deletions | additions       diff --git a/Molecular Vibrations 5.tex b/Molecular Vibrations 5.tex  index 4a96929..f6ff31d 100644  --- a/Molecular Vibrations 5.tex  +++ b/Molecular Vibrations 5.tex 
...
\item Calculate approximate vibrational modes using molecular mechanics or another relatively cheap method.  \item Transform the approximate modes using the discrete cosine transform matrix.  \item Randomly select a few of the transformed modes.  \item Calculate the energy second  derivatives along these random modes to yield random columns of the true quantum mechanical Hessian. \item Apply compressed sensing to rebuild the full quantum mechanical Hessian in the (sparse) basis of approximate vibrational modes.  \item Transform the full quantum mechanical Hessian back into the atomic coordinate basis.  \item Diagonalize the quantum mechanical Hessian to obtain thetrue  vibrational modes and frequencies. \end{enumerate}  As discussed above, a key aspect of the compressed sensing procedure is incoherent sampling. The key step which must be implemented in the quantum mechanical code to enable incoherent sampling is step 4. In particular, columns of the true quantum mechanical Hessian must be computed in the basis that is the discrete cosine transform of the molecular mechanics normal modes. In principle, computing each column is no more expensive than computing columns in the traditional basis of atomic coordinates, and the computational advantage stems from the fact that substantial undersampling of columns is possible in this discrete cosine transform basis.