We have $$ \varepsilon_{p}^{\mathrm{HF}}=\varepsilon_p+ \sum_{i\le F} \langle pi \vert \hat{V} \vert pi\rangle_{AS}, $$ where the single-particle states \( p=[n_p,j_p,m_p,t_{z_{p}}] \). Let us assume that \( p \) is a state above the Fermi level. The quantity \( \varepsilon_p \) could represent the harmonic oscillator single-particle energies.

Let \( p\rightarrow a \).

The energies, as we have seen, are independent of \( m_a \) and \( m_i \). We sum now over all \( m_a \) on both sides of the above equation and divide by \( 2j_a+1 \), recalling that \( \sum_{m_a}=2j_a+1 \). This results in $$ \varepsilon_{a}^{\mathrm{HF}}=\varepsilon_a+ \frac{1}{2j_a+1}\sum_{i\le F}\sum_{m_a} \langle ai \vert \hat{V}\vert ai\rangle_{AS}, $$