| \Psi _{b}  \rangle -  \langle \Phi ^{\prime} _{b} | H _{b} | \Phi _{b} ^{\prime}  \rangle$. The non-interacting part of $H_{b}$ induces the correlation energy $ tr H_{sb} d ^{(1)} = \langle \Psi _{b} | (H _{b} - \hat{U}) | \Psi _{b}  \rangle - \langle \Phi _{b} | (H _{b} - \hat{U}) | \Phi _{b}  \rangle = \langle \Psi _{b} | (H _{b} - \hat{U}) | \Psi _{b}  \rangle - \langle \Phi ^{\prime} _{b} | (H _{b} - \hat{U}) | \Phi ^{\prime} _{b}  \rangle $. We can take $I = | A \rangle \langle A| +  | B \rangle \langle B | $, which is the identity operator on $\mathbb{C} ^{2} _{\Omega}$, in Eq. (3) to introduce the density-fluctuation operator $ d ^{(1)} $. The operator $d ^{(1)}$ results from the factor $\alpha _{i} \alpha _{c} ^{*}  \langle \Psi _{c} | (H _{b} - \hat{U}) | \Psi _{i}  \rangle + \alpha _{c} \alpha _{i} ^{*}  \langle \Psi _{i} | (H _{b} - \hat{U}) | \Psi _{c}  \rangle  $ when we calculate $\langle \Psi _{b} | (H _{b} - \hat{U}) | \Psi _{b}  \rangle $, and becomes zero if the ionic and covalent components do not coexist. The up- or 