The radial distribution function

Throughout this chapter, we shall suppose that the interactions between N particles of a liquid medium are additive and paired, meaning that the internal energy due to these interactions is merely the sum of the interactions between molecules, two by two. Thus, the internal energy is the sum of the energies between the molecules taken two by two ij r.jY This energy 

The volumetric density p is defined as the ratio of the total number of molecules in the liquid in question to the volume of that liquid, i.e.  

We define the paired correlation factor or the radial distribution function 

As we can see, this function is the ratio of the mean value of the local density of molecules (mean calculated at the positions, at a given time and over a period of time) to the volumetric density of molecules. The correlation factor g(r) is proportional to the probability of finding a molecule at a distance r + dr from another molecule. Thus, we can write the relation  

This ratio [1.7] quantifies the local structure - in other words, the way in which the molecules are arranged in relation to one another. 

Let us now consider how we might represent atomic orbitals in three-dimensional space. We said earlier that a useful description of an electron in an atom is the probability of finding the electron in a given volume of space. The function ij (see Box 1.4) is proportional to the probability density of the electron at a point in space. By considering values of at points around the nucleus, we can define a surface boundary which encloses the volume of space in which the electron will spend, say, 95% of its time. This effectively gives us a physical representation of the atomic orbital, since ij may be described in terms of the radial and angular components R r) and A 0, 

First consider the radial components. A useful way of depicting the probability density is to plot a radial distribution 

Although we use ij in the text, it should strictly be written as ipip where 0 is the complex conjugate of ip. In the x-direction, the probability of finding the electron between the limits x and (x + dx) is proportional to ip x)ip (x) dx. In three-dimensional space this is expressed as ipip dr in which we are considering the probability of finding the electron in a volume element dr. For just the radial part of the wavefunction, the function is R r)R r). 

In all of our mathematical manipulations, we must ensure that the result shows that the electron is somewhere 

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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. 

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.

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

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).

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... 

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

The radial distribution function can also be used to monitor the progress of the equilibration. This function is particularly useful for detecting the presence of two phases. Such a situation is characterised by a larger than expected first peak and by the fact that g r) does not decay towards a value of 1 at long distances. If two-phase behaviour is inappropriate then the simulation should probably be terminated and examined. If, however, a two-phase system is desired, then a long equilibration phase is usually required. 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

This factor is reminiscent of the radial distribution function for electron probability in an atom and the Maxwell distribution of molecular velocities in a gas, both of which pass through a maximum for similar reasons. 

Because the correlation of atomic positions decreases as r — co, = 1. The function 47T p (, the radial distribution function (RDF), may also be... 

Pings [106], and a neutron diffraction study by Mildner and Carpenter [107], both concluded that there is no clear evidence for sp carbon and that the radial distribution functions can be satisfactorily indexed to a hexagonal arrays of carbon atoms. A similar conclusion was reached in a recent neutron diffraction study of activated carbons by Gardner et al [108]. 

The case A = 2 is of greatest interest. Since the force is central, it is not necessary to use rj and ri as variables. The single variable r 2 is sufficient since the position of the center of mass is irrelevant. Thus, we have the radial distribution function (RDF), g r 12). 

Distribution functions measure the (average) value of a property as a function of an independent variable. A typical example is the radial distribution function g r) which measmes the probability of finding a particle as a function of distance from a typical ... 

For molecules the radial distribution function can be extended with orientational degrees of freedom to characterize the angular distribution. 

The degree of ordering of the microspheres was estimated by using the radial distribution function g(D) of the P4VP cores of the microspheres (Fig. 11). As previously described, for hexagonal packed spheres, the ratio of the peaks of the distances between the centers of the cores would be For the film at r = 0.5, the... 

The desired average is simply obtained by a time average of the given property. For example, one of the interesting properties of bulk solvents is the radial distribution function (rdf), which expresses the probability of finding a given atom type around a reference atom by... 

This subroutine calculates the three radial distribution functions for the solvent. The radial distribution functions provide information on the solvent structure. Specially, the function g-AB(r) is die average number of type B atoms within a spherical shell at a radius r centered on an aibitaiy type A atom, divided by the number of type B atoms that one would expect to find in the shell based cm the hulk solvent density. 

The radial distribution function, P, is closely related to the wavefunction... 

A note on good practice Be careful to distinguish the radial distribution function from the wavefunction and its square, the probability density ... 

The radial distribution function tells us, through P(r)f>r, the probability of finding the electron in the range of radii 8r, at a particular value of the radius, summed over all values of 0 and < >. 

The radial distribution function for the population of the Earth, for instance, is zero up to about 6400 km from the center of the Earth, rises sharply, and then falls back to almost zero (the almost takes into account the small number of people who are on mountains or flying in airplanes). 

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

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