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

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 Γ⁽ᵖ⁾(12⋯p|1^'2^'⋯p^'), there exists an antisymmetric N-particle wave function Ψ(12⋯N) such that ${{Γ}}⁽ᵖ⁾(12{⋯}p|{1}^{{'}}{2}^{{'}}{⋯}{p}^{{'}})=({N}{p}){∫}{Ψ}(12{⋯}N)×{}{{Ψ}}*({1}^{{'}}{2}^{{'}}{⋯}{p}^{{'}},p+1{⋯}N){d}_{{{τ}}ₚ₊₁}{⋯}{d}_{{{τ}}N}$. 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.