deletions | additions       diff --git a/Molecular Vibrations 2.tex b/Molecular Vibrations 2.tex  index 41c9ead..27b5da0 100644  --- a/Molecular Vibrations 2.tex  +++ b/Molecular Vibrations 2.tex 
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For a quantum mechanical Hessian, our method for cheaply estimating the eigenvectors is simply to use a molecular mechanics computation, which is essentially instantaneous for moderately-sized organic molecules. Diagonalizing the molecular mechanics Hessian yields a cheap approximation to eigenvectors of the true quantum mechanical 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 the simple case of benzene (\(\textrm{C}_{6}\textrm{H}_{6}\)). The left side depicts the quantum mechanical Hessian in the basis of atomic Cartesian coordinates, while the right side shows the same matrix in the approximate eigenbasis obtained via an auxiliary molecular mechanics computation. As the figure shows, the matrix on the right in the basis of molecular mechanics normal modes is much sparser than the matrix on the left, and is therefore well-suited to recovery via compressed sensing. In particular, we expect compressed sensing to require less sampling to  recover the matrix on the rightwith far less sampling  than would be required to recover the matrix on the left.