deletions | additions       diff --git a/Molecular Vibrations 3.tex b/Molecular Vibrations 3.tex  index cfb119e..f3f1df2 100644  --- a/Molecular Vibrations 3.tex  +++ b/Molecular Vibrations 3.tex 
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This scaling result illustrates the critical importance of recovering the Hessian in a sparse basis, with \(S\) as small as possible. The sparsest possible basis for the Hessian is its eigenbasis, with \(S = N\), but of course knowing the exact eigenbasis corresponds to already knowing the Hessian we are trying to find. So what is the smallest \(S\) that can reasonably be achieved?  For many large systems, the Hessian is already sparse directly in the basis of atomic Cartesian coordinates. Since the elements of the Hessian are \emph{partial} second derivatives of the energy with respect to the positions of two atoms, only direct interactions between the two atoms,  with the positions of all other atoms held fixed, the elements do not take must be taken  into account interactions mediated by the movement of other atoms. account.  For most systems we expect that this direct interaction has a finite range or decays strongly with distance. Note that this does not preclude collective vibrational modes, which can still emerge when the Hessian is diagonalized as a result of "chaining together" ``chaining together''  direct interactions between nearby atoms. 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.