We can now use the above relations to compute the Hartre-Fock energy in \( j \)-scheme. In \( m \)-scheme we defined the Hartree-Fock energy as $$ \varepsilon_{pq}^{\mathrm{HF}}=\delta_{pq}\varepsilon_p+ \sum_{i\le F} \langle pi \vert \hat{V} \vert qi\rangle_{AS}, $$ where the single-particle states \( pqi \) point to the quantum numbers in \( m \)-scheme. For a state with for example \( j=5/2 \), this results in six identical values for the above potential. We would obviously like to reduce this to one only by rewriting our equations in \( j \)-scheme.

Our Hartree-Fock basis is orthogonal by definition, meaning that we have $$ \varepsilon_{p}^{\mathrm{HF}}=\varepsilon_p+ \sum_{i\le F} \langle pi \vert \hat{V} \vert pi\rangle_{AS}, $$