Orthogonality does not diagonalize a varying potential ($⟨{\varphi }_1,{\varphi }_2⟩=0\nRightarrow ⟨{\varphi }_1,V{\varphi }_2⟩=0$)

In the Hilbert space $L^2([0,2\pi ])$, choose orthonormal functions ${\varphi }_1(x)=\cos x/\sqrt{\pi }$ and ${\varphi }_2(x)=\sin x/\sqrt{\pi }$. Their inner product is zero, but for the real potential energy $V(x)=V_0\sin 2x$,Thus ordinary orthogonality alone does not remove an off-diagonal matrix element of a spatially varying potential energy. A constant scalar potential does give zero between distinct orthonormal states. The example separates these two claims; it is not a model of localized atomic orbitals in a particular crystal.