The 1-DM
The one-electron density matrix, or 1-DM, \(\require{physics} \vb{D}\) is given by its elements \[\begin{equation} D_{PQ} = \ev{\hat{E}_{PQ}}{\Psi} % \thinspace , \end{equation}\] from which immediately follows that it is Hermitian: \[\begin{equation} D^*_{PQ} = D_{QP} % \thinspace . \end{equation}\] Furthermore, since its elements are either \(0\) or positive (cfr. norms between states), it is a positive semi-definite matrix. Its diagonal elements are called the occupation numbers: \[\begin{equation} \omega_P = D_{PP} = \ev{\hat{N}_P}{\Psi} = \sum_{\vb{k}} k_P |c_{\vb{k}}|^2 \thinspace , \end{equation}\] and its trace is equal to the number of electrons in the reference state \(\ket{\Psi}\): \[\begin{equation} \tr \vb{D} = \sum_{P}^M D_{PP} = N \thinspace . \end{equation}\]
Since the 1-DM is Hermitian, it can be diagonalized with a unitary matrix: \[\begin{equation} \vb{D} = \bar{\vb{U}} \thinspace \boldsymbol{\eta} \thinspace \bar{\vb{U}}^\dagger \thinspace , \end{equation}\] where \(\boldsymbol{\eta}\) is a diagonal matrix collecting natural orbital occupation numbers \(\eta_P\), whose values are in the interval (T. Helgaker, Jørgensen, and Olsen 2000) \[\begin{equation} 0 \leq \eta_P \leq 1 \thinspace . \end{equation}\]
Since, for an 1-DM to be N-representable, its eigenvalues should be in the interval \([0,1]\) (Lanssens et al. 2018) (Coleman 1963), the previously defined 1-DM is N-representable.