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

At a Hartree-Fock stationary point the reference-single terms vanish. Expanding the normalized expectation through second order givesThe reference-double term arises twice in the numerator and cancels the factor $1/2$ in the wavefunction. The normalization subtracts $E_0{\sum }_{ia}({\lambda }_i^a{)}^2$.

Using Slater-Condon rules in canonical orbitals,and $⟨{\Phi }_0|\hat{H}|{\Phi }_{ij}^{ab}⟩=⟨ab|ij⟩-⟨ab|ji⟩$. Combining them gives the requested $X_{ij}^{ab}$. The orbital energies arefor either occupied or virtual $p$. Diagonalizing the occupied and virtual Fock blocks is the suitable orbital choice that makes the delta form valid for coincident as well as distinct indices.