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If we assume that a system has a finite range interaction between atoms, and since each atom has an approximately constant number of neighbors, irrespective of the total number of atoms in the molecule, the number of non-zero elements in a single column of the Hessian is constant. Hence, for large molecules, the sparsity \(S\) of the Hessian scales linearly with the number of atoms \(N\). Putting this result into \(O(S/N \log N) \times OE\) yields a best-case final scaling result of \(O\left(\log N \right)\times OE\), which is a significant improvement over the original \(O(N)\times OE\) in the absence of compressed sensing. {\color{red} However, we should note that eq.~\ref{eq:csscaling} is only valid for a random  sampling, and it is not necessarily valid for a column sampling.}  To study the validity of our scaling result we have performed numerical calculations...Although for large systems the Hessian is expected to be sparse in the  molecular coordinate basis, for more moderately-sized molecules all  the atoms are relatively close to each other so we cannot rely on the  Hessian being sparse. Instead, we pursue an alternative strategy for  sparsifying the Hessian, namely guessing the basis of eigenvectors in  which the Hessian is diagonal. We believe this procedure of  estimating the eigenvectors and then recovering the sparse matrix in  this approximate eigenbasis via compressed sensing will generalize to  a wide range of scientific applications beyond molecular vibrations.  For a quantum mechanical Hessian, our method for cheaply estimating  the eigenvectors involves a simple molecular mechanics computation  which is essentially instantaneous for moderately-sized organic  molecules. Molecular mechanics computations approximate the  interactions between pairs of atoms via a force field, which is a set  of empirically-derived pair potentials between all pairs of atoms in  the molecule. The energy derivatives which appear in the Hessian may  be computed as analytical derivatives of these pair potentials, and  diagonalizing this molecular mechanics Hessian yields a cheap  approximation to eigenvectors of the true quantum mechanical Hessian.  Fig.~\ref{fig:HessianScheme} illustrates the viability of this procedure for anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), an aromatic molecule consisting of three fused benzene rings. The left side depicts the quantum mechanical Hessian of anthracene in the basis of molecular coordinates (\(H\)), while the right side shows the same matrix in the approximate eigenbasis obtained via an auxiliary molecular mechanics computation (\(A\)). The matrix \(A\) is nearly diagonal and quite sparse, and is therefore well-suited to recovery via compressed sensing.