_{A \sigma} ,c _{A \sigma ^{\prime} } \} = \{ c _{A \sigma} ,c _{B \sigma ^{\prime} } \} = \{ c _{A \sigma} ,c _{B \sigma ^{\prime} } ^{\dag} \} =0$ and $\{ c _{A \sigma} ,c _{A \sigma ^{\prime} } ^{\dag} \} = \{ c _{B \sigma} ,c _{B \sigma ^{\prime} } ^{\dag} \} = \delta _{ \sigma \sigma ^{\prime} } $, the factor $\sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} +t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma})$ is responsible for the intra-bond hopping, and $\hat{U}$ denotes effective e-e interaction potential. To model the AF states, we shall note that such states occur as the up- and down-spin electrons repel each other when they are at the same sites. We can take $$\begin{aligned}
\hat{U} = \int d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{ \prime} - {\bf r} ^{ \prime \prime} | ) \psi _{ \uparrow } ^{\dag} ({\bf r} ^{\prime}) \psi _{ \uparrow } ( {\bf r} ^{\prime} ) \psi _{ \downarrow } ^{\dag} ({\bf r} ^{ \prime \prime}) \psi _{ \downarrow } ({\bf r} ^{ \prime \prime})\end{aligned}$$ as the short-range repulsive potential 