The 2-DM

The elements of the two-electron density matrix (2-DM) are \[\begin{equation} \require{physics} d_{PQRS} = \ev{\hat{e}_{PQRS}}{\Psi} \thinspace , \end{equation}\] from which follows that some of the elements are related by complex conjugation: \[\begin{equation} d_{PQRS}^* = d_{QPSR} \thinspace , \end{equation}\] some are related through their sign: \[\begin{equation} d_{PQRS} = d_{RSPQ} = - d_{PSRQ} = - d_{RQPS} \thinspace , \end{equation}\] and some necessarily vanish \[\begin{equation} d_{PQPS} = d_{PQRQ} = d_{PQPQ} = 0 \end{equation}\] because of the anticommutation relations of the elementary second-quantization operators.

The diagonal elements of the 2-DM are \[\begin{equation} \omega_{PQ} = d_{PQQP} = \ev{ \hat{a}^\dagger_P \hat{a}^\dagger_Q \hat{a}_Q \hat{a}_P }{\Psi} = \sum_{\vb{k}} k_P k_Q |c_{\vb{k}}|^2 - \delta_{PQ} \omega_P \thinspace , \end{equation}\] so that the following trace becomes two times the number of pairs in the reference state: \[\begin{equation} \tr \bar{\vb{d}} = \sum_{PQ}^M \bar{\omega}_{PQ} = N (N-1) \thinspace . \end{equation}\]

We can show that the 1-DM can be calculated from the 2-DM in the following way: \[\begin{equation} d_{PQ} = \frac{1}{N-1} \sum_R^M d_{PQRR} \thinspace . \end{equation}\]