Radial distribution function

Density Expansion of the radial distribution function

The radial distribution function of a compressed gas may be expanded in powers of the density (Ref. 1)

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

where $\rho$ is the number of molecules per unit volume. 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

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

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

where $r_{ik}$ is the distance $|r_i-r_k|$, where $f(r)$ is the Mayer f-function

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

and

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

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

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

References

1. 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)