Radial distribution function

The radial distribution function is a special case of the pair distribution function for an isotropic system. A Fourier transform of the radial distribution function results in the structure factor, which is experimentally measurable. The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid.

Typical radial distribution function for the monatomic Lennard-Jones liquid.

Density Expansion of the radial distribution function

The radial distribution function of a compressed gas may be expanded in powers of the density (Ref. 2)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. {\rm g}(r) \right. = e^{-\beta \Phi(r)} (1 + \rho {\rm g}_1 (r) + \rho^2 {\rm g}_2 (r) + ...)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is the number of molecules per unit volume and $\Phi (r)$ is the intermolecular pair potential. The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}(r)} is normalized to the value 1 for large distances. As is known, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}_1 (r)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}_2 (r)} , ... can be expressed by cluster integrals in which the position of of two particles is kept fixed. In classical mechanics, and on the assumption of additivity of intermolecular forces, one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}_1 (r_{12})= \int f (r_{13}) f(r_{23}) ~{\rm d}{\mathbf r}_3}

${\rm {g}}_{2}(r_{12})={\frac {1}{2}}({\rm {g}}_{1}(r_{12}))^{2}+\varphi (r_{12})+2\psi (r_{12})+{\frac {1}{2}}\chi (r_{12})$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{ik}} is the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |{\mathbf r}_i -{\mathbf r}_k|} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(r)} is the Mayer f-function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. f(r) \right. = e^{-\beta \Phi(r)} -1}

and

$\varphi (r_{12})=\int f(r_{13})f(r_{24})f(r_{34})~{\rm {d}}{\mathbf {r} }_{3}{\rm {d}}{\mathbf {r} }_{4}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{24}) f (r_{34}) ~ {\rm d}{\mathbf r}_3 {\rm d}{\mathbf r}_4}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{14}) f (r_{24}) f (r_{34}) ~ {\rm d}{\mathbf r}_3 {\rm d}{\mathbf r}_4}

References

1. John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics 10 pp. 394-402 (1942)
2. B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review 85 pp. 777 - 783 (1952)
3. J. L. Lebowitz and J. K. Percus "Asymptotic Behavior of the Radial Distribution Function", Journal of Mathematical Physics 4 pp. 248-254 (1963)
4. B. Widom "On the Radial Distribution Function in Fluids", Journal of Chemical Physics 41 pp. 74-77 (1964)
5. J. G. Malherbe and W. Krauth "Selective-pivot sampling of radial distribution functions in asymmetric liquid mixtures", Molecular Physics (preprint)