= Square-lattice band-overlap criterion with diagonal Fourier coupling
{title2=$\Delta_{\rm ind}=-E_X-V_{11}-3V_{10}$}
For a divalent square <Bravais lattice>, let $E_X=\hbar^2\pi^2/(2m_ea^2)$ and real <periodic potential> coefficients $V_{10}<V_{11}<0$. The leading two-wave edge levels are $E_X\pm V_{10}$. The four-wave corner levels are $2E_X+V_{11}\pm2V_{10}$ and a doublet at $2E_X-V_{11}$. Comparing the lowest corner level with the upper edge level gives the displayed <indirect band gap>. The first spin-degenerate <energy band> can hold the two <electrons> per <primitive unit cell>; it becomes isolated when $-E_X-V_{11}-3V_{10}>0$ in this truncation. This follows from resonant <degenerate perturbation theory>, and higher <plane waves> can shift the threshold.
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