deletions | additions       diff --git a/Magnetic response.tex b/Magnetic response.tex  index 2d6157d..58f571b 100644  --- a/Magnetic response.tex  +++ b/Magnetic response.tex 
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\label{eq:mag2}  \delta^2{V}^{\mathrm{mag}} = \frac1{8c^2}(\vec{B}\times{\vec{r}})^2\ .  \end{equation}  As in the electric case with the dipole, the induced magnetic moment  can be expanded in terms of the external magnetic field which, to first  order, reads  %  \begin{equation}  \label{eq:bchi}  m_\ii=m^0_\ii+\sum_\ji\chi_{ij}B^{ext}_\ji\ ,  \end{equation}  %  where \(\vec{\chi}\) is the magnetic susceptibility tensor. The  permanent magnetic moment can be calculated directly from the  ground-state wave-functions as  %  \begin{equation}  \label{eq:magneticmoment}  \vec{m}^0=  \sum_\io\bra{\varphi_\io}\delta\op{\vec{\vv}}^{\mathrm{mag}}\ket{\varphi_{\io}}\ .  \end{equation}  %  For the susceptibility, we need to calculate the first order response  functions in presence of a magnetic field. This can be done in  practice by using the magnetic perturbation, Eq.~(\ref{eq:mag1}), in  the Sternheimer equation, Eqs.~(\ref{eq:sternheimer})  and~(\ref{eq:h1}). Since in this case the perturbation is purely  imaginary\footnote{Since in real space  \(\op{\vec{\pp}}=-\imi\vec{\nabla}\).}, it does not induce a change  in the density and the Sternheimer equation is not  self-consistent.\footnote{This is only valid for a static magnetic  field. If we consider a time-dependent magnetic field a variation in  the density can appear. This is expected, as the variation of the  density and the self-consistency process is necessary to shift from  the Kohn-Sham excitation energies to the real ones.} We label the  variation of the orbital \(\io\) in the direction  \(\ii\) as \(\ket{\delta\varphi_{\io\,,\,\ii}}\). From there, the  magnetic susceptibility tensor \(\vec{\chi}\) is  %  \begin{equation}  \label{eq:susc}  \chi_{ij} = \sum_\io\Big\{  \bra{\varphi_\io}\delta\op{\vv}^{\mathrm{mag}}_\ji\ket{\delta\varphi_{\io\,,\,i}}  +\bra{\delta\varphi_{\io\,,\,i}}\delta\op{\vv}^{\mathrm{mag}}_\ji\ket{\varphi_{\io}}  +\bra{\varphi_\io}\delta^2\op{\vv}^{\mathrm{mag}}_{ij}\ket{\varphi_{\io}}  \Big\}\ .  \end{equation}