We have defined the correlated function as
$$ \Psi_C=\prod_{i < j}g(r_{ij})=\prod_{i < j}^Ng(r_{ij})= \prod_{i=1}^N\prod_{j=i+1}^Ng(r_{ij}), $$with \( r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2} \) in three dimensions or \( r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2} \) if we work with two-dimensional systems.
In our particular case we have
$$ \Psi_C=\prod_{i < j}g(r_{ij})=\exp{\left\{\sum_{i < j}f(r_{ij})\right\}}. $$