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and \(i\) yields the derivatives for all values of \(B\) and \(j\), a  whole column of the Hessian. An equivalent result is obtained for  analytic derivatives obtained from perturbation  theory~\cite{Baroni1994,Kadantsev2005}. theory~\cite{Baroni2001,Kadantsev2005}.  Since we obtain the Hessian one column at a time, only \(O(N)\) calculations are required instead  of the \(O(N^2)\) calculations required for an element by element  calculation.