Brillouin's theorem
= Brillouin's theorem
{c}
At a stationary <Hartree-Fock wavefunction> $|\Phi_0\rangle$, the <Hamiltonian> has zero <matrix element> to every singly excited <Slater determinant> $|\Phi_i^a\rangle$ formed by replacing an occupied orbital $i$ with an unoccupied orbital $a$:
$$
\langle\Phi_i^a|H|\Phi_0\rangle=0.
$$
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.