The general derivative formula of the Jastrow factor (or the ansatz for the correlated part of the wave function) is (the subscript \( C \) stands for Correlation)
$$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} + \sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_k} $$However, with our written in way which can be reused later as
$$ \Psi_C=\prod_{i < j}g(r_{ij})= \exp{\left\{\sum_{i < j}f(r_{ij})\right\}}, $$the gradient needed for the quantum force and local energy is easy to compute. The function \( f(r_{ij}) \) will depends on the system under study. In the equations below we will keep this general form.