Importance sampling, programming elements

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.

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