...

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