d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime \prime} \rangle \langle {\bf r} ^{\prime \prime} | d ^{(2)} | {\bf r} ^{\prime} \rangle \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$ from Eq. (12) when the ionic-covalent chemical bond in Fig. 1 (a) is half-filled. In the above equation, the factors in the second and third lines are due to the bonding correlation. The above equation provides the energy form of the density matrix $\rho  _{ea}$, which can be decomposed into $\rho ^{(1)} = e _{1} ^{\dag} \rho  _{ea} e _{1}$ and $\rho ^{(2)} = e _{2} ^{\dag} \rho  _{ea} e _{2}$, for the one-bond system.

III-B Compound system {#iii-b-compound-system .unnumbered}
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Consider the binary compound where the anions and cations are connected by the identical ionic-covalent bonds, and assume that all the bonds are well-separated without overlap. Figure 1 (b) shows such bonds in the zincblende-structure crystal [@Grosso], in which each atom provides 4 site 