Singlet ground state of the Hubbard dimer ($E_g-2E_0=(U-\sqrt{U^2+16t^2})/2$)

For real hopping $t$ and repulsion $U>0$, the covalent singlet and symmetric ionic state have relative Hamiltonian $\left(\begin{array}{cc}0& 2t\\ 2t& U\end{array}\right)$. 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.