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

A Hartree-Fock wavefunction is stationary in energy with respect to all infinitesimal occupied-virtual orbital rotations, subject to orbital orthonormality. The determinant then satisfies the self-consistent Hartree-Fock equations. Stationarity alone does not guarantee that it is the lowest-energy or even a locally stable determinant.

${\Phi }_i^a$ is the normalized determinant obtained by replacing occupied spin-orbital $i$ in ${\Phi }_0$ with previously unoccupied orbital $a$, using a consistent antisymmetric determinant ordering. Thus $i=1,\dots ,N$ labels occupied orbitals and $a=N+1,\dots ,2M$ labels virtual orbitals. Occupied-to-occupied rotations only change the representation of the same occupied subspace; the occupied-virtual rotations describe distinct nearby determinants.

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 $\hat{F}{\varphi }_p={\gamma }_p{\varphi }_p$.