Let us now derive the expression for our two-body interaction part, which also conserves the number of particles. We can proceed in exactly the same way as for the one-body operator. In the coordinate representation our two-body interaction part takes the following expression $$ \begin{equation} \hat{H}_I = \sum_{i < j} V(x_i,x_j) \tag{43} \end{equation} $$ where the summation runs over distinct pairs. The term \( V \) can be an interaction model for the nucleon-nucleon interaction or the interaction between two electrons. It can also include additional two-body interaction terms.

The action of this operator on a product of two single-particle functions is defined as $$ \begin{equation} V(x_i,x_j) \psi_{\alpha_k}(x_i) \psi_{\alpha_l}(x_j) = \sum_{\alpha_k'\alpha_l'} \psi_{\alpha_k}'(x_i)\psi_{\alpha_l}'(x_j) \langle \alpha_k'\alpha_l'|\hat{v}|\alpha_k\alpha_l\rangle \tag{44} \end{equation} $$