deletions | additions       diff --git a/Magnetic response.tex b/Magnetic response.tex  index 0cc42dc..2d8ed02 100644  --- a/Magnetic response.tex  +++ b/Magnetic response.tex 
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\end{equation}  where \(\epsilon_\io\) is the eigenvalue of the state \(\varphi_\io\).  When working with a real-space mesh, meshes,  this problem also appears, though milder, because the standard operator representation in the grid is  not gauge-invariant. However, the error can beeasily  controlled by reducing the spacing of the mesh. On the other hand, our methods real-space grids  typically require the use of the pseudo-potential approximation (discussed in detail in section~\ref{sec:pseudos}), where the electron-ion interaction is  described by a non-local potential  $\vv_{\text{nl}}(\vec{r},\vec{r'})$. \(\V_{\text{nl}}(\vec{r},\vec{r'})\).  This, or any other non-local potential, introduces a fundamental problem when describing the  interaction with magnetic fields or vector potentials in general. To  preserve gauge invariance, this term must be adequately coupled to the