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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.

Bloch's theorem states that an eigenfunction in a periodic potential can be written ${\psi }_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{\mathbf{k}}(\mathbf{r})$, with $u_{\mathbf{k}}$ lattice-periodic. In the orthogonal tight-binding approximation, choose localized orthonormal orbitals $|j⟩$ with on-site matrix element $E_0$ and real nearest-neighbour hopping integral $t$. The Bloch sum $|k⟩=N^{-1/2}{\sum }_j{e}^{ikja}|j⟩$ is an eigenstate because each neighbour shift contributes a phase $e^{\pm ika}$. ThusThe simple-cubic nearest-neighbour tight-binding band has six neighbours, giving $E(\mathbf{k})=E_0+2t[\cos (k_{x}a)+\cos (k_{y}a)+\cos (k_{z}a)]$. The cosine sum ranges from $-3$ to $3$, so the positive bandwidth is $W=12|t|$. The stated hopping formula is negative when $E_1>0$, so the printed $12t$ must be read as a bandwidth magnitude.

Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$. To leading order the constant-energy surfaces are spheres, and comparison with $E_{\Gamma }+{\hslash }^2{k}^2/(2m^{\ast })$ givesThis is the effective mass for overlap-weighted exponential hopping. With $s=a/a_1$, its logarithmic derivative is $d\ln m^{\ast }/ds=1-2/s-1/(1+s)$. The positive stationary point is $s=1+\sqrt{3}$, and it is a minimum. The formula grows at both very small and very large $s$; the small-separation divergence is a mathematical extrapolation into a regime where localized tight binding itself becomes unreliable.

For the bandwidth, $W=24E_1(1+s)e^{-s}$ and $d\ln W/d\ln a=-s^2/(1+s)$. At $s=1$, a fractional spacing change $-0.01$ therefore givesThe exact ratio $W(0.99a_1)/W(a_1)=(1.99/2)e^{0.01}=1.004999916\dots$ checks the first-order result. Compression strengthens the hopping and widens the band.