 Effective mass for overlap-weighted exponential hopping

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 $m^{\ast }=-{\hslash }^2/(2t{a}^2)$ gives the displayed result. With $s=a/a_1$, its logarithmic derivative is $1-2/s-1/(1+s)$, which vanishes only at $s=1+\sqrt{3}>0$. 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.