N-Representability Problem for Fermion Density Matrices. II. The First-Order Density Matrix withNEven

For a quantum-mechanical system of $N$ identical fermions, the $N$-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\Gamma }^{\mathrm{}\left(p\right)\mathrm{}}\left(12\mathrm{}\cdots p\right|1^{\prime }{2}^{\prime }\cdots p^{\prime })$, there exists an antisymmetric $N$-particle wave function $\Psi \left(12\mathrm{}\cdots N\right)$ such that ${\Gamma }^{\mathrm{}\left(p\right)\mathrm{}}\left(12\mathrm{}\cdots p\right|1^{\prime }{2}^{\prime }\cdots p^{\prime })=(\left\{N\right\}\left\{p\right\})\int \Psi \left(12\mathrm{}\cdots N\right)\times {\Psi }^{*}(1^{\prime }{2}^{\prime }\cdots p^{\prime },p+1\mathrm{}\cdots N)d_{{\tau }_{p+1\mathrm{}}}\cdots d_{{\tau }_N}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, $N$, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be $N$-representable is that its natural spin-orbital occupation numbers be equal in pairs.