deletions | additions       diff --git a/Molecular Vibrations 2.tex b/Molecular Vibrations 2.tex  index f555391..55f14fc 100644  --- a/Molecular Vibrations 2.tex  +++ b/Molecular Vibrations 2.tex 
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The second derivatives of the energy required for the Hessian, eq.~\ref{eq:hessian}, can be calculated either via finite differences, generating  what is are  known as \textit{numerical} derivatives, or using perturbation theory, usually called generating so-called  \textit{analytical} derivatives~\cite{Pulay1969,Jorgensen1983,Baroni2001,Kadantsev2005}. A property of the calculations of the energy derivatives is that the numerical cost does not depend on the direction they are calculated. This can be readily seen in the case of finite differences, as the cost of calculating \(E(\vec{R}^1,\ldots,\vec{R}^j+\Delta,\ldots,\vec{R}^N)\) \(E(\vec{R}^1,\ldots,\vec{R}^j+\Delta^j,\ldots,\vec{R}^N)\)  is essentially the same as computing \(E(\vec{R}^1 + \Delta^1,\ldots,\vec{R}^j+\Delta,\ldots,\vec{R}^N \Delta^1,\ldots,\vec{R}^j+\Delta^j,\ldots,\vec{R}^N  + \Delta^N)\). As discussed previously, this ability to compute matrix elements at a comparable numerical cost in any desired basis  is an essential requirement of our method that, for example, a pair-potential approximation would not fulfill. method.  %If on the contrary, there is a preferred basis where the matrix is cheaper to compute, to %compute it in a different basis might offset the reduction in cost offered by compressed %sensing.  
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- \frac{\partial F^B_j(\vec{R}^1,\ldots,\vec{R}^N)}{\partial R^A_i} \ .  \end{equation}  %  By the Hellman-Feynman theorem~\cite{Hellmann1937,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 \(i\) 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{Baroni2001,Kadantsev2005}. Thus, our compressed sensing procedure for this particular application  focuses on measuring random columns of the quantum mechanical Hessian rather than individual random entries. Thus, the The  full compressed sensing procedure applied to the calculation of a quantum mechanicalcalculation of the  Hessian is therefore  implemented as the following: \begin{enumerate}  \item Calculate approximate vibrational modes using molecular mechanics. 
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\item Diagonalize the quantum mechanical Hessian to obtain the vibrational modes and frequencies.  \end{enumerate}  Fig.~\ref{fig:Anthracene} illustrates the results of applying our Hessian recovery procedure to anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), a moderately-sized polyacene consisting of three linearly fused benzene rings. The top panel illustrates the vibrational frequencies obtained by the compressed sensing procedure outlined above for different extents of undersampling of the true quantum mechanical Hessian. Even sampling only 25\% of the columns yields vibrational frequencies that are close to the true quantum mechanical frequencies, and much closer than the molecular mechanics frequencies. The middle panel illustrates the error in the vibrational frequencies from the true quantum mechanical frequencies. Sampling only 30\% of the columns gives rise to a maximum frequency error of less than 3 cm\(^{-1}\), and sampling 35\% of the columns yields nearly exact recovery. The bottom panel illustrates the error in the normal modes. Once again, sampling only 30\% of the columns gives accurate recovery of all vibrational normal modes to within 1\%. In short, our compressed sensing procedure applied to anthracene reduces the number of expensive quantum mechanical computations by a factor of three. The additional cost of the molecular mechanics computation and the compressed sensing procedure, which take a few seconds, is negligible in comparison with this reduction in computational time in the computation of the Hessian that for anthracene takes of on  the order of hours.