deletions | additions       diff --git a/Molecular Vibrations 5.tex b/Molecular Vibrations 5.tex  index d55b413..0f9ea4b 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 are already quite cheap to compute in the first place.Since the interactions between pairs of atoms are approximated via a set of empirically-derived pair potentials, the energy derivatives in the Hessian may be directly computed as analytical derivatives of these pair potentials.  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 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.