Brillouin's theorem

At a stationary Hartree-Fock wavefunction $|{\Phi }_0⟩$, the Hamiltonian has zero matrix element to every singly excited Slater determinant $|{\Phi }_i^a⟩$ formed by replacing an occupied orbital $i$ with an unoccupied orbital $a$:An infinitesimal occupied-to-unoccupied orbital rotation adds precisely such singly excited components to the reference. Stationarity of the energy with respect to that rotation forces their linear coefficients to vanish. The theorem explains why single excitations do not provide the usual second-order reference-energy contribution in perturbation theory built on optimized Hartree-Fock orbitals; double excitations can still contribute. It does not state that the reference is an exact many-electron energy eigenstate, or that all stationary Hartree-Fock solutions are minima.