## Definitions and notations

The expectation value of the two-body part of the Hamiltonian is obtained in a similar manner. We have $$\int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = A! \int \Phi_H^*\hat{A}\hat{H}_I\hat{A}\Phi_H d\mathbf{\tau},$$ which reduces to $$\int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \sum_{i\le j=1}^A \sum_{p} (-)^p\int \Phi_H^*\hat{v}(r_{ij})\hat{P}\Phi_H d\mathbf{\tau},$$ by following the same arguments as for the one-body Hamiltonian.