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Calculating the Hessian requires a method for obtaining the energy of a given nuclear configuration, and many methods exist which offer a trade-off between accuracy and computational cost. Molecular mechanics approaches, which model the interactions between atoms via empirical pair potentials~\cite{MM}, are computationally cheap, while more accurate and expensive methods explicitly model the electronic degrees of freedom at some level of approximated quantum mechanics, such as density functional theory (DFT)~\cite{Hohenberg_1964,Kohn_1965} or wavefunction methods~\cite{Szabo}. We focus on these latter approaches, since it is here that the computation time is dominated by the calculation of the Hessian matrix, making it an ideal application for our compressed sensing method.  The second derivatives required for the Hessian can be calculated analytically via perturbation theory or numerically via finite differences~\cite{Pulay1969,doi:10.1063/1.445528,Baroni2001,Kadantsev2005}. differences~\cite{Pulay1969,Jo_rgensen_1983,Baroni2001,Kadantsev2005}.  If the entries of the Hessian must be calculated independently, the cost of calculating the Hessian would scale as \(O(N^2) \times OE\), where \(OE\) is the cost of computing the energy of a given nuclear configuration. [For example, for a DFT-based calculation, \(OE\) is typically \(O(N^3)\).] However, for methods based on quantum mechanics, the cost of calculating the Hessian actually scales more favorably as \(O(N)\times OE\). This fact is most easily demonstrated for a finite difference calculation. We can write the second derivative of the energy as a first derivative  of the force