We evaluate the derivatives ∂ ∂ ε ε n ∂ ∂ k and ∂ ∂ 2 ε ε n ( k ) ∂ ∂ k i ∂ ∂ k j given they are the coefficients of the following expansion in q where q is considered small with respect to k ε ε n ( k + q ) = ε ε n ( k ) + ∑ ∑ i ∂ ∂ ε ε n ∂ ∂ k i q i + 1 2 ∑ ∑ i j ∂ ∂ 2 ε ε n ∂ ∂ k i ∂ ∂ k j q i q j + O ( q 3 ) Given ε ε n ( k + q ) are eigenvalues of H ^ ^ k + q We can consider the following perturbation problem in q: H ^ ^ k + q = H ^ ^ k + ℏ ℏ 2 m q ⋅ ⋅ ( − − i ∇ ∇ + k ) + ℏ ℏ 2 2 m q 2 Perturbation theory of the second order states that E n = E n 0 + ∫ ∫ d r ψ ψ n ∗ ∗ V ^ ^ ψ ψ n + ∑ ∑ n ′ ≠ ≠ n | ∫ ∫ d r ψ ψ n ∗ ∗ V ^ ^ ψ ψ n | 2 E n 0 − − E n ′ 0 + . . . To compute to linear order in q ∑ ∑ i ∂ ∂ ε ε n ∂ ∂ k i q i = ∑ ∑ i ∫ ∫ d r u n k ∗ ∗ ℏ ℏ 2 m ( − − i ∇ ∇ + k ) i q i u n k where the integrations are over a primitive cell or the entire crystal, given if the integral ∫ ∫ d r u n k ∗ ∗ u n k is normalized across the cell or the crystal.