deletions | additions       diff --git a/Molecular Vibrations 2.tex b/Molecular Vibrations 2.tex  index 2e725cd..7db2b54 100644  --- a/Molecular Vibrations 2.tex  +++ b/Molecular Vibrations 2.tex 
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The second derivatives required for the Hessian can be calculated either via finite differences, what is known as \texit{numerical} derivatives, or using perturbation theory, usually called \texit{analytical} derivatives~\cite{Pulay1969,Jorgensen_1983,Baroni2001,Kadantsev2005}.   A property of these the  calculations of the energy derivatives  is that the numerical cost does not depend on thein which the  direction of the derivatives that they  are calculated. This can be readily seen in the case of finite differences, as the cost of calculating \(E(\vec{R}^1,\ldots,\vec{R}^j+\Delta,\vec{R}^N)\) \(E(\vec{R}^1,\ldots,\vec{R}^j+\Delta,\ldots,\vec{R}^N)\)  is essentially the same as computing \(E(\vec{R}^1 + \Delta^1,\ldots,\vec{R}^j+\Delta,\vec{R}^N \Delta^1,\ldots,\vec{R}^j+\Delta,\ldots,\vec{R}^N  + \Delta^N)\) \Delta^N)\). This is an essential requirement of our method. If on the contrary, there is a preferred basis where the matrix is cheaper to compute, to compute it in a different basis might offset the reduction in cost offered by compressed sensing.  In A second property of  bothcases, the derivatives are usually calculated along the cartesian coordinates, however the  numerical cost and analytical derivatives that appears in variational quantum chemistry formalisms like DFT or Hartree-Fock  is essentially that each calculation yields a full column of  the same Hessian, rather than a single matrix element. Again, this is easy to see in finite difference computations.  We can write the second derivative of the energy as a first derivative of the force %  \begin{equation}  \label{eq:force} 
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- \frac{\partial F^B_j(\vec{R}^1,\ldots,\vec{R}^N)}{\partial R^A_i} \ .  \end{equation}  %  By the Hellman-Feynman theorem~\cite{Feynman1939}, a single energy calculation yields the forces acting over \emph{all} atoms, so the evaluation of eq.~\ref{eq:force} by finite differences for fixed \(A\) and \(i\) yields the derivatives for all values of \(B\) and \(j\), a whole column of the Hessian. An equivalent result holds for analytic derivatives obtained via perturbation theory~\cite{Baroni1994,Kadantsev2005}. Thus, our compressed sensing procedure for this particular  focuses on measuring random columns of the quantum mechanical Hessian rather than individual random entries. A key aspect of compressed sensing is that measurements must be taken in a basis that is as incoherent as possible with respect to the sparse basis. Thus, we want to compute columns of the quantum mechanical Hessian in the basis that is the discrete cosine transform (DCT) of the molecular mechanics normal modes. In principle, As stated previously,  computing each column in the DCT basis is in principle  no more expensive than computing columns in the traditional basis of atomic coordinates, and the computational advantage stems from the fact that substantial undersampling of columns is possible in the DCT basis. Thus, the full compressed sensing procedure to be implemented is the following: