deletions | additions       diff --git a/Magnetic response.tex b/Magnetic response.tex  index 6da5fe2..a835171 100644  --- a/Magnetic response.tex  +++ b/Magnetic response.tex 
...
%  \begin{equation}  \label{eq:magnetic}  \hat{H} = \frac12\left({\hat{\vec{p}}} -\frac1c{\vec{\mathrm{A}}}\right)^2 + \hat{v} +  \vec{\mathrm{B}}\cdot{\hat{\vec{S}}}\ . \end{equation}  %  The first part describes the orbital interaction with the field, and the 
...
electronic spin with the magnetic field.  As our main interest is the evaluation of the magnetic susceptibility,   in the following following,  we consider a perturbative uniform static magnetic field \(\vec{\mathrm{B}}\) and applied to a finite system with  zero total spin. In the Coulomb gauge the corresponding vector potential, \(\vec{A}\), is given as  %  \begin{equation} 
...
\sum_n\langle\varphi_n|\delta{\hat{\vec{v}}}^{\mathrm{mag}}|\varphi_{n}\rangle\ .  \end{equation}  %  For the susceptibility, we need to calculate the first order 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}), in  the Sternheimer formalism described in   section \ref{sec:sternheimer}. Since in this case If the system is time-reversal symmetric, since  the perturbation is purely  imaginary, antisymmetric 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  %  \begin{multline}  \label{eq:susc}  \chi_{ij} = \sum_k\Big[  \langle{\varphi_n}|\delta\hat{v}^{\mathrm{mag}}_j|{\delta\varphi_{n,i}}\rangle +\langle{\delta\varphi_{n\,,\,i}}|\delta\hat{v}^{\mathrm{mag}}_j|{\varphi_{n}}\rangle\\ + {\rm cc.}  %+\langle{\delta\varphi_{n\,,\,i}}|\delta\hat{v}^{\mathrm{mag}}_j|{\varphi_{n}}\rangle\\  +\langle{\varphi_n}|\delta^2\hat{v}^{\mathrm{mag}}_{ij}|{\varphi_{n}}\rangle  \Big]\ .  \end{multline} 
...
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,  the error can be traced to the finite basis set finite-basis-set  representation of the wave-functions~\cite{Wolinski_1990,Schindler_1982}. A simple measure of the error  is to check for the fulfillment of the hyper-virial  relation~\cite{Bouman_1977}.