Variational Principle for Many-Fermion Systems

If $\psi$ is a determinantal eigenfunction for the $N$-fermion Hamiltonian, $H$, with one- and two-body terms, then $e_0<~〈\psi ,H\psi 〉=E\left(K\right)\mathrm{}$, where $e_0$ is the ground-state energy, $K$ is the one-body reduced density matrix of $\psi$, and $E\left(K\right)\mathrm{}$ is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body $K$ is given, which does not come from a determinantal $\psi$, then $E>~e_0$ does not necessarily hold. This Letter proves, however, that if the two-body part of $H$ is positive, then in fact $e_0<~e_{\mathrm{HF}}<~E\left(K\right)\mathrm{}$, where $e_{\mathrm{HF}}$ is the Hartree-Fock ground-state energy.