deletions | additions       diff --git a/Molecular Vibrations 1.tex b/Molecular Vibrations 1.tex  index 314e607..0c9ba35 100644  --- a/Molecular Vibrations 1.tex  +++ b/Molecular Vibrations 1.tex 
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If we assume that a system has a finite range interaction between atoms, and since each atom has an approximately constant number of neighbors, irrespective of the total number of atoms in the molecule, the number of non-zero elements in a single column of the Hessian is constant. Hence, for large molecules, the sparsity \(S\) of the Hessian scales linearly with the number of atoms \(N\). Putting this result into \(O(S/N \log N) \times OE\) yields a best-case final scaling result of \(O\left(\log N \right)\times OE\), which is a significant improvement over the original \(O(N)\times OE\) in the absence of compressed sensing. \textbf{However, we should note that eq.~\ref{eq:csscaling} is only valid for a random  sampling, and it is not necessarily valid for a column sampling.}  To study the validity of our scaling results we have performed numerical calculations on a series of molecules known as polyacenes, which are aromatic compounds made of linearly fused benzene rings. For polyacenes ranging from 1 to 15 rings, Fig.~\ref{fig:HessSparsity} illustrates the average number of non-zeros per column in the Hessian matrices obtained via molecular mechanics and quantum mechanical calculations. calculations in the basis of atomic coordinates.  In the molecular mechanics Hessians, the average sparsity per column approaches a constant value as the size of the polyacene increases, consistent with each atom having a roughly constant number of other atoms close enough for direct interaction. (In fact, in molecular mechanics, the interaction energy for two atoms beyond a certain cut-off radius is set to zero.)