adding units

Manuscript/ufGW-SI.tex

@@ -60,23 +60,11 @@
 \maketitle
 
 \begin{figure}
-   \includegraphics[width=0.8\linewidth]{h2_kappa_1}
+	\includegraphics[width=0.6\linewidth]{h2_eta_100_meV}
+	\includegraphics[width=0.6\linewidth]{h2_kappa_1}
+	\includegraphics[width=0.6\linewidth]{h2_eta_1}
 	\caption{
-	Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\kappa = 1$.
-}
-\end{figure}
-
-\begin{figure}
-	\includegraphics[width=0.8\linewidth]{h2_eta_1}
-	 \caption{
-	Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = 1$.
-}
-\end{figure}
-
-\begin{figure} 
-    \includegraphics[width=0.8\linewidth]{h2_eta_100_meV}
-     \caption{
-	Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = \SI{100}{\milli\eV}$.
+	Regularized quasiparticle energies $\reps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level for $\eta = \SI{100}{\milli\eV}$ (top), $\eta = \SI{1}{\hartree}$ (center), and $\kappa = \SI{1}{\hartree}$ (bottom).
 }
 \end{figure}
 
@@ -84,14 +72,14 @@
 	\includegraphics[width=0.33\linewidth]{eta_0_1}
 	\includegraphics[width=0.33\linewidth]{eta_1}
 	\includegraphics[width=0.33\linewidth]{eta_10}
-	\caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = 0.1$ (left), $\eta = 1$ (center), and  $\eta = 10$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. }
+	\caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with $\eta = \SI{0.1}{\hartree}$ (left), $\eta = \SI{1}{\hartree}$ (center), and  $\eta = \SI{10}{\hartree}$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. }
 \end{figure}		
 
 \begin{figure}
 	\includegraphics[width=0.33\linewidth]{kappa_0_1}
 	\includegraphics[width=0.33\linewidth]{kappa_1}
 	\includegraphics[width=0.33\linewidth]{kappa_10}
-	\caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with computed with $\kappa = 0.1$ (left), $\kappa = 1$ (center), and  $\kappa = 10$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. }
+	\caption{Difference between non-regularized and regularized quasiparticle energies $\eps{p}{\GW}-\reps{p}{\GW}$ computed with computed with $\kappa = \SI{0.1}{\hartree}$ (left), $\kappa = \SI{1}{\hartree}$ (center), and  $\kappa = \SI{10}{\hartree}$ (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level. }
 \end{figure}	
 
 \begin{figure}
@@ -100,7 +88,7 @@
 	\includegraphics[width=0.3\linewidth]{f2_eta_1}
 	\hspace{0.03\textwidth}
 	\includegraphics[width=0.3\linewidth]{f2_eta_10}
-	\caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = 0.1$ (left), $\eta = 1$ (center), and $\eta = 10$ (right).}
+	\caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = \SI{0.1}{\hartree}$ (left), $\eta = \SI{1}{\hartree}$ (center), and $\eta = \SI{10}{\hartree}$ (right).}
 \end{figure}		
 
 \begin{figure}
@@ -109,9 +97,9 @@
 	\includegraphics[width=0.3\linewidth]{f2_kappa_1}
 	\hspace{0.03\textwidth}
 	\includegraphics[width=0.3\linewidth]{f2_kappa_10}
-	\caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = 0.1$ (left), $\kappa = 1$ (center), and $\kappa = 10$ (right).
-	For $\kappa = 0.1$, the BSE@ev$GW$@HF calculations do not converge for numerous values of $R_{\ce{F-F}}$ and are not shown in this figure.
-	For $\kappa = 10$, the black and gray curves are superposed.}
+	\caption{Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set for $\kappa = \SI{0.1}{\hartree}$ (left), $\kappa = \SI{1}{\hartree}$ (center), and $\kappa = \SI{10}{\hartree}$ (right).
+	For $\kappa = \SI{0.1}{\hartree}$, the BSE@ev$GW$@HF calculations do not converge for numerous values of $R_{\ce{F-F}}$ and are not shown in this figure.
+	For $\kappa = \SI{10}{\hartree}$, the black and gray curves are superposed.}
 \end{figure}	
 	
 

Manuscript/ufGW.tex

@@ -374,8 +374,8 @@ 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 $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\alert{\kappa} = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level.
-	\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ are reported as {\SupMat}.}
+	Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\alert{\kappa} = \SI{1}{\hartree}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level.
+	\alert{Similar graphs for $\kappa = \SI{0.1}{\hartree}$ and $\kappa = \SI{10}{\hartree}$ are reported as {\SupMat}.}
 }
 \end{figure}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -388,7 +388,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
 	\caption{
 	\label{fig:F2}
 	Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.
-	\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ are reported as {\SupMat}.}
+	\alert{Similar graphs for $\kappa = \SI{0.1}{\hartree}$ and $\kappa = \SI{10}{\hartree}$ are reported as {\SupMat}.}
 }
 \end{figure}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -457,14 +457,14 @@ However, it can be certainly refined for specific applications.
 To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized (computed at $\alert{\kappa} = \SI{1.0}{\hartree}$ with the SRG-based regularizer) 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.
-\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ [and the simple regularizer given in Eq.~\eqref{eq:simple_reg}] are reported as {\SupMat}, where one clearly sees that the larger the value of $\kappa$, the larger the difference between regularized and non-regularizer quasiparticle energies.}
+\alert{Similar graphs for $\kappa = \SI{0.1}{\hartree}$ and $\kappa = \SI{10}{\hartree}$ [and the simple regularizer given in Eq.~\eqref{eq:simple_reg}] are reported as {\SupMat}, where one clearly sees that the larger the value of $\kappa$, the larger the difference between regularized and non-regularizer quasiparticle energies.}
 
 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 as detailed in Ref.~\onlinecite{Loos_2020e}.
 These results are compared to high-level coupled-cluster \alert{(CC)} calculations extracted from the same work: \alert{CC with singles and doubles (CCSD) \cite{Purvis_1982} and the non-perturbative third-order approximate CC method (CC3). \cite{Christiansen_1995b}}
-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 $\alert{\kappa} = 1$) 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 $\alert{\kappa} = \SI{1}{\hartree}$) 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). 
-\alert{For the sake of completeness, a similar graph for $\kappa = 10$ is reported as {\SupMat}.
+\alert{For the sake of completeness, a similar graph for $\kappa = \SI{10}{\hartree}$ is reported as {\SupMat}.
 Interestingly, for this rather large value of $\kappa$, the smooth BSE@{\GOWO}@HF and BSE@ev$GW$@HF curves are superposed, and of very similar quality as CCSD.
 It is therefore clear that a smaller value of $\kappa$ is more suitable.}