deletions | additions       diff --git a/Molecular Vibrations 2.tex b/Molecular Vibrations 2.tex  index 6264e9c..7f1a39a 100644  --- a/Molecular Vibrations 2.tex  +++ b/Molecular Vibrations 2.tex 
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Although for large systems the Hessian is expected to be sparse in While  the recovery of  molecular coordinate basis, for more moderately-sized molecules all mechanics Hessians provides a clear illustration of  the atoms scaling of our compressed sensing procedure, molecular mechanics Hessians  are relatively close already quite cheap  to each other so we cannot rely on the  Hessian being sparse. Instead, we pursue an alternative strategy for  sparsifying compute in  the Hessian, namely guessing first place. Since  the basis interactions between pairs of atoms are approximated via a set  of eigenvectors empirically-derived pair potentials, the energy derivatives  inwhich  the Hessian is diagonal. We believe this procedure may be directly computed as analytical derivatives  of estimating the eigenvectors and then recovering these pair potentials. Hence, from a computational standpoint,  the sparse matrix in  this approximate eigenbasis via real challenge is to apply our  compressed sensing will generalize procedure  to a wide range the computational  of scientific applications beyond molecular vibrations. 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, although for large molecules this is once again expected to be the case~\cite{}. Instead, we must 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 is simply to use  asimple  molecular mechanics computation computation,  which is essentially instantaneous for moderately-sized organic molecules. Molecular mechanics computations approximate Diagonalizing  theinteractions 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. Hessian, and hence the quantum mechanical Hessian is expected to be sparse in the basis of molecular mechanics normal modes.  Fig.~\ref{fig:HessianScheme} illustrates the viability of this procedure for anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), an aromatic molecule consisting the simple case  ofthree fused  benzene rings. (\(\textrm{C}_{6}\textrm{H}_{6}\)).  The left side depicts the quantum mechanical Hessianof anthracene  in the basis of molecular coordinates (\(H\)), atomic Cartesian coordinates,  while the right side shows the same matrix in the approximate eigenbasis obtained via an auxiliary molecular mechanics computation (\(A\)). The computation. As the figure shows, the  matrix \(A\) on the right in the basis of molecular mechanics normal modes  is nearly diagonal and quite sparse, much sparser than the matrix on the left,  and is therefore well-suited to recovery via our compressed sensing procedure. In particular, we expect  compressed sensing. sensing to recover the matrix on the right with far less sampling than would be required to recover the matrix on the left.