Singlet ground state of the Hubbard dimer
= Singlet ground state of the Hubbard dimer
{title2=$E_g-2E_0=(U-\sqrt{U^2+16t^2})/2$}
For real hopping $t$ and repulsion $U>0$, the covalent singlet and symmetric ionic state have relative <Hamiltonian> $\bigl(\begin{smallmatrix}0&2t\\2t&U\end{smallmatrix}\bigr)$. The lower energy is $(U-\sqrt{U^2+16t^2})/2$, relative to the common one-electron energy. It approaches $-4t^2/U$ at small hopping, and its state continuously mixes covalent and ionic configurations.