deletions | additions       diff --git a/Molecular Vibrations 2.tex b/Molecular Vibrations 2.tex  index aaa9391..427692f 100644  --- a/Molecular Vibrations 2.tex  +++ b/Molecular Vibrations 2.tex 
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As a final illustration, We must discuss one more aspect of computing quantum mechanical Hessian matrices before  we can  outline the entire procedure procedure, namely that they can be computed an entire \emph{column} at a time, rather than element-by-element. The second derivatives required for the Hessian can be calculated analytically via perturbation theory or numerically via finite differences~\cite{Pulay1969,Jo_rgensen_1983,Baroni2001,Kadantsev2005}. We can write the second derivative of the energy as a first derivative of the force  \begin{equation}  \label{eq:force}  \frac{\partial^2E(\vec{R}^1,\ldots,\vec{R}^N)}{\partial R^A_i \partial R^B_j} =  - \frac{\partial F^B_j(\vec{R}^1,\ldots,\vec{R}^N)}{\partial R^A_i} \ .  \end{equation}  By the Hellman-Feynman theorem~\cite{Feynman1939}, a single energy calculation yields the forces acting over \emph{all} atoms, so the evaluation of eq.~\ref{eq:force} by finite differences for fixed \(A\)  and perform it \(i\) yields the derivatives for all values of \(B\) and \(j\), a whole column of the Hessian. An equivalent result holds for analytic derivatives obtained via perturbation theory~\cite{Baroni1994,Kadantsev2005}. Thus, our compressed sensing procedure focuses  on anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), measuring random columns of the quantum mechanical Hessian rather than individual random entries.  A key aspect of compressed sensing is that measurements must be taken in  a moderately-sized polyacene consisting of three linearly fused benzene rings, basis that is as incoherent as possible with respect  to show the sparse basis. Thus, we want to compute columns of the quantum mechanical Hessian  in practice what kinds the basis that is the discrete cosine transform (DCT)  of results the molecular mechanics normal modes. In principle, computing each column in the DCT basis is no more expensive than computing columns in the traditional basis of atomic coordinates,  and speed-ups may be obtained. The the computational advantage stems from the fact that substantial undersampling of columns is possible in the DCT basis.  Thus, the full compressed sensing  procedure which must to  be implemented is as follows: the following:  \begin{enumerate}  \item Calculate approximate vibrational modes using molecular mechanics or another relatively cheap method. 
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\item Diagonalize the quantum mechanical Hessian to obtain the vibrational modes and frequencies.  \end{enumerate}  Asdiscussed above,  a key aspect of final illustration, we outline  the compressed sensing entire  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 and perform it on anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), a moderately-sized polyacene consisting  of the molecular mechanics normal modes. In principle, computing each column is no more expensive than computing columns three linearly fused benzene rings, to show  in the traditional basis practice what kinds  of atomic coordinates, results  and the computational advantage stems from the fact that substantial undersampling of columns is possible in this discrete cosine transform basis. speed-ups may be obtained.  Figs.~\ref{fig:AnthraceneFreqError} and \ref{fig:AnthraceneNormModeError} illustrate Fig.~\ref{fig:Anthracene} illustrates  the results of applying our Hessian recovery procedure to anthracene. Fig.~\ref{fig:AnthraceneFreqError} anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), a moderately-sized polyacene consisting of three linearly fused benzene rings. The middle panel  illustrates the error in the vibrational frequencies obtained by our compressed sensing procedure for different extents of undersampling. Sampling only 30\% of the columns gives rise to a maximum frequency error of less than 3 cm\(^{-1}\). Fig.~\ref{fig:AnthraceneNormModeError} The bottom panel  illustrates the error in the vibrational normal modes; this error is calculated as one minus the overlap (dot product) between the exact quantum mechanical normal mode and the normal mode recovered via compressed sensing. Once again, sampling only 30\% of the columns gives accurate recovery of all vibrational normal modes to within 1\%. In short, for the specific case of anthracene, the compressed sensing procedure we have devised reduces the number of expensive quantum mechanical computations by a factor of three, resulting in a 3x speed-up.