$$\begin{aligned}
H _{b} = \sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} + t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{A}  c _{A \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{B}  c _{B \sigma} ^{\dag} c _{B \sigma}) + \hat{U},\end{aligned}$$ which is suitable to model the polar molecule [@Prasad], to introduce the on-site energies $\varepsilon _{A}$ and $\varepsilon _{B}$. In the above equation, $\sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} + t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{A}  c _{A \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{B}  c _{B \sigma} ^{\dag} c _{B \sigma}) = H _{b} - \hat{U} $ is the non-interacting part of $H _{b}$ while the e-e potential $\hat{U}$ serves as the interacting part. In the following, assume that $\varepsilon _{B} > \varepsilon _{A}$ such that atom $A$ has the higher electronegativity. The two-electron wavefunction becomes the uncorrelated state $ | \Psi _{i} \rangle = c _{A \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle$ in the ionic limit, so we shall take both $ | \Psi _{i} \rangle$ and $ | \Psi _{c} \rangle$ in general 