deletions | additions       diff --git a/Molecular Vibrations 1.tex b/Molecular Vibrations 1.tex  index 4beb49f..1ec377b 100644  --- a/Molecular Vibrations 1.tex  +++ b/Molecular Vibrations 1.tex 
...
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 Hessianmatrix  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 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 that provide a cheap approximation to the eigenvectors of thetrue  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 thebasis of  molecular mechanics normal modes basis  is much sparser, and is therefore better suited to recovery via compressed sensing.