this is for holding javascript data
Jacob Sanders deleted file Molecular Vibrations 5.tex
over 9 years ago
Commit id: ccfe54c935609118ab0c4abb654f23e7ace7dd30
deletions | additions
diff --git a/Molecular Vibrations 5.tex b/Molecular Vibrations 5.tex
deleted file mode 100644
index aaa9391..0000000
--- a/Molecular Vibrations 5.tex
+++ /dev/null
...
As a final illustration, we outline the entire procedure and perform it on anthracene (\(\textrm{C}_{14}\textrm{H}_{10}\)), a moderately-sized polyacene consisting of three linearly fused benzene rings, to show in practice what kinds of results and speed-ups may be obtained. The procedure which must be implemented is as follows:
\begin{enumerate}
\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 the 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.
Figs.~\ref{fig:AnthraceneFreqError} and \ref{fig:AnthraceneNormModeError} illustrate the results of applying our Hessian recovery procedure to anthracene. Fig.~\ref{fig:AnthraceneFreqError} 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} 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.
