= Momentum matrix-element expansion of band energy
For a real scalar lattice-periodic potential, let $|n\rangle$ be a nondegenerate zone-centre periodic eigenstate with energy $E_n$. Conjugation by $e^{i\mathbf k\cdot\mathbf r}$ gives $H(\mathbf k)=H(0)+\hbar\mathbf k\cdot\mathbf p/m+\hbar^2k^2/(2m)$. The zone-centre state can be chosen real, so $\langle n|\mathbf p|n\rangle=0$. <Second-order nondegenerate perturbation theory> then gives
$$
E_n(\mathbf k)=E_n+\frac{\hbar^2k^2}{2m}+\frac{\hbar^2}{m^2}\sum_{l\ne n}\frac{|\langle n|\mathbf k\cdot\mathbf p|l\rangle|^2}{E_n-E_l}+O(|\mathbf k|^4).
$$
Time-reversal symmetry makes this isolated spinless band even in $\mathbf k$. In a direction $\mathbf e$, the curvature gives
$$
\frac{m}{m^*_{\mathbf e}}=1+\frac2m\sum_{l\ne n}\frac{|\langle n|\mathbf e\cdot\mathbf p|l\rangle|^2}{E_n-E_l}.
$$
A degeneracy requires a matrix treatment before extracting curvatures; a nonlocal potential or spin-orbit terms can also alter the simple momentum vertex.
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