## Definitions and notations

Because of the dependence on the inter-particle distance $$r_{ij}$$, permutations of any two particles no longer vanish, and we get $$\int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \sum_{i < j=1}^A \int \Phi_H^*\hat{v}(r_{ij})(1-P_{ij})\Phi_H d\mathbf{\tau}.$$ where $$P_{ij}$$ is the permutation operator that interchanges particle $$i$$ and particle $$j$$. Again we use the assumption that the single-particle wave functions are orthogonal.