minor modifs

Manuscript/ufGW.tex

@@ -321,7 +321,7 @@ Similar notations will be employed for the $(\Ne\pm1)$-electron configurations.
 
 In Fig.~\ref{fig:H2}, we report the variation of the quasiparticle energies of the four orbitals as functions of the internuclear distance $\RHH$. 
 One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{1.2}{\angstrom}$ for the LUMO$+1$ ($p = 3$) and $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ ($p = 4$).
-As thoroughly explained in Ref.~\onlinecite{Veril_2018}, if one relies on the linearization of the quasiparticle equation \eqref{eq:qp_eq} to compute the quasiparticle energies, \ie, $\eps{p}{\GW} = \eps{p}{\HF} + Z_{p} \SigC{p}(\eps{p}{\HF})$, these discontinuities are transformed into irregularities as the renormalization factor cancels out the singularities of the self-energy. 
+As thoroughly explained in Ref.~\onlinecite{Veril_2018}, if one relies on the linearization of the quasiparticle equation \eqref{eq:qp_eq} to compute the quasiparticle energies, \ie, $\eps{p}{\GW} \approx \eps{p}{\HF} + Z_{p} \SigC{p}(\eps{p}{\HF})$, these discontinuities are transformed into irregularities as the renormalization factor cancels out the singularities of the self-energy. 
 
 Figure \ref{fig:H2_zoom} shows the evolution of the quasiparticle energy, the energetically close-by satellites and their corresponding weights as functions of $\RHH$.
 Let us first look more closely at the region around $\RHH = \SI{1.2}{\angstrom}$ involving the LUMO$+1$ (left panel of Fig.~\ref{fig:H2_zoom}).
@@ -331,8 +331,8 @@ By construction, the quasiparticle solution diabatically follows the reference d
 
 A similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$).
 The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{2}3}$ and $\ket*{12\Bar{3}}$.
-These forms two avoided crossings in rapid successions, which create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}).
-In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron state involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations. 
+These states form two avoided crossings in rapid successions, which create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}).
+In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron wave function involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations. 
 Therefore, one can conclude that this downfall of $GW$ is a key signature of strong correlation in the $(\Ne\pm1)$-electron states that yields a significant redistribution of weights amongst electronic configurations.
 
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@@ -360,7 +360,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
 	\includegraphics[width=\linewidth]{fig4}
 	\caption{
 	\label{fig:H2reg}
-	Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level.
+	Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\eta = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level.
 }
 \end{figure}	
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@@ -418,14 +418,14 @@ For $\eta = 0.1$, we have the opposite scenario where $\eta$ is too small and so
 We have found that $\eta = 1$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions.
 This value can be certainly refined for specific applications.
 
-To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital.
-The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} for the HOMO and LUMO ($p = 1$ and $p = 2$), which is practically viable.
-Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brought by the regularization procedure is larger but it has the undeniable advantage to provide smooth curves.
+To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized (computed at $\eta = 1$) and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital.
+The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} for the HOMO ($p = 1$) and LUMO ($p = 2$), which is practically viable.
+Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brought by the regularization procedure is larger (as it should) but it has the undeniable advantage to provide smooth curves.
 
 As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis.
 In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol detailed in Ref.~\onlinecite{Loos_2020e}.
 These results are compared to high-level coupled-cluster calculations \cite{Purvis_1982,Christiansen_1995b} extracted from the same work.
-As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve) allows to smooth it out without significantly altering the overall accuracy.
+As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve computed with $\eta = 1$) allows to smooth it out without significantly altering the overall accuracy.
 Moreover, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface (gray curve). 
 
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