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 measure able.

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)

$\mathrm{g}(r)=e^{-\beta \mathrm{\Phi }(r)}(1+\rho {\mathrm{g}}_1(r)+{\rho }^2{\mathrm{g}}_2(r)+...)$

where $\rho$ is the number of molecules per unit volume and $\mathrm{\Phi }(r)$ is the intermolecular pair potential. The function $\mathrm{g}(r)$ is normalized to the value 1 for large distances. As is known, ${\mathrm{g}}_1(r)$, ${\mathrm{g}}_2(r)$, ... 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

${\mathrm{g}}_1(r_{12})=\int f(r_{13})f(r_{23})\mathrm{d}{\mathbf{r}}_3$

${\mathrm{g}}_2(r_{12})=\frac{1}{2}({\mathrm{g}}_1(r_{12}){)}^2+\phi (r_{12})+2\psi (r_{12})+\frac{1}{2}\chi (r_{12})$

where $r_{ik}$ is the distance $|{\mathbf{r}}_i-{\mathbf{r}}_k|$, where $f(r)$ is the Mayer f-function

$f(r)=e^{-\beta \mathrm{\Phi }(r)}-1$

and

$\phi (r_{12})=\int f(r_{13})f(r_{24})f(r_{34})\mathrm{d}{\mathbf{r}}_3\mathrm{d}{\mathbf{r}}_4$

$\psi (r_{12})=\int f(r_{13})f(r_{23})f(r_{24})f(r_{34})\mathrm{d}{\mathbf{r}}_3\mathrm{d}{\mathbf{r}}_4$

$\chi (r_{12})=\int f(r_{13})f(r_{23})f(r_{14})f(r_{24})f(r_{34})\mathrm{d}{\mathbf{r}}_3\mathrm{d}{\mathbf{r}}_4$

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)