Pair density spherically averaged

Then, they proposed to examine the Taylor series expansion of the spherical average conditional pair density in the vicinity of the point s = 0 where one is measuring the short-range behaviour of the electron at point r2 approaching the reference point rl. The leading term of the Taylor series expansion is given by... 

Around the same time, Becke [184] proposed a real-space normalized model for the correlation hole. He started from the observation that, for a given coupling strength A, the spin-polarized components of the spherically averaged pair densities have well-defined short-range behavior near the reference point [185] ... 

Nagy, A. Spherically and system-averaged pair density functional theory. J. Chem. Phys. 2006,125, 184104. 

In the same manner one can define the second-order density matrix of a given symmetry and also the conditional pair density. Hence, using the Taylor series expansion of the spherical average conditional pair density of a given symmetry is as follows ... 

The situation is even more complicated for explicitly correlated wave functions, like those from a configuration interaction (Cl) calculation. To extend the approach of Becke and Edgecombe to the correlated level, the Laplacian of spherically averaged correlated conditional same-spin pair density needs to be computed. Consistently, at the same time the calibration should be performed using the correlated uniform electron gas as well, an aspect that was not taken into account in any ELF analysis of correlated wave functions. Of course, the expression for fuUy correlated ELFbe would deviate from the original (HF) ELFbe formula. 

As noted above, the function g(R, R") is a function of the scalar distance R = ] R" — R. (This is true both for spherical particles and for molecular fluids for which an orientational average has been carried out.) Because of the spherical symmetry of the locational pair correlation function, the local density has the same value for any point on the spherical shell of radius R from the center of the fixed particle at R. It is also convenient to choose as an element of volume a spherical shell of width dR and radius R. The average number of particles in this element of volume is... 

The second approach is to extend the simple two-parameter corresponding-states principle at its molecular origin. This is accomplished by making the intermolecular potential parameters functions of the additional characterization parameters /I, and the thermodynamic state, for example, the density p and temperature T. This can be justified theoretically on the basis of results obtained by performing angle averaging on a non-spherical model potential and by apparent three-body effects in the intermolecular pair potential. The net result of this substitution is a corresponding-states model that has the same mathematical form as the simple two-parameter model, but the definitions of the dimensionless volume and temperature are more complex. In particular the... 

Pair correlation function (r) can be found from experiments using the scattering of electromagnetic radiation. From this function, you can get other quantities, such as the average number of nearest neighbors of any molecule. Because the volume of a spherical shell at radius r is Anr dr, the number of molecules in the first neighbor shell is density x (volume of the shell) = pg r)4nr-dr. The total number of molecules in the first shell of a molecule is the integral... 

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