= Cube nearest-neighbour Hückel model
{title2=$E-\alpha\in\{3\beta,\beta,-\beta,-3\beta\}$}
With one orthonormal site orbital per cube vertex, common site energy $\alpha$, and coupling $\beta$ along every edge, the one-electron <Hamiltonian> is $\alpha I+\beta A$, where $A$ is the cube <adjacency matrix>. Its <eigenvalues> are $\alpha+3\beta,\alpha+\beta,\alpha-\beta,\alpha-3\beta$ with <multiplicities> $1,3,3,1$.
One proof labels each vertex by three signs. The adjacency operator flips one sign at a time, so a product of $k$ coordinate signs is an <eigenfunction> of $A$ with <eigenvalue> $3-2k$. There are $\binom3k$ such products. For $\beta<0$, eight <electrons> fill the lowest four spatial <molecular orbitals>, yielding total one-electron energy $8\alpha+12\beta$.
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