The second order term 1 2 ∑ ∑ i j ∂ ∂ 2 ε ε n ∂ ∂ k i ∂ ∂ k j q i q j = ℏ ℏ 2 2 m q 2 + ∑ ∑ n ′ ≠ ≠ n | ∫ ∫ d r u n k ∗ ∗ ℏ ℏ 2 m q ⋅ ⋅ ( − − i ∇ ∇ + k ) u n ′ k | 2 ε ε n k − − ε ε n ′ k Again with ψ ψ n k = | n k ⟩ ⟩ = e i k x u n k 1 2 ∑ ∑ i j ∂ ∂ 2 ε ε n ∂ ∂ k i ∂ ∂ k j q i q j = ℏ ℏ 2 2 m q 2 + ∑ ∑ n ′ ≠ ≠ n | ⟨ ⟨ n k | ℏ ℏ 2 m q ⋅ ⋅ ( − − i ∇ ∇ ) | n ′ k ⟩ ⟩ | 2 ε ε n k − − ε ε n ′ k Eliminating q i and q j we have the theorem ∂ ∂ 2 ε ε n ( k ) ∂ ∂ k i ∂ ∂ k j = ℏ ℏ 2 m δ δ i j + ( ℏ ℏ 2 m ) 2 ∑ ∑ n ′ ≠ ≠ n ⟨ ⟨ n k | − − i ∇ ∇ i | n ′ k ⟩ ⟩ ⟨ ⟨ n ′ k | − − i ∇ ∇ j | n k ⟩ ⟩ + ⟨ ⟨ n k | − − i ∇ ∇ j | n ′ k ⟩ ⟩ ⟨ ⟨ n ′ k | − − i ∇ ∇ i | n k ⟩ ⟩ ε ε n ( k ) − − ε ε n ′ ( k )