done with GW section

Manuscript/sfBSE.tex

@@ -81,12 +81,12 @@ Unless otherwise stated, atomic units are used.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons, respectively) and $N$ one-electron basis functions.
 The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming the absence of linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
-The number of spin-conserved (sc) single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
-Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
+The number of spin-conserved (sc) single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip (sf) excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
+Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital with spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
 It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$.
-In a spin-flip (sf) excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.
-In the following, we assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
-Moreover, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
+In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.
+We assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
+Moreover, we consider systems with collinear spins and a spin-independent Hamiltonian without contributions such as spin-orbit interaction.
 
 %================================
 \subsection{The dynamical screening}
@@ -101,7 +101,7 @@ The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBo
 \end{equation}
 where $\eta$ is a positive infinitesimal.
 As readily seen in Eq.~\eqref{eq:G}, the Green's function can be evaluated at different levels of theory depending on the choice of orbitals and energies, $\MO{p_\sig}$ and $\e{p_\sig}{}$. 
-For example, $G_{\KS}^{\sig}$ is the independent-particle Green's function built with KS orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$. \cite{Hohenberg_1964,Kohn_1965,ParrBook} 
+For example, $G_{\KS}^{\sig}$ is the independent-particle Green's function built with Kohn-Sham (KS) orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$. \cite{Hohenberg_1964,Kohn_1965,ParrBook} 
 Within self-consistent schemes, these quantities can be replaced by quasiparticle energies and orbitals evaluated within the $GW$ approximation (see below). \cite{Hedin_1965,Golze_2019}
 
 Based on the spin-up and spin-down components of $G$ defined in Eq.~\eqref{eq:G}, one can easily compute the non-interacting polarizability (which is a sum over spins)
@@ -308,7 +308,7 @@ In addition to the principal quasiparticle peak which, in a well-behaved case, c
 Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018}  several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the KS level).
 
 Finally, within the quasiparticle self-consistent $GW$ (qs$GW$) scheme, \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} both the one-electron energies and the orbitals are updated until convergence is reached.
-These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and hermitian self-energy defined as
+These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and Hermitian self-energy defined as
 \begin{equation}
 	\Tilde{\Sigma}_{p_\sig q_\sig}^{\xc} = \frac{1}{2} \qty[ \Sig{p_\sig q_\sig}{\xc}(\e{p_\sig}{}) + \Sig{q_\sig p_\sig}{\xc}(\e{p_\sig}{}) ]
 \end{equation}