deletions | additions       diff --git a/Molecular Vibrations 1.tex b/Molecular Vibrations 1.tex  index 9574799..0c9ba35 100644  --- a/Molecular Vibrations 1.tex  +++ b/Molecular Vibrations 1.tex 
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Our goal, therefore, is to understand how our compressed sensing approach can reduce to cost of computing the Hessian matrix of a molecule. We achieve this understanding in two complementary ways. First, 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. Second, for a moderately-sized molecule, we perform the entire numerical procedure and show in practice what kinds of speed-ups may be obtained.  Calculating the Hessian requires a method for obtaining the energy of a given nuclear configuration, and many methods exist which offer a trade-off between accuracy and computational cost. Molecular mechanics approaches, which model the interactions between atoms via empirical pair potentials~\cite{MM}, are computationally cheap, while more accurate and expensive methods explicitly model the electronic degrees of freedom at some level of approximated quantum mechanics, such as density functional theory (DFT)~\cite{Hohenberg_1964,doi:10.1103/PhysRev.140.A1133} (DFT)~\cite{Hohenberg_1964,Kohn_1965}  or wavefunction methods~\cite{Szabo}. We focus on these latter approaches, since it is here that the computation time is dominated by the calculation of the Hessian matrix, making it an ideal application for our compressed sensing method. The second derivatives required for the Hessian can be calculated analytically via perturbation theory or numerically via finite differences~\cite{Pulay1969,Jorgensen1983,Baroni2001,Kadantsev2005}. If the entries of the Hessian must be calculated independently, the cost of calculating the Hessian would scale as \(O(N^2) \times OE\), where \(OE\) is the cost of computing the energy of a given nuclear configuration. [For example, for a DFT-based calculation, \(OE\) is typically \(O(N^3)\).]