\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.  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~\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
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\begin{equation}
\label{eq:hessian}
H_{Ai,Bj} = \frac{1}{\sqrt{M_A M_B}}\frac{\partial^2E(\vec{R}^1,\ldots,\vec{R}^N)}{\partial R^A_i \partial R^B_j}\,
\end{equation}
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where \(E(\vec{R}^1,\ldots,\vec{R}^N)\) is the ground-state energy of the molecule, \(R^A_i\) is coordinate \(i\) of atom \(A\), and \(M_A\) is its mass.  Hence, the Hessian is a real \(3N \times 3N\) matrix where \(N\) is the number of atoms in the molecule.  When the molecule is in a minimum energy conformation, the eigenvectors of the Hessian correspond to the vibrational modes of the molecule, and the square root of the eigenvalues correspond to the vibrational frequencies~\cite{Cramer2004}.

Our goal, therefore, is to understand how our approach can reduce the cost of computing the Hessian matrix of a molecule.  We achieve this understanding in two complementary ways.  First, for a moderately-sized molecule, we outline and perform the entire numerical procedure to show in practice what kinds of speed-ups may be obtained.  Second, for large systems, we investigate the ability of compressed sensing to improve how the cost of computing the Hessian scales with the number of atoms.

Calculating the Hessian requires a method for obtaining the energy of a given nuclear configuration. There exist many methods to chose from, which offer a trade-off between accuracy and computational cost.  Molecular mechanics approaches, which model the interactions between atoms via empirical potentials~\cite{Cramer2004}, are computationally cheap for systems of hundreds or thousands of atoms, while more accurate and expensive methods explicitly model the electronic degrees of freedom at some level of approximated quantum mechanics, such as methods based on density functional theory (DFT)~\cite{Hohenberg_1964,Kohn_1965} or wavefunction methods~\cite{Szabo1996}.  We focus on these quantum mechanical approaches, since in that type of calculations the computation time is dominated by the calculation of the elements of the Hessian matrix, making it an ideal application for our  matrix-recovery method.

To recover a quantum mechanical Hessian efficiently with compressed sensing, we need to find a basis in which the matrix is sparse. While we might expect to the Hessian to have some degree of sparsity in the space of atomic Cartesian coordinates, especially for large molecules, we have found that it is possible to find a better basis. The approach we take is to use a basis of approximated eigenvectors generated by a molecular mechanics computation that provide a cheap approximation to the eigenvectors of the quantum mechanical Hessian. This is illustrated in Fig.~\ref{fig:HessianScheme} for the case of the benzene molecule (\(\textrm{C}_{6}\textrm{H}_{6}\)). The figure compares the quantum mechanical Hessian in the basis of atomic Cartesian coordinates with the same matrix in the approximate eigenbasis obtained via an auxiliary molecular mechanics computation.  As the figure shows, the matrix in the molecular mechanics basis is much sparser, and is therefore better suited to recovery via compressed sensing.