figures and Manuscript

Manuscript/sfBSE.tex

@@ -597,7 +597,7 @@ As a first example, we consider the simple case of the beryllium atom which was
 Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration. 
 The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $P$ and $D$ spatial symmetries (respectively), obtained with the 6-31G basis set are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}.
 
-In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where from SF-TD-BLYP to SF-TD-BH\&HLYP there is an increase of the exact exchange in the functional. As noticed in \cite{Casanova_2020}, for the BLYP functional we have the $^3P(1s^22s2p)$ and the $^1P(1s^22s2p)$ states that appear to be degenerated because their energies are given by the $2s/2p$ gap. The exact exchange break this degeneracy. However even with the increase of the exact exchange in the functionals, SF-TD-DFT excitation energy for the $^1P(1s^22s2p)$ state is at 1.6 eV from the FCI reference. For the other states, SF-TD-DFT excitation energies are in much better agreement with FCI. The middle part shows the SF-BSE results. They are close the FCI results with an error of 0.02-0.6 eV depending on the scheme of SF-BSE. For the last excited state $^1D(1s^22p^2)$ the largest error is 0.85 eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination. Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies.
+In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where from SF-TD-BLYP to SF-TD-BH\&HLYP there is an increase of the exact exchange in the functional. As noticed in \cite{Casanova_2020}, for the BLYP functional we have the $^3P(1s^22s2p)$ and the $^1P(1s^22s2p)$ states that appear to be degenerated because their energies are given by the $2s/2p$ gap. The exact exchange break this degeneracy. However even with the increase of the exact exchange in the functionals, SF-TD-BH\&HLYP excitation energy for the $^1P(1s^22s2p)$ state is at 1.6 eV from the FCI reference. For the other states, SF-TD-BH\&HLYP excitation energies are in much better agreement with FCI. The middle part shows the SF-BSE results. They are close the FCI results, because of the fact that there is one hundred percent exact exchange in the BSE method, with an error of 0.02-0.6 eV depending on the scheme of SF-BSE. For the last excited state $^1D(1s^22p^2)$ the largest error is 0.85 eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination. Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies.
 
 %%% TABLE I %%%
 \begin{squeezetable}
@@ -756,21 +756,20 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f
 \label{sec:H2}
 %===============================
 
-The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BHHLYP and BSE and are  compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule. 
+The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. The three lowest singlets states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BH\&HLYP and BSE and are  compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule. 
-Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit with an error of more than 1 eV. The same analysis can be done for the F${}^1 \Sigma_g^+$ state at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing. SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit. In general, SF-CIS does not give a good description of the double excitation. As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state. The rigth panel gives results of the TD-BHHLYP calculation with and without spin-flip. TD-BHHLYP shows bad results for all the states of interest with and without spin-flip. Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip. 
+Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit with an error of more than 1 eV. The same analysis can be done for the F${}^1 \Sigma_g^+$ state at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing. SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit. In general, SF-CIS does not give a good description of the double excitation. As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state. The rigth panel gives results of the TD-BH\&HLYP calculation with and without spin-flip. TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip. Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip. 
 In the last panel we have results for BSE calculation with and without spin-flip. SF-BSE gives a good representation of the B${}^1 \Sigma_u^+$ state with error of  0.05-0.3 eV. However SF-BSE does not describe well the E${}^1 \Sigma_g^+$ state with error of 0.5-1.6 eV. SF-BSE shows a good agreement with the EOM-CCSD reference for the double excitation to the F${}^1 \Sigma_g^+$ state, indeed we have an error of 0.008-0.6 eV. BSE results for the B${}^1 \Sigma_u^+$ state are close to the reference until 2.0 \AA and the give bad agreement for the dissociation limit. For the E${}^1 \Sigma_g^+$ state BSE gives closer results to the reference than SF-BSE. However BSE does not retrieve the double excitation as it was pointed out in the theoretical section.
 
 
-\begin{figure*}
+\begin{figure}
-\includegraphics[width=0.49\linewidth]{H2_CIS.pdf}
+\includegraphics[width=1\linewidth]{H2_CIS.pdf}
-\hspace{0.05cm}
+\includegraphics[width=1\linewidth]{H2_BHHLYP.pdf}
-\includegraphics[width=0.49\linewidth]{H2_TDDFT.pdf}
+\includegraphics[width=1\linewidth]{H2_BSE.pdf}
-\includegraphics[width=0.49\linewidth]{H2_BSE_RHF.pdf}
 \caption{
-	Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with spin-flip method and the dotted curves are obtained without using spin-flip. The left panel shows CIS results, the right panel shows TD-BHHLYP results and the last panel shows the BSE results. 
+	Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with spin-flip method and the dotted curves are obtained without using spin-flip. The top panel shows CIS results, the center panel shows TD-BH\&HLYP results and the bottom panel shows the BSE results. 
 	All the spin-flip calculations have been performed with a UHF reference.
 	\label{fig:H2}}
-\end{figure*}
+\end{figure}
 
 
 
@@ -806,10 +805,13 @@ Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-A
 									\cline{2-4}
 			Method					&	$1\,{}^3B_{1g}$	&	$1\,{}^1B_{1g}$	&	$2\,{}^1A_{1g}$	\\
 			\hline
-			EOM-SF-CIS\fnm[1]			&	&	&	\\
+			SF-TD-BLYP \fnm[3] & & & \\
-			EOM-SF-CCSD\fnm[1]			&	&	&	\\
+			SF-TD-B3LYP \fnm[3] & & & \\
-			EOM-SF-CCSD(fT)\fnm[1]		&	&	&	\\
+			SF-TD-BH\&HLYP\fnm[3] &1.583 &2.813 & 4.528\\
-			EOM-SF-CCSD(dT)\fnm[1]		&	&	&	\\
+			SF-CIS\fnm[1]			&	&	&	\\
+			EOM-SF-CCSD\fnm[1]			&1.654	&	3.416&4.360	\\
+			EOM-SF-CCSD(fT)\fnm[1]		&	1.516&	3.260&4.205	\\
+			EOM-SF-CCSD(dT)\fnm[1]		&1.475	&3.215	&4.176	\\
 			SF-ADC(2)-s\fnm[2]			&	1.572&	3.201&	4.241\\
 			SF-ADC(2)-x\fnm[2]			&1.576	&3.134	&3.792	\\
 			SF-ADC(3)\fnm[2]			&	1.455&3.276	&4.328	\\
@@ -835,13 +837,16 @@ Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-A
 									\cline{2-4}
 			Method					&	$1\,{}^3A_{2g}$	&	$2\,{}^1A_{1g}$	&	$1\,{}^1B_{2g}$	\\
 			\hline
-			EOM-SF-CIS\fnm[1]			&	&	&	\\
+			SF-TD-BLYP \fnm[3] & & & \\
-			EOM-SF-CCSD\fnm[1]			&	&	&	\\
+			SF-TD-B3LYP \fnm[3] & & & \\
-			EOM-SF-CCSD(fT)\fnm[1]		&	&	&	\\
+			SF-TD-BH\&HLYP\fnm[3] & & & \\
-			EOM-SF-CCSD(dT)\fnm[1]		&	&	&	\\
+			SF-CIS\fnm[1]			&0.317	&	3.125&2.650	\\
+			EOM-SF-CCSD\fnm[1]			&0.369	&	1.824&	2.143\\
+			EOM-SF-CCSD(fT)\fnm[1]		&	0.163&1.530	&1.921	\\
+			EOM-SF-CCSD(dT)\fnm[1]		&0.098	&1.456	&1.853	\\
 			SF-ADC(2)-s\fnm[2]			& 0.265	&	1.658& 1.904	\\
-			SF-ADC(2)-x\fnm[2]			&	&	&	\\
+			SF-ADC(2)-x\fnm[2]			&	0.217&1.123	&1.799	\\
-			SF-ADC(3)\fnm[2]			&	&	&	\\
+			SF-ADC(3)\fnm[2]			&0.083	&1.621	&1.930	\\
 			SF-BSE@{\GOWO}\fnm[3] 		& -0.049	& 1.189	& 1.480	\\
 			SF-dBSE@{\GOWO}\fnm[3]		& 0.012	& 1.507	& 1.841	\\
 		\end{tabular}