deletions | additions       diff --git a/Molecular Vibrations 1.tex b/Molecular Vibrations 1.tex  index 729a925..f462469 100644  --- a/Molecular Vibrations 1.tex  +++ b/Molecular Vibrations 1.tex 
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Our goal, therefore, is to understand how our approach can reduce to 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 compressed sensing matrix-recovery  method. To recover a quantum mechanical Hessian matrix efficiently with compressed sensing, we need to find a basis in which the matrix is sparse. While might expect to the Hessian to have some degree of sparsity in the coordinate space, 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, whose cost is negligibe in comparison with the cost of quantum mechanical calculations. Diagonalizing the molecular mechanics Hessian yields computation that provide  a cheap approximation to the 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 mode. 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 firgure 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 basis of molecular mechanics normal modes is much sparser, and is therefore better suited to recovery via compressed sensing.