 Rectangular free-electron band-overlap scale

For rectangular real-space periods $a>b$, the free first-zone corner has energy ${\hslash }^2{\pi }^2(a^{-2}+b^{-2})/(2m)$, whereas the free second-zone minimum is ${\hslash }^2{\pi }^2/(2m{a}^2)$. Their difference is the displayed overlap. A divalent spin-degenerate system is a band insulator only when the actual first-band maximum lies below the second-band minimum. A rigid symmetric splitting $\Delta$ removes the overlap if $\Delta >{\Delta }_{\mathrm{overlap}}$, but an arbitrary periodic potential does not shift every corner and face by the same amount. The exact criterion uses the indirect band gap, not a geometry-only local direct-gap width.