= Three-site tight-binding ring
In an <orthonormal basis> of localized <atomic orbitals>, a ring of three equivalent sites with real hopping $t$ has <Hamiltonian> $H=(E_0-t)I+tJ$, where $J$ is the three-by-three <all-ones matrix>. The normalized symmetric state $(|1\rangle+|2\rangle+|3\rangle)/\sqrt3$ has energy $E_0+2t$. Its two-dimensional <orthogonal complement> has energy $E_0-t$. Thus the symmetric state is the unique spatial <ground state> for $t<0$, with an <energy> difference $3|t|$ to the excited <eigenstates>. Equivalently, the three discrete <Bloch states> have energies $E_0+2t\cos(2\pi l/3)$, $l=0,\pm1$. Nonorthogonal <atomic orbitals> instead require a <generalized eigenvalue problem>.
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