Finally, by using ⟨ n | ∂ ∂ H / ∂ ∂ R | n ′ ⟩ = ⟨ ∂ ∂ n / ∂ ∂ R | n ′ ⟩ ( ε ε n − − ε ε n ′ ) for n ≠ ≠ n ′ , the Berry curvature can also be written as a summation over all the other eigenstates in the form Ω Ω n , μ μ ν ν ( R ) = i ∑ ∑ n ′ ≠ ≠ n 1 ( ε ε n − − ε ε n ′ ) 2 ( ⟨ n | ∂ ∂ H ∂ ∂ R μ μ | n ′ ⟩ ⟨ n ′ | ∂ ∂ H ∂ ∂ R ν ν | n ⟩ − − ⟨ n | ∂ ∂ H ∂ ∂ R ν ν | n ′ ⟩ ⟨ n ′ | ∂ ∂ H ∂ ∂ R μ μ | n ⟩ ) . Note that the curvature of the nth energy level is contributed by all the other energy levels. That is, the Berry curvature can be viewed as the result of the residual interaction of those projected-out eigenstates. This gives the local conservation law for the Berry curvature, ∑ ∑ n Ω Ω n , μ μ ν ν ( R ) = 0 , if we sum over all possible energy levels for each value of R . This equation also offers the advantage that no differentiation on the eigenstates is involved, and thus it can be computed under any gauge choice.