Expand the definition of the pair density (n_{2}left(boldsymbol{r}, boldsymbol{r}^{prime} ight)) in powers of the fugacity (z)

Expand the definition of the pair density \(n_{2}\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} \right)\) in powers of the fugacity \(z\) using the grand canonical partition function and the Mayer functions \(f_{i j}=\exp \left(-\beta u\left(r_{i j} \right) \right)-1\).

\[
\begin{aligned}
n_{2}\left(r_{12} \right) & =\frac{1}{\mathscr{Q}(\mu, V, T)} \sum_{N=2}^{\infty} \frac{z^{N}}{(N-2) !} \int d \boldsymbol{r}_{3} \cdots d \boldsymbol{r}_{N} \exp \left(-\beta u\left(r_{12} \right)-\beta u\left(r_{13} \right)-\cdots \right) \\
& =e^{-\beta u\left(r_{12} \right)}\left[z^{2}+z^{3} \int\left(1+f_{13}+f_{23}+f_{13} f_{23} \right) d \boldsymbol{r}_{3}-z^{3} Q_{1}+\cdots \right] \\
& =e^{-\beta u\left(r_{12} \right)}\left[\left(z^{2}+2 z^{3} \int f(r) d \boldsymbol{r} \right)+z^{3} \int f_{13} f_{23} d \boldsymbol{r}_{3}+\cdots \right]
\end{aligned}
\]

Note every term includes the factor \(e^{-\beta u\left(r_{12} \right)}\). The coefficients of those terms are integrals over the Mayer functions that are continuous functions of \(r_{12}\) even for the infinite step function potential; see equation (10.3.19) and discussion in Hansen and McDonald (1986) Chapter 5.