^{\prime \prime} ) | \Psi _{b} \rangle =  \langle \Psi _{b} | \psi _{ \downarrow } ^{ \dagger } ({\bf r} ^{ \prime } ) \psi _{ \downarrow } ({\bf r} ^{\prime \prime} ) | \Psi _{b} \rangle  $ and thus serves as the one-electron density matrix, which is spin-degenerate because $\langle \Psi _{b} | \psi _{ \sigma } ^{ \dagger } ({\bf r} ^{\prime}) \psi _{ - \sigma } ({\bf r}^{\prime \prime} ) | \Psi _{b} \rangle=0$ for $\sigma = \uparrow$ and $\downarrow$.

Quasielectrons in ionic-covalent bonds
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In this section, the energy forms are constructed for the quasielectrons in the half-filled bonding systems. For convenience, I discuss the one-bond system based on the 4-level dimer [@Matlak1] in subsection III-A, and extend the results to the compound system composed of identical dimers [@Matlak2] in subsection III-B.

III-A One-bond system {#iii-a-one-bond-system .unnumbered}
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Consider the two uncorrelated states $| \Phi _{b} \rangle = c _{A \uparrow} ^{\dag} c _{ L \downarrow} ^{\dag} | 0 \rangle = \alpha _{i} | \Psi _{i} \rangle + \alpha _{c} c _{ A \uparrow} ^{\dag} c _{B \downarrow} ^{\dag} | 0 \rangle  $ and $| \Phi _{b} ^{\prime} \rangle = c _{L \uparrow} ^{\dag} c 