The pair distribution function

Clearly, P(2) (X, X ) dX dX is the probability of finding a specific particle, say 1, in dX at X and another specific particle, say 2, in dX at X . The same probability applies for any specific pair of two different particles. 

As in the case of the singlet MDF, here we also start with the specific pair distribution function defined in (2.28). To get the generic pair distribution function, consider the list of events and their corresponding probabilities in table 2.2. Note that the probabilities of all the events on the left-hand column of table 2.2 are equal. 

The last equality in (2.30) follows from the equivalence of all the N(N— 1) pairs of specific and different particles. Using the definition of P(2) (X, X ) in (2.28), we can transform the definition of p(2) (X, X ) into an expression which may be interpreted as an average quantity  

In the second form of the rhs of (2.31), we employ the basic property of the Dirac delta function, so that integration is now extended over all the vectors Xx. XN. In the third form we have used the equivalence of the N particles, as we have done in (2.30), to get an average of the quantity 

This can be viewed as a counting function, i.e., for any specific configuration XN, this quantity counts the number of pairs of particles occupying the elements dX and dX . Hence, the integral (2.31) is the average number of pairs occupying dX and dX . The normalization of pl2)(X, X ) follows directly from (2.31)  

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Between the limits of small and large r, the pair distribution function g(r) of a monatomic fluid is detemrined by the direct interaction between the two particles, and by the indirect interaction between the same two particles tlirough other particles. At low densities, it is only the direct interaction that operates through the Boltzmaim distribution and... 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

Let us proceed with the description of the results from theory and simulation. First, consider the case of a narrow barrier, w = 0.5, and discuss the pair distribution functions (pdfs) of fluid species with respect to a matrix particle, gfm r). This pdf has been a main focus of previous statistical mechanical investigations of simple fluids in contact with an individual permeable barrier via integral equations and density functional methodology [49-52]. 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

In order to get the pair distribution functions gjj, which satisfy the symmetry constraints, so-called minor iterations must be converged. When we use gij(r >r j) of (fl) in (2), we obtain the minor equations regarding the rotation operator R of the angle tt/2 ... 

Although we work with the pair distribution functions, what we are to solve are essentially the point probability functions fj(rj), i=A and B. 

We have solved the set of integral equations on the pair distribution functions by discretizing them. This is equivalent to allowing the atomic displacements to finite number of points. When they are discretized, to solve them is a straightforward application of the exsisting CVM. 

In conclusion, we have presented a new formulation of the CVM which allows continuous atomic displacement from lattice point and applied the scheme to the calculations of the phase diagrams of binary alloy systems. For treating 3D systems, the memory space can be reduced by storing only point distribution function f(r), but not the pair distribution function g(r,r ). Therefore, continuous CVM scheme can be applicable for the calculations of phase diagrams of 3D alloy systems [6,7], with the use of the standard type of computers. 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

Figure 5. The Fourier transformed signal AS[r, i] of I2/CCI4. The pump-probe delay times are I = 200 ps, 1 ns, and 1 ps. The green bars indicate the bond lengths of iodine in the X and A/A states. The blue bars show the positions of the first two intermolecular peaks in the pair distribution function gci-ci- (See color insert.)...

Figure 6. The Fourier transformed signal AS[r, i] of CH2I2/CH3OH. The pump-probe time delays vary between i = —250 ps and 1 ps. The pair distribution function gl-I peaks in the 3 A region. If T < 50 ns, the I—I bond corresponds to the short-lived intermediate (CH2ri), and if x > 100 ns it belongs to the (I3") ion. Red curves indicate the theory, and black curves describe the experiment.

Almarza, N. G. Lomba, E., Determination of the interaction potential from the pair distribution function an inverse Monte Carlo technique, Phys. Rev. E 2003, 68, 011202... 

At infinite dilution, g(R) — exp (be KRjR). Linearizing the inner exponential and neglecting the second term in the denominator of the last equation we recover the Bjerrum result (Eq. (185)). However, at finite concentrations even if we retain terms to the same order in log y1 and g(R), Eqs. (183) and (186) will not in general give the same value of p. The use of a mass action formalism as a means both of calculating activity coefficients and of studying the pair distribution function via the degree of association p at finite concentrations is not done in a self-consistent manner in the Bjerrum type of treatment. 

In the particular case of electrolytes, we may use Eq. (134) for the pair distribution function. We thus have ... 

The value of the peaks and troughs in the pair distribution function represent the fluctuation in number density. The peaks represent regions where the concentrations are in excess of the average value while the troughs represent a deficit. As the volume fraction is increased, the peaks and troughs grow, reflecting the increase in order with concentration. We... 

Figure 5.7 The pair distribution function g(r) for hard spheres as a function of the dimensionless centre-to-centre separation. This was calculated using the algorithm from McQuarrie1 at a range of volume fractions...

The major difficulty in predicting the viscosity of these systems is due to the interplay between hydrodynamics, the colloid pair interaction energy and the particle microstructure. Whilst predictions for atomic fluids exist for the contribution of the microstructural properties of the system to the rheology, they obviously will not take account of the role of the solvent medium in colloidal systems. Many of these models depend upon the notion that the applied shear field distorts the local microstructure. The mathematical consequence of this is that they rely on the rate of change of the pair distribution function with distance over longer length scales than is the case for the shear modulus. Thus... 

The structure of a liquid is conventionally described by the set of distributions of relative separations of atom pairs, atom triplets, etc. The fundamental basis for X-ray and neutron diffraction studies of liquids is the observation that in the absence of multiple scattering the diffraction pattern is completely determined by the pair distribution function. 

For the case of a pure monatomic liquid, in the limit that there are only pair interactions, the pair distribution function provides a complete microscopic specification from which all thermodynamic properties can be calculated 2>. If there are (excess) three molecule interactions, then one must also know the triplet distribution function to complete the microscopic description the extension to still higher order (excess) interactions is obvious. 

The diffusion-reaction equation for the pair distribution function g(r,t) of the reactants, which react with a rate constant which at any r is k(r), is given by (29-32)... 

One of the desirable features of compact wavefunctions is the ability to use them to examine additional features of the electron distribution without the necessity of repeating extensive computations to recreate complicated wavefunctions. We illustrate this point, and also exhibit the similarity of our wavefunctions with those of the 66-configuration study of Thakkar and Smith [15] by looking at the pair distribution functions. It is most instructive to present these as Z-scaled quantities Figure 3 contains the electron-nuclear distributions D r ) and p(ri) for clarity we only plot data for H, He, Li, and Ne. Even after Z scaling, a small but systematic narrowing of the distributions with increasing Z is still in process at Z-10. 

This function is the integrally normalized probability for each water molecule being oriented such that it makes an angle B between its OH bond vectors and the vector from the water oxygen to the carbon atom. This function is calculated for those molecules within 4.9 A of the carbon atom (nearest neighbors), as this distance marks the first minimum in the pair distribution function for that atom. The curve in Figure 10 is typical for hydrophobic hydration (22). 

Using the separation of the pair-distribution function nf into its uncorrelated and correlated parts, Eq. (284), and noting that the combination... 

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