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Jacob Sanders renamed Hence_our_approach_t.tex to Molecular Vibrations 3.tex
over 9 years ago
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Hence, our approach to reduce the cost of computing a quantum mechanical Hessian is to apply precisely the compressed sensing procedure outlined in the previous section, and illustrated in Fig.~\ref{fig:NumericsScheme}, to recover the sparse matrix \(A\). After the sparse matrix \(A\) has been recovered, we can transform it back to the molecular coordinate basis or, more usefully, we can diagonalize it to obtain the quantum mechanical normal modes as eigenvectors and the quantum mechanical frequencies as the square root of the eigenvalues.
Just as in the numerical examples of the previous section, a key aspect of the compressed sensing procedure is incoherent sampling. In particular, we do not measure the sparse Hessian \(A\) directly in the basis of molecular mechanics normal modes. Instead, just as before, we apply the discrete cosine transform (\(P\)) to the molecular mechanics normal modes, and it is in this transformed basis that we actually sample the Hessian (\(B\)). Hence, what must actually be implemented into a quantum chemistry software package is the ability to compute entries of the Hessian in this discrete cosine transform basis. In principle, this is no more expensive than computing entries in the traditional molecular coordinate basis, and the computational advantage comes from the fact that substantial undersampling is possible in the discrete cosine transform basis. Since, as previously discussed, the computational cost scales with computing each \emph{column} of the Hessian, we actually sample full columns rather than individual entries. In addition, we take advantage of the symmetry of the matrix to cut the problem in half, considering only entries along and above the diagonal.
\begin{enumerate}
\item Calculate approximated vibrational modes using molecular
mechanics or some other relatively cheap method.
\item Mix .
\item Select a few of those directions at random.
\item Calculate the derivatives along those random modes
\item Apply compressed sensing to rebuild the Hessian in the
\end{enumerate}
Figs.~\ref{fig:AnthraceneFreqError} and \ref{fig:AnthraceneNormModeError} illustrate the results of applying our Hessian recovery procedure to anthracene. By sampling only 30\% of the columns, we are able to accurately recover \emph{all} the vibrational normal modes and frequencies of anthracene. As compared to measuring the \emph{entire} Hessian, sampling only 30\% of the columns gives rise to a maximum frequency error of less than 7 cm\(^{-1}\). The vibrational normal modes are essentially recovered exactly, with the minor exception that normal modes corresponding to nearly degenerate frequencies occasionally get mixed into each other. When 40\% of the columns are sampled, the error in frequency disappears completely and we obtain a virtually exact quantum mechanical vibrational spectrum of anthracene at just 40\% of the usual computational cost.
