Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 2 / A9 / b / ii

Let the common on-site energy be zero and the common hopping magnitude be $t=|\beta |$. Consistent with the axial-$p$ convention and the formula in part (iii), the hopping signs are ${\beta }_s=-t$ and ${\beta }_p=+t$.

Without mixing,The curves cross at $k=\pi /(2a)$, $E=0$, with no gap. The lower-left panel shows the continuous pure-$s$ and pure-$p$ branches. Their identities remain fixed through the crossing; the energy-ordered lower state changes from $s$ to $p$ character on passing through it.

With mixing and $\gamma =|{\beta }_{sp}|$, the equal-on-site-energy result isFor the regime $0<\gamma <t$ considered in part (iii), the crossing becomes an avoided crossing with gap $4\gamma$ at $k=\pi /(2a)$. The lower band has its maximum there and the upper band its minimum. Their orbital characters exchange smoothly across this region, with equal $s/p$ weights at its centre. The lower-right panel gives the sketch.

The zero-mixing limit recovers the crossing of case (1). A useful special limiting case is $\gamma =t$, where the square root is independent of wavevector and both bands are flat at $E=\pm 2t$. This flat-band limit is outside the strict $\gamma <t$ assumption made in part (iii).