deletions | additions       diff --git a/Molecular Vibrations 5.tex b/Molecular Vibrations 5.tex  index 0f9ea4b..3da4648 100644  --- a/Molecular Vibrations 5.tex  +++ b/Molecular Vibrations 5.tex 
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While the recovery of molecular mechanics Hessians provides a clear illustration of the scaling of our compressed sensing procedure, molecular mechanics Hessians matrix elements  are already quite cheap not expensive  to compute in comparison with rest of the linear algebra operations required to diagonalize  the first place. Hessian.  Hence, from a computational standpoint, the real challenge is to apply our compressed sensing procedure to the computational of quantum mechanical Hessians, and it is to this problem which we now turn. As Fig.~\ref{fig:HessSparsity} shows, for moderately-sized molecules, the sparsity \(S\) of a quantum mechanical Hessian does not scale linearly with the number of atoms \(N\) in the molecule. Fig.~\ref{fig:NColumnsVsNRings} illustrates the results of applying compressed sensing to the recovery of the quantum mechanical Hessians of polyacenes. We recover the quantum mechanical Hessians in a variety of sparse bases and, in all cases, columns are sampled in the Fourier DCT  basis with respect to the recovery basis. We plot the minimum number of columns which must be sampled to achieve a relative Frobenius norm error less than \(10^{-3}\) as a function of both the recovery basis and the size of the polyacene. Recovering the Hessian in the atomic coordinate basis already provides a considerable computational advantage over measuring directly computing  the entire Hessian. In fact, this curve mirrors the sparsity per column curve for quantum mechanical Hessians in Fig.~\ref{fig:HessSparsity}, consistent with our prediction that the number of sampled columns scales as \(O(S/N \log N) \times OE\). More significantly, recovering the quantum mechanical Hessian in the basis of molecular mechanics normal modes provides a substantial advantage over recovery in the basis of atomic coordinates, reducing the number of columns which must be sampled approximately by a factor of two. This is consistent with the fact that the quantum mechanical Hessian is considerably sparser in the approximate eigenbasis of molecular mechanics normal modes. (Of Of  course, nothing beats recovery in the exact eigenbasis, which is as sparse as possible, but knowing this basis that  requires knowing the exact Hessian in the first place.) place.  In short, the take-home message of Fig.~\ref{fig:NColumnsVsNRings} is that using compressed sensing to recover a quantum mechanical Hessian in its basis of molecular mechanics normal modes is a practical procedure which substantially reduces the computational cost. cost of the procedure.