\rangle \langle B | + t _{AB} ^{*} | B \rangle \langle A | + \varepsilon _{A}  |  A \rangle \langle A | + \varepsilon _{B} | B  \rangle  \langle B |$ results from the non-interacting part of $H_{b}$. The factors $t _{AB}  | B \rangle \langle A | + t _{AB} ^{*} | A \rangle \langle B |$ and $\varepsilon _{A}  |  A \rangle \langle A | + \varepsilon _{B} | B  \rangle  \langle B |$ of $H _{sb}$ are responsible for the intra-bond hopping and on-site energy difference, respectively.

While we may construct the qausielectron density matrices by Eq. (11), the wavefunction $| \Psi _{b} \rangle$ in Eq. (1) is different from the uncorrelated functions $| \Phi _{b} \rangle$ and $| \Phi _{b} ^{\prime} \rangle$ when $ \alpha _{c} \neq 0$. To obtain the bonding energy $E _{b} = \langle \Psi _{b} | H _{b} | \Psi _{b}  \rangle$, therefore, we shall include the correlation contributions corresponding to the blue dash curve in Fig. 2 (a) by calculating the difference $\langle \Psi _{b} | H _{b} | \Psi _{b}  \rangle - \langle \Phi _{b} | H _{b} | \Phi _{b}  \rangle = \langle \Psi _{b} | H _{b} 