Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 60 / b / iii / Solution

Orthogonality of the determinants gives $⟨{\Phi }^{(1)}|{\Phi }^{(1)}⟩=1+O({\lambda }^2)$. To first order the normalized energy is thereforeSlater-Condon rules give the matrix element as the occupied-virtual Fock elementStationarity for independently variable real ${\lambda }_i^a$ requires every $F_{ia}=0$. This is Brillouin's theorem. Because $F$ is Hermitian, occupied and virtual blocks can separately be diagonalized; canonical orbitals then satisfy $\hat{F}{\varphi }_p={\gamma }_p{\varphi }_p$.