$${\mathcal{H}}_{AA}=\frac{1}{N}\sum _{i=1}^N\sum _{j=1}^N{e}^{i\vec{k}.\left({\vec{R}}_{Aj}-{\vec{R}}_{Ai}\right)}⟨{\phi }_A\left(\vec{r}-{\vec{R}}_{Ai}\right)\left|\mathrm{H}\right|{\phi }_A\left(\vec{r}-{\vec{R}}_{Aj}\right)⟩$$ (9)