= Effective mass for overlap-weighted exponential hopping
{title2=$m^*(a)=\hbar^2e^{a/a_1}/[4E_1a^2(1+a/a_1)]$}
Take the nearest-neighbour hopping $t(a)=-2E_1(1+a/a_1)e^{-a/a_1}$, with $E_1,a_1>0$, in a <simple-cubic nearest-neighbour tight-binding band>. Substituting into $m^*=-\hbar^2/(2ta^2)$ gives the displayed result. With $s=a/a_1$, its logarithmic derivative is $1-2/s-1/(1+s)$, which vanishes only at $s=1+\sqrt3>0$. The mass diverges at both positive-domain ends in this formula, so that point is its unique minimum. Its bandwidth has derivative $d\ln W/d\ln a=-s^2/(1+s)$: a small compression increases $W$. The small-$a$ continuation lies beyond reliable localized-orbital tight binding.
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