How do I find $g(r)$ from the Ornstein-Zernike equation for a hard-sphere fluid in Percus-Yevick Approximation? I have been reading this paper from Thiele [J. Chem. Phys. 39, 474 (1963)], who has obtained the direct correlation function $c(r)$ for a hard-sphere system using the Percus-Yevick approximation.
My question is, how do I find $g(r)$ from this?
In Torquato's Random Heterogeneous Materials, he has written
$$\frac{p}{\rho kT} = 1+2^{d-1}\eta g_2 (D^{+})$$
where $g_2(D^+)$ is the contact value from the right-side of the radial distribution function, and $\eta$ is a dimensionless reduced density.
After a couple lines, he states that for hard spheres, via the Ornstein-Zernike equation, we can rewrite the above equation in terms of the direct correlation function $c(r)$ as
$$\frac{p}{\rho kT} = 1+2^{d-1}\eta [c(D^+)-c(D^-)]$$
How does he reach this conclusion?
Ornstein-Zernike states that
$$h(r_{12}) = c(r_{12}) + \rho \int d\mathbf{r}_3 c(r_{13})h(r_{32})$$
which after a Fourier transform becomes
$$\hat{C} (\mathbf{k}) = \frac{\hat{H}(\mathbf{k})}{1+\rho \hat{H}(\mathbf{k})}$$
However, I don't see how to simplify this to the second equation he has. I would appreciate any advice you have.
What I want to do is tabulate values of $g(r)$, for different values of $r$ greater than $\sigma$. Using $g(r)$ I want to calculate reduced density, $p/\rho k T$, and compare it to the values I get from Stirling-Carnahan for the reduced density.
 A: Finding the explicit analytic form of $g(r)$ for all distances is feasible but not straightforward. The basic step is in Wertheim's solution ( Wertheim, M. S. (1963). Exact solution of the Percus-Yevick integral equation for hard spheres. Physical Review Letters, 10(8), 321 ).
However, if the problem is only the contact value of $g(r)$, the solution is much simpler. It is based on the fact that, although $g(r)$ and $c(r)$ are discontinuous at the diameter distance $r=\sigma$, their difference must be continuous. This is a trivial consequence of the Ornstein-Zernike equation: $h(r)-c(r)$ is a convolution of two functions with a discontinuity at $\sigma$. Thus, it must be continuous at $\sigma$ (a possible way to get convinced of this fact is from the Fourier representation of OZ showing that the leading term of the asymptotic behavior of $\hat{C} (\mathbf{k}) $ and $\hat{H} (\mathbf{k}) $ must be the same).
Therefore, $g(\sigma^+)-g(\sigma^-)= c(\sigma^+)-c(\sigma^-)$. But, since $g(\sigma^-)=0$ (core condition) and $c(\sigma^+)=0$ (Percus-Yevick approximation), from the knowledge of $c(r)$ inside the core it is possible to get the contact value of $g(r)$.
