deletions | additions       diff --git a/Magnetic response.tex b/Magnetic response.tex  index d66ef15..15244c5 100644  --- a/Magnetic response.tex  +++ b/Magnetic response.tex 
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%  For the susceptibility, we need to calculate the first-order response  functions in the presence of a magnetic field. This can be done in  practice by using the magnetic perturbation, Eq.~(\ref{eq:mag1}), eq.~(\ref{eq:mag1}),  in the Sternheimer formalism described in   section \ref{sec:sternheimer}. If the system is time-reversal symmetric, since the perturbation is antisymmetric anti-symmetric  under time-reversal (anti-Hermitian), it does not induce a change in the density and the Sternheimer equation does not need to be solved  self-consistently. From there we find  % 
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potential might lead to poor convergence with the quality of the  discretization, and to a dependence of the magnetic response on the  origin of the simulation cell. In other words, an arbitrary  translation of the molecule could introduce an unphysical nonphysical  change in the calculated observables. This broken gauge-invariance is well known  in molecular calculations with all-electron methods that make use of  localized basis sets. In this case, 
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%  where \(\vec{R}_\alpha\) and \(\hat{v}^\alpha_{\text{nl}}\) are, respectively, the position and non-local potential of atom \(\alpha\).  %  With the inclusion of either one of these methods, both implemented in Octopus, we recover gauge invariance in our formalism when pseudo-potentials are used. This allows us to predict the magnetic susceptibility and other properites properties  that depend on magnetic observables, like optical activity~\cite{Varsano_2009}. A class of systems with interesting magnetic susceptibilities are fullerenes. For example, it is know that the C\(_{60}\) fullerene has a very small magnetic susceptibility due to the cancellation of the paramagnetic and diamagnetic responses~\cite{Haddon_1991,Haddon_1995}. Botti~\emph{et al.}~\cite{Botti_2009} used the real-space implementation of Octopus to study the magnetic response of the boron fullerenes depicted in Fig.~\ref{fig:boronfullerenes}. As shown in table~\ref{tab:boronmagnetic}, they found that, while most clusters are diamagnetic, B\(_{80}\) is paramagnetic, with a strong cancellation of the paramagnetic and diamagnetic terms.