_{ A \downarrow} ^{\dag} | 0 \rangle = \alpha _{i} | \Psi _{i} \rangle + \alpha _{c} c _{ B \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle  $, which can be obtained from $| \Psi _{b} \rangle$ by substituting the AF states $ c _{ A \uparrow} ^{\dag} c _{B \downarrow} ^{\dag} | 0 \rangle  $ and $c _{ B \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle$, respectively, for the covalent component $| \Psi _{c} \rangle$ in Eq. (1). Here $c _{L \sigma}$ is the operator to annihilate the electron with the spin orientation $\sigma$ at $| L \rangle$ in the bonding region $\Omega$ in Fig. 1 (a). The matrices $\rho ^{(1)}$ and $\rho ^{(2)}$ in Eq. (11) correspond to the up- and down-spin electrons in $| \Phi _{b} \rangle$, and correspond to the down- and up-spin ones in $ | \Phi _{b} ^{\prime} \rangle$. So $\rho ^{(1)}$ and $\rho ^{(2)}$ are for the quasielectrons with the opposite spin orientations in the one-bond system, and can represent the occupied levels of the half-filled 4-level dimer in Fig. 2 (a) if $| 1 \rangle \otimes | \sigma \rangle = |A \rangle \otimes | \sigma \rangle $ and 