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

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.