 On-site Coulomb repulsion

The energy cost $U=\iint |a({\mathbf{r}}_1){|}^2|a({\mathbf{r}}_2){|}^2{e}^2/[4\pi {ϵ}_0|{\mathbf{r}}_1-{\mathbf{r}}_2|]{\mathrm{d}}^3{r}_1{\mathrm{d}}^3{r}_2$ for double occupancy of a localized orbital. It is positive for an ordinary nonzero orbital. A model retaining only this integral also neglects intersite and mixed Coulomb terms; orbital orthogonality alone does not make those terms vanish.