Coulson-Rushbrooke pairing theorem ($PKP=-K$)

For a bipartite pi system with a common diagonal energy $\alpha$, the orbital spectrum is paired about $\alpha$. If $H=\alpha I+K$ and $P$ changes the sign on one sublattice, then $PKP=-K$; an eigenvector of energy $\alpha +ϵ$ is mapped to one of energy $\alpha -ϵ$. Their site-probability magnitudes are equal.