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\section{Application: molecular vibrations}  Calculating the vibrations of a molecule, both the frequencies and associated normal modes, is one of the most ubiquitous tasks in computational chemistry~\cite{Wilson1955}. Integrated into nearly all computational chemistry packages, molecular vibrations are routinely computed by theoretical and experimental chemists alike. Synthetic chemists routinely optimize molecular geometries to find minimal energy conformations; computing and confirming the positivity of all vibrational frequencies is the standard method of assuring that a local minimum has been found. Another common task is to find the transition state for a proposed reaction: here it is also necessary to compute the molecular vibrations to find one mode with an imaginary frequency, confirming the existence of a local maximum along the reaction coordinate. coordinate~\cite{Cramer2004}.  Despite the centrality of molecular vibrations in computational chemistry, it remains one of the most expensive computations routinely performed by chemists. The core of the technique lies in calculating the matrix of the mass-weighted second derivatives of the energy with respect to the atomic positions  \begin{equation}