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Jacob Sanders deleted file Molecular Vibrations 2.tex
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While the recovery of molecular mechanics Hessians provides a clear illustration of the scaling of our compressed sensing procedure, molecular mechanics Hessians are already quite cheap to compute in the first place. Since the interactions between pairs of atoms are approximated via a set of empirically-derived pair potentials, the energy derivatives in the Hessian may be directly computed as analytical derivatives of these pair potentials. Hence, from a computational standpoint, the real challenge is to apply our compressed sensing procedure to the computational of quantum mechanical Hessians, and it is to this problem which we now turn.
As Fig.~\ref{fig:HessSparsity} shows, for moderately-sized molecules, the sparsity \(S\) of a quantum mechanical Hessian does not scale linearly with the number of atoms \(N\) in the molecule, although for large molecules this is once again expected to be the case~\cite{}. Instead, we must pursue an alternative strategy for sparsifying the Hessian, namely guessing the basis of eigenvectors in which the Hessian is diagonal. We believe this procedure of estimating the eigenvectors and then recovering the sparse matrix in this approximate eigenbasis via compressed sensing will generalize to a wide range of scientific applications beyond molecular vibrations.
For a quantum mechanical Hessian, our method for cheaply estimating the eigenvectors is simply to use a molecular mechanics computation, which is essentially instantaneous for moderately-sized organic molecules. Diagonalizing the molecular mechanics Hessian yields a cheap approximation to eigenvectors of the true quantum mechanical Hessian, and hence the quantum mechanical Hessian is expected to be sparse in the basis of molecular mechanics normal modes.
Fig.~\ref{fig:HessianScheme} illustrates the viability of this procedure for the simple case of benzene (\(\textrm{C}_{6}\textrm{H}_{6}\)). The left side depicts the quantum mechanical Hessian in the basis of atomic Cartesian coordinates, while the right side shows the same matrix in the approximate eigenbasis obtained via an auxiliary molecular mechanics computation. As the figure shows, the matrix on the right in the basis of molecular mechanics normal modes is much sparser than the matrix on the left, and is therefore well-suited to recovery via compressed sensing. In particular, we expect compressed sensing to require less sampling to recover the matrix on the right than would be required to recover the matrix on the left.
