adding figures

BSEdyn.tex

@@ -736,7 +736,7 @@ Although it might be reduced to $\order*{\Norb^4}$ operations with standard reso
 \section{Computational details}
 \label{sec:compdet}
 %%%%%%%%%%%%%%%%%%%%%%%%
-All systems under investigation have closed-shell singlet ground states.
+All systems under investigation have a closed-shell singlet ground state.
 We then adopt a restricted formalism throughout this work.
 The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point.
 Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
@@ -744,7 +744,9 @@ These quasiparticle energies are obtained by linearizing the frequency-dependent
 Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
 Note that, for the present (small) molecular systems, {\GOWO}@HF and ev$GW$@HF yield similar quasiparticle energies and fundamental gap. 
 Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
-As one-electron basis sets, we employ the Dunning families (cc-pVXZ and aug-cc-pVXZ) defined with cartesian Gaussian functions. 
+In the present study, the zeroth-order Hamiltonian [see Eq.~\eqref{eq:LR-PT}] is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
+The dynamical correction, however, is computed in the dTDA throughout.
+As one-electron basis sets, we employ the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions. 
 Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
 
 For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted. 
@@ -760,7 +762,6 @@ All the static and dynamic BSE calculations have been performed with the softwar
 \begin{table*}
 	\caption{
 	Singlet and triplet excitation energies (in eV) of \ce{N2} computed at the BSE@{\GOWO}@HF level for various basis sets.
-	The dynamical correction is computed in the dTDA.
 	\label{tab:N2}
 	}
 	\begin{ruledtabular}
@@ -814,24 +815,21 @@ All the static and dynamic BSE calculations have been performed with the softwar
 \end{squeezetable}
 
 First, we investigate the basis set dependency of the dynamical correction.
-Note that, in the present calculations, the zeroth-order Hamiltonian is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
+The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the cc-pVXZ and aug-cc-pVXZ families of basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that corrected transitions are usually well below this gap.
-The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that corrected transitions are usually well below this gap.
+The \ce{N2} molecule is a very convenient example for this kind of study as it contains $n \ra \pis$ and $\pi \ra \pis$ valence excitations as well as Rydberg transitions.
-The \ce{N2} molecule is a very convenient example as it contains $n \ra \pis$ and $\pi \ra \pis$ valence excitations as well as Rydberg transitions.
 As we shall further illustrate below, the magnitude of the dynamical correction is characteristic of the type of transitions.
 One key result of the present investigation is that the dynamical correction is quite basis set insensitive with a maximum variation of $0.03$ eV between in smallest (aug-cc-pVDZ) and largest (aug-cc-pVQZ) basis sets.
-This is quite a nice feature as one does not need to compute this more expensive correction in a very large basis.
+This is quite a nice feature as it means that one does not need to compute the dynamical correction in a very large basis to get a meaningful estimate of its magnitude.
 %The second important observation extracted from the results gathered in Table \ref{tab:N2} is that the dTDA is a rather satisfactory approximation, especially for the singlet states where one observes a maximum discrepancy of $0.03$ eV between the ``full'' and dTDA excitation energies.
 %The story is different for the triplet states for which deviations of the order of $0.3$ eV is observed between the two sets, the dTDA of excitation energies being, as we shall see later on, more accurate.
 %Indeed, although the dynamical correction systematically red-shift the excitation energies (as anticipated in Sec.~\ref{sec:dynW}), taking into account the coupling between the resonant and anti-resonant parts of the BSE Hamiltonian [see Eq.~\eqref{eq:BSE-final}] yields a systematic blue-shift of the correction, the overall correction remaining negative but by a smaller amount.
 %This outcome is similar to the conclusions of several benchmark studies \cite{Jacquemin_2017b,Rangel_2017} which clearly concluded that static BSE triplet excitations are notably too low in energy and that the use of the TDA is able to partly reduce this error. 
-%In accordance with the success of the dTDA, the remaining calculations of the present study are performed within this approximation.
 
 %%% TABLE I %%%
 \begin{squeezetable}
 \begin{table*}
 	\caption{
 	Singlet excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set computed at various levels of theory.
-	The dynamical correction is computed in the dTDA.
 	CT stands for charge transfer.
 	\label{tab:BigTabSi}
 	}
@@ -916,6 +914,12 @@ This is quite a nice feature as one does not need to compute this more expensive
 																		&	0.41	&	0.24		&	0.14		&	0.25	&	0.00		\\
 			MSE			&									&			&			&	0.65		&	0.48		&			&			
 																		&	0.12	&	0.00		&	0.13		&	0.00	&	0.00		\\
+			RMSE		&									&			&			&	0.71		&	0.58		&			&			
+																		&	0.54	&	0.34		&	0.19		&	0.33	&	0.00		\\
+			Max(+)		&									&			&			&	1.08		&	0.91		&			&			
+																		&	1.06	&	0.54		&	0.44		&	0.57	&	0.00		\\
+			Max(-)		&									&			&			&	0.20		&	-0.22		&			&			
+																		&	-1.77	&	-0.76		&	-0.02		&	-0.71	&	0.00		\\
 		\end{tabular} 
 	\end{ruledtabular}
 \end{table*}
@@ -926,7 +930,6 @@ This is quite a nice feature as one does not need to compute this more expensive
 \begin{table*}
 	\caption{
 	Triplet excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set computed at various levels of theory.
-	The dynamical correction is computed in the dTDA.
 	\label{tab:BigTabTr}
 	}
 	\begin{ruledtabular}
@@ -998,6 +1001,25 @@ This is quite a nice feature as one does not need to compute this more expensive
 \end{table*}
 \end{squeezetable}
 
+%%% FIG I %%%
+\begin{figure*}
+	\includegraphics[width=\linewidth]{fig1a}
+	\includegraphics[width=\linewidth]{fig1b}
+	\caption{Error (in eV) with respect to the TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} for singlet (top) and triplet (bottom) excitation energies of various molecules obtained with the aug-cc-pVTZ basis set computed within the static (white) and dynamic (colored) BSE formalism. 
+	CT and R stand for charge transfer and Rydberg state, respectively.
+	See Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} for raw data.
+	\label{fig:SiTr-SmallMol}}
+\end{figure*}
+
+%%% FIG II %%%
+\begin{figure*}
+	\includegraphics[width=\linewidth]{fig2}
+	\caption{Error (in eV) with respect to CC3 for singlet and triplet excitation energies of various molecules obtained with the aug-cc-pVDZ basis set computed within the static (white) and dynamic (colored) BSE formalism. 
+	R stands for Rydberg state.
+	See Table \ref{tab:BigMol} for raw data.
+	\label{fig:SiTr-BigMol}}
+\end{figure*}
+
 Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set.
 For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CCSD, and CC2] are also reported.
 Finally, the highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference.
@@ -1005,12 +1027,11 @@ For each excitation, we report the static and dynamic excitation energies, $\Om{
 As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_S$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result.
 Moreover, we have observed that an iterative, self-consistent resolution [where the dynamically-corrected excitation energies are re-injected in Eq.~\eqref{eq:Om1}] yields basically the same results as its (cheaper) renormalized version.
 
-%%% TABLE I %%%
+%%% TABLE III %%%
 \begin{squeezetable}
 \begin{table}
 	\caption{
 	Singlet and triplet excitation energies (in eV) for various molecules obtained with the aug-cc-pVDZ basis set computed at various levels of theory.
-	The dynamical correction is computed in the dTDA.
 	\label{tab:BigMol}
 	}
 	\begin{ruledtabular}