Reduced Hamiltonians
Since the 1-DM can be calculated from the 2-DM, we can write the energy as follows: \[\begin{align} \require{physics} E &= \frac{1}{2} \sum_{PQRS}^M K_{PQRS} d_{PQRS} \\ &= \frac{1}{4} \sum_{PQRS} \tilde{K}_{PQRS} d_{PQRS} \thinspace . \end{align}\] Here, \(\vb{K}\) and \(\tilde{\vb{K}}\) are called reduced Hamiltonians (Poelmans 2015) (Lanssens 2016), with the following matrix elements: \[\begin{equation} K_{PQRS} = \frac{1}{N-1} \qty( h_{PQ} \delta_{RS} + h_{RS} \delta_{PQ} ) + g_{PQRS} \end{equation}\] and \[\begin{equation} \tilde{K}_{PQRS} = \frac{1}{N-1} \qty( h_{PQ} \delta_{RS} + h_{RS} \delta_{PQ} - h_{PS} \delta_{RQ} - h_{RQ} \delta_{SP} ) + \tilde{g}_{PQRS} \thinspace . \end{equation}\]