deletions | additions       diff --git a/Molecular Vibrations 4.tex b/Molecular Vibrations 4.tex  index 15368c7..f8b8b66 100644  --- a/Molecular Vibrations 4.tex  +++ b/Molecular Vibrations 4.tex 
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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 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 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 shown in Fig.~\ref{fig:HessScheme} Fig.~\ref{fig:HessianScheme}  that the quantum mechanical Hessian is considerably sparser in the approximate eigenbasis of molecular mechanics normal modes. (Of course, nothing beats recovery in the exact eigenbasis, which is as sparse as possible, but knowing this basis requires knowing the exact Hessian in the first place.)