Using the definition of the single-particle energy in Eq. (23), the definition of the monopole matrix element in Eqs. (20) or (22) and Eq. (21), we can rewrite Eq. (23) as $$ \begin{equation} \epsilon_{\alpha_{p}}=\langle \alpha_p|\hat{h}_0|\alpha_p\rangle+\sum_{\alpha_i\le \alpha_F}N_{\alpha_i}\bar{V}_{\alpha_p\alpha_i}, \tag{25} \end{equation} $$ with \( N_{\alpha_i}=2\alpha_i+1 \), and Eq. (24) as $$ \begin{equation} \epsilon_{\alpha_{p}}=\langle \alpha_p|\hat{h}_0|\alpha_p\rangle+\sum_{\alpha_i\le \alpha_F}N_{\alpha_i}\tilde{V}_{\alpha_p\alpha_i}. \tag{26} \end{equation} $$