+ \Delta = e _{1} ^{\dag} \rho _{ea} e _{1} $ and $\rho ^{(2)} = \rho - \Delta = e _{2} ^{\dag} \rho _{ea} e _{2} $, where $e _{1}  ^{\dag} = \frac{1}{ \sqrt{2} } (1,1) $ and $e _{2} ^{\dag} = \frac{1}{ \sqrt{2} } (1,-1) $. The effective Hamiltonian $\hat{H_{D}}$ given by Eq. (49) in Ref. [@Huang1] can be obtained by generalizing $H _{ddw}$ for the extended AF density matrix. In addition to the orbital antiferromagnetism, the order parameter $\Delta _{ {\bf k} }$ is taken into account for d-wave superconductivity [@Chakravarty; @Bena; @Kee]. Moreover, we may use $\rho ^{(1)}$ and $\rho ^{(2)}$ to represent the quasielectrons in the one-bond system [@Huang1].

The 4-level dimer composed of the two spatial sites with up- and down-spin orientations has been introduced by considering the dimer Hamiltonian $H _{t-U} = \sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} +t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma}) + \hat{U}$ [@Matlak1]. Here $A$ and $B$ represent the two spatial sites, $t _{AB}$ is the hopping coefficient, the annihilators $c _{A \sigma}$ and $c _{B \sigma}$ follow $\{ c _{B \sigma} ,c _{B \sigma ^{\prime} } \} = \{ c 