But we have a simple form for the function, namely
$$ \Psi_{C}=\prod_{i < j}\exp{f(r_{ij})}, $$and it is easy to see that for particle \( k \) we have
$$ \frac{\mathbf{\nabla}^2_k \Psi_C}{\Psi_C }= \sum_{ij\ne k}\frac{(\mathbf{r}_k-\mathbf{r}_i)(\mathbf{r}_k-\mathbf{r}_j)}{r_{ki}r_{kj}}f'(r_{ki})f'(r_{kj})+ \sum_{j\ne k}\left( f''(r_{kj})+\frac{2}{r_{kj}}f'(r_{kj})\right) $$